Subgroups — #3334

Order 128.

210 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#547	C₂	1	2	yes	maximal, complemented
2	#500	C₂	1	1	yes	maximal, characteristic, complemented
3	#480	C₂	1	4	yes	maximal, complemented
4	#474	C₂	1	2	yes	maximal, complemented
5	#466	C₂	1	2	yes	maximal, complemented
6	C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#107	C₂	1	2	yes	maximal, complemented
9	C₂×#41	—	2	2	no	—
10	C₂×D₈	C₂²	1	1	yes	characteristic
11	C₂²×C₈	C₂²	1	2	yes	—
12	C₄⋊Q₈	C₂²	1	2	yes	—
13	#123	—	2	4	no	—
14	C₂²⋊Q₈	—	2	2	no	—
15	#121	C₂²	1	4	yes	—
16	C₄×D₄	C₂²	1	2	yes	—
17	C₄×D₄	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄	C₂²	1	4	yes	—
19	#107	C₂²	1	1	yes	characteristic
20	#107	C₂²	1	2	yes	—
21	#107	C₂²	1	4	yes	—
22	#107	C₂²	1	2	yes	—
23	C₄⋊C₈	C₂²	1	1	yes	characteristic
24	#102	C₂²	1	2	yes	—
25	#102	C₂²	1	4	yes	—
26	C₂²⋊C₈	C₂²	1	2	yes	—
27	C₄×C₈	C₂²	1	1	yes	characteristic
28	#41	—	4	8	no	—
29	C₂×Q₈	—	2	2	no	—
30	C₂×D₄	—	2	4	no	—
31	C₂×D₄	D₄	1	1	yes	characteristic
32	C₂×D₄	C₂³	1	2	yes	—
33	C₂²×C₄	—	2	4	no	—
34	C₂²×C₄	—	2	2	no	—
35	C₂²×C₄	C₂³	1	2	yes	—
36	D₈	—	2	2	no	—
37	C₂×C₈	—	2	4	no	—
38	C₂×C₈	C₂³	1	1	yes	characteristic
39	C₂×C₈	C₂³	1	2	yes	—
40	C₂×C₈	C₂³	1	1	yes	characteristic
41	C₄⋊C₄	—	2	4	no	—
42	C₄⋊C₄	—	2	2	no	—
43	C₄⋊C₄	D₄	1	1	yes	characteristic
44	C₄⋊C₄	C₂³	1	2	yes	—
45	C₄⋊C₄	C₂³	1	4	yes	—
46	C₂²⋊C₄	—	2	2	no	—
47	C₂²⋊C₄	—	2	4	no	—
48	C₂²⋊C₄	D₄	1	2	yes	—
49	C₄²	C₂³	1	1	yes	characteristic
50	C₂³	—	2	2	no	—
51	C₂³	C₂×D₄	1	2	yes	—
52	Q₈	—	4	4	no	—
53	D₄	—	2	2	no	—
54	D₄	—	2	2	no	—
55	D₄	—	4	8	no	—
56	D₄	D₈	1	4	yes	complemented
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	—	4	8	no	—
62	C₂×C₄	—	4	4	no	—
63	C₂×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄	C₂×D₄	1	2	yes	—
65	C₂×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄	C₂⁴	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₈	—	2	1	no	—
69	C₈	#41	1	2	yes	complemented
70	C₂²	—	2	2	no	—
71	C₂²	—	2	2	no	—
72	C₂²	—	4	2	no	—
73	C₂²	—	4	2	no	—
74	C₂²	—	4	2	no	—
75	C₂²	C₂×D₈	1	4	yes	—
76	C₂²	C₂²×D₄	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	2	1	no	—
79	C₄	—	4	2	no	—
80	C₄	—	4	4	no	—
81	C₄	C₂×D₈	1	2	yes	—
82	C₄	C₂×#41	1	1	yes	characteristic
83	C₄	2₋¹⁺⁴	1	1	yes	characteristic
84	C₂	—	2	4	no	—
85	C₂	—	8	2	no	—
86	C₂	#547	1	1	yes	characteristic
87	C₂	C₂²×D₈	1	1	yes	characteristic
88	C₂	#578	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #547	C₂	semidirect
2. #500	C₂	semidirect
3. #480	C₂	semidirect
4. #474	C₂	semidirect
5. #466	C₂	semidirect
6. C₄×D₈	C₂	semidirect
7. C₈×D₄	C₂	semidirect
8. C₂×#107	C₂	semidirect
56. D₄	D₈	semidirect
69. C₈	#41	semidirect

Maximal subgroups

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

subgroup	index
#547	2
#500	2
#480	2
#474	2
#466	2
C₄×D₈	2
C₈×D₄	2
C₂×#107	2