Finite Group Databasebeta

← #26423283

Subgroups — #26423283

Order 12032.

2450 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689749C211yesmaximal, characteristic, complemented
2#20251675C212yesmaximal, complemented
3#20249803C218yesmaximal, complemented
4#20249725C218yesmaximal, complemented
5#20249201C212yesmaximal, complemented
6#20248546C212yesmaximal, complemented
7#20247862C212yesmaximal, complemented
8#20247666C212yesmaximal, complemented
9#20246711C212yesmaximal
10#20245986C211yesmaximal, characteristic, complemented
11C22×#123×C47C211yesmaximal, characteristic
12#15689749—28no—
13C2×#13094005C2214yes—
14C2×#13094003C2212yes—
15C2×#13094003C2212yes—
16C2×#13094003C2214yescomplemented
17C2×#13094001C2214yescomplemented
18C2×#13094001C2214yescomplemented
19C22×C22⋊Dic47C2212yes—
20#15689062C2214yescomplemented
21#15689056C2218yescomplemented
22#15689054C2218yescomplemented
23#15689051C2218yescomplemented
24#15689048C2214yescomplemented
25#15689043C2214yescomplemented
26#15689037C2218yes—
27#15689035C2214yescomplemented
28#15689028C2214yescomplemented
29#15689017C2218yescomplemented
30#15689016C2218yescomplemented
31#15689003C2214yes—
32#15688996C2214yes—
33#15688996C2214yes—
34#15688996C2218yescomplemented
35#15688995C2214yescomplemented
36#15688992C2214yescomplemented
37#15688986C2214yes—
38#15688984C2214yescomplemented
39#15688983C2212yes—
40#15688983C2214yescomplemented
41#15688979C2214yes—
42#15688979C2212yes—
43C4×C22⋊Dic47C2212yes—
44C23×D4×C47C2211yescharacteristic
45C24×C4×C47C2211yescharacteristic
46C2×#123×C47—28no—
47C2×#123×C47C2218yes—
48C22×C4⋊C4×C47C2211yescharacteristic
49C22×C4⋊C4×C47C2211yescharacteristic
50C22×C22⋊C4×C47C2211yescharacteristic
51C22×C22⋊C4×C47C2211yescharacteristic
52C22×C22⋊C4×C47C2211yescharacteristic
53C22×Dic94C2312yes—
54C22×Dic94C2312yes—
55C23×Dic47C2312yes—
56C23×Dic47C2312yes—
57#13094005—48no—
58#13094003—28no—
59#13094003—28no—
60#13094003—48no—
61#13094001—216no—
62#13094001—216no—
63C2×C4⋊Dic47C2312yes—
64C2×C4⋊Dic47C2312yes—
65C2×C4⋊Dic47C2314yes—
66C2×Dic47⋊C4C2314yes—
67C2×Dic47⋊C4C2314yes—
68C2×Dic47⋊C4C2318yes—
69C2×Dic47⋊C4C2318yes—
70C2×C22⋊Dic47—28no—
71C2×C22⋊Dic47C2314yes—
72C2×C22⋊Dic47C2314yes—
73C2×C22⋊Dic47C2318yes—
74C2×C22⋊Dic47C2314yes—
75C2×C22⋊Dic47C2318yes—
76C2×C22⋊Dic47C2314yes—
77C2×C22⋊Dic47C2314yes—
78C2×C4×Dic47C2314yes—
79C47⋊#95C2314yes—
80C47⋊#95C2314yes—
81C47⋊#95C2318yes—
82C47⋊#95C2318yes—
83C47⋊#95C2318yes—
84C47⋊#95C2318yes—
85C25×C47C2311yescharacteristic
86C25×C47C2311yescharacteristic
87C22×D4×C47—24no—
88C22×D4×C47—44no—
89C22×D4×C47C2314yes—
90C23×C4×C47—24no—
91C23×C4×C47—44no—
92C23×C4×C47C2311yescharacteristic
93C23×C4×C47C2314yes—
94C23×C4×C47C2311yescharacteristic
95C23×C4×C47C2311yescharacteristic
96C23×C4×C47C2311yescharacteristic
97C23×C4×C47C2311yescharacteristic
98#123×C47—232no—
99C2×C4⋊C4×C47—24no—
100C2×C4⋊C4×C47—24no—
101C2×C4⋊C4×C47C2314yes—
102C2×C4⋊C4×C47C2314yes—
103C2×C22⋊C4×C47—24no—
104C2×C22⋊C4×C47—24no—
105C2×C22⋊C4×C47D418yes—
106C2×C22⋊C4×C47C2314yes—
107C2×C22⋊C4×C47C2314yes—
108C2×C22⋊C4×C47C2314yes—
109C24×C47—22no—
110C24×C47—24no—
111C24×C47—24no—
112C24×C47—24no—
113C24×C47—42no—
114C24×C47—42no—
115C24×C47—42no—
116C24×C47C2×D414yes—
117C24×C47C2412yes—
118C24×C47C2412yes—
119C24×C47C2411yescharacteristic
120C2×D4×C47—28no—
121C2×D4×C47—28no—
122C2×D4×C47—44no—
123C2×D4×C47—416no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—28no—
128C22×C4×C47—24no—
129C22×C4×C47—28no—
130C22×C4×C47—42no—
131C22×C4×C47—44no—
132C22×C4×C47—42no—
133C22×C4×C47—42no—
134C22×C4×C47—42no—
135C22×C4×C47—416no—
136C22×C4×C47C2×D414yes—
137C22×C4×C47C2×D414yes—
138C22×C4×C47C2412yes—
139C22×C4×C47C2412yes—
140C22×C4×C47C2412yes—
141C22×C4×C47C2412yes—
142C22×C4×C47C2412yes—
143C22×Dic47—24no—
144C22×Dic47—24no—
145C22×Dic47—44no—
146C22×Dic47—44no—
147C22×Dic47C2418yes—
148C22×Dic47C2414yes—
149C22×Dic47C2414yes—
150C2×Dic94—24no—
151C2×Dic94—24no—
152C2×Dic94—44no—
153C2×Dic94—44no—
154C4⋊C4×C47—28no—
155C4⋊C4×C47—28no—
156C22⋊C4×C47—28no—
157C22⋊C4×C47—28no—
158C22⋊C4×C47C2×D4116yes—
159C22⋊Dic47—28no—
160C22⋊Dic47—28no—
161C22⋊Dic47—216no—
162C22⋊Dic47—28no—
163C22⋊Dic47—44no—
164C22⋊Dic47—48no—
165C22⋊Dic47—44no—
166C4⋊Dic47—24no—
167C4⋊Dic47—24no—
168C4⋊Dic47—44no—
169Dic47⋊C4—28no—
170Dic47⋊C4—28no—
171Dic47⋊C4—48no—
172Dic47⋊C4—48no—
173C4×Dic47—44no—
174C23×C47—22no—
175C23×C47—22no—
176C23×C47—28no—
177C23×C47—22no—
178C23×C47—24no—
179C23×C47—22no—
180C23×C47—24no—
181C23×C47—24no—
182C23×C47—24no—
183C23×C47—24no—
184C23×C47—28no—
185C23×C47—44no—
186C23×C47—44no—
187C23×C47—416no—
188C23×C47—44no—
189C23×C47—44no—
190C23×C47—88no—
191C23×C47C22×D414yes—
192C23×C47C22×D412yes—
193C23×C47H(5,𝔽2)14yes—
194C23×C472−1+414yes—
195C23×C47C2511yescharacteristic
196D4×C47—416no—
197C2×C4×C47—24no—
198C2×C4×C47—24no—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—22no—
202C2×C4×C47—24no—
203C2×C4×C47—24no—
204C2×C4×C47—24no—
205C2×C4×C47—44no—
206C2×C4×C47—44no—
207C2×C4×C47—416no—
208C2×C4×C47—44no—
209C2×C4×C47—44no—
210C2×C4×C47—44no—
211C2×C4×C47C22×D414yes—
212C2×C4×C47C22×D414yes—
213C2×Dic47—24no—
214C2×Dic47—24no—
215C2×Dic47—28no—
216C2×Dic47—28no—
217C2×Dic47—28no—
218C2×Dic47—28no—
219C2×Dic47—28no—
220C2×Dic47—84no—
221C2×Dic47—84no—
222Dic94—84no—
223Dic94—84no—
224#36720—471nomaximal, Sylow
225C22×C47—22no—
226C22×C47—22no—
227C22×C47—28no—
228C22×C47—24no—
229C22×C47—24no—
230C22×C47—24no—
231C22×C47—24no—
232C22×C47—28no—
233C22×C47—416no—
234C22×C47—42no—
235C22×C47—42no—
236C22×C47—44no—
237C22×C47—48no—
238C22×C47—88no—
239C22×C47#53614yes—
240C22×C47C23×D411yescharacteristic
241C22×C47C2×H(5,𝔽2)12yes—
242C22×C47C2×2−1+412yes—
243C22×C47#58512yes—
244C4×C47—24no—
245C4×C47—24no—
246C4×C47—42no—
247C4×C47—42no—
248C4×C47—42no—
249Dic47—44no—
250Dic47—44no—
251Dic47—48no—
252C2×#543—471no—
253C22×#123D4711yescharacteristic
254#2848—472no—
255#2687—478no—
256#2680—478no—
257#2627—472no—
258#2561—472no—
259#2502—472no—
260#2483—472no—
261#2405—472no—
262#2312—471no—
263C2×C47—24no—
264C2×C47—44no—
265C2×C47—44no—
266C2×C47C2×#53611yescharacteristic
267C2×C47#348411yescharacteristic
268C2×C47#348511yescharacteristic
269C2×C47#353514yes—
270C23×D4C2×D4711yescharacteristic
271C24×C4C2×D4711yescharacteristic
272#543—948no—
273C2×#123—28no—
274C2×#123—474no—
275C2×#123C2×D4718yes—
276C2×C22⋊Q8—472no—
277C2×C22⋊Q8—472no—
278C2×C22⋊Q8—474no—
279C2×C22⋊Q8—474no—
280C2×C22⋊Q8—474no—
281C22×C4⋊C4C2×D4711yescharacteristic
282C22×C4⋊C4C2×D4711yescharacteristic
283C22×C22⋊C4—472no—
284C22×C22⋊C4C2×D4711yescharacteristic
285C22×C22⋊C4C2×D4711yescharacteristic
286C22×C22⋊C4C2×D4711yescharacteristic
287#400—474no—
288#398—478no—
289#398—478no—
290#397—474no—
291#397—478no—
292#395—474no—
293#394—478no—
294#393—474no—
295#391—474no—
296#388—478no—
297#388—478no—
298#386—474no—
299#385—474no—
300#385—478no—
301#385—474no—
302#385—474no—
303#384—474no—
304#381—474no—
305#381—474no—
306#380—472no—
307#380—474no—
308#379—474no—
309#379—472no—
310C4×C22⋊C4—472no—
311C47#3672011yescharacteristic, complemented, Sylow
312C25C22×D4711yescharacteristic
313C25C22×D4711yescharacteristic
314C22×Q8—472no—
315C22×Q8—472no—
316C22×D4—24no—
317C22×D4—44no—
318C22×D4C22×D4714yes—
319C23×C4—24no—
320C23×C4—44no—
321C23×C4—472no—
322C23×C4—472no—
323C23×C4C22×D4711yescharacteristic
324C23×C4C22×D4714yes—
325C23×C4C22×D4711yescharacteristic
326C23×C4C22×D4711yescharacteristic
327C23×C4C22×D4711yescharacteristic
328C23×C4C22×D4711yescharacteristic
329#123—232no—
330#123—1888no—
331C22⋊Q8—948no—
332C22⋊Q8—948no—
333C22⋊Q8—9416no—
334C22⋊Q8—9416no—
335C22⋊Q8—1888no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—474no—
339C2×C4⋊C4—474no—
340C2×C4⋊C4—472no—
341C2×C4⋊C4—472no—
342C2×C4⋊C4—478no—
343C2×C4⋊C4—474no—
344C2×C4⋊C4—478no—
345C2×C4⋊C4C22×D4714yes—
346C2×C4⋊C4C22×D4714yes—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—474no—
350C2×C22⋊C4—474no—
351C2×C22⋊C4—478no—
352C2×C22⋊C4—474no—
353C2×C22⋊C4—474no—
354C2×C22⋊C4—478no—
355C2×C22⋊C4—474no—
356C2×C22⋊C4—948no—
357C2×C22⋊C4C22⋊D4718yes—
358C2×C22⋊C4C22×D4714yes—
359C2×C22⋊C4C22×D4714yes—
360C2×C22⋊C4C22×D4714yes—
361C2×C42—474no—
362#95—474no—
363#95—478no—
364#95—474no—
365#95—478no—
366#95—478no—
367#95—478no—
368C24—22no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24C2×C22⋊D4714yes—
376C24C23×D4712yes—
377C24C23×D4712yes—
378C24C23×D4711yescharacteristic
379C2×Q8—944no—
380C2×Q8—944no—
381C2×Q8—1884no—
382C2×Q8—1884no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—44no—
386C2×D4—416no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—416no—
399C22×C4—478no—
400C22×C4—474no—
401C22×C4—474no—
402C22×C4—944no—
403C22×C4—944no—
404C22×C4—1884no—
405C22×C4—1884no—
406C22×C4C2×C22⋊D4714yes—
407C22×C4C2×C22⋊D4714yes—
408C22×C4C23×D4712yes—
409C22×C4C23×D4712yes—
410C22×C4C23×D4712yes—
411C22×C4C23×D4712yes—
412C22×C4C23×D4712yes—
413C4⋊C4—28no—
414C4⋊C4—28no—
415C4⋊C4—948no—
416C4⋊C4—948no—
417C4⋊C4—944no—
418C4⋊C4—944no—
419C4⋊C4—1888no—
420C4⋊C4—1888no—
421C4⋊C4—1884no—
422C22⋊C4—28no—
423C22⋊C4—28no—
424C22⋊C4—948no—
425C22⋊C4—948no—
426C22⋊C4—9416no—
427C22⋊C4—948no—
428C22⋊C4—1884no—
429C22⋊C4—1888no—
430C22⋊C4—1884no—
431C22⋊C4C2×C22⋊D47116yes—
432C42—1884no—
433C23—22no—
434C23—22no—
435C23—28no—
436C23—22no—
437C23—24no—
438C23—22no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—28no—
444C23—44no—
445C23—44no—
446C23—416no—
447C23—44no—
448C23—44no—
449C23—88no—
450C23C22×C22⋊D4714yes—
451C23C22×C22⋊D4712yes—
452C23#1309406914yes—
453C23#1309407114yes—
454C23C24×D4711yescharacteristic
455Q8—3764no—
456Q8—3764no—
457D4—416no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—416no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—944no—
473C2×C4—944no—
474C2×C4—948no—
475C2×C4—948no—
476C2×C4—948no—
477C2×C4—948no—
478C2×C4—948no—
479C2×C4—3764no—
480C2×C4—3764no—
481C2×C4C22×C22⋊D4714yes—
482C2×C4C22×C22⋊D4714yes—
483C22—22no—
484C22—28no—
485C22—22no—
486C22—24no—
487C22—28no—
488C22—24no—
489C22—24no—
490C22—24no—
491C22—416no—
492C22—42no—
493C22—44no—
494C22—48no—
495C22—42no—
496C22—88no—
497C22#1568969414yes—
498C22C23×C22⋊D4711yescharacteristic
499C22C2×#1309406912yes—
500C22C2×#1309407112yes—
501C22#1568995912yes—
502C4—24no—
503C4—24no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—1884no—
508C4—1888no—
509C4—1884no—
510C2—24no—
511C2—44no—
512C2—44no—
513C2C2×#1568969411yescharacteristic
514C2#2025702311yescharacteristic
515C2#2025703211yescharacteristic
516C2#2025760114yes—
subgroupG/H
center, FrattiniC23C24×D47
commutator, socle, cosocleC23×C47C25
FittingC22×#123×C47C2
prime radicalC23×D4×C47C22
hypercenterC22×#123D47
hypocenterC47#36720
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. C2×#15689749C2semidirect
2. #20251675C2semidirect
3. #20249803C2semidirect
4. #20249725C2semidirect
5. #20249201C2semidirect
6. #20248546C2semidirect
7. #20247862C2semidirect
8. #20247666C2semidirect
10. #20245986C2semidirect
16. C2×#13094003C22semidirect
17. C2×#13094001C22semidirect
18. C2×#13094001C22semidirect
20. #15689062C22semidirect
21. #15689056C22semidirect
22. #15689054C22semidirect
23. #15689051C22semidirect
24. #15689048C22semidirect
25. #15689043C22semidirect
27. #15689035C22semidirect
28. #15689028C22semidirect
29. #15689017C22semidirect
30. #15689016C22semidirect
34. #15688996C22semidirect
35. #15688995C22semidirect
36. #15688992C22semidirect
38. #15688984C22semidirect
40. #15688983C22semidirect
311. C47#36720semidirect

Maximal subgroups

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

subgroupindex
C2×#156897492
#202516752
#202498032
#202497252
#202492012
#202485462
#202478622
#202476662
#202467112
#202459862
C22×#123×C472
#3672047