Subgroups — #135679

Order 512.

184 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12592	C₂	1	1	yes	maximal, characteristic
2	#8780	C₂	1	2	yes	maximal
3	#7157	C₂	1	2	yes	maximal
4	#7155	C₂	1	2	yes	maximal
5	C₂×#346	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
7	#1726	C₂²	1	1	yes	characteristic
8	#1564	C₂²	1	4	yes	—
9	#1374	C₄	1	4	yes	—
10	#1281	—	2	2	no	—
11	#1274	—	2	2	no	—
12	C₂²×C₁₆	C₂×C₄	1	2	yes	—
13	C₂²×C₁₆	C₂×C₄	1	2	yes	—
14	#431	—	2	4	no	—
15	C₂×#97	D₄	1	1	yes	characteristic
16	C₂×#97	Q₈	1	2	yes	—
17	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
19	#376	D₄	1	1	yes	characteristic
20	#346	—	2	2	no	—
21	C₄×C₁₆	—	2	2	no	—
22	#335	—	2	4	no	—
23	#334	—	2	4	no	—
24	C₂²×C₈	C₄⋊C₄	1	2	yes	—
25	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₄²	C₄⋊C₄	1	2	yes	—
28	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
29	C₂×C₁₆	—	2	2	no	—
30	C₂×C₁₆	—	2	2	no	—
31	C₂×C₁₆	—	2	4	no	—
32	C₂×C₁₆	—	2	4	no	—
33	C₄⋊C₈	—	4	4	no	—
34	C₂²⋊C₈	—	4	4	no	—
35	#97	—	2	2	no	—
36	#97	—	4	2	no	—
37	C₄×C₈	—	2	4	no	—
38	C₄×C₈	—	4	4	no	—
39	C₄×C₈	#41	1	2	yes	—
40	C₄×C₈	#41	1	2	yes	—
41	#95	—	2	2	no	—
42	C₂²×C₄	—	2	2	no	—
43	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄	#135	1	2	yes	—
45	C₂×C₈	—	2	2	no	—
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	8	4	no	—
50	C₂×C₈	#117	1	1	yes	characteristic
51	C₂×C₈	#117	1	1	yes	characteristic
52	C₂×C₈	#125	1	2	yes	—
53	C₂×C₈	#131	1	4	yes	—
54	C₄²	—	2	2	no	—
55	C₄²	—	2	2	no	—
56	C₄²	#124	1	2	yes	—
57	C₄²	#135	1	2	yes	—
58	C₁₆	—	4	2	no	—
59	C₁₆	—	4	2	no	—
60	C₂³	#427	1	1	yes	characteristic
61	C₂×C₄	—	4	2	no	—
62	C₂×C₄	—	4	4	no	—
63	C₂×C₄	—	4	2	no	—
64	C₂×C₄	#382	1	1	yes	characteristic
65	C₂×C₄	#424	1	2	yes	—
66	C₂×C₄	#427	1	1	yes	characteristic
67	C₂×C₄	#430	1	2	yes	—
68	C₂×C₄	#488	1	4	yes	—
69	C₈	—	2	2	no	—
70	C₈	—	4	2	no	—
71	C₈	—	16	4	no	—
72	C₈	#433	1	2	yes	—
73	C₈	#433	1	2	yes	—
74	C₂²	—	2	2	no	—
75	C₂²	#1572	1	2	yes	—
76	C₂²	#1835	1	1	yes	characteristic
77	C₄	—	2	4	no	—
78	C₄	—	8	2	no	—
79	C₄	#1572	1	2	yes	—
80	C₄	#1832	1	1	yes	characteristic
81	C₄	#1838	1	1	yes	characteristic
82	C₂	—	2	2	no	—
83	C₂	#8788	1	1	yes	characteristic
84	C₂	#12893	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1835
prime radical	C₂³	#427
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#12592	2
#8780	2
#7157	2
#7155	2