Finite Group Databasebeta

← #302412

Subgroups — #302412

Order 512.

740 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#125C211yesmaximal, characteristic
2#33256C211yesmaximal, characteristic
3#33067C211yesmaximal, characteristic
4#28691C211yesmaximal, characteristic
5#10582C212yesmaximal
6#10580C212yesmaximal
7#9897C211yesmaximal, characteristic
8#9871C211yesmaximal, characteristic
9#8892C212yesmaximal
10#8888C212yesmaximal
11C4×Q8⋊C8C211yesmaximal, characteristic
12C2×#499—21no—
13#2859—21no—
14#2859C2211yescharacteristic
15#2859C2211yescharacteristic
16#2552—22no—
17#2552C2212yes—
18#2507—24no—
19#2507—22no—
20Q8×C4⋊C4—22no—
21Q8×C4⋊C4C2212yes—
22C4×#125—21no—
23C4×#125C2211yescharacteristic
24C42×Q8C2211yescharacteristic
25#2085—22no—
26#1991—21no—
27#1948C2212yes—
28#1948C2212yes—
29#1947C2212yes—
30#1941—24no—
31#1919—22no—
32#1865—22no—
33#1844C2212yes—
34#1834C2212yes—
35#1831C2211yescharacteristic
36#1803C2212yes—
37C4×C4⋊C8C2211yescharacteristic
38C42×C8C2211yescharacteristic
39C2×Q8⋊C8C2211yescharacteristic
40C2×Q8⋊C8C2211yescharacteristic
41#1296C2212yes—
42#1296C2212yes—
43#1279C2211yescharacteristic
44#1279C2212yes—
45#1279C2211yescharacteristic
46#1279C2212yes—
47C2×#125—22no—
48C2×#125—21no—
49C2×#125—21no—
50C2×#125D411yescharacteristic
51C2×C4×Q8—22no—
52C2×C4×Q8D412yes—
53C2×C4×Q8C2311yescharacteristic
54#499—44no—
55C2×#107—22no—
56C2×#106—22no—
57C2×#106—24no—
58C2×#106—22no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2311yescharacteristic
62C2×#103—22no—
63C2×C4×C8—21no—
64C2×C4×C8—22no—
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67#398—24no—
68#398—24no—
69#398—24no—
70#389—24no—
71#389—24no—
72#389—24no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—24no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389D412yes—
81#389D414yes—
82#387—24no—
83#387—22no—
84#384—24no—
85#384—22no—
86#384—22no—
87#384C2312yes—
88#384C2312yes—
89#382—21no—
90#382—21no—
91#382D411yescharacteristic
92C4×C4⋊C4—24no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4D412yes—
98C4×C4⋊C4C2312yes—
99C4×C4⋊C4C2311yescharacteristic
100C43—21no—
101C43C2311yescharacteristic
102#340—22no—
103#340—22no—
104#340—22no—
105#340—24no—
106#340—22no—
107#336—21no—
108#336—22no—
109Q8⋊C8Q814yes—
110Q8⋊C8Q814yes—
111C22×Q8C2×D411yescharacteristic
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—22no—
117#125—22no—
118#125—42no—
119#125—42no—
120#125—81no—
121#125—81no—
122#125—82no—
123#125—82no—
124#125SD1614yes—
125C4×Q8—21no—
126C4×Q8—21no—
127C4×Q8—24no—
128C4×Q8—42no—
129C4×Q8—44no—
130C4×Q8—44no—
131C4×Q8C2×Q812yes—
132C4×Q8C2×Q812yes—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4C2×D412yes—
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C4⋊C4C2×D412yes—
153C2×C4⋊C4C2×D414yes—
154C2×C42—21no—
155C2×C42—24no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—21no—
159C2×C42—22no—
160C2×C42C2×D412yes—
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D412yes—
163C2×C42C2×D411yescharacteristic
164C2×C42C2411yescharacteristic
165#107—44no—
166#106—44no—
167#106—48no—
168#106—44no—
169C4⋊C8—21no—
170C4⋊C8—21no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8C2×Q812yes—
174C4⋊C8C2×Q812yes—
175#103—44no—
176C4×C8—21no—
177C4×C8—21no—
178C4×C8—44no—
179C4×C8—42no—
180C4×C8C2×Q812yes—
181C4×C8C2×Q812yes—
182#95—24no—
183#95—24no—
184#95—22no—
185#95—24no—
186#95—22no—
187#95—22no—
188#95—21no—
189#95—22no—
190#95—42no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8C4⋊Q812yes—
194C2×Q8#13514yes—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—42no—
202C22×C4#12012yes—
203C22×C4#12012yes—
204C22×C4C22×D411yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D411yescharacteristic
207C2×C8—24no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—42no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—28no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—48no—
232C4⋊C4—44no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—82no—
242C4⋊C4—84no—
243C4⋊C4—84no—
244C4⋊C4—84no—
245C4⋊C4—84no—
246C4⋊C4—84no—
247C4⋊C4—84no—
248C4⋊C4C4⋊Q812yes—
249C4⋊C4C2×SD1614yes—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—21no—
255C42—21no—
256C42—42no—
257C42—44no—
258C42—48no—
259C42—42no—
260C42—42no—
261C42—41no—
262C42—41no—
263C42—82no—
264C42C2×SD1612yes—
265C42#13712yes—
266C42C22×Q811yescharacteristic
267C42C22×Q811yescharacteristic
268C422−1+411yescharacteristic
269C422−1+411yescharacteristic
270C23C2×#12011yescharacteristic
271Q8—22no—
272Q8—24no—
273Q8#49914yescomplemented
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4#44814yes—
300C2×C4C2×C4⋊Q811yescharacteristic
301C2×C4#53611yescharacteristic
302C2×C4D4×Q811yescharacteristic
303C2×C4D4×Q811yescharacteristic
304C2×C4D4×Q811yescharacteristic
305C2×C4D4×Q811yescharacteristic
306C2×C4C22×SD1611yescharacteristic
307C2×C4C2×#13512yes—
308C2×C4C2×#13711yescharacteristic
309C2×C4#57512yes—
310C2×C4#57711yescharacteristic
311C2×C4#57711yescharacteristic
312C8—24no—
313C8—42no—
314C8—42no—
315C8—42no—
316C8—82no—
317C22#243411yescharacteristic
318C22C2×#44811yescharacteristic
319C22#300212yes—
320C22#300911yescharacteristic
321C22#301812yes—
322C4—24no—
323C4—24no—
324C4—21no—
325C4—21no—
326C4—44no—
327C4—42no—
328C4—42no—
329C4—82no—
330C4—82no—
331C4—84no—
332C4—162no—
333C4C2×#49911yescharacteristic
334C4#316611yescharacteristic
335C4#323212yes—
336C4Q8×SD1611yescharacteristic
337C4Q8×SD1611yescharacteristic
338C4#338711yescharacteristic
339C4#338711yescharacteristic
340C2#1851812yes—
341C2#1867112yes—
342C2#2581511yescharacteristic
343C2#2583111yescharacteristic
344C2#2583711yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
273. Q8#499semidirect

Maximal subgroups

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

subgroupindex
Q8×#1252
#332562
#330672
#286912
#105822
#105802
#98972
#98712
#88922
#88882
C4×Q8⋊C82