Finite Group Databasebeta

Subgroups — #507605

Order 512.

417 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 417 conjugacy classes of subgroups — too many to draw. Only its 144 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	#21736	C₂	1	1	yes	maximal, characteristic
2	#21728	C₂	1	1	yes	maximal, characteristic
3	#13541	C₂	1	2	yes	maximal
4	#13454	C₂	1	2	yes	maximal
5	#13273	C₂	1	2	yes	maximal
6	#13180	C₂	1	2	yes	maximal
7	#13119	C₂	1	4	yes	maximal
8	#12882	C₂	1	1	yes	maximal, characteristic
9	#3162	C₂²	1	1	yes	characteristic
10	C₂×#499	C₂²	1	1	yes	characteristic
11	#2961	—	2	1	no	—
12	C₄×#125	—	2	1	no	—
13	#2159	C₂²	1	4	yes	—
14	C₂×#364	C₂²	1	2	yes	—
15	#1994	—	2	1	no	—
16	#1991	—	2	1	no	—
17	#1946	C₂²	1	2	yes	—
18	#1945	—	2	1	no	—
19	#1944	C₂²	1	2	yes	—
20	#1943	—	2	1	no	—
21	#1849	C₂²	1	1	yes	characteristic
22	#1846	C₂²	1	1	yes	characteristic
23	#1776	—	2	1	no	—
24	#1776	C₂²	1	2	yes	—
25	#1773	—	2	1	no	—
26	#1772	C₂²	1	2	yes	—
27	C₄×#97	C₂²	1	1	yes	characteristic
28	#1387	C₂²	1	4	yes	—
29	#1386	C₂²	1	4	yes	—
30	#1383	C₂²	1	4	yes	—
31	#1377	C₂²	1	2	yes	—
32	#1375	C₂²	1	2	yes	—
33	#533	—	2	1	no	—
34	C₂×#125	—	2	1	no	—
35	C₂×M₃₂	C₂³	1	4	yes	—
36	#501	—	2	4	no	—
37	#499	—	2	2	no	—
38	#499	Q₈	1	4	yes	complemented
39	#446	—	4	4	no	—
40	C₂×#108	—	2	1	no	—
41	C₂×#108	C₂³	1	2	yes	—
42	#427	—	2	1	no	—
43	#427	C₂³	1	2	yes	—
44	C₂×#107	—	2	1	no	—
45	C₂×#107	C₂³	1	2	yes	—
46	C₂×#106	—	2	1	no	—
47	C₂×#106	C₂³	1	2	yes	—
48	#424	—	2	2	no	—
49	#423	—	2	1	no	—
50	#405	—	2	2	no	—
51	C₂×#97	D₄	1	1	yes	characteristic
52	C₂×#97	D₄	1	1	yes	characteristic
53	C₂×#97	C₂³	1	1	yes	characteristic
54	C₂×#97	C₂³	1	1	yes	characteristic
55	C₂×C₄×C₈	D₄	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
57	#389	—	2	2	no	—
58	#387	—	4	2	no	—
59	#382	—	2	1	no	—
60	C₄×C₄⋊C₄	—	2	2	no	—
61	C₄×C₄⋊C₄	—	4	2	no	—
62	C₄³	D₄	1	1	yes	characteristic
63	#367	—	2	4	no	—
64	#366	—	2	4	no	—
65	#364	—	2	4	no	—
66	#340	—	2	2	no	—
67	#340	—	2	2	no	—
68	#339	—	2	2	no	—
69	#339	—	2	2	no	—
70	C₂×M₁₆	—	2	2	no	—
71	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₈	C₂⁴	1	1	yes	characteristic
75	C₄⋊Q₈	—	2	2	no	—
76	#125	—	2	2	no	—
77	#125	—	2	4	no	—
78	#125	—	4	2	no	—
79	C₂²⋊Q₈	—	4	2	no	—
80	C₄×Q₈	—	4	2	no	—
81	#117	—	2	2	no	—
82	C₂×C₄⋊C₄	—	2	2	no	—
83	C₂×C₄⋊C₄	—	2	2	no	—
84	C₂×C₄⋊C₄	—	4	2	no	—
85	C₂×C₄⋊C₄	—	4	2	no	—
86	C₂×C₄²	—	2	2	no	—
87	C₂×C₄²	—	4	2	no	—
88	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
91	M₃₂	—	2	4	no	—
92	M₃₂	—	2	4	no	—
93	C₂×C₁₆	—	2	4	no	—
94	#108	—	2	4	no	—
95	#108	—	4	2	no	—
96	#107	—	2	2	no	—
97	#107	—	2	4	no	—
98	#107	—	2	4	no	—
99	#107	#41	1	4	yes	—
100	#106	—	2	2	no	—
101	#106	—	2	2	no	—
102	#106	—	2	2	no	—
103	#106	—	4	2	no	—
104	#106	C₂×Q₈	1	4	yes	—
105	C₄⋊C₈	—	4	2	no	—
106	C₄⋊C₈	—	4	4	no	—
107	#97	—	2	2	no	—
108	#97	—	2	2	no	—
109	#97	—	2	2	no	—
110	#97	—	4	2	no	—
111	#97	Q₁₆	1	4	yes	—
112	C₄×C₈	—	4	1	no	—
113	C₄×C₈	—	4	2	no	—
114	C₄×C₈	C₂×Q₈	1	2	yes	—
115	C₄×C₈	#41	1	2	yes	—
116	#95	—	4	2	no	—
117	#95	—	4	2	no	—
118	#95	—	4	2	no	—
119	C₂×Q₈	—	4	2	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
125	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
126	M₁₆	—	4	2	no	—
127	M₁₆	—	4	2	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₈	C₂×Q₁₆	1	2	yes	—
136	C₂×C₈	C₂×Q₁₆	1	2	yes	—
137	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₈	C₂×#41	1	1	yes	characteristic
140	C₂×C₈	C₂×#41	1	2	yes	—
141	C₂×C₈	2₋¹⁺⁴	1	2	yes	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	8	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	8	4	no	—
151	C₂²⋊C₄	—	4	2	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	4	2	no	—
157	C₄²	—	8	2	no	—
158	C₄²	—	8	4	no	—
159	C₄²	C₂²⋊Q₈	1	2	yes	—
160	C₄²	C₂×Q₁₆	1	2	yes	—
161	C₁₆	—	4	4	no	—
162	C₂³	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
163	Q₈	—	16	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	8	2	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
178	C₂×C₄	#525	1	1	yes	characteristic
179	C₂×C₄	#536	1	1	yes	characteristic
180	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
181	C₂×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂×C₄	#575	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	4	1	no	—
185	C₈	—	8	2	no	—
186	C₈	#477	1	4	yes	—
187	C₈	#556	1	2	yes	—
188	C₈	#557	1	2	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	#3077	1	1	yes	characteristic
191	C₂²	#3109	1	1	yes	characteristic
192	C₂²	#3230	1	1	yes	characteristic
193	C₄	—	4	2	no	—
194	C₄	—	4	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	8	4	no	—
197	C₄	—	8	4	no	—
198	C₄	—	16	2	no	—
199	C₄	#2581	1	1	yes	characteristic
200	C₄	C₂×#477	1	1	yes	characteristic
201	C₄	#3104	1	1	yes	characteristic
202	C₄	#3235	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	#29277	1	1	yes	characteristic
206	C₂	#33690	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
38. #499	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#21736	2
#21728	2
#13541	2
#13454	2
#13273	2
#13180	2
#13119	2
#12882	2