Finite Group Databasebeta

← #5482440

Subgroups — #5482440

Order 512.

1803 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17229C211yesmaximal, characteristic, complemented
2#17092C214yesmaximal
3#16918C214yesmaximal, complemented
4#16531C214yesmaximal, complemented
5#16247C211yesmaximal, characteristic, complemented
6#15907C214yesmaximal, complemented
7#15629C214yesmaximal, complemented
8#15384C212yesmaximal
9#14753C214yesmaximal, complemented
10#14652C212yesmaximal, complemented
11#14167C211yesmaximal, characteristic
12C22×#123C2211yescharacteristic
13C2×#399C2212yes—
14C2×#399C2214yes—
15C2×#398C2214yes—
16C2×#398C2214yes—
17C2×#397C2212yes—
18C2×#394C2214yes—
19C2×#389C2214yes—
20C2×#387C2214yes—
21C2×#385C2214yes—
22C2×#384C2212yes—
23C2×C4×C22⋊C4C2212yes—
24#1445C2214yes—
25#1445C2212yes—
26#1445C2214yes—
27#1444C2214yes—
28#1444C2214yes—
29#1443C2214yes—
30#1442C2214yes—
31#1442C2214yes—
32#1442C2214yes—
33#1442C2214yes—
34#1442C2214yes—
35#1441C2214yes—
36#1441C2214yes—
37#1441C2214yes—
38#1440C2214yes—
39#1440C2214yes—
40#1440C2211yescharacteristic
41#1440C2211yescharacteristic
42#1439C2214yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1438C2212yes—
49#1437C2212yes—
50#1437C2212yes—
51#1437C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57C23×D4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C2×#123—24no—
60C2×#123—44no—
61C22×C4⋊C4C2314yes—
62C22×C4⋊C4C2314yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C22⋊C4C2312yes—
71C22×C22⋊C4C2312yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2311yescharacteristic
79C22×C42C2312yes—
80C22×C42C2314yes—
81#399—44no—
82#399—48no—
83#398—48no—
84#398—48no—
85#397—28no—
86#394—48no—
87#389—48no—
88#387—48no—
89#385—48no—
90#384—44no—
91C4×C22⋊C4—44no—
92C2×#95C2314yes—
93C2×#95C2312yes—
94C2×#95C2314yes—
95C2×#95C2314yes—
96C2×#95C2314yes—
97C2×#95C2314yes—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2312yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C25C2411yescharacteristic
116C25C2411yescharacteristic
117C22×D4—22no—
118C22×D4—42no—
119C22×D4—44no—
120C23×C4—22no—
121C23×C4—24no—
122C23×C4—44no—
123C23×C4C2414yes—
124C23×C4C2414yes—
125C23×C4C2414yes—
126C23×C4C2414yes—
127C23×C4C2411yescharacteristic
128C23×C4C2412yes—
129C23×C4C2414yes—
130C23×C4C2412yes—
131C23×C4C2414yes—
132#123—416no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—44no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4#4114yes—
184C2×C42—42no—
185C2×C42—44no—
186C2×C42—44no—
187C2×C42—44no—
188C2×C42—44no—
189#95—24no—
190#95—28no—
191#95—28no—
192#95—28no—
193#95—28no—
194#95—28no—
195#95—28no—
196#95—28no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—44no—
201#95—44no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—42no—
212C24—24no—
213C24—22no—
214C24—24no—
215C24—24no—
216C24—21no—
217C24—22no—
218C24—24no—
219C24—24no—
220C24—24no—
221C24C2×#4112yes—
222C24C2511yescharacteristic
223C2×D4—44no—
224C2×D4—84no—
225C2×D4—88no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—28no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—21no—
250C22×C4—24no—
251C22×C4—24no—
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—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—88no—
279C22×C4—84no—
280C22×C4C2×#4114yes—
281C22×C4H(5,𝔽2)18yes—
282C22×C4H(5,𝔽2)14yes—
283C22×C42−1+418yes—
284C4⋊C4—416no—
285C4⋊C4—416no—
286C4⋊C4—48no—
287C4⋊C4—84no—
288C4⋊C4—88no—
289C4⋊C4—88no—
290C4⋊C4—88no—
291C4⋊C4—88no—
292C4⋊C4—88no—
293C22⋊C4—44no—
294C22⋊C4—48no—
295C22⋊C4—48no—
296C22⋊C4—416no—
297C22⋊C4—48no—
298C22⋊C4—88no—
299C22⋊C4—88no—
300C22⋊C4—88no—
301C22⋊C4—88no—
302C42—48no—
303C42—88no—
304C23—22no—
305C23—24no—
306C23—28no—
307C23—42no—
308C23—44no—
309C23—44no—
310C23—44no—
311C23—42no—
312C23—44no—
313C23—44no—
314C23—44no—
315C23—41no—
316C23—42no—
317C23—44no—
318C23—42no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—42no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23C22×#4111yescharacteristic
327C23C2×H(5,𝔽2)12yes—
328C23C2×H(5,𝔽2)14yes—
329C23C2×2−1+414yes—
330C23#58514yes—
331D4—88no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—28no—
335C2×C4—28no—
336C2×C4—28no—
337C2×C4—28no—
338C2×C4—28no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—28no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—88no—
388C22—42no—
389C22—44no—
390C22—48no—
391C22—84no—
392C22—84no—
393C22—81no—
394C22—84no—
395C22—82no—
396C22—84no—
397C22—84no—
398C22#349011yescharacteristic
399C22#350812yes—
400C22#351514yes—
401C22#352912yes—
402C22#353314yes—
403C22#353514yes—
404C22#353714yes—
405C22#353812yes—
406C22#355214yes—
407C22#355414yes—
408C22#355414yes—
409C4—44no—
410C4—48no—
411C4—44no—
412C4—48no—
413C4—84no—
414C4—81no—
415C4—84no—
416C4—84no—
417C4—84no—
418C2—82no—
419C2—161no—
420C2#4574212yes—
421C2#4645411yescharacteristic
422C2#4908714yes—
423C2#5247814yes—
424C2#5362314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17229C2semidirect
3. #16918C2semidirect
4. #16531C2semidirect
5. #16247C2semidirect
6. #15907C2semidirect
7. #15629C2semidirect
9. #14753C2semidirect
10. #14652C2semidirect

Maximal subgroups

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

subgroupindex
#172292
#170922
#169182
#165312
#162472
#159072
#156292
#153842
#147532
#146522
#141672