Finite Group Databasebeta

← #309185

Subgroups — #309185

Order 512.

525 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28830C211yesmaximal, characteristic
2#26527C212yesmaximal
3#23161C211yesmaximal, characteristic
4#10592C212yesmaximal
5#10353C212yesmaximal
6#9836C211yesmaximal, characteristic
7#9731C211yesmaximal, characteristic
8#9618C211yesmaximal, characteristic
9#9596C211yesmaximal, characteristic
10#8923C212yesmaximal
11#8291C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13#2571C2212yes—
14#2365C2212yes—
15#2097—22no—
16#2085—22no—
17#2066—22no—
18#1990C2212yes—
19#1988C2212yes—
20#1981C2212yes—
21#1956C2212yes—
22#1948C2212yes—
23#1947C2212yes—
24#1943—21no—
25#1941—21no—
26#1939C2212yes—
27#1921—22no—
28#1863—22no—
29#1846—21no—
30#1844C2211yescharacteristic
31#1805—22no—
32#1776—21no—
33C4×#107C2211yescharacteristic
34C4×#106C2211yescharacteristic
35#1770C2211yescharacteristic
36C8×C4⋊C4C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#335C2211yescharacteristic
39C2×#332C2212yes—
40C2×#322C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2211yescharacteristic
44#1279C2212yes—
45C2×C4⋊Q8C2312yes—
46C2×C4×Q8—22no—
47#500—22no—
48#500Q814yes—
49C2×#107—21no—
50C2×#107—21no—
51C2×#107—22no—
52C2×#107D412yes—
53C2×#106—21no—
54C2×#106—21no—
55C2×#106—22no—
56C2×#106D411yescharacteristic
57C2×#106D411yescharacteristic
58C2×C4⋊C8—22no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2311yescharacteristic
62C2×#103—22no—
63C2×#103—22no—
64C2×#103—22no—
65C2×#103—22no—
66C2×#97—21no—
67C2×C4×C8—21no—
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71#400—42no—
72#391—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#387—42no—
77#384C2312yes—
78#382—22no—
79#382—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4C2311yescharacteristic
82C4×C4⋊C4C2312yes—
83C4×C4⋊C4C2311yescharacteristic
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#340—42no—
89#340D412yes—
90#340D412yes—
91#336—21no—
92#336—22no—
93#336—21no—
94#335—22no—
95#335—22no—
96#332—24no—
97#322—22no—
98C22×Q8—22no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—22no—
102C22×C8—21no—
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8C2×Q814yes—
111C4×Q8—44no—
112C4×Q8—82no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4C2×D412yes—
121C2×C4⋊C4C2×D412yes—
122C2×C4⋊C4C2×D412yes—
123C2×C4⋊C4C2×D412yes—
124C2×C42—22no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—42no—
128C2×C42C2411yescharacteristic
129#107—24no—
130#107—24no—
131#107—42no—
132#107—82no—
133#106—22no—
134#106—22no—
135#106—42no—
136#106—82no—
137#106—81no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142C4⋊C8—21no—
143C4⋊C8—82no—
144C4⋊C8#4112yes—
145#103—44no—
146#103—82no—
147#103—82no—
148#103—82no—
149#97—42no—
150C4×C8—21no—
151C4×C8—21no—
152C4×C8—21no—
153C4×C8—21no—
154C4×C8—42no—
155C4×C8C2×Q812yes—
156C4×C8#4112yes—
157#95—21no—
158#95—22no—
159#95—21no—
160#95—42no—
161#95—42no—
162#95—42no—
163#95—42no—
164#95—42no—
165C2×Q8—24no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4C22×D411yescharacteristic
178C22×C4C22×D411yescharacteristic
179C22×C4H(5,𝔽2)11yescharacteristic
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—41no—
197C2×C8C22⋊Q812yes—
198C2×C8C22⋊Q812yes—
199C2×C8#13512yes—
200C2×C8#13712yes—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—82no—
218C4⋊C4—84no—
219C4⋊C4—162no—
220C42—41no—
221C42—41no—
222C42—42no—
223C42—42no—
224C42—41no—
225C42—41no—
226C42—84no—
227C42C22×Q811yescharacteristic
228C42C2×#4111yescharacteristic
229C42C2×#4111yescharacteristic
230C422−1+411yescharacteristic
231C23D4211yescharacteristic
232Q8—44no—
233Q8—44no—
234Q8—82no—
235Q8—82no—
236C2×C4—22no—
237C2×C4—44no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4C2×C22⋊Q811yescharacteristic
256C2×C4#53211yescharacteristic
257C2×C4#54211yescharacteristic
258C2×C4#54811yescharacteristic
259C2×C4D4×Q811yescharacteristic
260C2×C4#55411yescharacteristic
261C2×C4C2×#13511yescharacteristic
262C2×C4C2×#13711yescharacteristic
263C2×C4#57512yes—
264C2×C4#57612yes—
265C2×C4#57711yescharacteristic
266C2×C4#57811yescharacteristic
267C8—22no—
268C8—42no—
269C8—41no—
270C8—41no—
271C8—81no—
272C8—81no—
273C8—82no—
274C8#47914yes—
275C22#245211yescharacteristic
276C22#326812yes—
277C22#327811yescharacteristic
278C22#328311yescharacteristic
279C22#328912yes—
280C4—21no—
281C4—21no—
282C4—82no—
283C4—84no—
284C4—82no—
285C4—82no—
286C4—82no—
287C4—162no—
288C4C2×#47911yescharacteristic
289C4#308211yescharacteristic
290C4#317611yescharacteristic
291C4#318311yescharacteristic
292C4#336011yescharacteristic
293C4#337411yescharacteristic
294C4#338912yes—
295C2#1986312yes—
296C2#2026812yes—
297C2#2646111yescharacteristic
298C2#2649211yescharacteristic
299C2#2650511yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ 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
#288302
#265272
#231612
#105922
#103532
#98362
#97312
#96182
#95962
#89232
#82912