Subgroups — #3338

Order 128.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#554	C₂	1	2	yes	maximal
2	#500	C₂	1	1	yes	maximal, characteristic, complemented
3	#484	C₂	1	4	yes	maximal, complemented
4	#477	C₂	1	2	yes	maximal, complemented
5	#467	C₂	1	2	yes	maximal, complemented
6	C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄	C₂	1	1	yes	maximal, characteristic
8	C₂×#107	C₂	1	2	yes	maximal, complemented
9	C₂×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₈	C₂²	1	2	yes	—
11	C₄⋊Q₈	C₂²	1	2	yes	—
12	#125	—	2	2	no	—
13	C₂²⋊Q₈	—	2	4	no	—
14	C₂²⋊Q₈	C₂²	1	4	yes	—
15	C₄×Q₈	C₂²	1	2	yes	—
16	C₄×D₄	—	2	2	no	—
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	#103	C₂²	1	2	yes	—
25	#103	C₂²	1	4	yes	—
26	C₂²⋊C₈	C₂²	1	2	yes	—
27	C₄×C₈	C₂²	1	1	yes	characteristic
28	C₂×Q₈	—	2	2	no	—
29	C₂×Q₈	C₂³	1	2	yes	—
30	C₂×D₄	D₄	1	1	yes	characteristic
31	C₂²×C₄	—	2	4	no	—
32	C₂²×C₄	C₂³	1	2	yes	—
33	Q₁₆	—	2	2	no	—
34	C₂×C₈	—	2	4	no	—
35	C₂×C₈	C₂³	1	1	yes	characteristic
36	C₂×C₈	C₂³	1	2	yes	—
37	C₂×C₈	C₂³	1	1	yes	characteristic
38	C₄⋊C₄	—	2	4	no	—
39	C₄⋊C₄	—	2	2	no	—
40	C₄⋊C₄	—	2	2	no	—
41	C₄⋊C₄	—	2	4	no	—
42	C₄⋊C₄	D₄	1	1	yes	characteristic
43	C₄⋊C₄	C₂³	1	2	yes	—
44	C₄⋊C₄	C₂³	1	4	yes	—
45	C₂²⋊C₄	—	2	4	no	—
46	C₂²⋊C₄	D₄	1	2	yes	—
47	C₄²	—	2	2	no	—
48	C₄²	C₂³	1	1	yes	characteristic
49	C₂³	C₂×D₄	1	2	yes	—
50	Q₈	—	2	2	no	—
51	Q₈	—	2	2	no	—
52	Q₈	—	8	2	no	—
53	D₄	Q₁₆	1	4	yes	complemented
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	4	no	—
59	C₂×C₄	—	8	4	no	—
60	C₂×C₄	C₂×D₄	1	1	yes	characteristic
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	2	no	—
65	C₈	—	2	1	no	—
66	C₈	#41	1	2	yes	—
67	C₂²	—	2	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	C₂×Q₁₆	1	4	yes	—
70	C₂²	C₂²×D₄	1	1	yes	characteristic
71	C₄	—	2	2	no	—
72	C₄	—	2	1	no	—
73	C₄	—	4	2	no	—
74	C₄	—	4	2	no	—
75	C₄	—	4	4	no	—
76	C₄	C₂×Q₁₆	1	2	yes	—
77	C₄	C₂×#41	1	1	yes	characteristic
78	C₄	2₋¹⁺⁴	1	1	yes	characteristic
79	C₂	—	2	4	no	—
80	C₂	#547	1	1	yes	characteristic
81	C₂	C₂²×Q₁₆	1	1	yes	characteristic
82	C₂	#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂×D₄	D₄
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
2. #500	C₂	semidirect
3. #484	C₂	semidirect
4. #477	C₂	semidirect
5. #467	C₂	semidirect
6. C₄×Q₁₆	C₂	semidirect
8. C₂×#107	C₂	semidirect
53. D₄	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#554	2
#500	2
#484	2
#477	2
#467	2
C₄×Q₁₆	2
C₈×D₄	2
C₂×#107	2