Finite Group Databasebeta

← #26370762

Subgroups — #26370762

Order 12032.

686 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689564C211yesmaximal, characteristic, complemented
2C22×#13093964C211yesmaximal, characteristic, complemented
3#20252969C211yesmaximal, characteristic
4#20245503C212yesmaximal, complemented
5#20245494C212yesmaximal, complemented
6#20245486C212yesmaximal, complemented
7#20245477C212yesmaximal, complemented
8#20245174C211yesmaximal, characteristic, complemented
9#20245108C211yesmaximal, characteristic, complemented
10#20245098C211yesmaximal, characteristic, complemented
11C2×C47×#467C211yesmaximal, characteristic, complemented
12C2×#13094046C2211yescharacteristic
13#15689634—21no—
14#15689564—24no—
15C2×#13093970C2212yescomplemented
16C2×#13093967C2212yescomplemented
17C2×#13093964—24no—
18C2×#13093964C2211yescharacteristic
19C2×#13093964C2212yes—
20C2×#13093964C2211yescharacteristic
21C22×C47⋊C16C2212yes—
22#15689364C2211yescharacteristic
23#15689364C2211yescharacteristic
24#15689359C2211yescharacteristic
25#15689359C2211yescharacteristic
26#15689352—21no—
27#15689352—21no—
28#15689161C2211yescharacteristic
29#15689120—21no—
30#15688950C2212yescomplemented
31#15688948C2212yescomplemented
32#15688927C2212yescomplemented
33#15688927C2212yescomplemented
34#15688926C2212yescomplemented
35#15688923C2212yescomplemented
36C22×Q16×C47C2211yescharacteristic
37C23×C8×C47C2211yescharacteristic
38C2×C22⋊Q8×C47—21no—
39C47×#467—21no—
40C47×#467—21no—
41C47×#467C2212yes—
42C47×#467C2212yes—
43C2×#107×C47C2211yescharacteristic
44C2×#103×C47—21no—
45C2×#10625350—21no—
46#13094046—22no—
47#13094046—22no—
48C2×Dic188C2311yescharacteristic
49C2×#10625325—21no—
50C2×D376C2311yescharacteristic
51#13094007—41no—
52#13094003—21no—
53#13094003—21no—
54#13093998—21no—
55#13093998—21no—
56#13093993—41no—
57C4×C22⋊D47—41no—
58C4×C22⋊D47—41no—
59#13093983—21no—
60#13093970—24no—
61#13093967—24no—
62#13093964—22no—
63#13093964—24no—
64#13093964D414yescomplemented
65C2×C47⋊C16—24no—
66C2×C47⋊C16C2312yes—
67C2×C47⋊C16C2312yes—
68#13093958C2311yescharacteristic
69#13093958C2311yescharacteristic
70#13093956—21no—
71#13093949—21no—
72#13093949—21no—
73#13093946—21no—
74#13093946—21no—
75C22×Q8×C47—21no—
76C23×C4×C47D411yescharacteristic
77C2×Q16×C47—22no—
78C2×Q16×C47—21no—
79C2×Q16×C47—22no—
80C2×Q16×C47C2312yes—
81C22×C8×C47—22no—
82C22×C8×C47—22no—
83C22×C8×C47D411yescharacteristic
84C22×C8×C47D411yescharacteristic
85C22×C8×C47D411yescharacteristic
86C22×C8×C47C2311yescharacteristic
87C22×C8×C47C2311yescharacteristic
88C22×C8×C47C2311yescharacteristic
89C22⋊Q8×C47—22no—
90C22⋊Q8×C47—22no—
91C22⋊Q8×C47—41no—
92C22⋊Q8×C47—41no—
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47—41no—
95C2×C22⋊C4×C47—41no—
96#107×C47—21no—
97#107×C47C2312yes—
98#103×C47—22no—
99#103×C47—22no—
100C24×C47D811yescharacteristic
101C2×Q8×C47—22no—
102C2×Q8×C47—41no—
103C2×Q8×C47—42no—
104C2×Q8×C47—42no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—41no—
108C22×C4×C47—41no—
109C22×C4×C47D811yescharacteristic
110C22×C4×C47D811yescharacteristic
111C22×C4×C47D811yescharacteristic
112C22×C4×C47C2×D411yescharacteristic
113C22×C4×C47C2×D411yescharacteristic
114C22×C4×C47C2×D411yescharacteristic
115C2×C22⋊D47—41no—
116C2×C22⋊D47—41no—
117C2×C22⋊D47—41no—
118#10625350—42no—
119#10625350—42no—
120C2×D188—21no—
121C2×C4×D47—41no—
122C2×Dic94—21no—
123Q16×C47—22no—
124Q16×C47—22no—
125Q16×C47—42no—
126C2×C8×C47—22no—
127C2×C8×C47—22no—
128C2×C8×C47—22no—
129C2×C8×C47—21no—
130C2×C8×C47—21no—
131C2×C8×C47—21no—
132C2×C8×C47—21no—
133C2×C8×C47C2×D411yescharacteristic
134C2×C8×C47C2×D411yescharacteristic
135C2×C8×C47C2×D411yescharacteristic
136C2×C8×C47C2×D412yes—
137C2×C8×C47#4112yes—
138C2×C8×C47C2411yescharacteristic
139C4⋊C4×C47—22no—
140C4⋊C4×C47—41no—
141C4⋊C4×C47—42no—
142C4⋊C4×C47—42no—
143C22⋊C4×C47—42no—
144C22⋊C4×C47—42no—
145C22⋊Dic47—41no—
146C22⋊Dic47—41no—
147C22⋊Dic47—41no—
148(C2×D47)⋊C4—41no—
149(C2×D47)⋊C4—41no—
150C4⋊Dic47—21no—
151C4⋊Dic47—21no—
152Dic47⋊C4—41no—
153Dic47⋊C4—41no—
154C4×Dic47—41no—
155Dic188—21no—
156Dic188—21no—
157D376—21no—
158D376—21no—
159#10625325—42no—
160C47⋊C16—22no—
161C47⋊C16C2×D414yes—
162C23×C47—22no—
163C23×C47—22no—
164C23×C47—41no—
165C23×C47C2×D811yescharacteristic
166C23×C47C2×D811yescharacteristic
167C23×C47C2×D811yescharacteristic
168C22×D47—41no—
169Q8×C47—42no—
170Q8×C47—42no—
171Q8×C47—82no—
172C2×C4×C47—22no—
173C2×C4×C47—21no—
174C2×C4×C47—21no—
175C2×C4×C47—21no—
176C2×C4×C47—21no—
177C2×C4×C47—22no—
178C2×C4×C47—22no—
179C2×C4×C47—42no—
180C2×C4×C47—42no—
181C2×C4×C47—82no—
182C2×C4×C47—82no—
183C2×C4×C47#12112yes—
184C2×C4×C47C2×D811yescharacteristic
185C2×C4×C47C2×D811yescharacteristic
186C2×C4×C47C2×D811yescharacteristic
187C2×C4×C47#13512yes—
188C2×C4×C47C22×D411yescharacteristic
189C22⋊D47—82no—
190C22⋊D47—82no—
191C22⋊D47—82no—
192C2×Dic47—41no—
193C2×Dic47—41no—
194C2×Dic47—41no—
195D188—41no—
196D188—41no—
197C4×D47—82no—
198Dic94—41no—
199Dic94—41no—
200C8×C47—21no—
201C8×C47—21no—
202C8×C47#12112yes—
203C8×C47C22×D411yescharacteristic
204C8×C47C2×#4111yescharacteristic
205#33606—471nomaximal, Sylow
206C22×C47—21no—
207C22×C47—21no—
208C22×C47—21no—
209C22×C47—21no—
210C22×C47—21no—
211C22×C47—21no—
212C22×C47—21no—
213C22×C47—44no—
214C22×C47#46612yes—
215C22×C47#51012yescomplemented
216C22×C47C22×D811yescharacteristic
217C2×D47—81no—
218C2×D47—81no—
219C2×D47—81no—
220C4×C47—21no—
221C4×C47—21no—
222C4×C47—82no—
223C4×C47—82no—
224C4×C47#46612yes—
225C4×C47C2×#12111yescharacteristic
226C4×C47C2×#13511yescharacteristic
227Dic47—81no—
228Dic47—81no—
229Dic47—81no—
230C2×#510—471no—
231C22×M32—471no—
232#3050—471no—
233C2×#467D4711yescharacteristic, complemented
234#2220—472no—
235#2219—472no—
236#2217—472no—
237#2216—472no—
238#2159—471no—
239#2141—471no—
240#2139—471no—
241C2×C47—22no—
242C2×C47—22no—
243C2×C47—41no—
244C2×C47C2×#46611yescharacteristic
245C2×C47C2×#51011yescharacteristic
246C2×C47#341411yescharacteristic
247D47—161no—
248C2×#135—471no—
249C22×Q16C2×D4711yescharacteristic
250C23×C8C2×D4711yescharacteristic
251#525—941no—
252C2×C22⋊Q8—21no—
253#510—944no—
254C2×Q32—472no—
255C2×SD32—472no—
256C2×M32—471no—
257C2×M32—471no—
258C2×M32—472no—
259C2×M32—944no—
260C22×C16—472no—
261#467—21no—
262#467—21no—
263#467—471no—
264#467—471no—
265#467C2×D4712yes—
266#467C2×D4712yes—
267#466—471no—
268#466—471no—
269#465—941no—
270#465—941no—
271#427—471no—
272C2×#107C2×D4711yescharacteristic
273#416—941no—
274C2×#103—21no—
275#367—472no—
276#366—472no—
277#358—472no—
278#358—472no—
279#358—472no—
280#357—472no—
281C47#3360611yescharacteristic, complemented, Sylow
282C2×#41—941no—
283C22×Q8—21no—
284C23×C4C22⋊D4711yescharacteristic
285#135—942no—
286#135—942no—
287C2×Q16—22no—
288C2×Q16—22no—
289C2×Q16—21no—
290C2×Q16—471no—
291C2×Q16C22×D4712yes—
292C2×SD16—941no—
293C2×D8—471no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8C22⋊D4711yescharacteristic
297C22×C8C22⋊D4711yescharacteristic
298C22×C8C22⋊D4711yescharacteristic
299C22×C8C22×D4711yescharacteristic
300C22×C8C22×D4711yescharacteristic
301C22×C8C22×D4711yescharacteristic
302#123—1881no—
303C22⋊Q8—22no—
304C22⋊Q8—22no—
305C22⋊Q8—41no—
306C22⋊Q8—41no—
307C22⋊Q8—941no—
308C22⋊Q8—941no—
309#121—941no—
310#121—941no—
311#120—1881no—
312C4×D4—1881no—
313C4×D4—1881no—
314#117—941no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—41no—
317C2×C22⋊C4—41no—
318Q32—944no—
319SD32—944no—
320M32—474no—
321M32—944no—
322M32—942no—
323C2×C16—472no—
324C2×C16—472no—
325C2×C16—944no—
326#107—21no—
327#107—471no—
328#107—471no—
329#107C22×D4712yes—
330#106—941no—
331#103—22no—
332#103—22no—
333#103—941no—
334#103—941no—
335#102—941no—
336#102—941no—
337C24#1062533411yescharacteristic
338#41—1882no—
339#41—1882no—
340C2×Q8—22no—
341C2×Q8—41no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—941no—
345C2×D4—941no—
346C2×D4—1881no—
347C2×D4—1881no—
348C2×D4—1881no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—1881no—
354C22×C4#1062533411yescharacteristic
355C22×C4#1062533411yescharacteristic
356C22×C4#1062533411yescharacteristic
357C22×C4C2×C22⋊D4711yescharacteristic
358C22×C4C2×C22⋊D4711yescharacteristic
359C22×C4C2×C22⋊D4711yescharacteristic
360Q16—22no—
361Q16—22no—
362Q16—42no—
363Q16—941no—
364Q16—941no—
365SD16—1882no—
366D8—941no—
367D8—941no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8D4×D4712yes—
376C2×C8#1062535212yes—
377C2×C8C2×C22⋊D4711yescharacteristic
378C2×C8C2×C22⋊D4711yescharacteristic
379C2×C8C2×C22⋊D4711yescharacteristic
380C2×C8C23×D4711yescharacteristic
381C4⋊C4—22no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—941no—
386C4⋊C4—941no—
387C4⋊C4—1881no—
388C4⋊C4—1881no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—1881no—
392C22⋊C4—1881no—
393C22⋊C4—1881no—
394C22⋊C4—1881no—
395C22⋊C4—1881no—
396C42—1881no—
397C16—474no—
398C16—942no—
399C23—22no—
400C23—22no—
401C23—41no—
402C23—1881no—
403C23C2×#1062533411yescharacteristic
404C23C2×#1062533411yescharacteristic
405C23C2×#1062533411yescharacteristic
406Q8—42no—
407Q8—42no—
408Q8—82no—
409Q8—1881no—
410Q8—1881no—
411D4—3762no—
412D4—3762no—
413D4—3762no—
414D4—1881no—
415D4—1881no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—3762no—
428C2×C4—1881no—
429C2×C4—1881no—
430C2×C4—1881no—
431C2×C4#1309399612yes—
432C2×C4C2×#1062533411yescharacteristic
433C2×C4C2×#1062533411yescharacteristic
434C2×C4C2×#1062533411yescharacteristic
435C2×C4#1309404412yes—
436C2×C4C22×C22⋊D4711yescharacteristic
437C8—21no—
438C8—21no—
439C8#1309399612yes—
440C8C2×D4×D4711yescharacteristic
441C8C2×#1062535211yescharacteristic
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—44no—
450C22—3761no—
451C22—3761no—
452C22—3761no—
453C22#1568935712yes—
454C22#1568956412yescomplemented
455C22C22×#1062533411yescharacteristic
456C4—21no—
457C4—21no—
458C4—82no—
459C4—82no—
460C4—3761no—
461C4—3761no—
462C4—3761no—
463C4#1568935712yes—
464C4C2×#1309399611yescharacteristic
465C4C2×#1309404411yescharacteristic
466C2—22no—
467C2—22no—
468C2—41no—
469C2—7521no—
470C2C2×#1568935711yescharacteristic
471C2C2×#1568956411yescharacteristic
472C2#2025652211yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC2×C47×#467C2
socleC22×C47C22×D8
prime radical#20252969C2
hypercenterC2×#467D47
hypocenterC47#33606
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
1. C2×#15689564C2semidirect
2. C22×#13093964C2semidirect
4. #20245503C2semidirect
5. #20245494C2semidirect
6. #20245486C2semidirect
7. #20245477C2semidirect
8. #20245174C2semidirect
9. #20245108C2semidirect
10. #20245098C2semidirect
11. C2×C47×#467C2semidirect
15. C2×#13093970C22semidirect
16. C2×#13093967C22semidirect
30. #15688950C22semidirect
31. #15688948C22semidirect
32. #15688927C22semidirect
33. #15688927C22semidirect
34. #15688926C22semidirect
35. #15688923C22semidirect
64. #13093964D4semidirect
215. C22×C47#510semidirect
233. C2×#467D47semidirect
281. C47#33606semidirect
454. C22#15689564semidirect

Maximal subgroups

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

subgroupindex
C2×#156895642
C22×#130939642
#202529692
#202455032
#202454942
#202454862
#202454772
#202451742
#202451082
#202450982
C2×C47×#4672
#3360647