Subgroups — C₂²⋊C₈×#92420 #30597410

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#92420	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊#92420	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#92420	C₂	1	2	yes	maximal, complemented
4	#21460641	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×#92420	C₄	1	2	yes	—
7	C₂×C₂²⋊#92420	C₄	1	2	yes	—
8	C₂×C₄×#92420	—	2	4	no	—
9	C₂×C₄×#92420	C₄	1	2	yes	—
10	C₂×C₄×#92420	C₂²	1	2	yes	—
11	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
12	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
13	#16840628	C₄	1	2	yes	—
14	C₈×#92420	—	2	4	no	—
15	C₂²×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊#92418	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
18	#16839516	C₄	1	2	yes	—
19	C₂×C₂²⋊C₈×C₆₁	C₄	1	1	yes	characteristic
20	C₂²×#92420	—	2	4	no	—
21	C₂²×#92420	—	2	4	no	—
22	C₂²×#92420	C₈	1	8	yes	complemented
23	C₂²×#92420	C₂×C₄	1	2	yes	—
24	C₂²×#92420	C₂×C₄	1	4	yes	—
25	C₂²×#92420	C₂×C₄	1	2	yes	—
26	C₂²⋊#92420	C₈	1	8	yes	complemented
27	C₄×#92420	—	2	4	no	—
28	C₄×#92420	—	2	2	no	—
29	C₄×#92420	—	2	2	no	—
30	C₄×#92420	—	2	2	no	—
31	C₄×#92420	D₄	1	4	yes	—
32	C₂³×Dic₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂³×Dic₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×#92418	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₆₁	—	2	2	no	—
36	C₂×C₄×Dic₆₁	—	2	2	no	—
37	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
41	C₂²⋊#92418	C₈	1	4	yes	complemented
42	C₈×Dic₆₁	—	2	2	no	—
43	C₈×Dic₆₁	—	2	2	no	—
44	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
46	C₂²⋊C₈×C₆₁	C₈	1	4	yes	complemented
47	C₂⁴×C₆₁	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	—	2	2	no	—
49	C₂²×C₄×C₆₁	—	2	2	no	—
50	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×Dic₆₁	—	2	1	no	—
55	C₂²×Dic₆₁	—	2	1	no	—
56	C₂²×Dic₆₁	—	2	2	no	—
57	C₂²×Dic₆₁	—	2	2	no	—
58	C₂²×Dic₆₁	—	2	2	no	—
59	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
60	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
61	C₂²×Dic₆₁	C₂×C₈	1	4	yes	—
62	C₂²×Dic₆₁	C₂×C₈	1	4	yes	—
63	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×#92420	—	2	2	no	—
66	C₂×#92420	—	2	4	no	—
67	C₂×#92420	—	2	2	no	—
68	C₂×#92420	—	2	4	no	—
69	C₂×#92420	—	2	16	no	—
70	C₂×#92420	C₂²⋊C₄	1	4	yes	—
71	C₂×#92420	C₂×C₈	1	4	yes	—
72	C₂×#92420	C₂×C₈	1	4	yes	—
73	C₂×#92420	C₂×C₈	1	8	yes	complemented
74	C₂×#92420	M₁₆	1	4	yes	—
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	—	2	2	no	—
77	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	complemented
78	C₄×Dic₆₁	—	2	2	no	—
79	C₄×Dic₆₁	—	2	1	no	—
80	C₄×Dic₆₁	—	2	1	no	—
81	C₄×Dic₆₁	—	2	2	no	—
82	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
83	C₄×Dic₆₁	#41	1	2	yes	—
84	C₂×#92418	—	2	2	no	—
85	C₂×#92418	—	2	2	no	—
86	C₂×#92418	C₂×C₈	1	4	yes	complemented
87	C₂³×C₆₁	—	2	1	no	—
88	C₂³×C₆₁	—	2	1	no	—
89	C₂³×C₆₁	—	2	1	no	—
90	C₂³×C₆₁	—	2	1	no	—
91	C₂³×C₆₁	C₄×C₈	1	2	yes	—
92	C₂³×C₆₁	C₄×C₈	1	2	yes	—
93	C₂³×C₆₁	C₄×C₈	1	2	yes	—
94	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₆₁	—	2	1	no	—
96	C₂×C₄×C₆₁	—	2	1	no	—
97	C₂×C₄×C₆₁	—	2	4	no	—
98	C₂×C₄×C₆₁	—	2	4	no	—
99	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
100	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₆₁	#131	1	2	yes	—
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	4	no	—
109	C₂×Dic₆₁	—	2	4	no	—
110	C₂×Dic₆₁	—	2	8	no	—
111	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
112	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
113	C₂×Dic₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×Dic₆₁	#117	1	1	yes	characteristic
115	C₂×Dic₆₁	C₄×D₄	1	2	yes	—
116	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
117	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
118	C₂×Dic₆₁	C₂×M₁₆	1	2	yes	—
119	C₂×Dic₆₁	#131	1	2	yes	—
120	#92420	—	2	4	no	—
121	#92420	—	2	8	no	—
122	#92420	C₂²⋊C₈	1	8	yes	direct product factor
123	C₈×C₆₁	—	2	4	no	—
124	#92418	—	2	4	no	—
125	C₈×C₂²⋊C₈	—	61	1	no	maximal, Sylow
126	C₂²×C₆₁	—	2	2	no	—
127	C₂²×C₆₁	—	2	2	no	—
128	C₂²×C₆₁	—	2	2	no	—
129	C₂²×C₆₁	—	2	2	no	—
130	C₂²×C₆₁	—	2	2	no	—
131	C₂²×C₆₁	—	2	2	no	—
132	C₂²×C₆₁	C₈²	1	4	yes	complemented
133	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
135	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
136	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
137	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
138	C₂²×C₆₁	#405	1	1	yes	characteristic
139	C₂²×C₆₁	#405	1	1	yes	characteristic
140	C₄×C₆₁	—	2	2	no	—
141	C₄×C₆₁	C₈×D₄	1	2	yes	—
142	C₄×C₆₁	C₈×D₄	1	2	yes	—
143	Dic₆₁	—	2	2	no	—
144	Dic₆₁	—	2	2	no	—
145	Dic₆₁	C₂×C₂²⋊C₈	1	2	yes	—
146	Dic₆₁	#431	1	2	yes	—
147	Dic₆₁	C₈×D₄	1	4	yes	—
148	C₂²×C₄×C₈	—	61	1	no	—
149	C₄×C₂²⋊C₈	—	61	1	no	—
150	C₄×C₂²⋊C₈	—	61	1	no	—
151	C₂×C₈²	—	61	2	no	—
152	#1273	—	61	2	no	—
153	C₂×C₆₁	—	2	4	no	—
154	C₂×C₆₁	C₂×C₈²	1	1	yes	characteristic
155	C₂×C₆₁	C₈×M₁₆	1	1	yes	characteristic
156	C₂×C₆₁	C₈×M₁₆	1	1	yes	characteristic
157	C₂×C₆₁	#1450	1	1	yes	characteristic
158	C₂×C₆₁	C₄×C₂²⋊C₈	1	1	yes	characteristic
159	C₂×C₆₁	C₈×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₆₁	C₈×C₂²⋊C₄	1	1	yes	characteristic
161	C₂³×C₈	—	61	2	no	—
162	C₂²×C₄²	—	61	1	no	—
163	C₂×C₂²⋊C₈	—	61	2	no	—
164	C₂×C₂²⋊C₈	—	61	1	no	—
165	C₂×C₂²⋊C₈	#6923	1	1	yes	characteristic
166	C₂×C₄×C₈	—	61	1	no	—
167	C₂×C₄×C₈	—	61	2	no	—
168	C₂×C₄×C₈	—	61	1	no	—
169	C₂×C₄×C₈	—	61	2	no	—
170	C₂×C₄×C₈	—	61	2	no	—
171	C₂×C₄×C₈	—	122	4	no	—
172	#336	—	61	2	no	—
173	#336	—	61	2	no	—
174	C₈²	—	122	4	no	—
175	C₆₁	C₈×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	—	61	1	no	—
177	C₂³×C₄	—	61	1	no	—
178	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
179	C₂²×C₈	—	61	2	no	—
180	C₂²×C₈	—	61	8	no	—
181	C₂²×C₈	—	61	2	no	—
182	C₂²×C₈	—	61	4	no	—
183	C₂²×C₈	—	61	2	no	—
184	C₂²×C₈	—	122	4	no	—
185	C₂²×C₈	—	122	4	no	—
186	C₂²×C₈	C₂×#6923	1	2	yes	—
187	C₂×C₄²	—	61	1	no	—
188	C₂×C₄²	—	61	1	no	—
189	C₂×C₄²	—	61	1	no	—
190	C₂×C₄²	—	61	1	no	—
191	C₂×C₄²	—	122	2	no	—
192	C₂×C₄²	—	122	2	no	—
193	C₂²⋊C₈	—	61	8	no	—
194	C₂²⋊C₈	—	61	4	no	—
195	C₂²⋊C₈	#92420	1	4	yes	direct product factor
196	C₄×C₈	—	61	4	no	—
197	C₄×C₈	—	122	2	no	—
198	C₄×C₈	—	122	2	no	—
199	C₄×C₈	—	122	2	no	—
200	C₄×C₈	—	122	2	no	—
201	C₄×C₈	—	122	2	no	—
202	C₄×C₈	—	122	4	no	—
203	C₂⁴	C₄×#6923	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	61	1	no	—
207	C₂²×C₄	—	61	1	no	—
208	C₂²×C₄	—	61	4	no	—
209	C₂²×C₄	—	61	4	no	—
210	C₂²×C₄	—	61	1	no	—
211	C₂²×C₄	—	61	1	no	—
212	C₂²×C₄	—	122	1	no	—
213	C₂²×C₄	—	122	1	no	—
214	C₂²×C₄	—	122	2	no	—
215	C₂²×C₄	—	122	2	no	—
216	C₂²×C₄	—	122	2	no	—
217	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
218	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
219	C₂²×C₄	C₂×#92420	1	2	yes	—
220	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	61	4	no	—
224	C₂×C₈	—	61	4	no	—
225	C₂×C₈	—	61	4	no	—
226	C₂×C₈	—	61	8	no	—
227	C₂×C₈	—	61	4	no	—
228	C₂×C₈	—	61	4	no	—
229	C₂×C₈	—	122	16	no	—
230	C₂×C₈	—	122	2	no	—
231	C₂×C₈	—	122	4	no	—
232	C₂×C₈	—	122	2	no	—
233	C₂×C₈	—	122	4	no	—
234	C₂×C₈	—	122	2	no	—
235	C₂×C₈	—	122	2	no	—
236	C₂×C₈	C₂×#92420	1	4	yes	complemented
237	C₄²	—	61	2	no	—
238	C₄²	—	61	2	no	—
239	C₄²	—	122	1	no	—
240	C₄²	—	122	1	no	—
241	C₄²	—	122	2	no	—
242	C₄²	—	122	2	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	C₈×#6923	1	2	yes	—
248	C₂³	C₈×#6923	1	2	yes	—
249	C₂³	C₄×#92420	1	2	yes	—
250	C₂³	C₂×C₄×#6923	1	1	yes	characteristic
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	61	2	no	—
256	C₂×C₄	—	61	2	no	—
257	C₂×C₄	—	61	2	no	—
258	C₂×C₄	—	61	2	no	—
259	C₂×C₄	—	61	1	no	—
260	C₂×C₄	—	61	1	no	—
261	C₂×C₄	—	61	2	no	—
262	C₂×C₄	—	61	2	no	—
263	C₂×C₄	—	61	2	no	—
264	C₂×C₄	—	122	4	no	—
265	C₂×C₄	—	122	4	no	—
266	C₂×C₄	—	122	8	no	—
267	C₂×C₄	—	122	1	no	—
268	C₂×C₄	—	122	1	no	—
269	C₂×C₄	—	122	1	no	—
270	C₂×C₄	—	122	1	no	—
271	C₂×C₄	C₄×#92420	1	2	yes	—
272	C₂×C₄	D₄×#6923	1	2	yes	—
273	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
274	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
275	C₂×C₄	#14519864	1	2	yes	—
276	C₈	—	2	4	no	—
277	C₈	—	61	8	no	—
278	C₈	—	122	8	no	—
279	C₈	—	122	4	no	—
280	C₈	—	122	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	C₈×#92420	1	4	yes	complemented
288	C₂²	C₂²⋊C₄×#6923	1	1	yes	characteristic
289	C₂²	C₂×C₈×#6923	1	1	yes	characteristic
290	C₂²	C₂×C₈×#6923	1	1	yes	characteristic
291	C₂²	C₂×C₄×#92420	1	1	yes	characteristic
292	C₂²	M₁₆×#6923	1	1	yes	characteristic
293	C₂²	#16840726	1	1	yes	characteristic
294	C₂²	#16840728	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	61	2	no	—
297	C₄	—	61	2	no	—
298	C₄	—	61	4	no	—
299	C₄	—	122	2	no	—
300	C₄	—	122	2	no	—
301	C₄	D₄×#92420	1	2	yes	—
302	C₄	#16840789	1	2	yes	—
303	C₂	—	2	4	no	—
304	C₂	C₂×C₈×#92420	1	1	yes	characteristic
305	C₂	M₁₆×#92420	1	1	yes	characteristic
306	C₂	#21461020	1	1	yes	characteristic
307	C₂	#21461024	1	1	yes	characteristic
308	C₂	C₂²⋊C₈×#6923	1	1	yes	characteristic
309	C₂	C₂²⋊C₄×#92420	1	1	yes	characteristic
310	C₂	#21461554	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6923
commutator	C₂×C₆₁	C₂×C₈²
Fitting	C₂×C₂²⋊C₈×C₆₁	C₄
socle	C₂³×C₆₁	C₂×C₄²
prime radical	C₂⁴×C₆₁	C₄²
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×C₂²⋊C₈	#6923
hypocenter	C₆₁	C₈×C₂²⋊C₈
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₈×#92420	C₂	semidirect
4. #21460641	C₂	semidirect
22. C₂²×#92420	C₈	semidirect
26. C₂²⋊#92420	C₈	semidirect
41. C₂²⋊#92418	C₈	semidirect
46. C₂²⋊C₈×C₆₁	C₈	semidirect
73. C₂×#92420	C₂×C₈	semidirect
77. C₂×C₈×C₆₁	C₂×C₈	semidirect
86. C₂×#92418	C₂×C₈	semidirect
122. #92420	C₂²⋊C₈	direct (N × H)
132. C₂²×C₆₁	C₈²	semidirect
175. C₆₁	C₈×C₂²⋊C₈	semidirect
195. C₂²⋊C₈	#92420	direct (N × H)
236. C₂×C₈	C₂×#92420	semidirect
287. C₂²	C₈×#92420	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×#92420	2
C₄×C₂²⋊#92420	2
C₂×C₈×#92420	2
#21460641	2
C₂²⋊C₈×Dic₆₁	2
C₈×C₂²⋊C₈	61

Subgroups — C22⋊C8×#92420 #30597410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊C₈×#92420 #30597410