Finite Group Databasebeta

← #309191

Subgroups — #309191

Order 512.

553 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28811C211yesmaximal, characteristic
2#26528C212yesmaximal
3#23161C211yesmaximal, characteristic
4#10588C212yesmaximal
5#10357C212yesmaximal
6#9836C211yesmaximal, characteristic
7#9731C211yesmaximal, characteristic
8#9618C211yesmaximal, characteristic
9#9596C211yesmaximal, characteristic
10#8923C212yesmaximal
11#8290C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#499—21no—
14#2552C2212yes—
15#2365C2212yes—
16#2096—22no—
17#2066—22no—
18#2059—22no—
19#1990C2212yes—
20#1988C2212yes—
21#1981C2212yes—
22#1956C2212yes—
23#1948C2212yes—
24#1947C2212yes—
25#1943—21no—
26#1941—21no—
27#1939C2212yes—
28#1921—22no—
29#1865—22no—
30#1844C2211yescharacteristic
31#1803—22no—
32#1776—21no—
33C4×#106C2211yescharacteristic
34C4×#106C2211yescharacteristic
35#1770C2211yescharacteristic
36C8×C4⋊C4C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#334C2211yescharacteristic
39C2×#332C2212yes—
40C2×#322C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2211yescharacteristic
44#1279C2212yes—
45C2×C4⋊Q8C2312yes—
46C2×#125—22no—
47C2×C4×Q8—22no—
48#500—22no—
49#500Q814yes—
50#499—44no—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107—22no—
54C2×#106—21no—
55C2×#106—21no—
56C2×#106—22no—
57C2×#106D411yescharacteristic
58C2×#106D412yes—
59C2×#106D411yescharacteristic
60C2×C4⋊C8—22no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2311yescharacteristic
63C2×C4⋊C8C2311yescharacteristic
64C2×#103—22no—
65C2×#103—22no—
66C2×#103—22no—
67C2×#103—22no—
68C2×#97—21no—
69C2×C4×C8—21no—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#398—42no—
74#391—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#387—42no—
79#384—22no—
80#384C2312yes—
81C4×C4⋊C4—22no—
82C4×C4⋊C4C2311yescharacteristic
83C4×C4⋊C4C2312yes—
84C4×C4⋊C4C2311yescharacteristic
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—42no—
90#340D412yes—
91#340D412yes—
92#336—21no—
93#336—22no—
94#336—21no—
95#335—22no—
96#334—22no—
97#332—24no—
98#322—22no—
99C22×Q8—22no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—22no—
103C22×C8—21no—
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8C2×Q814yes—
112#125—44no—
113#125—82no—
114C4×Q8—44no—
115C4×Q8—82no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4C2×D412yes—
126C2×C4⋊C4C2×D412yes—
127C2×C4⋊C4C2×D412yes—
128C2×C4⋊C4C2×D412yes—
129C2×C42—22no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—42no—
133C2×C42C2411yescharacteristic
134#107—24no—
135#107—24no—
136#107—44no—
137#106—22no—
138#106—42no—
139#106—42no—
140#106—82no—
141#106—81no—
142#106SD1614yes—
143C4⋊C8—22no—
144C4⋊C8—21no—
145C4⋊C8—22no—
146C4⋊C8—21no—
147C4⋊C8—21no—
148C4⋊C8—82no—
149C4⋊C8#4112yes—
150#103—44no—
151#103—82no—
152#103—82no—
153#103—82no—
154#97—42no—
155C4×C8—21no—
156C4×C8—21no—
157C4×C8—21no—
158C4×C8—21no—
159C4×C8—42no—
160C4×C8C2×Q812yes—
161C4×C8#4112yes—
162#95—21no—
163#95—22no—
164#95—21no—
165#95—42no—
166#95—42no—
167#95—42no—
168C2×Q8—24no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4C22×D411yescharacteristic
181C22×C4C22×D411yescharacteristic
182C22×C4H(5,𝔽2)11yescharacteristic
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8C22⋊Q812yes—
201C2×C8C22⋊Q812yes—
202C2×C8C2×SD1612yes—
203C2×C8#13712yes—
204C4⋊C4—22no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—84no—
220C4⋊C4—84no—
221C4⋊C4—82no—
222C4⋊C4—84no—
223C4⋊C4—162no—
224C4⋊C4C2×SD1614yes—
225C42—41no—
226C42—41no—
227C42—42no—
228C42—42no—
229C42—41no—
230C42—41no—
231C42—84no—
232C42C22×Q811yescharacteristic
233C42C2×#4111yescharacteristic
234C42C2×#4111yescharacteristic
235C422−1+411yescharacteristic
236C23D4211yescharacteristic
237Q8—44no—
238Q8—44no—
239Q8—82no—
240Q8—82no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4C2×C22⋊Q811yescharacteristic
261C2×C4#53211yescharacteristic
262C2×C4#54211yescharacteristic
263C2×C4#54811yescharacteristic
264C2×C4D4×Q811yescharacteristic
265C2×C4#55411yescharacteristic
266C2×C4C22×SD1611yescharacteristic
267C2×C4C2×#13711yescharacteristic
268C2×C4#57512yes—
269C2×C4#57612yes—
270C2×C4#57711yescharacteristic
271C2×C4#57811yescharacteristic
272C8—22no—
273C8—42no—
274C8—41no—
275C8—41no—
276C8—81no—
277C8—81no—
278C8—82no—
279C8#47514yes—
280C22#245211yescharacteristic
281C22#326812yes—
282C22#327611yescharacteristic
283C22D4×SD1611yescharacteristic
284C22#328912yes—
285C4—21no—
286C4—21no—
287C4—44no—
288C4—82no—
289C4—82no—
290C4—82no—
291C4—84no—
292C4—162no—
293C4C2×#47511yescharacteristic
294C4#308211yescharacteristic
295C4#317611yescharacteristic
296C4#318311yescharacteristic
297C4#335911yescharacteristic
298C4#336911yescharacteristic
299C4#338912yes—
300C2#1976712yes—
301C2#2026312yes—
302C2#2646211yescharacteristic
303C2#2649211yescharacteristic
304C2#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
#288112
#265282
#231612
#105882
#103572
#98362
#97312
#96182
#95962
#89232
#82902