Subgroups — C₈₉×#60458 #36438306

Order 22784.

2824 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#576×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#131×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3488	C₂	1	2	yes	maximal, complemented
4	C₈₉×#3062	C₂	1	6	yes	maximal, complemented
5	C₈₉×#3061	C₂	1	3	yes	maximal, complemented
6	C₈₉×#3058	C₂	1	3	yes	maximal, complemented
7	C₈₉×#3055	C₂	1	6	yes	maximal, complemented
8	C₈₉×#3050	C₂	1	3	yes	maximal, complemented
9	C₂×#466×C₈₉	C₂	1	3	yes	maximal, complemented
10	C₈₉×#2913	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2897	C₂	1	2	yes	maximal, complemented
12	C₂×H(5,𝔽₂)×C₈₉	C₂²	1	2	yes	—
13	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
14	#576×C₈₉	—	2	8	no	—
15	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	6	yes	complemented
16	C₂×#135×C₈₉	C₂²	1	3	yes	complemented
17	C₂²×D₈×C₈₉	C₂²	1	3	yes	complemented
18	C₂×#131×C₈₉	—	2	8	no	—
19	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×#131×C₈₉	C₂²	1	3	yes	complemented
22	C₂×#131×C₈₉	C₂²	1	3	yes	complemented
23	C₂²×M₁₆×C₈₉	C₂²	1	3	yes	complemented
24	C₂³×C₈×C₈₉	C₂²	1	3	yes	complemented
25	#550×C₈₉	—	2	2	no	—
26	#548×C₈₉	—	2	2	no	—
27	#546×C₈₉	—	2	6	no	—
28	D₄²×C₈₉	—	2	6	no	—
29	#537×C₈₉	C₂²	1	6	yes	complemented
30	#535×C₈₉	C₂²	1	6	yes	complemented
31	#525×C₈₉	C₂²	1	6	yes	complemented
32	C₂×#121×C₈₉	C₂²	1	6	yes	complemented
33	#520×C₈₉	C₂²	1	2	yes	—
34	#469×C₈₉	C₂²	1	12	yes	complemented
35	#468×C₈₉	C₂²	1	12	yes	complemented
36	#467×C₈₉	C₂²	1	1	yes	characteristic, complemented
37	#467×C₈₉	C₂²	1	3	yes	complemented
38	#466×C₈₉	C₂²	1	6	yes	complemented
39	#466×C₈₉	C₂²	1	3	yes	complemented
40	#466×C₈₉	C₂²	1	6	yes	complemented
41	#466×C₈₉	C₂²	1	6	yes	complemented
42	#466×C₈₉	C₂²	1	3	yes	complemented
43	#465×C₈₉	C₂²	1	6	yes	complemented
44	#465×C₈₉	C₂²	1	6	yes	complemented
45	#428×C₈₉	C₂²	1	6	yes	complemented
46	#427×C₈₉	C₂²	1	3	yes	complemented
47	C₂×#107×C₈₉	C₂²	1	3	yes	complemented
48	#418×C₈₉	C₂²	1	6	yes	complemented
49	#417×C₈₉	C₂²	1	6	yes	complemented
50	#416×C₈₉	C₂²	1	6	yes	complemented
51	C₂×#102×C₈₉	C₂²	1	6	yes	complemented
52	H(5,𝔽₂)×C₈₉	—	2	8	no	—
53	H(5,𝔽₂)×C₈₉	—	2	8	no	—
54	C₂×#41×C₈₉	—	2	4	no	—
55	C₂×#41×C₈₉	—	2	4	no	—
56	C₂×#41×C₈₉	—	2	2	no	—
57	C₂×#41×C₈₉	D₄	1	3	yes	—
58	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
59	C₂×#41×C₈₉	C₂³	1	6	yes	—
60	C₂×#41×C₈₉	C₂³	1	3	yes	—
61	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
63	C₂²×D₄×C₈₉	—	2	6	no	—
64	C₂²×D₄×C₈₉	—	2	6	no	—
65	C₂²×D₄×C₈₉	D₄	1	3	yes	—
66	C₂²×D₄×C₈₉	C₂³	1	6	yes	—
67	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
68	AGL(1,ℤ/8)×C₈₉	—	2	24	no	—
69	#135×C₈₉	—	2	12	no	—
70	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×SD₁₆×C₈₉	C₂³	1	6	yes	—
72	C₂×D₈×C₈₉	—	2	12	no	—
73	C₂×D₈×C₈₉	C₂³	1	3	yes	—
74	C₂×D₈×C₈₉	C₂³	1	6	yes	—
75	#131×C₈₉	—	2	4	no	—
76	#131×C₈₉	—	2	12	no	—
77	#131×C₈₉	—	2	12	no	—
78	#131×C₈₉	D₄	1	8	yes	complemented
79	C₂×M₁₆×C₈₉	—	2	12	no	—
80	C₂×M₁₆×C₈₉	C₂³	1	3	yes	—
81	C₂×M₁₆×C₈₉	C₂³	1	6	yes	—
82	C₂×M₁₆×C₈₉	C₂³	1	3	yes	—
83	C₂²×C₈×C₈₉	—	2	12	no	—
84	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
85	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
86	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
87	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
88	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
89	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
90	#127×C₈₉	—	2	6	no	—
91	#124×C₈₉	—	2	6	no	—
92	#123×C₈₉	—	2	6	no	—
93	C₂²⋊Q₈×C₈₉	—	2	6	no	—
94	C₂²⋊Q₈×C₈₉	C₂³	1	2	yes	—
95	C₂²⋊Q₈×C₈₉	C₂³	1	6	yes	—
96	#121×C₈₉	—	2	12	no	—
97	#121×C₈₉	—	2	6	no	—
98	#121×C₈₉	—	2	6	no	—
99	#121×C₈₉	—	2	6	no	—
100	#121×C₈₉	C₂³	1	6	yes	—
101	#121×C₈₉	C₂³	1	6	yes	—
102	#121×C₈₉	C₂³	1	12	yes	—
103	#120×C₈₉	—	2	6	no	—
104	#120×C₈₉	—	2	12	no	—
105	C₄×Q₈×C₈₉	—	2	2	no	—
106	C₄×D₄×C₈₉	—	2	6	no	—
107	C₄×D₄×C₈₉	—	2	6	no	—
108	C₄×D₄×C₈₉	—	2	6	no	—
109	#117×C₈₉	C₂³	1	6	yes	—
110	C₂×C₄⋊C₄×C₈₉	C₂³	1	6	yes	—
111	C₂×C₂²⋊C₄×C₈₉	—	2	6	no	—
112	#107×C₈₉	C₂³	1	3	yes	—
113	#107×C₈₉	C₂³	1	6	yes	—
114	#107×C₈₉	C₂³	1	1	yes	characteristic
115	#106×C₈₉	C₂³	1	6	yes	—
116	#103×C₈₉	C₂³	1	2	yes	—
117	#103×C₈₉	C₂³	1	6	yes	—
118	#102×C₈₉	C₂³	1	6	yes	—
119	#102×C₈₉	C₂³	1	12	yes	—
120	#102×C₈₉	C₂³	1	6	yes	—
121	C₂⁴×C₈₉	—	2	6	no	—
122	C₂⁴×C₈₉	C₂×D₄	1	3	yes	—
123	#41×C₈₉	—	2	4	no	—
124	#41×C₈₉	—	2	12	no	—
125	#41×C₈₉	—	2	6	no	—
126	#41×C₈₉	—	2	6	no	—
127	#41×C₈₉	—	2	12	no	—
128	#41×C₈₉	—	2	12	no	—
129	#41×C₈₉	—	4	8	no	—
130	#41×C₈₉	C₂×D₄	1	4	yes	—
131	#41×C₈₉	#41	1	4	yes	—
132	C₂×Q₈×C₈₉	—	2	4	no	—
133	C₂×Q₈×C₈₉	C₂×D₄	1	3	yes	—
134	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
136	C₂×D₄×C₈₉	—	2	12	no	—
137	C₂×D₄×C₈₉	—	2	6	no	—
138	C₂×D₄×C₈₉	—	2	6	no	—
139	C₂×D₄×C₈₉	—	2	6	no	—
140	C₂×D₄×C₈₉	—	2	12	no	—
141	C₂×D₄×C₈₉	—	2	12	no	—
142	C₂×D₄×C₈₉	—	2	6	no	—
143	C₂×D₄×C₈₉	—	2	12	no	—
144	C₂×D₄×C₈₉	—	4	24	no	—
145	C₂×D₄×C₈₉	—	8	12	no	—
146	C₂×D₄×C₈₉	C₂×D₄	1	6	yes	—
147	C₂×D₄×C₈₉	C₂×D₄	1	3	yes	—
148	C₂×D₄×C₈₉	C₂×D₄	1	3	yes	—
149	C₂×D₄×C₈₉	C₂⁴	1	6	yes	—
150	C₂²×C₄×C₈₉	—	2	6	no	—
151	C₂²×C₄×C₈₉	—	2	6	no	—
152	C₂²×C₄×C₈₉	—	2	6	no	—
153	C₂²×C₄×C₈₉	—	2	6	no	—
154	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
155	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
156	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
157	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
158	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
159	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
160	Q₁₆×C₈₉	—	2	2	no	—
161	SD₁₆×C₈₉	—	2	12	no	—
162	D₈×C₈₉	—	2	6	no	—
163	D₈×C₈₉	—	2	12	no	—
164	M₁₆×C₈₉	—	2	6	no	—
165	M₁₆×C₈₉	—	2	12	no	—
166	M₁₆×C₈₉	C₂×D₄	1	12	yes	complemented
167	C₂×C₈×C₈₉	—	2	6	no	—
168	C₂×C₈×C₈₉	—	2	2	no	—
169	C₂×C₈×C₈₉	—	2	6	no	—
170	C₂×C₈×C₈₉	—	2	6	no	—
171	C₂×C₈×C₈₉	—	2	6	no	—
172	C₂×C₈×C₈₉	C₂×D₄	1	12	yes	complemented
173	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
174	C₂×C₈×C₈₉	C₂⁴	1	3	yes	—
175	C₂×C₈×C₈₉	C₂⁴	1	3	yes	—
176	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
177	C₄⋊C₄×C₈₉	—	2	6	no	—
178	C₄⋊C₄×C₈₉	—	2	2	no	—
179	C₄⋊C₄×C₈₉	—	2	6	no	—
180	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
181	C₄⋊C₄×C₈₉	C₂⁴	1	6	yes	—
182	C₂²⋊C₄×C₈₉	—	2	6	no	—
183	C₂²⋊C₄×C₈₉	—	2	2	no	—
184	C₂²⋊C₄×C₈₉	—	2	6	no	—
185	C₂²⋊C₄×C₈₉	—	2	6	no	—
186	C₂²⋊C₄×C₈₉	—	2	12	no	—
187	C₂²⋊C₄×C₈₉	—	2	6	no	—
188	C₄²×C₈₉	—	2	6	no	—
189	C₂³×C₈₉	—	2	6	no	—
190	C₂³×C₈₉	—	2	6	no	—
191	C₂³×C₈₉	—	2	6	no	—
192	C₂³×C₈₉	—	4	3	no	—
193	C₂³×C₈₉	—	4	12	no	—
194	C₂³×C₈₉	—	8	6	no	—
195	C₂³×C₈₉	—	8	6	no	—
196	C₂³×C₈₉	C₂²×D₄	1	3	yes	—
197	C₂³×C₈₉	C₂²×D₄	1	3	yes	—
198	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
199	Q₈×C₈₉	—	2	6	no	—
200	Q₈×C₈₉	—	2	2	no	—
201	Q₈×C₈₉	—	2	2	no	—
202	Q₈×C₈₉	#121	1	4	yes	—
203	D₄×C₈₉	—	2	12	no	—
204	D₄×C₈₉	—	2	6	no	—
205	D₄×C₈₉	—	2	6	no	—
206	D₄×C₈₉	—	2	6	no	—
207	D₄×C₈₉	—	4	12	no	—
208	D₄×C₈₉	—	4	12	no	—
209	D₄×C₈₉	—	4	12	no	—
210	D₄×C₈₉	—	8	6	no	—
211	D₄×C₈₉	—	8	12	no	—
212	D₄×C₈₉	#121	1	12	yes	—
213	C₂×C₄×C₈₉	—	2	6	no	—
214	C₂×C₄×C₈₉	—	2	6	no	—
215	C₂×C₄×C₈₉	—	2	6	no	—
216	C₂×C₄×C₈₉	—	2	1	no	—
217	C₂×C₄×C₈₉	—	2	3	no	—
218	C₂×C₄×C₈₉	—	2	1	no	—
219	C₂×C₄×C₈₉	—	2	3	no	—
220	C₂×C₄×C₈₉	—	2	2	no	—
221	C₂×C₄×C₈₉	—	2	6	no	—
222	C₂×C₄×C₈₉	—	2	2	no	—
223	C₂×C₄×C₈₉	—	4	12	no	—
224	C₂×C₄×C₈₉	—	8	6	no	—
225	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
227	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
228	C₂×C₄×C₈₉	C₂²×D₄	1	6	yes	—
229	C₂×C₄×C₈₉	C₂×#41	1	6	yes	—
230	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
231	C₈×C₈₉	—	2	1	no	—
232	C₈×C₈₉	—	2	3	no	—
233	C₈×C₈₉	C₂²×D₄	1	2	yes	—
234	C₈×C₈₉	C₂²×D₄	1	6	yes	—
235	C₂²×C₈₉	—	2	1	no	—
236	C₂²×C₈₉	—	2	3	no	—
237	C₂²×C₈₉	—	2	1	no	—
238	C₂²×C₈₉	—	2	3	no	—
239	C₂²×C₈₉	—	2	3	no	—
240	C₂²×C₈₉	—	2	3	no	—
241	C₂²×C₈₉	—	2	3	no	—
242	C₂²×C₈₉	—	4	12	no	—
243	C₂²×C₈₉	—	4	6	no	—
244	C₂²×C₈₉	—	4	6	no	—
245	C₂²×C₈₉	—	4	6	no	—
246	C₂²×C₈₉	—	16	12	no	—
247	C₂²×C₈₉	C₂×#121	1	6	yes	—
248	C₂²×C₈₉	#576	1	2	yes	complemented
249	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
250	C₄×C₈₉	—	2	1	no	—
251	C₄×C₈₉	—	2	3	no	—
252	C₄×C₈₉	—	4	2	no	—
253	C₄×C₈₉	—	4	2	no	—
254	C₄×C₈₉	—	4	6	no	—
255	C₄×C₈₉	C₂×#121	1	6	yes	—
256	C₄×C₈₉	C₂³×D₄	1	1	yes	characteristic
257	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
258	#60458	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
259	C₂×C₈₉	—	2	2	no	—
260	C₂×C₈₉	—	2	6	no	—
261	C₂×C₈₉	—	4	3	no	—
262	C₂×C₈₉	—	8	6	no	—
263	C₂×C₈₉	C₂²×#121	1	1	yes	characteristic
264	C₂×C₈₉	C₂×#576	1	1	yes	characteristic
265	C₂×C₈₉	#3584	1	1	yes	characteristic
266	C₂×#576	C₂×C₈₉	1	1	yes	characteristic, complemented
267	C₂²×#131	C₂×C₈₉	1	1	yes	characteristic, complemented
268	#3488	C₂×C₈₉	1	2	yes	complemented
269	#3062	C₂×C₈₉	1	6	yes	complemented
270	#3061	C₂×C₈₉	1	3	yes	complemented
271	#3058	C₂×C₈₉	1	3	yes	complemented
272	#3055	C₂×C₈₉	1	6	yes	complemented
273	#3050	C₂×C₈₉	1	3	yes	complemented
274	C₂×#466	C₂×C₈₉	1	3	yes	complemented
275	#2913	C₂×C₈₉	1	1	yes	characteristic, complemented
276	#2897	C₂×C₈₉	1	2	yes	complemented
277	C₈₉	#60458	1	1	yes	characteristic, direct product factor, Sylow
278	C₂×H(5,𝔽₂)	C₂²×C₈₉	1	2	yes	—
279	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
280	#576	—	2	8	no	—
281	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	6	yes	complemented
282	C₂×#135	C₂²×C₈₉	1	3	yes	complemented
283	C₂²×D₈	C₂²×C₈₉	1	3	yes	complemented
284	C₂×#131	—	2	8	no	—
285	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
286	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
287	C₂×#131	C₂²×C₈₉	1	3	yes	complemented
288	C₂×#131	C₂²×C₈₉	1	3	yes	complemented
289	C₂²×M₁₆	C₂²×C₈₉	1	3	yes	complemented
290	C₂³×C₈	C₂²×C₈₉	1	3	yes	complemented
291	#550	—	2	2	no	—
292	#548	—	2	2	no	—
293	#546	—	2	6	no	—
294	D₄²	—	2	6	no	—
295	#537	C₂²×C₈₉	1	6	yes	complemented
296	#535	C₂²×C₈₉	1	6	yes	complemented
297	#525	C₂²×C₈₉	1	6	yes	complemented
298	C₂×#121	C₂²×C₈₉	1	6	yes	complemented
299	#520	C₂²×C₈₉	1	2	yes	—
300	#469	C₂²×C₈₉	1	12	yes	complemented
301	#468	C₂²×C₈₉	1	12	yes	complemented
302	#467	C₂²×C₈₉	1	1	yes	characteristic, complemented
303	#467	C₂²×C₈₉	1	3	yes	complemented
304	#466	C₂²×C₈₉	1	3	yes	complemented
305	#466	C₂²×C₈₉	1	6	yes	complemented
306	#466	C₂²×C₈₉	1	6	yes	complemented
307	#466	C₂²×C₈₉	1	6	yes	complemented
308	#466	C₂²×C₈₉	1	3	yes	complemented
309	#465	C₂²×C₈₉	1	6	yes	complemented
310	#465	C₂²×C₈₉	1	6	yes	complemented
311	#428	C₂²×C₈₉	1	6	yes	complemented
312	#427	C₂²×C₈₉	1	3	yes	complemented
313	C₂×#107	C₂²×C₈₉	1	3	yes	complemented
314	#418	C₂²×C₈₉	1	6	yes	complemented
315	#417	C₂²×C₈₉	1	6	yes	complemented
316	#416	C₂²×C₈₉	1	6	yes	complemented
317	C₂×#102	C₂²×C₈₉	1	6	yes	complemented
318	H(5,𝔽₂)	—	2	8	no	—
319	H(5,𝔽₂)	—	2	8	no	—
320	C₂×#41	—	2	4	no	—
321	C₂×#41	—	2	4	no	—
322	C₂×#41	—	2	2	no	—
323	C₂×#41	D₄×C₈₉	1	3	yes	—
324	C₂×#41	D₄×C₈₉	1	1	yes	characteristic
325	C₂×#41	C₂³×C₈₉	1	6	yes	—
326	C₂×#41	C₂³×C₈₉	1	3	yes	—
327	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
328	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
329	C₂²×D₄	—	2	6	no	—
330	C₂²×D₄	—	2	6	no	—
331	C₂²×D₄	D₄×C₈₉	1	3	yes	—
332	C₂²×D₄	C₂³×C₈₉	1	6	yes	—
333	C₂³×C₄	C₂³×C₈₉	1	3	yes	—
334	AGL(1,ℤ/8)	—	2	24	no	—
335	#135	—	2	12	no	—
336	C₂×Q₁₆	C₂³×C₈₉	1	1	yes	characteristic
337	C₂×SD₁₆	C₂³×C₈₉	1	6	yes	—
338	C₂×D₈	—	2	12	no	—
339	C₂×D₈	C₂³×C₈₉	1	3	yes	—
340	C₂×D₈	C₂³×C₈₉	1	6	yes	—
341	#131	—	2	4	no	—
342	#131	—	2	12	no	—
343	#131	—	2	12	no	—
344	#131	D₄×C₈₉	1	8	yes	complemented
345	C₂×M₁₆	—	2	12	no	—
346	C₂×M₁₆	C₂³×C₈₉	1	3	yes	—
347	C₂×M₁₆	C₂³×C₈₉	1	6	yes	—
348	C₂×M₁₆	C₂³×C₈₉	1	3	yes	—
349	C₂²×C₈	—	2	12	no	—
350	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
351	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
352	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
353	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
354	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
355	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
356	#127	—	2	6	no	—
357	#124	—	2	6	no	—
358	#123	—	2	6	no	—
359	C₂²⋊Q₈	—	2	6	no	—
360	C₂²⋊Q₈	C₂³×C₈₉	1	2	yes	—
361	C₂²⋊Q₈	C₂³×C₈₉	1	6	yes	—
362	#121	—	2	12	no	—
363	#121	—	2	6	no	—
364	#121	—	2	6	no	—
365	#121	—	2	6	no	—
366	#121	C₂³×C₈₉	1	6	yes	—
367	#121	C₂³×C₈₉	1	12	yes	—
368	#121	C₂³×C₈₉	1	6	yes	—
369	#120	—	2	6	no	—
370	#120	—	2	12	no	—
371	C₄×Q₈	—	2	2	no	—
372	C₄×D₄	—	2	6	no	—
373	C₄×D₄	—	2	6	no	—
374	C₄×D₄	—	2	6	no	—
375	#117	C₂³×C₈₉	1	6	yes	—
376	C₂×C₄⋊C₄	C₂³×C₈₉	1	6	yes	—
377	C₂×C₂²⋊C₄	—	2	6	no	—
378	#107	C₂³×C₈₉	1	3	yes	—
379	#107	C₂³×C₈₉	1	6	yes	—
380	#107	C₂³×C₈₉	1	1	yes	characteristic
381	#106	C₂³×C₈₉	1	6	yes	—
382	#103	C₂³×C₈₉	1	2	yes	—
383	#103	C₂³×C₈₉	1	6	yes	—
384	#102	C₂³×C₈₉	1	6	yes	—
385	#102	C₂³×C₈₉	1	12	yes	—
386	#102	C₂³×C₈₉	1	6	yes	—
387	C₂⁴	—	2	6	no	—
388	C₂⁴	C₂×D₄×C₈₉	1	3	yes	—
389	#41	—	2	4	no	—
390	#41	—	2	12	no	—
391	#41	—	2	6	no	—
392	#41	—	2	6	no	—
393	#41	—	2	12	no	—
394	#41	—	2	12	no	—
395	#41	—	4	8	no	—
396	#41	C₂×D₄×C₈₉	1	4	yes	—
397	#41	#41×C₈₉	1	4	yes	—
398	C₂×Q₈	—	2	4	no	—
399	C₂×Q₈	C₂×D₄×C₈₉	1	3	yes	—
400	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
401	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
402	C₂×D₄	—	2	12	no	—
403	C₂×D₄	—	2	6	no	—
404	C₂×D₄	—	2	6	no	—
405	C₂×D₄	—	2	6	no	—
406	C₂×D₄	—	2	12	no	—
407	C₂×D₄	—	2	12	no	—
408	C₂×D₄	—	2	6	no	—
409	C₂×D₄	—	2	12	no	—
410	C₂×D₄	—	4	24	no	—
411	C₂×D₄	—	8	12	no	—
412	C₂×D₄	C₂×D₄×C₈₉	1	6	yes	—
413	C₂×D₄	C₂×D₄×C₈₉	1	3	yes	—
414	C₂×D₄	C₂×D₄×C₈₉	1	3	yes	—
415	C₂×D₄	C₂⁴×C₈₉	1	6	yes	—
416	C₂²×C₄	—	2	6	no	—
417	C₂²×C₄	—	2	6	no	—
418	C₂²×C₄	—	2	6	no	—
419	C₂²×C₄	—	2	6	no	—
420	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
421	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
422	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
423	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
424	C₂²×C₄	C₂⁴×C₈₉	1	3	yes	—
425	C₂²×C₄	C₂⁴×C₈₉	1	3	yes	—
426	Q₁₆	—	2	2	no	—
427	SD₁₆	—	2	12	no	—
428	D₈	—	2	6	no	—
429	D₈	—	2	12	no	—
430	M₁₆	—	2	6	no	—
431	M₁₆	—	2	12	no	—
432	M₁₆	C₂×D₄×C₈₉	1	12	yes	complemented
433	C₂×C₈	—	2	6	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	6	no	—
436	C₂×C₈	—	2	6	no	—
437	C₂×C₈	—	2	6	no	—
438	C₂×C₈	C₂×D₄×C₈₉	1	12	yes	complemented
439	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
440	C₂×C₈	C₂⁴×C₈₉	1	3	yes	—
441	C₂×C₈	C₂⁴×C₈₉	1	3	yes	—
442	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
443	C₄⋊C₄	—	2	6	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	6	no	—
446	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
447	C₄⋊C₄	C₂⁴×C₈₉	1	6	yes	—
448	C₂²⋊C₄	—	2	6	no	—
449	C₂²⋊C₄	—	2	2	no	—
450	C₂²⋊C₄	—	2	6	no	—
451	C₂²⋊C₄	—	2	6	no	—
452	C₂²⋊C₄	—	2	12	no	—
453	C₂²⋊C₄	—	2	6	no	—
454	C₄²	—	2	6	no	—
455	C₂³	—	2	6	no	—
456	C₂³	—	2	6	no	—
457	C₂³	—	2	6	no	—
458	C₂³	—	4	3	no	—
459	C₂³	—	4	12	no	—
460	C₂³	—	8	6	no	—
461	C₂³	—	8	6	no	—
462	C₂³	C₂²×D₄×C₈₉	1	3	yes	—
463	C₂³	C₂²×D₄×C₈₉	1	3	yes	—
464	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
465	Q₈	—	2	6	no	—
466	Q₈	—	2	2	no	—
467	Q₈	—	2	2	no	—
468	Q₈	#121×C₈₉	1	4	yes	—
469	D₄	—	2	12	no	—
470	D₄	—	2	6	no	—
471	D₄	—	2	6	no	—
472	D₄	—	2	6	no	—
473	D₄	—	4	12	no	—
474	D₄	—	4	12	no	—
475	D₄	—	4	12	no	—
476	D₄	—	8	6	no	—
477	D₄	—	8	12	no	—
478	D₄	#121×C₈₉	1	12	yes	—
479	C₂×C₄	—	2	6	no	—
480	C₂×C₄	—	2	6	no	—
481	C₂×C₄	—	2	6	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	3	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	3	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	6	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	4	12	no	—
490	C₂×C₄	—	8	6	no	—
491	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
492	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
493	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
494	C₂×C₄	C₂²×D₄×C₈₉	1	6	yes	—
495	C₂×C₄	C₂×#41×C₈₉	1	6	yes	—
496	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
497	C₈	—	2	1	no	—
498	C₈	—	2	3	no	—
499	C₈	C₂²×D₄×C₈₉	1	2	yes	—
500	C₈	C₂²×D₄×C₈₉	1	6	yes	—
501	C₂²	—	2	3	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	2	3	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	2	3	no	—
506	C₂²	—	2	3	no	—
507	C₂²	—	2	3	no	—
508	C₂²	—	4	12	no	—
509	C₂²	—	4	6	no	—
510	C₂²	—	4	6	no	—
511	C₂²	—	4	6	no	—
512	C₂²	—	16	12	no	—
513	C₂²	C₂×#121×C₈₉	1	6	yes	—
514	C₂²	#576×C₈₉	1	2	yes	complemented
515	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
516	C₄	—	2	1	no	—
517	C₄	—	2	3	no	—
518	C₄	—	4	6	no	—
519	C₄	—	4	2	no	—
520	C₄	—	4	2	no	—
521	C₄	C₂×#121×C₈₉	1	6	yes	—
522	C₄	C₂³×D₄×C₈₉	1	1	yes	characteristic
523	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
524	C₂	—	2	2	no	—
525	C₂	—	2	6	no	—
526	C₂	—	4	3	no	—
527	C₂	—	8	6	no	—
528	C₂	C₂²×#121×C₈₉	1	1	yes	characteristic
529	C₂	C₂×#576×C₈₉	1	1	yes	characteristic
530	C₂	C₈₉×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂²×#41×C₈₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#576×C₈₉	C₂	semidirect
2. C₂²×#131×C₈₉	C₂	semidirect
3. C₈₉×#3488	C₂	semidirect
4. C₈₉×#3062	C₂	semidirect
5. C₈₉×#3061	C₂	semidirect
6. C₈₉×#3058	C₂	semidirect
7. C₈₉×#3055	C₂	semidirect
8. C₈₉×#3050	C₂	semidirect
9. C₂×#466×C₈₉	C₂	semidirect
10. C₈₉×#2913	C₂	semidirect
11. C₈₉×#2897	C₂	semidirect
15. C₂×AGL(1,ℤ/8)×C₈₉	C₂²	semidirect
16. C₂×#135×C₈₉	C₂²	semidirect
17. C₂²×D₈×C₈₉	C₂²	semidirect
21. C₂×#131×C₈₉	C₂²	semidirect
22. C₂×#131×C₈₉	C₂²	semidirect
23. C₂²×M₁₆×C₈₉	C₂²	semidirect
24. C₂³×C₈×C₈₉	C₂²	semidirect
29. #537×C₈₉	C₂²	semidirect
30. #535×C₈₉	C₂²	semidirect
31. #525×C₈₉	C₂²	semidirect
32. C₂×#121×C₈₉	C₂²	semidirect
34. #469×C₈₉	C₂²	semidirect
35. #468×C₈₉	C₂²	semidirect
36. #467×C₈₉	C₂²	semidirect
37. #467×C₈₉	C₂²	semidirect
38. #466×C₈₉	C₂²	semidirect
39. #466×C₈₉	C₂²	semidirect
40. #466×C₈₉	C₂²	semidirect
41. #466×C₈₉	C₂²	semidirect
42. #466×C₈₉	C₂²	semidirect
43. #465×C₈₉	C₂²	semidirect
44. #465×C₈₉	C₂²	semidirect
45. #428×C₈₉	C₂²	semidirect
46. #427×C₈₉	C₂²	semidirect
47. C₂×#107×C₈₉	C₂²	semidirect
48. #418×C₈₉	C₂²	semidirect
49. #417×C₈₉	C₂²	semidirect
50. #416×C₈₉	C₂²	semidirect
51. C₂×#102×C₈₉	C₂²	semidirect
78. #131×C₈₉	D₄	semidirect
166. M₁₆×C₈₉	C₂×D₄	semidirect
172. C₂×C₈×C₈₉	C₂×D₄	semidirect
248. C₂²×C₈₉	#576	semidirect
258. #60458	C₈₉	direct (N × H)
266. C₂×#576	C₂×C₈₉	semidirect
267. C₂²×#131	C₂×C₈₉	semidirect
268. #3488	C₂×C₈₉	semidirect
269. #3062	C₂×C₈₉	semidirect
270. #3061	C₂×C₈₉	semidirect
271. #3058	C₂×C₈₉	semidirect
272. #3055	C₂×C₈₉	semidirect
273. #3050	C₂×C₈₉	semidirect
274. C₂×#466	C₂×C₈₉	semidirect
275. #2913	C₂×C₈₉	semidirect
276. #2897	C₂×C₈₉	semidirect
277. C₈₉	#60458	direct (N × H)
281. C₂×AGL(1,ℤ/8)	C₂²×C₈₉	semidirect
282. C₂×#135	C₂²×C₈₉	semidirect
283. C₂²×D₈	C₂²×C₈₉	semidirect
287. C₂×#131	C₂²×C₈₉	semidirect
288. C₂×#131	C₂²×C₈₉	semidirect
289. C₂²×M₁₆	C₂²×C₈₉	semidirect
290. C₂³×C₈	C₂²×C₈₉	semidirect
295. #537	C₂²×C₈₉	semidirect
296. #535	C₂²×C₈₉	semidirect
297. #525	C₂²×C₈₉	semidirect
298. C₂×#121	C₂²×C₈₉	semidirect
300. #469	C₂²×C₈₉	semidirect
301. #468	C₂²×C₈₉	semidirect
302. #467	C₂²×C₈₉	semidirect
303. #467	C₂²×C₈₉	semidirect
304. #466	C₂²×C₈₉	semidirect
305. #466	C₂²×C₈₉	semidirect
306. #466	C₂²×C₈₉	semidirect
307. #466	C₂²×C₈₉	semidirect
308. #466	C₂²×C₈₉	semidirect
309. #465	C₂²×C₈₉	semidirect
310. #465	C₂²×C₈₉	semidirect
311. #428	C₂²×C₈₉	semidirect
312. #427	C₂²×C₈₉	semidirect
313. C₂×#107	C₂²×C₈₉	semidirect
314. #418	C₂²×C₈₉	semidirect
315. #417	C₂²×C₈₉	semidirect
316. #416	C₂²×C₈₉	semidirect
317. C₂×#102	C₂²×C₈₉	semidirect
344. #131	D₄×C₈₉	semidirect
432. M₁₆	C₂×D₄×C₈₉	semidirect
438. C₂×C₈	C₂×D₄×C₈₉	semidirect
514. C₂²	#576×C₈₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#576×C₈₉	2
C₂²×#131×C₈₉	2
C₈₉×#3488	2
C₈₉×#3062	2
C₈₉×#3061	2
C₈₉×#3058	2
C₈₉×#3055	2
C₈₉×#3050	2
C₂×#466×C₈₉	2
C₈₉×#2913	2
C₈₉×#2897	2
#60458	89

Subgroups — C89×#60458 #36438306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60458 #36438306