Finite Group Databasebeta

Subgroups — #513042

Order 512.

394 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2253	C₂	1	1	yes	maximal, characteristic, complemented
2	#33713	C₂	1	1	yes	maximal, characteristic
3	#13724	C₂	1	2	yes	maximal, complemented
4	#13723	C₂	1	2	yes	maximal, complemented
5	#13721	C₂	1	4	yes	maximal, complemented
6	#13616	C₂	1	4	yes	maximal, complemented
7	C₄×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#508	—	2	1	no	—
9	C₂×#508	C₂²	1	2	yes	—
10	C₂×#499	—	2	1	no	—
11	#3147	—	2	1	no	—
12	C₂×M₆₄	C₂²	1	2	yes	—
13	#2253	—	2	1	no	—
14	#2253	—	2	1	no	—
15	#2253	C₂²	1	2	yes	—
16	#2253	C₂²	1	2	yes	—
17	#2249	—	2	1	no	—
18	#2249	C₂²	1	2	yes	—
19	#2248	—	2	1	no	—
20	#2247	C₂²	1	1	yes	characteristic
21	#2246	C₂²	1	1	yes	characteristic
22	C₂×#367	—	2	1	no	—
23	C₂×#366	C₂²	1	2	yes	—
24	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
25	#1416	C₂²	1	8	yes	—
26	#1415	C₂²	1	8	yes	—
27	#1397	C₂²	1	2	yes	—
28	C₄×C₃₂	C₂²	1	2	yes	—
29	C₂×#135	—	2	2	no	—
30	C₂×#135	—	4	1	no	—
31	#532	—	4	1	no	—
32	C₂×#125	—	4	1	no	—
33	#508	—	2	8	no	—
34	#508	—	4	4	no	—
35	C₂×Q₃₂	—	2	1	no	—
36	C₂×Q₃₂	C₂³	1	2	yes	—
37	C₂×SD₃₂	—	2	2	no	—
38	C₂×SD₃₂	—	2	2	no	—
39	C₂×D₁₆	—	2	1	no	—
40	C₂×D₁₆	C₂³	1	2	yes	—
41	C₂²×C₁₆	D₄	1	1	yes	characteristic
42	C₂²×C₁₆	C₂³	1	1	yes	characteristic
43	#499	—	2	2	no	—
44	#499	—	2	2	no	—
45	#499	—	4	1	no	—
46	#499	—	4	1	no	—
47	#495	—	2	2	no	—
48	#495	—	4	1	no	—
49	#494	—	2	1	no	—
50	#493	—	2	1	no	—
51	#492	—	4	1	no	—
52	C₂×#107	—	2	2	no	—
53	C₂×#106	—	4	1	no	—
54	C₂×C₄×C₈	D₄	1	1	yes	characteristic
55	M₆₄	—	2	4	no	—
56	C₂×C₃₂	C₂³	1	4	yes	—
57	#367	—	2	2	no	—
58	#367	—	2	2	no	—
59	#366	—	2	2	no	—
60	#366	C₂³	1	4	yes	—
61	C₄×C₁₆	D₄	1	1	yes	characteristic
62	C₄×C₁₆	D₄	1	1	yes	characteristic
63	C₄×C₁₆	C₂³	1	1	yes	characteristic
64	C₄×C₁₆	C₂³	1	1	yes	characteristic
65	C₂×#41	—	4	2	no	—
66	#135	—	4	4	no	—
67	#135	—	4	4	no	—
68	#135	—	8	4	no	—
69	C₂×Q₁₆	—	2	2	no	—
70	C₂×Q₁₆	—	4	1	no	—
71	C₂×SD₁₆	—	4	2	no	—
72	C₂×SD₁₆	—	4	2	no	—
73	C₂×D₈	—	2	2	no	—
74	C₂×D₈	—	4	1	no	—
75	C₂²×C₈	D₈	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
77	C₄⋊Q₈	—	4	1	no	—
78	#127	—	4	1	no	—
79	#125	—	4	2	no	—
80	#125	—	4	2	no	—
81	#125	—	8	2	no	—
82	#124	—	4	2	no	—
83	#121	—	8	2	no	—
84	C₄×D₄	—	8	2	no	—
85	C₂×C₄⋊C₄	—	4	2	no	—
86	C₂×C₄⋊C₄	—	8	2	no	—
87	C₂×C₄²	D₈	1	1	yes	characteristic
88	Q₃₂	—	2	4	no	—
89	Q₃₂	—	4	2	no	—
90	SD₃₂	—	4	4	no	—
91	SD₃₂	—	4	4	no	—
92	D₁₆	—	2	4	no	—
93	D₁₆	—	4	2	no	—
94	C₂×C₁₆	SD₁₆	1	4	yes	—
95	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
96	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
97	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
98	C₂×C₁₆	#41	1	4	yes	—
99	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
100	#107	—	2	4	no	—
101	#107	—	4	2	no	—
102	#106	—	4	2	no	—
103	#106	—	4	2	no	—
104	C₄×C₈	D₈	1	1	yes	characteristic
105	C₄×C₈	D₈	1	1	yes	characteristic
106	C₄×C₈	C₂×D₄	1	1	yes	characteristic
107	C₄×C₈	C₂×D₄	1	1	yes	characteristic
108	C₃₂	—	2	4	no	—
109	#41	—	8	4	no	—
110	#41	—	8	4	no	—
111	C₂×Q₈	—	4	2	no	—
112	C₂×D₄	—	4	2	no	—
113	C₂×D₄	—	8	2	no	—
114	C₂×D₄	—	8	2	no	—
115	C₂²×C₄	—	8	2	no	—
116	C₂²×C₄	—	8	2	no	—
117	C₂²×C₄	D₁₆	1	2	yes	—
118	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
119	Q₁₆	—	4	2	no	—
120	Q₁₆	—	4	2	no	—
121	Q₁₆	—	8	2	no	—
122	SD₁₆	—	8	4	no	—
123	SD₁₆	—	8	4	no	—
124	D₈	—	4	2	no	—
125	D₈	—	4	2	no	—
126	D₈	—	8	2	no	—
127	C₂×C₈	SD₃₂	1	4	yes	—
128	C₂×C₈	#124	1	2	yes	—
129	C₂×C₈	C₂×D₈	1	1	yes	characteristic
130	C₂×C₈	C₂×D₈	1	1	yes	characteristic
131	C₂×C₈	C₂×D₈	1	1	yes	characteristic
132	C₂×C₈	C₂×SD₁₆	1	2	yes	—
133	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	8	2	no	—
136	C₄⋊C₄	—	8	4	no	—
137	C₄⋊C₄	—	8	4	no	—
138	C₄⋊C₄	—	8	2	no	—
139	C₂²⋊C₄	—	8	4	no	—
140	C₄²	D₁₆	1	2	yes	—
141	C₄²	C₂×D₈	1	1	yes	characteristic
142	C₄²	C₂×D₈	1	1	yes	characteristic
143	C₁₆	#124	1	2	yes	—
144	C₁₆	C₂×SD₁₆	1	2	yes	—
145	C₁₆	C₂×SD₁₆	1	2	yes	—
146	C₁₆	C₂×#41	1	2	yes	—
147	C₂³	—	8	2	no	—
148	C₂³	C₂×D₁₆	1	1	yes	characteristic
149	Q₈	—	8	2	no	—
150	Q₈	—	8	2	no	—
151	D₄	—	8	2	no	—
152	D₄	—	8	2	no	—
153	D₄	—	16	4	no	—
154	D₄	—	16	4	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	4	no	—
158	C₂×C₄	—	16	4	no	—
159	C₂×C₄	—	16	4	no	—
160	C₂×C₄	#486	1	2	yes	—
161	C₂×C₄	C₂×D₁₆	1	2	yes	—
162	C₂×C₄	C₂×D₁₆	1	2	yes	—
163	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
164	C₂×C₄	C₂×SD₃₂	1	2	yes	—
165	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
166	C₈	#486	1	2	yes	—
167	C₈	C₂×SD₃₂	1	2	yes	—
168	C₈	C₂×SD₃₂	1	2	yes	—
169	C₈	C₂×#124	1	1	yes	characteristic
170	C₈	C₂²×SD₁₆	1	1	yes	characteristic
171	C₂²	—	2	2	no	—
172	C₂²	—	16	2	no	—
173	C₂²	—	16	4	no	—
174	C₂²	#2240	1	2	yes	—
175	C₂²	C₂²×D₁₆	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	16	2	no	—
179	C₄	—	16	4	no	—
180	C₄	#2240	1	2	yes	—
181	C₄	C₂×#486	1	1	yes	characteristic
182	C₄	C₂²×SD₃₂	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	32	2	no	—
185	C₂	C₂×#2240	1	1	yes	characteristic
186	C₂	#33981	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#33713	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#2253	C₂	semidirect
3. #13724	C₂	semidirect
4. #13723	C₂	semidirect
5. #13721	C₂	semidirect
6. #13616	C₂	semidirect
7. C₄×M₆₄	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2253	2
#33713	2
#13724	2
#13723	2
#13721	2
#13616	2
C₄×M₆₄	2