Finite Group Databasebeta

← #26187704

Subgroups — #26187704

Order 12032.

656 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252402C211yesmaximal, characteristic, complemented
2#20252400C211yesmaximal, characteristic, complemented
3C4×#13094044C211yesmaximal, characteristic
4#20242282C212yesmaximal, complemented
5#20242227C212yesmaximal, complemented
6#20242221C212yesmaximal, complemented
7#20242180C212yesmaximal, complemented
8#20242149C211yesmaximal, characteristic, complemented
9#20241823C211yesmaximal, characteristic, complemented
10#20241669C211yesmaximal, characteristic, complemented
11C47×#1488C211yesmaximal, characteristic, complemented
12C2×#13094044C411yescharacteristic
13C4×#10625350C2211yescharacteristic
14#15689254C2212yes—
15#15689250C2212yes—
16C4×#10625337C411yescharacteristic
17C4×#10625337C2211yescharacteristic
18C4×#10625336C411yescharacteristic
19C4×#10625336C2211yescharacteristic
20C4×#10625335C411yescharacteristic
21C4×#10625335C2211yescharacteristic
22C4×#10625334C411yescharacteristic
23C4×#10625334C2211yescharacteristic
24#15689197—22no—
25#15689195—22no—
26D4×C47⋊C8C2212yes—
27#15689188C2212yes—
28#15689180—21no—
29#15689180C2211yescharacteristic
30#15689178—21no—
31#15689159C411yescharacteristic
32#15689119C411yescharacteristic
33#15689117C411yescharacteristic
34C4×#10625329C2211yescharacteristic
35C2×#13093935—21no—
36#97×D47—21no—
37C2×C4×C47⋊C8C2211yescharacteristic
38#15688875C2212yes—
39#15688873C2212yes—
40#15688855C2212yes—
41#15688852C2212yes—
42#15688845C2212yes—
43C8×C47⋊C8C2212yes—
44C4×#41×C47C2211yescharacteristic
45C47×#432C2211yescharacteristic
46C47×#431C2211yescharacteristic
47C47×Q8⋊C8C2212yes—
48C47×D4⋊C8C2212yes—
49C2×#10625350C2×C411yescharacteristic
50#13094044—24no—
51C2×#10625337C2×C411yescharacteristic
52C2×#10625335C2×C411yescharacteristic
53C2×#10625336C2×C411yescharacteristic
54C2×#10625334C2×C411yescharacteristic
55C2×#10625329D412yes—
56C22×C47⋊C8—22no—
57C22×C47⋊C8C2×C411yescharacteristic
58C22×C47⋊C8C2×C411yescharacteristic
59C2×C8×D47—22no—
60C4×Dic94C2×C411yescharacteristic
61C4×Dic94C2311yescharacteristic
62C4×D188C2×C411yescharacteristic
63C4×D188C2311yescharacteristic
64C4×C22⋊D47—21no—
65C4×C22⋊D47—21no—
66#13093983C2×C411yescharacteristic
67#13093981—21no—
68C42×D47—21no—
69#13093957C2×C411yescharacteristic
70#13093957C2×C411yescharacteristic
71#13093955C2×C411yescharacteristic
72#13093955C2×C411yescharacteristic
73C4⋊(C47⋊C8)—22no—
74C4⋊(C47⋊C8)C2312yes—
75Dic47⋊C8—22no—
76#13093948C2×C411yescharacteristic
77#13093948C2×C411yescharacteristic
78#13093947C2×C411yescharacteristic
79#13093947C2×C411yescharacteristic
80#13093945C2×C411yescharacteristic
81#13093945C2×C411yescharacteristic
82#13093944C2×C411yescharacteristic
83#13093944C2×C411yescharacteristic
84C22⋊(C47⋊C8)—21no—
85C22⋊(C47⋊C8)—22no—
86(C2×D47)⋊C8—22no—
87#13093935—21no—
88#13093935—22no—
89#13093935D411yescharacteristic
90#13093935D411yescharacteristic
91#13093934—22no—
92C4×C47⋊C8C2×C411yescharacteristic
93C4×C47⋊C8C2×C411yescharacteristic
94C4×C47⋊C8C2311yescharacteristic
95C4×C47⋊C8C2311yescharacteristic
96C4×C47⋊C8C2312yes—
97C2×#41×C47C2×C411yescharacteristic
98C4×Q8×C47C2×C411yescharacteristic
99C4×Q8×C47C2311yescharacteristic
100C4×D4×C47—21no—
101C4×D4×C47—21no—
102C4×D4×C47C2×C411yescharacteristic
103C4×D4×C47C2311yescharacteristic
104#117×C47—21no—
105#117×C47C2×C411yescharacteristic
106C2×C42×C47—21no—
107C2×C42×C47C2311yescharacteristic
108C4⋊C8×C47—21no—
109C4⋊C8×C47C2312yes—
110C22⋊C8×C47—21no—
111C22⋊C8×C47—21no—
112#97×C47—21no—
113C4×C8×C47C2312yes—
114#41×C47—22no—
115#41×C47—41no—
116C2×Q8×C47C22×C411yescharacteristic
117C2×D4×C47—21no—
118C2×D4×C47C22×C411yescharacteristic
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47C22×C411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47#4111yescharacteristic
125C2×C22⋊D47—21no—
126#10625350—22no—
127#10625350—41no—
128C2×D188C22×C411yescharacteristic
129C2×C4×D47—21no—
130C2×C4×D47—21no—
131C2×C4×D47—21no—
132C2×Dic94C22×C411yescharacteristic
133C2×C8×C47—22no—
134C2×C8×C47—22no—
135C4⋊C4×C47—21no—
136C4⋊C4×C47—21no—
137C4⋊C4×C47C22×C411yescharacteristic
138C4⋊C4×C47C22×C411yescharacteristic
139C22⋊C4×C47—21no—
140C22⋊C4×C47—21no—
141C22⋊C4×C47—21no—
142C42×C47—21no—
143C42×C47—21no—
144C42×C47—21no—
145C42×C47C22×C411yescharacteristic
146C42×C47C2×D411yescharacteristic
147C42×C47#4111yescharacteristic
148C42×C47C2411yescharacteristic
149C22⋊Dic47—21no—
150#10625337—22no—
151#10625336—22no—
152#10625335—22no—
153#10625334M1614yescomplemented
154(C2×D47)⋊C4—21no—
155(C2×D47)⋊C4—21no—
156C4⋊Dic47C22×C411yescharacteristic
157C4⋊Dic47C22×C411yescharacteristic
158Dic47⋊C4—21no—
159Dic47⋊C4—21no—
160C4×Dic47—21no—
161C4×Dic47—21no—
162C4×Dic47—21no—
163#10625329—24no—
164C2×C47⋊C8—22no—
165C2×C47⋊C8—22no—
166C2×C47⋊C8—41no—
167C2×C47⋊C8—44no—
168C2×C47⋊C8C22×C411yescharacteristic
169C2×C47⋊C8C22×C411yescharacteristic
170C2×C47⋊C8C22×C411yescharacteristic
171C2×C47⋊C8C22×C411yescharacteristic
172C2×C47⋊C8C2×D412yes—
173C2×C47⋊C8C2×D412yes—
174C8×D47—44no—
175C23×C47—21no—
176C23×C47C4×D411yescharacteristic
177C22×D47—21no—
178Q8×C47—21no—
179Q8×C47—21no—
180D4×C47—21no—
181D4×C47—42no—
182D4×C47C2×M1612yes—
183C2×C4×C47—21no—
184C2×C4×C47—21no—
185C2×C4×C47—22no—
186C2×C4×C47—21no—
187C2×C4×C47—21no—
188C2×C4×C47—21no—
189C2×C4×C47—41no—
190C2×C4×C47—42no—
191C2×C4×C47—42no—
192C2×C4×C47—81no—
193C2×C4×C47C4×D411yescharacteristic
194C2×C4×C47C4×D411yescharacteristic
195C2×C4×C47C4×D411yescharacteristic
196C2×C4×C47C23×C411yescharacteristic
197C2×C4×C47C22×D411yescharacteristic
198C2×C4×C47C2×#4111yescharacteristic
199C22⋊D47—42no—
200C2×Dic47—21no—
201C2×Dic47—21no—
202C2×Dic47—21no—
203D188—21no—
204D188C2×M1612yes—
205C4×D47—42no—
206C4×D47—42no—
207C4×D47—81no—
208Dic94—21no—
209Dic94—21no—
210C8×C47—42no—
211C8×C47—42no—
212C47⋊C8—21no—
213C47⋊C8—21no—
214C47⋊C8—21no—
215C47⋊C8—22no—
216C47⋊C8—42no—
217C47⋊C8C2×M1612yes—
218C47⋊C8#13514yescomplemented
219#18035—471nomaximal, Sylow
220C22×C47—21no—
221C22×C47—22no—
222C22×C47—21no—
223C22×C47—21no—
224C22×C47—41no—
225C22×C47—41no—
226C22×C47C2×C4×D411yescharacteristic
227C2×D47—22no—
228C2×D47—41no—
229C2×D47—41no—
230C4×C47—21no—
231C4×C47—21no—
232C4×C47—21no—
233C4×C47—21no—
234C4×C47—41no—
235C4×C47—41no—
236C4×C47—41no—
237C4×C47C22×M1611yescharacteristic
238C4×C47#56811yescharacteristic
239C4×C47C2×#13512yes—
240Dic47—41no—
241Dic47—41no—
242Dic47—41no—
243#2970—471no—
244#2970—471no—
245C4×#135—471no—
246#1591—472no—
247#1582—472no—
248#1582—472no—
249C8×D8—472no—
250#1568—471no—
251#1488—471no—
252#1488D4711yescharacteristic, complemented
253C8×M16—471no—
254C2×C47—41no—
255C2×C47—42no—
256C2×C47D4×M1611yescharacteristic
257C2×C47C4×#13511yescharacteristic
258C2×C47#295711yescharacteristic
259D47—42no—
260C2×#135—471no—
261C4×#41—471no—
262C4×#41C2×D4711yescharacteristic
263#446—472no—
264#446—472no—
265C4×Q16—471no—
266C4×Q16—471no—
267C4×SD16—471no—
268C4×SD16—471no—
269C4×SD16—471no—
270C4×SD16—471no—
271C4×D8—471no—
272C4×D8—471no—
273#435—942no—
274#435—942no—
275C8×D4—472no—
276C8×D4—472no—
277#432—471no—
278#432—941no—
279#432—941no—
280#432C2×D4711yescharacteristic
281#431C2×D4711yescharacteristic
282#427—471no—
283#416—471no—
284#416—471no—
285C4×M16—471no—
286C2×#97—941no—
287C2×#97—941no—
288C2×C4×C8—471no—
289#335—472no—
290#334—472no—
291Q8⋊C8—472no—
292Q8⋊C8C2×D4712yes—
293D4⋊C8—472no—
294D4⋊C8C2×D4712yes—
295#322—472no—
296C82—472no—
297C47#1803511yescharacteristic, complemented, Sylow
298C2×#41—471no—
299C2×#41C4×D4711yescharacteristic
300#135—944no—
301C2×Q16—471no—
302C2×SD16—471no—
303C2×SD16—471no—
304C2×D8—471no—
305C2×M16—472no—
306C22×C8—471no—
307C22×C8—471no—
308C22×C8—942no—
309C22×C8—942no—
310C4×Q8—471no—
311C4×Q8—471no—
312C4×Q8C4×D4711yescharacteristic
313C4×Q8C22×D4711yescharacteristic
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—471no—
317C4×D4—471no—
318C4×D4—941no—
319C4×D4—941no—
320C4×D4C4×D4711yescharacteristic
321C4×D4C22×D4711yescharacteristic
322#117—21no—
323#117—471no—
324#117—941no—
325#117C4×D4711yescharacteristic
326C2×C42—21no—
327C2×C42—941no—
328C2×C42C22×D4711yescharacteristic
329#107—471no—
330#107—471no—
331#106—471no—
332#106—471no—
333C4⋊C8—21no—
334C4⋊C8—472no—
335C4⋊C8—942no—
336C4⋊C8—942no—
337C4⋊C8C22×D4712yes—
338#103—471no—
339#103—471no—
340#103—471no—
341#103—471no—
342#102—471no—
343#102—471no—
344#102—471no—
345#102—471no—
346C22⋊C8—21no—
347C22⋊C8—21no—
348C22⋊C8—942no—
349C22⋊C8—942no—
350C22⋊C8—941no—
351#97—21no—
352#97—471no—
353#97—471no—
354#97—942no—
355#97—942no—
356#97—941no—
357C4×C8—471no—
358C4×C8—471no—
359C4×C8—471no—
360C4×C8—472no—
361C4×C8—471no—
362C4×C8C22×D4712yes—
363#41—22no—
364#41—41no—
365#41—942no—
366#41—1881no—
367C2×Q8—471no—
368C2×Q8C2×C4×D4711yescharacteristic
369C2×D4—21no—
370C2×D4—471no—
371C2×D4—941no—
372C2×D4C2×C4×D4711yescharacteristic
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—941no—
377C22×C4—941no—
378C22×C4—941no—
379C22×C4C2×C4×D4711yescharacteristic
380C22×C4D4×D4711yescharacteristic
381C22×C4#1062535211yescharacteristic
382Q16—942no—
383SD16—942no—
384SD16—942no—
385D8—474no—
386M16—944no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—471no—
390C2×C8—472no—
391C2×C8—471no—
392C2×C8—471no—
393C2×C8—471no—
394C2×C8—472no—
395C2×C8—942no—
396C2×C8—942no—
397C2×C8—1884no—
398C2×C8—1884no—
399C2×C8—1881no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—471no—
403C4⋊C4—471no—
404C4⋊C4—941no—
405C4⋊C4—941no—
406C4⋊C4C2×C4×D4711yescharacteristic
407C4⋊C4C2×C4×D4711yescharacteristic
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—941no—
412C22⋊C4—941no—
413C22⋊C4—941no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—941no—
418C42—941no—
419C42—941no—
420C42C2×C4×D4711yescharacteristic
421C42D4×D4711yescharacteristic
422C42#1062535211yescharacteristic
423C42C23×D4711yescharacteristic
424C23—21no—
425C23—941no—
426C23(C22⋊D47)⋊C411yescharacteristic
427Q8—21no—
428Q8—21no—
429Q8—941no—
430Q8—941no—
431D4—21no—
432D4—42no—
433D4—472no—
434D4—941no—
435D4—1882no—
436D4M16×D4712yes—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—81no—
447C2×C4—941no—
448C2×C4—941no—
449C2×C4—941no—
450C2×C4—3761no—
451C2×C4—1882no—
452C2×C4—1882no—
453C2×C4(C22⋊D47)⋊C411yescharacteristic
454C2×C4(C22⋊D47)⋊C411yescharacteristic
455C2×C4(C22⋊D47)⋊C411yescharacteristic
456C2×C4C22×C4×D4711yescharacteristic
457C2×C4C2×D4×D4711yescharacteristic
458C2×C4C2×#1062535211yescharacteristic
459C8—42no—
460C8—42no—
461C8—472no—
462C8—474no—
463C8—941no—
464C8—941no—
465C8—941no—
466C8—942no—
467C8—1882no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—942no—
475C22—1881no—
476C22—1881no—
477C22C2×(C22⋊D47)⋊C411yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—1881no—
486C4—1881no—
487C4—1881no—
488C4C2×M16×D4711yescharacteristic
489C4#1568988911yescharacteristic
490C4#135×D4712yes—
491C2—41no—
492C2—42no—
493C2—1882no—
494C2#2025212011yescharacteristic
495C2#2025215611yescharacteristic
496C2#2025230111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1488C2
socleC22×C47C2×C4×D4
prime radicalC2×#13094044C4
cosocleC42×C47C24
hypercenter#1488D47
hypocenterC47#18035
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1488

Complements

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

NH ≅ G/Ntype
1. #20252402C2semidirect
2. #20252400C2semidirect
4. #20242282C2semidirect
5. #20242227C2semidirect
6. #20242221C2semidirect
7. #20242180C2semidirect
8. #20242149C2semidirect
9. #20241823C2semidirect
10. #20241669C2semidirect
11. C47×#1488C2semidirect
153. #10625334M16semidirect
218. C47⋊C8#135semidirect
252. #1488D47semidirect
297. C47#18035semidirect

Maximal subgroups

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

subgroupindex
#202524022
#202524002
C4×#130940442
#202422822
#202422272
#202422212
#202421802
#202421492
#202418232
#202416692
C47×#14882
#1803547