Subgroups — C₈×#92497 #16854666

Order 3968.

456 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#92497	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521173	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₂²⋊D₃₁	C₂	1	2	yes	maximal, complemented
6	(C₂²⋊D₃₁)⋊C₈	C₂	1	2	yes	maximal, complemented
7	#14521101	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521099	C₂	1	2	yes	maximal, complemented
9	C₂×C₈×Dic₃₁	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#92497	C₄	1	1	yes	characteristic
13	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
14	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	D₄×Dic₃₁	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊D₃₁	C₂²	1	2	yes	—
17	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
18	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
19	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
21	#11758469	C₄	1	1	yes	characteristic
22	#11758464	C₄	1	1	yes	characteristic
23	(C₂²⋊D₃₁)⋊C₄	C₄	1	2	yes	—
24	#11758454	C₄	1	2	yes	—
25	C₄²×D₃₁	C₂²	1	1	yes	characteristic
26	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
27	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
31	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
32	Dic₃₁⋊C₈	C₂²	1	2	yes	—
33	Dic₃₁⋊C₈	C₂²	1	2	yes	—
34	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
35	C₈×Dic₃₁	C₂²	1	2	yes	—
36	C₈×Dic₃₁	C₂²	1	2	yes	—
37	C₈×Dic₃₁	C₂²	1	2	yes	—
38	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
39	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
40	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₃₁	C₂×C₄	1	2	yes	—
44	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
45	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
46	#92497	C₈	1	8	yes	direct product factor
47	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
50	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×C₃₁	—	2	4	no	—
52	C₂×C₈×C₃₁	C₂³	1	2	yes	—
53	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
57	C₄²×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
59	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
61	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
62	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
63	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
66	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
67	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
68	C₄×Dic₃₁	C₂³	1	2	yes	—
69	C₂×C₃₁⋊C₈	—	2	4	no	—
70	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
72	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₈×D₃₁	—	2	2	no	—
74	C₈×D₃₁	—	2	2	no	—
75	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
76	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
77	D₄×C₃₁	C₂×C₈	1	4	yes	complemented
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
84	C₂²⋊D₃₁	C₂×C₈	1	8	yes	complemented
85	C₂×Dic₃₁	—	2	4	no	—
86	C₂×Dic₃₁	C₂×C₈	1	8	yes	complemented
87	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
89	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
91	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
92	C₄×D₃₁	—	2	1	no	—
93	C₄×D₃₁	—	2	1	no	—
94	C₄×D₃₁	—	2	2	no	—
95	C₄×D₃₁	C₂×C₈	1	4	yes	complemented
96	Dic₆₂	C₂×C₈	1	4	yes	complemented
97	C₈×C₃₁	—	2	1	no	—
98	C₈×C₃₁	—	2	2	no	—
99	C₈×C₃₁	#41	1	2	yes	complemented
100	C₃₁⋊C₈	—	2	1	no	—
101	C₃₁⋊C₈	—	2	2	no	—
102	C₃₁⋊C₈	#41	1	2	yes	complemented
103	C₈×#41	—	31	1	no	maximal, Sylow
104	C₂²×C₃₁	—	2	2	no	—
105	C₂²×C₃₁	—	2	2	no	—
106	C₂²×C₃₁	C₂²×C₈	1	4	yes	—
107	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
108	C₂×D₃₁	—	2	1	no	—
109	C₂×D₃₁	—	2	1	no	—
110	C₂×D₃₁	C₂²×C₈	1	2	yes	—
111	C₄×C₃₁	—	2	2	no	—
112	C₄×C₃₁	—	2	1	no	—
113	C₄×C₃₁	C₂²×C₈	1	2	yes	—
114	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
115	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
116	Dic₃₁	—	2	1	no	—
117	Dic₃₁	—	2	2	no	—
118	Dic₃₁	C₂²×C₈	1	4	yes	—
119	Dic₃₁	C₂²×C₈	1	2	yes	—
120	Dic₃₁	#131	1	2	yes	—
121	C₄×#41	—	31	1	no	—
122	C₈×Q₈	—	31	1	no	—
123	C₈×Q₈	—	31	1	no	—
124	C₈×D₄	—	31	2	no	—
125	C₈×D₄	—	31	1	no	—
126	C₈×D₄	—	31	2	no	—
127	C₈×D₄	D₃₁	1	1	yes	characteristic, complemented
128	#431	—	31	1	no	—
129	#431	—	31	2	no	—
130	C₂×C₄×C₈	—	31	2	no	—
131	C₂×C₄×C₈	—	31	1	no	—
132	C₂×C₃₁	—	2	4	no	—
133	C₂×C₃₁	C₄×#41	1	1	yes	characteristic
134	C₂×C₃₁	C₂³×C₈	1	1	yes	characteristic
135	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
136	D₃₁	—	2	2	no	—
137	C₂×#41	—	31	1	no	—
138	C₂²×C₈	—	31	2	no	—
139	C₂²×C₈	—	31	1	no	—
140	C₂²×C₈	—	31	1	no	—
141	C₂²×C₈	C₂×D₃₁	1	2	yes	—
142	C₄×Q₈	—	31	1	no	—
143	C₄×Q₈	—	31	1	no	—
144	C₄×D₄	—	31	1	no	—
145	C₄×D₄	—	31	2	no	—
146	C₄×D₄	—	31	2	no	—
147	C₄×D₄	C₂×D₃₁	1	1	yes	characteristic
148	#117	—	31	1	no	—
149	#117	—	31	2	no	—
150	C₂×C₄²	—	31	2	no	—
151	C₂×C₄²	—	31	1	no	—
152	C₄⋊C₈	—	31	2	no	—
153	C₄⋊C₈	—	31	2	no	—
154	C₄⋊C₈	—	31	1	no	—
155	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
156	C₂²⋊C₈	—	31	2	no	—
157	C₂²⋊C₈	—	31	2	no	—
158	C₂²⋊C₈	C₂×D₃₁	1	2	yes	—
159	C₄×C₈	—	31	2	no	—
160	C₄×C₈	—	31	1	no	—
161	C₄×C₈	—	31	2	no	—
162	C₄×C₈	—	31	1	no	—
163	C₄×C₈	—	31	2	no	—
164	C₄×C₈	—	31	1	no	—
165	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
166	C₃₁	C₈×#41	1	1	yes	characteristic, complemented, Sylow
167	#41	—	31	8	no	—
168	C₂×Q₈	—	31	1	no	—
169	C₂×D₄	—	31	2	no	—
170	C₂×D₄	C₄×D₃₁	1	1	yes	characteristic
171	C₂²×C₄	—	31	2	no	—
172	C₂²×C₄	—	31	1	no	—
173	C₂²×C₄	—	31	2	no	—
174	C₂²×C₄	—	31	1	no	—
175	C₂²×C₄	—	31	1	no	—
176	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	31	1	no	—
179	C₂×C₈	—	31	1	no	—
180	C₂×C₈	—	31	2	no	—
181	C₂×C₈	—	62	2	no	—
182	C₂×C₈	—	62	4	no	—
183	C₂×C₈	—	62	2	no	—
184	C₂×C₈	C₂²×D₃₁	1	2	yes	—
185	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
186	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
187	C₄⋊C₄	—	31	2	no	—
188	C₄⋊C₄	—	31	2	no	—
189	C₄⋊C₄	—	31	1	no	—
190	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
191	C₂²⋊C₄	—	31	2	no	—
192	C₂²⋊C₄	—	31	2	no	—
193	C₂²⋊C₄	C₄×D₃₁	1	2	yes	—
194	C₄²	—	31	2	no	—
195	C₄²	—	31	1	no	—
196	C₄²	—	31	2	no	—
197	C₄²	—	31	1	no	—
198	C₄²	—	31	1	no	—
199	C₄²	—	31	2	no	—
200	C₄²	C₂²×D₃₁	1	1	yes	characteristic
201	C₂³	—	31	1	no	—
202	C₂³	C₂×C₄×D₃₁	1	2	yes	—
203	Q₈	—	31	4	no	—
204	D₄	—	31	8	no	—
205	D₄	C₈×D₃₁	1	4	yes	complemented
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	31	8	no	—
209	C₂×C₄	—	31	1	no	—
210	C₂×C₄	—	31	1	no	—
211	C₂×C₄	—	31	2	no	—
212	C₂×C₄	—	31	4	no	—
213	C₂×C₄	—	31	1	no	—
214	C₂×C₄	—	31	2	no	—
215	C₂×C₄	—	62	2	no	—
216	C₂×C₄	—	62	4	no	—
217	C₂×C₄	—	62	1	no	—
218	C₂×C₄	—	62	1	no	—
219	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
222	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	1	no	—
225	C₈	—	31	2	no	—
226	C₈	—	62	1	no	—
227	C₈	—	62	2	no	—
228	C₈	#92497	1	2	yes	direct product factor
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	31	2	no	—
232	C₂²	—	62	1	no	—
233	C₂²	—	62	1	no	—
234	C₂²	C₂×C₈×D₃₁	1	4	yes	—
235	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	31	2	no	—
239	C₄	—	31	4	no	—
240	C₄	—	31	2	no	—
241	C₄	—	62	1	no	—
242	C₄	—	62	2	no	—
243	C₄	C₂×C₈×D₃₁	1	2	yes	—
244	C₄	C₂×#92497	1	1	yes	characteristic
245	C₄	#41×D₃₁	1	1	yes	characteristic
246	C₂	—	2	4	no	—
247	C₂	—	62	2	no	—
248	C₂	C₄×#92497	1	1	yes	characteristic
249	C₂	C₂²×C₈×D₃₁	1	1	yes	characteristic
250	C₂	#131×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₃₁
commutator	C₂×C₃₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₈×D₄×C₃₁	C₂
socle	C₂²×C₃₁	C₂³×C₄
prime radical	C₂×#92497	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₈×D₄	D₃₁
hypocenter	C₃₁	C₈×#41
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₆₂	C₂	semidirect
3. #14521173	C₂	semidirect
4. D₄×C₃₁⋊C₈	C₂	semidirect
5. C₈×C₂²⋊D₃₁	C₂	semidirect
6. (C₂²⋊D₃₁)⋊C₈	C₂	semidirect
7. #14521101	C₂	semidirect
8. #14521099	C₂	semidirect
9. C₂×C₈×Dic₃₁	C₂	semidirect
10. C₄×C₈×D₃₁	C₂	semidirect
11. C₈×D₄×C₃₁	C₂	semidirect
46. #92497	C₈	direct (N × H)
77. D₄×C₃₁	C₂×C₈	semidirect
84. C₂²⋊D₃₁	C₂×C₈	semidirect
86. C₂×Dic₃₁	C₂×C₈	semidirect
95. C₄×D₃₁	C₂×C₈	semidirect
96. Dic₆₂	C₂×C₈	semidirect
99. C₈×C₃₁	#41	semidirect
102. C₃₁⋊C₈	#41	semidirect
127. C₈×D₄	D₃₁	semidirect
166. C₃₁	C₈×#41	semidirect
205. D₄	C₈×D₃₁	semidirect
228. C₈	#92497	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#92497	2
C₈×Dic₆₂	2
#14521173	2
D₄×C₃₁⋊C₈	2
C₈×C₂²⋊D₃₁	2
(C₂²⋊D₃₁)⋊C₈	2
#14521101	2
#14521099	2
C₂×C₈×Dic₃₁	2
C₄×C₈×D₃₁	2
C₈×D₄×C₃₁	2
C₈×#41	31

Subgroups — C8×#92497 #16854666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#92497 #16854666