Finite Group Databasebeta

Subgroups — #326494

Order 512.

437 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 138 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	#33257	C₂	1	1	yes	maximal, characteristic, complemented
2	#21658	C₂	1	1	yes	maximal, characteristic
3	#10368	C₂	1	2	yes	maximal
4	#10367	C₂	1	2	yes	maximal, complemented
5	#9757	C₂	1	2	yes	maximal
6	#8791	C₂	1	2	yes	maximal, complemented
7	#8677	C₂	1	2	yes	maximal, complemented
8	#8676	C₂	1	2	yes	maximal
9	#8266	C₂	1	1	yes	maximal, characteristic, complemented
10	#3131	C₂²	1	1	yes	characteristic
11	#2960	C₂²	1	1	yes	characteristic
12	#2931	C₂²	1	1	yes	characteristic
13	#2864	C₂²	1	2	yes	—
14	#1994	C₂²	1	1	yes	characteristic
15	#1962	C₂²	1	1	yes	characteristic
16	#1942	—	2	2	no	—
17	#1941	—	2	2	no	—
18	#1937	—	2	2	no	—
19	#1846	C₂²	1	2	yes	—
20	#1837	C₂²	1	1	yes	characteristic
21	#1835	C₂²	1	2	yes	—
22	#1764	—	2	2	no	—
23	C₄²×C₈	C₂²	1	1	yes	characteristic
24	#1564	C₂²	1	2	yes	—
25	#1518	C₂²	1	4	yes	—
26	#1517	C₂²	1	4	yes	—
27	#1450	C₂²	1	2	yes	—
28	#1286	C₂²	1	4	yes	—
29	#1286	C₂²	1	4	yes	—
30	#1276	C₂²	1	2	yes	—
31	C₂×#131	—	2	1	no	—
32	#533	C₂³	1	1	yes	characteristic
33	#532	C₂³	1	1	yes	characteristic
34	#488	—	2	1	no	—
35	#488	—	2	1	no	—
36	#487	Q₈	1	2	yes	—
37	#486	Q₈	1	2	yes	—
38	#446	—	2	2	no	—
39	C₈×Q₈	—	2	2	no	—
40	#436	—	2	2	no	—
41	C₈×D₄	—	2	2	no	—
42	#431	C₂³	1	4	yes	—
43	C₂×#107	—	2	2	no	—
44	C₂×#106	—	2	2	no	—
45	#424	—	2	1	no	—
46	#423	C₂³	1	1	yes	characteristic
47	#423	C₂³	1	1	yes	characteristic
48	C₂×#104	—	2	2	no	—
49	#416	—	2	2	no	—
50	#408	—	2	1	no	—
51	#405	—	2	1	no	—
52	C₂×C₄×C₈	D₄	1	2	yes	—
53	C₂×C₄×C₈	D₄	1	2	yes	—
54	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
56	#389	—	2	4	no	—
57	#387	—	4	4	no	—
58	#382	C₂³	1	4	yes	—
59	C₄³	C₂³	1	1	yes	characteristic
60	#341	—	2	2	no	—
61	#340	—	2	4	no	—
62	#339	—	2	2	no	—
63	#335	—	2	2	no	—
64	#334	—	2	2	no	—
65	#328	—	4	4	no	—
66	#327	—	4	4	no	—
67	#322	—	2	2	no	—
68	C₈²	—	2	2	no	—
69	C₂×#41	—	2	1	no	—
70	#131	—	4	4	no	—
71	C₂×M₁₆	—	2	1	no	—
72	C₂×M₁₆	—	2	1	no	—
73	C₂×M₁₆	—	4	1	no	—
74	C₂²×C₈	—	4	1	no	—
75	C₂²×C₈	C₂×D₄	1	2	yes	—
76	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
78	C₄⋊Q₈	C₂×Q₈	1	1	yes	characteristic
79	C₄⋊Q₈	C₂×Q₈	1	2	yes	—
80	#127	C₂×Q₈	1	1	yes	characteristic
81	#125	—	2	1	no	—
82	#124	—	2	1	no	—
83	C₂²⋊Q₈	—	2	2	no	—
84	#121	—	2	2	no	—
85	C₄×Q₈	—	2	2	no	—
86	C₄×D₄	—	2	2	no	—
87	#117	—	2	1	no	—
88	C₂×C₄⋊C₄	—	2	4	no	—
89	C₂×C₄⋊C₄	—	4	4	no	—
90	C₂×C₄²	—	2	2	no	—
91	C₂×C₄²	C₂×D₄	1	2	yes	—
92	C₂×C₄²	C₂⁴	1	1	yes	characteristic
93	#107	—	4	4	no	—
94	#106	—	4	4	no	—
95	C₄⋊C₈	—	2	4	no	—
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	4	2	no	—
99	#104	—	4	4	no	—
100	#103	—	2	2	no	—
101	#103	—	2	2	no	—
102	#102	—	2	2	no	—
103	#102	—	2	2	no	—
104	C₂²⋊C₈	—	2	4	no	—
105	C₂²⋊C₈	—	4	2	no	—
106	#97	—	4	1	no	—
107	C₄×C₈	—	2	4	no	—
108	C₄×C₈	—	2	4	no	—
109	C₄×C₈	—	4	2	no	—
110	C₄×C₈	—	4	1	no	—
111	C₄×C₈	C₂×Q₈	1	2	yes	—
112	C₄×C₈	#41	1	1	yes	characteristic
113	C₄×C₈	#41	1	1	yes	characteristic
114	C₄×C₈	#41	1	2	yes	—
115	C₄×C₈	#41	1	2	yes	—
116	#95	—	2	4	no	—
117	#95	—	2	4	no	—
118	#95	—	4	4	no	—
119	#41	—	4	2	no	—
120	#41	—	4	2	no	—
121	C₂×Q₈	—	2	1	no	—
122	C₂×Q₈	—	2	2	no	—
123	C₂×D₄	—	2	1	no	—
124	C₂×D₄	—	2	2	no	—
125	C₂×D₄	—	4	1	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	4	1	no	—
128	C₂²×C₄	—	4	4	no	—
129	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
130	C₂²×C₄	H(5,𝔽₂)	1	2	yes	—
131	M₁₆	—	4	2	no	—
132	M₁₆	—	4	2	no	—
133	M₁₆	—	8	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	4	4	no	—
140	C₂×C₈	—	8	2	no	—
141	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
142	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₈	#124	1	2	yes	—
144	C₂×C₈	#124	1	1	yes	characteristic
145	C₂×C₈	#124	1	1	yes	characteristic
146	C₂×C₈	#135	1	4	yes	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₄⋊C₄	—	4	4	no	—
153	C₄⋊C₄	—	8	8	no	—
154	C₂²⋊C₄	—	4	1	no	—
155	C₂²⋊C₄	—	4	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	4	4	no	—
159	C₄²	—	4	1	no	—
160	C₄²	C₂²×Q₈	1	1	yes	characteristic
161	C₄²	C₂×#41	1	1	yes	characteristic
162	C₄²	C₂×#41	1	2	yes	—
163	C₂³	—	4	1	no	—
164	C₂³	#535	1	1	yes	characteristic
165	Q₈	—	4	2	no	—
166	Q₈	—	8	2	no	—
167	D₄	—	4	2	no	—
168	D₄	—	8	2	no	—
169	D₄	—	8	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	8	1	no	—
177	C₂×C₄	—	8	1	no	—
178	C₂×C₄	—	8	8	no	—
179	C₂×C₄	—	8	4	no	—
180	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
181	C₂×C₄	C₂×#124	1	1	yes	characteristic
182	C₂×C₄	#554	1	2	yes	—
183	C₂×C₄	#555	1	2	yes	—
184	C₂×C₄	C₂×#135	1	2	yes	—
185	C₂×C₄	#575	1	2	yes	—
186	C₈	—	2	2	no	—
187	C₈	—	2	2	no	—
188	C₈	—	2	2	no	—
189	C₈	—	8	4	no	—
190	C₈	—	8	1	no	—
191	C₈	—	8	1	no	—
192	C₈	#488	1	2	yes	—
193	C₈	#488	1	2	yes	—
194	C₂²	—	2	2	no	—
195	C₂²	—	8	1	no	—
196	C₂²	—	8	2	no	—
197	C₂²	#2563	1	1	yes	characteristic
198	C₂²	#3212	1	2	yes	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	4	4	no	—
202	C₄	—	8	1	no	—
203	C₄	—	8	2	no	—
204	C₄	—	16	4	no	—
205	C₄	#3076	1	2	yes	—
206	C₄	C₂×#488	1	1	yes	characteristic
207	C₄	#3142	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	—	16	1	no	—
210	C₂	#29140	1	1	yes	characteristic
211	C₂	#29154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#2563
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #33257	C₂	semidirect
4. #10367	C₂	semidirect
6. #8791	C₂	semidirect
7. #8677	C₂	semidirect
9. #8266	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33257	2
#21658	2
#10368	2
#10367	2
#9757	2
#8791	2
#8677	2
#8676	2
#8266	2