Finite Group Databasebeta

← #304731

Subgroups — #304731

Order 512.

722 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 166 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#36600C211yesmaximal, characteristic
2#33198C211yesmaximal, characteristic
3#30220C211yesmaximal, characteristic
4#29226C211yesmaximal, characteristic
5#10355C214yesmaximal
6#9904C211yesmaximal, characteristic
7#9872C211yesmaximal, characteristic
8#8974C214yesmaximal
9#8175C211yesmaximal, characteristic
10C2×#477C2212yes—
11#2868C2211yescharacteristic
12#2859—21no—
13#2859C2211yescharacteristic
14#2585—24no—
15#2570—22no—
16#2552C2212yes—
17#2453—22no—
18#2370—22no—
19#2355C2212yes—
20C4×#125—21no—
21C4×#125C2211yescharacteristic
22C42×Q8C2211yescharacteristic
23#2099—22no—
24#2043—24no—
25#1992—21no—
26#1987C2214yes—
27#1987C2212yes—
28#1956C2214yes—
29#1947—22no—
30#1842C2211yescharacteristic
31#1836—21no—
32#1803—22no—
33#1767C2211yescharacteristic
34#1744C2211yescharacteristic
35C2×Q8⋊C8C2212yes—
36#1296C2214yes—
37#1279C2214yes—
38#1279C2212yes—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8C2312yes—
41C2×#125—22no—
42C2×#125—21no—
43C2×#125D411yescharacteristic
44C2×C4×Q8D412yes—
45C2×C4×Q8C2311yescharacteristic
46#477—28no—
47C2×#107—22no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8C2312yes—
50C2×C4⋊C8C2312yes—
51C2×#103—22no—
52C2×#103—28no—
53C2×#97—21no—
54C2×C4×C8C2312yes—
55#400—24no—
56#398—24no—
57#398—24no—
58#391—22no—
59#389—24no—
60#389—24no—
61#389—22no—
62#389—22no—
63#389—22no—
64#389D412yes—
65#389D414yes—
66#387—24no—
67#387—24no—
68#387—24no—
69#387—24no—
70#387—22no—
71#384—22no—
72#384—22no—
73#384—21no—
74#384—21no—
75#384C2312yes—
76#382—22no—
77#382—21no—
78#382D411yescharacteristic
79C4×C4⋊C4—24no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4D412yes—
86C4×C4⋊C4C2312yes—
87C4×C4⋊C4C2311yescharacteristic
88C43—21no—
89C43C2311yescharacteristic
90#340—24no—
91#340—24no—
92#340—24no—
93#336—21no—
94#336—22no—
95Q8⋊C8—24no—
96C22×Q8—22no—
97C22×Q8C2×D411yescharacteristic
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—22no—
101C4⋊Q8—24no—
102C4⋊Q8—24no—
103C4⋊Q8—84no—
104#125—22no—
105#125—22no—
106#125—44no—
107#125—44no—
108#125—42no—
109#125—42no—
110C4×Q8—22no—
111C4×Q8—24no—
112C4×Q8—42no—
113C4×Q8#4114yes—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4C2×D412yes—
129C2×C4⋊C4C2×D412yes—
130C2×C4⋊C4C2×D411yescharacteristic
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2×D414yes—
133C2×C42—21no—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42—21no—
137C2×C42—24no—
138C2×C42—22no—
139C2×C42C2×D412yes—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D412yes—
142C2×C42C2×D411yescharacteristic
143C2×C42C2411yescharacteristic
144#107—44no—
145C4⋊C8—22no—
146C4⋊C8—24no—
147C4⋊C8—22no—
148C4⋊C8—44no—
149#103—44no—
150#103—88no—
151#97—42no—
152C4×C8—22no—
153C4×C8—22no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—22no—
159#95—22no—
160#95—22no—
161#95—24no—
162#95—21no—
163#95—22no—
164C2×Q8—21no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—48no—
168C2×Q8—44no—
169C2×Q8#12412yes—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—42no—
177C22×C4#12012yes—
178C22×C4#12012yes—
179C22×C4C22×D411yescharacteristic
180C22×C4C22×D411yescharacteristic
181C22×C4C22×D411yescharacteristic
182C2×C8—44no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—44no—
188C2×C8—42no—
189C4⋊C4—21no—
190C4⋊C4—24no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—24no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—48no—
198C4⋊C4—48no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—44no—
204C4⋊C4—48no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—44no—
208C4⋊C4—82no—
209C4⋊C4—84no—
210C4⋊C4—84no—
211C4⋊C4—84no—
212C4⋊C4#12412yes—
213C4⋊C4#13714yes—
214C42—24no—
215C42—22no—
216C42—22no—
217C42—22no—
218C42—42no—
219C42—44no—
220C42—42no—
221C42—42no—
222C42—42no—
223C42—82no—
224C42—84no—
225C42#13512yes—
226C42#13712yes—
227C42C2×#4112yes—
228C422−1+412yes—
229C23C2×#12011yescharacteristic
230Q8—22no—
231Q8—22no—
232Q8—42no—
233Q8—88no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—82no—
257C2×C4—84no—
258C2×C4#45214yes—
259C2×C4C2×#12411yescharacteristic
260C2×C4#53611yescharacteristic
261C2×C4#54812yes—
262C2×C4#54812yes—
263C2×C4C2×#13511yescharacteristic
264C2×C4C2×#13711yescharacteristic
265C2×C4C2×#13712yes—
266C2×C4#57512yes—
267C2×C4#57812yes—
268C8—82no—
269C8—82no—
270C8—82no—
271C22#243611yescharacteristic
272C22C2×#45211yescharacteristic
273C22#300712yes—
274C22#301111yescharacteristic
275C22#302112yes—
276C4—22no—
277C4—24no—
278C4—42no—
279C4—42no—
280C4—42no—
281C4—44no—
282C4—82no—
283C4—84no—
284C4—82no—
285C4—162no—
286C4#313511yescharacteristic
287C4#313911yescharacteristic
288C4#323712yes—
289C4#336412yes—
290C4#337412yes—
291C2#1879112yes—
292C2#1883712yes—
293C2#2591011yescharacteristic
294C2#2592811yescharacteristic
295C2#2593111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#366002
#331982
#302202
#292262
#103552
#99042
#98722
#89742
#81752