Finite Group Databasebeta

← #26200250

Subgroups — #26200250

Order 12032.

754 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689470C211yesmaximal, characteristic, complemented
2#20252875C211yesmaximal, characteristic, complemented
3#20252868C211yesmaximal, characteristic
4#20242832C214yesmaximal, complemented
5#20242825C214yesmaximal, complemented
6#20242147C211yesmaximal, characteristic, complemented
7#20242059C211yesmaximal, characteristic, complemented
8#20242055C211yesmaximal, characteristic, complemented
9C8×#41×C47C211yesmaximal, characteristic, complemented
10C2×#13094054—21no—
11C2×#13094049—21no—
12C2×#13094024C2211yescharacteristic
13C2×#13094021C2211yescharacteristic
14#15689470—21no—
15#15689470—21no—
16#15689470C2214yes—
17#15689333—21no—
18#15689333C2212yescomplemented
19#15689328—21no—
20#15689328C2212yescomplemented
21#15689323—21no—
22#15689323C2212yes—
23#15689318—21no—
24#15689318C2212yes—
25#15689147C2211yescharacteristic
26#15689147C2211yescharacteristic
27#15689131—21no—
28#15689127—21no—
29C2×#13093946—22no—
30#15688876C2214yescomplemented
31#15688868C2214yescomplemented
32#15688866C2214yes—
33C4×#41×C47C2211yescharacteristic
34C8×Q8×C47C2212yescomplemented
35C8×D4×C47—22no—
36C8×D4×C47C2212yescomplemented
37C47×#431—21no—
38C47×#431C2211yescharacteristic
39C2×C4×C8×C47—21no—
40C2×C4×C8×C47C2211yescharacteristic
41C22×Dic94—21no—
42C22×D188—21no—
43C22×D188—21no—
44C22×D188—21no—
45#13094054—44no—
46#13094049—44no—
47C2×#10625337—22no—
48C2×#10625335—22no—
49C2×#10625336—22no—
50C2×#10625334—22no—
51C2×#10625329—21no—
52C2×#10625329—21no—
53#13094024—21no—
54#13094024—21no—
55#13094024—21no—
56#13094024C2312yes—
57#13094021—21no—
58#13094021—21no—
59#13094021—21no—
60#13094021C2312yes—
61C2×C4⋊Dic47—21no—
62C2×C4⋊Dic47—21no—
63C4⋊(C47⋊C8)—21no—
64C4⋊(C47⋊C8)—21no—
65C4⋊(C47⋊C8)C2312yes—
66C4⋊(C47⋊C8)C2312yes—
67#13093947—22no—
68#13093947—22no—
69#13093946—24no—
70#13093946—24no—
71#13093944—22no—
72#13093944—22no—
73C2×#41×C47D411yescharacteristic
74C22×C8×C47—22no—
75C22×C8×C47D412yes—
76C4×Q8×C47D411yescharacteristic
77C4×Q8×C47C2311yescharacteristic
78C4×D4×C47—21no—
79C4×D4×C47—21no—
80C4×D4×C47D411yescharacteristic
81C4×D4×C47C2311yescharacteristic
82#117×C47—21no—
83#117×C47D411yescharacteristic
84C2×C42×C47—21no—
85C2×C42×C47C2311yescharacteristic
86C4⋊C8×C47—22no—
87C4⋊C8×C47C2312yes—
88C22⋊C8×C47—21no—
89C22⋊C8×C47—22no—
90C4×C8×C47—22no—
91C4×C8×C47—21no—
92C4×C8×C47D411yescharacteristic
93C4×C8×C47D411yescharacteristic
94C4×C8×C47C2312yes—
95C23×D47—41no—
96#41×C47—22no—
97#41×C47D814yes—
98C2×Q8×C47C2×D411yescharacteristic
99C2×D4×C47—21no—
100C2×D4×C47C2×D411yescharacteristic
101C22×C4×C47—21no—
102C22×C4×C47—21no—
103C22×C4×C47—21no—
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47#4111yescharacteristic
107C22×Dic47—41no—
108C2×D188—22no—
109C2×D188—22no—
110C2×D188—22no—
111C2×D188—44no—
112C2×D188—42no—
113C2×D188—42no—
114C2×D188—41no—
115C2×D188—41no—
116C2×D188—41no—
117C2×D188—82no—
118C2×Dic94—22no—
119C2×Dic94—42no—
120C2×Dic94—42no—
121C2×Dic94—41no—
122C2×C8×C47—22no—
123C2×C8×C47—42no—
124C2×C8×C47—44no—
125C2×C8×C47C2×D412yes—
126C2×C8×C47C2×D412yes—
127C4⋊C4×C47—21no—
128C4⋊C4×C47—21no—
129C4⋊C4×C47C2×D411yescharacteristic
130C4⋊C4×C47C2×D411yescharacteristic
131C22⋊C4×C47—21no—
132C22⋊C4×C47—21no—
133C22⋊C4×C47—21no—
134C42×C47—21no—
135C42×C47—21no—
136C42×C47—21no—
137C42×C47C2×D411yescharacteristic
138C42×C47C2×D411yescharacteristic
139C42×C47#4111yescharacteristic
140C42×C47C2411yescharacteristic
141#10625337—44no—
142#10625336—44no—
143#10625335—44no—
144#10625334—44no—
145C4⋊Dic47—22no—
146C4⋊Dic47—22no—
147C4⋊Dic47—41no—
148C4⋊Dic47—41no—
149#10625329—42no—
150#10625329—42no—
151C2×C47⋊C8—22no—
152C2×C47⋊C8—22no—
153C23×C47—21no—
154C23×C47#12111yescharacteristic
155C22×D47—42no—
156C22×D47—82no—
157C22×D47—82no—
158C22×D47—82no—
159Q8×C47—21no—
160Q8×C47C2×D812yes—
161D4×C47—24no—
162D4×C47—21no—
163D4×C47C2×D812yes—
164C2×C4×C47—21no—
165C2×C4×C47—22no—
166C2×C4×C47—24no—
167C2×C4×C47—22no—
168C2×C4×C47—21no—
169C2×C4×C47—21no—
170C2×C4×C47—42no—
171C2×C4×C47—42no—
172C2×C4×C47#12111yescharacteristic
173C2×C4×C47#12111yescharacteristic
174C2×C4×C47#12111yescharacteristic
175C2×C4×C47C2×D812yes—
176C2×C4×C47#13512yes—
177C2×C4×C47C22×D411yescharacteristic
178C2×C4×C47C22×D411yescharacteristic
179C2×C4×C47C2×#4111yescharacteristic
180C2×Dic47—42no—
181C2×Dic47—82no—
182D188—42no—
183D188—42no—
184D188—44no—
185D188—82no—
186D188—82no—
187D188—82no—
188D188—82no—
189Dic94—42no—
190Dic94—42no—
191Dic94—82no—
192C8×C47—22no—
193C8×C47—22no—
194C8×C47—42no—
195C47⋊C8—42no—
196C47⋊C8—42no—
197#19276—471nomaximal, Sylow
198C22×C47—21no—
199C22×C47—21no—
200C22×C47—22no—
201C22×C47—41no—
202C22×C47—41no—
203C22×C47#46612yes—
204C22×C47C2×#12111yescharacteristic
205C2×D47—82no—
206C2×D47—82no—
207C2×D47—82no—
208C2×D47—164no—
209C4×C47—21no—
210C4×C47—22no—
211C4×C47—21no—
212C4×C47—41no—
213C4×C47—41no—
214C4×C47#46612yes—
215C4×C47#47112yes—
216C4×C47C22×D811yescharacteristic
217C4×C47C2×#13511yescharacteristic
218C4×C47#57512yes—
219Dic47—82no—
220C2×#486—471no—
221#3039—471no—
222#3038—471no—
223C8×#41D4711yescharacteristic, complemented
224#1682—474no—
225#1681—474no—
226#1567—471no—
227#1546—471no—
228#1545—471no—
229C2×C47—22no—
230C2×C47—42no—
231C2×C47#304011yescharacteristic
232C2×C47C2×#46611yescharacteristic
233C2×C47C2×#47111yescharacteristic
234D47—162no—
235C2×#137—941no—
236C2×AGL(1,ℤ/8)—941no—
237C2×C4⋊Q8—471no—
238C2×#127—471no—
239C4×#41C2×D4711yescharacteristic
240#486—474no—
241#486—941no—
242#486—941no—
243#462—472no—
244#462—941no—
245#461—472no—
246#461—941no—
247#460—472no—
248#460—941no—
249#459—472no—
250#459—941no—
251C8×Q8C2×D4712yescomplemented
252C8×D4—22no—
253C8×D4C2×D4712yescomplemented
254#431—21no—
255#431C2×D4711yescharacteristic
256#423—471no—
257#423—471no—
258#419—941no—
259#418—941no—
260C2×#102—942no—
261C2×C4×C8—21no—
262C2×C4×C8C2×D4711yescharacteristic
263#335—474no—
264#332—474no—
265#331—474no—
266C47#1927611yescharacteristic, complemented, Sylow
267C2×#41D18811yescharacteristic
268C22×Q8—941no—
269C22×D4—941no—
270C22×D4—941no—
271C22×D4—941no—
272#137—1884no—
273AGL(1,ℤ/8)—1884no—
274C2×Q16—942no—
275C2×SD16—942no—
276C2×SD16—942no—
277C2×D8—942no—
278C2×M16—941no—
279C2×M16—941no—
280C22×C8—22no—
281C22×C8C22⋊D4712yes—
282C4⋊Q8—472no—
283C4⋊Q8—941no—
284C4⋊Q8—941no—
285C4⋊Q8—941no—
286#127—472no—
287#127—941no—
288#127—941no—
289#127—941no—
290C4×Q8D18811yescharacteristic
291C4×Q8C22×D4711yescharacteristic
292C4×D4—21no—
293C4×D4—21no—
294C4×D4D18811yescharacteristic
295C4×D4C22×D4711yescharacteristic
296#117—21no—
297#117D18811yescharacteristic
298C2×C4⋊C4—941no—
299C2×C4⋊C4—941no—
300C2×C42—21no—
301C2×C42C22×D4711yescharacteristic
302C4⋊C8—22no—
303C4⋊C8—472no—
304C4⋊C8—472no—
305C4⋊C8—941no—
306C4⋊C8—941no—
307C4⋊C8C22×D4712yes—
308#103—942no—
309#103—942no—
310#102—944no—
311#102—944no—
312#102—942no—
313#102—942no—
314C22⋊C8—21no—
315C22⋊C8—22no—
316C4×C8—22no—
317C4×C8—21no—
318C4×C8C22⋊D4711yescharacteristic
319C4×C8C22⋊D4711yescharacteristic
320C4×C8C22×D4712yes—
321C24—1881no—
322#41—22no—
323#41D37614yes—
324C2×Q8—942no—
325C2×Q8—1882no—
326C2×Q8—1882no—
327C2×Q8—1881no—
328C2×Q8C2×D18811yescharacteristic
329C2×D4—21no—
330C2×D4—942no—
331C2×D4—942no—
332C2×D4—942no—
333C2×D4—3762no—
334C2×D4—1884no—
335C2×D4—1881no—
336C2×D4—1881no—
337C2×D4—1881no—
338C2×D4—1882no—
339C2×D4—1882no—
340C2×D4C2×D18811yescharacteristic
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—1881no—
345C22×C4C2×D18811yescharacteristic
346C22×C4#1062535011yescharacteristic
347C22×C4C2×C22⋊D4711yescharacteristic
348Q16—1884no—
349SD16—1884no—
350SD16—1884no—
351D8—1884no—
352M16—1882no—
353M16—1882no—
354C2×C8—22no—
355C2×C8—42no—
356C2×C8—44no—
357C2×C8—942no—
358C2×C8—942no—
359C2×C8C2×C22⋊D4712yes—
360C2×C8C2×C22⋊D4712yes—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—942no—
364C4⋊C4—942no—
365C4⋊C4—1881no—
366C4⋊C4—1881no—
367C4⋊C4C2×D18811yescharacteristic
368C4⋊C4C2×D18811yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—21no—
371C22⋊C4—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42C2×D18811yescharacteristic
376C42#1062535011yescharacteristic
377C42C2×C22⋊D4711yescharacteristic
378C42C23×D4711yescharacteristic
379C23—21no—
380C23—3762no—
381C23—3762no—
382C23—3762no—
383C23—1882no—
384C23#1309399811yescharacteristic
385Q8—21no—
386Q8—3762no—
387Q8—1882no—
388Q8—1882no—
389Q8C2×D37612yes—
390D4—24no—
391D4—21no—
392D4—3762no—
393D4—3762no—
394D4—3762no—
395D4—3762no—
396D4—1882no—
397D4—1882no—
398D4—1884no—
399D4C2×D37612yes—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—3762no—
409C2×C4—1882no—
410C2×C4#1309399811yescharacteristic
411C2×C4#1309399811yescharacteristic
412C2×C4#1309399811yescharacteristic
413C2×C4C2×D37612yes—
414C2×C4#1309404612yes—
415C2×C4C22×C22⋊D4711yescharacteristic
416C2×C4C22×D18811yescharacteristic
417C2×C4C2×#1062535011yescharacteristic
418C8—22no—
419C8—22no—
420C8—42no—
421C8—1882no—
422C8—1882no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—3762no—
429C22—3762no—
430C22—3762no—
431C22—7524no—
432C22#1568935912yes—
433C22C2×#1309399811yescharacteristic
434C4—22no—
435C4—21no—
436C4—21no—
437C4—41no—
438C4—41no—
439C4—3762no—
440C4#1568935912yes—
441C4#1568938612yes—
442C4C22×D37611yescharacteristic
443C4C2×#1309404611yescharacteristic
444C4#1568992112yes—
445C2—22no—
446C2—42no—
447C2—7522no—
448C2#2025288211yescharacteristic
449C2C2×#1568935911yescharacteristic
450C2C2×#1568938611yescharacteristic
subgroupG/H
centerC22C2×#13093998
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC8×#41×C47C2
socleC22×C47C2×#121
prime radical#20252868C2
hypercenterC8×#41D47
hypocenterC47#19276
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
1. C2×#15689470C2semidirect
2. #20252875C2semidirect
4. #20242832C2semidirect
5. #20242825C2semidirect
6. #20242147C2semidirect
7. #20242059C2semidirect
8. #20242055C2semidirect
9. C8×#41×C47C2semidirect
18. #15689333C22semidirect
20. #15689328C22semidirect
30. #15688876C22semidirect
31. #15688868C22semidirect
34. C8×Q8×C47C22semidirect
36. C8×D4×C47C22semidirect
223. C8×#41D47semidirect
251. C8×Q8C2×D47semidirect
253. C8×D4C2×D47semidirect
266. C47#19276semidirect

Maximal subgroups

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

subgroupindex
C2×#156894702
#202528752
#202528682
#202428322
#202428252
#202421472
#202420592
#202420552
C8×#41×C472
#1927647