Finite Group Databasebeta

← #26199894

Subgroups — #26199894

Order 12032.

636 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254730C211yesmaximal, characteristic, complemented
2#20254326C211yesmaximal, characteristic
3#20253763C211yesmaximal, characteristic
4#20242956C212yesmaximal, complemented
5#20242928C212yesmaximal, complemented
6#20242832C212yesmaximal, complemented
7#20242804C212yesmaximal, complemented
8#20242140C211yesmaximal, characteristic, complemented
9#20242046C211yesmaximal, characteristic, complemented
10C2×#15688866C211yesmaximal, characteristic
11C47×#2962C211yesmaximal, characteristic, complemented
12#15689680C2211yescharacteristic
13C2×#13094021C2211yescharacteristic
14#15689497—21no—
15#15689488—22no—
16#15689471—21no—
17#15689470C2212yes—
18#15689470C2212yes—
19#15689418C2212yescomplemented
20#15689415—21no—
21#15689362—22no—
22#15689333C2212yescomplemented
23#15689330—21no—
24#15689323C2212yes—
25#15689323C2212yes—
26#15689273—22no—
27#15689176C2211yescharacteristic
28C2×C4⋊(C47⋊C8)C2211yescharacteristic
29#15689129—21no—
30#15689129—21no—
31#15688876C2212yescomplemented
32#15688874C2212yescomplemented
33#15688871C2211yescharacteristic, complemented
34#15688871C2211yescharacteristic, complemented
35#15688868C2211yescharacteristic, complemented
36#15688868C2211yescharacteristic, complemented
37#15688866C2212yes—
38#15688866C2212yes—
39C47×#533C2211yescharacteristic
40C47×#446—22no—
41C47×#446C2212yescomplemented
42C8×Q8×C47C2212yescomplemented
43C47×#424—21no—
44C2×C4⋊C8×C47C2211yescharacteristic
45C47×#405—21no—
46C4×M16×C47C2211yescharacteristic
47C22×D188—21no—
48C22×D188—21no—
49C22×D188—21no—
50C2×#10625337—22no—
51C2×#10625336—22no—
52C2×#10625336—22no—
53C22×C47⋊C8—21no—
54#13094024C2311yescharacteristic
55#13094024C2311yescharacteristic
56#13094021—21no—
57#13094021—21no—
58#13094021—21no—
59#13094021C2312yes—
60#13094016—21no—
61#13094003—21no—
62#13094003—21no—
63C4×Dic94—21no—
64C4×Dic94—21no—
65#13093983—21no—
66#13093957—22no—
67C4⋊(C47⋊C8)—21no—
68C4⋊(C47⋊C8)C2311yescharacteristic
69C4⋊(C47⋊C8)C2311yescharacteristic
70C4⋊(C47⋊C8)C2312yes—
71#13093948—22no—
72#13093947—22no—
73#13093946—22no—
74#13093946—22no—
75#13093946—22no—
76#13093946—22no—
77#13093944—22no—
78#13093944—22no—
79C22⋊(C47⋊C8)—21no—
80C4×C47⋊C8—21no—
81C2×M16×C47D411yescharacteristic
82C2×M16×C47D411yescharacteristic
83C22×C8×C47—21no—
84C4⋊Q8×C47D411yescharacteristic
85C4⋊Q8×C47D411yescharacteristic
86#125×C47—21no—
87C22⋊Q8×C47—21no—
88C22⋊Q8×C47D412yes—
89C4×Q8×C47—21no—
90C4×Q8×C47C2312yes—
91#117×C47—21no—
92C2×C42×C47C2311yescharacteristic
93C4⋊C8×C47—22no—
94C4⋊C8×C47—22no—
95C4⋊C8×C47—21no—
96C4⋊C8×C47C2311yescharacteristic
97C4⋊C8×C47C2311yescharacteristic
98#97×C47—22no—
99#97×C47D412yes—
100C4×C8×C47—22no—
101C4×C8×C47C2312yes—
102C23×D47—41no—
103C2×Q8×C47—21no—
104C2×Q8×C47C2×D412yes—
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47#4111yescharacteristic
108C2×D188—22no—
109C2×D188—22no—
110C2×D188—22no—
111C2×D188—42no—
112C2×D188—42no—
113C2×D188—41no—
114C2×D188—41no—
115C2×D188—41no—
116C2×D188—81no—
117C2×D188—81no—
118C2×D188—82no—
119C2×Dic94—21no—
120C2×Dic94—21no—
121M16×C47—22no—
122M16×C47—22no—
123C2×C8×C47—21no—
124C2×C8×C47—21no—
125C2×C8×C47—42no—
126C2×C8×C47C2×D412yes—
127C2×C8×C47C2×D412yes—
128C4⋊C4×C47—21no—
129C4⋊C4×C47—21no—
130C4⋊C4×C47—21no—
131C4⋊C4×C47—22no—
132C4⋊C4×C47—22no—
133C4⋊C4×C47C2×D412yes—
134C22⋊C4×C47—22no—
135C42×C47—22no—
136C42×C47C2×D411yescharacteristic
137C42×C47C2×D411yescharacteristic
138C42×C47#4111yescharacteristic
139C42×C47C2411yescharacteristic
140C22⋊Dic47—41no—
141#10625337—82no—
142#10625336—82no—
143#10625336—82no—
144C4⋊Dic47—21no—
145C4⋊Dic47—21no—
146C4⋊Dic47—21no—
147C4⋊Dic47—21no—
148Dic47⋊C4—41no—
149Dic47⋊C4—41no—
150C4×Dic47—41no—
151C2×C47⋊C8—21no—
152C2×C47⋊C8—21no—
153C2×C47⋊C8—22no—
154C2×C47⋊C8—81no—
155C23×C47#12111yescharacteristic
156C22×D47—42no—
157C22×D47—82no—
158C22×D47—82no—
159C22×D47—82no—
160Q8×C47—22no—
161Q8×C47—22no—
162Q8×C47—81no—
163C2×C4×C47—21no—
164C2×C4×C47—21no—
165C2×C4×C47—22no—
166C2×C4×C47—22no—
167C2×C4×C47—41no—
168C2×C4×C47#12111yescharacteristic
169C2×C4×C47#12111yescharacteristic
170C2×C4×C47#12111yescharacteristic
171C2×C4×C47#13512yes—
172C2×C4×C47#13712yes—
173C2×C4×C47C22×D411yescharacteristic
174C2×C4×C47C22×D411yescharacteristic
175C2×C4×C47C2×#4111yescharacteristic
176C2×Dic47—41no—
177C2×Dic47—41no—
178D188—42no—
179D188—42no—
180D188—42no—
181D188—42no—
182D188—82no—
183D188—82no—
184D188—82no—
185D188—82no—
186Dic94—41no—
187Dic94—41no—
188Dic94—81no—
189C8×C47—22no—
190C8×C47—22no—
191C8×C47—41no—
192C8×C47—41no—
193C47⋊C8—42no—
194C47⋊C8—41no—
195C47⋊C8—41no—
196#19242—471nomaximal, Sylow
197C22×C47—21no—
198C22×C47—21no—
199C22×C47#46412yes—
200C22×C47C2×#12111yescharacteristic
201C2×D47—82no—
202C2×D47—82no—
203C2×D47—82no—
204C2×D47—164no—
205C4×C47—21no—
206C4×C47—21no—
207C4×C47—21no—
208C4×C47—42no—
209C4×C47—42no—
210C4×C47#46412yes—
211C4×C47C2×#13511yescharacteristic
212C4×C47C2×#13711yescharacteristic
213C4×C47#57611yescharacteristic
214C4×C47#57711yescharacteristic
215Dic47—81no—
216Dic47—81no—
217#3249—471no—
218#3208—471no—
219#3138—471no—
220#2962D4711yescharacteristic, complemented
221#1702—472no—
222#1698—472no—
223#1682—472no—
224#1678—472no—
225#1565—471no—
226#1542—471no—
227C2×#331—471no—
228C2×C47—22no—
229C2×C47C2×#46411yescharacteristic
230C2×C47#306211yescharacteristic
231C2×C47#306811yescharacteristic
232D47—162no—
233#533—471no—
234#533C2×D4711yescharacteristic
235C2×#127—471no—
236#492—941no—
237#490—942no—
238#486—472no—
239#486—472no—
240#486—941no—
241#477—472no—
242#477—941no—
243#467—942no—
244#462—472no—
245#462—941no—
246#460—472no—
247#460—472no—
248#450—942no—
249#446—22no—
250#446C2×D4712yescomplemented
251C8×Q8C2×D4712yescomplemented
252#431—471no—
253#424—21no—
254C2×C4⋊C8—471no—
255C2×C4⋊C8C2×D4711yescharacteristic
256#418—941no—
257#418—941no—
258#405—21no—
259C4×M16C2×D4711yescharacteristic
260#335—472no—
261#334—472no—
262#333—471no—
263#333—471no—
264#332—471no—
265#332—471no—
266#331—472no—
267#331—472no—
268C47#1924211yescharacteristic, complemented, Sylow
269C22×D4—941no—
270C22×D4—941no—
271C22×D4—941no—
272C2×Q16—942no—
273C2×SD16—942no—
274C2×SD16—942no—
275C2×M16C22⋊D4711yescharacteristic
276C2×M16C22⋊D4711yescharacteristic
277C22×C8—21no—
278C22×C8—941no—
279C4⋊Q8—471no—
280C4⋊Q8—471no—
281C4⋊Q8D18811yescharacteristic
282C4⋊Q8D18811yescharacteristic
283#127—472no—
284#127—941no—
285#127—941no—
286#127—941no—
287#125—21no—
288#125—941no—
289C22⋊Q8—21no—
290C22⋊Q8—941no—
291C22⋊Q8—941no—
292C22⋊Q8D18812yes—
293C4×Q8—21no—
294C4×Q8—941no—
295C4×Q8—941no—
296C4×Q8C22×D4712yes—
297#117—21no—
298#117—941no—
299C2×C42C22×D4711yescharacteristic
300#107—942no—
301C4⋊C8—22no—
302C4⋊C8—22no—
303C4⋊C8—21no—
304C4⋊C8—472no—
305C4⋊C8—471no—
306C4⋊C8—471no—
307C4⋊C8—941no—
308C4⋊C8C22×D4711yescharacteristic
309C4⋊C8C22×D4711yescharacteristic
310#103—942no—
311#103—942no—
312#102—942no—
313#102—942no—
314#102—942no—
315#102—942no—
316#102—942no—
317#102—942no—
318C22⋊C8—941no—
319#97—22no—
320#97C22⋊D4712yes—
321C4×C8—22no—
322C4×C8—941no—
323C4×C8C22×D4712yes—
324C24—1881no—
325C2×Q8—21no—
326C2×Q8—941no—
327C2×Q8—941no—
328C2×Q8C2×D18812yes—
329C2×D4—942no—
330C2×D4—942no—
331C2×D4—942no—
332C2×D4—3761no—
333C2×D4—3761no—
334C2×D4—3762no—
335C2×D4—1881no—
336C2×D4—1881no—
337C2×D4—1881no—
338C2×D4—1882no—
339C2×D4—1882no—
340C22×C4C2×D18811yescharacteristic
341C22×C4#1062535011yescharacteristic
342C22×C4C2×C22⋊D4711yescharacteristic
343Q16—3762no—
344SD16—3762no—
345SD16—3762no—
346M16—22no—
347M16—22no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—42no—
351C2×C8—941no—
352C2×C8—941no—
353C2×C8—942no—
354C2×C8—3761no—
355C2×C8C2×C22⋊D4712yes—
356C2×C8C2×C22⋊D4712yes—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—941no—
363C4⋊C4—941no—
364C4⋊C4—941no—
365C4⋊C4—941no—
366C4⋊C4—1881no—
367C4⋊C4—1881no—
368C4⋊C4C2×D18812yes—
369C22⋊C4—22no—
370C22⋊C4—1881no—
371C42—22no—
372C42—1881no—
373C42C2×D18811yescharacteristic
374C42#1062535011yescharacteristic
375C42C2×C22⋊D4711yescharacteristic
376C42C23×D4711yescharacteristic
377C23—3762no—
378C23—3762no—
379C23—3762no—
380C23—1882no—
381C23#1309399811yescharacteristic
382Q8—22no—
383Q8—22no—
384Q8—81no—
385Q8—3761no—
386Q8—1881no—
387Q8—1881no—
388D4—3762no—
389D4—3762no—
390D4—3762no—
391D4—3762no—
392D4—1882no—
393D4—1882no—
394D4—1882no—
395D4—1882no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—1881no—
402C2×C4—1881no—
403C2×C4#1309399811yescharacteristic
404C2×C4#1309399811yescharacteristic
405C2×C4#1309399811yescharacteristic
406C2×C4#1309404412yes—
407C2×C4#1309405512yes—
408C2×C4C22×C22⋊D4711yescharacteristic
409C2×C4C22×D18811yescharacteristic
410C2×C4C2×#1062535011yescharacteristic
411C8—22no—
412C8—22no—
413C8—41no—
414C8—41no—
415C8—1882no—
416C8—1881no—
417C8—1881no—
418C22—21no—
419C22—21no—
420C22—3762no—
421C22—3762no—
422C22—3762no—
423C22—7524no—
424C22#1568934512yes—
425C22C2×#1309399811yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—42no—
431C4—3761no—
432C4—3761no—
433C4#1568934512yes—
434C4C2×#1309404411yescharacteristic
435C4C2×#1309405511yescharacteristic
436C4#1568992811yescharacteristic
437C4#1568993511yescharacteristic
438C2—22no—
439C2—7522no—
440C2C2×#1568934511yescharacteristic
441C2#2025306811yescharacteristic
442C2#2025311611yescharacteristic
subgroupG/H
centerC22C2×#13093998
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#2962C2
socleC22×C47C2×#121
prime radicalC2×#13094021C22
hypercenter#2962D47
hypocenterC47#19242
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
1. #20254730C2semidirect
4. #20242956C2semidirect
5. #20242928C2semidirect
6. #20242832C2semidirect
7. #20242804C2semidirect
8. #20242140C2semidirect
9. #20242046C2semidirect
11. C47×#2962C2semidirect
19. #15689418C22semidirect
22. #15689333C22semidirect
31. #15688876C22semidirect
32. #15688874C22semidirect
33. #15688871C22semidirect
34. #15688871C22semidirect
35. #15688868C22semidirect
36. #15688868C22semidirect
41. C47×#446C22semidirect
42. C8×Q8×C47C22semidirect
220. #2962D47semidirect
250. #446C2×D47semidirect
251. C8×Q8C2×D47semidirect
268. C47#19242semidirect

Maximal subgroups

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

subgroupindex
#202547302
#202543262
#202537632
#202429562
#202429282
#202428322
#202428042
#202421402
#202420462
C2×#156888662
C47×#29622
#1924247