Subgroups — #140341

Order 512.

268 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#366	C₂	1	1	yes	maximal, characteristic
2	C₂×#1371	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#9594	C₂	1	2	yes	maximal
5	#7171	C₂	1	2	yes	maximal
6	C₂×#366	C₄	1	1	yes	characteristic
7	C₂×#366	C₄	1	1	yes	characteristic
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	#1939	—	2	2	no	—
11	C₄×#107	C₂²	1	2	yes	—
12	C₂×#335	C₂²	1	2	yes	—
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	#1380	C₄	1	2	yes	—
15	#1371	—	2	2	no	—
16	C₈×C₁₆	—	2	2	no	—
17	#1279	—	2	2	no	—
18	C₂²×C₁₆	—	2	1	no	—
19	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×#107	C₂×C₄	1	2	yes	—
22	C₂×#107	C₂×C₄	1	2	yes	—
23	C₂×C₄⋊C₈	—	2	2	no	—
24	C₂×C₄⋊C₈	—	4	2	no	—
25	C₂×C₄×C₈	D₄	1	1	yes	characteristic
26	C₂×C₄×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	C₄×C₄⋊C₄	—	2	2	no	—
29	#366	—	2	2	no	—
30	#366	C₈	1	4	yes	complemented
31	C₄×C₁₆	—	2	1	no	—
32	C₄×C₁₆	D₄	1	2	yes	—
33	C₄×C₁₆	Q₈	1	2	yes	—
34	C₄×C₁₆	Q₈	1	2	yes	—
35	#340	—	2	2	no	—
36	#336	—	4	2	no	—
37	#335	—	2	2	no	—
38	#335	—	2	2	no	—
39	C₈²	—	2	1	no	—
40	C₈²	—	2	1	no	—
41	C₂²×C₈	—	2	1	no	—
42	C₂²×C₈	—	4	2	no	—
43	C₂²×C₈	D₈	1	1	yes	characteristic
44	C₂²×C₈	D₈	1	1	yes	characteristic
45	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₈	#41	1	1	yes	characteristic
47	C₂×C₄⋊C₄	—	2	2	no	—
48	C₂×C₄⋊C₄	—	2	2	no	—
49	C₂×C₄²	—	4	2	no	—
50	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
51	C₂×C₁₆	—	2	2	no	—
52	C₂×C₁₆	—	2	2	no	—
53	C₂×C₁₆	—	2	2	no	—
54	C₂×C₁₆	—	2	1	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
57	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
58	C₂×C₁₆	C₂×C₈	1	2	yes	—
59	C₂×C₁₆	M₁₆	1	2	yes	—
60	#107	—	2	2	no	—
61	#107	—	2	2	no	—
62	#107	—	2	2	no	—
63	#107	C₂×C₈	1	4	yes	—
64	C₄⋊C₈	—	4	2	no	—
65	C₄⋊C₈	—	4	2	no	—
66	C₄⋊C₈	—	8	4	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	Q₁₆	1	2	yes	—
71	C₄×C₈	C₂×D₄	1	1	yes	characteristic
72	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
73	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
74	C₄×C₈	#41	1	1	yes	characteristic
75	#95	—	4	2	no	—
76	C₂²×C₄	—	4	2	no	—
77	C₂²×C₄	—	4	2	no	—
78	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
79	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
80	C₂²×C₄	#135	1	1	yes	characteristic
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	8	2	no	—
91	C₂×C₈	—	8	2	no	—
92	C₂×C₈	—	8	2	no	—
93	C₂×C₈	D₁₆	1	2	yes	—
94	C₂×C₈	Q₃₂	1	2	yes	—
95	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₈	#125	1	1	yes	characteristic
99	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₈	#131	1	1	yes	characteristic
102	C₂×C₈	#131	1	1	yes	characteristic
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄²	—	8	2	no	—
108	C₄²	—	8	2	no	—
109	C₄²	#121	1	1	yes	characteristic
110	C₄²	C₄⋊Q₈	1	1	yes	characteristic
111	C₄²	C₂×Q₁₆	1	1	yes	characteristic
112	C₄²	#135	1	1	yes	characteristic
113	C₁₆	—	2	4	no	—
114	C₁₆	—	2	2	no	—
115	C₁₆	C₄⋊C₈	1	4	yes	complemented
116	C₂³	C₄×D₈	1	1	yes	characteristic
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	4	4	no	—
120	C₂×C₄	—	8	2	no	—
121	C₂×C₄	—	8	2	no	—
122	C₂×C₄	—	8	2	no	—
123	C₂×C₄	—	8	2	no	—
124	C₂×C₄	—	8	2	no	—
125	C₂×C₄	#389	1	1	yes	characteristic
126	C₂×C₄	C₈×D₄	1	1	yes	characteristic
127	C₂×C₄	#436	1	1	yes	characteristic
128	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
129	C₂×C₄	#443	1	1	yes	characteristic
130	C₂×C₄	#443	1	1	yes	characteristic
131	C₂×C₄	#466	1	1	yes	characteristic
132	C₂×C₄	#467	1	1	yes	characteristic
133	C₂×C₄	#499	1	1	yes	characteristic
134	C₂×C₄	#500	1	1	yes	characteristic
135	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
136	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
137	C₂×C₄	#508	1	1	yes	characteristic
138	C₂×C₄	#508	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	2	no	—
143	C₈	—	2	2	no	—
144	C₈	—	16	2	no	—
145	C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
146	C₈	#424	1	1	yes	characteristic
147	C₈	C₈×Q₈	1	1	yes	characteristic
148	C₈	#446	1	1	yes	characteristic
149	C₂²	C₈×D₈	1	1	yes	characteristic
150	C₂²	#1589	1	1	yes	characteristic
151	C₂²	#1942	1	1	yes	characteristic
152	C₂²	C₄×D₁₆	1	1	yes	characteristic
153	C₂²	C₄×Q₃₂	1	1	yes	characteristic
154	C₂²	#2178	1	1	yes	characteristic
155	C₂²	#2178	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	8	4	no	—
159	C₄	—	16	2	no	—
160	C₄	C₈×Q₁₆	1	1	yes	characteristic
161	C₄	#1591	1	1	yes	characteristic
162	C₄	#1939	1	1	yes	characteristic
163	C₄	#1945	1	1	yes	characteristic
164	C₄	#2215	1	1	yes	characteristic
165	C₄	#2216	1	1	yes	characteristic
166	C₄	#2252	1	1	yes	characteristic
167	C₄	#2253	1	1	yes	characteristic
168	C₂	#9594	1	1	yes	characteristic
169	C₂	C₈×D₁₆	1	1	yes	characteristic
170	C₂	C₈×Q₃₂	1	1	yes	characteristic
171	C₂	#12081	1	1	yes	characteristic
172	C₂	#12083	1	1	yes	characteristic
173	C₂	#13165	1	1	yes	characteristic
174	C₂	#13169	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ 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
30. #366	C₈	semidirect
115. C₁₆	C₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#366	2
C₂×#1371	2
C₂×C₈×C₁₆	2
#9594	2
#7171	2