Subgroups — #12425

Order 256.

137 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2207	C₂	1	2	yes	maximal, complemented
2	#1712	C₂	1	1	yes	maximal, characteristic
3	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#1370	C₂	1	1	yes	maximal, characteristic, complemented
5	#1361	C₂	1	2	yes	maximal, complemented
6	C₂×SD₃₂	—	2	2	no	—
7	#493	—	2	1	no	—
8	#493	—	2	1	no	—
9	#493	C₂²	1	2	yes	—
10	C₈×Q₈	—	2	1	no	—
11	C₄×Q₁₆	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆	C₂²	1	2	yes	—
13	#357	—	2	2	no	—
14	#335	C₂²	1	1	yes	characteristic
15	Q₈⋊C₈	—	2	1	no	—
16	C₈²	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆	D₄	1	1	yes	characteristic
18	C₂×D₈	—	2	2	no	—
19	C₂×D₈	—	2	2	no	—
20	C₂×D₈	—	4	2	no	—
21	C₂×D₈	—	4	1	no	—
22	C₂×D₈	—	4	1	no	—
23	#127	—	2	2	no	—
24	C₄×Q₈	—	2	1	no	—
25	SD₃₂	—	4	4	no	—
26	C₂×C₁₆	—	2	2	no	—
27	#107	D₄	1	1	yes	characteristic
28	C₄⋊C₈	—	2	1	no	—
29	C₄⋊C₈	—	4	1	no	—
30	#103	—	2	1	no	—
31	C₄×C₈	—	4	1	no	—
32	C₄×C₈	D₄	1	1	yes	characteristic
33	C₄×C₈	D₄	1	1	yes	characteristic
34	C₄×C₈	C₂³	1	1	yes	characteristic
35	C₂×Q₈	—	2	1	no	—
36	C₂×D₄	—	4	2	no	—
37	C₂×D₄	—	4	2	no	—
38	C₂×D₄	—	4	2	no	—
39	Q₁₆	—	2	1	no	—
40	Q₁₆	D₈	1	2	yes	complemented
41	D₈	—	4	2	no	—
42	D₈	—	4	2	no	—
43	D₈	—	8	2	no	—
44	D₈	—	8	4	no	—
45	D₈	—	8	2	no	—
46	D₈	—	8	2	no	—
47	C₂×C₈	—	2	1	no	—
48	C₂×C₈	—	4	1	no	—
49	C₂×C₈	D₈	1	1	yes	characteristic
50	C₂×C₈	D₈	1	1	yes	characteristic
51	C₂×C₈	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈	#41	1	1	yes	characteristic
53	C₄⋊C₄	—	2	1	no	—
54	C₄⋊C₄	—	4	1	no	—
55	C₄²	—	4	1	no	—
56	C₄²	C₂×D₄	1	1	yes	characteristic
57	C₁₆	—	4	2	no	—
58	C₂³	—	8	2	no	—
59	Q₈	—	2	2	no	—
60	Q₈	—	4	1	no	—
61	D₄	—	8	4	no	—
62	D₄	—	8	2	no	—
63	D₄	—	8	2	no	—
64	D₄	—	8	2	no	—
65	D₄	—	8	2	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	—	4	1	no	—
68	C₂×C₄	#121	1	1	yes	characteristic
69	C₂×C₄	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₈	—	2	2	no	—
72	C₈	—	2	1	no	—
73	C₈	—	2	1	no	—
74	C₈	—	8	1	no	—
75	C₈	SD₃₂	1	2	yes	complemented
76	C₈	C₂×D₈	1	1	yes	characteristic
77	C₈	#135	1	1	yes	characteristic
78	C₂²	—	16	2	no	—
79	C₂²	—	16	2	no	—
80	C₂²	—	16	2	no	—
81	C₂²	#459	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	4	2	no	—
84	C₄	—	8	1	no	—
85	C₄	#466	1	1	yes	characteristic
86	C₄	#472	1	1	yes	characteristic
87	C₄	C₂×SD₃₂	1	1	yes	characteristic
88	C₄	#509	1	1	yes	characteristic
89	C₂	—	32	2	no	—
90	C₂	#1667	1	1	yes	characteristic
91	C₂	#2207	1	1	yes	characteristic
92	C₂	#2213	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³
prime radical	#1712	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ 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
1. #2207	C₂	semidirect
3. C₈×Q₁₆	C₂	semidirect
4. #1370	C₂	semidirect
5. #1361	C₂	semidirect
40. Q₁₆	D₈	semidirect
75. C₈	SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
#2207	2
#1712	2
C₈×Q₁₆	2
#1370	2
#1361	2