Subgroups — #2355

Order 128.

240 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	2	yes	maximal
3	#387	C₂	1	4	yes	maximal
4	#384	C₂	1	1	yes	maximal, characteristic
5	#382	C₂	1	2	yes	maximal
6	C₄×C₄⋊C₄	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄	C₂	1	2	yes	maximal
9	C₂²×Q₈	C₂²	1	1	yes	characteristic
10	C₄×Q₈	C₄	1	8	yes	complemented
11	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₄²	C₂²	1	2	yes	—
19	C₂×C₄²	C₂²	1	4	yes	—
20	C₂×C₄²	C₂²	1	2	yes	—
21	C₂×C₄²	C₂²	1	1	yes	characteristic
22	#95	C₂²	1	4	yes	—
23	#95	C₂²	1	2	yes	—
24	#95	C₂²	1	4	yes	—
25	#95	C₂²	1	4	yes	—
26	C₂×Q₈	—	2	2	no	—
27	C₂×Q₈	—	2	2	no	—
28	C₂×Q₈	C₂×C₄	1	4	yes	—
29	C₂²×C₄	C₂³	1	2	yes	—
30	C₂²×C₄	C₂³	1	4	yes	—
31	C₂²×C₄	C₂³	1	2	yes	—
32	C₂²×C₄	C₂³	1	2	yes	—
33	C₂²×C₄	C₂³	1	1	yes	characteristic
34	C₂²×C₄	C₂³	1	2	yes	—
35	C₂²×C₄	C₂³	1	1	yes	characteristic
36	C₂²×C₄	C₂³	1	1	yes	characteristic
37	C₄⋊C₄	—	2	4	no	—
38	C₄⋊C₄	—	2	2	no	—
39	C₄⋊C₄	—	2	4	no	—
40	C₄⋊C₄	—	4	2	no	—
41	C₄⋊C₄	C₂×C₄	1	8	yes	—
42	C₄⋊C₄	C₂×C₄	1	4	yes	—
43	C₄⋊C₄	Q₈	1	4	yes	complemented
44	C₄²	—	2	4	no	—
45	C₄²	—	4	4	no	—
46	C₄²	C₂×C₄	1	8	yes	—
47	C₄²	C₂×C₄	1	4	yes	—
48	C₂³	C₂⁴	1	1	yes	characteristic
49	Q₈	—	2	8	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	—	2	4	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	C₂²×C₄	1	4	yes	—
65	C₂×C₄	C₂²×C₄	1	2	yes	—
66	C₂×C₄	C₂²×C₄	1	4	yes	—
67	C₂×C₄	C₂²×C₄	1	2	yes	—
68	C₂×C₄	C₂²×C₄	1	2	yes	—
69	C₂×C₄	C₂×Q₈	1	4	yes	—
70	C₂×C₄	C₂×Q₈	1	2	yes	—
71	C₂×C₄	#41	1	2	yes	—
72	C₂×C₄	#41	1	2	yes	—
73	C₂×C₄	#41	1	2	yes	—
74	C₂²	C₂³×C₄	1	1	yes	characteristic
75	C₂²	C₂²×Q₈	1	1	yes	characteristic
76	C₂²	C₂×#41	1	1	yes	characteristic
77	C₂²	C₂×#41	1	1	yes	characteristic
78	C₂²	C₂×#41	1	1	yes	characteristic
79	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
80	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
81	C₄	—	2	4	no	—
82	C₄	—	2	4	no	—
83	C₄	—	2	2	no	—
84	C₄	—	2	4	no	—
85	C₄	—	4	2	no	—
86	C₄	—	4	4	no	—
87	C₄	#117	1	4	yes	—
88	C₄	C₄×Q₈	1	4	yes	—
89	C₂	C₂×#117	1	1	yes	characteristic
90	C₂	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₂	#520	1	1	yes	characteristic
92	C₂	#554	1	1	yes	characteristic
93	C₂	#556	1	1	yes	characteristic
94	C₂	#557	1	1	yes	characteristic
95	C₂	#559	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
10. C₄×Q₈	C₄	semidirect
43. 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
#387	2
#384	2
#382	2
C₄×C₄⋊C₄	2