Finite Group Databasebeta

Subgroups — #508475

Order 512.

455 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 150 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×#2254	C₂	1	1	yes	maximal, characteristic, complemented
2	#33420	C₂	1	1	yes	maximal, characteristic
3	#33222	C₂	1	2	yes	maximal
4	#13518	C₂	1	4	yes	maximal, complemented
5	#13496	C₂	1	4	yes	maximal
6	#13290	C₂	1	2	yes	maximal, complemented
7	#12891	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#500	C₂²	1	2	yes	—
9	#3079	—	2	2	no	—
10	C₄×#131	C₂²	1	1	yes	characteristic
11	#2254	—	2	2	no	—
12	#2254	—	2	2	no	—
13	#2164	—	2	2	no	—
14	#2162	—	2	2	no	—
15	C₂×#367	C₂²	1	4	yes	—
16	#2151	C₂²	1	2	yes	—
17	#2113	C₂²	1	2	yes	—
18	C₄×M₃₂	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
20	#1997	—	2	2	no	—
21	#1952	—	2	2	no	—
22	#1952	C₂²	1	4	yes	—
23	#1848	C₂²	1	2	yes	—
24	#1816	—	2	2	no	—
25	#1816	C₂²	1	4	yes	—
26	#1387	C₂²	1	8	yes	—
27	#1381	C₂²	1	4	yes	—
28	C₂×#131	D₄	1	1	yes	characteristic
29	C₂×#131	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Q₈	—	2	2	no	—
31	C₄×#41	D₄	1	1	yes	characteristic
32	C₂×M₃₂	C₂³	1	2	yes	—
33	C₂²×C₁₆	C₂³	1	2	yes	—
34	#500	—	2	2	no	—
35	#500	—	2	2	no	—
36	#500	—	2	2	no	—
37	#500	—	2	2	no	—
38	#477	—	4	2	no	—
39	#476	—	4	2	no	—
40	#475	—	4	2	no	—
41	#474	—	4	2	no	—
42	C₂×#108	—	2	2	no	—
43	C₂×#108	C₂³	1	4	yes	—
44	#428	—	2	4	no	—
45	C₂×#107	—	2	2	no	—
46	C₂×#107	C₂³	1	4	yes	—
47	#423	—	2	2	no	—
48	#417	—	2	4	no	—
49	#405	—	2	1	no	—
50	#405	—	2	1	no	—
51	#405	D₄	1	1	yes	characteristic
52	#405	C₂³	1	1	yes	characteristic
53	C₄×M₁₆	—	2	1	no	—
54	C₄×M₁₆	D₄	1	1	yes	characteristic
55	C₂×C₄×C₈	—	2	1	no	—
56	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
57	#367	—	2	8	no	—
58	C₄⋊C₁₆	—	2	4	no	—
59	C₂²⋊C₁₆	—	2	4	no	—
60	#346	—	2	1	no	—
61	C₄×C₁₆	—	2	1	no	—
62	C₄×C₁₆	D₄	1	2	yes	—
63	C₄×C₁₆	D₄	1	2	yes	—
64	#343	—	2	4	no	—
65	#343	—	2	4	no	—
66	C₂×#41	C₂×D₄	1	1	yes	characteristic
67	C₂²×Q₈	—	4	2	no	—
68	#131	—	2	4	no	—
69	#131	—	2	4	no	—
70	C₂×M₁₆	—	2	1	no	—
71	C₂×M₁₆	—	2	1	no	—
72	C₂×M₁₆	—	2	4	no	—
73	C₂×M₁₆	C₂×D₄	1	1	yes	characteristic
74	C₂×M₁₆	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₈	—	2	1	no	—
76	C₂²×C₈	—	2	1	no	—
77	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₈	C₂⁴	1	1	yes	characteristic
79	C₄⋊Q₈	—	4	2	no	—
80	C₄⋊Q₈	—	4	2	no	—
81	C₄⋊Q₈	—	4	2	no	—
82	C₄⋊Q₈	—	4	2	no	—
83	C₄×Q₈	D₈	1	2	yes	—
84	C₄×D₄	—	2	2	no	—
85	C₄×D₄	D₈	1	2	yes	—
86	#117	—	2	1	no	—
87	#117	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	2	4	no	—
89	C₂×C₄⋊C₄	—	4	2	no	—
90	C₂×C₄²	—	2	1	no	—
91	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
92	M₃₂	—	2	4	no	—
93	C₂×C₁₆	—	2	4	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	C₂×D₄	1	4	yes	—
96	#108	—	2	8	no	—
97	#108	—	4	4	no	—
98	#107	—	2	4	no	—
99	#107	—	2	4	no	—
100	#107	—	4	2	no	—
101	#107	—	4	2	no	—
102	#107	—	4	4	no	—
103	#106	—	4	4	no	—
104	C₄⋊C₈	—	4	2	no	—
105	C₄⋊C₈	—	4	2	no	—
106	#103	—	4	4	no	—
107	#102	—	4	4	no	—
108	#97	—	2	1	no	—
109	#97	—	2	1	no	—
110	#97	—	2	1	no	—
111	C₄×C₈	—	2	1	no	—
112	C₄×C₈	—	2	1	no	—
113	C₄×C₈	—	2	1	no	—
114	C₄×C₈	C₂×D₄	1	1	yes	characteristic
115	C₄×C₈	C₂×D₄	1	1	yes	characteristic
116	C₄×C₈	#41	1	1	yes	characteristic
117	C₄×C₈	#41	1	1	yes	characteristic
118	#41	—	2	2	no	—
119	#41	—	2	2	no	—
120	C₂×Q₈	—	8	2	no	—
121	C₂×Q₈	—	8	2	no	—
122	C₂×Q₈	—	16	4	no	—
123	C₂×Q₈	C₂×D₈	1	1	yes	characteristic
124	C₂×D₄	—	2	1	no	—
125	C₂×D₄	C₂×D₈	1	1	yes	characteristic
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	—	4	4	no	—
129	C₂²×C₄	#135	1	2	yes	—
130	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
131	M₁₆	—	2	2	no	—
132	M₁₆	—	2	2	no	—
133	M₁₆	—	4	2	no	—
134	M₁₆	—	4	2	no	—
135	M₁₆	—	4	4	no	—
136	M₁₆	—	4	4	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	4	no	—
142	C₂×C₈	#121	1	1	yes	characteristic
143	C₂×C₈	#121	1	1	yes	characteristic
144	C₂×C₈	#137	1	4	yes	—
145	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₈	C₂×#41	1	1	yes	characteristic
147	C₂×C₈	2₋¹⁺⁴	1	4	yes	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	8	2	no	—
152	C₄⋊C₄	—	8	2	no	—
153	C₄⋊C₄	C₂×D₈	1	2	yes	—
154	C₂²⋊C₄	—	2	1	no	—
155	C₂²⋊C₄	—	2	2	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	2	no	—
158	C₄²	#121	1	1	yes	characteristic
159	C₄²	#121	1	1	yes	characteristic
160	C₄²	C₂×D₈	1	2	yes	—
161	C₁₆	—	2	4	no	—
162	C₁₆	—	4	2	no	—
163	C₂³	—	2	1	no	—
164	C₂³	C₂×#135	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	Q₈	—	16	4	no	—
167	Q₈	—	16	4	no	—
168	D₄	—	2	2	no	—
169	D₄	—	4	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	8	4	no	—
177	C₂×C₄	—	16	4	no	—
178	C₂×C₄	#466	1	2	yes	—
179	C₂×C₄	#466	1	2	yes	—
180	C₂×C₄	C₂×#121	1	1	yes	characteristic
181	C₂×C₄	#536	1	2	yes	—
182	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
183	C₂×C₄	C₂×#137	1	2	yes	—
184	C₈	—	4	1	no	—
185	C₈	—	4	1	no	—
186	C₈	—	8	4	no	—
187	C₈	#470	1	2	yes	—
188	C₈	#470	1	2	yes	—
189	C₈	#536	1	2	yes	—
190	C₈	#541	1	2	yes	—
191	C₂²	—	2	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	—	4	1	no	—
194	C₂²	C₂×#466	1	1	yes	characteristic
195	C₂²	#3231	1	2	yes	—
196	C₄	—	2	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	4	1	no	—
199	C₄	—	16	4	no	—
200	C₄	#2223	1	4	yes	—
201	C₄	#2621	1	1	yes	characteristic
202	C₄	C₂×#470	1	1	yes	characteristic
203	C₄	#3226	1	2	yes	—
204	C₂	—	2	2	no	—
205	C₂	—	8	1	no	—
206	C₂	#29876	1	1	yes	characteristic
207	C₂	C₂×#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#2254	C₂	semidirect
4. #13518	C₂	semidirect
6. #13290	C₂	semidirect
7. #12891	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2254	2
#33420	2
#33222	2
#13518	2
#13496	2
#13290	2
#12891	2