Subgroups — #94408

Order 512.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13642	C₂	1	1	yes	maximal, characteristic
2	#12020	C₂	1	1	yes	maximal, characteristic
3	#12011	C₂	1	1	yes	maximal, characteristic
4	C₂×#364	—	2	1	no	—
5	C₄×M₃₂	C₂²	1	1	yes	characteristic
6	#1566	—	2	1	no	—
7	#1422	—	2	1	no	—
8	#1400	C₄	1	2	yes	—
9	#1398	—	2	1	no	—
10	#1373	—	2	1	no	—
11	#1373	C₄	1	2	yes	—
12	#1368	—	2	1	no	—
13	#1368	C₄	1	2	yes	—
14	C₂×M₃₂	—	4	1	no	—
15	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
16	C₂×M₃₂	Q₈	1	1	yes	characteristic
17	#423	—	4	1	no	—
18	C₂×C₄×C₈	D₄	1	1	yes	characteristic
19	M₆₄	—	2	2	no	—
20	#364	—	2	2	no	—
21	#364	—	4	1	no	—
22	#346	C₂×C₄	1	1	yes	characteristic
23	#346	D₄	1	1	yes	characteristic
24	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₄×C₁₆	D₄	1	1	yes	characteristic
26	#334	—	2	2	no	—
27	#334	—	4	1	no	—
28	C₂×M₁₆	—	8	1	no	—
29	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₈	SD₁₆	1	1	yes	characteristic
31	C₂×C₄²	SD₁₆	1	1	yes	characteristic
32	M₃₂	—	2	2	no	—
33	M₃₂	—	4	1	no	—
34	M₃₂	—	4	2	no	—
35	M₃₂	—	8	1	no	—
36	C₂×C₁₆	—	8	1	no	—
37	C₂×C₁₆	C₄²	1	1	yes	characteristic
38	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
41	C₄⋊C₈	—	4	2	no	—
42	C₄⋊C₈	—	8	1	no	—
43	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
44	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
45	C₄×C₈	D₈	1	1	yes	characteristic
46	C₄×C₈	Q₁₆	1	1	yes	characteristic
47	C₃₂	—	4	2	no	—
48	C₂²×C₄	—	2	1	no	—
49	C₂²×C₄	#106	1	1	yes	characteristic
50	M₁₆	—	16	2	no	—
51	C₂×C₈	—	2	1	no	—
52	C₂×C₈	—	2	1	no	—
53	C₂×C₈	—	2	1	no	—
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	—	8	2	no	—
56	C₂×C₈	#95	1	1	yes	characteristic
57	C₂×C₈	#102	1	1	yes	characteristic
58	C₂×C₈	#103	1	1	yes	characteristic
59	C₂×C₈	#107	1	1	yes	characteristic
60	C₄²	—	2	1	no	—
61	C₄²	#102	1	1	yes	characteristic
62	C₄²	#103	1	1	yes	characteristic
63	C₁₆	—	2	1	no	—
64	C₁₆	—	2	1	no	—
65	C₁₆	—	2	1	no	—
66	C₁₆	—	8	2	no	—
67	C₁₆	#108	1	2	yes	—
68	C₂³	#365	1	1	yes	characteristic
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	#340	1	1	yes	characteristic
75	C₂×C₄	#365	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	2	1	no	—
78	C₈	—	16	2	no	—
79	C₈	#341	1	1	yes	characteristic
80	C₈	#344	1	1	yes	characteristic
81	C₈	#361	1	1	yes	characteristic
82	C₈	#362	1	1	yes	characteristic
83	C₂²	—	2	1	no	—
84	C₂²	—	4	1	no	—
85	C₂²	#1383	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	1	no	—
88	C₄	—	4	1	no	—
89	C₄	#1301	1	1	yes	characteristic
90	C₄	#1387	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	4	1	no	—
93	C₂	#7292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#7292
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₄×M₃₂	C₂²
prime radical	C₂³	#365
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×M₃₂ ▷ C₂×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
#13642	2
#12020	2
#12011	2