Subgroups — #33300

Order 256.

867 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#492	C₂	1	12	yes	maximal, complemented
2	#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	#2867	C₂	1	1	yes	maximal, characteristic
4	C₄²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×SD₁₆	C₂²	1	16	yes	—
6	C₂×C₄⋊Q₈	C₂²	1	6	yes	—
7	C₂×#127	—	2	4	no	—
8	C₂×#127	C₂²	1	6	yes	—
9	#492	—	2	48	no	—
10	C₂×C₄×C₈	C₂²	1	6	yes	—
11	#384	—	2	4	no	—
12	C₄³	C₂²	1	1	yes	characteristic
13	C₂²×Q₈	C₂³	1	4	yes	—
14	C₂²×D₄	—	2	12	no	—
15	C₂²×D₄	C₂³	1	4	yes	—
16	C₂×SD₁₆	—	2	96	no	—
17	C₂²×C₈	C₂³	1	4	yes	—
18	C₄⋊Q₈	—	2	24	no	—
19	#127	—	2	24	no	—
20	#127	—	4	16	no	—
21	C₂×C₄⋊C₄	—	2	12	no	—
22	C₂×C₄²	D₄	1	4	yes	—
23	C₂×C₄²	C₂³	1	3	yes	—
24	C₄×C₈	D₄	1	24	yes	complemented
25	C₂⁴	—	2	4	no	—
26	C₂×Q₈	—	2	24	no	—
27	C₂×D₄	—	2	24	no	—
28	C₂×D₄	—	4	24	no	—
29	C₂×D₄	—	4	48	no	—
30	C₂²×C₄	—	2	4	no	—
31	C₂²×C₄	C₂×D₄	1	6	yes	—
32	C₂²×C₄	C₂⁴	1	1	yes	characteristic
33	SD₁₆	—	4	64	no	—
34	C₂×C₈	C₂×D₄	1	24	yes	—
35	C₄⋊C₄	—	4	24	no	—
36	C₄²	SD₁₆	1	16	yes	complemented
37	C₄²	C₂×D₄	1	12	yes	—
38	C₂³	—	4	12	no	—
39	C₂³	—	4	16	no	—
40	C₂³	C₂²×D₄	1	1	yes	characteristic
41	Q₈	—	4	16	no	—
42	D₄	—	4	16	no	—
43	D₄	—	8	48	no	—
44	C₂×C₄	—	4	12	no	—
45	C₂×C₄	#127	1	12	yes	—
46	C₂×C₄	C₂×SD₁₆	1	24	yes	—
47	C₂×C₄	C₂²×D₄	1	6	yes	—
48	C₈	#127	1	16	yes	complemented
49	C₂²	—	8	4	no	—
50	C₂²	—	8	24	no	—
51	C₂²	C₂×#127	1	3	yes	—
52	C₂²	C₂²×SD₁₆	1	4	yes	—
53	C₄	—	8	4	no	—
54	C₄	#492	1	24	yes	complemented
55	C₄	C₂×#127	1	4	yes	—
56	C₂	—	16	4	no	—
57	C₂	#2867	1	1	yes	characteristic
58	C₂	C₂×#492	1	6	yes	—

	subgroup	G/H
center, socle	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	#2867	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#492	C₂	semidirect
2. #2868	C₂	semidirect
4. C₄²×C₈	C₂	semidirect
24. C₄×C₈	D₄	semidirect
36. C₄²	SD₁₆	semidirect
48. C₈	#127	semidirect
54. C₄	#492	semidirect

Maximal subgroups

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

subgroup	index
C₂×#492	2
#2868	2
#2867	2
C₄²×C₈	2