Subgroups — Q₈×#135×C₈₉ #36439205

Order 22784.

1786 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₈₉	C₂	1	2	yes	maximal, complemented
2	C₈₉×#3383	C₂	1	3	yes	maximal, complemented
3	Q₈×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#3381	C₂	1	6	yes	maximal, complemented
5	C₈₉×#3380	C₂	1	3	yes	maximal, complemented
6	Q₈×SD₁₆×C₈₉	C₂	1	2	yes	maximal, complemented
7	Q₈×D₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#3160	C₂	1	3	yes	maximal, complemented
9	C₈₉×#3076	C₂	1	6	yes	maximal, complemented
10	C₂×C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#135×C₈₉	C₂	1	3	yes	maximal
12	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
13	Q₈²×C₈₉	C₂²	1	2	yes	—
14	#557×C₈₉	C₂²	1	6	yes	—
15	#554×C₈₉	—	2	6	no	—
16	#554×C₈₉	C₂²	1	6	yes	—
17	D₄×Q₈×C₈₉	—	2	2	no	—
18	D₄×Q₈×C₈₉	C₂²	1	2	yes	—
19	#533×C₈₉	—	2	6	no	—
20	#533×C₈₉	C₂²	1	6	yes	—
21	C₄×#41×C₈₉	C₂²	1	6	yes	—
22	C₂×C₄×Q₈×C₈₉	—	2	2	no	—
23	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
24	#500×C₈₉	C₂²	1	3	yes	—
25	#500×C₈₉	C₂²	1	3	yes	—
26	#499×C₈₉	C₂²	1	6	yes	—
27	#499×C₈₉	C₂²	1	6	yes	—
28	#498×C₈₉	C₂²	1	6	yes	—
29	#479×C₈₉	C₂²	1	12	yes	—
30	#478×C₈₉	C₂²	1	12	yes	—
31	#477×C₈₉	C₂²	1	6	yes	—
32	#476×C₈₉	C₂²	1	6	yes	—
33	#475×C₈₉	C₂²	1	6	yes	—
34	#474×C₈₉	C₂²	1	6	yes	—
35	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₈×Q₈×C₈₉	C₂²	1	3	yes	—
38	C₈×Q₈×C₈₉	C₂²	1	3	yes	—
39	C₄×Q₁₆×C₈₉	C₂²	1	3	yes	—
40	C₄×Q₁₆×C₈₉	C₂²	1	3	yes	—
41	C₄×SD₁₆×C₈₉	C₂²	1	6	yes	—
42	C₄×SD₁₆×C₈₉	C₂²	1	6	yes	—
43	C₄×D₈×C₈₉	C₂²	1	3	yes	—
44	C₄×D₈×C₈₉	C₂²	1	3	yes	—
45	#427×C₈₉	C₂²	1	6	yes	—
46	#427×C₈₉	C₂²	1	3	yes	—
47	C₂×C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
48	#416×C₈₉	C₂²	1	6	yes	—
49	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
50	C₂×#41×C₈₉	C₂³	1	2	yes	—
51	C₂²×Q₈×C₈₉	—	2	2	no	—
52	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
53	#135×C₈₉	Q₈	1	8	yes	direct product factor
54	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
55	C₂×SD₁₆×C₈₉	C₂³	1	2	yes	—
56	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
58	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₈₉	—	2	6	no	—
60	C₄⋊Q₈×C₈₉	—	2	6	no	—
61	C₄⋊Q₈×C₈₉	C₂³	1	6	yes	—
62	C₄⋊Q₈×C₈₉	C₂³	1	6	yes	—
63	#125×C₈₉	—	2	12	no	—
64	#125×C₈₉	C₂³	1	12	yes	—
65	C₂²⋊Q₈×C₈₉	—	2	6	no	—
66	C₂²⋊Q₈×C₈₉	—	2	6	no	—
67	C₂²⋊Q₈×C₈₉	—	2	6	no	—
68	C₂²⋊Q₈×C₈₉	—	2	6	no	—
69	C₂²⋊Q₈×C₈₉	—	2	12	no	—
70	C₄×Q₈×C₈₉	—	2	2	no	—
71	C₄×Q₈×C₈₉	—	2	6	no	—
72	C₄×Q₈×C₈₉	—	2	6	no	—
73	C₄×Q₈×C₈₉	—	2	6	no	—
74	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
75	C₄×Q₈×C₈₉	D₄	1	3	yes	—
76	C₄×Q₈×C₈₉	C₂³	1	6	yes	—
77	C₄×Q₈×C₈₉	C₂³	1	6	yes	—
78	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₄×Q₈×C₈₉	C₂³	1	3	yes	—
80	C₄×D₄×C₈₉	—	2	6	no	—
81	C₄×D₄×C₈₉	—	2	6	no	—
82	C₄×D₄×C₈₉	C₂³	1	6	yes	—
83	C₄×D₄×C₈₉	C₂³	1	6	yes	—
84	#117×C₈₉	—	2	6	no	—
85	#117×C₈₉	C₂³	1	6	yes	—
86	C₂×C₄⋊C₄×C₈₉	—	2	6	no	—
87	C₂×C₄⋊C₄×C₈₉	D₄	1	3	yes	—
88	C₂×C₄²×C₈₉	—	2	6	no	—
89	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
90	#107×C₈₉	C₂³	1	3	yes	—
91	#107×C₈₉	C₂³	1	3	yes	—
92	#107×C₈₉	C₂³	1	6	yes	—
93	#107×C₈₉	C₂³	1	6	yes	—
94	#106×C₈₉	C₂³	1	6	yes	—
95	#106×C₈₉	C₂³	1	12	yes	—
96	C₄⋊C₈×C₈₉	C₂³	1	3	yes	—
97	C₄⋊C₈×C₈₉	C₂³	1	3	yes	—
98	C₄⋊C₈×C₈₉	C₂³	1	6	yes	—
99	#103×C₈₉	C₂³	1	6	yes	—
100	#103×C₈₉	C₂³	1	6	yes	—
101	#102×C₈₉	C₂³	1	6	yes	—
102	#102×C₈₉	C₂³	1	6	yes	—
103	C₄×C₈×C₈₉	C₂³	1	3	yes	—
104	C₄×C₈×C₈₉	C₂³	1	3	yes	—
105	C₄×C₈×C₈₉	C₂³	1	3	yes	—
106	C₄×C₈×C₈₉	C₂³	1	3	yes	—
107	#41×C₈₉	—	2	4	no	—
108	#41×C₈₉	C₂×Q₈	1	8	yes	complemented
109	C₂×Q₈×C₈₉	—	2	4	no	—
110	C₂×Q₈×C₈₉	—	2	6	no	—
111	C₂×Q₈×C₈₉	—	2	6	no	—
112	C₂×Q₈×C₈₉	—	4	8	no	—
113	C₂×Q₈×C₈₉	C₂×D₄	1	3	yes	—
114	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
116	C₂×D₄×C₈₉	—	2	2	no	—
117	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
118	C₂²×C₄×C₈₉	—	2	2	no	—
119	C₂²×C₄×C₈₉	—	2	6	no	—
120	C₂²×C₄×C₈₉	—	2	6	no	—
121	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
122	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
123	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
124	Q₁₆×C₈₉	C₂×Q₈	1	4	yes	complemented
125	SD₁₆×C₈₉	C₂×Q₈	1	8	yes	complemented
126	D₈×C₈₉	C₂×Q₈	1	4	yes	complemented
127	C₂×C₈×C₈₉	—	2	6	no	—
128	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	complemented
129	C₂×C₈×C₈₉	C₂⁴	1	3	yes	—
130	C₂×C₈×C₈₉	C₂⁴	1	3	yes	—
131	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
132	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
133	C₄⋊C₄×C₈₉	—	2	12	no	—
134	C₄⋊C₄×C₈₉	—	2	6	no	—
135	C₄⋊C₄×C₈₉	—	2	6	no	—
136	C₄⋊C₄×C₈₉	—	2	6	no	—
137	C₄⋊C₄×C₈₉	—	2	6	no	—
138	C₄⋊C₄×C₈₉	—	2	12	no	—
139	C₄⋊C₄×C₈₉	C₂×D₄	1	3	yes	—
140	C₄⋊C₄×C₈₉	C₂×D₄	1	6	yes	—
141	C₄⋊C₄×C₈₉	C₂×D₄	1	3	yes	—
142	C₄⋊C₄×C₈₉	C₂⁴	1	6	yes	—
143	C₄⋊C₄×C₈₉	C₂⁴	1	6	yes	—
144	C₂²⋊C₄×C₈₉	—	2	6	no	—
145	C₂²⋊C₄×C₈₉	—	2	6	no	—
146	C₂²⋊C₄×C₈₉	—	2	6	no	—
147	C₄²×C₈₉	—	2	6	no	—
148	C₄²×C₈₉	—	2	6	no	—
149	C₄²×C₈₉	—	2	6	no	—
150	C₄²×C₈₉	C₂×D₄	1	3	yes	—
151	C₄²×C₈₉	C₂×D₄	1	3	yes	—
152	C₄²×C₈₉	C₂⁴	1	3	yes	—
153	C₄²×C₈₉	C₂⁴	1	3	yes	—
154	C₂³×C₈₉	—	2	2	no	—
155	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
156	Q₈×C₈₉	—	2	6	no	—
157	Q₈×C₈₉	—	2	2	no	—
158	Q₈×C₈₉	—	4	12	no	—
159	Q₈×C₈₉	—	8	6	no	—
160	Q₈×C₈₉	#135	1	4	yes	direct product factor
161	Q₈×C₈₉	C₂²×Q₈	1	4	yes	—
162	D₄×C₈₉	—	2	8	no	—
163	D₄×C₈₉	—	2	2	no	—
164	D₄×C₈₉	C₂²×Q₈	1	4	yes	—
165	C₂×C₄×C₈₉	—	2	2	no	—
166	C₂×C₄×C₈₉	—	2	8	no	—
167	C₂×C₄×C₈₉	—	2	6	no	—
168	C₂×C₄×C₈₉	—	2	6	no	—
169	C₂×C₄×C₈₉	—	2	6	no	—
170	C₂×C₄×C₈₉	—	4	3	no	—
171	C₂×C₄×C₈₉	—	4	12	no	—
172	C₂×C₄×C₈₉	—	8	6	no	—
173	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
174	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
175	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
176	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
177	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
179	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
180	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
181	C₈×C₈₉	—	2	3	no	—
182	C₈×C₈₉	—	2	3	no	—
183	C₈×C₈₉	C₂²×Q₈	1	2	yes	—
184	C₈×C₈₉	C₂²×Q₈	1	2	yes	—
185	C₂²×C₈₉	—	2	1	no	—
186	C₂²×C₈₉	—	2	1	no	—
187	C₂²×C₈₉	—	2	4	no	—
188	C₂²×C₈₉	—	4	2	no	—
189	C₂²×C₈₉	—	4	2	no	—
190	C₂²×C₈₉	D₄×Q₈	1	2	yes	—
191	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
192	C₄×C₈₉	—	2	3	no	—
193	C₄×C₈₉	—	2	3	no	—
194	C₄×C₈₉	—	2	4	no	—
195	C₄×C₈₉	—	2	3	no	—
196	C₄×C₈₉	—	4	6	no	—
197	C₄×C₈₉	—	4	6	no	—
198	C₄×C₈₉	D₄×Q₈	1	2	yes	—
199	C₄×C₈₉	C₂×#135	1	6	yes	—
200	C₄×C₈₉	C₂³×Q₈	1	1	yes	characteristic
201	C₄×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
202	Q₈×#135	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂×C₈₉	—	2	2	no	—
204	C₂×C₈₉	—	4	4	no	—
205	C₂×C₈₉	C₂×D₄×Q₈	1	1	yes	characteristic
206	C₂×C₈₉	C₂²×#135	1	1	yes	characteristic
207	C₂×C₈₉	#3584	1	1	yes	characteristic
208	Q₈×#41	C₂×C₈₉	1	2	yes	complemented
209	#3383	C₂×C₈₉	1	3	yes	complemented
210	Q₈×Q₁₆	C₂×C₈₉	1	1	yes	characteristic, complemented
211	#3381	C₂×C₈₉	1	6	yes	complemented
212	#3380	C₂×C₈₉	1	3	yes	complemented
213	Q₈×SD₁₆	C₂×C₈₉	1	2	yes	complemented
214	Q₈×D₈	C₂×C₈₉	1	1	yes	characteristic, complemented
215	#3160	C₂×C₈₉	1	3	yes	complemented
216	#3076	C₂×C₈₉	1	6	yes	complemented
217	C₂×C₈×Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
218	C₄×#135	C₂×C₈₉	1	3	yes	—
219	C₈₉	Q₈×#135	1	1	yes	characteristic, direct product factor, Sylow
220	C₂×#135	C₂²×C₈₉	1	1	yes	characteristic
221	Q₈²	C₂²×C₈₉	1	2	yes	—
222	#557	C₂²×C₈₉	1	6	yes	—
223	#554	—	2	6	no	—
224	#554	C₂²×C₈₉	1	6	yes	—
225	D₄×Q₈	—	2	2	no	—
226	D₄×Q₈	C₂²×C₈₉	1	2	yes	—
227	#533	—	2	6	no	—
228	#533	C₂²×C₈₉	1	6	yes	—
229	C₄×#41	C₂²×C₈₉	1	6	yes	—
230	C₂×C₄×Q₈	—	2	2	no	—
231	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
232	#500	C₂²×C₈₉	1	3	yes	—
233	#500	C₂²×C₈₉	1	3	yes	—
234	#499	C₂²×C₈₉	1	6	yes	—
235	#499	C₂²×C₈₉	1	6	yes	—
236	#498	C₂²×C₈₉	1	6	yes	—
237	#479	C₂²×C₈₉	1	12	yes	—
238	#478	C₂²×C₈₉	1	12	yes	—
239	#477	C₂²×C₈₉	1	6	yes	—
240	#476	C₂²×C₈₉	1	6	yes	—
241	#475	C₂²×C₈₉	1	6	yes	—
242	#474	C₂²×C₈₉	1	6	yes	—
243	C₈×Q₈	C₂²×C₈₉	1	1	yes	characteristic
244	C₈×Q₈	C₂²×C₈₉	1	1	yes	characteristic
245	C₈×Q₈	C₂²×C₈₉	1	3	yes	—
246	C₈×Q₈	C₂²×C₈₉	1	3	yes	—
247	C₄×Q₁₆	C₂²×C₈₉	1	3	yes	—
248	C₄×Q₁₆	C₂²×C₈₉	1	3	yes	—
249	C₄×SD₁₆	C₂²×C₈₉	1	6	yes	—
250	C₄×SD₁₆	C₂²×C₈₉	1	6	yes	—
251	C₄×D₈	C₂²×C₈₉	1	3	yes	—
252	C₄×D₈	C₂²×C₈₉	1	3	yes	—
253	#427	C₂²×C₈₉	1	6	yes	—
254	#427	C₂²×C₈₉	1	3	yes	—
255	C₂×C₄⋊C₈	C₂²×C₈₉	1	3	yes	—
256	#416	C₂²×C₈₉	1	6	yes	—
257	C₂×C₄×C₈	C₂²×C₈₉	1	3	yes	—
258	C₂×#41	C₂³×C₈₉	1	2	yes	—
259	C₂²×Q₈	—	2	2	no	—
260	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
261	#135	Q₈×C₈₉	1	8	yes	direct product factor
262	C₂×Q₁₆	C₂³×C₈₉	1	1	yes	characteristic
263	C₂×SD₁₆	C₂³×C₈₉	1	2	yes	—
264	C₂×D₈	C₂³×C₈₉	1	1	yes	characteristic
265	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
266	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
267	C₄⋊Q₈	—	2	6	no	—
268	C₄⋊Q₈	—	2	6	no	—
269	C₄⋊Q₈	C₂³×C₈₉	1	6	yes	—
270	C₄⋊Q₈	C₂³×C₈₉	1	6	yes	—
271	#125	—	2	12	no	—
272	#125	C₂³×C₈₉	1	12	yes	—
273	C₂²⋊Q₈	—	2	6	no	—
274	C₂²⋊Q₈	—	2	6	no	—
275	C₂²⋊Q₈	—	2	6	no	—
276	C₂²⋊Q₈	—	2	6	no	—
277	C₂²⋊Q₈	—	2	12	no	—
278	C₄×Q₈	—	2	2	no	—
279	C₄×Q₈	—	2	6	no	—
280	C₄×Q₈	—	2	6	no	—
281	C₄×Q₈	—	2	6	no	—
282	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
283	C₄×Q₈	D₄×C₈₉	1	3	yes	—
284	C₄×Q₈	C₂³×C₈₉	1	6	yes	—
285	C₄×Q₈	C₂³×C₈₉	1	6	yes	—
286	C₄×Q₈	C₂³×C₈₉	1	1	yes	characteristic
287	C₄×Q₈	C₂³×C₈₉	1	3	yes	—
288	C₄×D₄	—	2	6	no	—
289	C₄×D₄	—	2	6	no	—
290	C₄×D₄	C₂³×C₈₉	1	6	yes	—
291	C₄×D₄	C₂³×C₈₉	1	6	yes	—
292	#117	—	2	6	no	—
293	#117	C₂³×C₈₉	1	6	yes	—
294	C₂×C₄⋊C₄	—	2	6	no	—
295	C₂×C₄⋊C₄	D₄×C₈₉	1	3	yes	—
296	C₂×C₄²	—	2	6	no	—
297	C₂×C₄²	C₂³×C₈₉	1	3	yes	—
298	#107	C₂³×C₈₉	1	3	yes	—
299	#107	C₂³×C₈₉	1	3	yes	—
300	#107	C₂³×C₈₉	1	6	yes	—
301	#107	C₂³×C₈₉	1	6	yes	—
302	#106	C₂³×C₈₉	1	6	yes	—
303	#106	C₂³×C₈₉	1	12	yes	—
304	C₄⋊C₈	C₂³×C₈₉	1	3	yes	—
305	C₄⋊C₈	C₂³×C₈₉	1	3	yes	—
306	C₄⋊C₈	C₂³×C₈₉	1	6	yes	—
307	#103	C₂³×C₈₉	1	6	yes	—
308	#103	C₂³×C₈₉	1	6	yes	—
309	#102	C₂³×C₈₉	1	6	yes	—
310	#102	C₂³×C₈₉	1	6	yes	—
311	C₄×C₈	C₂³×C₈₉	1	3	yes	—
312	C₄×C₈	C₂³×C₈₉	1	3	yes	—
313	C₄×C₈	C₂³×C₈₉	1	3	yes	—
314	C₄×C₈	C₂³×C₈₉	1	3	yes	—
315	#41	—	2	4	no	—
316	#41	C₂×Q₈×C₈₉	1	8	yes	complemented
317	C₂×Q₈	—	2	4	no	—
318	C₂×Q₈	—	2	6	no	—
319	C₂×Q₈	—	2	6	no	—
320	C₂×Q₈	—	4	8	no	—
321	C₂×Q₈	C₂×D₄×C₈₉	1	3	yes	—
322	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
323	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	6	no	—
328	C₂²×C₄	—	2	6	no	—
329	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
330	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
331	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
332	Q₁₆	C₂×Q₈×C₈₉	1	4	yes	complemented
333	SD₁₆	C₂×Q₈×C₈₉	1	8	yes	complemented
334	D₈	C₂×Q₈×C₈₉	1	4	yes	complemented
335	C₂×C₈	—	2	6	no	—
336	C₂×C₈	C₂×Q₈×C₈₉	1	4	yes	complemented
337	C₂×C₈	C₂⁴×C₈₉	1	3	yes	—
338	C₂×C₈	C₂⁴×C₈₉	1	3	yes	—
339	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
340	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
341	C₄⋊C₄	—	2	12	no	—
342	C₄⋊C₄	—	2	6	no	—
343	C₄⋊C₄	—	2	6	no	—
344	C₄⋊C₄	—	2	6	no	—
345	C₄⋊C₄	—	2	6	no	—
346	C₄⋊C₄	—	2	12	no	—
347	C₄⋊C₄	C₂×D₄×C₈₉	1	3	yes	—
348	C₄⋊C₄	C₂×D₄×C₈₉	1	6	yes	—
349	C₄⋊C₄	C₂×D₄×C₈₉	1	3	yes	—
350	C₄⋊C₄	C₂⁴×C₈₉	1	6	yes	—
351	C₄⋊C₄	C₂⁴×C₈₉	1	6	yes	—
352	C₂²⋊C₄	—	2	6	no	—
353	C₂²⋊C₄	—	2	6	no	—
354	C₂²⋊C₄	—	2	6	no	—
355	C₄²	—	2	6	no	—
356	C₄²	—	2	6	no	—
357	C₄²	—	2	6	no	—
358	C₄²	C₂×D₄×C₈₉	1	3	yes	—
359	C₄²	C₂×D₄×C₈₉	1	3	yes	—
360	C₄²	C₂⁴×C₈₉	1	3	yes	—
361	C₄²	C₂⁴×C₈₉	1	3	yes	—
362	C₂³	—	2	2	no	—
363	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
364	Q₈	—	2	6	no	—
365	Q₈	—	2	2	no	—
366	Q₈	—	4	12	no	—
367	Q₈	—	8	6	no	—
368	Q₈	#135×C₈₉	1	4	yes	direct product factor
369	Q₈	C₂²×Q₈×C₈₉	1	4	yes	—
370	D₄	—	2	8	no	—
371	D₄	—	2	2	no	—
372	D₄	C₂²×Q₈×C₈₉	1	4	yes	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	8	no	—
375	C₂×C₄	—	2	6	no	—
376	C₂×C₄	—	2	6	no	—
377	C₂×C₄	—	2	6	no	—
378	C₂×C₄	—	4	3	no	—
379	C₂×C₄	—	4	12	no	—
380	C₂×C₄	—	8	6	no	—
381	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
382	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
383	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
384	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
385	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
386	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
387	C₂×C₄	2₋¹⁺⁴×C₈₉	1	2	yes	—
388	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
389	C₈	—	2	3	no	—
390	C₈	—	2	3	no	—
391	C₈	C₂²×Q₈×C₈₉	1	2	yes	—
392	C₈	C₂²×Q₈×C₈₉	1	2	yes	—
393	C₂²	—	2	1	no	—
394	C₂²	—	2	1	no	—
395	C₂²	—	2	4	no	—
396	C₂²	—	4	2	no	—
397	C₂²	—	4	2	no	—
398	C₂²	D₄×Q₈×C₈₉	1	2	yes	—
399	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
400	C₄	—	2	3	no	—
401	C₄	—	2	3	no	—
402	C₄	—	2	3	no	—
403	C₄	—	2	4	no	—
404	C₄	—	4	6	no	—
405	C₄	—	4	6	no	—
406	C₄	D₄×Q₈×C₈₉	1	2	yes	—
407	C₄	C₂×#135×C₈₉	1	6	yes	—
408	C₄	C₂³×Q₈×C₈₉	1	1	yes	characteristic
409	C₄	C₂×2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
410	C₂	—	2	2	no	—
411	C₂	—	4	4	no	—
412	C₂	C₂×D₄×Q₈×C₈₉	1	1	yes	characteristic
413	C₂	C₂²×#135×C₈₉	1	1	yes	characteristic
414	C₂	C₈₉×#3584	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
socle	C₂²×C₈₉	C₂³×D₄
prime radical	C₂×#135×C₈₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈₉ ◁ C₂×C₄×Q₈×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×#41×C₈₉	C₂	semidirect
2. C₈₉×#3383	C₂	semidirect
3. Q₈×Q₁₆×C₈₉	C₂	semidirect
4. C₈₉×#3381	C₂	semidirect
5. C₈₉×#3380	C₂	semidirect
6. Q₈×SD₁₆×C₈₉	C₂	semidirect
7. Q₈×D₈×C₈₉	C₂	semidirect
8. C₈₉×#3160	C₂	semidirect
9. C₈₉×#3076	C₂	semidirect
10. C₂×C₈×Q₈×C₈₉	C₂	semidirect
53. #135×C₈₉	Q₈	direct (N × H)
108. #41×C₈₉	C₂×Q₈	semidirect
124. Q₁₆×C₈₉	C₂×Q₈	semidirect
125. SD₁₆×C₈₉	C₂×Q₈	semidirect
126. D₈×C₈₉	C₂×Q₈	semidirect
128. C₂×C₈×C₈₉	C₂×Q₈	semidirect
160. Q₈×C₈₉	#135	direct (N × H)
202. Q₈×#135	C₈₉	direct (N × H)
208. Q₈×#41	C₂×C₈₉	semidirect
209. #3383	C₂×C₈₉	semidirect
210. Q₈×Q₁₆	C₂×C₈₉	semidirect
211. #3381	C₂×C₈₉	semidirect
212. #3380	C₂×C₈₉	semidirect
213. Q₈×SD₁₆	C₂×C₈₉	semidirect
214. Q₈×D₈	C₂×C₈₉	semidirect
215. #3160	C₂×C₈₉	semidirect
216. #3076	C₂×C₈₉	semidirect
217. C₂×C₈×Q₈	C₂×C₈₉	semidirect
219. C₈₉	Q₈×#135	direct (N × H)
261. #135	Q₈×C₈₉	direct (N × H)
316. #41	C₂×Q₈×C₈₉	semidirect
332. Q₁₆	C₂×Q₈×C₈₉	semidirect
333. SD₁₆	C₂×Q₈×C₈₉	semidirect
334. D₈	C₂×Q₈×C₈₉	semidirect
336. C₂×C₈	C₂×Q₈×C₈₉	semidirect
368. Q₈	#135×C₈₉	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#41×C₈₉	2
C₈₉×#3383	2
Q₈×Q₁₆×C₈₉	2
C₈₉×#3381	2
C₈₉×#3380	2
Q₈×SD₁₆×C₈₉	2
Q₈×D₈×C₈₉	2
C₈₉×#3160	2
C₈₉×#3076	2
C₂×C₈×Q₈×C₈₉	2
C₄×#135×C₈₉	2
Q₈×#135	89

Subgroups — Q8×#135×C89 #36439205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#135×C₈₉ #36439205