Subgroups — #13617

Order 256.

165 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#2158	C₂	1	1	yes	maximal, characteristic, complemented
4	#1415	C₂	1	4	yes	maximal, complemented
5	C₂²×D₈	—	2	1	no	—
6	C₂×D₁₆	—	2	1	no	—
7	C₂×D₁₆	—	2	2	no	—
8	C₂×D₁₆	C₄	1	4	yes	complemented
9	C₂×D₁₆	C₂²	1	2	yes	—
10	C₂²×C₁₆	C₂²	1	1	yes	characteristic
11	#427	—	2	1	no	—
12	M₆₄	—	2	2	no	—
13	C₂×C₃₂	C₂²	1	2	yes	complemented
14	#367	—	2	1	no	—
15	#366	C₂²	1	2	yes	complemented
16	C₂²×D₄	—	4	1	no	—
17	C₂×D₈	—	2	4	no	—
18	C₂×D₈	—	2	2	no	—
19	C₂×D₈	—	4	1	no	—
20	C₂×D₈	—	4	2	no	—
21	C₂²×C₈	D₄	1	1	yes	characteristic
22	#117	—	4	1	no	—
23	D₁₆	—	2	4	no	—
24	D₁₆	—	2	2	no	—
25	D₁₆	C₂×C₄	1	4	yes	—
26	C₂×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₆	D₄	1	1	yes	characteristic
28	C₂×C₁₆	D₄	1	2	yes	—
29	C₂×C₁₆	C₂³	1	1	yes	characteristic
30	#107	—	2	2	no	—
31	#106	—	4	1	no	—
32	C₃₂	—	2	2	no	—
33	C₂⁴	—	8	1	no	—
34	C₂×D₄	—	4	4	no	—
35	C₂×D₄	—	4	2	no	—
36	C₂×D₄	—	8	1	no	—
37	C₂×D₄	—	8	2	no	—
38	C₂²×C₄	D₈	1	1	yes	characteristic
39	D₈	—	2	4	no	—
40	D₈	—	4	4	no	—
41	D₈	—	4	2	no	—
42	C₂×C₈	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈	D₈	1	1	yes	characteristic
44	C₂×C₈	SD₁₆	1	2	yes	—
45	C₂×C₈	C₂×D₄	1	1	yes	characteristic
46	C₄⋊C₄	—	4	2	no	—
47	C₂²⋊C₄	—	8	1	no	—
48	C₄²	—	8	1	no	—
49	C₁₆	C₂²⋊C₄	1	2	yes	—
50	C₁₆	C₂²×C₄	1	1	yes	characteristic
51	C₁₆	C₂×D₄	1	1	yes	characteristic
52	C₂³	—	8	4	no	—
53	C₂³	—	8	2	no	—
54	C₂³	—	16	4	no	—
55	C₂³	D₁₆	1	1	yes	characteristic
56	D₄	—	4	4	no	—
57	D₄	—	8	4	no	—
58	D₄	—	8	2	no	—
59	C₂×C₄	—	8	2	no	—
60	C₂×C₄	#102	1	2	yes	—
61	C₂×C₄	D₁₆	1	1	yes	characteristic
62	C₂×C₄	SD₃₂	1	2	yes	—
63	C₂×C₄	C₂×D₈	1	1	yes	characteristic
64	C₈	#102	1	2	yes	—
65	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₈	C₂×SD₁₆	1	1	yes	characteristic
67	C₂²	—	2	2	no	—
68	C₂²	—	8	4	no	—
69	C₂²	—	16	8	no	—
70	C₂²	—	16	4	no	—
71	C₂²	#357	1	2	yes	—
72	C₂²	C₂×D₁₆	1	1	yes	characteristic
73	C₄	—	16	2	no	—
74	C₄	#357	1	2	yes	—
75	C₄	C₂×#102	1	1	yes	characteristic
76	C₄	C₂×SD₃₂	1	1	yes	characteristic
77	C₂	—	2	2	no	—
78	C₂	—	16	4	no	—
79	C₂	C₂×#357	1	1	yes	characteristic
80	C₂	#2263	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×D₁₆
commutator	C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×M₆₄	C₂	semidirect
3. #2158	C₂	semidirect
4. #1415	C₂	semidirect
8. C₂×D₁₆	C₄	semidirect
13. C₂×C₃₂	C₂²	semidirect
15. #366	C₂²	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₆	2
C₂×M₆₄	2
#2158	2
#1415	2