Finite Group Databasebeta

← #27080847

Subgroups — #27080847

Order 12032.

1878 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
1#20256780C211yesmaximal, characteristic
2#20252394C218yesmaximal, complemented
3#20252381C218yesmaximal, complemented
4#20252323C211yesmaximal, characteristic, complemented
5#20252107C211yesmaximal, characteristic, complemented
6#20252099C211yesmaximal, characteristic, complemented
7#20252090C212yesmaximal, complemented
8#20252084C212yesmaximal, complemented
9#20252031C212yesmaximal, complemented
10#20252025C211yesmaximal, characteristic, complemented
11#20251815C212yesmaximal, complemented
12C2×#15689080C211yesmaximal, characteristic, complemented
13C47×#2961C211yesmaximal, characteristic, complemented
14C2×#13094032C2212yes—
15C2×#13094030C2212yes—
16C22×#10625324C2211yescharacteristic
17C22×C8×D47C2211yescharacteristic
18#15689747C414yes—
19#15689736C412yes—
20#15689729C412yes—
21#15689722C414yes—
22#15689713C414yes—
23#15689673C2211yescharacteristic
24#15689660C2218yes—
25#15689635C2212yes—
26#15689591C2212yes—
27C2×#13093981C2211yescharacteristic
28#15689254C2214yescomplemented
29Q8×C47⋊C8C2214yescomplemented
30#15689200C2218yes—
31#15689200C2214yescomplemented
32#15689196C2218yes—
33#15689196C2218yescomplemented
34#15689193C2218yes—
35C8×D188C2218yes—
36#15689188C2214yescomplemented
37#15689183C2218yes—
38#15689183C2218yescomplemented
39#15689175C2218yescomplemented
40#15689150C2212yes—
41#15689147C2211yescharacteristic
42#15689091C2212yes—
43#15689089C2212yes—
44#15689086C2212yes—
45#15689080C2214yes—
46#15689080C2214yes—
47#15689080C2214yescomplemented
48#15689079C2212yes—
49#15689077C2214yescomplemented
50M16×Dic47C2212yes—
51C4×#10625324C2214yescomplemented
52C2×#13093934C2211yescharacteristic
53#97×D47C2214yescomplemented
54C2×C4×C47⋊C8C2211yescharacteristic
55C2×C8×Dic47C2211yescharacteristic
56C47×#533C2211yescharacteristic
57C47×#446C2218yescomplemented
58C47×#424C2212yes—
59C47×#423C2211yescharacteristic
60C47×#405C2212yes—
61C2×#97×C47C2211yescharacteristic
62C2×#10625350C2312yes—
63C22×C4×D47C2311yescharacteristic
64#13094032—28no—
65#13094030—28no—
66C2×#10625329C2312yes—
67C2×#10625324—24no—
68C2×#10625324C2314yes—
69C2×#10625324C2314yes—
70C2×#10625324C2312yes—
71C2×#10625324C2312yes—
72C22×C47⋊C8C2311yescharacteristic
73C22×C47⋊C8C2311yescharacteristic
74C2×C8×D47—24no—
75C2×C8×D47C2314yes—
76C2×C8×D47C2314yes—
77C2×C8×D47C2314yes—
78#13094024C2×C411yescharacteristic
79#13094021C2×C411yescharacteristic
80#13094019C2×C418yes—
81#13094019C2×C418yes—
82#13094015C2×C418yes—
83#13094013C2×C418yes—
84#13094012C2×C412yes—
85#13094009C2×C418yes—
86#13094009C2×C418yes—
87#13094008C2×C414yes—
88#13094004C2×C414yes—
89#13094000C2×C418yes—
90#13093999C2×C414yes—
91#13093999C2×C418yes—
92#13093998C2×C414yes—
93#13093994C2×C414yes—
94#13093989C2×C414yes—
95C4×D188C2318yes—
96C4×D188C2314yes—
97#13093986C2×C414yes—
98(C22⋊D47)⋊C4C2×C418yes—
99#13093981C2×C412yes—
100#13093981C2×C412yes—
101#13093981C2314yes—
102#13093981C2314yes—
103C2×Dic47⋊C4C2×C412yes—
104C2×(C2×D47)⋊C4C2×C412yes—
105C2×C4×Dic47C2311yescharacteristic
106C42×D47C2314yes—
107C4⋊(C47⋊C8)C2314yes—
108C4⋊(C47⋊C8)C2318yes—
109(C2×D47)⋊C8C2318yes—
110(C2×D47)⋊C8C2318yes—
111(C2×D47)⋊C8C2318yes—
112#13093935C2314yes—
113#13093934C2312yes—
114#13093934C2314yes—
115#13093934C2312yes—
116C4×C47⋊C8C2312yes—
117C4×C47⋊C8C2314yes—
118C4×C47⋊C8C2312yes—
119C8×Dic47C2314yes—
120C8×Dic47C2314yes—
121C2×M16×C47C2312yes—
122C22×C8×C47C2311yescharacteristic
123C22×C8×C47C2311yescharacteristic
124C4⋊Q8×C47C2×C412yes—
125#125×C47C2×C412yes—
126C22⋊Q8×C47C2×C414yes—
127C4×Q8×C47C2314yes—
128#117×C47C2312yes—
129C2×C42×C47C2311yescharacteristic
130C4⋊C8×C47C2318yes—
131C4⋊C8×C47C2314yes—
132#97×C47C2314yes—
133#97×C47C2314yes—
134C4×C8×C47C2314yes—
135C23×D47C22×C411yescharacteristic
136C2×Q8×C47C22×C414yes—
137C22×C4×C47C22×C412yes—
138C22×C4×C47C2411yescharacteristic
139C2×C22⋊D47C22×C414yes—
140C22×Dic47C22×C411yescharacteristic
141C22×Dic47C22×C412yes—
142#10625350—24no—
143#10625350—24no—
144C2×D188C22×C412yes—
145C2×D188C22×C418yes—
146C2×D188C22×C414yes—
147C2×C4×D47—22no—
148C2×C4×D47—22no—
149C2×C4×D47C22×C412yes—
150C2×C4×D47C22×C418yes—
151C2×C4×D47C2414yes—
152C2×C4×D47C2412yes—
153C2×Dic94C22×C412yes—
154M16×C47—24no—
155C2×C8×C47—22no—
156C2×C8×C47#4114yes—
157C2×C8×C47C2412yes—
158C2×C8×C47C2414yes—
159C2×C8×C47C2412yes—
160C4⋊C4×C47C22×C414yes—
161C4⋊C4×C47C22×C418yes—
162C4⋊C4×C47C22×C414yes—
163C22⋊C4×C47C22×C414yes—
164C42×C47C22×C411yescharacteristic
165C42×C47C22×C411yescharacteristic
166C42×C47C2412yes—
167C42×C47C2414yes—
168(C2×D47)⋊C4C22×C414yes—
169(C2×D47)⋊C4C22×C414yes—
170(C2×D47)⋊C4C22×C418yes—
171(C2×D47)⋊C4C22×C418yes—
172(C2×D47)⋊C4C22×C418yes—
173(C2×D47)⋊C4C22×C418yes—
174C4⋊Dic47C22×C414yes—
175C4⋊Dic47C22×C418yes—
176Dic47⋊C4C22×C414yes—
177Dic47⋊C4C22×C414yes—
178Dic47⋊C4C22×C418yes—
179C4×Dic47C22×C418yes—
180C4×Dic47C22×C412yes—
181C4×Dic47C2412yes—
182#10625329—24no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8#4114yes—
185C2×C47⋊C8C2412yes—
186C2×C47⋊C8C2414yes—
187C2×C47⋊C8C2411yescharacteristic
188C2×C47⋊C8C2411yescharacteristic
189#10625324—28no—
190#10625324—24no—
191#10625324—24no—
192#10625324—28no—
193C8×D47—28no—
194C8×D47—28no—
195C8×D47—28no—
196C23×C47C23×C411yescharacteristic
197C22×D47—22no—
198C22×D47—42no—
199C22×D47C23×C412yes—
200C22×D47C23×C414yes—
201Q8×C47—44no—
202C2×C4×C47—42no—
203C2×C4×C47C23×C418yes—
204C2×C4×C47C23×C412yes—
205C2×C4×C47C23×C412yes—
206C2×C4×C47C23×C411yescharacteristic
207C2×C4×C47C2×#4112yes—
208C2×C4×C47C2×#4112yes—
209C2×C4×C47C2511yescharacteristic
210C22⋊D47—28no—
211C2×Dic47—22no—
212C2×Dic47—42no—
213C2×Dic47C23×C414yes—
214C2×Dic47C23×C414yes—
215C2×Dic47C23×C412yes—
216D188—24no—
217D188—48no—
218D188—44no—
219C4×D47—24no—
220C4×D47—22no—
221C4×D47—24no—
222C4×D47—24no—
223C4×D47—22no—
224C4×D47—48no—
225Dic94—24no—
226C8×C47—24no—
227C8×C47—22no—
228C8×C47C2×#4114yes—
229C47⋊C8—22no—
230C47⋊C8—24no—
231C47⋊C8C2×#4112yes—
232C47⋊C8C2×#4112yes—
233#60209—471nomaximal, Sylow
234C22×C47—22no—
235C22×C47C4×#4112yes—
236C22×C47C24×C411yescharacteristic
237C2×D47—22no—
238C2×D47—24no—
239C2×D47—28no—
240C2×D47—24no—
241C2×D47—44no—
242C4×C47—28no—
243C4×C47—22no—
244C4×C47—22no—
245C4×C47C4×#4112yes—
246C4×C47C22×#4111yescharacteristic
247C4×C47C22×#4111yescharacteristic
248Dic47—24no—
249Dic47—22no—
250Dic47—24no—
251#3455—471no—
252#2969—478no—
253#2968—478no—
254#2961D4711yescharacteristic, complemented
255#2960—471no—
256#2932—471no—
257#2931—471no—
258#2930—472no—
259#2929—472no—
260#2922—472no—
261#2921—471no—
262#2875—472no—
263C2×#405—471no—
264C2×C47—22no—
265C2×C47C2×C4×#4111yescharacteristic
266C2×C47#357312yes—
267D47—44no—
268D47—42no—
269C2×#131—472no—
270C2×#131—472no—
271C22×M16—471no—
272C23×C8—471no—
273#542—474no—
274#541—472no—
275#540—472no—
276#539—474no—
277#538—474no—
278#533C2×D4711yescharacteristic
279#532—471no—
280#529—478no—
281#525—472no—
282C4×#41—472no—
283C2×#117—471no—
284#446—474no—
285#446C2×D4718yescomplemented
286C8×Q8—474no—
287#436—478no—
288#436—474no—
289#435—478no—
290#435—478no—
291#435—478no—
292C8×D4—474no—
293C8×D4—478no—
294#433—478no—
295#433—478no—
296#431—478no—
297#424—472no—
298#424C2×D4712yes—
299#423—471no—
300#423C2×D4711yescharacteristic
301#408—472no—
302#408—472no—
303#407—472no—
304#405—472no—
305#405—474no—
306#405—474no—
307#405—474no—
308#405—474no—
309#405C2×D4712yes—
310C4×M16—472no—
311C4×M16—474no—
312C2×#97—471no—
313C2×#97—474no—
314C2×#97C2×D4711yescharacteristic
315C2×C4×C8—471no—
316C2×C4×C8—471no—
317C47#6020911yescharacteristic, complemented, Sylow
318C2×#41—472no—
319C23×C4—471no—
320#131—948no—
321#131—948no—
322C2×M16—472no—
323C2×M16—472no—
324C2×M16—472no—
325C2×M16—474no—
326C2×M16—474no—
327C2×M16—944no—
328C2×M16C22×D4712yes—
329C22×C8—471no—
330C22×C8—471no—
331C22×C8—474no—
332C22×C8—474no—
333C22×C8—474no—
334C22×C8—944no—
335C22×C8C22×D4711yescharacteristic
336C22×C8C22×D4711yescharacteristic
337C4⋊Q8—471no—
338C4⋊Q8C4×D4712yes—
339#127—471no—
340#126—478no—
341#126—478no—
342#125—478no—
343#125C4×D4712yes—
344#124—478no—
345#124—472no—
346#123—478no—
347#123—478no—
348#123—474no—
349C22⋊Q8—478no—
350C22⋊Q8—474no—
351C22⋊Q8C4×D4714yes—
352#121—474no—
353#121—474no—
354#121—478no—
355#120—474no—
356C4×Q8—474no—
357C4×Q8C22×D4714yes—
358C4×D4—474no—
359C4×D4—478no—
360C4×D4—478no—
361C4×D4—474no—
362#117—472no—
363#117—472no—
364#117—474no—
365#117—474no—
366#117C22×D4712yes—
367C2×C4⋊C4—472no—
368C2×C22⋊C4—472no—
369C2×C42—471no—
370C2×C42—474no—
371C2×C42C22×D4711yescharacteristic
372C4⋊C8—474no—
373C4⋊C8—478no—
374C4⋊C8C22×D4714yes—
375C4⋊C8C22×D4718yes—
376C22⋊C8—478no—
377C22⋊C8—478no—
378C22⋊C8—478no—
379#97—472no—
380#97—472no—
381#97—474no—
382#97—474no—
383#97C22×D4714yes—
384#97C22×D4714yes—
385C4×C8—474no—
386C4×C8—474no—
387C4×C8—472no—
388C4×C8—472no—
389C4×C8—474no—
390C4×C8C22×D4714yes—
391C24—471no—
392#41—944no—
393#41—944no—
394C2×Q8—472no—
395C2×Q8C2×C4×D4714yes—
396C2×D4—474no—
397C2×D4—472no—
398C2×D4—478no—
399C2×D4—474no—
400C22×C4—471no—
401C22×C4—472no—
402C22×C4—472no—
403C22×C4—474no—
404C22×C4—472no—
405C22×C4—478no—
406C22×C4—942no—
407C22×C4—942no—
408C22×C4C2×C4×D4712yes—
409C22×C4C23×D4711yescharacteristic
410M16—24no—
411M16—944no—
412M16—944no—
413M16—948no—
414M16—944no—
415M16—948no—
416C2×C8—22no—
417C2×C8—474no—
418C2×C8—471no—
419C2×C8—471no—
420C2×C8—472no—
421C2×C8—474no—
422C2×C8—948no—
423C2×C8—942no—
424C2×C8—948no—
425C2×C8—948no—
426C2×C8#1062535414yes—
427C2×C8C23×D4712yes—
428C2×C8C23×D4714yes—
429C2×C8C23×D4712yes—
430C4⋊C4—474no—
431C4⋊C4—474no—
432C4⋊C4—478no—
433C4⋊C4—474no—
434C4⋊C4—478no—
435C4⋊C4C2×C4×D4714yes—
436C4⋊C4C2×C4×D4718yes—
437C4⋊C4C2×C4×D4714yes—
438C22⋊C4—474no—
439C22⋊C4—474no—
440C22⋊C4—478no—
441C22⋊C4—478no—
442C22⋊C4—478no—
443C22⋊C4—478no—
444C22⋊C4C2×C4×D4714yes—
445C42—472no—
446C42—478no—
447C42—472no—
448C42C2×C4×D4711yescharacteristic
449C42C2×C4×D4711yescharacteristic
450C42C23×D4712yes—
451C42C23×D4714yes—
452C23—472no—
453C23—474no—
454C23—942no—
455C23—1882no—
456C23C22×C4×D4711yescharacteristic
457Q8—44no—
458Q8—944no—
459D4—948no—
460D4—944no—
461D4—1884no—
462D4—1888no—
463C2×C4—42no—
464C2×C4—474no—
465C2×C4—474no—
466C2×C4—472no—
467C2×C4—942no—
468C2×C4—944no—
469C2×C4—944no—
470C2×C4—942no—
471C2×C4—942no—
472C2×C4—944no—
473C2×C4—1888no—
474C2×C4—1882no—
475C2×C4C22×C4×D4718yes—
476C2×C4C22×C4×D4712yes—
477C2×C4C22×C4×D4712yes—
478C2×C4C22×C4×D4711yescharacteristic
479C2×C4#41×D4712yes—
480C2×C4C2×#1062535412yes—
481C2×C4C24×D4711yescharacteristic
482C8—22no—
483C8—24no—
484C8—472no—
485C8—472no—
486C8—942no—
487C8—944no—
488C8C2×#1062535414yes—
489C22—22no—
490C22—948no—
491C22—944no—
492C22—942no—
493C22—944no—
494C22—1884no—
495C22C4×#1062535412yes—
496C22C23×C4×D4711yescharacteristic
497C4—22no—
498C4—22no—
499C4—28no—
500C4—944no—
501C4—942no—
502C4—944no—
503C4C4×#1062535412yes—
504C4C2×#41×D4711yescharacteristic
505C4C22×#1062535411yescharacteristic
506C2—22no—
507C2—1882no—
508C2—1884no—
509C2C2×C4×#1062535411yescharacteristic
510C2#2025810112yes—
subgroupG/H
center, FrattiniC2×C4C24×D47
commutator, socleC22×C47C24×C4
FittingC47×#2961C2
prime radical#20256780C2
cosocleC2×C4×C47C25
hypercenter#2961D47
hypocenterC47#60209
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ #2961

Complements

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

NH ≅ G/Ntype
2. #20252394C2semidirect
3. #20252381C2semidirect
4. #20252323C2semidirect
5. #20252107C2semidirect
6. #20252099C2semidirect
7. #20252090C2semidirect
8. #20252084C2semidirect
9. #20252031C2semidirect
10. #20252025C2semidirect
11. #20251815C2semidirect
12. C2×#15689080C2semidirect
13. C47×#2961C2semidirect
28. #15689254C22semidirect
29. Q8×C47⋊C8C22semidirect
31. #15689200C22semidirect
33. #15689196C22semidirect
36. #15689188C22semidirect
38. #15689183C22semidirect
39. #15689175C22semidirect
47. #15689080C22semidirect
49. #15689077C22semidirect
51. C4×#10625324C22semidirect
53. #97×D47C22semidirect
57. C47×#446C22semidirect
254. #2961D47semidirect
285. #446C2×D47semidirect
317. C47#60209semidirect

Maximal subgroups

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

subgroupindex
#202567802
#202523942
#202523812
#202523232
#202521072
#202520992
#202520902
#202520842
#202520312
#202520252
#202518152
C2×#156890802
C47×#29612
#6020947