Finite Group Databasebeta

← #364055

Subgroups — #364055

Order 512.

1274 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34177C211yesmaximal, characteristic
2C2×#1820C211yesmaximal, characteristic
3C2×#1740C211yesmaximal, characteristic
4#11351C212yesmaximal
5#11351C212yesmaximal
6#10829C212yesmaximal
7#10829C212yesmaximal
8#10736C212yesmaximal
9#10726C212yesmaximal
10C22×#117C2212yes—
11C22×#117C2211yescharacteristic
12#2343—24no—
13#2343—24no—
14#2315C2211yescharacteristic
15#2315C2211yescharacteristic
16#2313C414yescomplemented
17#2283C414yescomplemented
18C2×#381C2212yes—
19C2×#381C2211yescharacteristic
20C2×#381C2211yescharacteristic
21#2273C2211yescharacteristic
22#2273C2211yescharacteristic
23#1820—24no—
24#1740—24no—
25C2×#342C2212yes—
26C2×#342C2212yes—
27C2×#342C2214yes—
28C2×#342C2214yes—
29C2×#342C2212yes—
30C2×#342C2212yes—
31C2×#342C2214yes—
32C2×#342C2214yes—
33C24×C4C2311yescharacteristic
34C2×#117—22no—
35C2×#117—24no—
36C2×#117—22no—
37C2×#117—24no—
38C2×#117—24no—
39C2×#117—28no—
40C2×#117—44no—
41C2×#117D414yes—
42C2×#117D414yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4C2×C412yes—
45C22×C22⋊C4C2314yes—
46C22×C22⋊C4C2311yescharacteristic
47C22×C22⋊C4C2311yescharacteristic
48C22×C22⋊C4C2311yescharacteristic
49C22×C22⋊C4C2311yescharacteristic
50C22×C22⋊C4C2314yes—
51C22×C22⋊C4C2311yescharacteristic
52C22×C22⋊C4C2311yescharacteristic
53C22×C42—22no—
54C22×C42C2×C412yes—
55#387—24no—
56#387—24no—
57#387—24no—
58#387—24no—
59#385—22no—
60#385—22no—
61#385—22no—
62#385—22no—
63#383—24no—
64#382—24no—
65#381—22no—
66#381—22no—
67#381—24no—
68#381—22no—
69#381—22no—
70#381C2×C412yes—
71#381C2×C412yes—
72#381C2×C412yes—
73#381C2×C412yes—
74#380C2×C414yes—
75#380C2×C414yes—
76C4×C4⋊C4—24no—
77C4×C22⋊C4—22no—
78C4×C22⋊C4—22no—
79C4×C22⋊C4—22no—
80C4×C22⋊C4—22no—
81C4×C22⋊C4C2×C414yes—
82C4×C22⋊C4C2×C414yes—
83#376—24no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—24no—
87#342—24no—
88#342—24no—
89#342—24no—
90#342—24no—
91#342—28no—
92#342—28no—
93#342—44no—
94#342—44no—
95C25C2411yescharacteristic
96C23×C4—24no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—24no—
104C23×C4—24no—
105C23×C4C22×C411yescharacteristic
106C23×C4C22×C411yescharacteristic
107C23×C4C22×C412yes—
108C23×C4C22×C412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4#4112yes—
112C23×C4#4114yes—
113C23×C4#4112yes—
114#117—28no—
115#117—28no—
116#117—28no—
117#117—28no—
118#117—216no—
119#117—416no—
120#117—48no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—48no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4C22⋊C414yes—
128C2×C4⋊C4C22⋊C414yes—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4C22×C412yes—
155C2×C22⋊C4C22×C412yes—
156C2×C22⋊C4C22×C412yes—
157C2×C22⋊C4C22×C412yes—
158C2×C22⋊C4C2×D412yes—
159C2×C22⋊C4C2×D412yes—
160C2×C22⋊C4C2×D412yes—
161C2×C22⋊C4C2×D412yes—
162C2×C22⋊C4#4118yes—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—24no—
168C2×C42—24no—
169C2×C42—48no—
170C2×C42—44no—
171C2×C42C22⋊C414yes—
172C2×C42C22⋊C414yes—
173#95—24no—
174#95—24no—
175#95—24no—
176#95—24no—
177#95—24no—
178#95—24no—
179#95—24no—
180#95—24no—
181#95—44no—
182#95—44no—
183C24—82no—
184C24C23×C411yescharacteristic
185C24C22×D411yescharacteristic
186C24C22×D411yescharacteristic
187C24C2×#4112yes—
188C24C2×#4112yes—
189C24C2×#4114yes—
190C24H(5,𝔽2)12yes—
191C242−1+412yes—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—28no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—28no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—28no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—88no—
233C22×C4—88no—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C414yes—
237C22×C4C2×C22⋊C414yes—
238C22×C4#11711yescharacteristic
239C22×C4#11712yes—
240C22×C4#11711yescharacteristic
241C22×C4#11712yes—
242C22×C4#11712yes—
243C22×C4#12314yes—
244C22×C4#12314yes—
245C22×C4#12314yes—
246C22×C4#12314yes—
247C4⋊C4—28no—
248C4⋊C4—28no—
249C4⋊C4—416no—
250C4⋊C4—84no—
251C4⋊C4—84no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—24no—
260C22⋊C4—28no—
261C22⋊C4—28no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—41no—
265C22⋊C4—44no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C42—24no—
274C42—24no—
275C42—24no—
276C42—24no—
277C42—416no—
278C42—84no—
279C42—84no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—44no—
284C23—44no—
285C23—84no—
286C23—82no—
287C23—82no—
288C23C22×C22⋊C411yescharacteristic
289C23C2×#11711yescharacteristic
290C23C2×#11711yescharacteristic
291C23C4×#4112yes—
292C23#52012yes—
293C23C2×#12312yes—
294C23C2×#12312yes—
295C23#52512yes—
296C23#53212yes—
297C23#53511yescharacteristic
298C23#53611yescharacteristic
299C23#53712yes—
300C23#55114yes—
301C23#55114yes—
302C23#55214yes—
303C23#55214yes—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—28no—
316C2×C4—28no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—48no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—88no—
337C2×C4—88no—
338C2×C4—88no—
339C2×C4—88no—
340C2×C4—84no—
341C2×C4—162no—
342C2×C4—162no—
343C2×C4—161no—
344C2×C4—161no—
345C2×C4—162no—
346C2×C4—164no—
347C2×C4—164no—
348C2×C4#38112yes—
349C2×C4#38112yes—
350C2×C4#38114yes—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—24no—
355C22—24no—
356C22—44no—
357C22—42no—
358C22—42no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—44no—
363C22—44no—
364C22—42no—
365C22—42no—
366C22—44no—
367C22C2×#38111yescharacteristic
368C22#229711yescharacteristic
369C22#231111yescharacteristic
370C22#234312yes—
371C22#234312yes—
372C22#248212yes—
373C22#248412yes—
374C22#249814yes—
375C4—22no—
376C4—24no—
377C4—22no—
378C4—44no—
379C4—44no—
380C4—82no—
381C4—82no—
382C4—82no—
383C4—82no—
384C4—88no—
385C4—164no—
386C2—21no—
387C2—22no—
388C2—21no—
389C2—24no—
390C2—22no—
391C2—22no—
392C2—22no—
393C2#1395811yescharacteristic
394C2#2071012yes—
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2313C4semidirect
17. #2283C4semidirect

Maximal subgroups

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

subgroupindex
#341772
C2×#18202
C2×#17402
#113512
#108292
#107362
#107262