Subgroups — #1803

Order 128.

154 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⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#103	C₂	1	2	yes	maximal
4	#384	C₂	1	1	yes	maximal, characteristic
5	#340	C₂	1	2	yes	maximal
6	C₂²×Q₈	C₂²	1	1	yes	characteristic
7	C₂²×C₈	C₂²	1	2	yes	—
8	C₄×Q₈	—	2	2	no	—
9	C₄×Q₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄	—	2	1	no	—
11	C₂×C₄⋊C₄	—	2	1	no	—
12	C₂×C₄⋊C₄	—	2	1	no	—
13	C₂×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄²	—	2	1	no	—
16	C₂×C₄²	C₂²	1	1	yes	characteristic
17	C₄⋊C₈	—	2	2	no	—
18	#103	—	2	4	no	—
19	C₂×Q₈	—	2	1	no	—
20	C₂×Q₈	—	2	1	no	—
21	C₂×Q₈	—	2	1	no	—
22	C₂×Q₈	C₂×C₄	1	2	yes	—
23	C₂×Q₈	D₄	1	2	yes	—
24	C₂×Q₈	Q₈	1	2	yes	—
25	C₂²×C₄	—	2	1	no	—
26	C₂²×C₄	—	2	1	no	—
27	C₂²×C₄	—	2	2	no	—
28	C₂²×C₄	D₄	1	1	yes	characteristic
29	C₂²×C₄	D₄	1	1	yes	characteristic
30	C₂²×C₄	C₂³	1	1	yes	characteristic
31	C₂×C₈	—	2	2	no	—
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	—	2	2	no	—
34	C₄⋊C₄	—	2	4	no	—
35	C₄⋊C₄	—	2	1	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	2	2	no	—
38	C₄⋊C₄	—	4	2	no	—
39	C₄⋊C₄	—	4	2	no	—
40	C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₄²	—	2	4	no	—
42	C₄²	C₂×C₄	1	2	yes	—
43	C₄²	D₄	1	2	yes	—
44	C₂³	C₂×D₄	1	1	yes	characteristic
45	Q₈	—	2	2	no	—
46	Q₈	—	2	2	no	—
47	Q₈	—	2	2	no	—
48	Q₈	C₄⋊C₄	1	4	yes	complemented
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	4	1	no	—
56	C₂×C₄	—	4	1	no	—
57	C₂×C₄	—	4	2	no	—
58	C₂×C₄	—	4	2	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄	SD₁₆	1	2	yes	—
63	C₂×C₄	Q₁₆	1	2	yes	—
64	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₄	#41	1	1	yes	characteristic
69	C₂×C₄	#41	1	1	yes	characteristic
70	C₈	—	4	2	no	—
71	C₂²	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²	#121	1	1	yes	characteristic
73	C₂²	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²	C₂×SD₁₆	1	1	yes	characteristic
75	C₂²	C₂×Q₁₆	1	1	yes	characteristic
76	C₂²	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂²	#137	1	1	yes	characteristic
78	C₄	—	2	4	no	—
79	C₄	—	2	2	no	—
80	C₄	—	4	2	no	—
81	C₄	—	8	2	no	—
82	C₄	#103	1	4	yes	complemented
83	C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₄	#117	1	1	yes	characteristic
85	C₄	#121	1	1	yes	characteristic
86	C₄	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂	#380	1	1	yes	characteristic
88	C₂	C₂×#103	1	1	yes	characteristic
89	C₂	#417	1	1	yes	characteristic
90	C₂	#460	1	1	yes	characteristic
91	C₂	#462	1	1	yes	characteristic
92	C₂	#475	1	1	yes	characteristic
93	C₂	#477	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
9. C₄×Q₈	C₄	semidirect
48. Q₈	C₄⋊C₄	semidirect
82. C₄	#103	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⋊C₈	2
C₂×#103	2
#384	2
#340	2