Subgroups — #7075

Order 256.

67 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1591	C₂	1	1	yes	maximal, characteristic
2	#1367	C₂	1	1	yes	maximal, characteristic
3	#1311	C₂	1	1	yes	maximal, characteristic
4	#446	—	2	1	no	—
5	C₄×Q₁₆	C₄	1	1	yes	characteristic
6	C₄⋊C₁₆	—	2	1	no	—
7	C₄×C₁₆	—	2	1	no	—
8	#335	C₄	1	1	yes	characteristic
9	Q₈⋊C₈	—	2	1	no	—
10	C₈²	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆	C₈	1	1	yes	characteristic
12	C₄×Q₈	—	2	1	no	—
13	C₂×C₁₆	—	2	2	no	—
14	C₂×C₁₆	—	4	1	no	—
15	#107	C₈	1	1	yes	characteristic
16	C₄⋊C₈	—	2	1	no	—
17	C₄⋊C₈	—	4	1	no	—
18	#103	—	2	1	no	—
19	#97	—	4	1	no	—
20	C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈	D₄	1	1	yes	characteristic
22	C₄×C₈	D₄	1	1	yes	characteristic
23	C₂×Q₈	—	2	1	no	—
24	Q₁₆	—	4	1	no	—
25	C₂×C₈	—	2	1	no	—
26	C₂×C₈	—	4	1	no	—
27	C₂×C₈	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈	M₁₆	1	1	yes	characteristic
29	C₂×C₈	D₈	1	1	yes	characteristic
30	C₂×C₈	SD₁₆	1	1	yes	characteristic
31	C₄⋊C₄	—	2	1	no	—
32	C₄⋊C₄	—	4	1	no	—
33	C₄²	—	4	1	no	—
34	C₄²	C₂²⋊C₄	1	1	yes	characteristic
35	C₁₆	—	2	4	no	—
36	C₁₆	—	8	1	no	—
37	Q₈	—	4	1	no	—
38	Q₈	—	4	1	no	—
39	C₂×C₄	—	4	1	no	—
40	C₂×C₄	—	4	1	no	—
41	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
42	C₂×C₄	#102	1	1	yes	characteristic
43	C₂×C₄	#104	1	1	yes	characteristic
44	C₈	—	2	2	no	—
45	C₈	—	2	1	no	—
46	C₈	—	2	1	no	—
47	C₈	—	2	1	no	—
48	C₈	—	2	1	no	—
49	C₈	—	8	1	no	—
50	C₂²	D₄⋊C₈	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	—	8	1	no	—
53	C₄	—	8	1	no	—
54	C₄	#349	1	1	yes	characteristic
55	C₄	#350	1	1	yes	characteristic
56	C₄	#360	1	1	yes	characteristic
57	C₄	#362	1	1	yes	characteristic
58	C₂	#1330	1	1	yes	characteristic
59	C₂	#1334	1	1	yes	characteristic
60	C₂	#1335	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#102
commutator	C₂×C₈	C₂×C₈
Frattini, cosocle	C₈²	C₂²
socle, prime radical	C₂²	D₄⋊C₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
#1591	2
#1367	2
#1311	2