Subgroups — #3140

Order 128.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#124	C₂	1	1	yes	maximal, characteristic
3	#489	C₂	1	4	yes	maximal, complemented
4	#488	C₂	1	4	yes	maximal, complemented
5	#419	C₂	1	2	yes	maximal, complemented
6	#418	C₂	1	2	yes	maximal
7	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×D₄	C₂²	1	1	yes	characteristic
10	C₂×M₁₆	C₂²	1	2	yes	—
11	C₄⋊Q₈	C₂²	1	2	yes	complemented
12	#125	C₂²	1	2	yes	complemented
13	#124	—	2	2	no	—
14	#124	C₂²	1	2	yes	—
15	#124	C₂²	1	2	yes	—
16	C₂²⋊Q₈	—	2	2	no	—
17	C₄×Q₈	—	2	2	no	—
18	#117	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄	—	2	2	no	—
20	C₂×C₄²	C₂²	1	1	yes	characteristic
21	#103	C₂²	1	8	yes	complemented
22	#102	C₂²	1	8	yes	—
23	#97	C₂²	1	2	yes	complemented
24	C₄×C₈	C₂²	1	2	yes	complemented
25	C₂⁴	—	2	1	no	—
26	C₂×Q₈	—	2	2	no	—
27	C₂×Q₈	—	2	1	no	—
28	C₂×Q₈	—	2	4	no	—
29	C₂×Q₈	C₂³	1	2	yes	—
30	C₂×D₄	—	2	1	no	—
31	C₂×D₄	—	2	4	no	—
32	C₂×D₄	C₂³	1	2	yes	—
33	C₂²×C₄	—	2	1	no	—
34	C₂²×C₄	D₄	1	2	yes	—
35	C₂²×C₄	C₂³	1	1	yes	characteristic
36	M₁₆	—	2	4	no	—
37	C₂×C₈	C₂³	1	4	yes	—
38	C₄⋊C₄	—	2	4	no	—
39	C₄⋊C₄	—	2	2	no	—
40	C₄⋊C₄	C₂³	1	4	yes	—
41	C₂²⋊C₄	—	2	2	no	—
42	C₂²⋊C₄	—	2	4	no	—
43	C₂²⋊C₄	—	2	4	no	—
44	C₄²	—	2	2	no	—
45	C₄²	D₄	1	2	yes	—
46	C₄²	C₂³	1	1	yes	characteristic
47	C₄²	C₂³	1	1	yes	characteristic
48	C₂³	—	2	2	no	—
49	C₂³	—	4	4	no	—
50	C₂³	—	4	2	no	—
51	C₂³	C₂×D₄	1	1	yes	characteristic
52	Q₈	—	2	4	no	—
53	Q₈	—	4	4	no	—
54	Q₈	—	4	2	no	—
55	D₄	—	2	4	no	—
56	D₄	—	4	2	no	—
57	C₂×C₄	—	2	4	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	C₂×D₄	1	2	yes	—
62	C₂×C₄	C₂×D₄	1	2	yes	—
63	C₂×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄	#41	1	4	yes	—
65	C₂×C₄	C₂⁴	1	1	yes	characteristic
66	C₈	—	2	4	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	4	2	no	—
69	C₂²	—	4	4	no	—
70	C₂²	—	8	4	no	—
71	C₂²	#124	1	2	yes	—
72	C₂²	C₂²×D₄	1	1	yes	characteristic
73	C₄	—	2	2	no	—
74	C₄	—	2	2	no	—
75	C₄	—	4	2	no	—
76	C₄	—	4	4	no	—
77	C₄	#124	1	2	yes	—
78	C₄	C₂×#41	1	2	yes	—
79	C₂	—	2	2	no	—
80	C₂	—	8	2	no	—
81	C₂	C₂×#124	1	1	yes	characteristic
82	C₂	#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂²×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #533	C₂	semidirect
3. #489	C₂	semidirect
4. #488	C₂	semidirect
5. #419	C₂	semidirect
7. C₄×M₁₆	C₂	semidirect
11. C₄⋊Q₈	C₂²	semidirect
12. #125	C₂²	semidirect
21. #103	C₂²	semidirect
23. #97	C₂²	semidirect
24. C₄×C₈	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#533	2
C₂×#124	2
#489	2
#488	2
#419	2
#418	2
C₄×M₁₆	2