Subgroups — #12491

Order 256.

141 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2206	C₂	1	2	yes	maximal, complemented
2	#1712	C₂	1	1	yes	maximal, characteristic, complemented
3	#1589	C₂	1	1	yes	maximal, characteristic, complemented
4	#1367	C₂	1	1	yes	maximal, characteristic, complemented
5	#1361	C₂	1	2	yes	maximal, complemented
6	C₂×D₁₆	—	2	2	no	—
7	#493	—	2	1	no	—
8	#493	—	2	1	no	—
9	#493	C₂²	1	2	yes	—
10	C₄×D₈	C₂²	1	1	yes	characteristic
11	#436	—	2	1	no	—
12	C₄⋊C₁₆	C₂²	1	2	yes	—
13	#357	—	2	2	no	—
14	#335	C₂²	1	1	yes	characteristic
15	D₄⋊C₈	—	2	1	no	—
16	C₈²	C₂²	1	1	yes	characteristic
17	C₂×D₈	—	2	2	no	—
18	C₂×D₈	—	2	2	no	—
19	C₂×D₈	—	4	2	no	—
20	C₂×D₈	—	4	1	no	—
21	C₂×D₈	—	4	1	no	—
22	C₂×D₈	D₄	1	1	yes	characteristic
23	C₂×M₁₆	—	4	1	no	—
24	#127	—	2	2	no	—
25	C₄×D₄	—	2	1	no	—
26	D₁₆	—	4	4	no	—
27	C₂×C₁₆	—	2	2	no	—
28	#107	D₄	1	1	yes	characteristic
29	C₄⋊C₈	—	2	1	no	—
30	#102	—	2	1	no	—
31	C₂²⋊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×D₄	—	2	1	no	—
36	C₂×D₄	—	4	2	no	—
37	C₂×D₄	—	4	2	no	—
38	C₂×D₄	—	4	2	no	—
39	C₂²×C₄	—	4	1	no	—
40	D₈	—	2	2	no	—
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	M₁₆	—	8	2	no	—
48	C₂×C₈	—	2	1	no	—
49	C₂×C₈	—	4	1	no	—
50	C₂×C₈	D₈	1	1	yes	characteristic
51	C₂×C₈	D₈	1	1	yes	characteristic
52	C₂×C₈	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈	#41	1	1	yes	characteristic
54	C₄⋊C₄	—	2	1	no	—
55	C₂²⋊C₄	—	4	1	no	—
56	C₄²	C₂×D₄	1	1	yes	characteristic
57	C₁₆	—	4	2	no	—
58	C₂³	—	4	1	no	—
59	C₂³	—	8	2	no	—
60	D₄	—	4	1	no	—
61	D₄	—	4	1	no	—
62	D₄	—	8	4	no	—
63	D₄	—	8	2	no	—
64	D₄	—	8	2	no	—
65	D₄	—	8	2	no	—
66	D₄	—	8	2	no	—
67	C₂×C₄	—	4	1	no	—
68	C₂×C₄	—	8	1	no	—
69	C₂×C₄	—	8	1	no	—
70	C₂×C₄	#121	1	1	yes	characteristic
71	C₂×C₄	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₈	—	2	1	no	—
75	C₈	—	2	1	no	—
76	C₈	—	8	1	no	—
77	C₈	D₁₆	1	2	yes	complemented
78	C₈	AGL(1,ℤ/8)	1	2	yes	—
79	C₂²	—	8	1	no	—
80	C₂²	—	8	1	no	—
81	C₂²	—	8	1	no	—
82	C₂²	—	16	2	no	—
83	C₂²	—	16	2	no	—
84	C₂²	—	16	2	no	—
85	C₂²	#459	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	8	1	no	—
88	C₄	#469	1	1	yes	characteristic
89	C₄	#472	1	1	yes	characteristic
90	C₄	C₂×D₁₆	1	1	yes	characteristic
91	C₄	#509	1	1	yes	characteristic
92	C₂	—	16	1	no	—
93	C₂	—	32	2	no	—
94	C₂	#1687	1	1	yes	characteristic
95	C₂	#2206	1	1	yes	characteristic
96	C₂	#2213	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³
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. #2206	C₂	semidirect
2. #1712	C₂	semidirect
3. #1589	C₂	semidirect
4. #1367	C₂	semidirect
5. #1361	C₂	semidirect
77. C₈	D₁₆	semidirect

Maximal subgroups

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

subgroup	index
#2206	2
#1712	2
#1589	2
#1367	2
#1361	2