Subgroups — #7094

Order 256.

79 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1706	C₂	1	1	yes	maximal, characteristic
2	#1370	C₂	1	1	yes	maximal, characteristic, complemented
3	#1367	C₂	1	1	yes	maximal, characteristic, complemented
4	#500	C₄	1	1	yes	characteristic
5	#493	C₄	1	1	yes	characteristic
6	#492	—	2	1	no	—
7	#486	—	2	1	no	—
8	C₄⋊C₁₆	—	2	1	no	—
9	C₄⋊C₁₆	—	2	1	no	—
10	C₈²	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆	—	4	1	no	—
12	C₂×SD₁₆	—	4	1	no	—
13	C₂×D₈	—	4	1	no	—
14	C₄⋊Q₈	—	2	1	no	—
15	#127	—	2	1	no	—
16	C₂×C₁₆	—	4	1	no	—
17	C₂×C₁₆	—	4	1	no	—
18	#107	—	4	1	no	—
19	#102	—	4	1	no	—
20	#102	—	4	1	no	—
21	C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈	D₄	1	1	yes	characteristic
23	C₄×C₈	D₄	1	1	yes	characteristic
24	C₂×Q₈	—	4	1	no	—
25	C₂×D₄	—	4	1	no	—
26	C₂×D₄	—	4	1	no	—
27	C₂×D₄	—	4	1	no	—
28	SD₁₆	—	8	2	no	—
29	SD₁₆	—	8	1	no	—
30	SD₁₆	—	8	1	no	—
31	D₈	—	8	2	no	—
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	D₈	1	1	yes	characteristic
34	C₂×C₈	D₈	1	1	yes	characteristic
35	C₂×C₈	SD₁₆	1	1	yes	characteristic
36	C₂×C₈	SD₁₆	1	1	yes	characteristic
37	C₄⋊C₄	—	4	1	no	—
38	C₄⋊C₄	—	4	1	no	—
39	C₄²	C₂²⋊C₄	1	1	yes	characteristic
40	C₁₆	—	8	1	no	—
41	C₁₆	—	8	1	no	—
42	C₂³	—	8	1	no	—
43	Q₈	—	8	1	no	—
44	Q₈	—	8	1	no	—
45	D₄	—	8	2	no	—
46	D₄	—	8	1	no	—
47	D₄	—	8	1	no	—
48	D₄	—	8	1	no	—
49	D₄	—	8	1	no	—
50	C₂×C₄	—	8	1	no	—
51	C₂×C₄	#100	1	1	yes	characteristic
52	C₂×C₄	#102	1	1	yes	characteristic
53	C₂×C₄	#102	1	1	yes	characteristic
54	C₈	—	2	2	no	—
55	C₈	—	2	1	no	—
56	C₈	—	2	1	no	—
57	C₈	—	2	1	no	—
58	C₈	D₁₆	1	1	yes	characteristic
59	C₈	SD₃₂	1	1	yes	characteristic
60	C₂²	—	16	1	no	—
61	C₂²	—	16	1	no	—
62	C₂²	—	16	1	no	—
63	C₂²	#331	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	16	1	no	—
66	C₄	#357	1	1	yes	characteristic
67	C₄	#359	1	1	yes	characteristic
68	C₄	#360	1	1	yes	characteristic
69	C₄	#361	1	1	yes	characteristic
70	C₂	—	32	1	no	—
71	C₂	#1359	1	1	yes	characteristic
72	C₂	#1360	1	1	yes	characteristic
73	C₂	#1361	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#331
commutator	C₄×C₈	C₂×C₄
Frattini, cosocle	C₈²	C₂²
prime radical	#493	C₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #1370	C₂	semidirect
3. #1367	C₂	semidirect

Maximal subgroups

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

subgroup	index
#1706	2
#1370	2
#1367	2