Finite Group Databasebeta

← #351539

Subgroups — #351539

Order 512.

626 conjugacy classes of subgroups.

The subgroup lattice has 307 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33154C212yesmaximal
2#29808C211yesmaximal, characteristic
3#20543C211yesmaximal, characteristic
4#9733C212yesmaximal, complemented
5#9616C212yesmaximal, complemented
6#9562C212yesmaximal, complemented
7#9561C212yesmaximal, complemented
8#9131C212yesmaximal
9#8599C211yesmaximal, characteristic
10C2×#494C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2857C2212yes—
14C4×#117C2211yescharacteristic
15#2056—22no—
16#2001—22no—
17#1988C2212yes—
18#1988C2212yes—
19#1956C2212yes—
20#1948C2212yes—
21#1947C2212yes—
22#1921—22no—
23#1920—22no—
24#1854—22no—
25#1834C2211yescharacteristic
26#1825—22no—
27#1797C2212yes—
28C8×C4⋊C4C2212yes—
29C8×C4⋊C4C2212yes—
30C4×C4⋊C8—22no—
31C8×C22⋊C4—22no—
32C2×#335C2212yes—
33C2×#332C2212yes—
34C2×#332C2212yes—
35#1296C2212yes—
36#1296C2212yes—
37#1277C2211yescharacteristic
38#1277C2211yescharacteristic
39C22×Q16—22no—
40C23×C8—21no—
41C2×C4⋊Q8C2312yes—
42C2×C22⋊Q8—22no—
43C2×#117D411yescharacteristic
44C2×#117D411yescharacteristic
45C22×C42C2311yescharacteristic
46#494—24no—
47#431—24no—
48C2×#107—22no—
49C2×#106—22no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—24no—
52C2×C4⋊C8D412yes—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2312yes—
56C2×#103—22no—
57C2×#103—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8D412yes—
60C2×C4×C8—22no—
61C2×C4×C8—22no—
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2311yescharacteristic
64#391—22no—
65#389—22no—
66#389—22no—
67#384C2312yes—
68#380—22no—
69#380—22no—
70C4×C4⋊C4D412yes—
71C4×C4⋊C4C2312yes—
72C4×C22⋊C4—21no—
73C4×C22⋊C4—21no—
74#376—21no—
75C43—21no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—22no—
83#336—22no—
84#336—22no—
85#336—22no—
86#335—24no—
87#332—24no—
88#332—24no—
89C22×Q8—22no—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4#4111yescharacteristic
93C2×Q16—44no—
94C2×Q16—82no—
95C2×Q16—82no—
96C22×C8—22no—
97C22×C8—24no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—82no—
101C22×C8C2×D412yes—
102C22×C8C2×D412yes—
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C22⋊Q8—82no—
108C22⋊Q8—82no—
109#117—22no—
110#117—24no—
111#117Q1614yescomplemented
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4C2×D412yes—
119C2×C4⋊C4C2×D412yes—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—42no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42#4111yescharacteristic
132C2×C42C2411yescharacteristic
133#107—44no—
134#106—44no—
135C4⋊C8—24no—
136C4⋊C8—24no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8#4114yes—
143#103—82no—
144#103—82no—
145C22⋊C8—24no—
146C22⋊C8—44no—
147C4×C8—22no—
148C4×C8—82no—
149C4×C8—82no—
150C4×C8#4114yes—
151#95—22no—
152#95—22no—
153#95—42no—
154#95—42no—
155C24#12111yescharacteristic
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—42no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4#12111yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4C2×Q1612yes—
178C22×C4#13512yes—
179C22×C4C22×D411yescharacteristic
180C22×C4C22×D411yescharacteristic
181C22×C4C2×#4111yescharacteristic
182Q16—88no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—24no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—84no—
198C2×C8#12314yes—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—82no—
210C4⋊C4—84no—
211C4⋊C4C2×Q1614yes—
212C4⋊C4#13514yes—
213C22⋊C4—24no—
214C22⋊C4—24no—
215C22⋊C4—84no—
216C42—24no—
217C42—21no—
218C42—21no—
219C42—22no—
220C42—22no—
221C42—24no—
222C42—42no—
223C42—42no—
224C42—82no—
225C42C2×#4112yes—
226C42C2×#4112yes—
227C23—21no—
228C23—22no—
229C23—21no—
230C23—22no—
231C23#46712yes—
232C23C2×#12111yescharacteristic
233Q8—82no—
234Q8—82no—
235Q8—82no—
236Q8—82no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4#46712yes—
263C2×C4C2×#12311yescharacteristic
264C2×C4#52511yescharacteristic
265C2×C4#52912yes—
266C2×C4#53212yes—
267C2×C4C22×Q1611yescharacteristic
268C2×C4C2×#13511yescharacteristic
269C2×C4C2×#13512yes—
270C2×C4#57612yes—
271C2×C4#57712yes—
272C8—24no—
273C8—42no—
274C8—44no—
275C8—82no—
276C8—82no—
277C22—22no—
278C22—24no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22#285411yescharacteristic
283C22#304512yes—
284C22C2×#46711yescharacteristic
285C22#305011yescharacteristic
286C22#306912yes—
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—44no—
292C4—44no—
293C4—162no—
294C4—162no—
295C4#309612yes—
296C4#312211yescharacteristic
297C4#312611yescharacteristic
298C4#316812yes—
299C4#317012yes—
300C2—42no—
301C2#1941112yes—
302C2#1941312yes—
303C2#3311611yescharacteristic
304C2#3312611yescharacteristic
305C2#3313011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×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
4. #9733C2semidirect
5. #9616C2semidirect
6. #9562C2semidirect
7. #9561C2semidirect
111. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#331542
#298082
#205432
#97332
#96162
#95622
#95612
#91312
#85992