Finite Group Databasebeta

← #303009

Subgroups — #303009

Order 512.

1051 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1051 conjugacy classes of subgroups — too many to draw. Only its 180 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59817C211yesmaximal, characteristic
2#29320C212yesmaximal, complemented
3#29143C211yesmaximal, characteristic
4#10575C214yesmaximal, complemented
5#9868C212yesmaximal, complemented
6#8921C214yesmaximal
7#8231C211yesmaximal, characteristic
8C2×D4×Q8—22no—
9#2859C2211yescharacteristic
10#2683—22no—
11#2551C2212yes—
12#2538—22no—
13#2452—24no—
14#2409—22no—
15#2366C2212yes—
16C4×C4⋊Q8C2212yes—
17C4×#127C2211yescharacteristic
18#2083—22no—
19#2058—22no—
20#2025—22no—
21#1957C2212yes—
22#1948C2214yes—
23#1947C2214yes—
24#1944—24no—
25#1834C2212yes—
26#1804—22no—
27#1767C2212yes—
28C4×C4⋊C8C2211yescharacteristic
29C2×#331C2212yes—
30#1296C2212yes—
31#1279C2218yes—
32C23×Q8—22no—
33D4×Q8—416no—
34C2×C4⋊Q8—24no—
35C2×C4⋊Q8D412yes—
36C2×#127C2311yescharacteristic
37C2×#125—21no—
38C2×C22⋊Q8—24no—
39C2×C22⋊Q8—28no—
40C2×C4×Q8—22no—
41C2×C4×D4—24no—
42C2×C4×D4—22no—
43C2×C4×D4D412yes—
44C2×#107—24no—
45C2×#106—24no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8C2314yes—
48C2×#102—24no—
49C2×#97—22no—
50C2×C4×C8C2312yes—
51#396—28no—
52#395—24no—
53#393—24no—
54#391—22no—
55#390—24no—
56#390—22no—
57#390D412yes—
58#389—24no—
59#389—22no—
60#389—22no—
61#389—22no—
62#389D414yes—
63#386—24no—
64#385—24no—
65#384D412yes—
66#384C2312yes—
67#384C2311yescharacteristic
68#382—21no—
69#380—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4C2314yes—
72C43C2311yescharacteristic
73#340—24no—
74#340—24no—
75#340—24no—
76#336—22no—
77#336—21no—
78#331Q818yescomplemented
79C22×Q8—24no—
80C22×Q8—22no—
81C22×Q8—416no—
82C22×Q8—84no—
83C22×D4—22no—
84C22×D4C2×D412yes—
85C23×C4—24no—
86C23×C4—22no—
87C22×C8—24no—
88C22×C8—22no—
89C4⋊Q8—28no—
90C4⋊Q8—48no—
91C4⋊Q8—48no—
92#127—22no—
93#127C2×Q814yes—
94#125—82no—
95C22⋊Q8—432no—
96C22⋊Q8—416no—
97C4×Q8—48no—
98C4×D4—48no—
99C4×D4—44no—
100C4×D4—48no—
101C4×D4—48no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—28no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4C2×D414yes—
113C2×C4⋊C4C2×D414yes—
114C2×C4⋊C4C2×D412yes—
115C2×C22⋊C4—28no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—22no—
119C2×C42—24no—
120C2×C42C2×D412yes—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D412yes—
123C2×C42C2×D411yescharacteristic
124C2×C42C2411yescharacteristic
125#107—84no—
126#106—48no—
127C4⋊C8—24no—
128C4⋊C8—42no—
129C4⋊C8C2×Q818yescomplemented
130#102—84no—
131#97—82no—
132C4×C8—24no—
133#95—24no—
134#95—24no—
135#95—24no—
136#95—41no—
137#95—41no—
138C24—22no—
139C2×Q8—28no—
140C2×Q8—44no—
141C2×Q8—416no—
142C2×Q8—44no—
143C2×Q8—84no—
144C2×Q8—832no—
145C2×D4—22no—
146C2×D4—28no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—44no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4C4⋊Q814yes—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—48no—
157C22×C4—416no—
158C22×C4—44no—
159C22×C4—42no—
160C22×C4—84no—
161C22×C4#12011yescharacteristic
162C22×C4#12012yes—
163C22×C4#12011yescharacteristic
164C22×C4C22×D411yescharacteristic
165C22×C4C22×D412yes—
166C2×C8—28no—
167C2×C8—44no—
168C2×C8—44no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C4⋊C4—28no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—24no—
176C4⋊C4—42no—
177C4⋊C4—44no—
178C4⋊C4—48no—
179C4⋊C4—416no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—82no—
186C4⋊C4—84no—
187C4⋊C4AGL(1,ℤ/8)18yes—
188C22⋊C4—48no—
189C22⋊C4—416no—
190C22⋊C4—82no—
191C22⋊C4—84no—
192C42—24no—
193C42—22no—
194C42—22no—
195C42—22no—
196C42—41no—
197C42—44no—
198C42—44no—
199C42C22×Q812yes—
200C422−1+412yes—
201C23—24no—
202C23—44no—
203C23—44no—
204C23—42no—
205C23—42no—
206C23C2×#12011yescharacteristic
207Q8—416no—
208Q8—816no—
209D4—44no—
210D4—48no—
211D4—48no—
212D4—82no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—28no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—88no—
230C2×C4—816no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4#45512yes—
235C2×C4C2×C4⋊Q811yescharacteristic
236C2×C4#53611yescharacteristic
237C2×C4D4×Q814yes—
238C2×C4C2×AGL(1,ℤ/8)14yes—
239C2×C4#57712yes—
240C2×C4#57812yes—
241C8—48no—
242C8—82no—
243C22—44no—
244C22—48no—
245C22—82no—
246C22—82no—
247C22—84no—
248C22#243411yescharacteristic
249C22#300312yes—
250C22#301312yes—
251C22C2×#45511yescharacteristic
252C22#302111yescharacteristic
253C4—22no—
254C4—21no—
255C4—21no—
256C4—41no—
257C4—48no—
258C4—42no—
259C4—82no—
260C4—84no—
261C4—162no—
262C4#338414yes—
263C4#338614yes—
264C2—84no—
265C2#1852911yescharacteristic
266C2#1869311yescharacteristic
267C2#1885412yes—
268C2#2582311yescharacteristic
269C2#2584412yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #29320C2semidirect
4. #10575C2semidirect
5. #9868C2semidirect
78. #331Q8semidirect
129. C4⋊C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#598172
#293202
#291432
#105752
#98682
#89212
#82312