Subgroups — #30602124

Order 15616.

528 conjugacy classes of subgroups.

The subgroup lattice has 292 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	#21460655	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	C₂×Q₈×Dic₆₁	—	2	1	no	—
12	#16839669	—	2	1	no	—
13	#16839661	—	2	1	no	—
14	#16839661	C₄	1	2	yes	—
15	#16839632	C₄	1	1	yes	characteristic
16	C₄⋊C₄×Dic₆₁	—	2	1	no	—
17	C₄⋊C₄×Dic₆₁	C₂²	1	2	yes	—
18	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
20	C₂²×#92420	—	2	2	no	—
21	C₂²×#92420	—	2	2	no	—
22	C₄⋊#92420	—	2	4	no	—
23	C₄⋊#92420	—	2	2	no	—
24	C₄⋊#92420	Q₈	1	4	yes	complemented
25	C₄×#92420	—	2	4	no	—
26	C₄×#92420	—	4	2	no	—
27	Q₈×Dic₆₁	—	4	2	no	—
28	Q₈×Dic₆₁	—	4	2	no	—
29	C₂×C₄⋊Dic₆₁	—	2	1	no	—
30	C₂×C₄⋊Dic₆₁	—	2	1	no	—
31	C₂×C₄⋊Dic₆₁	—	2	2	no	—
32	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₆₁	—	2	2	no	—
34	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₆₁	D₄	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	C₂³	1	1	yes	characteristic
40	#14519684	—	2	2	no	—
41	#14519684	—	2	2	no	—
42	C₂²×Q₈×C₆₁	—	2	1	no	—
43	C₄⋊Q₈×C₆₁	—	2	2	no	—
44	C₄⋊Q₈×C₆₁	C₈	1	4	yes	complemented
45	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
46	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂×Q₈×C₆₁	—	2	4	no	—
49	C₂×Q₈×C₆₁	—	4	2	no	—
50	C₂×Q₈×C₆₁	—	4	2	no	—
51	C₂²×C₄×C₆₁	—	2	2	no	—
52	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₆₁	—	2	2	no	—
56	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂×#92420	—	2	4	no	—
61	C₂×#92420	—	2	4	no	—
62	C₂×#92420	—	4	2	no	—
63	C₂×#92420	—	4	2	no	—
64	C₂×#92420	—	4	2	no	—
65	C₂×#92420	—	4	2	no	—
66	C₄⋊C₄×C₆₁	—	2	4	no	—
67	C₄⋊C₄×C₆₁	—	2	2	no	—
68	C₄⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
69	C₄²×C₆₁	C₂×C₈	1	2	yes	—
70	C₄²×C₆₁	M₁₆	1	2	yes	—
71	C₄⋊Dic₆₁	—	2	4	no	—
72	C₄⋊Dic₆₁	—	4	2	no	—
73	C₄⋊Dic₆₁	—	4	2	no	—
74	C₄⋊Dic₆₁	—	4	4	no	—
75	C₄×Dic₆₁	—	2	2	no	—
76	C₄×Dic₆₁	—	2	2	no	—
77	C₄×Dic₆₁	—	2	2	no	—
78	C₄×Dic₆₁	—	4	1	no	—
79	C₄×Dic₆₁	—	4	2	no	—
80	C₄×Dic₆₁	—	4	2	no	—
81	C₄×Dic₆₁	D₈	1	2	yes	—
82	C₄×Dic₆₁	SD₁₆	1	2	yes	—
83	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
84	C₄×Dic₆₁	#41	1	2	yes	—
85	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	Q₈×C₆₁	—	4	8	no	—
87	C₂×C₄×C₆₁	—	2	1	no	—
88	C₂×C₄×C₆₁	—	2	1	no	—
89	C₂×C₄×C₆₁	—	2	4	no	—
90	C₂×C₄×C₆₁	—	4	2	no	—
91	C₂×C₄×C₆₁	—	4	2	no	—
92	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
93	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
94	C₂×C₄×C₆₁	#102	1	2	yes	—
95	C₂×C₄×C₆₁	#104	1	2	yes	—
96	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
97	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₆₁	#131	1	2	yes	—
100	C₂×Dic₆₁	—	2	1	no	—
101	C₂×Dic₆₁	—	2	1	no	—
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	—	4	2	no	—
108	C₂×Dic₆₁	—	4	2	no	—
109	C₂×Dic₆₁	—	4	2	no	—
110	C₂×Dic₆₁	#102	1	2	yes	—
111	C₂×Dic₆₁	C₂²⋊Q₈	1	2	yes	—
112	C₂×Dic₆₁	#124	1	1	yes	characteristic
113	C₂×Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
115	C₂×Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×Dic₆₁	#137	1	2	yes	—
117	#92420	—	4	4	no	—
118	#92420	—	4	4	no	—
119	#9865	—	61	1	no	maximal, Sylow
120	C₂²×C₆₁	#391	1	1	yes	characteristic
121	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₂²×C₆₁	#408	1	1	yes	characteristic
123	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
124	C₂²×C₆₁	#419	1	1	yes	characteristic
125	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
126	C₂²×C₆₁	#421	1	1	yes	characteristic
127	C₄×C₆₁	—	2	2	no	—
128	C₄×C₆₁	—	4	4	no	—
129	C₄×C₆₁	D₄⋊C₈	1	4	yes	—
130	C₄×C₆₁	C₈×Q₈	1	2	yes	—
131	C₄×C₆₁	#446	1	2	yes	—
132	Dic₆₁	—	2	1	no	—
133	Dic₆₁	—	2	1	no	—
134	Dic₆₁	—	2	2	no	—
135	Dic₆₁	—	4	1	no	—
136	Dic₆₁	—	8	2	no	—
137	Dic₆₁	#474	1	2	yes	—
138	Dic₆₁	#475	1	2	yes	—
139	C₄×C₄⋊Q₈	—	61	1	no	—
140	C₄×C₄⋊C₈	—	61	1	no	—
141	C₄×C₄⋊C₈	—	61	1	no	—
142	#1279	—	61	4	no	—
143	C₂×C₆₁	C₂×D₄⋊C₈	1	1	yes	characteristic
144	C₂×C₆₁	#1479	1	1	yes	characteristic
145	C₂×C₆₁	#1480	1	1	yes	characteristic
146	C₂×C₆₁	#1485	1	1	yes	characteristic
147	C₂×C₆₁	#1984	1	1	yes	characteristic
148	C₂×C₆₁	#1986	1	1	yes	characteristic
149	C₂×C₆₁	#1988	1	1	yes	characteristic
150	C₂×C₄⋊Q₈	#6923	1	1	yes	characteristic
151	C₂×C₄×Q₈	—	122	1	no	—
152	C₂×C₄⋊C₈	—	61	2	no	—
153	C₂×C₄⋊C₈	—	122	1	no	—
154	C₂×C₄×C₈	—	61	2	no	—
155	C₂×C₄×C₈	—	122	1	no	—
156	#391	—	122	1	no	—
157	#389	—	61	2	no	—
158	#389	—	122	1	no	—
159	#384	—	61	1	no	—
160	C₄×C₄⋊C₄	—	61	2	no	—
161	C₄×C₄⋊C₄	—	122	1	no	—
162	C₄³	—	61	1	no	—
163	#336	—	122	1	no	—
164	#336	—	122	1	no	—
165	C₆₁	#9865	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₈	—	2	1	no	—
167	C₂²×C₈	—	122	2	no	—
168	C₂²×C₈	—	122	2	no	—
169	C₄⋊Q₈	—	2	2	no	—
170	C₄⋊Q₈	#92420	1	4	yes	complemented
171	C₄×Q₈	—	244	2	no	—
172	C₄×Q₈	—	244	2	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	61	2	no	—
175	C₂×C₄⋊C₄	—	122	2	no	—
176	C₂×C₄⋊C₄	—	122	1	no	—
177	C₂×C₄⋊C₄	—	122	1	no	—
178	C₂×C₄⋊C₄	C₂×#6923	1	2	yes	—
179	C₂×C₄²	—	61	1	no	—
180	C₂×C₄²	—	61	1	no	—
181	C₂×C₄²	—	61	1	no	—
182	C₂×C₄²	—	61	1	no	—
183	C₂×C₄²	—	61	1	no	—
184	C₂×C₄²	—	61	1	no	—
185	C₂×C₄²	—	122	2	no	—
186	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
187	C₄⋊C₈	—	61	4	no	—
188	C₄⋊C₈	—	122	2	no	—
189	C₄⋊C₈	—	122	4	no	—
190	C₄×C₈	—	122	4	no	—
191	C₄×C₈	—	244	2	no	—
192	#95	—	122	2	no	—
193	#95	—	122	2	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	4	2	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	61	1	no	—
199	C₂²×C₄	—	61	1	no	—
200	C₂²×C₄	—	61	1	no	—
201	C₂²×C₄	—	61	1	no	—
202	C₂²×C₄	—	122	2	no	—
203	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
205	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
206	C₂×C₈	—	122	4	no	—
207	C₂×C₈	—	122	4	no	—
208	C₂×C₈	—	244	2	no	—
209	C₂×C₈	—	244	2	no	—
210	C₂×C₈	—	244	2	no	—
211	C₂×C₈	—	244	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	122	4	no	—
215	C₄⋊C₄	—	244	2	no	—
216	C₄⋊C₄	—	244	4	no	—
217	C₄⋊C₄	—	244	2	no	—
218	C₄⋊C₄	C₂×#92420	1	4	yes	—
219	C₄²	—	61	2	no	—
220	C₄²	—	61	2	no	—
221	C₄²	—	61	2	no	—
222	C₄²	—	61	2	no	—
223	C₄²	—	122	2	no	—
224	C₄²	—	122	2	no	—
225	C₄²	—	122	2	no	—
226	C₄²	—	244	1	no	—
227	C₄²	—	244	2	no	—
228	C₄²	—	244	2	no	—
229	C₄²	C₂×#92420	1	2	yes	—
230	C₄²	#11758217	1	2	yes	—
231	C₂³	C₂×C₂²⋊#6923	1	1	yes	characteristic
232	Q₈	—	4	8	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	61	2	no	—
239	C₂×C₄	—	61	1	no	—
240	C₂×C₄	—	61	1	no	—
241	C₂×C₄	—	61	2	no	—
242	C₂×C₄	—	61	1	no	—
243	C₂×C₄	—	61	1	no	—
244	C₂×C₄	—	61	2	no	—
245	C₂×C₄	—	122	1	no	—
246	C₂×C₄	—	122	1	no	—
247	C₂×C₄	—	122	1	no	—
248	C₂×C₄	—	122	1	no	—
249	C₂×C₄	—	122	1	no	—
250	C₂×C₄	—	122	1	no	—
251	C₂×C₄	—	122	1	no	—
252	C₂×C₄	—	244	2	no	—
253	C₂×C₄	—	244	2	no	—
254	C₂×C₄	—	244	2	no	—
255	C₂×C₄	C₂²⋊#92420	1	2	yes	—
256	C₂×C₄	C₂²⋊#92420	1	2	yes	—
257	C₂×C₄	#14519839	1	2	yes	—
258	C₂×C₄	#14519841	1	2	yes	—
259	C₂×C₄	Q₈×#6923	1	2	yes	—
260	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
261	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
262	C₂×C₄	#14519865	1	2	yes	—
263	C₈	—	244	4	no	—
264	C₈	—	244	4	no	—
265	C₂²	(C₂×#6923)⋊Q₈	1	1	yes	characteristic
266	C₂²	C₂×C₂²⋊#92420	1	1	yes	characteristic
267	C₂²	#16840737	1	1	yes	characteristic
268	C₂²	C₂×#14519839	1	1	yes	characteristic
269	C₂²	#16840755	1	1	yes	characteristic
270	C₂²	C₂×#14519841	1	1	yes	characteristic
271	C₂²	#16840761	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	4	4	no	—
274	C₄	—	61	2	no	—
275	C₄	—	61	2	no	—
276	C₄	—	122	2	no	—
277	C₄	—	122	1	no	—
278	C₄	—	122	1	no	—
279	C₄	—	488	2	no	—
280	C₄	—	244	1	no	—
281	C₄	D₄⋊#92420	1	4	yes	—
282	C₄	Q₈×#92420	1	2	yes	—
283	C₄	#16840813	1	2	yes	—
284	C₂	C₂×D₄⋊#92420	1	1	yes	characteristic
285	C₂	#21461121	1	1	yes	characteristic
286	C₂	#21461126	1	1	yes	characteristic
287	C₂	#21461143	1	1	yes	characteristic
288	C₂	#21461988	1	1	yes	characteristic
289	C₂	#21461995	1	1	yes	characteristic
290	C₂	#21461998	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	#9865
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
24. C₄⋊#92420	Q₈	semidirect
44. C₄⋊Q₈×C₆₁	C₈	semidirect
165. C₆₁	#9865	semidirect
170. 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
#21460655	2
C₄⋊Q₈×Dic₆₁	2
#9865	61