Finite Group Databasebeta

← #27080822

Subgroups — #27080822

Order 12032.

1962 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256685C211yesmaximal, characteristic
2#20252401C218yesmaximal, complemented
3#20252381C218yesmaximal, complemented
4#20252329C211yesmaximal, characteristic, complemented
5#20252107C211yesmaximal, characteristic, complemented
6#20252105C211yesmaximal, characteristic, complemented
7#20252084C212yesmaximal, complemented
8#20252081C212yesmaximal, complemented
9#20252038C212yesmaximal, complemented
10#20252020C211yesmaximal, characteristic, complemented
11#20251815C212yesmaximal, complemented
12C2×#97×D47C211yesmaximal, characteristic, complemented
13C47×#2961C211yesmaximal, characteristic, complemented
14C2×#13094032C2212yes—
15C2×#13094030C2212yes—
16C22×C8×D47C2211yescharacteristic
17C22×C8×D47C2211yescharacteristic
18#15689678C412yes—
19#15689674C412yes—
20#15689673C2211yescharacteristic
21#15689660C2214yes—
22#15689659C414yes—
23#15689658C414yes—
24#15689636C414yes—
25#15689635C2212yes—
26C4×#10625354C2214yes—
27#15689591C2212yes—
28C2×C42×D47C2211yescharacteristic
29#15689254C2218yescomplemented
30#15689196C2218yes—
31#15689196C2218yescomplemented
32#15689193C2218yes—
33#15689193C2218yescomplemented
34C8×D188C2218yes—
35#15689188C2218yes—
36#15689183C2218yescomplemented
37#15689179C2214yes—
38#15689179C2218yescomplemented
39#15689175C2214yes—
40#15689150C2212yes—
41#15689147C2211yescharacteristic
42#15689091C2212yes—
43#15689089C2212yes—
44#15689086C2212yes—
45#15689080C2214yescomplemented
46#15689079C2212yes—
47#15689077C2214yescomplemented
48M16×Dic47C2212yes—
49C4×#10625324C2214yescomplemented
50C2×#13093935C2211yescharacteristic
51C2×#13093934C2211yescharacteristic
52C2×#13093934C2211yescharacteristic
53#97×D47C2214yes—
54#97×D47C2214yes—
55#97×D47C2214yescomplemented
56C47×#533C2211yescharacteristic
57C47×#446C2218yescomplemented
58C47×#424C2212yes—
59C47×#423C2211yescharacteristic
60C47×#405C2212yes—
61C2×#97×C47C2211yescharacteristic
62C2×#10625354C2×C414yes—
63C2×#10625350C2312yes—
64C22×C4×D47C2×C412yes—
65C22×C4×D47C2311yescharacteristic
66#13094032—28no—
67#13094030—28no—
68C2×#10625329C2312yes—
69C2×#10625324C2314yes—
70C2×#10625324C2314yes—
71C22×C47⋊C8C2311yescharacteristic
72C22×C47⋊C8C2311yescharacteristic
73C2×C8×D47—24no—
74C2×C8×D47—24no—
75C2×C8×D47C2314yes—
76C2×C8×D47C2314yes—
77C2×C8×D47C2314yes—
78C2×C8×D47C2314yes—
79#13094024C2×C411yescharacteristic
80#13094021C2×C411yescharacteristic
81#13094019C2×C418yes—
82#13094015C2×C414yes—
83#13094013C2×C414yes—
84#13094012C2×C412yes—
85#13094009C2×C418yes—
86#13094008C2×C414yes—
87#13094004C2×C414yes—
88#13094000C2×C414yes—
89#13093999C2×C414yes—
90#13093999C2×C414yes—
91#13093998C2×C414yes—
92#13093994C2×C414yes—
93Q8×Dic47C2×C414yes—
94#13093989C2×C414yes—
95C4×D188C2318yes—
96C4×D188C2314yes—
97#13093986C2×C414yes—
98#13093986C2×C418yes—
99#13093986C2×C414yes—
100(C22⋊D47)⋊C4C2×C418yes—
101#13093982C2×C414yes—
102#13093982C2×C418yes—
103#13093981C2314yes—
104C2×C4×Dic47C2×C412yes—
105C2×C4×Dic47C2311yescharacteristic
106C42×D47C2×C412yes—
107C42×D47C2×C412yes—
108C42×D47C2312yes—
109C42×D47C2312yes—
110C42×D47C2314yes—
111C4⋊(C47⋊C8)C2314yes—
112C4⋊(C47⋊C8)C2318yes—
113(C2×D47)⋊C8C2318yes—
114(C2×D47)⋊C8C2318yes—
115(C2×D47)⋊C8C2318yes—
116#13093935C2314yes—
117#13093935C2314yes—
118#13093934C2312yes—
119#13093934C2312yes—
120#13093934C2312yes—
121#13093934C2312yes—
122#13093934C2314yes—
123C4×C47⋊C8C2314yes—
124C8×Dic47C2314yes—
125C2×M16×C47C2312yes—
126C22×C8×C47C2311yescharacteristic
127C22×C8×C47C2311yescharacteristic
128C4⋊Q8×C47C2×C412yes—
129#125×C47C2×C412yes—
130C22⋊Q8×C47C2×C414yes—
131C4×Q8×C47C2314yes—
132#117×C47C2312yes—
133C2×C42×C47C2311yescharacteristic
134C4⋊C8×C47C2314yes—
135C4⋊C8×C47C2318yes—
136#97×C47C2314yes—
137#97×C47C2314yes—
138C4×C8×C47C2314yes—
139C23×D47C22×C411yescharacteristic
140C2×Q8×C47C22×C414yes—
141C22×C4×C47C22×C412yes—
142C22×C4×C47C2411yescharacteristic
143C2×C22⋊D47C22×C414yes—
144C22×Dic47C22×C411yescharacteristic
145C22×Dic47C22×C412yes—
146#10625354—48no—
147#10625350—24no—
148#10625350—24no—
149C2×D188C22×C412yes—
150C2×D188C22×C418yes—
151C2×D188C22×C414yes—
152C2×C4×D47—22no—
153C2×C4×D47—22no—
154C2×C4×D47—44no—
155C2×C4×D47C22×C412yes—
156C2×C4×D47C22×C412yes—
157C2×C4×D47C22×C412yes—
158C2×C4×D47C22×C412yes—
159C2×C4×D47C22×C412yes—
160C2×C4×D47C22×C418yes—
161C2×C4×D47C2412yes—
162C2×C4×D47C2414yes—
163C2×Dic94C22×C412yes—
164M16×C47—24no—
165C2×C8×C47—22no—
166C2×C8×C47#4114yescomplemented
167C2×C8×C47C2412yes—
168C2×C8×C47C2414yes—
169C2×C8×C47C2412yes—
170C4⋊C4×C47C22×C414yes—
171C4⋊C4×C47C22×C418yes—
172C4⋊C4×C47C22×C414yes—
173C22⋊C4×C47C22×C414yes—
174C42×C47C22×C411yescharacteristic
175C42×C47C22×C411yescharacteristic
176C42×C47C2412yes—
177C42×C47C2414yes—
178(C2×D47)⋊C4C22×C418yes—
179(C2×D47)⋊C4C22×C418yes—
180(C2×D47)⋊C4C22×C418yes—
181(C2×D47)⋊C4C22×C418yes—
182C4⋊Dic47C22×C414yes—
183C4⋊Dic47C22×C418yes—
184Dic47⋊C4C22×C418yes—
185C4×Dic47C22×C412yes—
186C4×Dic47C22×C412yes—
187C4×Dic47C22×C412yes—
188C4×Dic47C22×C412yes—
189C4×Dic47C22×C412yes—
190C4×Dic47C22×C418yes—
191C4×Dic47C2411yescharacteristic
192C4×Dic47C2411yescharacteristic
193#10625329—24no—
194C2×C47⋊C8—22no—
195C2×C47⋊C8#4114yescomplemented
196C2×C47⋊C8C2414yes—
197C2×C47⋊C8C2412yes—
198C2×C47⋊C8C2412yes—
199#10625324—28no—
200#10625324—28no—
201C8×D47—28no—
202C8×D47—28no—
203C8×D47—28no—
204C8×D47—28no—
205C23×C47C23×C411yescharacteristic
206C22×D47—22no—
207C22×D47—24no—
208C22×D47C23×C414yes—
209C22×D47C23×C412yes—
210Q8×C47—44no—
211C2×C4×C47—42no—
212C2×C4×C47C23×C412yes—
213C2×C4×C47C23×C412yes—
214C2×C4×C47C23×C418yes—
215C2×C4×C47C23×C411yescharacteristic
216C2×C4×C47C2×#4112yes—
217C2×C4×C47C2×#4112yes—
218C2×C4×C47C2511yescharacteristic
219C22⋊D47—28no—
220C2×Dic47—22no—
221C2×Dic47—42no—
222C2×Dic47C23×C412yes—
223C2×Dic47C23×C412yes—
224C2×Dic47C23×C412yes—
225C2×Dic47C23×C414yes—
226D188—216no—
227D188—24no—
228D188—44no—
229C4×D47—24no—
230C4×D47—24no—
231C4×D47—24no—
232C4×D47—22no—
233C4×D47—22no—
234C4×D47—24no—
235C4×D47—24no—
236C4×D47—42no—
237C4×D47—42no—
238C4×D47—48no—
239Dic94—24no—
240C8×C47—22no—
241C8×C47—24no—
242C8×C47C2×#4114yes—
243C47⋊C8—22no—
244C47⋊C8—24no—
245C47⋊C8C2×#4114yes—
246#60207—471nomaximal, Sylow
247C22×C47—22no—
248C22×C47C4×#4112yes—
249C22×C47C24×C411yescharacteristic
250C2×D47—28no—
251C2×D47—22no—
252C2×D47—22no—
253C2×D47—24no—
254C2×D47—28no—
255C2×D47#56814yes—
256C4×C47—22no—
257C4×C47—22no—
258C4×C47—28no—
259C4×C47C4×#4112yes—
260C4×C47C22×#4111yescharacteristic
261C4×C47C22×#4111yescharacteristic
262Dic47—24no—
263Dic47—22no—
264Dic47—22no—
265Dic47—22no—
266#3444—471no—
267#2970—478no—
268#2968—478no—
269#2961—471no—
270#2961D4711yescharacteristic, complemented
271#2932—471no—
272#2932—471no—
273#2929—472no—
274#2929—472no—
275#2923—472no—
276#2920—471no—
277#2875—472no—
278C22×#97—471no—
279C2×C47—22no—
280C2×C47C2×C4×#4111yescharacteristic
281C2×C47C2×#56812yes—
282D47—24no—
283D47—44no—
284C2×#131—472no—
285C2×#131—472no—
286C23×C8—471no—
287C23×C8—471no—
288#533—472no—
289#533C2×D4711yescharacteristic
290#532—472no—
291#532—471no—
292#529—474no—
293#529—474no—
294#529—474no—
295#525—474no—
296#525—472no—
297C4×#41—474no—
298C4×#41—472no—
299C22×C42—471no—
300#446—478no—
301#446C2×D4718yescomplemented
302#435—478no—
303#435—478no—
304#435—478no—
305#435—478no—
306C8×D4—478no—
307C8×D4—478no—
308#433—478no—
309#432—474no—
310#432—478no—
311#431—474no—
312#424—472no—
313#424C2×D4712yes—
314#423—471no—
315#423C2×D4711yescharacteristic
316#408—472no—
317#408—472no—
318#407—472no—
319#405—472no—
320#405—474no—
321#405—474no—
322#405C2×D4712yes—
323C4×M16—472no—
324C4×M16—474no—
325C2×#97—471no—
326C2×#97—471no—
327C2×#97—471no—
328C2×#97—474no—
329C2×#97—474no—
330C2×#97—474no—
331C2×#97C2×D4711yescharacteristic
332C47#6020711yescharacteristic, complemented, Sylow
333C2×#41—474no—
334C2×#41—472no—
335C23×C4—471no—
336C23×C4—472no—
337#131—948no—
338#131—948no—
339C2×M16—472no—
340C2×M16—474no—
341C2×M16—474no—
342C2×M16C22×D4712yes—
343C22×C8—471no—
344C22×C8—471no—
345C22×C8—474no—
346C22×C8—474no—
347C22×C8—474no—
348C22×C8—474no—
349C22×C8—944no—
350C22×C8—944no—
351C22×C8C22×D4711yescharacteristic
352C22×C8C22×D4711yescharacteristic
353C4⋊Q8—471no—
354C4⋊Q8C4×D4712yes—
355#127—471no—
356#126—478no—
357#125—474no—
358#125C4×D4712yes—
359#124—474no—
360#124—472no—
361#123—478no—
362#123—474no—
363C22⋊Q8—474no—
364C22⋊Q8—474no—
365C22⋊Q8C4×D4714yes—
366#121—474no—
367#121—474no—
368#121—474no—
369#120—474no—
370C4×Q8—474no—
371C4×Q8—474no—
372C4×Q8C22×D4714yes—
373C4×D4—478no—
374C4×D4—474no—
375C4×D4—474no—
376C4×D4—478no—
377C4×D4—478no—
378C4×D4—474no—
379#117—474no—
380#117—478no—
381#117—474no—
382#117C22×D4712yes—
383C2×C42—472no—
384C2×C42—471no—
385C2×C42—472no—
386C2×C42—472no—
387C2×C42—474no—
388C2×C42—472no—
389C2×C42—472no—
390C2×C42C22×D4711yescharacteristic
391C4⋊C8—474no—
392C4⋊C8—478no—
393C4⋊C8C22×D4714yes—
394C4⋊C8C22×D4718yes—
395C22⋊C8—478no—
396C22⋊C8—478no—
397C22⋊C8—478no—
398#97—472no—
399#97—472no—
400#97—472no—
401#97—472no—
402#97—474no—
403#97—474no—
404#97—474no—
405#97C22×D4714yes—
406#97C22×D4714yes—
407C4×C8—474no—
408C4×C8—474no—
409C4×C8C22×D4714yes—
410C24—471no—
411#41—944no—
412#41—944no—
413#41—1888no—
414C2×Q8—472no—
415C2×Q8C2×C4×D4714yes—
416C2×D4—474no—
417C2×D4—478no—
418C2×D4—472no—
419C2×D4—474no—
420C22×C4—471no—
421C22×C4—472no—
422C22×C4—472no—
423C22×C4—472no—
424C22×C4—472no—
425C22×C4—472no—
426C22×C4—474no—
427C22×C4—472no—
428C22×C4—472no—
429C22×C4—478no—
430C22×C4—942no—
431C22×C4—942no—
432C22×C4—1884no—
433C22×C4C2×C4×D4712yes—
434C22×C4C23×D4711yescharacteristic
435M16—24no—
436M16—948no—
437M16—944no—
438M16—948no—
439C2×C8—22no—
440C2×C8—474no—
441C2×C8—472no—
442C2×C8—474no—
443C2×C8—472no—
444C2×C8—948no—
445C2×C8—948no—
446C2×C8—942no—
447C2×C8—948no—
448C2×C8—948no—
449C2×C8#1062535414yescomplemented
450C2×C8C23×D4714yes—
451C2×C8C23×D4712yes—
452C2×C8C23×D4712yes—
453C4⋊C4—478no—
454C4⋊C4—474no—
455C4⋊C4—478no—
456C4⋊C4C2×C4×D4714yes—
457C4⋊C4C2×C4×D4718yes—
458C4⋊C4C2×C4×D4714yes—
459C22⋊C4—478no—
460C22⋊C4—478no—
461C22⋊C4—478no—
462C22⋊C4—478no—
463C22⋊C4C2×C4×D4714yes—
464C42—472no—
465C42—472no—
466C42—472no—
467C42—472no—
468C42—472no—
469C42—478no—
470C42—471no—
471C42—471no—
472C42C2×C4×D4711yescharacteristic
473C42C2×C4×D4711yescharacteristic
474C42C23×D4714yes—
475C42C23×D4712yes—
476C23—472no—
477C23—474no—
478C23—944no—
479C23—942no—
480C23C22×C4×D4711yescharacteristic
481Q8—44no—
482Q8—944no—
483D4—9416no—
484D4—948no—
485D4—944no—
486D4—1884no—
487C2×C4—42no—
488C2×C4—472no—
489C2×C4—472no—
490C2×C4—472no—
491C2×C4—474no—
492C2×C4—944no—
493C2×C4—944no—
494C2×C4—942no—
495C2×C4—944no—
496C2×C4—942no—
497C2×C4—944no—
498C2×C4—942no—
499C2×C4—944no—
500C2×C4—1882no—
501C2×C4—1882no—
502C2×C4—1888no—
503C2×C4—1882no—
504C2×C4C22×C4×D4712yes—
505C2×C4C22×C4×D4712yes—
506C2×C4C22×C4×D4718yes—
507C2×C4C22×C4×D4711yescharacteristic
508C2×C4#41×D4712yes—
509C2×C4C2×#1062535412yes—
510C2×C4C24×D4711yescharacteristic
511C8—22no—
512C8—24no—
513C8—474no—
514C8—942no—
515C8—944no—
516C8C2×#1062535414yes—
517C22—22no—
518C22—474no—
519C22—948no—
520C22—942no—
521C22—942no—
522C22—948no—
523C22—944no—
524C22C4×#1062535412yes—
525C22C23×C4×D4711yescharacteristic
526C4—22no—
527C4—22no—
528C4—28no—
529C4—942no—
530C4—942no—
531C4—942no—
532C4—944no—
533C4C4×#1062535412yes—
534C4C2×#41×D4711yescharacteristic
535C4C22×#1062535411yescharacteristic
536C2—22no—
537C2—944no—
538C2—1884no—
539C2C2×C4×#1062535411yescharacteristic
540C2#568×D4712yes—
subgroupG/H
center, FrattiniC2×C4C24×D47
commutator, socleC22×C47C24×C4
FittingC47×#2961C2
prime radical#20256685C2
cosocleC2×C4×C47C25
hypercenter#2961D47
hypocenterC47#60207
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ #2961

Complements

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

NH ≅ G/Ntype
2. #20252401C2semidirect
3. #20252381C2semidirect
4. #20252329C2semidirect
5. #20252107C2semidirect
6. #20252105C2semidirect
7. #20252084C2semidirect
8. #20252081C2semidirect
9. #20252038C2semidirect
10. #20252020C2semidirect
11. #20251815C2semidirect
12. C2×#97×D47C2semidirect
13. C47×#2961C2semidirect
29. #15689254C22semidirect
31. #15689196C22semidirect
33. #15689193C22semidirect
36. #15689183C22semidirect
38. #15689179C22semidirect
45. #15689080C22semidirect
47. #15689077C22semidirect
49. C4×#10625324C22semidirect
55. #97×D47C22semidirect
57. C47×#446C22semidirect
166. C2×C8×C47#41semidirect
195. C2×C47⋊C8#41semidirect
270. #2961D47semidirect
301. #446C2×D47semidirect
332. C47#60207semidirect
449. C2×C8#10625354semidirect

Maximal subgroups

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

subgroupindex
#202566852
#202524012
#202523812
#202523292
#202521072
#202521052
#202520842
#202520812
#202520382
#202520202
#202518152
C2×#97×D472
C47×#29612
#6020747