Subgroups — #30237

Order 256.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2081	C₂	1	6	yes	maximal, complemented
2	#2069	C₂	1	2	yes	maximal, complemented
3	#1978	C₂	1	3	yes	maximal
4	#1953	C₂	1	3	yes	maximal, complemented
5	#1747	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈	C₂²	1	1	yes	characteristic
7	#430	C₂²	1	6	yes	—
8	C₂×#108	C₂²	1	3	yes	—
9	#418	C₂²	1	6	yes	—
10	C₂×#100	C₂²	1	6	yes	—
11	#405	C₂²	1	3	yes	—
12	#344	C₂²	1	6	yes	—
13	#338	C₂²	1	1	yes	characteristic
14	H(3,ℤ/4)	C₂²	1	3	yes	—
15	C₂²×D₄	C₂³	1	2	yes	—
16	C₂×D₈	—	2	3	no	—
17	C₂×D₈	—	2	3	no	—
18	C₂×M₁₆	C₂³	1	3	yes	—
19	C₂×M₁₆	C₂³	1	6	yes	—
20	C₂²×C₈	C₂³	1	1	yes	characteristic
21	#117	C₂³	1	3	yes	—
22	#108	—	2	3	no	—
23	#108	—	2	3	no	—
24	#108	—	4	6	no	—
25	#102	—	4	6	no	—
26	#100	—	4	6	no	—
27	#97	—	2	3	no	—
28	C₄×C₈	—	2	3	no	—
29	C₂⁴	—	2	2	no	—
30	C₂×D₄	—	2	6	no	—
31	C₂×D₄	—	2	6	no	—
32	C₂²×C₄	C₂⁴	1	1	yes	characteristic
33	D₈	—	4	1	no	—
34	D₈	—	4	3	no	—
35	M₁₆	—	2	6	no	—
36	M₁₆	—	2	3	no	—
37	M₁₆	—	2	6	no	—
38	M₁₆	—	2	3	no	—
39	C₂×C₈	—	2	3	no	—
40	C₂×C₈	—	2	6	no	—
41	C₂×C₈	#41	1	3	yes	—
42	C₂×C₈	#41	1	3	yes	—
43	C₄⋊C₄	—	2	3	no	—
44	C₂²⋊C₄	—	2	3	no	—
45	C₄²	—	2	3	no	—
46	C₂³	—	4	6	no	—
47	C₂³	—	16	2	no	—
48	C₂³	2₋¹⁺⁴	1	1	yes	characteristic
49	D₄	—	4	2	no	—
50	D₄	—	4	6	no	—
51	C₂×C₄	—	4	3	no	—
52	C₂×C₄	C₂×#41	1	3	yes	—
53	C₂×C₄	H(5,𝔽₂)	1	3	yes	—
54	C₈	—	4	3	no	—
55	C₈	—	4	6	no	—
56	C₈	#126	1	1	yes	characteristic
57	C₈	#126	1	3	yes	—
58	C₂²	—	4	1	no	—
59	C₂²	—	8	2	no	—
60	C₂²	—	16	6	no	—
61	C₂²	#542	1	3	yes	—
62	C₄	—	8	3	no	—
63	C₄	C₂×#126	1	1	yes	characteristic
64	C₄	#540	1	3	yes	—
65	C₂	—	2	3	no	—
66	C₂	—	16	2	no	—
67	C₂	#2644	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#2644
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂²×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #2081	C₂	semidirect
2. #2069	C₂	semidirect
4. #1953	C₂	semidirect
5. #1747	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2081	2
#2069	2
#1978	2
#1953	2
#1747	2