Subgroups — #3176

Order 128.

193 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#540	C₂	1	2	yes	maximal, complemented
2	#499	C₂	1	1	yes	maximal, characteristic, complemented
3	#493	C₂	1	1	yes	maximal, characteristic, complemented
4	#490	C₂	1	2	yes	maximal, complemented
5	#469	C₂	1	2	yes	maximal, complemented
6	#466	C₂	1	2	yes	maximal, complemented
7	#442	C₂	1	2	yes	maximal, complemented
8	C₄×D₈	C₂	1	2	yes	maximal, complemented
9	#405	C₂	1	1	yes	maximal, characteristic, 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	#127	C₂²	1	2	yes	—
15	#125	C₂²	1	2	yes	—
16	#123	—	2	4	no	—
17	#121	—	2	4	no	—
18	#121	C₂²	1	4	yes	—
19	C₄×D₄	C₂²	1	4	yes	—
20	#117	C₂²	1	1	yes	characteristic
21	#107	C₂²	1	2	yes	—
22	#106	C₂²	1	2	yes	—
23	#102	C₂²	1	4	yes	—
24	#102	C₂²	1	4	yes	—
25	#97	C₂²	1	1	yes	characteristic
26	#97	C₂²	1	1	yes	characteristic
27	C₄×C₈	C₂²	1	1	yes	characteristic
28	C₄×C₈	C₂²	1	1	yes	characteristic
29	C₂×D₄	—	2	4	no	—
30	C₂×D₄	—	2	4	no	—
31	C₂×D₄	C₂³	1	4	yes	—
32	C₂²×C₄	—	2	4	no	—
33	C₂²×C₄	C₂³	1	1	yes	characteristic
34	D₈	—	2	4	no	—
35	D₈	—	2	4	no	—
36	M₁₆	—	2	2	no	—
37	C₂×C₈	—	2	2	no	—
38	C₂×C₈	C₂³	1	1	yes	characteristic
39	C₂×C₈	C₂³	1	1	yes	characteristic
40	C₂×C₈	C₂³	1	1	yes	characteristic
41	C₂×C₈	C₂³	1	1	yes	characteristic
42	C₄⋊C₄	—	2	4	no	—
43	C₄⋊C₄	D₄	1	2	yes	—
44	C₄⋊C₄	C₂³	1	4	yes	—
45	C₂²⋊C₄	—	2	4	no	—
46	C₂²⋊C₄	—	2	4	no	—
47	C₂²⋊C₄	D₄	1	2	yes	—
48	C₄²	C₂³	1	1	yes	characteristic
49	C₄²	C₂³	1	1	yes	characteristic
50	C₂³	—	2	4	no	—
51	C₂³	C₂×D₄	1	1	yes	characteristic
52	D₄	—	2	4	no	—
53	D₄	—	2	4	no	—
54	D₄	—	8	4	no	—
55	D₄	—	8	4	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	4	no	—
59	C₂×C₄	—	8	4	no	—
60	C₂×C₄	C₂×D₄	1	2	yes	—
61	C₂×C₄	C₂×D₄	1	2	yes	—
62	C₂×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄	C₂⁴	1	1	yes	characteristic
64	C₈	—	2	1	no	—
65	C₈	—	2	1	no	—
66	C₈	#41	1	2	yes	—
67	C₈	#41	1	2	yes	—
68	C₂²	—	2	1	no	—
69	C₂²	—	2	2	no	—
70	C₂²	—	4	4	no	—
71	C₂²	—	4	4	no	—
72	C₂²	—	4	4	no	—
73	C₂²	C₂²×D₄	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	4	4	no	—
78	C₄	C₂×#41	1	1	yes	characteristic
79	C₄	C₂×#41	1	1	yes	characteristic
80	C₂	—	4	1	no	—
81	C₂	—	8	4	no	—
82	C₂	#532	1	1	yes	characteristic
83	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 ◁ #117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #540	C₂	semidirect
2. #499	C₂	semidirect
3. #493	C₂	semidirect
4. #490	C₂	semidirect
5. #469	C₂	semidirect
6. #466	C₂	semidirect
7. #442	C₂	semidirect
8. C₄×D₈	C₂	semidirect
9. #405	C₂	semidirect

Maximal subgroups

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

subgroup	index
#540	2
#499	2
#493	2
#490	2
#469	2
#466	2
#442	2
C₄×D₈	2
#405	2