Subgroups — #8792

Order 256.

131 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1837	C₂	1	1	yes	maximal, characteristic
2	#1835	C₂	1	1	yes	maximal, characteristic
3	#1450	C₂	1	1	yes	maximal, characteristic
4	#1286	C₂	1	4	yes	maximal
5	#431	C₂²	1	2	yes	—
6	#423	C₂²	1	2	yes	—
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	#382	C₂²	1	2	yes	—
9	#341	—	2	2	no	—
10	#340	—	2	2	no	—
11	#322	C₄	1	2	yes	complemented
12	C₈²	C₄	1	2	yes	complemented
13	C₂×M₁₆	—	2	2	no	—
14	C₂²×C₈	D₄	1	2	yes	—
15	C₂×C₄⋊C₄	—	2	2	no	—
16	C₂×C₄²	C₂³	1	1	yes	characteristic
17	C₄⋊C₈	—	2	2	no	—
18	C₄⋊C₈	—	2	4	no	—
19	C₂²⋊C₈	—	2	2	no	—
20	C₄×C₈	C₂×C₄	1	4	yes	—
21	C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈	Q₈	1	2	yes	—
24	#95	—	2	4	no	—
25	C₂²×C₄	—	4	2	no	—
26	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄	#41	1	2	yes	—
28	M₁₆	—	4	4	no	—
29	C₂×C₈	—	2	4	no	—
30	C₂×C₈	—	2	4	no	—
31	C₂×C₈	—	4	2	no	—
32	C₂×C₈	C₄⋊C₄	1	2	yes	—
33	C₂×C₈	C₄⋊C₄	1	2	yes	—
34	C₄⋊C₄	—	4	4	no	—
35	C₄²	C₂²×C₄	1	1	yes	characteristic
36	C₄²	C₂×Q₈	1	1	yes	characteristic
37	C₄²	#41	1	2	yes	—
38	C₂³	#124	1	1	yes	characteristic
39	C₂×C₄	—	4	2	no	—
40	C₂×C₄	—	8	4	no	—
41	C₂×C₄	—	8	2	no	—
42	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄	#117	1	2	yes	—
44	C₂×C₄	#124	1	1	yes	characteristic
45	C₂×C₄	#125	1	2	yes	—
46	C₂×C₄	#135	1	4	yes	—
47	C₈	—	2	2	no	—
48	C₈	—	2	2	no	—
49	C₈	—	4	4	no	—
50	C₈	—	8	2	no	—
51	C₂²	—	2	2	no	—
52	C₂²	#382	1	1	yes	characteristic
53	C₂²	#488	1	2	yes	—
54	C₄	—	2	2	no	—
55	C₄	—	2	2	no	—
56	C₄	—	16	2	no	—
57	C₄	#427	1	2	yes	—
58	C₄	#488	1	2	yes	—
59	C₂	—	2	2	no	—
60	C₂	#1833	1	2	yes	—
61	C₂	#1835	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#382
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
11. #322	C₄	semidirect
12. C₈²	C₄	semidirect

Maximal subgroups

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

subgroup	index
#1837	2
#1835	2
#1450	2
#1286	2