Subgroups — #2064

Order 128.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108	C₂	1	1	yes	maximal, characteristic
2	#428	C₂	1	1	yes	maximal, characteristic
3	#424	C₂	1	1	yes	maximal, characteristic
4	#419	C₂	1	1	yes	maximal, characteristic
5	C₂×#103	C₂	1	1	yes	maximal, characteristic
6	C₂×#101	C₂	1	1	yes	maximal, characteristic
7	#343	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈	C₂²	1	1	yes	characteristic
9	C₂×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₈	C₂²	1	1	yes	characteristic
13	#117	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	#108	—	2	2	no	—
16	#107	—	2	1	no	—
17	#106	—	2	1	no	—
18	C₄⋊C₈	—	2	2	no	—
19	#103	—	2	2	no	—
20	#103	—	2	1	no	—
21	#103	—	2	1	no	—
22	#101	—	2	2	no	—
23	C₂×Q₈	—	2	1	no	—
24	C₂×Q₈	—	2	1	no	—
25	C₂×Q₈	—	2	1	no	—
26	C₂×Q₈	—	2	1	no	—
27	C₂×Q₈	—	2	1	no	—
28	C₂×Q₈	Q₈	1	2	yes	—
29	C₂²×C₄	—	2	1	no	—
30	C₂²×C₄	—	2	1	no	—
31	C₂²×C₄	C₂³	1	1	yes	characteristic
32	M₁₆	—	2	1	no	—
33	M₁₆	—	2	1	no	—
34	M₁₆	—	2	1	no	—
35	M₁₆	—	2	1	no	—
36	M₁₆	—	2	1	no	—
37	M₁₆	D₄	1	2	yes	—
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	D₄	1	2	yes	—
44	C₄⋊C₄	—	2	1	no	—
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	D₄	1	1	yes	characteristic
47	C₄⋊C₄	D₄	1	1	yes	characteristic
48	C₂²⋊C₄	—	2	1	no	—
49	C₄²	—	2	1	no	—
50	C₂³	#41	1	1	yes	characteristic
51	Q₈	—	2	2	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	4	1	no	—
54	Q₈	—	4	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	8	1	no	—
62	C₂×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₄	#41	1	1	yes	characteristic
67	C₂×C₄	#41	1	1	yes	characteristic
68	C₈	—	2	2	no	—
69	C₈	—	2	2	no	—
70	C₈	—	4	1	no	—
71	C₈	—	4	1	no	—
72	C₂²	—	2	1	no	—
73	C₂²	—	2	1	no	—
74	C₂²	#121	1	1	yes	characteristic
75	C₂²	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²	#126	1	1	yes	characteristic
77	C₄	—	4	2	no	—
78	C₄	—	4	1	no	—
79	C₄	—	4	1	no	—
80	C₄	—	8	1	no	—
81	C₄	#121	1	1	yes	characteristic
82	C₄	#121	1	1	yes	characteristic
83	C₄	C₂²⋊Q₈	1	1	yes	characteristic
84	C₄	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂	—	2	1	no	—
86	C₂	—	2	1	no	—
87	C₂	#396	1	1	yes	characteristic
88	C₂	#471	1	1	yes	characteristic
89	C₂	#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³
prime radical	C₂³	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ 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×#108	2
#428	2
#424	2
#419	2
C₂×#103	2
C₂×#101	2
#343	2