Finite Group Databasebeta

← #305791

Subgroups — #305791

Order 512.

540 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28814C211yesmaximal, characteristic
2#26088C212yesmaximal
3#22727C211yesmaximal, characteristic
4#10654C212yesmaximal
5#10582C212yesmaximal
6#9615C211yesmaximal, characteristic
7#9594C211yesmaximal, characteristic
8#8891C212yesmaximal
9#8814C211yesmaximal, characteristic
10#8645C211yesmaximal, characteristic
11C8×#103C211yesmaximal, characteristic
12C2×#499—21no—
13C2×C8×Q8C2211yescharacteristic
14#2552C2212yes—
15#2355C2212yes—
16#2096—22no—
17#2066—22no—
18#2059—22no—
19#1949—22no—
20#1948—22no—
21#1948C2212yes—
22#1948C2212yes—
23#1948C2212yes—
24#1942—21no—
25#1939C2212yes—
26#1919—22no—
27#1846—21no—
28#1845C2211yescharacteristic
29#1845C2211yescharacteristic
30#1844C2212yes—
31#1834C2212yes—
32#1803C2212yes—
33C4×#107—21no—
34C4×#107C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C4×#103C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×Q8⋊C8C2211yescharacteristic
39C2×C82C2211yescharacteristic
40#1296C2212yes—
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2212yes—
44#1277C2211yescharacteristic
45C2×#125—22no—
46C2×C4×Q8C2311yescharacteristic
47#499—44no—
48C8×Q8—22no—
49C8×Q8Q814yes—
50C2×#107—21no—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107D411yescharacteristic
55C2×#107D411yescharacteristic
56C2×#106—22no—
57C2×#106—22no—
58C2×#106—22no—
59C2×#106—22no—
60C2×#106—22no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×#103—22no—
67C2×#103D411yescharacteristic
68C2×#103D411yescharacteristic
69C2×C4×C8—21no—
70C2×C4×C8—21no—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#398—42no—
75#391—22no—
76#389—22no—
77#389—22no—
78#387—42no—
79#384—22no—
80#384C2312yes—
81#384C2312yes—
82#382—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4C2312yes—
86C4×C4⋊C4C2311yescharacteristic
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—42no—
92#340D412yes—
93#340D411yescharacteristic
94#340D411yescharacteristic
95#336—21no—
96#336—22no—
97#335—22no—
98Q8⋊C8Q814yes—
99C82—22no—
100C22×Q8C2×D411yescharacteristic
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—22no—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109#125—44no—
110#125—82no—
111C4×Q8—21no—
112C4×Q8—21no—
113C4×Q8C2×Q812yes—
114C4×Q8C2×Q812yes—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4C2×D411yescharacteristic
126C2×C4⋊C4C2×D412yes—
127C2×C4⋊C4C2×D411yescharacteristic
128C2×C4⋊C4C2×D411yescharacteristic
129C2×C4⋊C4C2×D412yes—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—22no—
133C2×C42—42no—
134C2×C42C2411yescharacteristic
135#107—22no—
136#107—42no—
137#107—42no—
138#107—81no—
139#107—81no—
140#107SD1614yes—
141#106—44no—
142#106—44no—
143#106—82no—
144#106—82no—
145#106—82no—
146C4⋊C8—24no—
147C4⋊C8—21no—
148C4⋊C8—22no—
149C4⋊C8—21no—
150C4⋊C8—22no—
151C4⋊C8—44no—
152C4⋊C8C2×Q812yes—
153C4⋊C8C2×Q812yes—
154#103—22no—
155#103—22no—
156#103—82no—
157C4×C8—24no—
158C4×C8—21no—
159C4×C8—21no—
160C4×C8—21no—
161C4×C8—42no—
162C4×C8C2×Q812yes—
163C4×C8C2×Q812yes—
164C4×C8#4112yes—
165#95—22no—
166#95—21no—
167#95—42no—
168#95—42no—
169#95—42no—
170#95—42no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8C4⋊Q812yes—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4C22×D411yescharacteristic
186C22×C4C22×D411yescharacteristic
187C22×C4H(5,𝔽2)11yescharacteristic
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8C22⋊Q812yes—
207C2×C8C22⋊Q812yes—
208C2×C8C2×SD1612yes—
209C2×C8#13512yes—
210C2×C8#13512yes—
211C2×C8#13712yes—
212C4⋊C4—22no—
213C4⋊C4—21no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—22no—
219C4⋊C4—21no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—84no—
231C4⋊C4—84no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C4⋊C4—162no—
235C4⋊C4C4⋊Q812yes—
236C4⋊C4C2×SD1614yes—
237C42—22no—
238C42—22no—
239C42—41no—
240C42—42no—
241C42—41no—
242C42—42no—
243C42—162no—
244C42C22×Q811yescharacteristic
245C42C22×Q811yescharacteristic
246C42C2×#4111yescharacteristic
247C422−1+411yescharacteristic
248C23D4211yescharacteristic
249Q8—22no—
250Q8—22no—
251Q8—41no—
252Q8—41no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4C2×C22⋊Q811yescharacteristic
280C2×C4C2×C4⋊Q811yescharacteristic
281C2×C4#54411yescharacteristic
282C2×C4#54711yescharacteristic
283C2×C4D4×Q811yescharacteristic
284C2×C4#55411yescharacteristic
285C2×C4C22×SD1611yescharacteristic
286C2×C4C2×#13511yescharacteristic
287C2×C4C2×#13511yescharacteristic
288C2×C4C2×#13711yescharacteristic
289C2×C4#57611yescharacteristic
290C2×C4#57711yescharacteristic
291C2×C4#57711yescharacteristic
292C2×C4#57811yescharacteristic
293C8—22no—
294C8—22no—
295C8—22no—
296C8—42no—
297C8—42no—
298C8—42no—
299C8—82no—
300C8#47514yes—
301C22#244011yescharacteristic
302C22#327811yescharacteristic
303C22D4×SD1611yescharacteristic
304C22#328311yescharacteristic
305C22#328711yescharacteristic
306C22#329011yescharacteristic
307C22#329011yescharacteristic
308C4—21no—
309C4—21no—
310C4—44no—
311C4—42no—
312C4—42no—
313C4—81no—
314C4—81no—
315C4—82no—
316C4—162no—
317C4—162no—
318C4C2×#47511yescharacteristic
319C4#308411yescharacteristic
320C4#316011yescharacteristic
321C4#316411yescharacteristic
322C4#333611yescharacteristic
323C4#333911yescharacteristic
324C4Q8×SD1611yescharacteristic
325C4#338711yescharacteristic
326C2#1995911yescharacteristic
327C2#1996611yescharacteristic
328C2#1998011yescharacteristic
329C2#1998511yescharacteristic
330C2#2604211yescharacteristic
331C2#2606611yescharacteristic
332C2#2608211yescharacteristic
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
#288142
#260882
#227272
#106542
#105822
#96152
#95942
#88912
#88142
#86452
C8×#1032