Subgroups — #2567

Order 128.

250 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
3	#395	C₂	1	4	yes	maximal
4	#391	C₂	1	2	yes	maximal
5	#389	C₂	1	4	yes	maximal
6	#384	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×Q₈	C₂²	1	2	yes	—
10	C₄⋊Q₈	—	2	4	no	—
11	C₄×Q₈	—	2	4	no	—
12	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄	C₂²	1	8	yes	—
17	C₂×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄²	C₂²	1	2	yes	—
19	C₂×C₄²	C₂²	1	2	yes	—
20	C₂×C₄²	C₂²	1	1	yes	characteristic
21	#95	C₂²	1	2	yes	—
22	#95	C₂²	1	8	yes	—
23	C₂×Q₈	—	2	2	no	—
24	C₂×Q₈	—	2	4	no	—
25	C₂×Q₈	—	2	2	no	—
26	C₂×Q₈	—	4	4	no	—
27	C₂×Q₈	Q₈	1	4	yes	complemented
28	C₂²×C₄	C₂³	1	4	yes	—
29	C₂²×C₄	C₂³	1	2	yes	—
30	C₂²×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₄	C₂³	1	2	yes	—
32	C₂²×C₄	C₂³	1	4	yes	—
33	C₂²×C₄	C₂³	1	1	yes	characteristic
34	C₂²×C₄	C₂³	1	1	yes	characteristic
35	C₄⋊C₄	—	2	4	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	2	4	no	—
38	C₄⋊C₄	—	2	4	no	—
39	C₄⋊C₄	—	4	8	no	—
40	C₄⋊C₄	Q₈	1	8	yes	complemented
41	C₄²	—	2	4	no	—
42	C₄²	—	2	4	no	—
43	C₄²	D₄	1	4	yes	—
44	C₂³	C₂⁴	1	1	yes	characteristic
45	Q₈	—	2	8	no	—
46	Q₈	—	4	8	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	2	4	no	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	2	4	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	2	4	no	—
54	C₂×C₄	—	2	4	no	—
55	C₂×C₄	—	2	8	no	—
56	C₂×C₄	C₂×D₄	1	2	yes	—
57	C₂×C₄	C₂×D₄	1	2	yes	—
58	C₂×C₄	C₂×D₄	1	2	yes	—
59	C₂×C₄	C₂×Q₈	1	8	yes	—
60	C₂×C₄	C₂×Q₈	1	4	yes	—
61	C₂×C₄	C₂×Q₈	1	2	yes	—
62	C₂×C₄	C₂×Q₈	1	4	yes	—
63	C₂×C₄	#41	1	2	yes	—
64	C₂²	C₂²×D₄	1	1	yes	characteristic
65	C₂²	C₂²×Q₈	1	1	yes	characteristic
66	C₂²	C₂²×Q₈	1	2	yes	—
67	C₂²	C₂×#41	1	1	yes	characteristic
68	C₂²	H(5,𝔽₂)	1	2	yes	—
69	C₄	—	2	4	no	—
70	C₄	—	2	2	no	—
71	C₄	—	2	8	no	—
72	C₄	—	4	2	no	—
73	C₄	—	4	4	no	—
74	C₄	C₂²⋊Q₈	1	4	yes	—
75	C₄	C₄⋊Q₈	1	4	yes	—
76	C₂	C₂×C₂²⋊Q₈	1	1	yes	characteristic
77	C₂	C₂×C₄⋊Q₈	1	1	yes	characteristic
78	C₂	#535	1	1	yes	characteristic
79	C₂	#554	1	2	yes	—
80	C₂	Q₈²	1	2	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂³	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
27. C₂×Q₈	Q₈	semidirect
40. C₄⋊C₄	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈	2
C₂×C₄×Q₈	2
#395	2
#391	2
#389	2
#384	2
C₄×C₄⋊C₄	2