Finite Group Databasebeta

← #4893961

Subgroups — #4893961

Order 512.

2242 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17208C214yesmaximal, complemented
2#16873C214yesmaximal, complemented
3#16668C214yesmaximal, complemented
4#16419C214yesmaximal, complemented
5#16298C214yesmaximal
6#16245C211yesmaximal, characteristic, complemented
7#16208C212yesmaximal, complemented
8#15616C211yesmaximal, characteristic, complemented
9#14893C214yesmaximal, complemented
10#14341C211yesmaximal, characteristic
11#13868C212yesmaximal
12C22×#117C2212yes—
13C23×C22⋊C4C2211yescharacteristic
14C2×#400C2214yescomplemented
15C2×#399C2214yes—
16C2×#396C2214yes—
17C2×#396C2214yes—
18C2×#389C2214yescomplemented
19C2×#381C2212yes—
20C2×#379C2212yes—
21C2×#379C2214yes—
22#1445C2214yescomplemented
23#1445C2214yescomplemented
24#1444C2214yescomplemented
25#1444C2214yescomplemented
26#1442C2214yescomplemented
27#1442C2214yescomplemented
28#1441C2214yescomplemented
29#1441C2214yescomplemented
30#1441C2214yescomplemented
31#1441C2214yescomplemented
32#1441C2214yescomplemented
33#1440C2211yescharacteristic
34#1440C2214yes—
35#1440C2212yes—
36#1440C2211yescharacteristic
37#1439C2214yes—
38#1439C2214yes—
39#1439C2214yes—
40#1439C2214yes—
41#1439C2214yes—
42#1439C2214yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1437C2212yes—
49#1437C2212yes—
50#1437C2212yes—
51#1437C2212yes—
52#1436C2214yescomplemented
53#1436C2214yescomplemented
54#1434C2214yescomplemented
55C22⋊#95C2214yes—
56C22⋊#95C2214yes—
57C26C2311yescharacteristic
58C24×C4C2312yes—
59C2×#117—48no—
60C2×#117—44no—
61C22×C4⋊C4C2314yes—
62C22×C4⋊C4C2314yes—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C22⋊C4—44no—
69C22×C22⋊C4Q818yes—
70C22×C22⋊C4C2314yes—
71C22×C22⋊C4C2314yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2312yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2312yes—
80C22×C22⋊C4C2311yescharacteristic
81C22×C22⋊C4C2312yes—
82C22×C22⋊C4C2311yescharacteristic
83C22×C42C2312yes—
84C22×C42C2312yes—
85#400—48no—
86#399—48no—
87#396—48no—
88#396—48no—
89#389—48no—
90#381—44no—
91#379—48no—
92#379—44no—
93C2×#95C2314yes—
94C2×#95C2314yes—
95C2×#95C2314yes—
96C2×#95C2314yes—
97C2×#95C2314yes—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2312yes—
110C2×#95C2312yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C25—24no—
115C25—44no—
116C25—44no—
117C25—44no—
118C25C2×Q814yes—
119C25C2411yescharacteristic
120C25C2411yescharacteristic
121C25C2411yescharacteristic
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—44no—
125C23×C4—44no—
126C23×C4C2×Q818yes—
127C23×C4C2414yes—
128C23×C4C2414yes—
129C23×C4C2414yes—
130C23×C4C2414yes—
131C23×C4C2414yes—
132C23×C4C2412yes—
133C23×C4C2412yes—
134C23×C4C2414yes—
135#117—432no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—44no—
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×C22⋊C4—28no—
158C2×C22⋊C4—28no—
159C2×C22⋊C4—28no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—416no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
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—42no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4C2×Q814yes—
191C2×C22⋊C4C2×Q818yes—
192C2×C22⋊C4C2×Q814yes—
193C2×C42—28no—
194C2×C42—42no—
195C2×C42—44no—
196C2×C42—42no—
197#95—28no—
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—42no—
208#95—44no—
209#95—42no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—44no—
218C24—24no—
219C24—24no—
220C24—24no—
221C24—24no—
222C24—24no—
223C24—22no—
224C24—22no—
225C24—24no—
226C24—24no—
227C24—24no—
228C24—24no—
229C24—28no—
230C24—48no—
231C24—416no—
232C24—416no—
233C24—84no—
234C24—84no—
235C24—88no—
236C24—88no—
237C24—88no—
238C24—88no—
239C24—88no—
240C24C22×Q812yes—
241C24C22×Q812yes—
242C24C22×Q812yes—
243C24H(5,𝔽2)14yes—
244C24C2511yescharacteristic
245C22×C4—216no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
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—24no—
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—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—416no—
296C22×C4—48no—
297C22×C4—48no—
298C22×C4—416no—
299C22×C4C22×Q814yes—
300C22×C4C22×Q814yes—
301C22×C4H(5,𝔽2)18yes—
302C4⋊C4—416no—
303C4⋊C4—416no—
304C4⋊C4—88no—
305C4⋊C4—88no—
306C4⋊C4—88no—
307C4⋊C4—88no—
308C4⋊C4—88no—
309C22⋊C4—44no—
310C22⋊C4—48no—
311C22⋊C4—44no—
312C22⋊C4—416no—
313C22⋊C4—88no—
314C22⋊C4—88no—
315C22⋊C4—88no—
316C22⋊C4—88no—
317C22⋊C4—84no—
318C22⋊C4—84no—
319C22⋊C4—88no—
320C22⋊C4—88no—
321C22⋊C4—88no—
322C42—48no—
323C42—48no—
324C23—24no—
325C23—28no—
326C23—24no—
327C23—28no—
328C23—24no—
329C23—28no—
330C23—22no—
331C23—22no—
332C23—24no—
333C23—24no—
334C23—28no—
335C23—28no—
336C23—416no—
337C23—44no—
338C23—44no—
339C23—44no—
340C23—44no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—44no—
345C23—44no—
346C23—42no—
347C23—44no—
348C23—44no—
349C23—44no—
350C23—44no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23—816no—
358C23—832no—
359C23—832no—
360C23—832no—
361C23#54314yes—
362C23C23×Q811yescharacteristic
363C23C2×H(5,𝔽2)12yes—
364C23C2×H(5,𝔽2)14yes—
365C23#58514yes—
366C23#58514yes—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—28no—
370C2×C4—28no—
371C2×C4—28no—
372C2×C4—28no—
373C2×C4—28no—
374C2×C4—432no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C22—24no—
421C22—28no—
422C22—48no—
423C22—48no—
424C22—48no—
425C22—42no—
426C22—42no—
427C22—44no—
428C22—44no—
429C22—48no—
430C22—48no—
431C22—832no—
432C22—84no—
433C22—84no—
434C22—84no—
435C22—84no—
436C22C2×#54311yescharacteristic
437C22#345812yes—
438C22#350114yes—
439C22#353212yes—
440C22#353312yes—
441C22#353314yes—
442C22#354214yes—
443C22#354214yes—
444C22#356014yes—
445C22#356214yes—
446C22#356614yes—
447C4—44no—
448C4—44no—
449C4—48no—
450C4—84no—
451C4—84no—
452C4—84no—
453C4—84no—
454C4—84no—
455C2—44no—
456C2—82no—
457C2—82no—
458C2#3674712yes—
459C2#3747911yescharacteristic
460C2#4062314yes—
461C2#5629314yes—
462C2#5663514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17208C2semidirect
2. #16873C2semidirect
3. #16668C2semidirect
4. #16419C2semidirect
6. #16245C2semidirect
7. #16208C2semidirect
8. #15616C2semidirect
9. #14893C2semidirect
14. C2×#400C22semidirect
18. C2×#389C22semidirect
22. #1445C22semidirect
23. #1445C22semidirect
24. #1444C22semidirect
25. #1444C22semidirect
26. #1442C22semidirect
27. #1442C22semidirect
28. #1441C22semidirect
29. #1441C22semidirect
30. #1441C22semidirect
31. #1441C22semidirect
32. #1441C22semidirect
52. #1436C22semidirect
53. #1436C22semidirect
54. #1434C22semidirect

Maximal subgroups

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

subgroupindex
#172082
#168732
#166682
#164192
#162982
#162452
#162082
#156162
#148932
#143412
#138682