Subgroups — #36573323

Order 22784.

638 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#13083558	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151729	C₂	1	1	yes	maximal, characteristic
3	#26021027	C₂	1	2	yes	maximal
4	#26021027	C₂	1	1	yes	maximal, characteristic
5	#26021025	C₂	1	2	yes	maximal
6	#26021025	C₂	1	1	yes	maximal, characteristic
7	#26020426	C₂	1	2	yes	maximal
8	#26020423	C₂	1	1	yes	maximal, characteristic
9	#26020159	C₂	1	2	yes	maximal
10	C₄×#15667947	C₂	1	1	yes	maximal, characteristic
11	Q₈×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
12	#19151729	—	2	1	no	—
13	#19151729	—	2	2	no	—
14	#19151729	C₄	1	2	yes	—
15	#19151729	C₄	1	2	yes	—
16	#19151729	C₄	1	4	yes	—
17	Q₈×#10623494	—	2	1	no	—
18	C₂×#15667947	C₂²	1	1	yes	characteristic
19	C₂×#15667947	C₂²	1	2	yes	—
20	C₂×#15667947	C₂²	1	2	yes	—
21	C₂×#15667947	C₂²	1	2	yes	—
22	C₂×#15667947	C₂²	1	2	yes	—
23	C₂×#15667947	C₂²	1	1	yes	characteristic
24	#19151622	C₂²	1	1	yes	characteristic
25	#19151622	C₂²	1	2	yes	—
26	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
27	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
28	C₄×#13083558	C₂²	1	1	yes	characteristic
29	C₄×#13083558	C₂²	1	2	yes	—
30	C₂×#15667923	C₂²	1	2	yes	—
31	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
32	#19151362	C₂²	1	1	yes	characteristic
33	#19151362	C₂²	1	1	yes	characteristic
34	#19151362	C₂²	1	2	yes	—
35	#19151362	C₂²	1	2	yes	—
36	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
37	#446×C₈₉	—	2	1	no	—
38	#446×C₈₉	—	2	2	no	—
39	C₈×Q₈×C₈₉	—	2	1	no	—
40	#423×C₈₉	C₂²	1	1	yes	characteristic
41	#423×C₈₉	C₂²	1	2	yes	—
42	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
43	C₄×M₁₆×C₈₉	C₂²	1	2	yes	—
44	C₂×#13083558	C₂³	1	1	yes	characteristic
45	C₂×#13083558	C₂³	1	2	yes	—
46	C₂×#13083558	C₂³	1	1	yes	characteristic
47	C₂²×#10623494	C₂³	1	1	yes	characteristic
48	C₂²×#10623494	C₂³	1	2	yes	—
49	C₂²×#10623494	C₂³	1	1	yes	characteristic
50	#15667947	—	2	2	no	—
51	#15667947	—	2	4	no	—
52	#15667947	—	4	2	no	—
53	#15667947	—	4	2	no	—
54	#15667947	—	4	2	no	—
55	#15667947	Q₈	1	4	yes	complemented
56	C₄⋊#10623494	—	2	1	no	—
57	C₄⋊#10623494	—	2	2	no	—
58	C₄⋊#10623494	—	2	2	no	—
59	C₄⋊#10623494	—	2	1	no	—
60	C₄⋊#10623494	C₂×C₄	1	2	yes	—
61	C₄⋊#10623494	C₂×C₄	1	2	yes	—
62	C₄⋊#10623494	C₂×C₄	1	4	yes	—
63	C₄⋊#10623494	C₂×C₄	1	4	yes	—
64	#15667923	—	2	2	no	—
65	#15667923	—	2	1	no	—
66	#15667923	C₂×C₄	1	4	yes	—
67	#15667923	C₂×C₄	1	4	yes	—
68	C₄×#10623494	—	2	1	no	—
69	C₄×#10623494	—	2	2	no	—
70	C₄×#10623494	C₂×C₄	1	2	yes	—
71	C₄×#10623494	C₂×C₄	1	2	yes	—
72	C₂²×Q₈×C₈₉	Q₈	1	1	yes	characteristic
73	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
75	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
77	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
78	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
79	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
80	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
81	C₄×Q₈×C₈₉	D₄	1	2	yes	—
82	C₂×C₄⋊C₄×C₈₉	Q₈	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₈₉	Q₈	1	2	yes	—
84	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
86	C₄⋊C₈×C₈₉	—	2	1	no	—
87	C₄⋊C₈×C₈₉	—	2	2	no	—
88	C₄⋊C₈×C₈₉	—	2	2	no	—
89	C₄⋊C₈×C₈₉	—	2	1	no	—
90	#97×C₈₉	—	2	2	no	—
91	#97×C₈₉	—	2	1	no	—
92	C₄×C₈×C₈₉	—	2	1	no	—
93	C₄×C₈×C₈₉	—	2	2	no	—
94	#13083558	—	2	1	no	—
95	#13083558	—	2	1	no	—
96	#13083558	—	2	2	no	—
97	#13083558	—	2	1	no	—
98	#13083558	—	2	2	no	—
99	#13083558	C₂×Q₈	1	2	yes	—
100	C₂×#10623494	—	2	1	no	—
101	C₂×#10623494	—	2	1	no	—
102	C₂×#10623494	—	2	2	no	—
103	C₂×#10623494	—	2	1	no	—
104	C₂×#10623494	—	2	2	no	—
105	C₂×#10623494	—	2	1	no	—
106	C₂×#10623494	—	2	2	no	—
107	C₂×#10623494	C₂²×C₄	1	2	yes	—
108	C₂×#10623494	C₂²×C₄	1	4	yes	—
109	C₂×#10623494	C₂²×C₄	1	2	yes	—
110	C₂×#10623494	C₂×Q₈	1	2	yes	—
111	C₂×#10623494	#41	1	2	yes	—
112	C₂×Q₈×C₈₉	—	4	2	no	—
113	C₂×Q₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
114	C₂×Q₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
115	C₂×Q₈×C₈₉	C₄⋊C₄	1	2	yes	—
116	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
117	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
118	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
119	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
120	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
121	M₁₆×C₈₉	—	2	1	no	—
122	M₁₆×C₈₉	—	2	1	no	—
123	M₁₆×C₈₉	—	2	2	no	—
124	M₁₆×C₈₉	—	2	1	no	—
125	M₁₆×C₈₉	—	2	2	no	—
126	M₁₆×C₈₉	C₂×Q₈	1	2	yes	—
127	C₂×C₈×C₈₉	—	2	1	no	—
128	C₂×C₈×C₈₉	—	2	1	no	—
129	C₂×C₈×C₈₉	—	2	2	no	—
130	C₄⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
131	C₄⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
132	C₄⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
133	C₄⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
134	C₄⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
135	C₄⋊C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
136	C₄⋊C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
137	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
138	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
139	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
140	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
141	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
142	C₄²×C₈₉	C₂×D₄	1	2	yes	—
143	C₄²×C₈₉	C₂×D₄	1	2	yes	—
144	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
145	C₂³×C₈₉	C₂²×Q₈	1	1	yes	characteristic
146	Q₈×C₈₉	—	2	2	no	—
147	Q₈×C₈₉	—	2	2	no	—
148	Q₈×C₈₉	—	4	2	no	—
149	C₂×C₄×C₈₉	—	2	2	no	—
150	C₂×C₄×C₈₉	—	4	1	no	—
151	C₂×C₄×C₈₉	—	4	2	no	—
152	C₂×C₄×C₈₉	—	4	2	no	—
153	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
156	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
157	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
159	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
160	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
167	C₈×C₈₉	—	2	2	no	—
168	C₈×C₈₉	—	4	1	no	—
169	C₈×C₈₉	—	4	2	no	—
170	#10623494	—	2	2	no	—
171	#10623494	—	2	4	no	—
172	#10623494	—	2	2	no	—
173	#10623494	—	4	1	no	—
174	#10623494	—	4	2	no	—
175	#10623494	C₄×Q₈	1	4	yes	—
176	C₂²×C₈₉	—	2	1	no	—
177	C₂²×C₈₉	—	2	1	no	—
178	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
179	C₂²×C₈₉	#557	1	1	yes	characteristic
180	C₂²×C₈₉	Q₈²	1	1	yes	characteristic
181	C₄×C₈₉	—	2	1	no	—
182	C₄×C₈₉	—	2	2	no	—
183	C₄×C₈₉	—	2	2	no	—
184	C₄×C₈₉	—	2	1	no	—
185	C₄×C₈₉	—	2	1	no	—
186	C₄×C₈₉	—	2	2	no	—
187	C₄×C₈₉	—	2	2	no	—
188	C₄×C₈₉	#430	1	2	yes	—
189	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
190	C₄×C₈₉	#520	1	1	yes	characteristic
191	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
192	C₄×C₈₉	#550	1	1	yes	characteristic
193	#22733	—	89	1	no	maximal, Sylow
194	C₂×C₈₉	—	2	1	no	—
195	C₂×C₈₉	—	2	1	no	—
196	C₂×C₈₉	Q₈×C₄⋊C₄	1	1	yes	characteristic
197	C₂×C₈₉	C₂×#430	1	1	yes	characteristic
198	C₂×C₈₉	#2916	1	1	yes	characteristic
199	Q₈×M₁₆	—	89	1	no	—
200	Q₈×M₁₆	D₈₉	1	1	yes	characteristic
201	C₂×#446	—	89	1	no	—
202	#1950	—	89	2	no	—
203	#1950	—	89	1	no	—
204	#1950	—	89	2	no	—
205	#1950	—	89	1	no	—
206	#1849	—	89	2	no	—
207	#1848	—	89	1	no	—
208	#1778	—	89	2	no	—
209	C₄×#108	—	89	1	no	—
210	C₈₉	#22733	1	1	yes	characteristic, complemented, Sylow
211	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
212	#446	—	2	1	no	—
213	#446	—	2	2	no	—
214	#446	—	89	2	no	—
215	#446	—	89	2	no	—
216	#446	—	89	4	no	—
217	#446	—	178	1	no	—
218	#446	—	178	2	no	—
219	C₈×Q₈	—	2	1	no	—
220	C₈×Q₈	—	178	1	no	—
221	C₂×#108	—	89	1	no	—
222	C₂×#108	—	89	2	no	—
223	C₂×#108	—	89	2	no	—
224	C₂×#108	—	89	2	no	—
225	C₂×#108	—	89	2	no	—
226	C₂×#108	—	89	1	no	—
227	#423	—	89	1	no	—
228	#423	—	89	2	no	—
229	#423	C₂×D₈₉	1	1	yes	characteristic
230	#423	C₂×D₈₉	1	2	yes	—
231	C₂×C₄⋊C₈	—	89	1	no	—
232	C₂×C₄⋊C₈	—	89	2	no	—
233	C₄×M₁₆	—	89	1	no	—
234	C₄×M₁₆	—	89	2	no	—
235	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
236	C₄×M₁₆	C₂×D₈₉	1	2	yes	—
237	C₂×#97	—	89	2	no	—
238	C₂×C₄×C₈	—	89	1	no	—
239	#341	—	89	1	no	—
240	#341	—	89	1	no	—
241	#341	—	89	2	no	—
242	#341	—	89	2	no	—
243	C₂²×Q₈	Dic₁₇₈	1	1	yes	characteristic
244	C₂×M₁₆	—	89	1	no	—
245	C₂×M₁₆	—	89	1	no	—
246	C₂×M₁₆	—	89	2	no	—
247	C₂×M₁₆	C₂²×D₈₉	1	1	yes	characteristic
248	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
249	C₂×M₁₆	C₂²×D₈₉	1	1	yes	characteristic
250	C₂²×C₈	—	89	1	no	—
251	C₂²×C₈	—	89	1	no	—
252	C₂²×C₈	—	89	2	no	—
253	C₄×Q₈	C₄×D₈₉	1	1	yes	characteristic
254	C₄×Q₈	C₄×D₈₉	1	1	yes	characteristic
255	C₄×Q₈	C₄×D₈₉	1	2	yes	—
256	C₄×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
257	C₄×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
258	C₄×Q₈	C₂²⋊D₈₉	1	2	yes	—
259	C₂×C₄⋊C₄	Dic₁₇₈	1	1	yes	characteristic
260	C₂×C₄⋊C₄	Dic₁₇₈	1	2	yes	—
261	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
262	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
263	#108	—	89	4	no	—
264	#108	—	356	2	no	—
265	#108	—	356	2	no	—
266	#108	—	356	2	no	—
267	#108	—	178	4	no	—
268	#108	—	178	2	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	2	2	no	—
271	C₄⋊C₈	—	2	2	no	—
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	—	89	2	no	—
274	C₄⋊C₈	—	89	2	no	—
275	C₄⋊C₈	—	89	4	no	—
276	C₄⋊C₈	—	89	4	no	—
277	C₄⋊C₈	—	178	1	no	—
278	C₄⋊C₈	—	178	2	no	—
279	C₄⋊C₈	—	178	2	no	—
280	C₄⋊C₈	—	178	1	no	—
281	#97	—	2	2	no	—
282	#97	—	2	1	no	—
283	#97	—	89	4	no	—
284	#97	—	89	4	no	—
285	#97	—	178	2	no	—
286	#97	—	178	1	no	—
287	C₄×C₈	—	2	1	no	—
288	C₄×C₈	—	2	2	no	—
289	C₄×C₈	—	89	2	no	—
290	C₄×C₈	—	89	2	no	—
291	C₄×C₈	—	178	1	no	—
292	C₄×C₈	—	178	2	no	—
293	C₂×Q₈	—	4	2	no	—
294	C₂×Q₈	#13083553	1	1	yes	characteristic
295	C₂×Q₈	#13083553	1	1	yes	characteristic
296	C₂×Q₈	#13083553	1	2	yes	—
297	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
298	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
299	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
300	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
301	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
302	M₁₆	—	2	1	no	—
303	M₁₆	—	2	2	no	—
304	M₁₆	—	2	1	no	—
305	M₁₆	—	2	2	no	—
306	M₁₆	—	2	1	no	—
307	M₁₆	—	89	2	no	—
308	M₁₆	—	178	2	no	—
309	M₁₆	—	178	1	no	—
310	M₁₆	—	178	2	no	—
311	M₁₆	—	178	1	no	—
312	M₁₆	—	178	1	no	—
313	M₁₆	Q₈×D₈₉	1	2	yes	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	89	2	no	—
318	C₂×C₈	—	89	2	no	—
319	C₂×C₈	—	89	2	no	—
320	C₂×C₈	—	89	2	no	—
321	C₂×C₈	—	89	4	no	—
322	C₂×C₈	—	178	1	no	—
323	C₂×C₈	—	178	2	no	—
324	C₂×C₈	—	178	1	no	—
325	C₂×C₈	—	178	1	no	—
326	C₂×C₈	—	178	2	no	—
327	C₂×C₈	—	178	1	no	—
328	C₂×C₈	—	178	2	no	—
329	C₄⋊C₄	#13083553	1	2	yes	—
330	C₄⋊C₄	#13083553	1	2	yes	—
331	C₄⋊C₄	#13083553	1	2	yes	—
332	C₄⋊C₄	#13083553	1	2	yes	—
333	C₄⋊C₄	#13083553	1	2	yes	—
334	C₄⋊C₄	#13083553	1	1	yes	characteristic
335	C₄⋊C₄	#13083553	1	1	yes	characteristic
336	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
337	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
338	C₄²	C₂×C₄×D₈₉	1	2	yes	—
339	C₄²	C₂×C₄×D₈₉	1	2	yes	—
340	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
341	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
342	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
343	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
344	C₂³	C₂²×Dic₁₇₈	1	1	yes	characteristic
345	Q₈	—	2	2	no	—
346	Q₈	—	2	2	no	—
347	Q₈	—	4	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
353	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
354	C₂×C₄	C₂×#13083553	1	2	yes	—
355	C₂×C₄	C₂×#13083553	1	2	yes	—
356	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
357	C₂×C₄	C₂×#13083553	1	2	yes	—
358	C₂×C₄	C₂×#13083553	1	2	yes	—
359	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
360	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
361	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
362	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
363	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
364	C₂×C₄	#15668058	1	1	yes	characteristic
365	C₂×C₄	#15668060	1	1	yes	characteristic
366	C₈	—	2	2	no	—
367	C₈	—	4	1	no	—
368	C₈	—	4	2	no	—
369	C₈	—	89	4	no	—
370	C₈	—	356	1	no	—
371	C₈	—	356	2	no	—
372	C₈	—	178	2	no	—
373	C₈	—	178	2	no	—
374	C₈	—	178	4	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	C₂²×#13083553	1	1	yes	characteristic
378	C₂²	#19152324	1	1	yes	characteristic
379	C₂²	Q₈×Dic₁₇₈	1	1	yes	characteristic
380	C₄	—	2	1	no	—
381	C₄	—	2	2	no	—
382	C₄	—	2	2	no	—
383	C₄	—	2	1	no	—
384	C₄	—	2	1	no	—
385	C₄	—	2	2	no	—
386	C₄	—	2	2	no	—
387	C₄	#19151646	1	2	yes	—
388	C₄	C₄×Q₈×D₈₉	1	1	yes	characteristic
389	C₄	#19152080	1	1	yes	characteristic
390	C₄	Q₈×C₂²⋊D₈₉	1	1	yes	characteristic
391	C₄	#19152267	1	1	yes	characteristic
392	C₂	—	2	1	no	—
393	C₂	—	2	1	no	—
394	C₂	Q₈×#13083553	1	1	yes	characteristic
395	C₂	C₂×#19151646	1	1	yes	characteristic
396	C₂	#26028861	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	Q₈×M₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂³×C₈₉	C₂²×Q₈
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	Q₈×M₁₆	D₈₉
hypocenter	C₈₉	#22733
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
55. #15667947	Q₈	semidirect
210. C₈₉	#22733	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#13083558	2
C₂×#19151729	2
#26021027	2
#26021025	2
#26020426	2
#26020423	2
#26020159	2
C₄×#15667947	2
Q₈×M₁₆×C₈₉	2
#22733	89