Subgroups — #33999

Order 256.

1799 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41	C₂	1	3	yes	maximal
2	#2276	C₂	1	4	yes	maximal, complemented
3	#2275	C₂	1	12	yes	maximal, complemented
4	C₄×#117	C₂	1	9	yes	maximal, complemented
5	C₂×#376	C₂	1	3	yes	maximal, complemented
6	C₂²×#41	C₂²	1	1	yes	characteristic
7	C₄×#41	C₄	1	48	yes	complemented
8	C₂×C₄×Q₈	C₂²	1	6	yes	—
9	C₂×C₄×D₄	C₂²	1	18	yes	—
10	C₂×#117	C₂²	1	9	yes	—
11	C₂²×C₄²	C₂²	1	9	yes	—
12	C₄×C₄⋊C₄	C₂²	1	36	yes	—
13	C₄×C₂²⋊C₄	C₂²	1	36	yes	—
14	#376	C₂²	1	1	yes	characteristic
15	#376	C₂²	1	9	yes	—
16	#376	C₂²	1	12	yes	—
17	C₂×#95	C₂²	1	12	yes	—
18	C₄³	C₂²	1	6	yes	—
19	C₂×#41	C₂×C₄	1	24	yes	—
20	C₂²×Q₈	C₂³	1	1	yes	characteristic
21	C₂²×D₄	C₂³	1	3	yes	—
22	C₂³×C₄	C₂³	1	18	yes	—
23	C₂³×C₄	C₂³	1	3	yes	—
24	C₄×Q₈	C₂×C₄	1	48	yes	complemented
25	C₄×D₄	C₂×C₄	1	144	yes	complemented
26	#117	C₂×C₄	1	72	yes	complemented
27	C₂×C₄⋊C₄	C₂³	1	18	yes	—
28	C₂×C₂²⋊C₄	C₂³	1	18	yes	—
29	C₂×C₄²	C₂×C₄	1	72	yes	complemented
30	C₂×C₄²	C₂³	1	24	yes	—
31	C₂×C₄²	C₂³	1	18	yes	—
32	C₂×C₄²	C₂³	1	9	yes	—
33	C₂×C₄²	C₂³	1	3	yes	—
34	#95	C₂³	1	4	yes	—
35	#95	C₂³	1	36	yes	—
36	C₂⁴	C₂⁴	1	3	yes	—
37	#41	C₄²	1	64	yes	complemented
38	C₂×Q₈	C₂²×C₄	1	12	yes	—
39	C₂×D₄	C₂²×C₄	1	36	yes	—
40	C₂²×C₄	—	4	36	no	—
41	C₂²×C₄	C₂²×C₄	1	72	yes	—
42	C₂²×C₄	C₂²×C₄	1	36	yes	—
43	C₂²×C₄	C₂⁴	1	6	yes	—
44	C₂²×C₄	C₂⁴	1	18	yes	—
45	C₂²×C₄	C₂⁴	1	3	yes	—
46	C₂²×C₄	C₂⁴	1	1	yes	characteristic
47	C₄⋊C₄	C₂²×C₄	1	72	yes	—
48	C₂²⋊C₄	C₂²×C₄	1	72	yes	—
49	C₄²	—	4	24	no	—
50	C₄²	C₂²×C₄	1	12	yes	—
51	C₄²	C₂²×C₄	1	36	yes	—
52	C₄²	C₂²×C₄	1	72	yes	—
53	C₂³	—	2	12	no	—
54	C₂³	C₂³×C₄	1	18	yes	—
55	C₂³	C₂⁵	1	1	yes	characteristic
56	Q₈	C₂×C₄²	1	16	yes	—
57	D₄	C₂×C₄²	1	48	yes	—
58	C₂×C₄	—	2	36	no	—
59	C₂×C₄	—	2	12	no	—
60	C₂×C₄	—	4	72	no	—
61	C₂×C₄	C₂×C₄²	1	48	yes	—
62	C₂×C₄	C₂³×C₄	1	12	yes	—
63	C₂×C₄	C₂³×C₄	1	36	yes	—
64	C₂×C₄	C₂³×C₄	1	6	yes	—
65	C₂×C₄	C₂³×C₄	1	18	yes	—
66	C₂²	—	2	36	no	—
67	C₂²	C₂²×C₄²	1	12	yes	—
68	C₂²	C₂⁴×C₄	1	3	yes	—
69	C₂²	#585	1	4	yes	—
70	C₄	—	2	12	no	—
71	C₄	—	2	36	no	—
72	C₄	C₂²×C₄²	1	4	yes	—
73	C₄	C₂²×C₄²	1	12	yes	—
74	C₂	—	2	12	no	—
75	C₂	C₂³×C₄²	1	1	yes	characteristic
76	C₂	#3428	1	6	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂⁴
commutator	C₂	C₂³×C₄²
Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2276	C₂	semidirect
3. #2275	C₂	semidirect
4. C₄×#117	C₂	semidirect
5. C₂×#376	C₂	semidirect
7. C₄×#41	C₄	semidirect
24. C₄×Q₈	C₂×C₄	semidirect
25. C₄×D₄	C₂×C₄	semidirect
26. #117	C₂×C₄	semidirect
29. C₂×C₄²	C₂×C₄	semidirect
37. #41	C₄²	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41	2
#2276	2
#2275	2
C₄×#117	2
C₂×#376	2