Subgroups — #8262

Order 256.

247 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₄×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×#322	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈²	C₂	1	1	yes	maximal, characteristic
5	#1273	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈	C₄	1	4	yes	—
7	C₂×#97	C₄	1	2	yes	—
8	C₂×#97	C₄	1	2	yes	—
9	C₂×C₄×C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₄³	C₂²	1	1	yes	characteristic
14	#336	C₄	1	4	yes	—
15	#322	—	2	2	no	—
16	C₈²	—	2	2	no	—
17	C₂²×C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₈	C₂×C₄	1	2	yes	—
19	C₂²×C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₈	C₂×C₄	1	2	yes	—
21	C₂×C₄²	C₂×C₄	1	2	yes	—
22	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²	C₂×C₄	1	2	yes	—
24	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²	C₂³	1	1	yes	characteristic
26	C₄⋊C₈	—	2	8	no	—
27	#97	—	2	4	no	—
28	#97	C₈	1	8	yes	complemented
29	C₄×C₈	—	2	2	no	—
30	C₄×C₈	—	2	2	no	—
31	C₄×C₈	—	2	2	no	—
32	C₄×C₈	—	2	2	no	—
33	C₄×C₈	—	2	2	no	—
34	C₄×C₈	D₄	1	2	yes	—
35	C₄×C₈	Q₈	1	2	yes	—
36	C₂²×C₄	C₄²	1	1	yes	characteristic
37	C₂²×C₄	C₄²	1	1	yes	characteristic
38	C₂²×C₄	C₄²	1	2	yes	—
39	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄	C₂²×C₄	1	2	yes	—
41	C₂×C₈	—	2	2	no	—
42	C₂×C₈	—	2	4	no	—
43	C₂×C₈	—	2	2	no	—
44	C₂×C₈	—	2	4	no	—
45	C₂×C₈	—	2	4	no	—
46	C₂×C₈	—	2	4	no	—
47	C₂×C₈	—	2	4	no	—
48	C₂×C₈	C₄⋊C₄	1	4	yes	—
49	C₂×C₈	C₂×C₈	1	4	yes	—
50	C₂×C₈	C₂×C₈	1	4	yes	—
51	C₂×C₈	M₁₆	1	8	yes	—
52	C₂×C₈	M₁₆	1	4	yes	—
53	C₄²	—	2	2	no	—
54	C₄²	—	2	4	no	—
55	C₄²	—	2	4	no	—
56	C₄²	C₂×C₈	1	4	yes	—
57	C₄²	C₂×D₄	1	1	yes	characteristic
58	C₄²	C₂×Q₈	1	1	yes	characteristic
59	C₄²	#41	1	2	yes	—
60	C₂³	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	C₄×C₈	1	4	yes	—
68	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄	#117	1	1	yes	characteristic
70	C₂×C₄	C₄×D₄	1	2	yes	—
71	C₂×C₄	C₄×Q₈	1	2	yes	—
72	C₂×C₄	C₂²×C₈	1	2	yes	—
73	C₂×C₄	C₂×M₁₆	1	2	yes	—
74	C₂×C₄	C₂×M₁₆	1	4	yes	—
75	C₂×C₄	#131	1	4	yes	—
76	C₈	—	2	4	no	—
77	C₈	—	2	8	no	—
78	C₈	—	4	4	no	—
79	C₈	C₄⋊C₈	1	8	yes	complemented
80	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
82	C₂²	C₄×M₁₆	1	2	yes	—
83	C₂²	C₄×M₁₆	1	1	yes	characteristic
84	C₂²	#405	1	2	yes	—
85	C₄	—	2	4	no	—
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₄	C₂×C₄⋊C₈	1	2	yes	—
89	C₄	#431	1	2	yes	—
90	C₄	#435	1	4	yes	—
91	C₄	#446	1	4	yes	—
92	C₂	C₈×M₁₆	1	2	yes	—
93	C₂	#1453	1	1	yes	characteristic
94	C₂	#1455	1	1	yes	characteristic
95	C₂	C₄×C₄⋊C₈	1	1	yes	characteristic
96	C₂	#1771	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂³
commutator	C₂²	C₂×C₄×C₈
socle, prime radical	C₂³	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
28. #97	C₈	semidirect
79. C₈	C₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊C₈	2
C₄×#97	2
C₂×#322	2
C₂×C₈²	2
#1273	2