Subgroups — #7334

Order 256.

91 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊C₃₂	C₂	1	2	yes	maximal, complemented
3	C₂³×C₈	C₄	1	1	yes	characteristic
4	C₂²×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₃₂	—	2	2	no	—
7	C₂²⋊C₁₆	—	2	2	no	—
8	C₂³×C₄	C₈	1	1	yes	characteristic
9	C₂²×C₈	—	2	1	no	—
10	C₂²×C₈	—	2	1	no	—
11	C₂²×C₈	—	4	1	no	—
12	C₂²×C₈	—	4	1	no	—
13	C₂²×C₈	C₈	1	1	yes	characteristic
14	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂×C₁₆	—	2	1	no	—
16	C₂×C₁₆	—	4	1	no	—
17	C₂×C₁₆	—	4	1	no	—
18	C₂×C₁₆	D₄	1	2	yes	—
19	C₃₂	—	4	2	no	—
20	C₂⁴	C₁₆	1	1	yes	characteristic
21	C₂²×C₄	—	2	1	no	—
22	C₂²×C₄	—	2	1	no	—
23	C₂²×C₄	—	4	1	no	—
24	C₂²×C₄	—	4	1	no	—
25	C₂²×C₄	C₁₆	1	1	yes	characteristic
26	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈	—	2	1	no	—
28	C₂×C₈	—	2	1	no	—
29	C₂×C₈	—	2	1	no	—
30	C₂×C₈	—	4	2	no	—
31	C₂×C₈	—	4	1	no	—
32	C₂×C₈	—	4	1	no	—
33	C₂×C₈	—	4	1	no	—
34	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₈	M₁₆	1	1	yes	characteristic
36	C₁₆	—	2	2	no	—
37	C₁₆	—	4	1	no	—
38	C₂³	—	2	1	no	—
39	C₂³	—	2	1	no	—
40	C₂³	—	4	1	no	—
41	C₂³	—	4	1	no	—
42	C₂³	C₃₂	1	2	yes	complemented
43	C₂³	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄	—	2	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	4	2	no	—
48	C₂×C₄	—	4	1	no	—
49	C₂×C₄	—	4	1	no	—
50	C₂×C₄	—	4	1	no	—
51	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
52	C₂×C₄	M₃₂	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	4	1	no	—
55	C₈	#99	1	1	yes	characteristic
56	C₈	#100	1	1	yes	characteristic
57	C₂²	—	2	1	no	—
58	C₂²	—	2	1	no	—
59	C₂²	—	2	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	—	4	2	no	—
62	C₂²	—	4	1	no	—
63	C₂²	—	4	1	no	—
64	C₂²	C₂²⋊C₁₆	1	1	yes	characteristic
65	C₂²	C₂×C₃₂	1	1	yes	characteristic
66	C₂²	M₆₄	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	4	1	no	—
69	C₄	#323	1	1	yes	characteristic
70	C₄	#349	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	2	1	no	—
73	C₂	—	4	2	no	—
74	C₂	#1314	1	1	yes	characteristic
75	C₂	C₂²⋊C₃₂	1	1	yes	characteristic
76	C₂	#1400	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊C₄
commutator	C₂²	C₂×C₃₂
Frattini, cosocle	C₂²×C₁₆	C₂²
socle	C₂²	C₂²⋊C₁₆
prime radical	C₂⁴	C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²⋊C₃₂	C₂	semidirect
42. C₂³	C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊C₁₆	2
C₂²⋊C₃₂	2