Subgroups — #2571

Order 128.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#400	C₂	1	4	yes	maximal
2	#389	C₂	1	4	yes	maximal
3	#384	C₂	1	1	yes	maximal, characteristic
4	#382	C₂	1	4	yes	maximal
5	C₄×C₄⋊C₄	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₄	C₂²	1	2	yes	—
7	C₂×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₄²	C₂²	1	4	yes	—
12	C₂×C₄²	C₂²	1	1	yes	characteristic
13	#95	C₂²	1	4	yes	—
14	#95	C₂²	1	4	yes	—
15	#95	C₂²	1	8	yes	—
16	C₂²×C₄	C₂³	1	4	yes	—
17	C₂²×C₄	C₂³	1	4	yes	—
18	C₂²×C₄	C₂³	1	4	yes	—
19	C₂²×C₄	C₂³	1	1	yes	characteristic
20	C₂²×C₄	C₂³	1	2	yes	—
21	C₄⋊C₄	—	2	4	no	—
22	C₄⋊C₄	—	2	4	no	—
23	C₄⋊C₄	—	2	8	no	—
24	C₄⋊C₄	—	4	2	no	—
25	C₄⋊C₄	—	4	4	no	—
26	C₄²	—	2	8	no	—
27	C₄²	Q₈	1	4	yes	—
28	C₂³	C₂⁴	1	1	yes	characteristic
29	C₂×C₄	—	2	1	no	—
30	C₂×C₄	—	2	1	no	—
31	C₂×C₄	—	2	4	no	—
32	C₂×C₄	—	2	4	no	—
33	C₂×C₄	—	2	4	no	—
34	C₂×C₄	—	2	4	no	—
35	C₂×C₄	—	2	8	no	—
36	C₂×C₄	—	2	4	no	—
37	C₂×C₄	—	2	4	no	—
38	C₂×C₄	—	2	4	no	—
39	C₂×C₄	C₂×Q₈	1	2	yes	—
40	C₂×C₄	C₂×Q₈	1	4	yes	—
41	C₂×C₄	#41	1	4	yes	—
42	C₂×C₄	#41	1	4	yes	—
43	C₂²	C₂²×Q₈	1	1	yes	characteristic
44	C₂²	C₂×#41	1	2	yes	—
45	C₂²	C₂×#41	1	2	yes	—
46	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
47	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	4	4	no	—
50	C₄	—	4	4	no	—
51	C₄	—	4	4	no	—
52	C₄	#125	1	8	yes	—
53	C₂	C₂×#125	1	2	yes	—
54	C₂	#544	1	1	yes	characteristic
55	C₂	#555	1	1	yes	characteristic
56	C₂	#556	1	2	yes	—
57	C₂	#559	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂³	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Maximal subgroups

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

subgroup	index
#400	2
#389	2
#384	2
#382	2
C₄×C₄⋊C₄	2