Subgroups — #129140

Order 512.

290 conjugacy classes of subgroups.

The subgroup lattice has 159 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12662	C₂	1	2	yes	maximal
2	C₂×#1370	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₈×#107	C₂	1	1	yes	maximal, characteristic
5	#7167	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#346	—	2	2	no	—
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₄×#107	C₄	1	1	yes	characteristic
11	C₈×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×#335	C₄	1	1	yes	characteristic
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	#1374	—	2	2	no	—
15	#1370	C₄	1	4	yes	complemented
16	C₈×C₁₆	C₄	1	4	yes	complemented
17	#1279	C₄	1	2	yes	—
18	C₂²×C₁₆	—	2	1	no	—
19	C₂²×C₁₆	—	2	2	no	—
20	C₂²×C₁₆	D₄	1	1	yes	characteristic
21	C₂²×C₁₆	D₄	1	1	yes	characteristic
22	C₂×#107	C₈	1	2	yes	—
23	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₈	—	2	2	no	—
25	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
29	C₄⋊C₁₆	—	2	2	no	—
30	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
31	#346	—	4	4	no	—
32	C₄×C₁₆	—	2	1	no	—
33	C₄×C₁₆	C₂×C₄	1	2	yes	—
34	C₄×C₁₆	C₂×C₄	1	2	yes	—
35	C₄×C₁₆	Q₈	1	2	yes	—
36	#340	C₈	1	2	yes	—
37	#336	—	2	2	no	—
38	#335	C₈	1	4	yes	—
39	C₈²	C₂×C₄	1	2	yes	—
40	C₈²	C₂×C₄	1	2	yes	—
41	C₂²×C₈	—	2	1	no	—
42	C₂²×C₈	—	2	2	no	—
43	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₈	#41	1	1	yes	characteristic
47	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
48	C₂×C₄²	—	2	2	no	—
49	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₁₆	—	2	2	no	—
51	C₂×C₁₆	—	2	2	no	—
52	C₂×C₁₆	—	2	2	no	—
53	C₂×C₁₆	—	2	1	no	—
54	C₂×C₁₆	—	2	1	no	—
55	C₂×C₁₆	—	2	4	no	—
56	C₂×C₁₆	—	4	2	no	—
57	C₂×C₁₆	—	4	2	no	—
58	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
59	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
60	C₂×C₁₆	D₈	1	2	yes	—
61	C₂×C₁₆	Q₁₆	1	2	yes	—
62	#107	—	2	4	no	—
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	C₂×C₈	1	4	yes	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	—	4	2	no	—
67	C₄×C₈	—	4	2	no	—
68	C₄×C₈	C₄²	1	2	yes	—
69	C₄×C₈	C₄²	1	2	yes	—
70	C₄×C₈	C₂×C₈	1	2	yes	—
71	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
74	C₄×C₈	#41	1	1	yes	characteristic
75	#95	—	2	2	no	—
76	C₂²×C₄	—	2	4	no	—
77	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
78	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
79	C₂²×C₄	#131	1	1	yes	characteristic
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	4	no	—
86	C₂×C₈	—	4	2	no	—
87	C₂×C₈	—	4	2	no	—
88	C₂×C₈	C₄×C₈	1	2	yes	—
89	C₂×C₈	C₄×C₈	1	2	yes	—
90	C₂×C₈	M₃₂	1	4	yes	—
91	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₈	#117	1	1	yes	characteristic
93	C₂×C₈	C₄×D₄	1	1	yes	characteristic
94	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
95	C₂×C₈	C₂×D₈	1	1	yes	characteristic
96	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
97	C₂×C₈	#135	1	1	yes	characteristic
98	C₂×C₈	#135	1	1	yes	characteristic
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄²	—	4	2	no	—
102	C₄²	—	4	2	no	—
103	C₄²	C₂×C₄²	1	1	yes	characteristic
104	C₄²	C₄×Q₈	1	1	yes	characteristic
105	C₄²	C₂²×C₈	1	1	yes	characteristic
106	C₄²	#131	1	1	yes	characteristic
107	C₁₆	—	2	4	no	—
108	C₁₆	—	2	2	no	—
109	C₁₆	—	4	4	no	—
110	C₁₆	#107	1	4	yes	complemented
111	C₂³	C₈×D₄	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	4	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
118	C₂×C₄	#405	1	1	yes	characteristic
119	C₂×C₄	C₈×D₄	1	1	yes	characteristic
120	C₂×C₄	C₄×D₈	1	1	yes	characteristic
121	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
122	C₂×C₄	#443	1	1	yes	characteristic
123	C₂×C₄	#443	1	1	yes	characteristic
124	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
125	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
126	C₂×C₄	C₂×M₃₂	1	2	yes	—
127	C₂×C₄	#504	1	2	yes	—
128	C₈	—	2	4	no	—
129	C₈	—	2	2	no	—
130	C₈	—	2	1	no	—
131	C₈	—	2	1	no	—
132	C₈	—	4	4	no	—
133	C₈	#346	1	4	yes	complemented
134	C₈	C₂×#107	1	1	yes	characteristic
135	C₈	#427	1	1	yes	characteristic
136	C₈	C₄×D₈	1	1	yes	characteristic
137	C₈	C₄×Q₁₆	1	1	yes	characteristic
138	C₂²	C₈×D₈	1	1	yes	characteristic
139	C₂²	C₈×Q₁₆	1	1	yes	characteristic
140	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
141	C₂²	#2169	1	2	yes	—
142	C₂²	#2170	1	2	yes	—
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	8	4	no	—
146	C₄	C₈×D₈	1	1	yes	characteristic
147	C₄	C₈×Q₁₆	1	1	yes	characteristic
148	C₄	#1772	1	1	yes	characteristic
149	C₄	C₄×#107	1	1	yes	characteristic
150	C₄	C₂×#346	1	1	yes	characteristic
151	C₄	#2108	1	1	yes	characteristic
152	C₄	#2183	1	2	yes	—
153	C₂	C₈×#107	1	1	yes	characteristic
154	C₂	#12039	1	2	yes	—
155	C₂	#12054	1	2	yes	—
156	C₂	#12662	1	1	yes	characteristic
157	C₂	#12664	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
15. #1370	C₄	semidirect
16. C₈×C₁₆	C₄	semidirect
110. C₁₆	#107	semidirect
133. C₈	#346	semidirect

Maximal subgroups

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

subgroup	index
#12662	2
C₂×#1370	2
C₂×C₈×C₁₆	2
C₈×#107	2
#7167	2