Subgroups — #2948

Order 128.

264 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈	C₂	1	1	yes	maximal, characteristic
2	#518	C₂	1	2	yes	maximal, complemented
3	#442	C₂	1	4	yes	maximal, complemented
4	C₄×D₈	C₂	1	4	yes	maximal, complemented
5	#427	C₂	1	1	yes	maximal, characteristic, complemented
6	#418	C₂	1	2	yes	maximal, complemented
7	#405	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄	C₂²	1	2	yes	—
9	C₂×D₈	C₄	1	8	yes	complemented
10	C₂×D₈	C₂²	1	2	yes	—
11	C₂×D₈	C₂²	1	2	yes	—
12	C₂×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₈	C₂²	1	1	yes	characteristic
14	C₄×D₄	—	2	4	no	—
15	C₄×D₄	C₂²	1	8	yes	complemented
16	#117	C₂²	1	2	yes	—
17	#117	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄	—	2	4	no	—
19	#107	C₂²	1	2	yes	complemented
20	#106	C₂²	1	2	yes	complemented
21	#102	C₂²	1	8	yes	complemented
22	#97	C₂²	1	2	yes	complemented
23	C₄×C₈	C₂²	1	2	yes	complemented
24	C₂⁴	—	2	2	no	—
25	C₂×D₄	—	2	2	no	—
26	C₂×D₄	—	2	4	no	—
27	C₂×D₄	C₂×C₄	1	8	yes	complemented
28	C₂×D₄	C₂³	1	4	yes	—
29	C₂²×C₄	—	2	8	no	—
30	C₂²×C₄	C₂³	1	1	yes	characteristic
31	D₈	C₂×C₄	1	8	yes	complemented
32	D₈	C₂×C₄	1	8	yes	complemented
33	M₁₆	—	2	2	no	—
34	C₂×C₈	C₂×C₄	1	4	yes	complemented
35	C₂×C₈	C₂³	1	2	yes	—
36	C₂×C₈	C₂³	1	1	yes	characteristic
37	C₂×C₈	C₂³	1	1	yes	characteristic
38	C₄⋊C₄	D₄	1	2	yes	—
39	C₄⋊C₄	C₂³	1	4	yes	—
40	C₂²⋊C₄	—	2	2	no	—
41	C₂²⋊C₄	—	2	8	no	—
42	C₂²⋊C₄	D₄	1	2	yes	—
43	C₄²	—	2	2	no	—
44	C₄²	C₂³	1	2	yes	—
45	C₂³	—	2	4	no	—
46	C₂³	—	2	8	no	—
47	C₂³	—	8	4	no	—
48	C₂³	C₂×D₄	1	1	yes	characteristic
49	D₄	—	2	8	no	—
50	D₄	—	2	4	no	—
51	D₄	C₂²×C₄	1	8	yes	—
52	C₂×C₄	—	2	4	no	—
53	C₂×C₄	—	8	8	no	—
54	C₂×C₄	C₂²×C₄	1	2	yes	—
55	C₂×C₄	C₂×D₄	1	4	yes	—
56	C₂×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄	#41	1	2	yes	—
58	C₂×C₄	C₂⁴	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	C₂²×C₄	1	2	yes	—
61	C₈	C₂²×C₄	1	2	yes	—
62	C₂²	—	2	2	no	—
63	C₂²	—	2	8	no	—
64	C₂²	—	4	8	no	—
65	C₂²	—	8	8	no	—
66	C₂²	C₄×D₄	1	2	yes	—
67	C₂²	C₂²×D₄	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	4	4	no	—
70	C₄	C₄×D₄	1	2	yes	—
71	C₄	C₂³×C₄	1	1	yes	characteristic
72	C₄	C₂×#41	1	1	yes	characteristic
73	C₂	—	2	2	no	—
74	C₂	—	4	8	no	—
75	C₂	C₂×C₄×D₄	1	1	yes	characteristic
76	C₂	#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator	C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂²×D₈	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #518	C₂	semidirect
3. #442	C₂	semidirect
4. C₄×D₈	C₂	semidirect
5. #427	C₂	semidirect
6. #418	C₂	semidirect
7. #405	C₂	semidirect
9. C₂×D₈	C₄	semidirect
15. C₄×D₄	C₂²	semidirect
19. #107	C₂²	semidirect
20. #106	C₂²	semidirect
21. #102	C₂²	semidirect
22. #97	C₂²	semidirect
23. C₄×C₈	C₂²	semidirect
27. C₂×D₄	C₂×C₄	semidirect
31. D₈	C₂×C₄	semidirect
32. D₈	C₂×C₄	semidirect
34. C₂×C₈	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₈	2
#518	2
#442	2
C₄×D₈	2
#427	2
#418	2
#405	2