Finite Group Databasebeta

← #193296

Subgroups — #193296

Order 512.

948 conjugacy classes of subgroups.

The subgroup lattice has 296 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#8926C214yesmaximal, complemented
6#8923C214yesmaximal, 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#1279C2214yes—
31#1279C2214yes—
32C22×M16—22no—
33C23×C8—21no—
34#533—24no—
35#533C2×C414yes—
36#533Q814yes—
37C2×C4⋊Q8C2312yes—
38C2×#125D412yes—
39C2×C22⋊Q8—22no—
40C2×C4×Q8—24no—
41C2×C4×Q8—22no—
42C2×#117—22no—
43C2×#117—21no—
44C2×#117D412yes—
45C22×C42C2311yescharacteristic
46C2×#107—24no—
47C2×#106—24no—
48#423—24no—
49#423C2×C418yes—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2314yes—
52C2×C4×C8—24no—
53C2×C4×C8—21no—
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2311yescharacteristic
56#391—24no—
57#389—28no—
58#389—24no—
59#388—28no—
60#387—28no—
61#387—28no—
62#384—22no—
63#384D412yes—
64#384D412yes—
65#384C2312yes—
66#382—22no—
67#382—22no—
68#382—21no—
69#380—22no—
70#380—22no—
71#380D414yes—
72C4×C4⋊C4—24no—
73C4×C4⋊C4—24no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4C2314yes—
77#340—24no—
78#340—24no—
79#340—24no—
80#340—24no—
81#332C2×C4116yes—
82C22×Q8—22no—
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D412yes—
85C2×M16—28no—
86C2×M16—82no—
87C2×M16—82no—
88C22×C8—24no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—24no—
92C22×C8—42no—
93C22×C8—42no—
94C4⋊Q8—22no—
95C4⋊Q8—22no—
96C4⋊Q8C22×C414yes—
97C4⋊Q8C2×Q814yes—
98#125—24no—
99#125C4⋊C418yes—
100C22⋊Q8—28no—
101C22⋊Q8—28no—
102C4×Q8—28no—
103C4×Q8—28no—
104C4×Q8—84no—
105C4×Q8—84no—
106#117—24no—
107#117—44no—
108#117—82no—
109#117—82no—
110#117C4⋊C418yes—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4C2×D414yes—
122C2×C4⋊C4C2×D414yes—
123C2×C4⋊C4C2×D414yes—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—41no—
127C2×C42—22no—
128C2×C42—24no—
129C2×C42—28no—
130C2×C42—41no—
131C2×C42C22×C412yes—
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D412yes—
134C2×C42C2×Q812yes—
135C2×C42#4114yes—
136C2×C42C2411yescharacteristic
137#107—84no—
138#106—84no—
139C4⋊C8—28no—
140C4⋊C8—24no—
141C4⋊C8C22×C418yes—
142C4×C8—22no—
143C4×C8—22no—
144C4×C8—42no—
145#95—28no—
146#95—28no—
147#95—24no—
148#95—24no—
149#95—42no—
150C24#12011yescharacteristic
151C2×Q8—24no—
152C2×Q8—24no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—42no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—42no—
167C22×C4C2×C4⋊C412yes—
168C22×C4C4×D414yes—
169C22×C4#12012yes—
170C22×C4#12011yescharacteristic
171C22×C4C22⋊Q814yes—
172C22×C4#12312yes—
173C22×C4C22×D411yescharacteristic
174C22×C4C22×D412yes—
175M16—88no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—28no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—48no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—44no—
185C2×C8—44no—
186C4⋊C4—24no—
187C4⋊C4—24no—
188C4⋊C4—24no—
189C4⋊C4—216no—
190C4⋊C4—24no—
191C4⋊C4—24no—
192C4⋊C4—24no—
193C4⋊C4—24no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—82no—
202C4⋊C4—82no—
203C4⋊C4—88no—
204C4⋊C4C2×C4⋊C418yes—
205C22⋊C4—28no—
206C22⋊C4—84no—
207C22⋊C4—82no—
208C42—28no—
209C42—44no—
210C42—44no—
211C42—88no—
212C42—82no—
213C42C2×C4⋊C412yes—
214C42C4×D414yes—
215C42C22⋊Q814yes—
216C42#12312yes—
217C42C23×C411yescharacteristic
218C42C22×Q811yescharacteristic
219C42C2×#4112yes—
220C23—22no—
221C23—22no—
222C23—21no—
223C23—21no—
224C23#38512yes—
225C23C2×#12011yescharacteristic
226Q8—44no—
227Q8—44no—
228Q8—44no—
229Q8—44no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—28no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4#38514yes—
253C2×C4#38512yes—
254C2×C4#45412yes—
255C2×C4#45612yes—
256C2×C4C22×C4⋊C411yescharacteristic
257C2×C4C2×C4×D412yes—
258C2×C4C2×C22⋊Q812yes—
259C2×C4C2×#12311yescharacteristic
260C2×C4#57612yes—
261C2×C4#57712yes—
262C2×C4#57712yes—
263C2×C4#57812yes—
264C8—42no—
265C8—42no—
266C8—48no—
267C22—22no—
268C22—24no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22C2×#38511yescharacteristic
273C22#301212yes—
274C22#301312yes—
275C22C2×#45411yescharacteristic
276C22C2×#45611yescharacteristic
277C4—21no—
278C4—21no—
279C4—24no—
280C4—48no—
281C4—84no—
282C4—84no—
283C4—162no—
284C4#185714yes—
285C4#294612yes—
286C4#294712yes—
287C4#308312yes—
288C4#308412yes—
289C2—42no—
290C2#1869712yes—
291C2#1870111yescharacteristic
292C2#1870311yescharacteristic
293C2C2×#185711yescharacteristic
294C2#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. #8926C2semidirect
6. #8923C2semidirect
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
#89262
#89232
#78972