Finite Group Databasebeta

← #26216719

Subgroups — #26216719

Order 12032.

1070 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689164C211yesmaximal, characteristic
2C2×#15689162C211yesmaximal, characteristic
3C2×#15689160C211yesmaximal, characteristic
4C2×#106×D47C211yesmaximal, characteristic
5C2×#15689080C211yesmaximal, characteristic
6#20246013C211yesmaximal, characteristic
7#20246009C211yesmaximal, characteristic
8#20243556C211yesmaximal, characteristic, complemented
9#20243555C211yesmaximal, characteristic, complemented
10#20243553C211yesmaximal, characteristic, complemented
11#20243551C211yesmaximal, characteristic, complemented
12#20243550C211yesmaximal, characteristic, complemented
13#20243544C211yesmaximal, characteristic, complemented
14#20243543C211yesmaximal, characteristic, complemented
15C47×#1847C211yesmaximal, characteristic, complemented
16C22×#10625324C2211yescharacteristic
17C22×C8×D47C2211yescharacteristic
18C2×#13093982C2211yescharacteristic
19C2×#13093982C2211yescharacteristic
20C2×#13093981C2211yescharacteristic
21C2×C4⋊C4×D47C2211yescharacteristic
22C2×C4⋊C4×D47C2211yescharacteristic
23#15689164—24no—
24#15689162—24no—
25#15689160—24no—
26C2×#13093958C2211yescharacteristic
27C2×#13093958C2211yescharacteristic
28C2×#13093957C2211yescharacteristic
29C2×#13093957C2211yescharacteristic
30C2×#13093956C2211yescharacteristic
31C2×#13093956C2211yescharacteristic
32C2×#13093955C2211yescharacteristic
33C2×#13093955C2211yescharacteristic
34C2×#13093955C2211yescharacteristic
35C2×#13093955C2211yescharacteristic
36C2×#13093955C2211yescharacteristic
37C2×#13093955C2211yescharacteristic
38#106×D47—24no—
39#15689080C418yescomplemented
40C2×#13093934C2211yescharacteristic
41C2×C4×C47⋊C8C2211yescharacteristic
42C2×C8×Dic47C2211yescharacteristic
43#15689020—22no—
44#15689018—22no—
45#15688997—22no—
46#15688994—22no—
47#15688993C2211yescharacteristic
48#15688992C2211yescharacteristic
49#15688992C2211yescharacteristic
50#15688989C2211yescharacteristic
51#15688988C2211yescharacteristic
52#15688988C2211yescharacteristic
53C2×#107×C47C2211yescharacteristic
54C2×#107×C47C2211yescharacteristic
55C2×#106×C47C2211yescharacteristic
56C2×#106×C47C2211yescharacteristic
57C2×#97×C47C2211yescharacteristic
58C47×#384C2211yescharacteristic
59C47×#382C2211yescharacteristic
60C22×C4×D47—21no—
61C22×C4×D47—21no—
62C22×C4×D47C2311yescharacteristic
63C2×#10625324C2×C414yes—
64C2×#10625324D414yes—
65C2×#10625324Q814yes—
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C2×C8×D47C2×C414yes—
69C2×C8×D47D414yes—
70C2×C8×D47Q814yes—
71#13093982—22no—
72#13093982—22no—
73#13093982—22no—
74#13093982—22no—
75#13093981C2×C414yes—
76#13093981Q814yes—
77C2×C4⋊Dic47—22no—
78C2×C4⋊Dic47C2311yescharacteristic
79C2×C4⋊Dic47C2311yescharacteristic
80C2×C4⋊Dic47C2311yescharacteristic
81C2×C4⋊Dic47C2311yescharacteristic
82C2×Dic47⋊C4—21no—
83C2×Dic47⋊C4—21no—
84C2×Dic47⋊C4—22no—
85C2×Dic47⋊C4—22no—
86C2×Dic47⋊C4D412yes—
87C4⋊C4×D47—22no—
88C4⋊C4×D47—22no—
89C4⋊C4×D47—22no—
90C4⋊C4×D47—22no—
91C2×(C2×D47)⋊C4—21no—
92C2×(C2×D47)⋊C4—21no—
93C2×(C2×D47)⋊C4D412yes—
94C2×C4×Dic47—21no—
95C2×C4×Dic47—21no—
96C2×C4×Dic47C2311yescharacteristic
97#13093958—22no—
98#13093958—22no—
99#13093957—22no—
100#13093957—22no—
101#13093956—22no—
102#13093956—22no—
103#13093955—22no—
104#13093955—22no—
105#13093955—22no—
106#13093955—22no—
107#13093955—22no—
108#13093955—22no—
109#13093934C2×C414yescomplemented
110C4×C47⋊C8C2×C414yescomplemented
111C8×Dic47C2×C414yescomplemented
112C47⋊#95—22no—
113C47⋊#95—22no—
114C47⋊#95—22no—
115C22×C8×C47C2311yescharacteristic
116C22×C8×C47C2311yescharacteristic
117C2×C4⋊C4×C47—22no—
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C4⋊C4×C47C2311yescharacteristic
120C2×C4⋊C4×C47C2311yescharacteristic
121C2×C4⋊C4×C47C2311yescharacteristic
122C2×C42×C47C2311yescharacteristic
123#107×C47—22no—
124#107×C47—22no—
125#106×C47—22no—
126#106×C47—22no—
127#97×C47C2×C414yescomplemented
128#95×C47—22no—
129C23×D47C2×D411yescharacteristic
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47C2×D412yes—
135C22×C4×C47C2411yescharacteristic
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47C2×D412yes—
141C22×Dic47C2×D411yescharacteristic
142C2×C4×D47—42no—
143C2×C4×D47—42no—
144C2×C4×D47—82no—
145C2×C4×D47—82no—
146C2×C4×D47C22×C412yes—
147C2×C4×D47C2×D412yes—
148C2×C4×D47C2×D412yes—
149C2×C4×D47C2×Q812yes—
150C2×C4×D47C2×Q812yes—
151C2×C4×D47C2×Q812yes—
152C2×C8×C47C22×C412yes—
153C2×C8×C47C22×C412yes—
154C2×C8×C47C2×D412yes—
155C2×C8×C47C2×D412yes—
156C2×C8×C47C2×Q812yes—
157C2×C8×C47C2×Q812yes—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—82no—
167C42×C47C22×C412yes—
168C42×C47C2×Q812yes—
169(C2×D47)⋊C4—42no—
170(C2×D47)⋊C4—42no—
171(C2×D47)⋊C4C4⋊C418yes—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—82no—
181Dic47⋊C4—42no—
182Dic47⋊C4—42no—
183Dic47⋊C4—82no—
184Dic47⋊C4—82no—
185Dic47⋊C4C4⋊C418yes—
186C4×Dic47—42no—
187C4×Dic47—42no—
188C4×Dic47C22×C412yes—
189C4×Dic47C2×Q812yes—
190C2×C47⋊C8C22×C412yes—
191C2×C47⋊C8C22×C412yes—
192C2×C47⋊C8C2×D412yes—
193C2×C47⋊C8C2×D412yes—
194C2×C47⋊C8C2×Q812yes—
195C2×C47⋊C8C2×Q812yes—
196#10625324C4⋊C4116yescomplemented
197C8×D47C4⋊C4116yescomplemented
198C23×C47C22×D411yescharacteristic
199C22×D47—22no—
200C22×D47—22no—
201C22×D47C2×C4⋊C412yes—
202C22×D47#12712yes—
203C22×D47C4⋊Q812yes—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—41no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47C2×C4⋊C414yes—
215C2×C4×C47C23×C411yescharacteristic
216C2×C4×C47C22×D411yescharacteristic
217C2×C4×C47C22×D411yescharacteristic
218C2×C4×C47C22×Q811yescharacteristic
219C2×C4×C47C22×Q811yescharacteristic
220C2×C4×C47C22×Q811yescharacteristic
221C2×Dic47—22no—
222C2×Dic47—22no—
223C2×Dic47—41no—
224C2×Dic47—41no—
225C2×Dic47—41no—
226C2×Dic47—41no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47C2×C4⋊C414yes—
232C2×Dic47C2×C4⋊C412yes—
233C2×Dic47#12712yes—
234C2×Dic47C4⋊Q812yes—
235C4×D47—84no—
236C4×D47—84no—
237C4×D47C2×C4⋊C414yes—
238C4×D47C2×C4⋊C414yes—
239C4×D47C4⋊Q814yes—
240C4×D47C4⋊Q814yes—
241C8×C47C2×C4⋊C414yes—
242C8×C47C2×C4⋊C414yes—
243C47⋊C8C2×C4⋊C414yes—
244C47⋊C8C2×C4⋊C414yes—
245#20802—471nomaximal, Sylow
246C22×C47C22×C4⋊C411yescharacteristic
247C22×C47C2×#12711yescharacteristic
248C22×C47C2×C4⋊Q811yescharacteristic
249C22×C47#57712yes—
250C22×C47#57712yes—
251C2×D47—22no—
252C2×D47—22no—
253C2×D47—24no—
254C2×D47—24no—
255C2×D47#38414yes—
256C4×C47—24no—
257C4×C47—81no—
258C4×C47—81no—
259C4×C47—81no—
260C4×C47—81no—
261C4×C47C22×C4⋊C411yescharacteristic
262C4×C47C22×C4⋊C411yescharacteristic
263C4×C47C2×C4⋊Q811yescharacteristic
264C4×C47C2×C4⋊Q811yescharacteristic
265Dic47—24no—
266Dic47—81no—
267Dic47—81no—
268Dic47—81no—
269Dic47—81no—
270Dic47#38414yes—
271C2×#428—471no—
272C2×#428—471no—
273C2×#427—471no—
274C22×#106—471no—
275C2×#405—471no—
276#2317—471no—
277#2317—471no—
278#1847—471no—
279#1847—471no—
280#1847—471no—
281#1847D4711yescharacteristic, complemented
282#1846—471no—
283#1846—471no—
284#1844—471no—
285#1844—471no—
286C2×C47C2×#38411yescharacteristic
287C2×C47#291212yes—
288C2×C47#315411yescharacteristic
289C2×C47#315411yescharacteristic
290C2×C47#316312yes—
291D47—24no—
292C22×M16—471no—
293C23×C8—471no—
294C2×#117—471no—
295C2×#117—471no—
296C2×#117—471no—
297C22×C4⋊C4—471no—
298C22×C4⋊C4—471no—
299#428—944no—
300#428—944no—
301#427—944no—
302C2×#107—471no—
303C2×#107—471no—
304C2×#107—471no—
305C2×#107—471no—
306C2×#107C2×D4711yescharacteristic
307C2×#107C2×D4711yescharacteristic
308C2×#106—471no—
309C2×#106—471no—
310C2×#106—471no—
311C2×#106—471no—
312C2×#106—471no—
313C2×#106—471no—
314C2×#106—471no—
315C2×#106—471no—
316C2×#106—944no—
317C2×#106C2×D4711yescharacteristic
318C2×#106C2×D4711yescharacteristic
319#405—478no—
320C2×#97—471no—
321C2×#97C2×D4711yescharacteristic
322C2×C4×C8—471no—
323C2×C4×C8—471no—
324#389—942no—
325#389—942no—
326#385—942no—
327#385—942no—
328#384—471no—
329#384—471no—
330#384—471no—
331#384C2×D4711yescharacteristic
332#382—471no—
333#382—471no—
334#382—471no—
335#382C2×D4711yescharacteristic
336C47#2080211yescharacteristic, complemented, Sylow
337C23×C4—471no—
338C23×C4—941no—
339C23×C4—941no—
340C2×M16—474no—
341C2×M16—474no—
342C2×M16—474no—
343C22×C8—471no—
344C22×C8—471no—
345C22×C8—474no—
346C22×C8—474no—
347C22×C8—474no—
348C22×C8C22×D4711yescharacteristic
349C22×C8C22×D4711yescharacteristic
350#117—474no—
351#117—474no—
352#117—942no—
353#117—942no—
354#117—942no—
355#117—942no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—472no—
358C2×C4⋊C4—471no—
359C2×C4⋊C4—471no—
360C2×C4⋊C4—471no—
361C2×C4⋊C4—471no—
362C2×C4⋊C4—942no—
363C2×C4⋊C4—942no—
364C2×C4⋊C4—942no—
365C2×C4⋊C4—942no—
366C2×C4⋊C4—942no—
367C2×C4⋊C4—942no—
368C2×C4⋊C4—942no—
369C2×C4⋊C4—941no—
370C2×C4⋊C4—941no—
371C2×C4⋊C4C22×D4711yescharacteristic
372C2×C4⋊C4C22×D4711yescharacteristic
373C2×C4⋊C4C22×D4711yescharacteristic
374C2×C4⋊C4C22×D4711yescharacteristic
375C2×C22⋊C4—472no—
376C2×C22⋊C4—941no—
377C2×C22⋊C4—941no—
378C2×C42—471no—
379C2×C42—941no—
380C2×C42—941no—
381C2×C42C22×D4711yescharacteristic
382#107—22no—
383#107—22no—
384#107—942no—
385#107—942no—
386#107—942no—
387#107—942no—
388#106—22no—
389#106—22no—
390#106—942no—
391#106—942no—
392#106—942no—
393#106—942no—
394#106—942no—
395#106—942no—
396#106—942no—
397#106—942no—
398#97—474no—
399#97C4×D4714yescomplemented
400C4×C8—474no—
401C4×C8—474no—
402#95—22no—
403#95—942no—
404#95—942no—
405#95—942no—
406C24—471no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—471no—
412C22×C4—472no—
413C22×C4—472no—
414C22×C4—472no—
415C22×C4—472no—
416C22×C4—472no—
417C22×C4—472no—
418C22×C4—472no—
419C22×C4—941no—
420C22×C4—941no—
421C22×C4—941no—
422C22×C4—941no—
423C22×C4—3762no—
424C22×C4—3762no—
425C22×C4—1882no—
426C22×C4—1882no—
427C22×C4D4×D4712yes—
428C22×C4C23×D4711yescharacteristic
429M16—4716no—
430C2×C8—472no—
431C2×C8—472no—
432C2×C8—472no—
433C2×C8—472no—
434C2×C8—472no—
435C2×C8—472no—
436C2×C8—4716no—
437C2×C8C2×C4×D4712yes—
438C2×C8C2×C4×D4712yes—
439C2×C8D4×D4712yes—
440C2×C8D4×D4712yes—
441C2×C8Q8×D4712yes—
442C2×C8Q8×D4712yes—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—82no—
452C4⋊C4—478no—
453C4⋊C4—941no—
454C4⋊C4—941no—
455C4⋊C4—941no—
456C4⋊C4—941no—
457C4⋊C4—941no—
458C4⋊C4—941no—
459C4⋊C4—941no—
460C4⋊C4—941no—
461C4⋊C4—3762no—
462C4⋊C4—3762no—
463C4⋊C4—3762no—
464C4⋊C4—1882no—
465C4⋊C4—1882no—
466C22⋊C4—478no—
467C22⋊C4—1882no—
468C22⋊C4—1882no—
469C42—472no—
470C42—472no—
471C42—1882no—
472C42—1882no—
473C42C2×C4×D4712yes—
474C42Q8×D4712yes—
475C23—472no—
476C23—472no—
477C23—472no—
478C23—942no—
479C23—942no—
480C23C2×D4×D4711yescharacteristic
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—472no—
492C2×C4—472no—
493C2×C4—472no—
494C2×C4—474no—
495C2×C4—474no—
496C2×C4—474no—
497C2×C4—474no—
498C2×C4—474no—
499C2×C4—942no—
500C2×C4—942no—
501C2×C4—3764no—
502C2×C4—3764no—
503C2×C4—1881no—
504C2×C4—1881no—
505C2×C4—1881no—
506C2×C4—1881no—
507C2×C4—1882no—
508C2×C4—1882no—
509C2×C4—1882no—
510C2×C4—1882no—
511C2×C4C4⋊C4×D4714yes—
512C2×C4C22×C4×D4711yescharacteristic
513C2×C4C2×D4×D4711yescharacteristic
514C2×C4C2×D4×D4711yescharacteristic
515C2×C4C2×Q8×D4711yescharacteristic
516C2×C4C2×Q8×D4711yescharacteristic
517C2×C4C2×Q8×D4711yescharacteristic
518C8—474no—
519C8—474no—
520C8C4⋊C4×D4714yes—
521C8C4⋊C4×D4714yes—
522C22—474no—
523C22—944no—
524C22—942no—
525C22—942no—
526C22—944no—
527C22C2×C4⋊C4×D4711yescharacteristic
528C22#127×D4711yescharacteristic
529C22C4⋊Q8×D4711yescharacteristic
530C22#1568993212yes—
531C22#1568993312yes—
532C4—24no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—81no—
537C4—474no—
538C4—944no—
539C4—3761no—
540C4—3761no—
541C4—3761no—
542C4—3761no—
543C4C2×C4⋊C4×D4711yescharacteristic
544C4C2×C4⋊C4×D4711yescharacteristic
545C4C4⋊Q8×D4711yescharacteristic
546C4C4⋊Q8×D4711yescharacteristic
547C2—944no—
548C2#384×D4711yescharacteristic
549C2#2025198212yes—
550C2#2025386111yescharacteristic
551C2#2025386311yescharacteristic
552C2#2025391512yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1847C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1847D47
hypocenterC47#20802
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
8. #20243556C2semidirect
9. #20243555C2semidirect
10. #20243553C2semidirect
11. #20243551C2semidirect
12. #20243550C2semidirect
13. #20243544C2semidirect
14. #20243543C2semidirect
15. C47×#1847C2semidirect
39. #15689080C4semidirect
109. #13093934C2×C4semidirect
110. C4×C47⋊C8C2×C4semidirect
111. C8×Dic47C2×C4semidirect
127. #97×C47C2×C4semidirect
196. #10625324C4⋊C4semidirect
197. C8×D47C4⋊C4semidirect
281. #1847D47semidirect
336. C47#20802semidirect
399. #97C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156891642
C2×#156891622
C2×#156891602
C2×#106×D472
C2×#156890802
#202460132
#202460092
#202435562
#202435552
#202435532
#202435512
#202435502
#202435442
#202435432
C47×#18472
#2080247