Subgroups — #2369

Order 128.

248 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	4	yes	maximal
4	C₄×C₄⋊C₄	C₂	1	4	yes	maximal
5	C₄×C₄⋊C₄	C₂	1	2	yes	maximal
6	C₂²×Q₈	C₂²	1	2	yes	—
7	C₄⋊Q₈	C₄	1	8	yes	complemented
8	C₂×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₄²	C₂²	1	4	yes	—
13	C₂×C₄²	C₂²	1	4	yes	—
14	C₂×C₄²	C₂²	1	1	yes	characteristic
15	#95	C₂²	1	8	yes	—
16	#95	C₂²	1	4	yes	—
17	C₂×Q₈	—	4	2	no	—
18	C₂×Q₈	—	4	2	no	—
19	C₂×Q₈	C₂×C₄	1	8	yes	—
20	C₂²×C₄	C₂³	1	4	yes	—
21	C₂²×C₄	C₂³	1	1	yes	characteristic
22	C₂²×C₄	C₂³	1	4	yes	—
23	C₂²×C₄	C₂³	1	4	yes	—
24	C₂²×C₄	C₂³	1	2	yes	—
25	C₄⋊C₄	—	2	4	no	—
26	C₄⋊C₄	—	4	4	no	—
27	C₄⋊C₄	C₂×C₄	1	16	yes	—
28	C₄⋊C₄	Q₈	1	8	yes	complemented
29	C₄²	—	2	8	no	—
30	C₄²	—	4	4	no	—
31	C₄²	C₂×C₄	1	4	yes	—
32	C₂³	C₂⁴	1	1	yes	characteristic
33	Q₈	—	4	8	no	—
34	C₂×C₄	—	2	4	no	—
35	C₂×C₄	—	2	4	no	—
36	C₂×C₄	—	2	4	no	—
37	C₂×C₄	—	2	4	no	—
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	2	4	no	—
41	C₂×C₄	—	2	4	no	—
42	C₂×C₄	—	2	4	no	—
43	C₂×C₄	C₂²×C₄	1	2	yes	—
44	C₂×C₄	C₂²×C₄	1	8	yes	—
45	C₂×C₄	C₂²×C₄	1	4	yes	—
46	C₂×C₄	C₂×Q₈	1	8	yes	—
47	C₂×C₄	C₂×Q₈	1	4	yes	—
48	C₂×C₄	#41	1	4	yes	—
49	C₂²	C₂³×C₄	1	1	yes	characteristic
50	C₂²	C₂²×Q₈	1	2	yes	—
51	C₂²	C₂×#41	1	2	yes	—
52	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
53	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
54	C₄	—	2	8	no	—
55	C₄	—	2	2	no	—
56	C₄	—	2	8	no	—
57	C₄	—	4	4	no	—
58	C₄	C₄×Q₈	1	8	yes	—
59	C₂	C₂×C₄×Q₈	1	2	yes	—
60	C₂	#518	1	1	yes	characteristic
61	C₂	#554	1	2	yes	—
62	C₂	#555	1	1	yes	characteristic
63	C₂	Q₈²	1	1	yes	characteristic

	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
7. C₄⋊Q₈	C₄	semidirect
28. 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
#391	2
#389	2
C₄×C₄⋊C₄	2