Subgroups — #33404

Order 256.

318 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3407	C₂	1	1	yes	maximal, characteristic
2	#2916	C₂	1	1	yes	maximal, characteristic
3	#2912	C₂	1	1	yes	maximal, characteristic
4	#2155	C₂	1	6	yes	maximal, complemented
5	C₂×#365	C₂	1	6	yes	maximal, complemented
6	#568	—	2	1	no	—
7	C₂×#131	C₂²	1	1	yes	characteristic
8	#520	—	2	1	no	—
9	#504	C₄	1	8	yes	complemented
10	C₂×M₃₂	C₂²	1	6	yes	—
11	#430	—	2	3	no	—
12	C₂×#108	C₂²	1	6	yes	—
13	#428	—	2	3	no	—
14	#427	C₂²	1	3	yes	—
15	C₂×#106	C₂²	1	3	yes	—
16	#365	C₂²	1	12	yes	—
17	#365	C₂²	1	4	yes	—
18	C₂×#41	D₄	1	1	yes	characteristic
19	#131	—	4	4	no	—
20	#131	C₂×C₄	1	4	yes	—
21	#131	Q₈	1	4	yes	—
22	C₂×M₁₆	—	2	6	no	—
23	C₂×M₁₆	D₄	1	3	yes	—
24	C₂²×C₈	C₂³	1	3	yes	—
25	C₄×Q₈	—	4	1	no	—
26	C₄×D₄	—	4	3	no	—
27	#117	—	2	3	no	—
28	C₂×C₄⋊C₄	—	2	3	no	—
29	M₃₂	C₂×C₄	1	12	yes	complemented
30	M₃₂	C₂³	1	4	yes	—
31	C₂×C₁₆	C₂×C₄	1	12	yes	complemented
32	#108	—	2	6	no	—
33	#108	C₂³	1	4	yes	—
34	#107	—	2	3	no	—
35	#106	—	2	3	no	—
36	#106	C₂³	1	3	yes	—
37	#106	C₂³	1	1	yes	characteristic
38	#41	C₄⋊C₄	1	4	yes	—
39	#41	SD₁₆	1	4	yes	—
40	C₂×Q₈	SD₁₆	1	1	yes	characteristic
41	C₂×D₄	SD₁₆	1	3	yes	—
42	C₂²×C₄	—	4	3	no	—
43	C₂²×C₄	C₂×D₄	1	3	yes	—
44	M₁₆	—	2	4	no	—
45	M₁₆	—	4	6	no	—
46	M₁₆	C₄⋊C₄	1	12	yes	—
47	C₂×C₈	—	4	6	no	—
48	C₂×C₈	C₂²×C₄	1	6	yes	—
49	C₂×C₈	C₂×D₄	1	3	yes	—
50	C₂×C₈	C₂×Q₈	1	6	yes	—
51	C₂×C₈	C₂⁴	1	1	yes	characteristic
52	C₄⋊C₄	—	2	3	no	—
53	C₄⋊C₄	—	2	1	no	—
54	C₄⋊C₄	—	4	3	no	—
55	C₂²⋊C₄	—	4	3	no	—
56	C₄²	—	4	3	no	—
57	C₁₆	C₂²×C₄	1	8	yes	—
58	C₂³	C₂×SD₁₆	1	3	yes	—
59	Q₈	#106	1	4	yes	—
60	D₄	#106	1	12	yes	—
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	—	4	3	no	—
63	C₂×C₄	—	16	3	no	—
64	C₂×C₄	C₂×C₄⋊C₄	1	6	yes	—
65	C₂×C₄	C₂×SD₁₆	1	6	yes	—
66	C₂×C₄	C₂×SD₁₆	1	3	yes	—
67	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
68	C₈	—	4	4	no	—
69	C₈	C₂×C₄⋊C₄	1	6	yes	—
70	C₈	C₂³×C₄	1	1	yes	characteristic
71	C₈	C₂²×Q₈	1	1	yes	characteristic
72	C₂²	—	4	3	no	—
73	C₂²	C₂×#106	1	6	yes	—
74	C₂²	C₂²×SD₁₆	1	1	yes	characteristic
75	C₄	—	8	1	no	—
76	C₄	—	8	3	no	—
77	C₄	C₂×#106	1	6	yes	—
78	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
79	C₄	C₂²×SD₁₆	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	2	6	no	—
82	C₂	C₂²×#106	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂²×#106
commutator	C₈	C₂³×C₄
Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #2155	C₂	semidirect
5. C₂×#365	C₂	semidirect
9. #504	C₄	semidirect
29. M₃₂	C₂×C₄	semidirect
31. C₂×C₁₆	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
#3407	2
#2916	2
#2912	2
#2155	2
C₂×#365	2