Subgroups — #12009

Order 256.

121 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#1564	C₂	1	1	yes	maximal, characteristic
3	C₂×#335	C₂	1	1	yes	maximal, characteristic
4	#1372	C₂	1	2	yes	maximal, complemented
5	#1371	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂	C₄	1	1	yes	characteristic
7	C₂×M₃₂	C₄	1	1	yes	characteristic
8	#431	—	2	1	no	—
9	C₂×C₄⋊C₈	—	2	1	no	—
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	#346	C₄	1	2	yes	—
12	C₄×C₁₆	C₂²	1	2	yes	—
13	#335	—	2	1	no	—
14	#335	C₂²	1	2	yes	—
15	#335	C₂²	1	1	yes	characteristic
16	#335	C₂²	1	1	yes	characteristic
17	#334	—	2	1	no	—
18	C₂²×C₈	—	4	1	no	—
19	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈	Q₈	1	1	yes	characteristic
21	C₂×C₄²	D₄	1	1	yes	characteristic
22	M₃₂	—	2	2	no	—
23	M₃₂	C₈	1	4	yes	complemented
24	C₂×C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₁₆	C₂×C₄	1	2	yes	—
26	C₄⋊C₈	—	2	1	no	—
27	C₄⋊C₈	—	2	1	no	—
28	C₄⋊C₈	—	2	2	no	—
29	C₄⋊C₈	—	4	1	no	—
30	C₂²⋊C₈	—	4	1	no	—
31	C₄×C₈	—	4	1	no	—
32	C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈	D₄	1	1	yes	characteristic
34	C₄×C₈	Q₈	1	1	yes	characteristic
35	C₄×C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄	D₈	1	1	yes	characteristic
38	C₂²×C₄	Q₁₆	1	1	yes	characteristic
39	C₂×C₈	—	2	2	no	—
40	C₂×C₈	—	4	1	no	—
41	C₂×C₈	—	4	1	no	—
42	C₂×C₈	—	4	2	no	—
43	C₂×C₈	—	8	2	no	—
44	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈	C₂×C₈	1	2	yes	—
47	C₂×C₈	M₁₆	1	2	yes	—
48	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₄²	C₄⋊C₄	1	1	yes	characteristic
51	C₄²	D₈	1	1	yes	characteristic
52	C₄²	Q₁₆	1	1	yes	characteristic
53	C₄²	C₂×D₄	1	1	yes	characteristic
54	C₁₆	—	2	2	no	—
55	C₁₆	C₂×C₈	1	4	yes	complemented
56	C₂³	#107	1	1	yes	characteristic
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	C₄⋊C₈	1	2	yes	—
61	C₂×C₄	#107	1	1	yes	characteristic
62	C₂×C₄	#107	1	1	yes	characteristic
63	C₂×C₄	#107	1	1	yes	characteristic
64	C₂×C₄	#108	1	2	yes	—
65	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	—	2	1	no	—
70	C₈	—	8	1	no	—
71	C₈	—	8	2	no	—
72	C₈	—	8	1	no	—
73	C₈	C₄⋊C₈	1	2	yes	—
74	C₈	C₂²×C₈	1	1	yes	characteristic
75	C₈	C₂×M₁₆	1	1	yes	characteristic
76	C₂²	—	2	1	no	—
77	C₂²	—	2	1	no	—
78	C₂²	#335	1	2	yes	—
79	C₂²	C₂×#107	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	2	1	no	—
82	C₄	—	2	1	no	—
83	C₄	#335	1	2	yes	—
84	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₄	C₂×#108	1	1	yes	characteristic
86	C₄	#509	1	1	yes	characteristic
87	C₄	#510	1	1	yes	characteristic
88	C₂	—	2	2	no	—
89	C₂	C₂×#335	1	1	yes	characteristic
90	C₂	#2159	1	1	yes	characteristic
91	C₂	#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator	C₈	C₂²×C₈
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	C₂×#107
prime radical	C₂³	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ 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
4. #1372	C₂	semidirect
5. #1371	C₂	semidirect
23. M₃₂	C₈	semidirect
55. C₁₆	C₂×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₃₂	2
#1564	2
C₂×#335	2
#1372	2
#1371	2