Finite Group Databasebeta

Subgroups — #140271

Order 512.

244 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 84 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	#13279	C₂	1	1	yes	maximal, characteristic
2	#13183	C₂	1	1	yes	maximal, characteristic
3	#11940	C₂	1	1	yes	maximal, characteristic
4	#9836	C₂	1	1	yes	maximal, characteristic
5	#8295	C₂	1	1	yes	maximal, characteristic
6	#7278	C₂	1	1	yes	maximal, characteristic
7	#7152	C₂	1	1	yes	maximal, characteristic
8	C₂×#500	C₂²	1	1	yes	characteristic
9	C₂×#365	—	2	1	no	—
10	#2150	C₂²	1	1	yes	characteristic
11	#2150	C₂²	1	1	yes	characteristic
12	#1990	—	2	1	no	—
13	#1956	—	2	1	no	—
14	#1844	C₂²	1	1	yes	characteristic
15	C₄×#108	C₂²	1	1	yes	characteristic
16	#1564	C₂²	1	1	yes	characteristic
17	#1451	—	2	1	no	—
18	C₂×#322	C₂²	1	1	yes	characteristic
19	#1387	—	2	1	no	—
20	#1387	—	2	1	no	—
21	#1383	—	2	1	no	—
22	#1364	C₄	1	4	yes	complemented
23	#1278	—	2	1	no	—
24	C₂×C₄⋊Q₈	—	2	1	no	—
25	C₂×M₃₂	—	2	1	no	—
26	C₂×M₃₂	—	2	1	no	—
27	#500	—	2	1	no	—
28	#500	—	2	1	no	—
29	#500	—	2	1	no	—
30	#500	C₂×C₄	1	2	yes	—
31	#431	—	2	1	no	—
32	C₂×#108	D₄	1	1	yes	characteristic
33	C₂×#108	D₄	1	1	yes	characteristic
34	C₂×#107	—	2	1	no	—
35	C₂×#107	—	2	1	no	—
36	C₂×#106	—	2	1	no	—
37	C₂×#106	—	2	1	no	—
38	C₂×#103	—	4	2	no	—
39	C₄×M₁₆	—	2	1	no	—
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	#384	—	2	1	no	—
44	#365	—	4	2	no	—
45	C₄⋊C₁₆	—	2	1	no	—
46	C₄⋊C₁₆	—	2	1	no	—
47	C₄⋊C₁₆	—	2	1	no	—
48	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
49	#341	—	2	1	no	—
50	#340	—	4	2	no	—
51	#335	C₂×C₄	1	2	yes	—
52	#334	—	2	1	no	—
53	#322	—	2	1	no	—
54	#322	—	2	1	no	—
55	#322	—	4	2	no	—
56	C₂²×Q₈	—	4	1	no	—
57	C₂×M₁₆	—	2	1	no	—
58	C₂×M₁₆	—	2	1	no	—
59	C₂²×C₈	—	2	1	no	—
60	C₂²×C₈	SD₁₆	1	2	yes	—
61	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₈	#41	1	1	yes	characteristic
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	4	1	no	—
65	C₄⋊Q₈	—	4	1	no	—
66	C₄⋊Q₈	—	4	1	no	—
67	C₂×C₄⋊C₄	—	4	1	no	—
68	C₂×C₄⋊C₄	—	4	1	no	—
69	C₂×C₄⋊C₄	—	4	1	no	—
70	C₂×C₄⋊C₄	—	4	1	no	—
71	C₂×C₄⋊C₄	—	4	1	no	—
72	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
73	M₃₂	—	4	1	no	—
74	M₃₂	—	4	1	no	—
75	M₃₂	—	4	1	no	—
76	M₃₂	—	4	1	no	—
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	—	4	1	no	—
79	#108	—	2	1	no	—
80	#108	—	2	1	no	—
81	#108	—	2	1	no	—
82	#108	—	2	1	no	—
83	#107	—	2	2	no	—
84	#107	—	2	2	no	—
85	#107	—	4	1	no	—
86	#107	—	4	1	no	—
87	#106	—	4	1	no	—
88	#106	—	4	1	no	—
89	#106	—	8	1	no	—
90	C₄⋊C₈	—	2	2	no	—
91	#103	—	8	4	no	—
92	C₂²⋊C₈	—	4	1	no	—
93	#97	—	4	1	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	4	1	no	—
99	C₄×C₈	—	4	1	no	—
100	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₄×C₈	#41	1	1	yes	characteristic
102	C₄×C₈	#41	1	1	yes	characteristic
103	C₄×C₈	#41	1	1	yes	characteristic
104	C₂×Q₈	—	4	2	no	—
105	C₂×Q₈	—	8	2	no	—
106	C₂×Q₈	—	8	2	no	—
107	C₂×Q₈	—	8	1	no	—
108	C₂²×C₄	—	8	1	no	—
109	C₂²×C₄	—	8	1	no	—
110	C₂²×C₄	#121	1	1	yes	characteristic
111	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
112	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
113	M₁₆	—	4	1	no	—
114	M₁₆	—	4	1	no	—
115	M₁₆	—	4	1	no	—
116	M₁₆	—	4	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	4	1	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	#117	1	1	yes	characteristic
127	C₂×C₈	C₄×D₄	1	1	yes	characteristic
128	C₂×C₈	#124	1	1	yes	characteristic
129	C₂×C₈	#126	1	1	yes	characteristic
130	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₈	#137	1	1	yes	characteristic
132	C₂×C₈	#137	1	1	yes	characteristic
133	C₂×C₈	#137	1	1	yes	characteristic
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	8	2	no	—
137	C₄⋊C₄	—	8	2	no	—
138	C₄⋊C₄	—	8	1	no	—
139	C₄⋊C₄	—	8	1	no	—
140	C₄⋊C₄	—	8	1	no	—
141	C₄⋊C₄	—	8	1	no	—
142	C₄²	C₄×D₄	1	1	yes	characteristic
143	C₄²	#123	1	1	yes	characteristic
144	C₄²	#135	1	1	yes	characteristic
145	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₁₆	—	4	2	no	—
147	C₁₆	—	8	1	no	—
148	C₂³	#460	1	1	yes	characteristic
149	Q₈	—	8	4	no	—
150	Q₈	—	16	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	16	1	no	—
155	C₂×C₄	—	16	1	no	—
156	C₂×C₄	—	16	1	no	—
157	C₂×C₄	—	16	1	no	—
158	C₂×C₄	—	16	1	no	—
159	C₂×C₄	#388	1	1	yes	characteristic
160	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
161	C₂×C₄	#442	1	1	yes	characteristic
162	C₂×C₄	#463	1	1	yes	characteristic
163	C₂×C₄	#468	1	1	yes	characteristic
164	C₂×C₄	#470	1	1	yes	characteristic
165	C₂×C₄	#471	1	1	yes	characteristic
166	C₂×C₄	#473	1	1	yes	characteristic
167	C₂×C₄	#481	1	1	yes	characteristic
168	C₂×C₄	#482	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	4	2	no	—
172	C₈	—	8	1	no	—
173	C₈	—	8	2	no	—
174	C₈	—	8	1	no	—
175	C₈	—	8	1	no	—
176	C₈	#440	1	1	yes	characteristic
177	C₈	#441	1	1	yes	characteristic
178	C₈	#489	1	1	yes	characteristic
179	C₈	#491	1	1	yes	characteristic
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	#1686	1	1	yes	characteristic
183	C₂²	#1690	1	1	yes	characteristic
184	C₂²	#1926	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	16	2	no	—
188	C₄	—	32	1	no	—
189	C₄	#1699	1	1	yes	characteristic
190	C₄	#1703	1	1	yes	characteristic
191	C₄	#1930	1	1	yes	characteristic
192	C₄	#1935	1	1	yes	characteristic
193	C₄	AGL(1,ℤ/16)	1	2	yes	—
194	C₄	#2239	1	2	yes	—
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	#9570	1	1	yes	characteristic
198	C₂	#13139	1	1	yes	characteristic
199	C₂	#13142	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1926
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#460
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
22. #1364	C₄	semidirect

Maximal subgroups

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

subgroup	index
#13279	2
#13183	2
#11940	2
#9836	2
#8295	2
#7278	2
#7152	2