Finite Group Databasebeta

Subgroups — #513135

Order 512.

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

normal subgroups only

This group has 317 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	#33733	C₂	1	1	yes	maximal, characteristic
2	#33726	C₂	1	1	yes	maximal, characteristic, complemented
3	#13734	C₂	1	2	yes	maximal, complemented
4	#13723	C₂	1	2	yes	maximal, complemented
5	#13713	C₂	1	4	yes	maximal, complemented
6	#13705	C₂	1	2	yes	maximal, complemented
7	#13703	C₂	1	2	yes	maximal, complemented
8	#13640	C₂	1	1	yes	maximal, characteristic, complemented
9	#3169	—	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	#2246	C₂²	1	1	yes	characteristic
18	#2240	—	2	1	no	—
19	#2215	—	2	1	no	—
20	#2215	C₂²	1	2	yes	—
21	C₄×D₁₆	—	2	1	no	—
22	C₄×D₁₆	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	#1415	C₂²	1	4	yes	—
30	#1415	C₂²	1	4	yes	—
31	C₂²⋊C₃₂	C₂²	1	2	yes	—
32	#1397	C₂²	1	2	yes	—
33	#533	—	4	1	no	—
34	#532	—	4	1	no	—
35	C₂×D₁₆	—	2	1	no	—
36	C₂×D₁₆	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	#493	—	2	1	no	—
46	#486	—	4	1	no	—
47	#466	—	2	2	no	—
48	#466	—	4	1	no	—
49	C₄×D₈	—	2	2	no	—
50	C₄×D₈	—	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	#357	—	2	2	no	—
64	#357	—	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×#41	—	8	1	no	—
70	C₂×D₈	—	2	2	no	—
71	C₂×D₈	—	4	1	no	—
72	C₂²×C₈	D₈	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
74	C₄⋊Q₈	—	4	1	no	—
75	C₄⋊Q₈	—	4	1	no	—
76	C₄⋊Q₈	—	4	1	no	—
77	#127	—	4	1	no	—
78	#125	—	4	2	no	—
79	#124	—	8	1	no	—
80	C₂²⋊Q₈	—	8	1	no	—
81	C₂²⋊Q₈	—	8	1	no	—
82	#121	—	4	2	no	—
83	#121	—	8	1	no	—
84	C₄×Q₈	—	8	1	no	—
85	C₄×Q₈	—	8	1	no	—
86	C₄×D₄	—	4	2	no	—
87	C₄×D₄	—	8	1	no	—
88	#117	—	4	2	no	—
89	C₂×C₄²	D₈	1	1	yes	characteristic
90	D₁₆	—	2	4	no	—
91	D₁₆	—	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	#102	—	4	2	no	—
107	#102	—	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	#41	—	16	4	no	—
114	C₂×Q₈	—	8	1	no	—
115	C₂×Q₈	—	8	1	no	—
116	C₂×Q₈	—	8	1	no	—
117	C₂×D₄	—	4	2	no	—
118	C₂×D₄	—	8	2	no	—
119	C₂×D₄	—	8	1	no	—
120	C₂×D₄	—	8	1	no	—
121	C₂²×C₄	—	8	2	no	—
122	C₂²×C₄	D₁₆	1	1	yes	characteristic
123	C₂²×C₄	D₁₆	1	1	yes	characteristic
124	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
125	D₈	—	4	2	no	—
126	D₈	—	4	2	no	—
127	D₈	—	8	2	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	C₂×D₈	1	1	yes	characteristic
132	C₂×C₈	C₂×D₈	1	1	yes	characteristic
133	C₂×C₈	C₂×D₈	1	1	yes	characteristic
134	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	8	1	no	—
139	C₄⋊C₄	—	8	2	no	—
140	C₄⋊C₄	—	8	2	no	—
141	C₄⋊C₄	—	8	1	no	—
142	C₄⋊C₄	—	8	1	no	—
143	C₂²⋊C₄	—	8	2	no	—
144	C₂²⋊C₄	—	8	2	no	—
145	C₂²⋊C₄	—	8	2	no	—
146	C₄²	—	8	2	no	—
147	C₄²	D₁₆	1	1	yes	characteristic
148	C₄²	D₁₆	1	1	yes	characteristic
149	C₄²	C₂×D₈	1	1	yes	characteristic
150	C₄²	C₂×D₈	1	1	yes	characteristic
151	C₁₆	—	2	1	no	—
152	C₁₆	—	2	1	no	—
153	C₁₆	#137	1	2	yes	—
154	C₁₆	2₋¹⁺⁴	1	2	yes	—
155	C₂³	—	8	2	no	—
156	C₂³	C₂×D₁₆	1	1	yes	characteristic
157	Q₈	—	16	2	no	—
158	Q₈	—	16	2	no	—
159	Q₈	—	16	2	no	—
160	D₄	—	8	2	no	—
161	D₄	—	8	2	no	—
162	D₄	—	16	2	no	—
163	D₄	—	16	2	no	—
164	D₄	—	16	4	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	8	2	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	16	4	no	—
173	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
174	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
175	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
176	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
177	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
178	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	#510	1	2	yes	—
182	C₈	#536	1	1	yes	characteristic
183	C₈	C₂×#137	1	1	yes	characteristic
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	16	2	no	—
188	C₂²	—	16	2	no	—
189	C₂²	—	16	2	no	—
190	C₂²	C₂²×D₁₆	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	16	2	no	—
196	C₄	—	16	2	no	—
197	C₄	—	16	2	no	—
198	C₄	#2263	1	2	yes	—
199	C₄	#3226	1	1	yes	characteristic
200	C₄	C₂×#510	1	1	yes	characteristic
201	C₂	—	4	1	no	—
202	C₂	—	32	2	no	—
203	C₂	#33792	1	1	yes	characteristic
204	C₂	C₂×#2263	1	1	yes	characteristic
205	C₂	#33981	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#33733	C₂
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
2. #33726	C₂	semidirect
3. #13734	C₂	semidirect
4. #13723	C₂	semidirect
5. #13713	C₂	semidirect
6. #13705	C₂	semidirect
7. #13703	C₂	semidirect
8. #13640	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33733	2
#33726	2
#13734	2
#13723	2
#13713	2
#13705	2
#13703	2
#13640	2