Finite Group Databasebeta

Subgroups — #512993

Order 512.

358 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 128 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	#33611	C₂	1	1	yes	maximal, characteristic
2	#33414	C₂	1	1	yes	maximal, characteristic, complemented
3	#13718	C₂	1	4	yes	maximal, complemented
4	#13717	C₂	1	4	yes	maximal, complemented
5	C₂×#1426	C₂	1	2	yes	maximal, complemented
6	#13625	C₂	1	2	yes	maximal, complemented
7	#13556	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#508	C₂²	1	1	yes	characteristic
9	C₂×#504	C₂²	1	1	yes	characteristic
10	#3061	—	2	1	no	—
11	#2916	—	2	1	no	—
12	C₂×M₆₄	C₂²	1	2	yes	—
13	#2219	C₂²	1	2	yes	—
14	#2218	C₂²	1	2	yes	—
15	#2217	—	2	2	no	—
16	#2161	—	2	1	no	—
17	#2161	C₂²	1	4	yes	—
18	C₂×#368	—	2	2	no	—
19	C₂×#368	C₂²	1	2	yes	—
20	C₂×#367	C₂²	1	1	yes	characteristic
21	#2139	—	2	1	no	—
22	#1426	C₂²	1	8	yes	—
23	#1420	C₂²	1	4	yes	—
24	#1419	C₂²	1	4	yes	—
25	#1400	C₂²	1	4	yes	—
26	C₂×#135	—	2	1	no	—
27	#568	—	4	1	no	—
28	C₂×#131	D₄	1	1	yes	characteristic
29	#537	—	4	1	no	—
30	#508	—	2	2	no	—
31	#508	—	2	2	no	—
32	C₂×Q₃₂	C₂³	1	1	yes	characteristic
33	C₂×SD₃₂	—	2	1	no	—
34	C₂×D₁₆	C₂³	1	1	yes	characteristic
35	#504	—	2	2	no	—
36	#504	—	2	2	no	—
37	C₂×M₃₂	D₄	1	1	yes	characteristic
38	C₂×M₃₂	C₂³	1	2	yes	—
39	C₂²×C₁₆	D₄	1	2	yes	—
40	C₂²×C₁₆	C₂³	1	1	yes	characteristic
41	#468	—	4	2	no	—
42	#467	—	2	2	no	—
43	#466	—	2	2	no	—
44	#430	—	4	2	no	—
45	#430	—	4	1	no	—
46	C₂×#108	—	2	4	no	—
47	C₂×#108	—	2	2	no	—
48	C₂×#107	—	2	1	no	—
49	#417	—	4	1	no	—
50	M₆₄	—	2	2	no	—
51	M₆₄	C₂³	1	4	yes	—
52	C₂×C₃₂	—	2	2	no	—
53	#368	—	2	2	no	—
54	#368	—	2	4	no	—
55	#368	C₂³	1	4	yes	—
56	#367	—	2	1	no	—
57	#367	C₂³	1	2	yes	—
58	#358	—	2	2	no	—
59	#357	—	2	2	no	—
60	C₂×#41	—	4	1	no	—
61	C₂×#41	D₈	1	1	yes	characteristic
62	#135	—	4	2	no	—
63	#135	—	4	2	no	—
64	C₂×Q₁₆	—	2	1	no	—
65	C₂×SD₁₆	—	4	1	no	—
66	C₂×D₈	—	2	1	no	—
67	#131	—	2	2	no	—
68	#131	—	2	2	no	—
69	#131	—	8	4	no	—
70	C₂×M₁₆	—	4	4	no	—
71	C₂×M₁₆	—	4	2	no	—
72	C₂×M₁₆	D₈	1	2	yes	—
73	C₂×M₁₆	D₈	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₄	1	2	yes	—
75	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂²⋊Q₈	—	4	2	no	—
77	C₂²⋊Q₈	—	8	1	no	—
78	#121	—	4	2	no	—
79	#121	—	8	1	no	—
80	#121	—	8	2	no	—
81	C₂×C₄⋊C₄	—	4	1	no	—
82	Q₃₂	—	2	2	no	—
83	SD₃₂	—	4	1	no	—
84	SD₃₂	—	4	1	no	—
85	D₁₆	—	2	2	no	—
86	M₃₂	—	2	2	no	—
87	M₃₂	—	2	4	no	—
88	C₂×C₁₆	—	2	2	no	—
89	C₂×C₁₆	—	2	2	no	—
90	C₂×C₁₆	C₂×D₄	1	2	yes	—
91	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
92	C₂×C₁₆	#41	1	2	yes	—
93	C₂×C₁₆	#41	1	2	yes	—
94	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
95	#108	—	2	4	no	—
96	#108	—	4	4	no	—
97	#108	—	4	2	no	—
98	#107	—	2	2	no	—
99	#107	—	4	1	no	—
100	#103	—	4	2	no	—
101	#102	—	4	2	no	—
102	C₃₂	—	2	4	no	—
103	#41	—	2	2	no	—
104	#41	—	2	2	no	—
105	#41	—	8	2	no	—
106	#41	—	8	2	no	—
107	C₂×Q₈	—	4	1	no	—
108	C₂×Q₈	SD₃₂	1	1	yes	characteristic
109	C₂×D₄	—	4	1	no	—
110	C₂×D₄	—	8	1	no	—
111	C₂×D₄	—	8	2	no	—
112	C₂×D₄	SD₃₂	1	2	yes	—
113	C₂×D₄	SD₃₂	1	1	yes	characteristic
114	C₂²×C₄	—	8	1	no	—
115	C₂²×C₄	—	8	1	no	—
116	C₂²×C₄	C₂×D₈	1	2	yes	—
117	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
118	Q₁₆	—	4	1	no	—
119	Q₁₆	—	4	1	no	—
120	SD₁₆	—	8	2	no	—
121	D₈	—	4	1	no	—
122	D₈	—	4	1	no	—
123	M₁₆	—	2	2	no	—
124	M₁₆	—	2	4	no	—
125	M₁₆	—	4	4	no	—
126	M₁₆	—	8	4	no	—
127	M₁₆	—	8	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	8	4	no	—
131	C₂×C₈	—	8	2	no	—
132	C₂×C₈	#123	1	2	yes	—
133	C₂×C₈	C₂×D₈	1	2	yes	—
134	C₂×C₈	C₂×D₈	1	1	yes	characteristic
135	C₂×C₈	#137	1	2	yes	—
136	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	8	1	no	—
139	C₄⋊C₄	—	8	2	no	—
140	C₂²⋊C₄	—	8	2	no	—
141	C₂²⋊C₄	—	8	2	no	—
142	C₁₆	—	2	2	no	—
143	C₁₆	#123	1	1	yes	characteristic
144	C₁₆	#123	1	1	yes	characteristic
145	C₁₆	C₂×#41	1	2	yes	—
146	C₂³	—	8	1	no	—
147	C₂³	C₂×SD₃₂	1	2	yes	—
148	C₂³	C₂×SD₃₂	1	1	yes	characteristic
149	Q₈	—	2	2	no	—
150	Q₈	—	8	1	no	—
151	Q₈	—	8	1	no	—
152	D₄	—	2	4	no	—
153	D₄	—	2	2	no	—
154	D₄	—	8	1	no	—
155	D₄	—	8	1	no	—
156	D₄	—	16	2	no	—
157	D₄	—	32	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	16	2	no	—
163	C₂×C₄	—	32	1	no	—
164	C₂×C₄	#480	1	2	yes	—
165	C₂×C₄	C₂×SD₃₂	1	2	yes	—
166	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
167	C₂×C₄	#509	1	2	yes	—
168	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	8	4	no	—
171	C₈	#480	1	2	yes	—
172	C₈	C₂×#123	1	1	yes	characteristic
173	C₈	C₂×#137	1	1	yes	characteristic
174	C₂²	—	2	2	no	—
175	C₂²	—	4	1	no	—
176	C₂²	—	4	2	no	—
177	C₂²	—	16	1	no	—
178	C₂²	—	16	1	no	—
179	C₂²	—	16	1	no	—
180	C₂²	#2233	1	2	yes	—
181	C₂²	C₂²×SD₃₂	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	16	2	no	—
184	C₄	—	16	1	no	—
185	C₄	#2233	1	2	yes	—
186	C₄	C₂×#480	1	1	yes	characteristic
187	C₄	C₂×#509	1	1	yes	characteristic
188	C₂	—	2	1	no	—
189	C₂	—	2	2	no	—
190	C₂	—	4	2	no	—
191	C₂	—	32	1	no	—
192	C₂	C₂×#2233	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂×#2233
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#33611	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ 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
2. #33414	C₂	semidirect
3. #13718	C₂	semidirect
4. #13717	C₂	semidirect
5. C₂×#1426	C₂	semidirect
6. #13625	C₂	semidirect
7. #13556	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33611	2
#33414	2
#13718	2
#13717	2
C₂×#1426	2
#13625	2
#13556	2