Subgroups — #12004

Order 256.

135 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2150	C₂	1	2	yes	maximal
2	C₂×C₈²	C₂	1	1	yes	maximal, characteristic
3	#1370	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂	—	2	2	no	—
5	C₂×C₄×C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆	—	2	2	no	—
9	C₄⋊C₁₆	C₂²	1	4	yes	—
10	C₈²	C₄	1	2	yes	—
11	C₈²	C₂²	1	2	yes	—
12	C₂²×C₈	—	2	1	no	—
13	C₂²×C₈	C₈	1	2	yes	—
14	C₂²×C₈	D₄	1	1	yes	characteristic
15	C₂²×C₈	Q₈	1	1	yes	characteristic
16	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
17	M₃₂	—	4	4	no	—
18	C₂×C₁₆	—	2	4	no	—
19	C₄×C₈	—	2	1	no	—
20	C₄×C₈	C₈	1	1	yes	characteristic
21	C₄×C₈	C₈	1	1	yes	characteristic
22	C₄×C₈	C₂×C₄	1	2	yes	—
23	C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈	C₂×C₄	1	2	yes	—
25	C₄×C₈	D₄	1	1	yes	characteristic
26	C₄×C₈	Q₈	1	1	yes	characteristic
27	C₄×C₈	C₂³	1	1	yes	characteristic
28	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄	M₁₆	1	1	yes	characteristic
31	C₂×C₈	—	2	4	no	—
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	—	2	2	no	—
34	C₂×C₈	—	2	4	no	—
35	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈	C₂×C₈	1	2	yes	—
38	C₂×C₈	C₂×C₈	1	2	yes	—
39	C₂×C₈	D₈	1	2	yes	—
40	C₂×C₈	Q₁₆	1	2	yes	—
41	C₂×C₈	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
43	C₄²	C₄⋊C₄	1	1	yes	characteristic
44	C₄²	C₂×C₈	1	1	yes	characteristic
45	C₄²	M₁₆	1	1	yes	characteristic
46	C₄²	C₂²×C₄	1	1	yes	characteristic
47	C₁₆	—	4	4	no	—
48	C₂³	C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄	#107	1	2	yes	—
55	C₂×C₄	#108	1	2	yes	—
56	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
59	C₈	—	2	4	no	—
60	C₈	—	2	1	no	—
61	C₈	—	2	1	no	—
62	C₈	—	2	2	no	—
63	C₈	#107	1	2	yes	—
64	C₈	M₃₂	1	4	yes	complemented
65	C₈	C₂×D₈	1	1	yes	characteristic
66	C₈	C₂×Q₁₆	1	1	yes	characteristic
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	#335	1	2	yes	—
70	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	2	1	no	—
73	C₄	#335	1	2	yes	—
74	C₄	#364	1	2	yes	—
75	C₄	C₂×#107	1	1	yes	characteristic
76	C₄	C₂×#108	1	1	yes	characteristic
77	C₄	C₂×M₃₂	1	2	yes	—
78	C₂	—	2	2	no	—
79	C₂	C₂×#335	1	1	yes	characteristic
80	C₂	#2150	1	1	yes	characteristic
81	C₂	C₂×#364	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	C₂×C₄⋊C₈
prime radical	C₂³	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1370	C₂	semidirect
64. C₈	M₃₂	semidirect

Maximal subgroups

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

subgroup	index
#2150	2
C₂×C₈²	2
#1370	2