Subgroups — #9759

Order 256.

185 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3160	C₂	1	1	yes	maximal, characteristic
2	#1835	C₂	1	1	yes	maximal, characteristic
3	#1518	C₂	1	4	yes	maximal
4	#1452	C₂	1	1	yes	maximal, characteristic
5	#533	C₂²	1	2	yes	—
6	#500	C₄	1	2	yes	complemented
7	#499	—	2	2	no	—
8	#498	C₄	1	2	yes	complemented
9	#431	C₂²	1	2	yes	—
10	#427	—	2	2	no	—
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	#382	C₂²	1	2	yes	—
13	#340	—	2	2	no	—
14	#328	—	2	8	no	—
15	#322	—	2	2	no	—
16	C₂²×C₈	D₄	1	2	yes	—
17	C₄⋊Q₈	C₂×C₄	1	4	yes	—
18	#125	—	2	2	no	—
19	C₂²⋊Q₈	—	2	4	no	—
20	C₄×Q₈	—	2	4	no	—
21	#117	—	2	2	no	—
22	C₂×C₄⋊C₄	—	2	2	no	—
23	C₂×C₄²	C₂³	1	1	yes	characteristic
24	#107	—	2	4	no	—
25	#106	—	2	4	no	—
26	C₄⋊C₈	—	2	2	no	—
27	C₂²⋊C₈	D₄	1	4	yes	—
28	C₄×C₈	—	2	2	no	—
29	C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈	D₄	1	2	yes	—
32	#95	—	2	4	no	—
33	C₂×Q₈	—	2	4	no	—
34	C₂²×C₄	—	4	2	no	—
35	C₂²×C₄	C₂×D₄	1	2	yes	—
36	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈	—	2	4	no	—
38	C₂×C₈	—	2	4	no	—
39	C₂×C₈	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈	C₂²⋊C₄	1	2	yes	—
41	C₄⋊C₄	—	2	4	no	—
42	C₄⋊C₄	—	2	4	no	—
43	C₄⋊C₄	—	4	4	no	—
44	C₄⋊C₄	—	4	4	no	—
45	C₂²⋊C₄	—	4	2	no	—
46	C₄²	—	4	2	no	—
47	C₄²	C₂²×C₄	1	1	yes	characteristic
48	C₄²	C₂×D₄	1	1	yes	characteristic
49	C₄²	#41	1	2	yes	—
50	C₂³	#127	1	1	yes	characteristic
51	Q₈	—	8	4	no	—
52	C₂×C₄	—	4	2	no	—
53	C₂×C₄	—	4	4	no	—
54	C₂×C₄	—	8	4	no	—
55	C₂×C₄	—	8	2	no	—
56	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄	C₄×D₄	1	2	yes	—
58	C₂×C₄	#121	1	2	yes	—
59	C₂×C₄	#124	1	1	yes	characteristic
60	C₂×C₄	#137	1	4	yes	—
61	C₈	—	2	2	no	—
62	C₈	—	2	2	no	—
63	C₈	—	4	4	no	—
64	C₂²	—	2	2	no	—
65	C₂²	#390	1	1	yes	characteristic
66	C₂²	#497	1	2	yes	—
67	C₄	—	2	2	no	—
68	C₄	—	2	2	no	—
69	C₄	—	8	4	no	—
70	C₄	—	16	2	no	—
71	C₄	#419	1	2	yes	—
72	C₄	#491	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#1960	1	1	yes	characteristic
75	C₂	#1976	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#390
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#127
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
6. #500	C₄	semidirect
8. #498	C₄	semidirect

Maximal subgroups

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

subgroup	index
#3160	2
#1835	2
#1518	2
#1452	2