Finite Group Databasebeta

Subgroups — #326508

Order 512.

449 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 134 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	#33298	C₂	1	1	yes	maximal, characteristic, complemented
2	#21660	C₂	1	1	yes	maximal, characteristic
3	#10363	C₂	1	2	yes	maximal, complemented
4	#10362	C₂	1	2	yes	maximal
5	#9757	C₂	1	2	yes	maximal
6	#8791	C₂	1	2	yes	maximal, complemented
7	#8681	C₂	1	2	yes	maximal, complemented
8	#8678	C₂	1	2	yes	maximal
9	#8268	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	#2862	C₂²	1	2	yes	—
14	#1994	C₂²	1	1	yes	characteristic
15	#1993	—	2	2	no	—
16	#1993	—	2	2	no	—
17	#1962	C₂²	1	1	yes	characteristic
18	#1938	—	2	2	no	—
19	#1837	C₂²	1	1	yes	characteristic
20	#1835	C₂²	1	2	yes	—
21	#1835	C₂²	1	2	yes	—
22	#1765	—	2	2	no	—
23	C₄×#97	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	C₂×C₄⋊Q₈	—	2	2	no	—
35	#488	—	2	2	no	—
36	#487	Q₈	1	2	yes	complemented
37	#486	Q₈	1	2	yes	complemented
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	#424	—	2	1	no	—
44	#423	C₂³	1	1	yes	characteristic
45	#423	C₂³	1	1	yes	characteristic
46	C₂×#104	—	2	2	no	—
47	#416	—	2	2	no	—
48	#408	—	2	1	no	—
49	#405	—	2	1	no	—
50	C₂×#97	D₄	1	2	yes	—
51	C₂×#97	D₄	1	2	yes	—
52	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	#391	—	4	4	no	—
55	#382	—	2	2	no	—
56	#382	C₂³	1	4	yes	—
57	C₄³	C₂³	1	1	yes	characteristic
58	#341	—	2	2	no	—
59	#340	—	2	4	no	—
60	#340	—	2	2	no	—
61	#340	—	2	2	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	C₂²×Q₈	—	4	2	no	—
71	#131	—	4	4	no	—
72	C₂×M₁₆	—	2	1	no	—
73	C₂×M₁₆	—	2	1	no	—
74	C₂×M₁₆	—	4	1	no	—
75	C₂²×C₈	—	4	1	no	—
76	C₂²×C₈	C₂×D₄	1	2	yes	—
77	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
79	C₄⋊Q₈	—	4	4	no	—
80	C₄⋊Q₈	—	4	4	no	—
81	C₄⋊Q₈	C₂×Q₈	1	1	yes	characteristic
82	C₄⋊Q₈	C₂×Q₈	1	2	yes	—
83	#127	C₂×Q₈	1	1	yes	characteristic
84	#125	—	2	1	no	—
85	#124	—	2	1	no	—
86	C₂²⋊Q₈	—	2	2	no	—
87	#121	—	2	2	no	—
88	C₄×Q₈	—	2	2	no	—
89	C₄×D₄	—	2	2	no	—
90	#117	—	2	1	no	—
91	C₂×C₄⋊C₄	—	2	4	no	—
92	C₂×C₄⋊C₄	—	4	2	no	—
93	C₂×C₄²	—	2	2	no	—
94	C₂×C₄²	C₂×D₄	1	2	yes	—
95	C₂×C₄²	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₈	—	2	4	no	—
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	—	4	2	no	—
100	#104	—	4	4	no	—
101	#103	—	2	4	no	—
102	#102	—	2	4	no	—
103	C₂²⋊C₈	—	2	4	no	—
104	C₂²⋊C₈	—	4	2	no	—
105	#97	—	2	4	no	—
106	#97	—	4	2	no	—
107	#97	—	4	1	no	—
108	C₄×C₈	—	2	4	no	—
109	C₄×C₈	—	4	1	no	—
110	C₄×C₈	C₂×Q₈	1	2	yes	—
111	C₄×C₈	#41	1	2	yes	—
112	C₄×C₈	#41	1	2	yes	—
113	C₄×C₈	#41	1	2	yes	—
114	#95	—	2	4	no	—
115	#95	—	2	4	no	—
116	#95	—	4	4	no	—
117	#41	—	4	2	no	—
118	#41	—	4	2	no	—
119	C₂×Q₈	—	2	1	no	—
120	C₂×Q₈	—	2	2	no	—
121	C₂×Q₈	—	8	4	no	—
122	C₂×Q₈	—	8	8	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	—	2	2	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	4	4	no	—
141	C₂×C₈	—	8	2	no	—
142	C₂×C₈	C₂²⋊Q₈	1	4	yes	—
143	C₂×C₈	#124	1	1	yes	characteristic
144	C₂×C₈	#124	1	1	yes	characteristic
145	C₂×C₈	#124	1	2	yes	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₄⋊C₄	—	8	4	no	—
153	C₂²⋊C₄	—	4	1	no	—
154	C₂²⋊C₄	—	4	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	4	1	no	—
157	C₄²	—	8	2	no	—
158	C₄²	#135	1	4	yes	—
159	C₄²	C₂²×Q₈	1	1	yes	characteristic
160	C₄²	C₂×#41	1	1	yes	characteristic
161	C₄²	C₂×#41	1	2	yes	—
162	C₂³	—	4	1	no	—
163	C₂³	#535	1	1	yes	characteristic
164	Q₈	—	4	2	no	—
165	Q₈	—	8	2	no	—
166	Q₈	—	16	8	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	4	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	—	8	1	no	—
177	C₂×C₄	—	8	8	no	—
178	C₂×C₄	—	8	4	no	—
179	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
180	C₂×C₄	C₂×#124	1	1	yes	characteristic
181	C₂×C₄	#554	1	2	yes	—
182	C₂×C₄	#555	1	2	yes	—
183	C₂×C₄	C₂×#135	1	2	yes	—
184	C₂×C₄	#575	1	2	yes	—
185	C₈	—	2	4	no	—
186	C₈	—	2	2	no	—
187	C₈	—	2	1	no	—
188	C₈	—	2	1	no	—
189	C₈	—	8	4	no	—
190	C₈	—	8	1	no	—
191	C₈	—	8	1	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	8	1	no	—
194	C₂²	—	8	2	no	—
195	C₂²	#2563	1	1	yes	characteristic
196	C₂²	#3211	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	4	4	no	—
200	C₄	—	8	1	no	—
201	C₄	—	8	2	no	—
202	C₄	—	16	4	no	—
203	C₄	#3076	1	2	yes	—
204	C₄	#3131	1	1	yes	characteristic
205	C₄	#3140	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	—	16	1	no	—
208	C₂	#29142	1	1	yes	characteristic
209	C₂	#29155	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. #33298	C₂	semidirect
3. #10363	C₂	semidirect
6. #8791	C₂	semidirect
7. #8681	C₂	semidirect
9. #8268	C₂	semidirect
36. #487	Q₈	semidirect
37. #486	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33298	2
#21660	2
#10363	2
#10362	2
#9757	2
#8791	2
#8681	2
#8678	2
#8268	2