Finite Group Databasebeta

← #26232495

Subgroups — #26232495

Order 12032.

1474 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689776C211yesmaximal, characteristic
2C2×#15689188C211yesmaximal, characteristic, complemented
3C4⋊C4×D188C211yesmaximal, characteristic, complemented
4#20244350C211yesmaximal, characteristic, complemented
5#20243815C212yesmaximal, complemented
6#20243704C212yesmaximal, complemented
7#20243473C212yesmaximal, complemented
8#20243389C211yesmaximal, characteristic, complemented
9C4×#13093944C211yesmaximal, characteristic, complemented
10#340×D47C212yesmaximal, complemented
11C47×#1988C211yesmaximal, characteristic, complemented
12C22×#10625354—21no—
13C22×C8×D47C2212yes—
14#15689776—24no—
15#15689776C418yescomplemented
16C2×#13094009—22no—
17C2×#13094000—21no—
18C2×#13093999C2212yes—
19C2×C4×D188—21no—
20C2×C4×D188C2211yescharacteristic
21C2×C4×D188C2211yescharacteristic
22C2×C4⋊C4×D47C2212yes—
23C2×C4⋊C4×D47C2212yes—
24#15689188—24no—
25C2×#13093946C2212yes—
26C2×#13093944C2211yescharacteristic
27C2×#13093944C2211yescharacteristic
28C2×(C2×D47)⋊C8C2212yes—
29C2×C4×C47⋊C8C2211yescharacteristic
30#15689021—21no—
31#15688997—22no—
32#15688993C2211yescharacteristic
33#15688980C2212yes—
34#15688886C2212yes—
35#15688885C2211yescharacteristic
36#15688885C2211yescharacteristic
37#15688885C2212yes—
38#15688885C2212yes—
39C2×C4⋊Q8×C47C2211yescharacteristic
40C2×C4⋊C8×C47C2211yescharacteristic
41C4×C4⋊C4×C47C2211yescharacteristic
42C47×#340C2212yes—
43C47×#340C2212yes—
44C2×#10625354—28no—
45C2×#10625354—44no—
46C2×#10625354—44no—
47C22×D188—22no—
48C22×D188D411yescharacteristic
49C22×D188C2311yescharacteristic
50C22×C4×D47—22no—
51C22×C4×D47—21no—
52C22×C4×D47—22no—
53C22×C4×D47C2312yes—
54C22×C47⋊C8C2311yescharacteristic
55C22×C47⋊C8C2311yescharacteristic
56C2×C8×D47—24no—
57C2×C8×D47—24no—
58C2×C8×D47—42no—
59#13094009—216no—
60#13094000—28no—
61#13093999—24no—
62#13093999C2×C418yescomplemented
63C4×D188—22no—
64C4×D188—44no—
65C4×D188C2×C414yes—
66C4×D188C2×C414yescomplemented
67C4×D188D414yes—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47D411yescharacteristic
71C2×C4⋊Dic47C2312yes—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—22no—
74C2×Dic47⋊C4—22no—
75C4⋊C4×D47—24no—
76C4⋊C4×D47—24no—
77C4⋊C4×D47—42no—
78C4⋊C4×D47C2×C418yescomplemented
79C2×(C2×D47)⋊C4—21no—
80C2×(C2×D47)⋊C4—22no—
81C2×(C2×D47)⋊C4—22no—
82C2×(C2×D47)⋊C4D412yes—
83#13093946—42no—
84#13093944—22no—
85#13093944—22no—
86(C2×D47)⋊C8—24no—
87C4×C47⋊C8—22no—
88C47⋊#95—22no—
89C22×Q8×C47—21no—
90C22×C8×C47C2312yes—
91C4⋊Q8×C47—22no—
92C4⋊Q8×C47C2×C414yescomplemented
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47C2311yescharacteristic
95C2×C4⋊C4×C47C2312yes—
96C2×C4⋊C4×C47C2311yescharacteristic
97C2×C42×C47—21no—
98C2×C42×C47C2311yescharacteristic
99C4⋊C8×C47—22no—
100#95×C47—21no—
101C23×D47—21no—
102C23×D47C2×D412yes—
103C2×Q8×C47—24no—
104C2×Q8×C47—42no—
105C2×Q8×C47—42no—
106C22×C4×C47—21no—
107C22×C4×C47—22no—
108C22×C4×C47—21no—
109C22×C4×C47C2×D411yescharacteristic
110C22×C4×C47C2×D411yescharacteristic
111C22×C4×C47C2411yescharacteristic
112C22×Dic47—21no—
113C22×Dic47—22no—
114C22×Dic47C2×D412yes—
115#10625354—432no—
116C2×D188—21no—
117C2×D188—21no—
118C2×D188—21no—
119C2×D188—21no—
120C2×D188—28no—
121C2×D188—21no—
122C2×D188—44no—
123C2×D188—44no—
124C2×D188C22⋊C414yes—
125C2×D188D814yes—
126C2×D188SD1614yes—
127C2×D188C22×C412yes—
128C2×C4×D47—22no—
129C2×C4×D47—22no—
130C2×C4×D47—22no—
131C2×C4×D47—22no—
132C2×C4×D47—28no—
133C2×C4×D47—24no—
134C2×C4×D47—44no—
135C2×C4×D47—44no—
136C2×C4×D47—42no—
137C2×C4×D47—42no—
138C2×C4×D47—44no—
139C2×C4×D47—44no—
140C2×C4×D47—82no—
141C2×C4×D47C22×C414yes—
142C2×C4×D47C2×D414yes—
143C2×C8×C47—22no—
144C2×C8×C47—22no—
145C2×C8×C47—22no—
146C4⋊C4×C47—24no—
147C4⋊C4×C47—22no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47C22×C414yes—
152C4⋊C4×C47#4112yescomplemented
153C42×C47—42no—
154C42×C47C22×C412yes—
155C42×C47C2×D412yes—
156(C2×D47)⋊C4—24no—
157(C2×D47)⋊C4—28no—
158(C2×D47)⋊C4—44no—
159(C2×D47)⋊C4C22⋊C418yes—
160C4⋊Dic47—21no—
161C4⋊Dic47—22no—
162C4⋊Dic47—22no—
163C4⋊Dic47—42no—
164C4⋊Dic47—81no—
165C4⋊Dic47C22⋊C414yes—
166C4⋊Dic47C22×C412yes—
167Dic47⋊C4—28no—
168Dic47⋊C4—44no—
169C2×C47⋊C8—21no—
170C2×C47⋊C8—21no—
171C2×C47⋊C8—21no—
172C2×C47⋊C8—21no—
173C2×C47⋊C8—21no—
174C2×C47⋊C8#4112yescomplemented
175C8×D47—48no—
176C23×C47C22×D411yescharacteristic
177C22×D47—22no—
178C22×D47—22no—
179C22×D47—22no—
180C22×D47—22no—
181C22×D47—22no—
182C22×D47—41no—
183C22×D47—41no—
184C22×D47—41no—
185C22×D47—41no—
186C22×D47—41no—
187C22×D47—41no—
188C22×D47C2×C22⋊C414yes—
189C22×D47C2×D814yes—
190C22×D47C2×SD1614yes—
191Q8×C47—48no—
192C2×C4×C47—21no—
193C2×C4×C47—21no—
194C2×C4×C47—24no—
195C2×C4×C47—22no—
196C2×C4×C47—41no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47—41no—
200C2×C4×C47—41no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47C2×C22⋊C412yes—
204C2×C4×C47C2×C22⋊C412yes—
205C2×C4×C47C2×D812yes—
206C2×C4×C47C2×SD1612yes—
207C2×C4×C47C23×C411yescharacteristic
208C2×C4×C47C22×D411yescharacteristic
209C2×C4×C47C2×#4111yescharacteristic
210C2×C4×C47C2×#4111yescharacteristic
211C2×C4×C47H(5,𝔽2)11yescharacteristic
212C2×C4×C472−1+411yescharacteristic
213C2×Dic47—22no—
214C2×Dic47—22no—
215C2×Dic47—22no—
216C2×Dic47—41no—
217C2×Dic47—41no—
218C2×Dic47—42no—
219C2×Dic47—42no—
220C2×Dic47—42no—
221C2×Dic47C2×C22⋊C414yes—
222D188—22no—
223D188—22no—
224D188—416no—
225D188—41no—
226D188—41no—
227D188#102116yescomplemented
228C4×D47—24no—
229C4×D47—24no—
230C4×D47—24no—
231C4×D47—24no—
232C4×D47—416no—
233C4×D47—48no—
234C4×D47—88no—
235C8×C47—42no—
236C47⋊C8—22no—
237C47⋊C8—41no—
238#22485—471nomaximal, Sylow
239C22×C47C22×C22⋊C411yescharacteristic
240C22×C47#54611yescharacteristic
241C22×C47#54711yescharacteristic
242C22×C47C22×D811yescharacteristic
243C22×C47C22×SD1611yescharacteristic
244C22×C47#57711yescharacteristic
245C22×C47#57811yescharacteristic
246C2×D47—24no—
247C2×D47—24no—
248C2×D47—24no—
249C2×D47—24no—
250C2×D47—24no—
251C2×D47—24no—
252C2×D47—42no—
253C2×D47—42no—
254C2×D47—42no—
255C2×D47—81no—
256C2×D47—81no—
257C2×D47—81no—
258C2×D47—81no—
259C2×D47C2×#10218yes—
260C4×C47—22no—
261C4×C47—44no—
262C4×C47—42no—
263C4×C47—81no—
264C4×C47C2×#10214yes—
265C4×C47C4×#4111yescharacteristic
266C4×C47#52011yescharacteristic
267C4×C47#54611yescharacteristic
268C4×C47#54711yescharacteristic
269Dic47—24no—
270Dic47—42no—
271Dic47—82no—
272C2×#547—471no—
273C2×C8×D4—471no—
274D4×C4⋊C4—471no—
275#1990—471no—
276#1988D4711yescharacteristic, complemented
277#1893—472no—
278#1874—472no—
279#1823—472no—
280#1802—471no—
281C4×#102—471no—
282C2×#340—472no—
283C2×C47#233911yescharacteristic
284C2×C47C22×#10211yescharacteristic
285C2×C47#289811yescharacteristic
286C2×C47#329411yescharacteristic
287C2×C47#329811yescharacteristic
288C2×C47#333411yescharacteristic
289C2×C47#333511yescharacteristic
290D47—28no—
291D47—82no—
292C22×#41—941no—
293C23×C8—472no—
294#547—478no—
295#547—944no—
296C2×C4⋊Q8C2×D4711yescharacteristic
297C2×#123—942no—
298C2×C22⋊Q8—941no—
299C2×#121—472no—
300C2×C4×D4—471no—
301C2×C4×D4—471no—
302C2×C4×D4—941no—
303C22×C4⋊C4—472no—
304C22×C4⋊C4—472no—
305C8×D4—944no—
306C2×C4⋊C8C2×D4711yescharacteristic
307C2×#102—471no—
308C2×#102—471no—
309C2×#102—472no—
310C2×C22⋊C8—472no—
311C2×C4×C8—471no—
312#389—941no—
313#385—942no—
314#384—471no—
315#380—472no—
316C4×C4⋊C4C2×D4711yescharacteristic
317#340—472no—
318#340—471no—
319#340—472no—
320#340—472no—
321#340—471no—
322#340C2×D4712yes—
323#340C2×D4712yes—
324C47#2248511yescharacteristic, complemented, Sylow
325C2×#41—948no—
326C2×#41—1884no—
327C2×#41—1884no—
328C22×Q8—21no—
329C22×D4—471no—
330C22×D4—471no—
331C22×D4—942no—
332C23×C4—472no—
333C23×C4—942no—
334C23×C4—942no—
335C23×C4—941no—
336C22×C8—471no—
337C22×C8—471no—
338C22×C8—944no—
339C22×C8—944no—
340C22×C8—1882no—
341C22×C8C22×D4712yes—
342C4⋊Q8—22no—
343C4⋊Q8C4×D4714yescomplemented
344#123—9416no—
345C22⋊Q8—948no—
346#121—478no—
347#121—944no—
348C4×D4—474no—
349C4×D4—474no—
350C4×D4—474no—
351C4×D4—942no—
352C4×D4—1884no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—471no—
355C2×C4⋊C4—471no—
356C2×C4⋊C4—478no—
357C2×C4⋊C4—472no—
358C2×C4⋊C4—944no—
359C2×C4⋊C4—942no—
360C2×C4⋊C4—944no—
361C2×C4⋊C4—941no—
362C2×C4⋊C4—942no—
363C2×C4⋊C4—941no—
364C2×C4⋊C4—1882no—
365C2×C4⋊C4C22×D4711yescharacteristic
366C2×C4⋊C4C22×D4712yes—
367C2×C4⋊C4C22×D4711yescharacteristic
368C2×C22⋊C4—472no—
369C2×C22⋊C4—942no—
370C2×C22⋊C4—941no—
371C2×C22⋊C4—942no—
372C2×C42—21no—
373C2×C42C22×D4711yescharacteristic
374C4⋊C8—22no—
375#102—942no—
376#102—942no—
377#102—1882no—
378C22⋊C8—944no—
379C4×C8—942no—
380#95—21no—
381#95—942no—
382C24—472no—
383C24—941no—
384#41—18832no—
385C2×Q8—24no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×D4—474no—
389C2×D4—474no—
390C2×D4—474no—
391C2×D4—472no—
392C2×D4—948no—
393C2×D4—941no—
394C2×D4—941no—
395C2×D4—941no—
396C2×D4—941no—
397C2×D4—941no—
398C2×D4—1884no—
399C2×D4—1884no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—472no—
404C22×C4—474no—
405C22×C4—474no—
406C22×C4—948no—
407C22×C4—942no—
408C22×C4—942no—
409C22×C4—942no—
410C22×C4—942no—
411C22×C4—941no—
412C22×C4—944no—
413C22×C4—942no—
414C22×C4—3762no—
415C22×C4—1884no—
416C22×C4—1884no—
417C22×C4—1884no—
418C22×C4—1884no—
419C22×C4—1882no—
420C22×C4—1882no—
421C22×C4D4×D4711yescharacteristic
422C22×C4D4×D4711yescharacteristic
423C22×C4C23×D4711yescharacteristic
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—472no—
428C2×C8—941no—
429C2×C8—941no—
430C2×C8—941no—
431C2×C8—941no—
432C2×C8—941no—
433C2×C8—1888no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—474no—
440C4⋊C4—472no—
441C4⋊C4—948no—
442C4⋊C4—941no—
443C4⋊C4—942no—
444C4⋊C4—942no—
445C4⋊C4—3761no—
446C4⋊C4—1884no—
447C4⋊C4—1882no—
448C4⋊C4C2×C4×D4714yes—
449C4⋊C4#1062535412yescomplemented
450C22⋊C4—478no—
451C22⋊C4—948no—
452C22⋊C4—944no—
453C22⋊C4—1884no—
454C42—42no—
455C42C2×C4×D4712yes—
456C42D4×D4712yes—
457C23—474no—
458C23—474no—
459C23—474no—
460C23—942no—
461C23—942no—
462C23—942no—
463C23—942no—
464C23—942no—
465C23—1881no—
466C23—1881no—
467C23—1881no—
468C23—1881no—
469C23—1881no—
470C23—1881no—
471C23C2×D4×D4711yescharacteristic
472Q8—48no—
473D4—4716no—
474D4—942no—
475D4—942no—
476D4—18816no—
477D4—1881no—
478D4—1881no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—474no—
491C2×C4—942no—
492C2×C4—942no—
493C2×C4—944no—
494C2×C4—944no—
495C2×C4—944no—
496C2×C4—944no—
497C2×C4—942no—
498C2×C4—3768no—
499C2×C4—18816no—
500C2×C4—1882no—
501C2×C4—1882no—
502C2×C4—1882no—
503C2×C4—1881no—
504C2×C4—1881no—
505C2×C4—1888no—
506C2×C4C22⋊C4×D4712yes—
507C2×C4C22⋊C4×D4712yes—
508C2×C4D8×D4712yes—
509C2×C4SD16×D4712yes—
510C2×C4C22×C4×D4711yescharacteristic
511C2×C4C2×D4×D4711yescharacteristic
512C2×C4#41×D4711yescharacteristic
513C2×C4C2×#1062535411yescharacteristic
514C2×C4#1309407011yescharacteristic
515C2×C4#1309407211yescharacteristic
516C8—42no—
517C8—942no—
518C8—1881no—
519C22—478no—
520C22—944no—
521C22—944no—
522C22—944no—
523C22—944no—
524C22—944no—
525C22—944no—
526C22—3761no—
527C22—3761no—
528C22—3761no—
529C22—3761no—
530C22—1882no—
531C22—1882no—
532C22—1882no—
533C22C2×C22⋊C4×D4711yescharacteristic
534C22#1568976411yescharacteristic
535C22#1568977511yescharacteristic
536C22C2×D8×D4711yescharacteristic
537C22C2×SD16×D4711yescharacteristic
538C22#1568993211yescharacteristic
539C22#1568993911yescharacteristic
540C4—22no—
541C4—42no—
542C4—44no—
543C4—81no—
544C4—944no—
545C4—3762no—
546C4—1882no—
547C4#102×D4714yes—
548C4C4×#1062535411yescharacteristic
549C4#1568960511yescharacteristic
550C4#1568976611yescharacteristic
551C4#1568977611yescharacteristic
552C2—948no—
553C2—3762no—
554C2#2024621711yescharacteristic
555C2C2×#102×D4711yescharacteristic
556C2#2025192011yescharacteristic
557C2#2025519811yescharacteristic
558C2#2025525011yescharacteristic
559C2#2025573411yescharacteristic
560C2#2025574811yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1988C2
socleC23×C47C22×D4
prime radicalC2×#15689776C2
cosocleC22×C4×C47C24
hypercenter#1988D47
hypocenterC47#22485
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
2. C2×#15689188C2semidirect
3. C4⋊C4×D188C2semidirect
4. #20244350C2semidirect
5. #20243815C2semidirect
6. #20243704C2semidirect
7. #20243473C2semidirect
8. #20243389C2semidirect
9. C4×#13093944C2semidirect
10. #340×D47C2semidirect
11. C47×#1988C2semidirect
15. #15689776C4semidirect
62. #13093999C2×C4semidirect
66. C4×D188C2×C4semidirect
78. C4⋊C4×D47C2×C4semidirect
92. C4⋊Q8×C47C2×C4semidirect
152. C4⋊C4×C47#41semidirect
174. C2×C47⋊C8#41semidirect
227. D188#102semidirect
276. #1988D47semidirect
324. C47#22485semidirect
343. C4⋊Q8C4×D47semidirect
449. C4⋊C4#10625354semidirect

Maximal subgroups

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

subgroupindex
C2×#156897762
C2×#156891882
C4⋊C4×D1882
#202443502
#202438152
#202437042
#202434732
#202433892
C4×#130939442
#340×D472
C47×#19882
#2248547