Subgroups — #1503

Order 128.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533	C₂	1	1	yes	maximal, characteristic
2	#431	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	#328	C₂	1	4	yes	maximal
5	C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
6	C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
7	#125	—	2	1	no	—
8	C₂²⋊Q₈	—	2	1	no	—
9	C₂²⋊Q₈	C₂²	1	2	yes	—
10	C₄×Q₈	—	2	1	no	—
11	C₄×Q₈	C₄	1	2	yes	complemented
12	#117	—	2	1	no	—
13	C₂×C₄⋊C₄	—	2	1	no	—
14	C₂×C₄²	—	2	1	no	—
15	C₂×C₄²	C₂²	1	1	yes	characteristic
16	C₄⋊C₈	—	2	1	no	—
17	C₂²⋊C₈	C₂²	1	2	yes	—
18	C₄×C₈	—	2	1	no	—
19	#95	C₂²	1	2	yes	—
20	C₂×Q₈	—	2	1	no	—
21	C₂×Q₈	C₂×C₄	1	2	yes	—
22	C₂²×C₄	—	2	2	no	—
23	C₂²×C₄	D₄	1	1	yes	characteristic
24	C₂²×C₄	D₄	1	1	yes	characteristic
25	C₂²×C₄	C₂³	1	1	yes	characteristic
26	C₂×C₈	—	2	2	no	—
27	C₄⋊C₄	—	2	4	no	—
28	C₄⋊C₄	—	2	1	no	—
29	C₄⋊C₄	—	2	1	no	—
30	C₄⋊C₄	—	2	1	no	—
31	C₄⋊C₄	—	2	2	no	—
32	C₄⋊C₄	—	2	2	no	—
33	C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂²⋊C₄	—	2	2	no	—
35	C₄²	—	2	2	no	—
36	C₄²	—	4	2	no	—
37	C₄²	C₂×C₄	1	1	yes	characteristic
38	C₄²	C₂×C₄	1	1	yes	characteristic
39	C₄²	D₄	1	1	yes	characteristic
40	C₄²	D₄	1	1	yes	characteristic
41	C₂³	C₂×D₄	1	1	yes	characteristic
42	Q₈	—	4	2	no	—
43	Q₈	—	4	2	no	—
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	—	2	4	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	4	1	no	—
49	C₂×C₄	—	4	2	no	—
50	C₂×C₄	—	4	2	no	—
51	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄	SD₁₆	1	2	yes	—
56	C₂×C₄	Q₁₆	1	2	yes	—
57	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄	C₂×D₄	1	1	yes	characteristic
59	C₈	—	4	2	no	—
60	C₂²	—	2	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²	C₂×Q₁₆	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	2	1	no	—
67	C₄	—	2	1	no	—
68	C₄	—	4	4	no	—
69	C₄	—	4	2	no	—
70	C₄	—	4	2	no	—
71	C₄	#103	1	2	yes	—
72	C₄	#103	1	2	yes	—
73	C₄	#104	1	2	yes	—
74	C₂	—	2	1	no	—
75	C₂	—	2	1	no	—
76	C₂	#410	1	1	yes	characteristic
77	C₂	C₂×#103	1	1	yes	characteristic
78	C₂	C₂×#104	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂²	C₂×C₂²⋊C₄
prime radical	C₂³	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
5. C₄⋊Q₈	C₄	semidirect
6. C₄⋊Q₈	C₄	semidirect
11. C₄×Q₈	C₄	semidirect

Maximal subgroups

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

subgroup	index
#533	2
#431	2
C₄×C₄⋊C₄	2
#328	2