Subgroups — #36712337

Order 22784.

498 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022539	C₂	1	2	yes	maximal, complemented
3	#26022532	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022406	C₂	1	2	yes	maximal, complemented
5	#26022399	C₂	1	2	yes	maximal, complemented
6	#26022396	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022389	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022161	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022126	C₂	1	2	yes	maximal
10	C₄×#15667953	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#3380	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×D₃₅₆	—	2	1	no	—
13	C₂×#15667953	C₂²	1	1	yes	characteristic
14	#19152003	C₂²	1	1	yes	characteristic
15	#19152003	C₂²	1	2	yes	—
16	#19151879	—	2	1	no	—
17	#19151879	—	2	2	no	—
18	C₄×D₇₁₂	C₂²	1	2	yes	—
19	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
20	#19151425	C₂²	1	2	yes	—
21	#19151425	C₂²	1	2	yes	—
22	#19151425	C₂²	1	2	yes	—
23	#19151423	C₂²	1	2	yes	—
24	#19151423	C₂²	1	1	yes	characteristic
25	C₄⋊#13083549	C₂²	1	2	yes	—
26	C₄⋊#13083549	C₂²	1	1	yes	characteristic
27	#19151399	C₂²	1	1	yes	characteristic
28	#19151399	C₂²	1	2	yes	—
29	#19151398	C₂²	1	1	yes	characteristic
30	#19151398	C₂²	1	2	yes	—
31	#19151374	C₂²	1	2	yes	—
32	C₄×#13083549	C₂²	1	1	yes	characteristic
33	#554×C₈₉	—	2	1	no	—
34	#500×C₈₉	C₂²	1	1	yes	characteristic
35	#499×C₈₉	C₂²	1	2	yes	—
36	#478×C₈₉	—	2	2	no	—
37	#476×C₈₉	—	2	1	no	—
38	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₈₉	C₂²	1	2	yes	—
41	C₂×Q₈×D₈₉	—	4	1	no	—
42	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
43	#15668012	—	2	2	no	—
44	#15668012	—	2	1	no	—
45	#15667992	—	4	1	no	—
46	#15667992	—	4	2	no	—
47	C₄×D₃₅₆	—	2	2	no	—
48	C₄×D₃₅₆	—	2	1	no	—
49	#15667953	Q₈	1	4	yes	complemented
50	C₂×#13083549	C₂³	1	1	yes	characteristic
51	C₂×#13083549	C₂³	1	2	yes	—
52	C₂×#13083549	C₂³	1	1	yes	characteristic
53	#15667946	—	2	2	no	—
54	#15667946	—	2	1	no	—
55	#15667946	C₂³	1	1	yes	characteristic
56	#15667946	C₂³	1	2	yes	—
57	#15667934	—	2	2	no	—
58	#15667934	—	2	1	no	—
59	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₈₉	—	2	1	no	—
61	#125×C₈₉	—	2	2	no	—
62	C₂²⋊Q₈×C₈₉	—	4	1	no	—
63	C₂²⋊Q₈×C₈₉	—	4	2	no	—
64	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
65	C₄×D₄×C₈₉	—	2	1	no	—
66	C₄×D₄×C₈₉	—	2	2	no	—
67	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
68	#107×C₈₉	—	2	1	no	—
69	#107×C₈₉	C₂³	1	1	yes	characteristic
70	#107×C₈₉	C₂³	1	2	yes	—
71	#106×C₈₉	—	2	2	no	—
72	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
73	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
74	#102×C₈₉	—	2	2	no	—
75	#102×C₈₉	—	2	1	no	—
76	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₈₉	C₂³	1	2	yes	—
78	C₂×Dic₁₇₈	—	4	2	no	—
79	C₂×Dic₁₇₈	—	4	1	no	—
80	Q₈×D₈₉	—	8	4	no	—
81	C₂×D₃₅₆	—	2	1	no	—
82	C₂×C₄×D₈₉	—	4	2	no	—
83	C₂×C₄×D₈₉	—	4	1	no	—
84	D₇₁₂	—	2	1	no	—
85	D₇₁₂	C₂×Q₈	1	2	yes	—
86	C₄⋊Dic₈₉	—	2	2	no	—
87	C₄⋊Dic₈₉	—	2	1	no	—
88	C₄⋊Dic₈₉	—	4	1	no	—
89	C₄⋊Dic₈₉	—	4	2	no	—
90	#13083551	—	4	1	no	—
91	#13083551	—	4	2	no	—
92	#13083549	—	2	1	no	—
93	#13083549	—	2	2	no	—
94	#13083549	C₂×Q₈	1	2	yes	complemented
95	C₂×Q₈×C₈₉	—	4	1	no	—
96	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
97	C₂×D₄×C₈₉	—	2	1	no	—
98	C₂²×C₄×C₈₉	—	4	1	no	—
99	C₂²×C₄×C₈₉	—	4	2	no	—
100	D₈×C₈₉	—	2	1	no	—
101	D₈×C₈₉	C₂×Q₈	1	2	yes	complemented
102	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	—	2	1	no	—
106	C₄⋊C₄×C₈₉	—	2	2	no	—
107	C₄⋊C₄×C₈₉	—	4	2	no	—
108	C₄⋊C₄×C₈₉	—	4	1	no	—
109	C₄⋊C₄×C₈₉	—	4	2	no	—
110	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
112	C₂²⋊C₄×C₈₉	—	4	1	no	—
113	C₂²⋊C₄×C₈₉	—	4	2	no	—
114	C₄²×C₈₉	C₂×D₄	1	2	yes	—
115	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂³×C₈₉	—	4	1	no	—
117	C₂²×D₈₉	—	4	1	no	—
118	Q₈×C₈₉	—	2	2	no	—
119	Q₈×C₈₉	—	16	1	no	—
120	D₄×C₈₉	—	2	2	no	—
121	D₄×C₈₉	—	4	1	no	—
122	C₂×C₄×C₈₉	—	4	1	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	8	2	no	—
125	C₂×C₄×C₈₉	—	16	2	no	—
126	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×Dic₈₉	—	4	1	no	—
132	C₂×Dic₈₉	—	4	2	no	—
133	D₃₅₆	—	2	2	no	—
134	D₃₅₆	—	4	1	no	—
135	C₄×D₈₉	—	8	4	no	—
136	C₄×D₈₉	—	8	2	no	—
137	Dic₁₇₈	—	8	4	no	—
138	Dic₁₇₈	—	8	2	no	—
139	C₈×C₈₉	—	2	2	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
142	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂²×C₈₉	—	4	2	no	—
144	C₂²×C₈₉	—	8	1	no	—
145	C₂²×C₈₉	—	8	1	no	—
146	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₈₉	—	4	2	no	—
148	C₂×D₈₉	—	8	1	no	—
149	C₂×D₈₉	—	8	1	no	—
150	C₄×C₈₉	—	2	2	no	—
151	C₄×C₈₉	—	2	2	no	—
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	8	1	no	—
154	C₄×C₈₉	—	8	2	no	—
155	C₄×C₈₉	#509	1	2	yes	—
156	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
157	C₄×C₈₉	#578	1	1	yes	characteristic
158	Dic₈₉	—	8	1	no	—
159	Dic₈₉	—	8	2	no	—
160	#33951	—	89	1	no	maximal, Sylow
161	C₂×C₈₉	—	8	2	no	—
162	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
163	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
164	C₂×C₈₉	#3416	1	1	yes	characteristic
165	D₈₉	—	8	2	no	—
166	#3380	D₈₉	1	1	yes	characteristic, complemented
167	Q₈×D₈	—	89	1	no	—
168	#2255	—	89	2	no	—
169	#2254	—	89	1	no	—
170	#2228	—	89	2	no	—
171	#2226	—	89	1	no	—
172	#2226	—	89	2	no	—
173	#2224	—	89	1	no	—
174	#2183	—	89	1	no	—
175	#2177	—	89	2	no	—
176	C₄×D₁₆	—	89	1	no	—
177	C₈₉	#33951	1	1	yes	characteristic, complemented, Sylow
178	#554	—	2	1	no	—
179	D₄×Q₈	—	178	1	no	—
180	C₂×D₁₆	—	89	1	no	—
181	#500	—	89	1	no	—
182	#500	—	89	2	no	—
183	#500	C₂×D₈₉	1	1	yes	characteristic
184	#499	C₂×D₈₉	1	2	yes	—
185	#478	—	2	2	no	—
186	#476	—	2	1	no	—
187	#474	—	178	1	no	—
188	#474	—	178	2	no	—
189	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
190	C₄×D₈	—	89	1	no	—
191	C₄×D₈	—	89	2	no	—
192	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
193	C₄×D₈	C₂×D₈₉	1	2	yes	—
194	#367	—	89	2	no	—
195	#367	—	89	2	no	—
196	#367	—	89	2	no	—
197	#366	—	89	1	no	—
198	#366	—	89	2	no	—
199	C₄⋊C₁₆	—	89	1	no	—
200	C₄⋊C₁₆	—	89	2	no	—
201	#357	—	89	1	no	—
202	#357	—	89	2	no	—
203	#357	—	89	1	no	—
204	#357	—	89	2	no	—
205	#346	—	89	2	no	—
206	C₄×C₁₆	—	89	1	no	—
207	C₂²×Q₈	—	356	1	no	—
208	C₂×D₈	—	89	1	no	—
209	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	178	2	no	—
212	C₄⋊Q₈	—	178	1	no	—
213	#125	—	2	2	no	—
214	C₂²⋊Q₈	—	4	1	no	—
215	C₂²⋊Q₈	—	4	2	no	—
216	C₂²⋊Q₈	—	356	1	no	—
217	C₂²⋊Q₈	—	356	2	no	—
218	C₄×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
219	C₄×D₄	—	2	1	no	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	—	178	2	no	—
222	C₄×D₄	—	178	1	no	—
223	C₂×C₄⋊C₄	—	4	1	no	—
224	D₁₆	—	89	4	no	—
225	C₂×C₁₆	—	89	1	no	—
226	C₂×C₁₆	—	89	1	no	—
227	C₂×C₁₆	—	89	2	no	—
228	#107	—	2	1	no	—
229	#107	—	89	1	no	—
230	#107	—	89	2	no	—
231	#107	—	178	2	no	—
232	#107	—	178	1	no	—
233	#107	C₂²×D₈₉	1	1	yes	characteristic
234	#107	C₂²×D₈₉	1	2	yes	—
235	#106	—	2	2	no	—
236	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
237	C₄⋊C₈	C₂²⋊D₈₉	1	2	yes	—
238	#102	—	2	2	no	—
239	#102	—	2	1	no	—
240	#102	—	178	2	no	—
241	#102	—	178	1	no	—
242	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₈₉	1	2	yes	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	356	2	no	—
246	C₂×Q₈	—	356	1	no	—
247	C₂×Q₈	—	712	4	no	—
248	C₂×Q₈	#13083559	1	1	yes	characteristic
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	178	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	356	2	no	—
254	C₂²×C₄	—	356	1	no	—
255	D₈	—	2	1	no	—
256	D₈	—	89	2	no	—
257	D₈	—	178	1	no	—
258	D₈	Q₈×D₈₉	1	2	yes	complemented
259	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
260	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
261	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	356	1	no	—
268	C₄⋊C₄	—	356	2	no	—
269	C₄⋊C₄	—	178	2	no	—
270	C₄⋊C₄	—	178	1	no	—
271	C₄⋊C₄	#13083559	1	1	yes	characteristic
272	C₄⋊C₄	#13083559	1	2	yes	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	356	1	no	—
276	C₂²⋊C₄	—	356	2	no	—
277	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
278	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
279	C₁₆	—	89	2	no	—
280	C₁₆	—	178	1	no	—
281	C₁₆	—	178	2	no	—
282	C₂³	—	4	1	no	—
283	C₂³	—	356	1	no	—
284	Q₈	—	2	2	no	—
285	Q₈	—	16	1	no	—
286	Q₈	—	712	4	no	—
287	Q₈	—	712	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	1	no	—
290	D₄	—	356	1	no	—
291	D₄	—	178	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	8	2	no	—
295	C₂×C₄	—	16	2	no	—
296	C₂×C₄	—	356	2	no	—
297	C₂×C₄	—	356	1	no	—
298	C₂×C₄	—	712	4	no	—
299	C₂×C₄	—	712	2	no	—
300	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
301	C₂×C₄	C₂×#13083559	1	2	yes	—
302	C₂×C₄	C₂×#13083559	1	2	yes	—
303	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	2	1	no	—
307	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
308	C₈	#15668060	1	1	yes	characteristic
309	C₂²	—	4	2	no	—
310	C₂²	—	8	1	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	356	2	no	—
313	C₂²	—	712	1	no	—
314	C₂²	—	712	1	no	—
315	C₂²	C₂²×#13083559	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	8	2	no	—
321	C₄	—	712	2	no	—
322	C₄	—	712	1	no	—
323	C₄	#19152033	1	2	yes	—
324	C₄	Q₈×C₂²⋊D₈₉	1	1	yes	characteristic
325	C₄	#19152413	1	1	yes	characteristic
326	C₂	—	8	2	no	—
327	C₂	—	712	2	no	—
328	C₂	Q₈×#13083559	1	1	yes	characteristic
329	C₂	C₂×#19152033	1	1	yes	characteristic
330	C₂	#26033399	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3380	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#15667953	C₂²
hypercenter	#3380	D₈₉
hypocenter	C₈₉	#33951
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3380

Complements

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

N	H ≅ G/N	type
1. Q₈×D₇₁₂	C₂	semidirect
2. #26022539	C₂	semidirect
3. #26022532	C₂	semidirect
4. #26022406	C₂	semidirect
5. #26022399	C₂	semidirect
6. #26022396	C₂	semidirect
7. #26022389	C₂	semidirect
8. #26022161	C₂	semidirect
11. C₈₉×#3380	C₂	semidirect
49. #15667953	Q₈	semidirect
94. #13083549	C₂×Q₈	semidirect
101. D₈×C₈₉	C₂×Q₈	semidirect
166. #3380	D₈₉	semidirect
177. C₈₉	#33951	semidirect
258. D₈	Q₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₇₁₂	2
#26022539	2
#26022532	2
#26022406	2
#26022399	2
#26022396	2
#26022389	2
#26022161	2
#26022126	2
C₄×#15667953	2
C₈₉×#3380	2
#33951	89