Subgroups — #13518

Order 256.

125 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#367	C₂	1	1	yes	maximal, characteristic
2	#2151	C₂	1	1	yes	maximal, characteristic
3	#1952	C₂	1	2	yes	maximal
4	#1387	C₂	1	2	yes	maximal
5	#1381	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#108	C₂²	1	2	yes	—
9	#428	—	2	2	no	—
10	C₂×#107	C₂²	1	2	yes	—
11	#405	C₂²	1	1	yes	characteristic
12	#367	—	2	2	no	—
13	C₄⋊C₁₆	—	2	2	no	—
14	#343	—	2	2	no	—
15	C₂×M₁₆	—	2	2	no	—
16	C₂×M₁₆	D₄	1	1	yes	characteristic
17	C₂²×C₈	C₂³	1	1	yes	characteristic
18	#117	D₄	1	1	yes	characteristic
19	C₂×C₄⋊C₄	—	2	2	no	—
20	M₃₂	—	2	2	no	—
21	C₂×C₁₆	—	2	2	no	—
22	C₂×C₁₆	—	2	1	no	—
23	C₂×C₁₆	D₄	1	2	yes	—
24	#108	—	2	2	no	—
25	#108	Q₈	1	4	yes	—
26	#107	—	2	2	no	—
27	#107	—	2	2	no	—
28	#107	—	4	2	no	—
29	#106	—	4	2	no	—
30	#97	—	2	1	no	—
31	C₄×C₈	—	2	1	no	—
32	C₂²×C₄	—	4	2	no	—
33	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
34	M₁₆	—	2	4	no	—
35	M₁₆	—	2	2	no	—
36	M₁₆	—	4	2	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	4	2	no	—
39	C₂×C₈	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈	C₂×Q₈	1	2	yes	—
41	C₂×C₈	#41	1	2	yes	—
42	C₂×C₈	#41	1	1	yes	characteristic
43	C₄⋊C₄	—	4	2	no	—
44	C₄⋊C₄	—	4	2	no	—
45	C₄⋊C₄	D₈	1	2	yes	—
46	C₂²⋊C₄	—	2	1	no	—
47	C₄²	—	2	1	no	—
48	C₁₆	—	2	2	no	—
49	C₁₆	—	4	1	no	—
50	C₂³	#135	1	1	yes	characteristic
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	8	2	no	—
53	C₂×C₄	—	16	2	no	—
54	C₂×C₄	#121	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
56	C₂×C₄	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄	#137	1	2	yes	—
58	C₈	—	4	1	no	—
59	C₈	—	4	4	no	—
60	C₈	#123	1	1	yes	characteristic
61	C₈	#125	1	2	yes	—
62	C₈	C₄⋊Q₈	1	1	yes	characteristic
63	C₂²	—	2	2	no	—
64	C₂²	#466	1	1	yes	characteristic
65	C₂²	#479	1	2	yes	—
66	C₄	—	4	2	no	—
67	C₄	—	16	2	no	—
68	C₄	#398	1	1	yes	characteristic
69	C₄	#470	1	1	yes	characteristic
70	C₄	#474	1	2	yes	—
71	C₂	—	2	2	no	—
72	C₂	#2086	1	1	yes	characteristic
73	C₂	#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ 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×#367	2
#2151	2
#1952	2
#1387	2
#1381	2