Finite Group Databasebeta

← #26225732

Subgroups — #26225732

Order 12032.

990 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689752C211yesmaximal, characteristic
2#20245990C211yesmaximal, characteristic
3#20245988C211yesmaximal, characteristic
4#20244374C211yesmaximal, characteristic
5#20244373C211yesmaximal, characteristic
6#20244372C211yesmaximal, characteristic
7#20244371C214yesmaximal
8#20243464C214yesmaximal
9C47×#1996C211yesmaximal, characteristic
10#15689752—22no—
11#15689752—22no—
12#15689752—22no—
13#15689752—22no—
14C2×#13094023C412yescomplemented
15C2×#13094022C412yescomplemented
16C2×#13094014C412yescomplemented
17C2×#13094014C412yescomplemented
18C2×#13094003C2211yescharacteristic
19C2×#13094001C2212yes—
20C2×#13093980C2212yes—
21C22×Dic47⋊C4C2211yescharacteristic
22#15689010—22no—
23#15689007—22no—
24#15689005—22no—
25#15689004—22no—
26#15688984C2212yes—
27#15688984C2212yes—
28#15688983C2211yescharacteristic
29#15688981C2212yes—
30C22⋊C4×Dic47C2212yes—
31C4×C22⋊Dic47C2211yescharacteristic
32#15688890C2214yes—
33#15688889C2214yes—
34#15688889C2214yes—
35C2×C22⋊Q8×C47C2211yescharacteristic
36C47×#380C2211yescharacteristic
37C4×C22⋊C4×C47C2211yescharacteristic
38C47×#342C2214yes—
39C22×Dic94C2×C412yes—
40C22×Dic94C2×C411yescharacteristic
41C23×Dic47C2311yescharacteristic
42C23×Dic47C2311yescharacteristic
43#13094023—24no—
44#13094023—24no—
45#13094022—24no—
46#13094022—24no—
47#13094014—24no—
48#13094014—24no—
49#13094014—24no—
50#13094014—24no—
51#13094003—21no—
52#13094003—21no—
53#13094003—21no—
54#13094003—21no—
55#13094001—24no—
56#13094001—24no—
57#13093980—24no—
58#13093980—24no—
59C2×C4⋊Dic47—21no—
60C2×C4⋊Dic47C2×C412yes—
61C2×C4⋊Dic47C2×C411yescharacteristic
62C2×Dic47⋊C4—22no—
63C2×Dic47⋊C4C2×C412yes—
64C2×Dic47⋊C4C2×C412yes—
65C2×Dic47⋊C4C2×C412yes—
66C2×Dic47⋊C4C2×C412yes—
67C2×Dic47⋊C4C2×C412yes—
68C2×Dic47⋊C4C2×C412yes—
69C2×Dic47⋊C4C2×C412yes—
70C2×Dic47⋊C4D412yes—
71C2×Dic47⋊C4D412yes—
72C2×Dic47⋊C4D412yes—
73C2×Dic47⋊C4D412yes—
74C2×C22⋊Dic47C2312yes—
75C2×C22⋊Dic47C2312yes—
76C2×C22⋊Dic47C2312yes—
77C2×C4×Dic47—21no—
78C2×C4×Dic47—22no—
79C2×C4×Dic47C2×C412yes—
80C2×C4×Dic47C2×C412yes—
81C47⋊#95—21no—
82C47⋊#95—21no—
83C47⋊#95—22no—
84C47⋊#95—22no—
85C47⋊#95—24no—
86C47⋊#95—24no—
87C22×Q8×C47C2×C411yescharacteristic
88C23×C4×C47C2311yescharacteristic
89C22⋊Q8×C47—21no—
90C22⋊Q8×C47—21no—
91C22⋊Q8×C47—21no—
92C22⋊Q8×C47—21no—
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47C2×C411yescharacteristic
95C2×C4⋊C4×C47C2×C412yes—
96C2×C22⋊C4×C47C2312yes—
97C2×C22⋊C4×C47C2312yes—
98C2×C22⋊C4×C47C2312yes—
99C2×C42×C47—21no—
100#95×C47—21no—
101#95×C47—21no—
102C24×C47C2411yescharacteristic
103C2×Q8×C47—21no—
104C2×Q8×C47—21no—
105C2×Q8×C47—81no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—81no—
109C22×C4×C47C22×C412yes—
110C22×C4×C47C22×C412yes—
111C22×C4×C47C22×C411yescharacteristic
112C22×C4×C47C22×C411yescharacteristic
113C22×C4×C47C2×D411yescharacteristic
114C22×C4×C47C2×D411yescharacteristic
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2×D411yescharacteristic
117C22×Dic47—22no—
118C22×Dic47—22no—
119C22×Dic47—22no—
120C22×Dic47—22no—
121C22×Dic47—81no—
122C22×Dic47C22×C412yes—
123C22×Dic47C22×C412yes—
124C22×Dic47C22×C412yes—
125C22×Dic47C22×C412yes—
126C22×Dic47C2×D412yes—
127C22×Dic47C2×D412yes—
128C22×Dic47C2×D412yes—
129C22×Dic47C2×D412yes—
130C2×Dic94—21no—
131C2×Dic94—21no—
132C2×Dic94—24no—
133C2×Dic94—82no—
134C2×Dic94—81no—
135C4⋊C4×C47—21no—
136C4⋊C4×C47—21no—
137C4⋊C4×C47—22no—
138C4⋊C4×C47—22no—
139C4⋊C4×C47—81no—
140C22⋊C4×C47—22no—
141C22⋊C4×C47—22no—
142C22⋊C4×C47—24no—
143C22⋊C4×C47—22no—
144C22⋊C4×C47—22no—
145C42×C47—81no—
146C22⋊Dic47—22no—
147C22⋊Dic47—22no—
148C22⋊Dic47—22no—
149C22⋊Dic47—22no—
150C22⋊Dic47—24no—
151C4⋊Dic47—21no—
152C4⋊Dic47—21no—
153C4⋊Dic47—24no—
154C4⋊Dic47—81no—
155Dic47⋊C4—22no—
156Dic47⋊C4—22no—
157Dic47⋊C4—24no—
158Dic47⋊C4—24no—
159Dic47⋊C4—24no—
160Dic47⋊C4—24no—
161Dic47⋊C4—82no—
162Dic47⋊C4C22⋊C414yes—
163Dic47⋊C4C22⋊C414yes—
164Dic47⋊C4C22⋊C414yes—
165Dic47⋊C4C22⋊C414yes—
166C4×Dic47—24no—
167C4×Dic47—24no—
168C4×Dic47—82no—
169C4×Dic47—81no—
170C23×C47—41no—
171C23×C47—41no—
172C23×C47C23×C411yescharacteristic
173C23×C47C22×D411yescharacteristic
174C23×C47C22×D411yescharacteristic
175C23×C472−1+412yes—
176C23×C472−1+412yes—
177Q8×C47—81no—
178Q8×C47—81no—
179C2×C4×C47—22no—
180C2×C4×C47—22no—
181C2×C4×C47—41no—
182C2×C4×C47—41no—
183C2×C4×C47—41no—
184C2×C4×C47—41no—
185C2×C4×C47—41no—
186C2×C4×C47—41no—
187C2×C4×C47—42no—
188C2×C4×C47—42no—
189C2×C4×C47—42no—
190C2×C4×C47—42no—
191C2×C4×C47—42no—
192C2×C4×C47—42no—
193C2×C4×C47—42no—
194C2×C4×C47—42no—
195C2×C4×C47C2×C22⋊C411yescharacteristic
196C2×C4×C47C2×C22⋊C411yescharacteristic
197C2×C4×C47C2×C22⋊C411yescharacteristic
198C2×C4×C47C2×C22⋊C411yescharacteristic
199C2×Dic47—24no—
200C2×Dic47—22no—
201C2×Dic47—24no—
202C2×Dic47—22no—
203C2×Dic47—22no—
204C2×Dic47—22no—
205C2×Dic47—42no—
206C2×Dic47—42no—
207C2×Dic47—42no—
208C2×Dic47—42no—
209C2×Dic47—42no—
210C2×Dic47—42no—
211C2×Dic47—42no—
212C2×Dic47—42no—
213C2×Dic47—42no—
214C2×Dic47—42no—
215C2×Dic47—42no—
216C2×Dic47C2×C22⋊C414yes—
217C2×Dic47C2×C22⋊C414yes—
218Dic94—84no—
219Dic94—81no—
220Dic94—81no—
221#21828—471nomaximal, Sylow
222C22×C47—21no—
223C22×C47—21no—
224C22×C47—21no—
225C22×C47—21no—
226C22×C47—21no—
227C22×C47—21no—
228C22×C47—42no—
229C22×C47—42no—
230C22×C47C22×C22⋊C411yescharacteristic
231C22×C47#51912yes—
232C22×C47#53611yescharacteristic
233C22×C47#53612yes—
234C22×C47#53611yescharacteristic
235C4×C47—21no—
236C4×C47—21no—
237C4×C47—21no—
238C4×C47—21no—
239C4×C47—42no—
240C4×C47—42no—
241C4×C47—82no—
242C4×C47—82no—
243Dic47—24no—
244Dic47—24no—
245Dic47—42no—
246Dic47—42no—
247Dic47—82no—
248Dic47—82no—
249C2×#544—471no—
250#2313—471no—
251#2313—471no—
252#1996—471no—
253#1996—471no—
254#1996—474no—
255#1996—471no—
256#1996D4711yescharacteristic
257#1820—474no—
258C2×C47—21no—
259C2×C47—22no—
260C2×C47—21no—
261C2×C47—22no—
262C2×C47#231011yescharacteristic
263C2×C47#288311yescharacteristic
264C2×C47#288411yescharacteristic
265#544—942no—
266#544—942no—
267#544—942no—
268#544—942no—
269C2×C4⋊Q8—472no—
270C2×C4⋊Q8—472no—
271C2×#125—472no—
272C2×#125—472no—
273C2×C22⋊Q8—471no—
274C2×C22⋊Q8—472no—
275C2×C22⋊Q8C2×D4711yescharacteristic
276C2×#117—472no—
277C22×C4⋊C4—471no—
278#387—942no—
279#387—942no—
280#387—942no—
281#387—942no—
282#381—472no—
283#381—472no—
284#380—471no—
285#380—472no—
286#380C2×D4711yescharacteristic
287C4×C22⋊C4—471no—
288C4×C22⋊C4—472no—
289C4×C22⋊C4C2×D4711yescharacteristic
290#342—474no—
291#342—474no—
292#342—474no—
293#342C2×D4714yes—
294C47#2182811yescharacteristic, complemented, Sylow
295C22×Q8—472no—
296C22×Q8—471no—
297C22×Q8C4×D4711yescharacteristic
298C23×C4—471no—
299C23×C4—471no—
300C23×C4C22×D4711yescharacteristic
301C4⋊Q8—944no—
302C4⋊Q8—944no—
303C4⋊Q8—944no—
304C4⋊Q8—944no—
305#125—944no—
306#125—944no—
307#125—944no—
308#125—944no—
309C22⋊Q8—21no—
310C22⋊Q8—21no—
311C22⋊Q8—21no—
312C22⋊Q8—21no—
313C22⋊Q8—944no—
314C22⋊Q8—944no—
315C22⋊Q8—941no—
316C22⋊Q8—941no—
317C22⋊Q8—941no—
318C22⋊Q8—941no—
319#117—944no—
320#117—944no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—472no—
323C2×C4⋊C4—472no—
324C2×C4⋊C4—472no—
325C2×C4⋊C4—472no—
326C2×C4⋊C4—472no—
327C2×C4⋊C4—472no—
328C2×C4⋊C4—472no—
329C2×C4⋊C4—472no—
330C2×C4⋊C4—472no—
331C2×C4⋊C4—471no—
332C2×C4⋊C4—472no—
333C2×C4⋊C4—472no—
334C2×C4⋊C4—472no—
335C2×C4⋊C4—942no—
336C2×C4⋊C4—941no—
337C2×C4⋊C4C4×D4711yescharacteristic
338C2×C4⋊C4C4×D4712yes—
339C2×C22⋊C4—472no—
340C2×C22⋊C4—472no—
341C2×C22⋊C4—472no—
342C2×C22⋊C4C22×D4712yes—
343C2×C22⋊C4C22×D4712yes—
344C2×C22⋊C4C22×D4712yes—
345C2×C42—21no—
346C2×C42—472no—
347C2×C42—472no—
348C2×C42—942no—
349C2×C42—941no—
350#95—21no—
351#95—21no—
352#95—942no—
353#95—942no—
354#95—944no—
355#95—944no—
356#95—941no—
357#95—941no—
358C24C23×D4711yescharacteristic
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—81no—
362C2×Q8—944no—
363C2×Q8—941no—
364C2×Q8—941no—
365C2×Q8—3761no—
366C2×Q8—3762no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—81no—
370C22×C4—472no—
371C22×C4—472no—
372C22×C4—472no—
373C22×C4—472no—
374C22×C4—472no—
375C22×C4—472no—
376C22×C4—472no—
377C22×C4—472no—
378C22×C4—942no—
379C22×C4—942no—
380C22×C4—942no—
381C22×C4—942no—
382C22×C4—3761no—
383C22×C4C2×C4×D4712yes—
384C22×C4C2×C4×D4712yes—
385C22×C4C2×C4×D4711yescharacteristic
386C22×C4C2×C4×D4711yescharacteristic
387C22×C4D4×D4711yescharacteristic
388C22×C4D4×D4711yescharacteristic
389C22×C4D4×D4711yescharacteristic
390C22×C4D4×D4711yescharacteristic
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—81no—
396C4⋊C4—474no—
397C4⋊C4—474no—
398C4⋊C4—474no—
399C4⋊C4—474no—
400C4⋊C4—944no—
401C4⋊C4—944no—
402C4⋊C4—944no—
403C4⋊C4—944no—
404C4⋊C4—944no—
405C4⋊C4—942no—
406C4⋊C4—942no—
407C4⋊C4—941no—
408C4⋊C4—941no—
409C4⋊C4—3762no—
410C4⋊C4—3761no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—24no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—942no—
417C22⋊C4—942no—
418C22⋊C4—942no—
419C22⋊C4—942no—
420C22⋊C4—944no—
421C42—81no—
422C42—944no—
423C42—944no—
424C42—3761no—
425C42—3762no—
426C23—41no—
427C23—41no—
428C23C22×C4×D4711yescharacteristic
429C23C2×D4×D4711yescharacteristic
430C23C2×D4×D4711yescharacteristic
431C23#1309407112yes—
432C23#1309407212yes—
433Q8—81no—
434Q8—81no—
435Q8—3761no—
436Q8—3761no—
437Q8—3764no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—474no—
455C2×C4—474no—
456C2×C4—944no—
457C2×C4—942no—
458C2×C4—944no—
459C2×C4—942no—
460C2×C4—942no—
461C2×C4—942no—
462C2×C4—1882no—
463C2×C4—1882no—
464C2×C4—1882no—
465C2×C4—1882no—
466C2×C4—1882no—
467C2×C4—1882no—
468C2×C4—1882no—
469C2×C4—1882no—
470C2×C4—1882no—
471C2×C4—1882no—
472C2×C4—1882no—
473C2×C4C22⋊C4×D4711yescharacteristic
474C2×C4C22⋊C4×D4711yescharacteristic
475C2×C4C22⋊C4×D4711yescharacteristic
476C2×C4C22⋊C4×D4711yescharacteristic
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—42no—
484C22—42no—
485C22C2×C22⋊C4×D4711yescharacteristic
486C22#1568959912yes—
487C22#1568969311yescharacteristic
488C22#1568969612yes—
489C22#1568969811yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—42no—
495C4—42no—
496C4—82no—
497C4—82no—
498C4—944no—
499C4—944no—
500C4—3762no—
501C4—3762no—
502C4—1882no—
503C4—1882no—
504C2—21no—
505C2—22no—
506C2—21no—
507C2—22no—
508C2#2024597111yescharacteristic
509C2#2025184611yescharacteristic
510C2#2025185011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC47×#1996C2
socleC22×C47C22×C22⋊C4
prime radical, cosocleC24×C47C24
hypercenter#1996D47
hypocenterC47#21828
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
14. C2×#13094023C4semidirect
15. C2×#13094022C4semidirect
16. C2×#13094014C4semidirect
17. C2×#13094014C4semidirect
294. C47#21828semidirect

Maximal subgroups

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

subgroupindex
C2×#156897522
#202459902
#202459882
#202443742
#202443732
#202443722
#202443712
#202434642
C47×#19962
#2182847