Finite Group Databasebeta

Subgroups — #513137

Order 512.

295 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 116 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	#33734	C₂	1	1	yes	maximal, characteristic
2	#33726	C₂	1	1	yes	maximal, characteristic
3	#13734	C₂	1	2	yes	maximal, complemented
4	#13724	C₂	1	2	yes	maximal, complemented
5	#13716	C₂	1	4	yes	maximal
6	#13706	C₂	1	2	yes	maximal, complemented
7	#13704	C₂	1	2	yes	maximal, complemented
8	#13640	C₂	1	1	yes	maximal, characteristic
9	#3170	—	2	1	no	—
10	#3160	—	2	1	no	—
11	#2254	—	2	1	no	—
12	#2253	—	2	1	no	—
13	#2253	C₂²	1	2	yes	—
14	#2252	C₂²	1	1	yes	characteristic
15	#2252	C₂²	1	1	yes	characteristic
16	#2252	C₂²	1	1	yes	characteristic
17	#2247	C₂²	1	1	yes	characteristic
18	#2241	—	2	1	no	—
19	#2216	—	2	1	no	—
20	#2216	C₂²	1	2	yes	—
21	C₄×Q₃₂	—	2	1	no	—
22	C₄×Q₃₂	C₂²	1	2	yes	—
23	#2158	—	2	1	no	—
24	#2158	C₂²	1	2	yes	—
25	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
26	#1424	C₂²	1	4	yes	—
27	#1423	C₂²	1	4	yes	—
28	C₄⋊C₃₂	C₂²	1	2	yes	—
29	#1416	C₂²	1	4	yes	—
30	#1416	C₂²	1	4	yes	—
31	C₂²⋊C₃₂	C₂²	1	2	yes	—
32	#1397	C₂²	1	2	yes	—
33	#533	—	4	1	no	—
34	#533	—	4	1	no	—
35	C₂×Q₃₂	—	2	1	no	—
36	C₂×Q₃₂	C₂³	1	2	yes	—
37	C₂²×C₁₆	D₄	1	1	yes	characteristic
38	C₂²×C₁₆	C₂³	1	1	yes	characteristic
39	#500	—	2	1	no	—
40	#500	—	2	1	no	—
41	#500	—	2	1	no	—
42	#499	—	2	2	no	—
43	#499	—	4	1	no	—
44	#498	—	4	1	no	—
45	#494	—	2	1	no	—
46	#487	—	4	1	no	—
47	#467	—	2	2	no	—
48	#467	—	4	1	no	—
49	C₄×Q₁₆	—	2	2	no	—
50	C₄×Q₁₆	—	4	1	no	—
51	#427	—	2	2	no	—
52	#427	—	4	1	no	—
53	C₂×C₄×C₈	D₄	1	1	yes	characteristic
54	C₂×C₃₂	C₂³	1	4	yes	—
55	#367	—	2	2	no	—
56	#367	—	2	2	no	—
57	#366	—	2	1	no	—
58	#366	—	2	1	no	—
59	#366	—	2	1	no	—
60	#366	C₂³	1	2	yes	—
61	#366	C₂³	1	2	yes	—
62	#366	C₂³	1	2	yes	—
63	#358	—	2	2	no	—
64	#358	—	2	2	no	—
65	C₄×C₁₆	D₄	1	1	yes	characteristic
66	C₄×C₁₆	D₄	1	1	yes	characteristic
67	C₄×C₁₆	C₂³	1	1	yes	characteristic
68	C₄×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×Q₁₆	—	2	2	no	—
70	C₂×Q₁₆	—	4	1	no	—
71	C₂²×C₈	D₈	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
73	C₄⋊Q₈	—	4	1	no	—
74	C₄⋊Q₈	—	4	1	no	—
75	C₄⋊Q₈	—	4	1	no	—
76	C₄⋊Q₈	—	4	1	no	—
77	#125	—	4	2	no	—
78	#125	—	8	1	no	—
79	C₂²⋊Q₈	—	4	2	no	—
80	C₂²⋊Q₈	—	8	1	no	—
81	C₂²⋊Q₈	—	8	1	no	—
82	C₂²⋊Q₈	—	8	1	no	—
83	C₄×Q₈	—	4	2	no	—
84	C₄×Q₈	—	8	1	no	—
85	C₄×Q₈	—	8	1	no	—
86	C₄×Q₈	—	8	1	no	—
87	#117	—	4	2	no	—
88	#117	—	8	1	no	—
89	C₂×C₄²	D₈	1	1	yes	characteristic
90	Q₃₂	—	2	4	no	—
91	Q₃₂	—	4	2	no	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
95	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
96	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
97	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
98	#107	—	2	2	no	—
99	#107	—	2	2	no	—
100	#107	—	2	2	no	—
101	#107	—	4	1	no	—
102	#107	—	4	1	no	—
103	#107	—	4	1	no	—
104	#106	—	4	2	no	—
105	#106	—	4	2	no	—
106	#103	—	4	2	no	—
107	#103	—	4	2	no	—
108	C₄×C₈	D₈	1	1	yes	characteristic
109	C₄×C₈	D₈	1	1	yes	characteristic
110	C₄×C₈	C₂×D₄	1	1	yes	characteristic
111	C₄×C₈	C₂×D₄	1	1	yes	characteristic
112	C₃₂	—	2	4	no	—
113	C₂×Q₈	—	4	2	no	—
114	C₂×Q₈	—	8	1	no	—
115	C₂×Q₈	—	8	1	no	—
116	C₂×Q₈	—	8	1	no	—
117	C₂²×C₄	D₁₆	1	1	yes	characteristic
118	C₂²×C₄	D₁₆	1	1	yes	characteristic
119	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
120	Q₁₆	—	4	2	no	—
121	Q₁₆	—	4	2	no	—
122	Q₁₆	—	8	2	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	C₂×D₈	1	1	yes	characteristic
127	C₂×C₈	C₂×D₈	1	1	yes	characteristic
128	C₂×C₈	C₂×D₈	1	1	yes	characteristic
129	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	8	1	no	—
134	C₄⋊C₄	—	8	1	no	—
135	C₄⋊C₄	—	8	2	no	—
136	C₄⋊C₄	—	8	2	no	—
137	C₄⋊C₄	—	8	2	no	—
138	C₄⋊C₄	—	8	2	no	—
139	C₄⋊C₄	—	8	1	no	—
140	C₄⋊C₄	—	8	1	no	—
141	C₄⋊C₄	—	8	1	no	—
142	C₂²⋊C₄	—	8	2	no	—
143	C₂²⋊C₄	—	8	2	no	—
144	C₄²	—	8	2	no	—
145	C₄²	—	8	2	no	—
146	C₄²	D₁₆	1	1	yes	characteristic
147	C₄²	D₁₆	1	1	yes	characteristic
148	C₄²	C₂×D₈	1	1	yes	characteristic
149	C₄²	C₂×D₈	1	1	yes	characteristic
150	C₁₆	—	2	1	no	—
151	C₁₆	—	2	1	no	—
152	C₁₆	#137	1	2	yes	—
153	C₁₆	2₋¹⁺⁴	1	2	yes	—
154	C₂³	C₂×D₁₆	1	1	yes	characteristic
155	Q₈	—	8	2	no	—
156	Q₈	—	8	2	no	—
157	Q₈	—	16	2	no	—
158	Q₈	—	16	2	no	—
159	Q₈	—	16	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
169	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
170	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
171	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
172	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
173	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	#510	1	2	yes	—
177	C₈	#536	1	1	yes	characteristic
178	C₈	C₂×#137	1	1	yes	characteristic
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₂²×D₁₆	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	16	2	no	—
188	C₄	—	16	2	no	—
189	C₄	—	16	2	no	—
190	C₄	—	16	2	no	—
191	C₄	#2264	1	2	yes	—
192	C₄	#3226	1	1	yes	characteristic
193	C₄	C₂×#510	1	1	yes	characteristic
194	C₂	—	4	1	no	—
195	C₂	#33792	1	1	yes	characteristic
196	C₂	C₂×#2264	1	1	yes	characteristic
197	C₂	#33981	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂³	C₂×D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ 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
3. #13734	C₂	semidirect
4. #13724	C₂	semidirect
6. #13706	C₂	semidirect
7. #13704	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33734	2
#33726	2
#13734	2
#13724	2
#13716	2
#13706	2
#13704	2
#13640	2