Subgroups — #2088

Order 128.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#389	C₂	1	1	yes	maximal, characteristic
4	#389	C₂	1	1	yes	maximal, characteristic
5	#340	C₂	1	1	yes	maximal, characteristic
6	#340	C₂	1	1	yes	maximal, characteristic
7	#340	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄	—	2	1	no	—
11	C₂×C₄⋊C₄	—	2	1	no	—
12	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₄²	C₂²	1	1	yes	characteristic
17	#106	—	2	2	no	—
18	C₄⋊C₈	—	2	2	no	—
19	#95	—	2	1	no	—
20	#95	—	2	1	no	—
21	C₂²×C₄	—	2	1	no	—
22	C₂²×C₄	—	2	1	no	—
23	C₂²×C₄	—	2	1	no	—
24	C₂²×C₄	—	2	1	no	—
25	C₂²×C₄	D₄	1	1	yes	characteristic
26	C₂²×C₄	D₄	1	1	yes	characteristic
27	C₂²×C₄	C₂³	1	1	yes	characteristic
28	C₂×C₈	—	2	1	no	—
29	C₂×C₈	—	2	1	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	2	1	no	—
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	D₄	1	2	yes	—
34	C₄⋊C₄	—	2	1	no	—
35	C₄⋊C₄	—	2	1	no	—
36	C₄⋊C₄	—	2	1	no	—
37	C₄⋊C₄	—	2	1	no	—
38	C₄⋊C₄	—	2	1	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	4	2	no	—
41	C₄⋊C₄	—	4	2	no	—
42	C₄⋊C₄	Q₈	1	2	yes	—
43	C₄⋊C₄	Q₈	1	2	yes	—
44	C₄²	—	2	1	no	—
45	C₄²	—	2	1	no	—
46	C₂³	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	2	1	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	—	4	1	no	—
55	C₂×C₄	—	4	1	no	—
56	C₂×C₄	—	4	1	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	—	4	1	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	SD₁₆	1	2	yes	—
65	C₂×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄	C₂×Q₈	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	C₂×C₄	#41	1	1	yes	characteristic
71	C₈	—	2	2	no	—
72	C₈	—	4	1	no	—
73	C₂²	#121	1	1	yes	characteristic
74	C₂²	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂²	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²	C₂×SD₁₆	1	1	yes	characteristic
77	C₂²	#135	1	1	yes	characteristic
78	C₂²	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂²	#137	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	4	1	no	—
82	C₄	—	4	2	no	—
83	C₄	—	4	2	no	—
84	C₄	—	8	1	no	—
85	C₄	—	8	1	no	—
86	C₄	#123	1	1	yes	characteristic
87	C₄	#125	1	1	yes	characteristic
88	C₄	#125	1	1	yes	characteristic
89	C₄	C₄⋊Q₈	1	1	yes	characteristic
90	C₂	#398	1	1	yes	characteristic
91	C₂	#465	1	1	yes	characteristic
92	C₂	#468	1	1	yes	characteristic
93	C₂	#475	1	1	yes	characteristic
94	C₂	#476	1	1	yes	characteristic
95	C₂	#478	1	1	yes	characteristic
96	C₂	#479	1	1	yes	characteristic

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

Maximal subgroups

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

subgroup	index
C₂×#106	2
C₂×C₄⋊C₈	2
#389	2
#340	2