Finite Group Databasebeta

← #27080846

Subgroups — #27080846

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#20252393C218yesmaximal, complemented
2#20252384C218yesmaximal, complemented
3#20252329C211yesmaximal, characteristic, complemented
4#20252323C211yesmaximal, characteristic, complemented
5#20252106C211yesmaximal, characteristic, complemented
6#20252100C211yesmaximal, characteristic, complemented
7#20252092C212yesmaximal, complemented
8#20252083C212yesmaximal, complemented
9#20252032C212yesmaximal, complemented
10#20252026C211yesmaximal, characteristic, complemented
11#20251816C212yesmaximal, complemented
12C2×#15689079C211yesmaximal, characteristic, complemented
13C47×#3455C211yesmaximal, characteristic
14C2×#13094031C2212yes—
15C2×#13094031C2212yes—
16C22×#10625329C2211yescharacteristic
17C23×C47⋊C8C2211yescharacteristic
18#15689254C2214yescomplemented
19#15689254C2218yescomplemented
20Q8×C47⋊C8C2214yescomplemented
21#15689201C2218yes—
22#15689201C2214yescomplemented
23#15689195C2218yes—
24#15689195C2218yes—
25#15689195C2218yescomplemented
26D4×C47⋊C8C2218yes—
27D4×C47⋊C8C2214yescomplemented
28#15689184C2218yes—
29#15689184C2218yescomplemented
30#15689176C2218yescomplemented
31#15689150C2212yes—
32#15689150C2212yes—
33#15689147C2211yescharacteristic
34#15689147C2211yescharacteristic
35#15689090C2212yes—
36#15689090C2212yes—
37#15689087C2212yes—
38#15689079C2214yes—
39#15689079C2214yes—
40#15689079C2212yes—
41#15689079C2212yes—
42#15689079C2214yescomplemented
43#15689079C2214yescomplemented
44C4×#10625329C2212yes—
45C4×#10625329C2214yescomplemented
46C2×#13093935C2211yescharacteristic
47C2×#13093935C2211yescharacteristic
48C2×#13093935C2214yescomplemented
49C2×C4×C47⋊C8C2211yescharacteristic
50C2×C4×C47⋊C8C2211yescharacteristic
51C47×#542C414yes—
52C47×#541C412yes—
53C47×#540C412yes—
54C47×#539C414yes—
55C47×#538C414yes—
56C47×#533C2211yescharacteristic
57C47×#532C2211yescharacteristic
58C47×#529C2218yes—
59C47×#525C2212yes—
60C4×#41×C47C2212yes—
61C2×#117×C47C2211yescharacteristic
62#13094031—28no—
63#13094031—28no—
64C2×#10625329—24no—
65C2×#10625329C2312yes—
66C2×#10625329C2312yes—
67C2×#10625329C2314yes—
68C2×#10625329C2314yes—
69C2×#10625329C2312yes—
70C2×#10625329C2312yes—
71C22×C47⋊C8—24no—
72C22×C47⋊C8C2311yescharacteristic
73C22×C47⋊C8C2311yescharacteristic
74C22×C47⋊C8C2314yes—
75C22×C47⋊C8C2314yes—
76C22×C47⋊C8C2314yes—
77C22×C47⋊C8C2311yescharacteristic
78C22×C47⋊C8C2311yescharacteristic
79C4⋊(C47⋊C8)C2314yes—
80C4⋊(C47⋊C8)C2318yes—
81C4⋊(C47⋊C8)C2314yes—
82C4⋊(C47⋊C8)C2318yes—
83C22⋊(C47⋊C8)C2318yes—
84C22⋊(C47⋊C8)C2318yes—
85C22⋊(C47⋊C8)C2318yes—
86#13093935C2314yes—
87#13093935C2314yes—
88#13093935C2314yes—
89#13093935C2312yes—
90#13093935C2312yes—
91#13093935C2314yes—
92C4×C47⋊C8C2312yes—
93C4×C47⋊C8C2314yes—
94C4×C47⋊C8C2312yes—
95C4×C47⋊C8C2314yes—
96C4×C47⋊C8C2314yes—
97C4×C47⋊C8C2314yes—
98C2×#41×C47C2312yes—
99C23×C4×C47C2311yescharacteristic
100C4⋊Q8×C47C2×C411yescharacteristic
101C4⋊Q8×C47C2×C412yes—
102#127×C47C2×C411yescharacteristic
103#126×C47C2×C418yes—
104#126×C47C2×C418yes—
105#125×C47C2×C418yes—
106#125×C47C2×C412yes—
107#124×C47C2×C418yes—
108#124×C47C2×C412yes—
109#123×C47C2×C414yes—
110#123×C47C2×C418yes—
111#123×C47C2×C418yes—
112C22⋊Q8×C47C2×C414yes—
113C22⋊Q8×C47C2×C418yes—
114C22⋊Q8×C47C2×C414yes—
115#121×C47C2×C414yes—
116#121×C47C2×C418yes—
117#121×C47C2×C414yes—
118#120×C47C2×C414yes—
119C4×Q8×C47C2×C414yes—
120C4×Q8×C47C2314yes—
121C4×D4×C47C2×C418yes—
122C4×D4×C47C2×C414yes—
123C4×D4×C47C2318yes—
124C4×D4×C47C2314yes—
125#117×C47C2×C412yes—
126#117×C47C2×C412yes—
127#117×C47C2314yes—
128#117×C47C2312yes—
129#117×C47C2314yes—
130C2×C4⋊C4×C47C2×C412yes—
131C2×C22⋊C4×C47C2×C412yes—
132C2×C42×C47C2311yescharacteristic
133C2×C42×C47C2311yescharacteristic
134C2×C42×C47C2314yes—
135C24×C47C22×C411yescharacteristic
136#41×C47—24no—
137#41×C47—24no—
138C2×Q8×C47C22×C412yes—
139C2×Q8×C47C22×C414yes—
140C2×D4×C47C22×C414yes—
141C2×D4×C47C22×C412yes—
142C2×D4×C47C22×C418yes—
143C2×D4×C47C22×C414yes—
144C22×C4×C47—22no—
145C22×C4×C47—22no—
146C22×C4×C47C22×C411yescharacteristic
147C22×C4×C47C22×C412yes—
148C22×C4×C47C22×C412yes—
149C22×C4×C47C22×C412yes—
150C22×C4×C47C22×C418yes—
151C22×C4×C47C2414yes—
152C22×C4×C47C2412yes—
153C22×C4×C47C2411yescharacteristic
154C4⋊C4×C47C22×C414yes—
155C4⋊C4×C47C22×C414yes—
156C4⋊C4×C47C22×C414yes—
157C4⋊C4×C47C22×C418yes—
158C4⋊C4×C47C22×C414yes—
159C4⋊C4×C47C22×C418yes—
160C4⋊C4×C47C22×C418yes—
161C4⋊C4×C47C22×C414yes—
162C22⋊C4×C47C22×C414yes—
163C22⋊C4×C47C22×C414yes—
164C22⋊C4×C47C22×C414yes—
165C22⋊C4×C47C22×C418yes—
166C22⋊C4×C47C22×C418yes—
167C22⋊C4×C47C22×C418yes—
168C22⋊C4×C47C22×C418yes—
169C42×C47C22×C412yes—
170C42×C47C22×C411yescharacteristic
171C42×C47C22×C411yescharacteristic
172C42×C47C22×C418yes—
173C42×C47C2412yes—
174C42×C47C2412yes—
175C42×C47C2414yes—
176#10625329—24no—
177#10625329—24no—
178#10625329—24no—
179#10625329—24no—
180#10625329—28no—
181#10625329—28no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—28no—
185C2×C47⋊C8—28no—
186C2×C47⋊C8—28no—
187C2×C47⋊C8#4114yes—
188C2×C47⋊C8#4114yes—
189C2×C47⋊C8C2412yes—
190C2×C47⋊C8C2414yes—
191C2×C47⋊C8C2414yes—
192C2×C47⋊C8C2412yes—
193C2×C47⋊C8C2411yescharacteristic
194C2×C47⋊C8C2411yescharacteristic
195C2×C47⋊C8C2412yes—
196C23×C47—22no—
197C23×C47—42no—
198C23×C47C23×C412yes—
199C23×C47C23×C414yes—
200C23×C47C23×C411yescharacteristic
201Q8×C47—24no—
202Q8×C47—44no—
203D4×C47—28no—
204D4×C47—24no—
205D4×C47—44no—
206D4×C47—48no—
207C2×C4×C47—22no—
208C2×C4×C47—24no—
209C2×C4×C47—22no—
210C2×C4×C47—24no—
211C2×C4×C47—24no—
212C2×C4×C47—22no—
213C2×C4×C47—42no—
214C2×C4×C47—48no—
215C2×C4×C47—42no—
216C2×C4×C47C23×C414yes—
217C2×C4×C47C23×C414yes—
218C2×C4×C47C23×C412yes—
219C2×C4×C47C23×C418yes—
220C2×C4×C47C23×C412yes—
221C2×C4×C47C23×C412yes—
222C2×C4×C47C23×C411yescharacteristic
223C2×C4×C47C2×#4112yes—
224C2×C4×C47C2×#4112yes—
225C2×C4×C47C2511yescharacteristic
226C47⋊C8—22no—
227C47⋊C8—22no—
228C47⋊C8—24no—
229C47⋊C8—24no—
230C47⋊C8C2×#4114yes—
231C47⋊C8C2×#4112yes—
232C47⋊C8C2×#4112yes—
233#60209—471nomaximal, Sylow
234C22×C47—22no—
235C22×C47—22no—
236C22×C47—24no—
237C22×C47—28no—
238C22×C47—24no—
239C22×C47—44no—
240C22×C47C4×#4112yes—
241C22×C47C24×C411yescharacteristic
242C4×C47—22no—
243C4×C47—24no—
244C4×C47—24no—
245C4×C47—22no—
246C4×C47—28no—
247C4×C47—22no—
248C4×C47C4×#4112yes—
249C4×C47C22×#4111yescharacteristic
250C4×C47C22×#4111yescharacteristic
251#3455D4711yescharacteristic
252#2969—478no—
253#2968—478no—
254#2961—471no—
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×C47—44no—
266C2×C47—42no—
267C2×C47C2×C4×#4111yescharacteristic
268C2×C47#357312yes—
269C2×#131—472no—
270C2×#131—472no—
271C22×M16—471no—
272C23×C8—471no—
273#542Dic4714yes—
274#541Dic4712yes—
275#540Dic4712yes—
276#539Dic4714yes—
277#538Dic4714yes—
278#533C2×D4711yescharacteristic
279#532C2×D4711yescharacteristic
280#529C2×D4718yes—
281#525C2×D4712yes—
282C4×#41C2×D4712yes—
283C2×#117C2×D4711yescharacteristic
284#446—474no—
285#446—478no—
286C8×Q8—474no—
287#436—478no—
288#436—474no—
289#435—478no—
290#435—478no—
291#435—478no—
292C8×D4—478no—
293C8×D4—474no—
294#433—478no—
295#433—478no—
296#431—478no—
297#424—472no—
298#424—472no—
299#423—471no—
300#423—471no—
301#408—472no—
302#408—472no—
303#407—472no—
304#405—474no—
305#405—474no—
306#405—472no—
307#405—472no—
308#405—474no—
309#405—474no—
310C4×M16—472no—
311C4×M16—474no—
312C2×#97—471no—
313C2×#97—471no—
314C2×#97—474no—
315C2×C4×C8—471no—
316C2×C4×C8—471no—
317C47#6020911yescharacteristic, complemented, Sylow
318C2×#41C22×D4712yes—
319C23×C4C22×D4711yescharacteristic
320#131—948no—
321#131—948no—
322C2×M16—472no—
323C2×M16—472no—
324C2×M16—472no—
325C2×M16—472no—
326C2×M16—474no—
327C2×M16—474no—
328C2×M16—944no—
329C22×C8—471no—
330C22×C8—471no—
331C22×C8—471no—
332C22×C8—474no—
333C22×C8—471no—
334C22×C8—474no—
335C22×C8—474no—
336C22×C8—944no—
337C4⋊Q8C2×Dic4711yescharacteristic
338C4⋊Q8C2×Dic4712yes—
339#127C2×Dic4711yescharacteristic
340#126C2×Dic4718yes—
341#126C2×Dic4718yes—
342#125C2×Dic4718yes—
343#125C2×Dic4712yes—
344#124C2×Dic4718yes—
345#124C2×Dic4712yes—
346#123C2×Dic4714yes—
347#123C2×Dic4718yes—
348#123C2×Dic4718yes—
349C22⋊Q8C2×Dic4714yes—
350C22⋊Q8C2×Dic4718yes—
351C22⋊Q8C2×Dic4714yes—
352#121C2×Dic4714yes—
353#121C2×Dic4718yes—
354#121C2×Dic4714yes—
355#120C2×Dic4714yes—
356C4×Q8C2×Dic4714yes—
357C4×Q8C22×D4714yes—
358C4×D4C2×Dic4718yes—
359C4×D4C2×Dic4714yes—
360C4×D4C22×D4718yes—
361C4×D4C22×D4714yes—
362#117C2×Dic4712yes—
363#117C2×Dic4712yes—
364#117C22×D4714yes—
365#117C22×D4712yes—
366#117C22×D4714yes—
367C2×C4⋊C4C2×Dic4712yes—
368C2×C22⋊C4C2×Dic4712yes—
369C2×C42C22×D4711yescharacteristic
370C2×C42C22×D4711yescharacteristic
371C2×C42C22×D4714yes—
372C4⋊C8—474no—
373C4⋊C8—474no—
374C4⋊C8—478no—
375C4⋊C8—478no—
376C22⋊C8—478no—
377C22⋊C8—478no—
378C22⋊C8—478no—
379#97—472no—
380#97—472no—
381#97—474no—
382#97—474no—
383#97—474no—
384#97—474no—
385C4×C8—474no—
386C4×C8—472no—
387C4×C8—472no—
388C4×C8—474no—
389C4×C8—474no—
390C4×C8—474no—
391C24C22×Dic4711yescharacteristic
392#41—24no—
393#41—24no—
394C2×Q8C22×Dic4712yes—
395C2×Q8C22×Dic4714yes—
396C2×D4C22×Dic4714yes—
397C2×D4C22×Dic4712yes—
398C2×D4C22×Dic4718yes—
399C2×D4C22×Dic4714yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4C22×Dic4711yescharacteristic
403C22×C4C22×Dic4712yes—
404C22×C4C22×Dic4712yes—
405C22×C4C22×Dic4712yes—
406C22×C4C22×Dic4718yes—
407C22×C4C23×D4714yes—
408C22×C4C23×D4712yes—
409C22×C4C23×D4711yescharacteristic
410M16—944no—
411M16—944no—
412M16—948no—
413M16—948no—
414M16—944no—
415M16—944no—
416C2×C8—474no—
417C2×C8—474no—
418C2×C8—471no—
419C2×C8—471no—
420C2×C8—472no—
421C2×C8—472no—
422C2×C8—472no—
423C2×C8—474no—
424C2×C8—474no—
425C2×C8—948no—
426C2×C8—948no—
427C2×C8—948no—
428C2×C8—942no—
429C2×C8—942no—
430C4⋊C4C22×Dic4714yes—
431C4⋊C4C22×Dic4714yes—
432C4⋊C4C22×Dic4714yes—
433C4⋊C4C22×Dic4718yes—
434C4⋊C4C22×Dic4714yes—
435C4⋊C4C22×Dic4718yes—
436C4⋊C4C22×Dic4718yes—
437C4⋊C4C22×Dic4714yes—
438C22⋊C4C22×Dic4714yes—
439C22⋊C4C22×Dic4714yes—
440C22⋊C4C22×Dic4714yes—
441C22⋊C4C22×Dic4718yes—
442C22⋊C4C22×Dic4718yes—
443C22⋊C4C22×Dic4718yes—
444C22⋊C4C22×Dic4718yes—
445C42C22×Dic4712yes—
446C42C22×Dic4711yescharacteristic
447C42C22×Dic4711yescharacteristic
448C42C22×Dic4718yes—
449C42C23×D4712yes—
450C42C23×D4712yes—
451C42C23×D4714yes—
452C23—22no—
453C23—42no—
454C23C23×Dic4712yes—
455C23C23×Dic4714yes—
456C23C23×Dic4711yescharacteristic
457Q8—24no—
458Q8—44no—
459D4—28no—
460D4—24no—
461D4—44no—
462D4—48no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—48no—
471C2×C4—42no—
472C2×C4C23×Dic4714yes—
473C2×C4C23×Dic4714yes—
474C2×C4C23×Dic4712yes—
475C2×C4C23×Dic4718yes—
476C2×C4C23×Dic4712yes—
477C2×C4C23×Dic4712yes—
478C2×C4C23×Dic4711yescharacteristic
479C2×C4#41×D4712yes—
480C2×C4#41×D4712yes—
481C2×C4C24×D4711yescharacteristic
482C8—474no—
483C8—472no—
484C8—472no—
485C8—942no—
486C8—944no—
487C8—944no—
488C8—942no—
489C22—22no—
490C22—28no—
491C22—24no—
492C22—22no—
493C22—24no—
494C22—44no—
495C22#41×Dic4712yes—
496C22C24×Dic4711yescharacteristic
497C4—24no—
498C4—22no—
499C4—28no—
500C4—22no—
501C4—22no—
502C4—24no—
503C4#41×Dic4712yes—
504C4C2×#41×D4711yescharacteristic
505C4C2×#41×D4711yescharacteristic
506C2—22no—
507C2—42no—
508C2—44no—
509C2C2×#41×Dic4711yescharacteristic
510C2#2025810012yes—
subgroupG/H
center, FrattiniC2×C4C24×D47
commutator, socleC22×C47C24×C4
Fitting, prime radicalC47×#3455C2
cosocleC2×C4×C47C25
hypercenter#3455D47
hypocenterC47#60209
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ #3455

Complements

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

NH ≅ G/Ntype
1. #20252393C2semidirect
2. #20252384C2semidirect
3. #20252329C2semidirect
4. #20252323C2semidirect
5. #20252106C2semidirect
6. #20252100C2semidirect
7. #20252092C2semidirect
8. #20252083C2semidirect
9. #20252032C2semidirect
10. #20252026C2semidirect
11. #20251816C2semidirect
12. C2×#15689079C2semidirect
18. #15689254C22semidirect
19. #15689254C22semidirect
20. Q8×C47⋊C8C22semidirect
22. #15689201C22semidirect
25. #15689195C22semidirect
27. D4×C47⋊C8C22semidirect
29. #15689184C22semidirect
30. #15689176C22semidirect
42. #15689079C22semidirect
43. #15689079C22semidirect
45. C4×#10625329C22semidirect
48. C2×#13093935C22semidirect
317. C47#60209semidirect

Maximal subgroups

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

subgroupindex
#202523932
#202523842
#202523292
#202523232
#202521062
#202521002
#202520922
#202520832
#202520322
#202520262
#202518162
C2×#156890792
C47×#34552
#6020947