Finite Group Databasebeta

← #302415

Subgroups — #302415

Order 512.

722 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36584C211yesmaximal, characteristic
2#33258C211yesmaximal, characteristic
3#33067C211yesmaximal, characteristic
4#28691C211yesmaximal, characteristic
5#10578C212yesmaximal
6#10576C212yesmaximal
7#9899C211yesmaximal, characteristic
8#9871C211yesmaximal, characteristic
9#8896C212yesmaximal
10#8884C212yesmaximal
11C4×Q8⋊C8C211yesmaximal, characteristic
12C2×#499—21no—
13#2859—21no—
14#2859C2211yescharacteristic
15#2859C2211yescharacteristic
16#2571—22no—
17#2571C2212yes—
18#2509—24no—
19#2509—22no—
20#2351—22no—
21Q8×C4⋊C4C2212yes—
22C4×#125—21no—
23C4×#125C2211yescharacteristic
24C42×Q8C2211yescharacteristic
25#2085—22no—
26#1991—21no—
27#1948C2212yes—
28#1948C2212yes—
29#1947C2212yes—
30#1942—24no—
31#1919—22no—
32#1865—22no—
33#1845C2212yes—
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—21no—
48C2×#125—21no—
49C2×#125D411yescharacteristic
50C2×C4×Q8—22no—
51C2×C4×Q8D412yes—
52C2×C4×Q8C2311yescharacteristic
53#499—44no—
54C2×#107—22no—
55C2×#107—24no—
56C2×#106—22no—
57C2×#106—22no—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2311yescharacteristic
60C2×C4⋊C8C2311yescharacteristic
61C2×#103—22no—
62C2×C4×C8—21no—
63C2×C4×C8—22no—
64C2×C4×C8C2311yescharacteristic
65C2×C4×C8C2311yescharacteristic
66#400—24no—
67#400—24no—
68#400—24no—
69#389—24no—
70#389—24no—
71#389—24no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—24no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389D412yes—
80#389D414yes—
81#387—24no—
82#387—22no—
83#384—22no—
84#384C2312yes—
85#384C2312yes—
86#382—24no—
87#382—22no—
88#382—22no—
89#382—21no—
90#382—21no—
91#382D411yescharacteristic
92C4×C4⋊C4—24no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—22no—
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—22no—
119#125—42no—
120#125—42no—
121#125—42no—
122#125—42no—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×Q8—24no—
126C4×Q8—42no—
127C4×Q8—44no—
128C4×Q8—44no—
129C4×Q8C2×Q812yes—
130C4×Q8C2×Q812yes—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4C2×D412yes—
146C2×C4⋊C4C2×D412yes—
147C2×C4⋊C4C2×D412yes—
148C2×C4⋊C4C2×D411yescharacteristic
149C2×C4⋊C4C2×D414yes—
150C2×C42—21no—
151C2×C42—24no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—21no—
155C2×C42—22no—
156C2×C42C2×D412yes—
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D412yes—
159C2×C42C2×D411yescharacteristic
160C2×C42C2411yescharacteristic
161#107—44no—
162#107—48no—
163#106—44no—
164#106—44no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8C2×Q812yes—
170C4⋊C8C2×Q812yes—
171#103—44no—
172C4×C8—21no—
173C4×C8—21no—
174C4×C8—44no—
175C4×C8—42no—
176C4×C8C2×Q812yes—
177C4×C8C2×Q812yes—
178#95—24no—
179#95—24no—
180#95—24no—
181#95—24no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—24no—
186#95—21no—
187#95—22no—
188#95—42no—
189C2×Q8—21no—
190C2×Q8—22no—
191C2×Q8C4⋊Q812yes—
192C2×Q8#13514yes—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—42no—
200C22×C4#12012yes—
201C22×C4#12012yes—
202C22×C4C22×D411yescharacteristic
203C22×C4C22×D411yescharacteristic
204C22×C4C22×D411yescharacteristic
205C2×C8—24no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—42no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C4⋊C4—21no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—48no—
230C4⋊C4—44no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—44no—
236C4⋊C4—48no—
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⋊C4C4⋊Q812yes—
246C42—22no—
247C42—22no—
248C42—22no—
249C42—22no—
250C42—21no—
251C42—21no—
252C42—42no—
253C42—44no—
254C42—48no—
255C42—42no—
256C42—42no—
257C42—41no—
258C42—41no—
259C42—82no—
260C42#13512yes—
261C42AGL(1,ℤ/8)12yes—
262C42C22×Q811yescharacteristic
263C42C22×Q811yescharacteristic
264C422−1+411yescharacteristic
265C422−1+411yescharacteristic
266C23C2×#12011yescharacteristic
267Q8—22no—
268Q8—24no—
269Q8#49914yescomplemented
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4#44914yes—
296C2×C4C2×C4⋊Q811yescharacteristic
297C2×C4#53611yescharacteristic
298C2×C4D4×Q811yescharacteristic
299C2×C4D4×Q811yescharacteristic
300C2×C4D4×Q811yescharacteristic
301C2×C4D4×Q811yescharacteristic
302C2×C4C2×#13511yescharacteristic
303C2×C4C2×#13512yes—
304C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
305C2×C4#57512yes—
306C2×C4#57811yescharacteristic
307C2×C4#57811yescharacteristic
308C8—24no—
309C8—42no—
310C8—42no—
311C8—42no—
312C8—82no—
313C22#243411yescharacteristic
314C22C2×#44911yescharacteristic
315C22#300212yes—
316C22#301111yescharacteristic
317C22#301812yes—
318C4—24no—
319C4—24no—
320C4—21no—
321C4—21no—
322C4—42no—
323C4—42no—
324C4—82no—
325C4—84no—
326C4—82no—
327C4—82no—
328C4—162no—
329C4C2×#49911yescharacteristic
330C4#316611yescharacteristic
331C4#323212yes—
332C4#338311yescharacteristic
333C4#338311yescharacteristic
334C4#338611yescharacteristic
335C4#338611yescharacteristic
336C2#1852012yes—
337C2#1867312yes—
338C2#2581511yescharacteristic
339C2#2583111yescharacteristic
340C2#2584011yescharacteristic
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
269. Q8#499semidirect

Maximal subgroups

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

subgroupindex
#365842
#332582
#330672
#286912
#105782
#105762
#98992
#98712
#88962
#88842
C4×Q8⋊C82