Subgroups — #36453381

Order 22784.

568 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#10623494	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10623494	—	2	1	no	—
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	C₂×C₄×#10623494	—	2	1	no	—
8	C₂×C₄×#10623494	C₂²	1	2	yes	—
9	#19151353	—	2	1	no	—
10	#19151353	—	2	1	no	—
11	Q₈⋊#10623494	—	2	8	no	—
12	C₂×C₄⋊Q₈×C₈₉	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
14	#391×C₈₉	C₄	1	2	yes	—
15	#389×C₈₉	—	2	1	no	—
16	#389×C₈₉	—	2	1	no	—
17	#384×C₈₉	C₄	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
19	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
20	C₄³×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×#10623494	—	2	2	no	—
22	C₂²×#10623494	—	2	2	no	—
23	C₄⋊#10623494	—	2	4	no	—
24	C₄⋊#10623494	—	2	2	no	—
25	C₄⋊#10623494	D₄	1	4	yes	—
26	C₄×#10623494	—	2	4	no	—
27	C₄×#10623494	—	4	2	no	—
28	C₂²×Q₈×C₈₉	C₂×C₄	1	2	yes	—
29	C₄⋊Q₈×C₈₉	—	2	2	no	—
30	C₄⋊Q₈×C₈₉	C₈	1	4	yes	complemented
31	C₄×Q₈×C₈₉	—	2	4	no	—
32	C₄×Q₈×C₈₉	—	2	4	no	—
33	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₈₉	—	2	2	no	—
40	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
47	#95×C₈₉	—	2	2	no	—
48	#95×C₈₉	—	2	2	no	—
49	C₂×#10623494	—	2	4	no	—
50	C₂×#10623494	—	2	4	no	—
51	C₂×#10623494	—	4	2	no	—
52	C₂×#10623494	—	4	2	no	—
53	C₂×#10623494	—	4	2	no	—
54	C₂×#10623494	—	4	2	no	—
55	C₂×Q₈×C₈₉	—	2	2	no	—
56	C₂×Q₈×C₈₉	—	2	4	no	—
57	C₂×Q₈×C₈₉	—	2	4	no	—
58	C₂×Q₈×C₈₉	C₂×C₈	1	4	yes	—
59	C₂²×C₄×C₈₉	—	2	2	no	—
60	C₂²×C₄×C₈₉	—	2	2	no	—
61	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₈₉	—	2	4	no	—
69	C₄⋊C₄×C₈₉	—	2	4	no	—
70	C₄⋊C₄×C₈₉	—	2	4	no	—
71	C₄⋊C₄×C₈₉	—	4	2	no	—
72	C₄⋊C₄×C₈₉	—	4	2	no	—
73	C₄⋊C₄×C₈₉	—	4	2	no	—
74	C₄⋊C₄×C₈₉	—	4	2	no	—
75	C₄²×C₈₉	—	2	2	no	—
76	C₄²×C₈₉	—	2	2	no	—
77	C₄²×C₈₉	—	2	2	no	—
78	C₄²×C₈₉	—	4	1	no	—
79	C₄²×C₈₉	—	4	4	no	—
80	C₄²×C₈₉	C₂×C₈	1	2	yes	—
81	C₄²×C₈₉	M₁₆	1	2	yes	—
82	C₄²×C₈₉	SD₁₆	1	2	yes	—
83	C₄²×C₈₉	Q₁₆	1	2	yes	—
84	C₄²×C₈₉	C₂×D₄	1	2	yes	—
85	C₄²×C₈₉	#41	1	2	yes	—
86	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	Q₈×C₈₉	—	2	8	no	—
88	Q₈×C₈₉	—	4	4	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	1	no	—
91	C₂×C₄×C₈₉	—	2	1	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	4	no	—
96	C₂×C₄×C₈₉	—	2	1	no	—
97	C₂×C₄×C₈₉	—	2	1	no	—
98	C₂×C₄×C₈₉	—	2	1	no	—
99	C₂×C₄×C₈₉	—	4	2	no	—
100	C₂×C₄×C₈₉	—	4	2	no	—
101	C₂×C₄×C₈₉	—	4	2	no	—
102	C₂×C₄×C₈₉	—	4	2	no	—
103	C₂×C₄×C₈₉	—	4	2	no	—
104	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
105	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
106	C₂×C₄×C₈₉	#103	1	2	yes	—
107	C₂×C₄×C₈₉	#103	1	2	yes	—
108	C₂×C₄×C₈₉	#104	1	2	yes	—
109	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₈₉	#121	1	2	yes	—
111	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
112	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₈₉	#131	1	2	yes	—
116	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#137	1	2	yes	—
119	#10623494	—	4	4	no	—
120	#10623494	—	4	4	no	—
121	C₂²×C₈₉	#390	1	1	yes	characteristic
122	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₂²×C₈₉	#408	1	1	yes	characteristic
124	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
125	C₂²×C₈₉	#419	1	1	yes	characteristic
126	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
127	C₂²×C₈₉	#421	1	1	yes	characteristic
128	C₄×C₈₉	—	2	1	no	—
129	C₄×C₈₉	—	2	1	no	—
130	C₄×C₈₉	—	2	2	no	—
131	C₄×C₈₉	—	2	2	no	—
132	C₄×C₈₉	—	4	4	no	—
133	C₄×C₈₉	—	4	1	no	—
134	C₄×C₈₉	—	8	2	no	—
135	C₄×C₈₉	Q₈⋊C₈	1	4	yes	complemented
136	C₄×C₈₉	C₈×D₄	1	2	yes	—
137	C₄×C₈₉	#436	1	2	yes	—
138	C₄×C₈₉	#461	1	2	yes	—
139	C₄×C₈₉	#462	1	2	yes	—
140	#9673	—	89	1	no	maximal, Sylow
141	C₂×C₈₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
142	C₂×C₈₉	#1479	1	1	yes	characteristic
143	C₂×C₈₉	#1481	1	1	yes	characteristic
144	C₂×C₈₉	#1485	1	1	yes	characteristic
145	C₂×C₈₉	#1954	1	1	yes	characteristic
146	C₂×C₈₉	#1980	1	1	yes	characteristic
147	C₂×C₈₉	#1981	1	1	yes	characteristic
148	C₄×C₄⋊Q₈	D₈₉	1	1	yes	characteristic
149	C₄×C₄⋊C₈	—	89	1	no	—
150	C₄×C₄⋊C₈	—	89	1	no	—
151	C₂×Q₈⋊C₈	—	89	4	no	—
152	C₈₉	#9673	1	1	yes	characteristic, complemented, Sylow
153	C₂×C₄⋊Q₈	Dic₈₉	1	1	yes	characteristic
154	C₂×C₄×Q₈	C₂×D₈₉	1	2	yes	—
155	C₂×C₄⋊C₈	—	89	2	no	—
156	C₂×C₄⋊C₈	—	178	1	no	—
157	C₂×C₄×C₈	—	89	2	no	—
158	C₂×C₄×C₈	—	178	1	no	—
159	#391	Dic₈₉	1	2	yes	—
160	#389	—	2	1	no	—
161	#389	—	2	1	no	—
162	#384	Dic₈₉	1	1	yes	characteristic
163	C₄×C₄⋊C₄	—	2	1	no	—
164	C₄×C₄⋊C₄	—	2	1	no	—
165	C₄³	C₂×D₈₉	1	1	yes	characteristic
166	#336	—	178	1	no	—
167	#336	—	178	1	no	—
168	Q₈⋊C₈	—	178	8	no	—
169	C₂²×Q₈	C₂×Dic₈₉	1	2	yes	—
170	C₂²×C₈	—	178	2	no	—
171	C₂²×C₈	—	178	2	no	—
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	#10623494	1	4	yes	complemented
174	C₄×Q₈	—	2	4	no	—
175	C₄×Q₈	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
182	C₂×C₄²	—	2	2	no	—
183	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
184	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
186	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
188	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
189	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
190	C₄⋊C₈	—	89	4	no	—
191	C₄⋊C₈	—	178	4	no	—
192	C₄⋊C₈	—	178	2	no	—
193	C₄×C₈	—	356	2	no	—
194	C₄×C₈	—	178	4	no	—
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	C₂×#10623494	1	4	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
205	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
210	C₂×C₈	—	356	2	no	—
211	C₂×C₈	—	356	2	no	—
212	C₂×C₈	—	356	2	no	—
213	C₂×C₈	—	356	2	no	—
214	C₂×C₈	—	178	4	no	—
215	C₂×C₈	—	178	4	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	4	1	no	—
227	C₄²	—	4	4	no	—
228	C₄²	C₂×#10623494	1	2	yes	—
229	C₄²	#13083558	1	2	yes	—
230	C₄²	#13083561	1	2	yes	—
231	C₄²	#13083564	1	2	yes	—
232	C₄²	D₄×D₈₉	1	2	yes	—
233	C₄²	#13083573	1	2	yes	—
234	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
235	Q₈	—	2	8	no	—
236	Q₈	—	4	4	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	C₂²⋊#10623494	1	2	yes	—
253	C₂×C₄	C₂²⋊#10623494	1	2	yes	—
254	C₂×C₄	#15667936	1	2	yes	—
255	C₂×C₄	#15667936	1	2	yes	—
256	C₂×C₄	#15667939	1	2	yes	—
257	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
258	C₂×C₄	#15667983	1	2	yes	—
259	C₂×C₄	#15667999	1	1	yes	characteristic
260	C₂×C₄	#15668008	1	1	yes	characteristic
261	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
262	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
263	C₂×C₄	#15668019	1	2	yes	—
264	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
265	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
266	C₂×C₄	#15668043	1	2	yes	—
267	C₈	—	356	4	no	—
268	C₈	—	356	4	no	—
269	C₂²	#19151501	1	1	yes	characteristic
270	C₂²	C₂×C₂²⋊#10623494	1	1	yes	characteristic
271	C₂²	#19151565	1	1	yes	characteristic
272	C₂²	C₂×#15667936	1	1	yes	characteristic
273	C₂²	#19151607	1	1	yes	characteristic
274	C₂²	C₂×#15667939	1	1	yes	characteristic
275	C₂²	#19151615	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	4	no	—
282	C₄	—	8	2	no	—
283	C₄	Q₈⋊#10623494	1	4	yes	complemented
284	C₄	D₄×#10623494	1	2	yes	—
285	C₄	#19151676	1	2	yes	—
286	C₄	#19151800	1	2	yes	—
287	C₄	#19151805	1	2	yes	—
288	C₂	C₂×Q₈⋊#10623494	1	1	yes	characteristic
289	C₂	#26018647	1	1	yes	characteristic
290	C₂	#26018655	1	1	yes	characteristic
291	C₂	#26018673	1	1	yes	characteristic
292	C₂	#26021047	1	1	yes	characteristic
293	C₂	#26021172	1	1	yes	characteristic
294	C₂	#26021176	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×C₄⋊Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×C₄⋊Q₈	D₈₉
hypocenter	C₈₉	#9673
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
30. C₄⋊Q₈×C₈₉	C₈	semidirect
135. C₄×C₈₉	Q₈⋊C₈	semidirect
152. C₈₉	#9673	semidirect
173. C₄⋊Q₈	#10623494	semidirect
283. C₄	Q₈⋊#10623494	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#10623494	2
C₂×Q₈⋊#10623494	2
C₄×C₄⋊Q₈×C₈₉	2
#9673	89