Subgroups — C₂×#3458 #62684

Order 256.

3511 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3458	C₂	1	32	yes	maximal, direct product factor
2	C₂×#544	C₂	1	12	yes	maximal, complemented
3	C₂×#543	C₂	1	4	yes	maximal
4	C₂×#533	C₂	1	12	yes	maximal, complemented
5	C₂²×#117	C₂	1	3	yes	maximal
6	C₂⁴×C₄	C₂²	1	1	yes	characteristic
7	#544	C₂²	1	192	yes	complemented
8	#543	C₂²	1	64	yes	—
9	#533	C₂²	1	192	yes	complemented
10	C₂×C₄⋊Q₈	C₂²	1	24	yes	complemented
11	C₂×#125	C₂²	1	24	yes	complemented
12	C₂×C₂²⋊Q₈	C₂²	1	48	yes	—
13	C₂×C₄×Q₈	C₂²	1	16	yes	complemented
14	C₂×#117	C₂²	1	12	yes	—
15	C₂×#117	C₂²	1	12	yes	—
16	C₂×#117	C₂²	1	48	yes	—
17	C₂²×C₄⋊C₄	C₂²	1	6	yes	—
18	C₂²×C₂²⋊C₄	C₂²	1	6	yes	—
19	C₂²×C₄²	C₂²	1	6	yes	—
20	C₂⁵	C₂³	1	1	yes	characteristic
21	C₂²×Q₈	C₂³	1	16	yes	—
22	C₂³×C₄	Q₈	1	16	yes	complemented
23	C₂³×C₄	C₂³	1	12	yes	—
24	C₂³×C₄	C₂³	1	3	yes	—
25	C₂³×C₄	C₂³	1	3	yes	—
26	C₂³×C₄	C₂³	1	8	yes	—
27	C₄⋊Q₈	C₂³	1	192	yes	complemented
28	#125	C₂³	1	192	yes	complemented
29	C₂²⋊Q₈	C₂³	1	384	yes	—
30	C₄×Q₈	C₂³	1	128	yes	complemented
31	#117	C₂³	1	96	yes	—
32	#117	C₂³	1	96	yes	—
33	C₂×C₄⋊C₄	C₂³	1	24	yes	—
34	C₂×C₄⋊C₄	C₂³	1	48	yes	—
35	C₂×C₄⋊C₄	C₂³	1	48	yes	—
36	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
37	C₂×C₂²⋊C₄	C₂³	1	48	yes	—
38	C₂×C₄²	C₂³	1	12	yes	—
39	C₂×C₄²	C₂³	1	48	yes	—
40	C₂×C₄²	C₂³	1	12	yes	—
41	C₂⁴	—	4	4	no	—
42	C₂⁴	C₂×Q₈	1	8	yes	—
43	C₂⁴	C₂⁴	1	3	yes	—
44	C₂⁴	C₂⁴	1	4	yes	—
45	C₂×Q₈	—	4	32	no	—
46	C₂×Q₈	C₂⁴	1	64	yes	—
47	C₂²×C₄	—	4	24	no	—
48	C₂²×C₄	C₂×Q₈	1	24	yes	—
49	C₂²×C₄	C₂×Q₈	1	24	yes	—
50	C₂²×C₄	C₂×Q₈	1	64	yes	complemented
51	C₂²×C₄	C₂⁴	1	24	yes	—
52	C₂²×C₄	C₂⁴	1	48	yes	—
53	C₂²×C₄	C₂⁴	1	3	yes	—
54	C₂²×C₄	C₂⁴	1	12	yes	—
55	C₂²×C₄	C₂⁴	1	12	yes	—
56	C₂²×C₄	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄	C₂⁴	1	96	yes	—
58	C₄⋊C₄	C₂⁴	1	192	yes	—
59	C₂²⋊C₄	C₂⁴	1	96	yes	—
60	C₄²	C₂⁴	1	48	yes	—
61	C₄²	C₂⁴	1	48	yes	—
62	C₂³	—	2	12	no	—
63	C₂³	—	4	16	no	—
64	C₂³	—	4	8	no	—
65	C₂³	C₂²×Q₈	1	12	yes	—
66	C₂³	C₂²×Q₈	1	16	yes	—
67	C₂³	C₂⁵	1	6	yes	—
68	C₂³	C₂⁵	1	1	yes	characteristic
69	Q₈	—	4	64	no	—
70	C₂×C₄	—	2	48	no	—
71	C₂×C₄	—	4	48	no	—
72	C₂×C₄	C₂²×Q₈	1	12	yes	—
73	C₂×C₄	C₂²×Q₈	1	4	yes	—
74	C₂×C₄	C₂²×Q₈	1	48	yes	—
75	C₂×C₄	C₂²×Q₈	1	48	yes	—
76	C₂×C₄	C₂⁵	1	48	yes	—
77	C₂×C₄	C₂⁵	1	2	yes	—
78	C₂×C₄	C₂⁵	1	6	yes	—
79	C₂²	—	2	12	no	—
80	C₂²	—	2	24	no	—
81	C₂²	—	4	16	no	—
82	C₂²	C₂³×Q₈	1	12	yes	—
83	C₂²	C₂³×Q₈	1	2	yes	—
84	C₂²	#585	1	4	yes	—
85	C₂²	C₂⁶	1	1	yes	characteristic
86	C₄	—	2	48	no	—
87	C₄	C₂³×Q₈	1	12	yes	—
88	C₄	C₂³×Q₈	1	4	yes	—
89	C₂	—	2	12	no	—
90	C₂	#3458	1	4	yes	direct product factor
91	C₂	C₂⁴×Q₈	1	1	yes	characteristic
92	C₂	C₂×#585	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶
socle	C₂³	C₂⁵
prime radical	C₂⁵	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3458	C₂	direct (N × H)
2. C₂×#544	C₂	semidirect
4. C₂×#533	C₂	semidirect
7. #544	C₂²	semidirect
9. #533	C₂²	semidirect
10. C₂×C₄⋊Q₈	C₂²	semidirect
11. C₂×#125	C₂²	semidirect
13. C₂×C₄×Q₈	C₂²	semidirect
22. C₂³×C₄	Q₈	semidirect
27. C₄⋊Q₈	C₂³	semidirect
28. #125	C₂³	semidirect
30. C₄×Q₈	C₂³	semidirect
50. C₂²×C₄	C₂×Q₈	semidirect
90. C₂	#3458	direct (N × H)

Maximal subgroups

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

subgroup	index
#3458	2
C₂×#544	2
C₂×#543	2
C₂×#533	2
C₂²×#117	2

Subgroups — C2×#3458 #62684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3458 #62684