Finite Group Databasebeta

← #195715

Subgroups — #195715

Order 512.

1148 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 274 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
1C2×#3139C211yesmaximal, characteristic
2#34374C211yesmaximal, characteristic
3#21834C212yesmaximal
4C2×#1544C211yesmaximal, characteristic
5#9065C218yesmaximal, complemented
6#7896C212yesmaximal
7C2×#533C2211yescharacteristic
8C22×C4⋊Q8C2211yescharacteristic
9#3139—24no—
10#3139C418yescomplemented
11C2×#491—22no—
12C2×#487C2214yes—
13C2×#431C2212yes—
14C2×#419—22no—
15C2×C4×M16C2211yescharacteristic
16#2382—22no—
17#2365C2214yes—
18#2355—22no—
19Q8×C4⋊C4—22no—
20#2323—24no—
21#2314—22no—
22#2285C2212yes—
23#1990—22no—
24#1990—22no—
25#1987C2218yes—
26#1824—24no—
27#1544—24no—
28C2×#332C2214yes—
29#1279C2218yes—
30C23×Q8—21no—
31C22×M16—21no—
32#533—24no—
33#533C2×C414yes—
34#533D414yes—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—24no—
39C2×C4⋊Q8C2×C414yes—
40C2×C4⋊Q8C2312yes—
41C2×C4⋊Q8C2312yes—
42C2×#125D412yes—
43C2×C22⋊Q8—22no—
44C2×C4×Q8—24no—
45C2×C4×Q8—22no—
46C2×#117—22no—
47C2×#117D412yes—
48C22×C4⋊C4—22no—
49C22×C42C2311yescharacteristic
50#491—216no—
51#487—28no—
52#487C2×C4116yes—
53#431—24no—
54#431—24no—
55C2×C4⋊C8C2314yes—
56#419—216no—
57C2×#103—28no—
58C2×C22⋊C8—22no—
59C4×M16—22no—
60C4×M16C2×C414yes—
61C2×#97—21no—
62C2×C4×C8—22no—
63C2×C4×C8C2312yes—
64#391—24no—
65#389—28no—
66#389—24no—
67#388—28no—
68#387—28no—
69#387—28no—
70#384—22no—
71#384D412yes—
72#384D412yes—
73#382—22no—
74#382—22no—
75#380—22no—
76#380D414yes—
77C4×C4⋊C4—24no—
78C4×C4⋊C4—24no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4C2314yes—
81#340—28no—
82#340—28no—
83#332—28no—
84C22×Q8—22no—
85C22×Q8—24no—
86C22×Q8—22no—
87C22×Q8—44no—
88C22×Q8—44no—
89C22×Q8—41no—
90C22×Q8—42no—
91C23×C4—41no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D412yes—
94C2×M16—24no—
95C2×M16—42no—
96C2×M16—42no—
97C22×C8—22no—
98C22×C8—24no—
99C4⋊Q8—24no—
100C4⋊Q8—28no—
101C4⋊Q8—22no—
102C4⋊Q8—22no—
103C4⋊Q8—22no—
104C4⋊Q8—24no—
105C4⋊Q8—44no—
106C4⋊Q8—42no—
107C4⋊Q8C22×C414yes—
108C4⋊Q8C22×C414yes—
109C4⋊Q8C2×D414yes—
110#125—24no—
111#125C22⋊C418yes—
112C22⋊Q8—28no—
113C22⋊Q8—28no—
114C4×Q8—28no—
115C4×Q8—28no—
116C4×Q8—88no—
117#117—24no—
118#117—84no—
119#117C22⋊C418yes—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4C2×D414yes—
132C2×C4⋊C4C2×D414yes—
133C2×C4⋊C4C2×D414yes—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—24no—
136C2×C42—22no—
137C2×C42—24no—
138C2×C42—28no—
139C2×C42C22×C412yes—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D412yes—
142C2×C42C2×D412yes—
143C2×C42#4114yes—
144C2×C42C2411yescharacteristic
145C4⋊C8—24no—
146C4⋊C8—24no—
147#103—232no—
148C22⋊C8—44no—
149#97—24no—
150C4×C8—22no—
151C4×C8—44no—
152C4×C8C22×C414yes—
153#95—28no—
154#95—28no—
155#95—24no—
156#95—24no—
157C24#12011yescharacteristic
158C2×Q8—24no—
159C2×Q8—24no—
160C2×Q8—24no—
161C2×Q8—44no—
162C2×Q8—44no—
163C2×Q8—416no—
164C2×Q8—42no—
165C2×Q8—44no—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—88no—
169C2×Q8—82no—
170C2×Q8—82no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4C2×C22⋊C412yes—
183C22×C4C4×D414yes—
184C22×C4#12012yes—
185C22×C4#12011yescharacteristic
186C22×C4#12114yes—
187C22×C4#12312yes—
188C22×C4C22×D411yescharacteristic
189C22×C4C22×D412yes—
190M16—48no—
191C2×C8—24no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—44no—
195C2×C8—44no—
196C2×C8—44no—
197C4⋊C4—24no—
198C4⋊C4—24no—
199C4⋊C4—24no—
200C4⋊C4—216no—
201C4⋊C4—28no—
202C4⋊C4—28no—
203C4⋊C4—28no—
204C4⋊C4—44no—
205C4⋊C4—48no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—82no—
209C4⋊C4—82no—
210C4⋊C4—88no—
211C4⋊C4C2×C22⋊C418yes—
212C22⋊C4—28no—
213C22⋊C4—84no—
214C42—28no—
215C42—48no—
216C42—88no—
217C42C2×C22⋊C412yes—
218C42C4×D414yes—
219C42#12114yes—
220C42#12312yes—
221C42C23×C411yescharacteristic
222C42C22×D411yescharacteristic
223C42C2×#4112yes—
224C23—22no—
225C23—22no—
226C23—21no—
227C23—21no—
228C23#38612yes—
229C23C2×#12011yescharacteristic
230Q8—48no—
231Q8—48no—
232Q8—48no—
233Q8—84no—
234Q8—88no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—28no—
242C2×C4—28no—
243C2×C4—28no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—88no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4#38614yes—
257C2×C4#38612yes—
258C2×C4#45412yes—
259C2×C4#45412yes—
260C2×C4C22×C22⋊C411yescharacteristic
261C2×C4C2×C4×D412yes—
262C2×C4C2×#12112yes—
263C2×C4C2×#12311yescharacteristic
264C2×C4#57714yes—
265C2×C4#57814yes—
266C8—44no—
267C8—84no—
268C22—24no—
269C22—22no—
270C22—41no—
271C22—42no—
272C22—41no—
273C22C2×#38611yescharacteristic
274C22#300912yes—
275C22#301112yes—
276C22C2×#45411yescharacteristic
277C22C2×#45411yescharacteristic
278C4—24no—
279C4—21no—
280C4—21no—
281C4—48no—
282C4—84no—
283C4—84no—
284C4—84no—
285C4#188814yes—
286C4#294714yes—
287C4#304614yes—
288C2—42no—
289C2#1872811yescharacteristic
290C2#1873012yes—
291C2#1873211yescharacteristic
292C2#2093112yes—
293C2C2×#188811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9065C2semidirect
10. #3139C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31392
#343742
#218342
C2×#15442
#90652
#78962