Subgroups — #2331

Order 128.

230 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	#391	C₂	1	4	yes	maximal
3	#389	C₂	1	2	yes	maximal
4	#387	C₂	1	4	yes	maximal
5	#382	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊C₄	C₂	1	2	yes	maximal
7	#376	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈	C₂²	1	2	yes	—
9	C₄⋊Q₈	C₄	1	8	yes	complemented
10	C₂×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₄²	C₂²	1	1	yes	characteristic
15	C₂×C₄²	C₂²	1	4	yes	—
16	C₂×C₄²	C₂²	1	1	yes	characteristic
17	C₂×C₄²	C₂²	1	1	yes	characteristic
18	#95	C₂²	1	2	yes	—
19	#95	C₂²	1	2	yes	—
20	#95	C₂²	1	4	yes	—
21	#95	C₂²	1	2	yes	—
22	#95	C₂²	1	2	yes	—
23	#95	C₂²	1	4	yes	—
24	C₂×Q₈	—	4	4	no	—
25	C₂×Q₈	C₂×C₄	1	8	yes	—
26	C₂²×C₄	C₂³	1	2	yes	—
27	C₂²×C₄	C₂³	1	4	yes	—
28	C₂²×C₄	C₂³	1	1	yes	characteristic
29	C₂²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₄	C₂³	1	4	yes	—
31	C₂²×C₄	C₂³	1	2	yes	—
32	C₂²×C₄	C₂³	1	1	yes	characteristic
33	C₄⋊C₄	—	2	4	no	—
34	C₄⋊C₄	—	4	4	no	—
35	C₄⋊C₄	C₂×C₄	1	8	yes	—
36	C₄⋊C₄	C₂×C₄	1	8	yes	—
37	C₄²	—	2	2	no	—
38	C₄²	—	4	4	no	—
39	C₄²	C₂×C₄	1	4	yes	—
40	C₄²	Q₈	1	4	yes	complemented
41	C₂³	C₂⁴	1	1	yes	characteristic
42	Q₈	—	4	8	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	—	2	4	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	2	4	no	—
51	C₂×C₄	—	2	4	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	4	no	—
55	C₂×C₄	C₂²×C₄	1	2	yes	—
56	C₂×C₄	C₂²×C₄	1	2	yes	—
57	C₂×C₄	C₂²×C₄	1	8	yes	—
58	C₂×C₄	C₂²×C₄	1	2	yes	—
59	C₂×C₄	C₂×Q₈	1	4	yes	—
60	C₂×C₄	C₂×Q₈	1	2	yes	—
61	C₂×C₄	#41	1	2	yes	—
62	C₂²	C₂³×C₄	1	1	yes	characteristic
63	C₂²	C₂²×Q₈	1	1	yes	characteristic
64	C₂²	C₂×#41	1	1	yes	characteristic
65	C₂²	H(5,𝔽₂)	1	2	yes	—
66	C₂²	2₋¹⁺⁴	1	2	yes	—
67	C₄	—	2	8	no	—
68	C₄	—	2	2	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	4	no	—
71	C₄	—	4	2	no	—
72	C₄	—	4	4	no	—
73	C₄	C₄×Q₈	1	4	yes	—
74	C₂	C₂×C₄×Q₈	1	1	yes	characteristic
75	C₂	#518	1	1	yes	characteristic
76	C₂	#519	1	1	yes	characteristic
77	C₂	#542	1	2	yes	—
78	C₂	#544	1	2	yes	—

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂³	C₂⁴
commutator	C₂²	C₂³×C₄
LCS	G ▷ C₂² ▷ 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
9. C₄⋊Q₈	C₄	semidirect
40. C₄²	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
#391	2
#389	2
#387	2
#382	2
C₄×C₄⋊C₄	2
#376	2