Subgroups — #30601725

Order 15616.

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⋊#92420	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#92420	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#92420	C₂	1	4	yes	maximal
4	C₄⋊Q₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#92420	—	2	1	no	—
6	C₂×C₄⋊#92420	C₂²	1	2	yes	—
7	C₂×C₄×#92420	—	2	1	no	—
8	C₂×C₄×#92420	C₂²	1	2	yes	—
9	#16840628	—	2	1	no	—
10	#16840628	—	2	1	no	—
11	Q₈⋊#92420	—	2	8	no	—
12	C₂×Q₈×Dic₆₁	C₂²	1	2	yes	—
13	#16839669	C₄	1	2	yes	—
14	#16839661	—	2	1	no	—
15	#16839661	—	2	1	no	—
16	#16839632	C₄	1	1	yes	characteristic
17	C₄⋊C₄×Dic₆₁	—	2	1	no	—
18	C₄⋊C₄×Dic₆₁	—	2	1	no	—
19	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
21	C₂²×#92420	—	2	2	no	—
22	C₂²×#92420	—	2	2	no	—
23	C₄⋊#92420	—	2	4	no	—
24	C₄⋊#92420	—	2	2	no	—
25	C₄⋊#92420	D₄	1	4	yes	—
26	C₄×#92420	—	2	4	no	—
27	C₄×#92420	—	4	2	no	—
28	Q₈×Dic₆₁	—	2	4	no	—
29	Q₈×Dic₆₁	—	2	4	no	—
30	C₂×C₄⋊Dic₆₁	—	2	1	no	—
31	C₂×C₄⋊Dic₆₁	—	2	1	no	—
32	C₂×C₄⋊Dic₆₁	—	2	2	no	—
33	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₆₁	—	2	2	no	—
35	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
37	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
38	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
40	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
41	#14519684	—	2	2	no	—
42	#14519684	—	2	2	no	—
43	C₂²×Q₈×C₆₁	C₂×C₄	1	2	yes	—
44	C₄⋊Q₈×C₆₁	—	2	2	no	—
45	C₄⋊Q₈×C₆₁	C₈	1	4	yes	complemented
46	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
47	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
48	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₂×Q₈×C₆₁	—	2	2	no	—
50	C₂×Q₈×C₆₁	—	2	4	no	—
51	C₂×Q₈×C₆₁	—	2	4	no	—
52	C₂×Q₈×C₆₁	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₆₁	—	2	2	no	—
54	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₂²×Dic₆₁	—	2	2	no	—
58	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂×#92420	—	2	4	no	—
63	C₂×#92420	—	2	4	no	—
64	C₂×#92420	—	4	2	no	—
65	C₂×#92420	—	4	2	no	—
66	C₂×#92420	—	4	2	no	—
67	C₂×#92420	—	4	2	no	—
68	C₄⋊C₄×C₆₁	—	2	4	no	—
69	C₄⋊C₄×C₆₁	—	2	4	no	—
70	C₄²×C₆₁	C₂×C₈	1	2	yes	—
71	C₄²×C₆₁	M₁₆	1	2	yes	—
72	C₄⋊Dic₆₁	—	2	4	no	—
73	C₄⋊Dic₆₁	—	4	2	no	—
74	C₄⋊Dic₆₁	—	4	2	no	—
75	C₄⋊Dic₆₁	—	4	2	no	—
76	C₄⋊Dic₆₁	—	4	2	no	—
77	C₄×Dic₆₁	—	2	2	no	—
78	C₄×Dic₆₁	—	2	2	no	—
79	C₄×Dic₆₁	—	2	2	no	—
80	C₄×Dic₆₁	—	4	1	no	—
81	C₄×Dic₆₁	—	4	4	no	—
82	C₄×Dic₆₁	SD₁₆	1	2	yes	—
83	C₄×Dic₆₁	Q₁₆	1	2	yes	—
84	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
85	C₄×Dic₆₁	#41	1	2	yes	—
86	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	Q₈×C₆₁	—	2	8	no	—
88	Q₈×C₆₁	—	4	4	no	—
89	C₂×C₄×C₆₁	—	2	1	no	—
90	C₂×C₄×C₆₁	—	2	1	no	—
91	C₂×C₄×C₆₁	—	2	4	no	—
92	C₂×C₄×C₆₁	—	4	2	no	—
93	C₂×C₄×C₆₁	—	4	2	no	—
94	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₆₁	#103	1	2	yes	—
97	C₂×C₄×C₆₁	#104	1	2	yes	—
98	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₁	#131	1	2	yes	—
102	C₂×Dic₆₁	—	2	1	no	—
103	C₂×Dic₆₁	—	2	1	no	—
104	C₂×Dic₆₁	—	2	1	no	—
105	C₂×Dic₆₁	—	2	1	no	—
106	C₂×Dic₆₁	—	2	1	no	—
107	C₂×Dic₆₁	—	2	1	no	—
108	C₂×Dic₆₁	—	2	1	no	—
109	C₂×Dic₆₁	—	4	2	no	—
110	C₂×Dic₆₁	—	4	2	no	—
111	C₂×Dic₆₁	—	4	2	no	—
112	C₂×Dic₆₁	#103	1	2	yes	—
113	C₂×Dic₆₁	#121	1	2	yes	—
114	C₂×Dic₆₁	#124	1	1	yes	characteristic
115	C₂×Dic₆₁	#127	1	1	yes	characteristic
116	C₂×Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×Dic₆₁	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×Dic₆₁	#137	1	2	yes	—
119	#92420	—	4	4	no	—
120	#92420	—	4	4	no	—
121	#9673	—	61	1	no	maximal, Sylow
122	C₂²×C₆₁	#390	1	1	yes	characteristic
123	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	C₂²×C₆₁	#408	1	1	yes	characteristic
125	C₂²×C₆₁	C₂×#103	1	1	yes	characteristic
126	C₂²×C₆₁	#419	1	1	yes	characteristic
127	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
128	C₂²×C₆₁	#421	1	1	yes	characteristic
129	C₄×C₆₁	—	2	2	no	—
130	C₄×C₆₁	—	4	4	no	—
131	C₄×C₆₁	Q₈⋊C₈	1	4	yes	complemented
132	C₄×C₆₁	C₈×D₄	1	2	yes	—
133	C₄×C₆₁	#436	1	2	yes	—
134	Dic₆₁	—	2	1	no	—
135	Dic₆₁	—	2	1	no	—
136	Dic₆₁	—	2	2	no	—
137	Dic₆₁	—	4	1	no	—
138	Dic₆₁	—	8	2	no	—
139	Dic₆₁	#461	1	2	yes	—
140	Dic₆₁	#462	1	2	yes	—
141	C₄×C₄⋊Q₈	—	61	1	no	—
142	C₄×C₄⋊C₈	—	61	1	no	—
143	C₄×C₄⋊C₈	—	61	1	no	—
144	C₂×Q₈⋊C₈	—	61	4	no	—
145	C₂×C₆₁	C₂×Q₈⋊C₈	1	1	yes	characteristic
146	C₂×C₆₁	#1479	1	1	yes	characteristic
147	C₂×C₆₁	#1481	1	1	yes	characteristic
148	C₂×C₆₁	#1485	1	1	yes	characteristic
149	C₂×C₆₁	#1954	1	1	yes	characteristic
150	C₂×C₆₁	#1980	1	1	yes	characteristic
151	C₂×C₆₁	#1981	1	1	yes	characteristic
152	C₂×C₄⋊Q₈	#6923	1	1	yes	characteristic
153	C₂×C₄×Q₈	—	61	2	no	—
154	C₂×C₄⋊C₈	—	61	2	no	—
155	C₂×C₄⋊C₈	—	122	1	no	—
156	C₂×C₄×C₈	—	61	2	no	—
157	C₂×C₄×C₈	—	122	1	no	—
158	#391	—	61	2	no	—
159	#389	—	122	1	no	—
160	#389	—	122	1	no	—
161	#384	—	61	1	no	—
162	C₄×C₄⋊C₄	—	122	1	no	—
163	C₄×C₄⋊C₄	—	122	1	no	—
164	C₄³	—	61	1	no	—
165	#336	—	122	1	no	—
166	#336	—	122	1	no	—
167	Q₈⋊C₈	—	122	8	no	—
168	C₆₁	#9673	1	1	yes	characteristic, complemented, Sylow
169	C₂²×Q₈	C₂×#6923	1	2	yes	—
170	C₂²×C₈	—	122	2	no	—
171	C₂²×C₈	—	122	2	no	—
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	#92420	1	4	yes	complemented
174	C₄×Q₈	—	122	4	no	—
175	C₄×Q₈	—	122	4	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	61	2	no	—
179	C₂×C₄⋊C₄	—	122	2	no	—
180	C₂×C₄⋊C₄	—	122	1	no	—
181	C₂×C₄⋊C₄	—	122	1	no	—
182	C₂×C₄²	—	61	1	no	—
183	C₂×C₄²	—	61	1	no	—
184	C₂×C₄²	—	61	1	no	—
185	C₂×C₄²	—	61	1	no	—
186	C₂×C₄²	—	61	1	no	—
187	C₂×C₄²	—	61	1	no	—
188	C₂×C₄²	—	122	2	no	—
189	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
190	C₄⋊C₈	—	61	4	no	—
191	C₄⋊C₈	—	122	2	no	—
192	C₄⋊C₈	—	122	4	no	—
193	C₄×C₈	—	122	4	no	—
194	C₄×C₈	—	244	2	no	—
195	#95	—	122	2	no	—
196	#95	—	122	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	C₂×#92420	1	4	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	61	1	no	—
203	C₂²×C₄	—	61	1	no	—
204	C₂²×C₄	—	61	1	no	—
205	C₂²×C₄	—	61	1	no	—
206	C₂²×C₄	—	122	2	no	—
207	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
209	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
210	C₂×C₈	—	122	4	no	—
211	C₂×C₈	—	122	4	no	—
212	C₂×C₈	—	244	2	no	—
213	C₂×C₈	—	244	2	no	—
214	C₂×C₈	—	244	2	no	—
215	C₂×C₈	—	244	2	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	122	4	no	—
219	C₄⋊C₄	—	244	2	no	—
220	C₄⋊C₄	—	244	2	no	—
221	C₄⋊C₄	—	244	2	no	—
222	C₄⋊C₄	—	244	2	no	—
223	C₄²	—	61	2	no	—
224	C₄²	—	61	2	no	—
225	C₄²	—	61	2	no	—
226	C₄²	—	61	2	no	—
227	C₄²	—	122	2	no	—
228	C₄²	—	122	2	no	—
229	C₄²	—	122	2	no	—
230	C₄²	—	244	1	no	—
231	C₄²	—	244	4	no	—
232	C₄²	C₂×#92420	1	2	yes	—
233	C₄²	#11758217	1	2	yes	—
234	C₂³	C₂×C₂²⋊#6923	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	4	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	61	2	no	—
243	C₂×C₄	—	61	1	no	—
244	C₂×C₄	—	61	1	no	—
245	C₂×C₄	—	61	2	no	—
246	C₂×C₄	—	61	1	no	—
247	C₂×C₄	—	61	1	no	—
248	C₂×C₄	—	61	2	no	—
249	C₂×C₄	—	122	1	no	—
250	C₂×C₄	—	122	1	no	—
251	C₂×C₄	—	122	1	no	—
252	C₂×C₄	—	122	1	no	—
253	C₂×C₄	—	122	1	no	—
254	C₂×C₄	—	122	1	no	—
255	C₂×C₄	—	122	1	no	—
256	C₂×C₄	—	244	2	no	—
257	C₂×C₄	—	244	2	no	—
258	C₂×C₄	—	244	2	no	—
259	C₂×C₄	C₂²⋊#92420	1	2	yes	—
260	C₂×C₄	C₂²⋊#92420	1	2	yes	—
261	C₂×C₄	#14519840	1	2	yes	—
262	C₂×C₄	#14519842	1	2	yes	—
263	C₂×C₄	D₄×#6923	1	2	yes	—
264	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
265	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
266	C₂×C₄	#14519864	1	2	yes	—
267	C₈	—	244	4	no	—
268	C₈	—	244	4	no	—
269	C₂²	#16840714	1	1	yes	characteristic
270	C₂²	C₂×C₂²⋊#92420	1	1	yes	characteristic
271	C₂²	#16840736	1	1	yes	characteristic
272	C₂²	C₂×#14519840	1	1	yes	characteristic
273	C₂²	#16840755	1	1	yes	characteristic
274	C₂²	C₂×#14519842	1	1	yes	characteristic
275	C₂²	#16840761	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	4	4	no	—
278	C₄	—	61	2	no	—
279	C₄	—	61	2	no	—
280	C₄	—	122	2	no	—
281	C₄	—	122	1	no	—
282	C₄	—	122	1	no	—
283	C₄	—	488	2	no	—
284	C₄	—	244	1	no	—
285	C₄	Q₈⋊#92420	1	4	yes	complemented
286	C₄	D₄×#92420	1	2	yes	—
287	C₄	#16840797	1	2	yes	—
288	C₂	C₂×Q₈⋊#92420	1	1	yes	characteristic
289	C₂	#21461121	1	1	yes	characteristic
290	C₂	#21461128	1	1	yes	characteristic
291	C₂	#21461142	1	1	yes	characteristic
292	C₂	#21461940	1	1	yes	characteristic
293	C₂	#21461982	1	1	yes	characteristic
294	C₂	#21461985	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×C₄⋊Q₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×C₄⋊Q₈	#6923
hypocenter	C₆₁	#9673
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
45. C₄⋊Q₈×C₆₁	C₈	semidirect
131. C₄×C₆₁	Q₈⋊C₈	semidirect
168. C₆₁	#9673	semidirect
173. C₄⋊Q₈	#92420	semidirect
285. C₄	Q₈⋊#92420	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#92420	2
C₂×Q₈⋊#92420	2
C₄⋊Q₈×Dic₆₁	2
#9673	61