Subgroups — #7225

Order 256.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2165	C₂	1	1	yes	maximal, characteristic
2	#1841	C₂	1	1	yes	maximal, characteristic
3	#1455	C₂	1	1	yes	maximal, characteristic
4	C₄×M₁₆	—	2	1	no	—
5	C₄×M₁₆	C₂²	1	1	yes	characteristic
6	#364	—	2	1	no	—
7	#349	—	2	1	no	—
8	#347	C₄	1	2	yes	complemented
9	#344	—	2	1	no	—
10	#339	—	2	1	no	—
11	#338	C₄	1	1	yes	characteristic
12	H(3,ℤ/4)	C₄	1	1	yes	characteristic
13	#322	—	2	1	no	—
14	#322	C₄	1	1	yes	characteristic, complemented
15	C₈²	C₄	1	1	yes	characteristic, complemented
16	C₂×M₁₆	—	2	2	no	—
17	C₂×M₁₆	—	2	1	no	—
18	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆	D₄	1	1	yes	characteristic
20	#117	—	2	1	no	—
21	C₂×C₄²	D₄	1	1	yes	characteristic
22	M₃₂	—	2	2	no	—
23	#97	—	2	2	no	—
24	#97	C₂×C₄	1	1	yes	characteristic
25	#97	Q₈	1	1	yes	characteristic
26	C₄×C₈	—	2	1	no	—
27	C₄×C₈	—	2	1	no	—
28	C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₄	—	2	1	no	—
31	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	M₁₆	—	4	1	no	—
33	M₁₆	—	4	1	no	—
34	M₁₆	—	4	4	no	—
35	M₁₆	—	4	1	no	—
36	M₁₆	—	4	1	no	—
37	C₂×C₈	—	2	2	no	—
38	C₂×C₈	—	2	2	no	—
39	C₂×C₈	—	4	1	no	—
40	C₂×C₈	C₄²	1	1	yes	characteristic
41	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₄⋊C₄	—	4	1	no	—
45	C₂²⋊C₄	—	4	1	no	—
46	C₄²	—	2	1	no	—
47	C₄²	—	4	1	no	—
48	C₄²	C₂²⋊C₄	1	1	yes	characteristic
49	C₄²	C₄⋊C₄	1	1	yes	characteristic
50	C₁₆	—	4	2	no	—
51	C₂³	#104	1	1	yes	characteristic
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	4	2	no	—
57	C₂×C₄	—	8	1	no	—
58	C₂×C₄	#95	1	1	yes	characteristic
59	C₂×C₄	#104	1	1	yes	characteristic
60	C₈	—	2	4	no	—
61	C₈	—	2	1	no	—
62	C₈	—	2	1	no	—
63	C₈	—	2	1	no	—
64	C₈	—	2	1	no	—
65	C₈	—	4	2	no	—
66	C₈	—	8	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	—	4	1	no	—
69	C₂²	#339	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	16	1	no	—
74	C₄	H(3,ℤ/4)	1	1	yes	characteristic
75	C₄	#344	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	#1288	1	1	yes	characteristic

	subgroup	G/H
center	C₄	H(3,ℤ/4)
commutator	C₂×C₈	C₄²
Frattini, cosocle	C₄×M₁₆	C₂²
socle	C₂	#1288
prime radical	C₂³	#104
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
8. #347	C₄	semidirect
14. #322	C₄	semidirect
15. C₈²	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2165	2
#1841	2
#1455	2