Finite Group Databasebeta

← #26231737

Subgroups — #26231737

Order 12032.

1490 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689778C211yesmaximal, characteristic
2C2×#15689120C212yesmaximal, complemented
3#20246337C211yesmaximal, characteristic, complemented
4#20244310C211yesmaximal, characteristic, complemented
5#20244190C211yesmaximal, characteristic, complemented
6#20243811C212yesmaximal, complemented
7#20243698C212yesmaximal, complemented
8#20243389C211yesmaximal, characteristic, complemented
9#20243329C212yesmaximal, complemented
10C4×#13093946C211yesmaximal, characteristic, complemented
11C2×C8×D4×C47C211yesmaximal, characteristic, complemented
12C22×#10625350C2212yes—
13#15689778—24no—
14#15689778C418yescomplemented
15C2×#13094024C2211yescharacteristic
16C2×#13094006—22no—
17C2×#13094003C2212yes—
18C2×#13093995—21no—
19C2×C4×D188C2211yescharacteristic
20C2×#13093983C2212yes—
21C2×C4⋊C4×D47—21no—
22#15689120—28no—
23C2×#13093949C2212yes—
24C2×#13093949C2212yes—
25C2×#13093949C2212yes—
26C2×#13093949C2212yes—
27C2×#13093946C2211yescharacteristic
28C2×#13093946C2212yes—
29C2×#13093946C2211yescharacteristic
30#15689026—21no—
31#15689016—22no—
32#15688993C2211yescharacteristic
33#15688983C2212yes—
34C4⋊C4×Dic47—21no—
35C4×C4⋊Dic47C2211yescharacteristic
36#15688886C2211yescharacteristic
37#15688886C2211yescharacteristic
38#15688886C2212yes—
39C23×C8×C47C2212yes—
40C2×C4×D4×C47C2211yescharacteristic
41C8×D4×C47—24no—
42C2×C4⋊C8×C47C2211yescharacteristic
43C2×C22⋊C8×C47C2212yes—
44C2×C4×C8×C47C2211yescharacteristic
45C2×#10625350—24no—
46C2×#10625350—24no—
47C2×#10625350—24no—
48C2×#10625350—42no—
49C2×#10625350—42no—
50C2×#10625350C2×C418yes—
51C22×Dic94C2312yes—
52C22×D188C2311yescharacteristic
53C22×C22⋊D47—22no—
54C22×C4×D47—21no—
55C22×C4×D47—22no—
56#13094024—22no—
57#13094024C2×C414yes—
58#13094006—216no—
59#13094003—24no—
60#13094003C2×C418yes—
61#13093995—28no—
62C4×D188—22no—
63C4×D188C2×C414yes—
64#13093983—24no—
65#13093983—42no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2312yes—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×Dic47⋊C4—22no—
75C4⋊C4×D47—28no—
76C2×C22⋊Dic47—22no—
77C2×C22⋊Dic47—22no—
78C2×C22⋊Dic47—22no—
79C2×(C2×D47)⋊C4—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—22no—
82#13093949—24no—
83#13093949—24no—
84#13093949—42no—
85#13093949—42no—
86#13093946—24no—
87#13093946—22no—
88#13093946D414yescomplemented
89C47⋊#95—21no—
90C47⋊#95—22no—
91C22×D4×C47D411yescharacteristic
92C23×C4×C47C2312yes—
93C22×C8×C47—24no—
94C22×C8×C47—24no—
95C22×C8×C47—42no—
96C22×C8×C47C2312yes—
97C22×C8×C47C2311yescharacteristic
98C22×C8×C47C2311yescharacteristic
99C4×D4×C47C2×C414yes—
100C4×D4×C47D414yes—
101C2×C4⋊C4×C47D411yescharacteristic
102C2×C22⋊C4×C47D412yes—
103C2×C42×C47C2311yescharacteristic
104C4⋊C8×C47—22no—
105C22⋊C8×C47—24no—
106C4×C8×C47—22no—
107C24×C47C2×D412yes—
108C23×D47—21no—
109C2×D4×C47—22no—
110C2×D4×C47—22no—
111C2×D4×C47C22⋊C414yes—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47C22×C414yes—
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2×D412yes—
119C22×C4×C47C2×D414yes—
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2411yescharacteristic
122C2×C22⋊D47—28no—
123C2×C22⋊D47—44no—
124C2×C22⋊D47—44no—
125C22×Dic47—21no—
126C22×Dic47—21no—
127C22×Dic47—22no—
128C22×Dic47—21no—
129C22×Dic47—22no—
130#10625350—216no—
131#10625350—48no—
132#10625350—48no—
133#10625350—48no—
134C2×D188—21no—
135C2×D188—21no—
136C2×D188—21no—
137C2×D188C22×C412yes—
138C2×D188C2×D412yes—
139C2×D188#4112yes—
140C2×C4×D47—24no—
141C2×C4×D47—28no—
142C2×C4×D47—44no—
143C2×C4×D47—44no—
144C2×C4×D47—42no—
145C2×C4×D47—42no—
146C2×Dic94—22no—
147C2×Dic94—22no—
148C2×Dic94—22no—
149C2×Dic94—22no—
150C2×Dic94—22no—
151C2×Dic94C22×C414yes—
152C2×C8×C47—21no—
153C2×C8×C47—21no—
154C2×C8×C47—22no—
155C2×C8×C47—22no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47—22no—
159C2×C8×C47—21no—
160C2×C8×C47—48no—
161C2×C8×C47C2×D412yes—
162C4⋊C4×C47C22⋊C414yes—
163C22⋊C4×C47C22⋊C418yes—
164C42×C47C22×C412yes—
165C42×C47C2×D412yes—
166C22⋊Dic47—28no—
167C22⋊Dic47—44no—
168C22⋊Dic47—82no—
169(C2×D47)⋊C4—24no—
170C4⋊Dic47—24no—
171C4⋊Dic47—24no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—22no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—22no—
178C4⋊Dic47—42no—
179C4⋊Dic47—81no—
180C4⋊Dic47C22×C412yes—
181C4⋊Dic47C2×D412yes—
182Dic47⋊C4—28no—
183C4×Dic47—44no—
184C4×Dic47—42no—
185C23×C47—22no—
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47C2×C22⋊C414yes—
192C23×C47C22×D411yescharacteristic
193C22×D47—22no—
194C22×D47—22no—
195C22×D47—22no—
196C22×D47—41no—
197C22×D47—41no—
198C22×D47—41no—
199C22×D47—41no—
200D4×C47—28no—
201C2×C4×C47—22no—
202C2×C4×C47—22no—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—24no—
206C2×C4×C47—24no—
207C2×C4×C47—24no—
208C2×C4×C47—24no—
209C2×C4×C47C2×C22⋊C414yes—
210C2×C4×C47C2×C22⋊C412yes—
211C2×C4×C47C2×C22⋊C412yes—
212C2×C4×C47#13512yes—
213C2×C4×C47#13512yes—
214C2×C4×C47C23×C411yescharacteristic
215C2×C4×C47C22×D411yescharacteristic
216C2×C4×C47C22×D411yescharacteristic
217C2×C4×C47C2×#4111yescharacteristic
218C2×C4×C47H(5,𝔽2)11yescharacteristic
219C2×C4×C47H(5,𝔽2)11yescharacteristic
220C22⋊D47—416no—
221C2×Dic47—22no—
222C2×Dic47—24no—
223C2×Dic47—22no—
224C2×Dic47—42no—
225C2×Dic47—42no—
226C2×Dic47—41no—
227C2×Dic47—41no—
228C2×Dic47—41no—
229C2×Dic47—41no—
230C2×Dic47—42no—
231C2×Dic47—41no—
232C2×Dic47—41no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—41no—
236D188—22no—
237D188—22no—
238D188—22no—
239D188C4×D414yes—
240C4×D47—416no—
241C4×D47—48no—
242Dic94—24no—
243Dic94—24no—
244Dic94—42no—
245Dic94—42no—
246C8×C47—22no—
247C8×C47—42no—
248C8×C47—41no—
249#22417—471nomaximal, Sylow
250C22×C47—24no—
251C22×C47—24no—
252C22×C47—24no—
253C22×C47—42no—
254C22×C47—42no—
255C22×C47—42no—
256C22×C47—42no—
257C22×C47C22×C22⋊C411yescharacteristic
258C22×C47D4211yescharacteristic
259C22×C47#54611yescharacteristic
260C22×C47C2×#13511yescharacteristic
261C22×C47C2×#13511yescharacteristic
262C22×C47#57711yescharacteristic
263C22×C47#57811yescharacteristic
264C2×D47—24no—
265C2×D47—42no—
266C2×D47—42no—
267C2×D47—42no—
268C2×D47—42no—
269C2×D47—42no—
270C2×D47—42no—
271C4×C47—24no—
272C4×C47—22no—
273C4×C47#41614yescomplemented
274C4×C47C2×C4×D411yescharacteristic
275C4×C47#51811yescharacteristic
276C4×C47D4211yescharacteristic
277C4×C47#54611yescharacteristic
278Dic47—44no—
279Dic47—42no—
280Dic47—42no—
281Dic47—82no—
282Dic47—81no—
283C2×#547—471no—
284C2×C8×D4D4711yescharacteristic, complemented
285C2×#416—472no—
286#2354—471no—
287#1981—471no—
288#1956—471no—
289#1892—472no—
290#1873—472no—
291#1802—471no—
292#1788—472no—
293C4×#102—471no—
294C2×C47—44no—
295C2×C47D4×C22⋊C411yescharacteristic
296C2×C47C2×#41611yescharacteristic
297C2×C47#289811yescharacteristic
298C2×C47#328311yescharacteristic
299C2×C47#328411yescharacteristic
300C2×C47#330211yescharacteristic
301C2×C47#330311yescharacteristic
302D47—44no—
303C22×#41—472no—
304C23×C8C2×D4712yes—
305#547—478no—
306#547—944no—
307C2×C4⋊Q8—471no—
308C2×#123—942no—
309C2×C22⋊Q8—472no—
310C2×#121—941no—
311C2×C4×D4—471no—
312C2×C4×D4C2×D4711yescharacteristic
313C2×#117—472no—
314C22×C4⋊C4—941no—
315C8×D4—24no—
316C2×C4⋊C8C2×D4711yescharacteristic
317#416—948no—
318C2×#103—472no—
319C2×#103—472no—
320C2×#103—472no—
321C2×#103—472no—
322C2×#102—471no—
323C2×#102—472no—
324C2×#102—471no—
325C2×C22⋊C8C2×D4712yes—
326C2×C4×C8C2×D4711yescharacteristic
327#390—941no—
328#388—942no—
329#384—471no—
330#380—472no—
331C4×C4⋊C4—471no—
332C4×C4⋊C4—941no—
333#340—472no—
334#340—471no—
335#340—471no—
336C47#2241711yescharacteristic, complemented, Sylow
337C2×#41—478no—
338C2×#41—944no—
339C2×#41—944no—
340C2×#41—944no—
341C2×#41—1882no—
342C2×#41—1882no—
343C22×Q8—472no—
344C22×D4—471no—
345C22×D4—942no—
346C22×D4D18811yescharacteristic
347C23×C4—941no—
348C23×C4—942no—
349C23×C4C22×D4712yes—
350C22×C8—24no—
351C22×C8—24no—
352C22×C8—42no—
353C22×C8C22×D4712yes—
354C22×C8C22×D4711yescharacteristic
355C22×C8C22×D4711yescharacteristic
356C4⋊Q8—474no—
357C4⋊Q8—942no—
358#123—9416no—
359C22⋊Q8—478no—
360C22⋊Q8—944no—
361#121—948no—
362C4×D4—474no—
363C4×D4—942no—
364C4×D4C4×D4714yes—
365C4×D4C22⋊D4714yes—
366#117—944no—
367#117—1882no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—472no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4—942no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—948no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4—941no—
378C2×C4⋊C4D18811yescharacteristic
379C2×C22⋊C4—942no—
380C2×C22⋊C4—941no—
381C2×C22⋊C4—942no—
382C2×C22⋊C4—942no—
383C2×C22⋊C4D18812yes—
384C2×C42—942no—
385C2×C42—941no—
386C2×C42C22×D4711yescharacteristic
387C4⋊C8—22no—
388#103—944no—
389#103—944no—
390#103—1882no—
391#103—1882no—
392#102—474no—
393#102—942no—
394#102—944no—
395C22⋊C8—24no—
396C4×C8—22no—
397#95—941no—
398#95—942no—
399C24—941no—
400C24C2×D18812yes—
401#41—9416no—
402#41—1888no—
403#41—1888no—
404#41—1888no—
405C2×Q8—474no—
406C2×Q8—942no—
407C2×Q8—942no—
408C2×Q8—942no—
409C2×Q8—942no—
410C2×Q8—942no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—472no—
414C2×D4—472no—
415C2×D4—472no—
416C2×D4—948no—
417C2×D4—941no—
418C2×D4—941no—
419C2×D4—941no—
420C2×D4—1884no—
421C2×D4—1884no—
422C2×D4(C2×D47)⋊C414yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—941no—
428C22×C4—941no—
429C22×C4—942no—
430C22×C4—944no—
431C22×C4—948no—
432C22×C4—941no—
433C22×C4—942no—
434C22×C4—1884no—
435C22×C4—1884no—
436C22×C4—1882no—
437C22×C4—1882no—
438C22×C4C2×C4×D4714yes—
439C22×C4C2×D18812yes—
440C22×C4C2×D18811yescharacteristic
441C22×C4C2×D18811yescharacteristic
442C22×C4C2×C22⋊D4714yes—
443C22×C4C23×D4711yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—22no—
447C2×C8—21no—
448C2×C8—22no—
449C2×C8—21no—
450C2×C8—22no—
451C2×C8—21no—
452C2×C8—48no—
453C2×C8D4×D4712yes—
454C4⋊C4—472no—
455C4⋊C4—472no—
456C4⋊C4—948no—
457C4⋊C4—944no—
458C4⋊C4—944no—
459C4⋊C4—941no—
460C4⋊C4—941no—
461C4⋊C4—942no—
462C4⋊C4—941no—
463C4⋊C4—941no—
464C4⋊C4—942no—
465C4⋊C4—3761no—
466C4⋊C4—1882no—
467C4⋊C4(C2×D47)⋊C414yes—
468C22⋊C4—948no—
469C22⋊C4—944no—
470C22⋊C4—3762no—
471C22⋊C4—1884no—
472C22⋊C4(C2×D47)⋊C418yes—
473C42—1884no—
474C42—1882no—
475C42C2×C4×D4712yes—
476C42C2×C22⋊D4712yes—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—942no—
484C23—942no—
485C23—942no—
486C23—1881no—
487C23—1881no—
488C23—1881no—
489C23—1881no—
490C23C2×(C2×D47)⋊C414yes—
491C23C22×D18811yescharacteristic
492Q8—944no—
493Q8—944no—
494Q8—1882no—
495Q8—1882no—
496D4—28no—
497D4—474no—
498D4—942no—
499D4—942no—
500D4—942no—
501D4—18816no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—942no—
511C2×C4—942no—
512C2×C4—944no—
513C2×C4—18816no—
514C2×C4—1882no—
515C2×C4—1881no—
516C2×C4—1881no—
517C2×C4—1882no—
518C2×C4—1881no—
519C2×C4—1882no—
520C2×C4—1882no—
521C2×C4—1882no—
522C2×C4—1881no—
523C2×C4—1881no—
524C2×C4—1881no—
525C2×C4—1881no—
526C2×C4—1888no—
527C2×C4C2×(C2×D47)⋊C412yes—
528C2×C4C2×(C2×D47)⋊C414yes—
529C2×C4C2×(C2×D47)⋊C412yes—
530C2×C4#1309404612yes—
531C2×C4#1309404612yes—
532C2×C4C22×C4×D4711yescharacteristic
533C2×C4C2×D4×D4711yescharacteristic
534C2×C4C22×C22⋊D4711yescharacteristic
535C2×C4#41×D4711yescharacteristic
536C2×C4#1309406911yescharacteristic
537C2×C4#1309407011yescharacteristic
538C8—22no—
539C8—42no—
540C8—41no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—944no—
549C22—1882no—
550C22—1882no—
551C22—1882no—
552C22—1882no—
553C22—1882no—
554C22—1882no—
555C22C22×(C2×D47)⋊C411yescharacteristic
556C22D4×D18811yescharacteristic
557C22#1568976811yescharacteristic
558C22C2×#1309404611yescharacteristic
559C22C2×#1309404611yescharacteristic
560C22#1568993511yescharacteristic
561C22#1568994011yescharacteristic
562C4—22no—
563C4—24no—
564C4—3761no—
565C4—3762no—
566C4—1882no—
567C4—1884no—
568C4—1882no—
569C4#1568912014yescomplemented
570C4C4×D4×D4711yescharacteristic
571C4#1568959511yescharacteristic
572C4D4×C22⋊D4711yescharacteristic
573C4#1568976111yescharacteristic
574C2—44no—
575C2—1884no—
576C2D4×(C2×D47)⋊C411yescharacteristic
577C2C2×#1568912011yescharacteristic
578C2#2025192211yescharacteristic
579C2#2025511011yescharacteristic
580C2#2025512011yescharacteristic
581C2#2025530611yescharacteristic
582C2#2025531711yescharacteristic
subgroupG/H
centerC23C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C8×D4×C47C2
socleC23×C47C22×D4
prime radicalC2×#15689778C2
cosocleC22×C4×C47C24
hypercenterC2×C8×D4D47
hypocenterC47#22417
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
2. C2×#15689120C2semidirect
3. #20246337C2semidirect
4. #20244310C2semidirect
5. #20244190C2semidirect
6. #20243811C2semidirect
7. #20243698C2semidirect
8. #20243389C2semidirect
9. #20243329C2semidirect
10. C4×#13093946C2semidirect
11. C2×C8×D4×C47C2semidirect
14. #15689778C4semidirect
88. #13093946D4semidirect
273. C4×C47#416semidirect
284. C2×C8×D4D47semidirect
336. C47#22417semidirect
569. C4#15689120semidirect

Maximal subgroups

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

subgroupindex
C2×#156897782
C2×#156891202
#202463372
#202443102
#202441902
#202438112
#202436982
#202433892
#202433292
C4×#130939462
C2×C8×D4×C472
#2241747