Subgroups — #7418

Order 256.

81 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2215	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#1380	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆	C₄	1	1	yes	characteristic, complemented
5	C₂²×C₁₆	C₂²	1	1	yes	characteristic
6	#466	—	2	1	no	—
7	C₂×#107	—	2	1	no	—
8	C₂×C₃₂	—	2	1	no	—
9	#366	C₄	1	1	yes	characteristic, complemented
10	#357	—	2	1	no	—
11	C₂×D₈	—	2	1	no	—
12	C₂²×C₈	D₄	1	1	yes	characteristic
13	#121	—	4	1	no	—
14	C₂×C₄⋊C₄	—	4	1	no	—
15	D₁₆	—	4	1	no	—
16	C₂×C₁₆	—	4	1	no	—
17	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆	D₄	1	1	yes	characteristic
19	#107	—	2	2	no	—
20	#107	—	2	1	no	—
21	#107	—	4	1	no	—
22	#102	—	4	1	no	—
23	C₃₂	—	4	1	no	—
24	C₂×D₄	—	4	1	no	—
25	C₂×D₄	—	8	1	no	—
26	C₂²×C₄	—	8	1	no	—
27	C₂²×C₄	D₈	1	1	yes	characteristic
28	D₈	—	4	1	no	—
29	D₈	—	4	1	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	2	1	no	—
32	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₈	SD₁₆	1	1	yes	characteristic
34	C₄⋊C₄	—	4	2	no	—
35	C₄⋊C₄	—	4	1	no	—
36	C₄⋊C₄	—	8	1	no	—
37	C₂²⋊C₄	—	8	1	no	—
38	C₁₆	—	2	1	no	—
39	C₁₆	—	2	1	no	—
40	C₂³	—	8	1	no	—
41	C₂³	D₁₆	1	1	yes	characteristic
42	D₄	—	8	1	no	—
43	D₄	—	8	1	no	—
44	D₄	—	16	1	no	—
45	D₄	—	16	1	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	8	2	no	—
49	C₂×C₄	—	8	1	no	—
50	C₂×C₄	—	16	1	no	—
51	C₂×C₄	—	16	1	no	—
52	C₂×C₄	#102	1	1	yes	characteristic
53	C₂×C₄	SD₃₂	1	1	yes	characteristic
54	C₈	—	2	1	no	—
55	C₈	#99	1	1	yes	characteristic
56	C₈	#104	1	1	yes	characteristic
57	C₂²	—	2	1	no	—
58	C₂²	—	2	1	no	—
59	C₂²	—	16	1	no	—
60	C₂²	—	16	1	no	—
61	C₂²	—	16	1	no	—
62	C₂²	#357	1	1	yes	characteristic
63	C₂²	D₃₂	1	1	yes	characteristic
64	C₂²	SD₆₄	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	16	2	no	—
67	C₄	—	16	1	no	—
68	C₄	#327	1	1	yes	characteristic
69	C₄	#360	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	2	1	no	—
72	C₂	—	32	1	no	—
73	C₂	#1347	1	1	yes	characteristic
74	C₂	#1415	1	1	yes	characteristic
75	C₂	#1418	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#357
commutator	C₂×C₁₆	C₂×C₄
Frattini, cosocle	C₂²×C₁₆	C₂²
prime radical	#2215	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²⋊C₃₂	C₂	semidirect
3. #1380	C₂	semidirect
4. C₂×D₁₆	C₄	semidirect
9. #366	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2215	2
C₂²⋊C₃₂	2
#1380	2