Subgroups — #3058

Order 128.

262 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, complemented
2	C₂×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	#535	C₂	1	2	yes	maximal, complemented
4	#469	C₂	1	4	yes	maximal, complemented
5	#466	C₂	1	4	yes	maximal, complemented
6	#427	C₂	1	1	yes	maximal, characteristic, complemented
7	#418	C₂	1	2	yes	maximal, complemented
8	C₂×#41	C₂²	1	1	yes	characteristic
9	C₂²×D₄	C₂²	1	2	yes	—
10	C₂×D₈	—	2	4	no	—
11	C₂×D₈	C₂²	1	2	yes	—
12	C₂×D₈	C₂²	1	2	yes	—
13	#131	—	2	4	no	—
14	C₂×M₁₆	C₂²	1	1	yes	characteristic
15	C₂×M₁₆	C₂²	1	2	yes	complemented
16	C₂²×C₈	C₂²	1	1	yes	characteristic
17	C₂²×C₈	C₂²	1	2	yes	complemented
18	#127	—	2	2	no	—
19	#124	—	2	2	no	—
20	#121	C₂²	1	8	yes	complemented
21	#120	—	2	4	no	—
22	#117	C₂²	1	2	yes	—
23	#107	C₂²	1	2	yes	complemented
24	#106	C₂²	1	2	yes	complemented
25	#102	C₂²	1	8	yes	complemented
26	C₂⁴	—	2	2	no	—
27	#41	—	2	2	no	—
28	#41	—	2	2	no	—
29	C₂×Q₈	D₄	1	1	yes	characteristic
30	C₂×D₄	—	2	8	no	—
31	C₂×D₄	—	2	4	no	—
32	C₂×D₄	—	2	2	no	—
33	C₂×D₄	—	2	4	no	—
34	C₂×D₄	D₄	1	2	yes	—
35	C₂×D₄	D₄	1	1	yes	characteristic
36	C₂×D₄	C₂³	1	4	yes	—
37	C₂²×C₄	C₂³	1	2	yes	—
38	C₂²×C₄	C₂³	1	1	yes	characteristic
39	D₈	—	2	4	no	—
40	D₈	—	2	4	no	—
41	M₁₆	—	2	2	no	—
42	M₁₆	—	2	4	no	—
43	C₂×C₈	—	2	4	no	—
44	C₂×C₈	D₄	1	4	yes	complemented
45	C₂×C₈	C₂³	1	2	yes	—
46	C₂×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₈	C₂³	1	1	yes	characteristic
48	C₄⋊C₄	C₂³	1	4	yes	—
49	C₂²⋊C₄	—	2	2	no	—
50	C₂²⋊C₄	—	2	8	no	—
51	C₄²	—	2	2	no	—
52	C₂³	—	2	4	no	—
53	C₂³	—	4	4	no	—
54	C₂³	—	8	4	no	—
55	C₂³	C₂×D₄	1	2	yes	—
56	C₂³	C₂×D₄	1	1	yes	characteristic
57	Q₈	—	2	2	no	—
58	D₄	—	2	4	no	—
59	D₄	—	2	2	no	—
60	D₄	—	2	4	no	—
61	D₄	—	2	4	no	—
62	D₄	—	4	4	no	—
63	D₄	—	8	8	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	C₂×D₄	1	2	yes	—
68	C₂×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄	C₂×D₄	1	2	yes	—
70	C₂×C₄	#41	1	2	yes	—
71	C₂×C₄	C₂⁴	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₈	C₂×D₄	1	2	yes	—
74	C₈	C₂×D₄	1	2	yes	—
75	C₂²	—	2	2	no	—
76	C₂²	—	2	2	no	—
77	C₂²	—	2	4	no	—
78	C₂²	—	4	4	no	—
79	C₂²	—	4	8	no	—
80	C₂²	—	8	8	no	—
81	C₂²	#121	1	2	yes	—
82	C₂²	C₂²×D₄	1	1	yes	characteristic
83	C₄	—	2	2	no	—
84	C₄	—	4	4	no	—
85	C₄	#121	1	2	yes	—
86	C₄	C₂²×D₄	1	1	yes	characteristic
87	C₄	C₂×#41	1	1	yes	characteristic
88	C₂	—	2	2	no	—
89	C₂	—	4	2	no	—
90	C₂	—	8	4	no	—
91	C₂	C₂×#121	1	1	yes	characteristic
92	C₂	#576	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂²×D₈	C₂	semidirect
2. C₂×#131	C₂	semidirect
3. #535	C₂	semidirect
4. #469	C₂	semidirect
5. #466	C₂	semidirect
6. #427	C₂	semidirect
7. #418	C₂	semidirect
15. C₂×M₁₆	C₂²	semidirect
17. C₂²×C₈	C₂²	semidirect
20. #121	C₂²	semidirect
23. #107	C₂²	semidirect
24. #106	C₂²	semidirect
25. #102	C₂²	semidirect
44. C₂×C₈	D₄	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
C₂×#131	2
#535	2
#469	2
#466	2
#427	2
#418	2