Subgroups — #34552

Order 256.

981 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2402	C₂	1	6	yes	maximal, complemented
2	#2399	C₂	1	6	yes	maximal, complemented
3	#2396	C₂	1	12	yes	maximal
4	C₄²×Q₈	C₂	1	4	yes	maximal, complemented
5	C₄×#117	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈	C₂²	1	12	yes	—
7	C₂×#117	C₂²	1	6	yes	—
8	C₂²×C₄²	C₂²	1	1	yes	characteristic
9	#400	C₂²	1	12	yes	—
10	#399	C₂²	1	4	yes	—
11	#398	C₂²	1	24	yes	—
12	#396	C₂²	1	8	yes	—
13	#395	C₂²	1	12	yes	—
14	#393	C₂²	1	4	yes	—
15	C₄×C₄⋊C₄	C₂²	1	12	yes	—
16	C₄×C₄⋊C₄	C₂²	1	24	yes	—
17	C₄×C₄⋊C₄	C₂²	1	12	yes	—
18	C₄×C₂²⋊C₄	C₂²	1	12	yes	—
19	#376	C₂²	1	3	yes	—
20	#376	C₂²	1	3	yes	—
21	C₄³	C₂²	1	6	yes	—
22	C₂²×Q₈	C₂³	1	4	yes	—
23	C₂³×C₄	C₂³	1	3	yes	—
24	C₄×Q₈	—	2	48	no	—
25	#117	—	2	24	no	—
26	C₂×C₄⋊C₄	C₂³	1	12	yes	—
27	C₂×C₄⋊C₄	C₂³	1	24	yes	—
28	C₂×C₄⋊C₄	C₂³	1	12	yes	—
29	C₂×C₂²⋊C₄	C₂³	1	12	yes	—
30	C₂×C₄²	Q₈	1	8	yes	complemented
31	C₂×C₄²	C₂³	1	6	yes	—
32	C₂×C₄²	C₂³	1	24	yes	—
33	C₂×C₄²	C₂³	1	3	yes	—
34	C₂×C₄²	C₂³	1	6	yes	—
35	C₂×C₄²	C₂³	1	1	yes	characteristic
36	#95	C₂³	1	12	yes	—
37	#95	C₂³	1	12	yes	—
38	#95	C₂³	1	12	yes	—
39	#95	C₂³	1	4	yes	—
40	#95	C₂³	1	8	yes	—
41	C₂⁴	C₂⁴	1	1	yes	characteristic
42	C₂×Q₈	—	2	24	no	—
43	C₂²×C₄	—	2	12	no	—
44	C₂²×C₄	C₂×Q₈	1	12	yes	—
45	C₂²×C₄	C₂⁴	1	12	yes	—
46	C₂²×C₄	C₂⁴	1	3	yes	—
47	C₂²×C₄	C₂⁴	1	12	yes	—
48	C₂²×C₄	C₂⁴	1	3	yes	—
49	C₄⋊C₄	—	2	24	no	—
50	C₄⋊C₄	—	2	48	no	—
51	C₄⋊C₄	—	2	24	no	—
52	C₂²⋊C₄	—	2	24	no	—
53	C₄²	—	2	12	no	—
54	C₄²	—	2	48	no	—
55	C₄²	C₂×Q₈	1	12	yes	—
56	C₄²	C₂×Q₈	1	4	yes	—
57	C₄²	#41	1	24	yes	—
58	C₂³	—	2	6	no	—
59	C₂³	C₂²×Q₈	1	2	yes	—
60	C₂³	C₂⁵	1	1	yes	characteristic
61	Q₈	—	2	32	no	—
62	C₂×C₄	—	2	24	no	—
63	C₂×C₄	—	2	6	no	—
64	C₂×C₄	—	2	24	no	—
65	C₂×C₄	—	2	12	no	—
66	C₂×C₄	—	2	24	no	—
67	C₂×C₄	C₂²×Q₈	1	6	yes	—
68	C₂×C₄	C₂²×Q₈	1	6	yes	—
69	C₂×C₄	C₂×#41	1	24	yes	—
70	C₂×C₄	C₂×#41	1	12	yes	—
71	C₂²	—	2	6	no	—
72	C₂²	—	4	4	no	—
73	C₂²	C₂³×Q₈	1	1	yes	characteristic
74	C₂²	C₂²×#41	1	6	yes	—
75	C₄	—	2	6	no	—
76	C₄	—	2	24	no	—
77	C₄	—	4	12	no	—
78	C₄	#533	1	12	yes	—
79	C₂	—	4	2	no	—
80	C₂	C₂×#533	1	3	yes	—
81	C₂	#3444	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂³
commutator, Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂⁴	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #2402	C₂	semidirect
2. #2399	C₂	semidirect
4. C₄²×Q₈	C₂	semidirect
30. C₂×C₄²	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#2402	2
#2399	2
#2396	2
C₄²×Q₈	2
C₄×#117	2