Finite Group Databasebeta

← #193295

Subgroups — #193295

Order 512.

948 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34374C211yesmaximal, characteristic
2#21964C212yesmaximal
3#21318C211yesmaximal, characteristic
4C2×#1561C211yesmaximal, characteristic
5#8925C214yesmaximal, complemented
6#8922C214yesmaximal, complemented
7#7897C212yesmaximal
8C2×#533C2211yescharacteristic
9C2×#423C2212yes—
10C22×C4×C8C2211yescharacteristic
11#2382—22no—
12#2365C2214yes—
13#2355—22no—
14Q8×C4⋊C4—22no—
15#2323—24no—
16#2314—22no—
17#2285C2211yescharacteristic
18#2285C2212yes—
19#1991—22no—
20#1990C2212yes—
21#1990C2212yes—
22#1948—22no—
23#1948C2214yes—
24#1947—22no—
25#1947C2214yes—
26#1809—24no—
27#1735—22no—
28#1561C418yescomplemented
29C2×#332C2214yes—
30#1279C2218yes—
31C22×M16—22no—
32C23×C8—21no—
33#533—24no—
34#533C2×C414yes—
35#533Q814yes—
36C2×C4⋊Q8C2312yes—
37C2×#125D412yes—
38C2×C22⋊Q8—22no—
39C2×C4×Q8—24no—
40C2×C4×Q8—22no—
41C2×#117—22no—
42C2×#117—21no—
43C2×#117D412yes—
44C22×C42C2311yescharacteristic
45C2×#107—24no—
46C2×#106—24no—
47#423—24no—
48#423C2×C418yes—
49C2×C4⋊C8—22no—
50C2×C4⋊C8C2314yes—
51C2×C4×C8—24no—
52C2×C4×C8—21no—
53C2×C4×C8C2312yes—
54#391—24no—
55#389—28no—
56#389—24no—
57#388—28no—
58#387—28no—
59#387—28no—
60#384—22no—
61#384D412yes—
62#384D412yes—
63#384C2311yescharacteristic
64#384C2311yescharacteristic
65#382—22no—
66#382—22no—
67#382—21no—
68#380—22no—
69#380—22no—
70#380D414yes—
71C4×C4⋊C4—24no—
72C4×C4⋊C4—24no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4C2314yes—
76#340—24no—
77#340—24no—
78#340—24no—
79#340—24no—
80#332C2×C4116yes—
81C22×Q8—22no—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D412yes—
84C2×M16—28no—
85C2×M16—82no—
86C2×M16—82no—
87C22×C8—24no—
88C22×C8—22no—
89C22×C8—24no—
90C22×C8—42no—
91C22×C8—42no—
92C4⋊Q8—22no—
93C4⋊Q8—22no—
94C4⋊Q8C22×C414yes—
95C4⋊Q8C2×Q814yes—
96#125—24no—
97#125C4⋊C418yes—
98C22⋊Q8—28no—
99C22⋊Q8—28no—
100C4×Q8—28no—
101C4×Q8—28no—
102C4×Q8—84no—
103C4×Q8—84no—
104#117—24no—
105#117—44no—
106#117—82no—
107#117—82no—
108#117C4⋊C418yes—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4C2×D414yes—
121C2×C4⋊C4C2×D414yes—
122C2×C4⋊C4C2×D414yes—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—41no—
126C2×C42—22no—
127C2×C42—24no—
128C2×C42—28no—
129C2×C42—41no—
130C2×C42C22×C412yes—
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D412yes—
133C2×C42C2×Q812yes—
134C2×C42#4114yes—
135C2×C42C2411yescharacteristic
136#107—84no—
137#106—84no—
138C4⋊C8—28no—
139C4⋊C8—24no—
140C4⋊C8C22×C418yes—
141C4×C8—24no—
142C4×C8—42no—
143#95—28no—
144#95—28no—
145#95—24no—
146#95—24no—
147#95—42no—
148C24#12011yescharacteristic
149C2×Q8—24no—
150C2×Q8—24no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4C2×C4⋊C412yes—
167C22×C4C4×D414yes—
168C22×C4#12012yes—
169C22×C4#12011yescharacteristic
170C22×C4C22⋊Q814yes—
171C22×C4#12312yes—
172C22×C4C22×D411yescharacteristic
173C22×C4C22×D412yes—
174M16—88no—
175C2×C8—24no—
176C2×C8—28no—
177C2×C8—42no—
178C2×C8—48no—
179C2×C8—42no—
180C2×C8—44no—
181C2×C8—44no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—24no—
185C4⋊C4—216no—
186C4⋊C4—24no—
187C4⋊C4—24no—
188C4⋊C4—24no—
189C4⋊C4—24no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—82no—
199C4⋊C4—82no—
200C4⋊C4—88no—
201C4⋊C4C2×C4⋊C418yes—
202C22⋊C4—28no—
203C22⋊C4—84no—
204C22⋊C4—82no—
205C42—28no—
206C42—44no—
207C42—44no—
208C42—88no—
209C42—82no—
210C42C2×C4⋊C412yes—
211C42C4×D414yes—
212C42C22⋊Q814yes—
213C42#12312yes—
214C42C23×C411yescharacteristic
215C42C22×Q811yescharacteristic
216C42C2×#4112yes—
217C23—22no—
218C23—22no—
219C23—21no—
220C23—21no—
221C23#38512yes—
222C23C2×#12011yescharacteristic
223Q8—44no—
224Q8—44no—
225Q8—44no—
226Q8—44no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—28no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4#38514yes—
251C2×C4#38512yes—
252C2×C4#45412yes—
253C2×C4#45612yes—
254C2×C4C22×C4⋊C411yescharacteristic
255C2×C4C2×C4×D412yes—
256C2×C4C2×C22⋊Q812yes—
257C2×C4C2×#12311yescharacteristic
258C2×C4#57612yes—
259C2×C4#57712yes—
260C2×C4#57712yes—
261C2×C4#57812yes—
262C8—44no—
263C8—48no—
264C22—22no—
265C22—24no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22C2×#38511yescharacteristic
270C22#301212yes—
271C22#301312yes—
272C22C2×#45411yescharacteristic
273C22C2×#45611yescharacteristic
274C4—21no—
275C4—21no—
276C4—24no—
277C4—48no—
278C4—84no—
279C4—84no—
280C4—161no—
281C4—161no—
282C4#185714yes—
283C4#294612yes—
284C4#294712yes—
285C4#308312yes—
286C4#308412yes—
287C2—42no—
288C2#1869611yescharacteristic
289C2#1869811yescharacteristic
290C2#1870212yes—
291C2C2×#185711yescharacteristic
292C2#2087812yes—
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. #8925C2semidirect
6. #8922C2semidirect
28. #1561C4semidirect

Maximal subgroups

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

subgroupindex
#343742
#219642
#213182
C2×#15612
#89252
#89222
#78972