Finite Group Databasebeta

← #26141608

Subgroups — #26141608

Order 12032.

1758 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic47C211yesmaximal, characteristic
2C2×#15689042C211yesmaximal, characteristic
3C2×#15689010C212yesmaximal
4C2×#15688974C211yesmaximal, characteristic
5#20241644C211yesmaximal, characteristic
6#20241642C211yesmaximal, characteristic
7#20241623C212yesmaximal
8#20241622C212yesmaximal
9#20241619C212yesmaximal
10#20241613C211yesmaximal, characteristic
11C47×#1446C211yesmaximal, characteristic
12C2×Q8×Dic47—28no—
13C2×Q8×Dic47—24no—
14C22×C4⋊Dic47C2211yescharacteristic
15C22×C4⋊Dic47C2212yes—
16C22×Dic47⋊C4C2212yes—
17C22×Dic47⋊C4C2211yescharacteristic
18C22×Dic47⋊C4C2211yescharacteristic
19C22×C4×Dic47C2212yes—
20C22×C4×Dic47C2211yescharacteristic
21C22×C4×Dic47C2211yescharacteristic
22C22×C4×Dic47C2211yescharacteristic
23#15689042C418yescomplemented
24#15689010—28no—
25#15688974—24no—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2211yescharacteristic
33C2×C47⋊#95C2211yescharacteristic
34C2×C47⋊#95C2212yes—
35C23×Q8×C47C2211yescharacteristic
36C2×#95×C47C2212yes—
37C2×#95×C47C2211yescharacteristic
38C2×#95×C47C2212yes—
39C2×#95×C47C2211yescharacteristic
40C23×Dic47C2312yes—
41C23×Dic47C2311yescharacteristic
42C23×Dic47C2312yes—
43C23×Dic47C2311yescharacteristic
44C23×Dic47C2311yescharacteristic
45C23×Dic47C2311yescharacteristic
46Q8×Dic47—232no—
47C2×C4⋊Dic47—22no—
48C2×C4⋊Dic47—24no—
49C2×C4⋊Dic47—24no—
50C2×C4⋊Dic47—24no—
51C2×C4⋊Dic47D418yes—
52C2×Dic47⋊C4—22no—
53C2×Dic47⋊C4—24no—
54C2×Dic47⋊C4—22no—
55C2×Dic47⋊C4—24no—
56C2×Dic47⋊C4—22no—
57C2×Dic47⋊C4—22no—
58C2×Dic47⋊C4C2×C418yes—
59C2×Dic47⋊C4C2×C414yes—
60C2×Dic47⋊C4Q814yescomplemented
61C2×C4×Dic47—24no—
62C2×C4×Dic47—22no—
63C2×C4×Dic47—22no—
64C2×C4×Dic47—24no—
65C2×C4×Dic47—24no—
66C2×C4×Dic47—24no—
67C2×C4×Dic47—22no—
68C2×C4×Dic47—22no—
69C2×C4×Dic47—24no—
70C2×C4×Dic47D414yes—
71C47⋊#95—24no—
72C47⋊#95—22no—
73C47⋊#95—24no—
74C47⋊#95—24no—
75C47⋊#95—24no—
76C47⋊#95—24no—
77C47⋊#95—24no—
78C47⋊#95C2×C414yes—
79C47⋊#95C2×C418yes—
80C22×Q8×C47—24no—
81C22×Q8×C47—22no—
82C22×Q8×C47—22no—
83C22×Q8×C47—24no—
84C22×Q8×C47C2×C414yes—
85C23×C4×C47C2312yes—
86C23×C4×C47C2311yescharacteristic
87C23×C4×C47C2311yescharacteristic
88C23×C4×C47C2312yes—
89C23×C4×C47C2311yescharacteristic
90#95×C47—22no—
91#95×C47—24no—
92#95×C47—24no—
93#95×C47—22no—
94C24×C47C2411yescharacteristic
95C2×Q8×C47—28no—
96C2×Q8×C47—216no—
97C2×Q8×C47—28no—
98C2×Q8×C47—28no—
99C2×Q8×C47—216no—
100C22×C4×C47—22no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—22no—
106C22×C4×C47—21no—
107C22×C4×C47—22no—
108C22×C4×C47—21no—
109C22×C4×C47—21no—
110C22×C4×C47—21no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47C22×C414yes—
122C22×C4×C47C22×C412yes—
123C22×C4×C47C2×D414yes—
124C22×C4×C47C2×D412yes—
125C22×C4×C47C2×Q812yes—
126C22×C4×C47#4112yes—
127C22×C4×C47#4112yes—
128C22×C4×C47#4114yes—
129C22×Dic47—22no—
130C22×Dic47—21no—
131C22×Dic47—21no—
132C22×Dic47—21no—
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—21no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—21no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47C22×C414yes—
155C22×Dic47C22×C412yes—
156C22×Dic47C22×C412yes—
157C22×Dic47C2×D414yes—
158C22×Dic47C2×D414yes—
159C22×Dic47C2×D412yes—
160C22×Dic47C2×D412yes—
161C22×Dic47C2×Q812yes—
162C22×Dic47C2×Q812yes—
163C22×Dic47#4112yes—
164C22×Dic47#4112yes—
165C4⋊Dic47—28no—
166C4⋊Dic47—216no—
167Dic47⋊C4—28no—
168Dic47⋊C4—216no—
169Dic47⋊C4—28no—
170C4×Dic47—28no—
171C4×Dic47—216no—
172C4×Dic47—28no—
173C4×Dic47—44no—
174C23×C47C23×C411yescharacteristic
175C23×C47C22×D411yescharacteristic
176C23×C47C22×D412yes—
177C23×C47C22×Q811yescharacteristic
178C23×C47C2×#4111yescharacteristic
179C23×C47C2×#4111yescharacteristic
180C23×C47C2×#4111yescharacteristic
181C23×C47C2×#4112yes—
182C23×C47C2×#4111yescharacteristic
183C23×C472−1+412yes—
184C23×C472−1+412yes—
185Q8×C47—232no—
186C2×C4×C47—24no—
187C2×C4×C47—24no—
188C2×C4×C47—24no—
189C2×C4×C47—24no—
190C2×C4×C47—22no—
191C2×C4×C47—22no—
192C2×C4×C47—24no—
193C2×C4×C47—24no—
194C2×C4×C47—24no—
195C2×C4×C47—24no—
196C2×C4×C47—22no—
197C2×C4×C47—22no—
198C2×C4×C47—24no—
199C2×C4×C47—24no—
200C2×C4×C47—41no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47—41no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47C4×D414yes—
214C2×C4×C47C4×D418yes—
215C2×Dic47—22no—
216C2×Dic47—22no—
217C2×Dic47—22no—
218C2×Dic47—22no—
219C2×Dic47—22no—
220C2×Dic47—22no—
221C2×Dic47—22no—
222C2×Dic47—22no—
223C2×Dic47—22no—
224C2×Dic47—22no—
225C2×Dic47—24no—
226C2×Dic47—24no—
227C2×Dic47—24no—
228C2×Dic47—24no—
229C2×Dic47—24no—
230C2×Dic47—22no—
231C2×Dic47—24no—
232C2×Dic47—24no—
233C2×Dic47—24no—
234C2×Dic47—24no—
235C2×Dic47—24no—
236C2×Dic47—24no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47—42no—
242C2×Dic47—42no—
243C2×Dic47—42no—
244C2×Dic47—42no—
245C2×Dic47—42no—
246C2×Dic47—42no—
247C2×Dic47—42no—
248C2×Dic47C4×Q814yescomplemented
249C2×Dic47#12014yes—
250C2×Dic47C22⋊Q814yes—
251#14174—471nomaximal, Sylow
252C22×C47C2×C4×D411yescharacteristic
253C22×C47C2×C4×D412yes—
254C22×C47C2×C4×Q811yescharacteristic
255C22×C47C4×#4111yescharacteristic
256C22×C47#51912yes—
257C22×C47C2×#12011yescharacteristic
258C22×C47C2×C22⋊Q811yescharacteristic
259C22×C47#52511yescharacteristic
260C22×C47#52512yes—
261C22×C47#52511yescharacteristic
262C22×C47#52911yescharacteristic
263C22×C47#53311yescharacteristic
264C22×C47#53611yescharacteristic
265C22×C47#53611yescharacteristic
266C22×C47#54112yes—
267C22×C47#54712yes—
268C22×C47#54712yes—
269C22×C47#54812yes—
270C22×C47D4×Q812yes—
271C22×C47#55212yes—
272C22×C47#55612yes—
273C22×C47#55612yes—
274C22×C47#55712yes—
275C4×C47—28no—
276C4×C47—24no—
277C4×C47—42no—
278C4×C47—42no—
279C4×C47—82no—
280Dic47—24no—
281Dic47—24no—
282Dic47—24no—
283Dic47—44no—
284Dic47—44no—
285Dic47—42no—
286C22×C4×Q8—471no—
287C2×#395—471no—
288C2×#387—472no—
289C2×C4×C4⋊C4—471no—
290#1446D4711yescharacteristic
291#1442—471no—
292#1441—471no—
293#1436—472no—
294#1436—472no—
295#1436—472no—
296#1434—471no—
297C2×C47C4×#12011yescharacteristic
298C2×C47C4×C22⋊Q811yescharacteristic
299C2×C47#232711yescharacteristic
300C2×C47#236212yes—
301C2×C47#236512yes—
302C2×C47#240611yescharacteristic
303C2×C47#240911yescharacteristic
304C2×C47#246212yes—
305C2×C47#246912yes—
306C2×C47#258111yescharacteristic
307C2×C47#259911yescharacteristic
308C23×Q8C2×D4711yescharacteristic
309C2×C4×Q8—944no—
310C2×C4×Q8—948no—
311C22×C4⋊C4—472no—
312C22×C4⋊C4—471no—
313C22×C4⋊C4—472no—
314C22×C4⋊C4—471no—
315C22×C4⋊C4—471no—
316C22×C42—471no—
317C22×C42—472no—
318C22×C42—471no—
319C22×C42—471no—
320#395—478no—
321#387—948no—
322C4×C4⋊C4—944no—
323C2×#95—471no—
324C2×#95—472no—
325C2×#95—472no—
326C2×#95—472no—
327C2×#95—472no—
328C2×#95—472no—
329C2×#95—471no—
330C2×#95—472no—
331C2×#95—472no—
332C2×#95C2×D4711yescharacteristic
333C2×#95C2×D4712yes—
334C2×#95C2×D4712yes—
335C2×#95C2×D4711yescharacteristic
336C47#1417411yescharacteristic, complemented, Sylow
337C22×Q8—22no—
338C22×Q8—24no—
339C22×Q8—22no—
340C22×Q8—24no—
341C22×Q8C4×D4714yes—
342C23×C4—471no—
343C23×C4—471no—
344C23×C4—471no—
345C23×C4—471no—
346C23×C4—472no—
347C23×C4—472no—
348C23×C4C22×D4712yes—
349C23×C4C22×D4711yescharacteristic
350C23×C4C22×D4711yescharacteristic
351C23×C4C22×D4711yescharacteristic
352C23×C4C22×D4712yes—
353C4×Q8—9432no—
354C2×C4⋊C4—474no—
355C2×C4⋊C4—478no—
356C2×C4⋊C4—478no—
357C2×C4⋊C4—474no—
358C2×C4⋊C4—944no—
359C2×C4⋊C4—944no—
360C2×C4⋊C4—944no—
361C2×C4⋊C4—942no—
362C2×C4⋊C4—942no—
363C2×C4⋊C4—942no—
364C2×C4⋊C4—942no—
365C2×C4⋊C4—944no—
366C2×C4⋊C4—942no—
367C2×C4⋊C4—944no—
368C2×C42—474no—
369C2×C42—944no—
370C2×C42—944no—
371C2×C42—944no—
372C2×C42—942no—
373C2×C42—942no—
374C2×C42—944no—
375C2×C42—942no—
376C2×C42—942no—
377C2×C42—944no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95—478no—
383#95—474no—
384#95—944no—
385#95—944no—
386#95—942no—
387#95—944no—
388#95—944no—
389#95—944no—
390#95—944no—
391C24C23×D4711yescharacteristic
392C2×Q8—28no—
393C2×Q8—216no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—216no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—472no—
419C22×C4—472no—
420C22×C4—474no—
421C22×C4—472no—
422C22×C4—472no—
423C22×C4—472no—
424C22×C4—472no—
425C22×C4—472no—
426C22×C4—474no—
427C22×C4—472no—
428C22×C4—474no—
429C22×C4—942no—
430C22×C4—941no—
431C22×C4—941no—
432C22×C4—941no—
433C22×C4—941no—
434C22×C4—941no—
435C22×C4—941no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—941no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—941no—
447C22×C4—942no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4—942no—
451C22×C4—942no—
452C22×C4—942no—
453C22×C4—942no—
454C22×C4C2×C4×D4714yes—
455C22×C4C2×C4×D4712yes—
456C22×C4D4×D4712yes—
457C22×C4D4×D4714yes—
458C22×C4#1062535212yes—
459C22×C4#1062535214yes—
460C22×C4Q8×D4712yes—
461C22×C4#1062535412yes—
462C4⋊C4—948no—
463C4⋊C4—9416no—
464C4⋊C4—948no—
465C4⋊C4—948no—
466C4⋊C4—9416no—
467C42—948no—
468C42—948no—
469C42—9416no—
470C42—1884no—
471C23C22×C4×D4711yescharacteristic
472C23C2×D4×D4712yes—
473C23C2×D4×D4711yescharacteristic
474C23C2×Q8×D4711yescharacteristic
475C23#41×D4711yescharacteristic
476C23#41×D4711yescharacteristic
477C23C2×#1062535211yescharacteristic
478C23C2×#1062535212yes—
479C23C2×#1062535411yescharacteristic
480C23#1309407112yes—
481C23#1309407212yes—
482Q8—232no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—474no—
511C2×C4—474no—
512C2×C4—474no—
513C2×C4—944no—
514C2×C4—944no—
515C2×C4—944no—
516C2×C4—944no—
517C2×C4—942no—
518C2×C4—942no—
519C2×C4—942no—
520C2×C4—942no—
521C2×C4—942no—
522C2×C4—942no—
523C2×C4—942no—
524C2×C4—942no—
525C2×C4—942no—
526C2×C4—942no—
527C2×C4—944no—
528C2×C4—944no—
529C2×C4—942no—
530C2×C4—944no—
531C2×C4—944no—
532C2×C4—944no—
533C2×C4—944no—
534C2×C4—944no—
535C2×C4—1882no—
536C2×C4—1882no—
537C2×C4—1882no—
538C2×C4—1882no—
539C2×C4—1882no—
540C2×C4—1882no—
541C2×C4—1881no—
542C2×C4—1882no—
543C2×C4—1881no—
544C2×C4—1882no—
545C2×C4—1882no—
546C2×C4(C22⋊D47)⋊C414yes—
547C2×C4(C22⋊D47)⋊C418yes—
548C22C2×(C22⋊D47)⋊C411yescharacteristic
549C22C2×(C22⋊D47)⋊C412yes—
550C22C4×Q8×D4711yescharacteristic
551C22C4×#1062535411yescharacteristic
552C22#1568959912yes—
553C22#120×D4711yescharacteristic
554C22C22⋊Q8×D4711yescharacteristic
555C22#1568962912yes—
556C22#1568962911yescharacteristic
557C22#1568963611yescharacteristic
558C22#1568965211yescharacteristic
559C22#1568967811yescharacteristic
560C22#1568969311yescharacteristic
561C22#1568969811yescharacteristic
562C22#1568973312yes—
563C22#1568977512yes—
564C22#1568977612yes—
565C22#1568978112yes—
566C22#1568978612yes—
567C22#1568981512yes—
568C22#1568984212yes—
569C22#1568984412yes—
570C22#1568984812yes—
571C4—28no—
572C4—24no—
573C4—42no—
574C4—42no—
575C4—82no—
576C4—944no—
577C4—944no—
578C4—944no—
579C4—1882no—
580C4—1884no—
581C4—1884no—
582C2#2024588711yescharacteristic
583C2#2024592611yescharacteristic
584C2#2024610611yescharacteristic
585C2#2024642012yes—
586C2#2024644512yes—
587C2#2024672411yescharacteristic
588C2#2024674611yescharacteristic
589C2#2024736512yes—
590C2#2024746412yes—
591C2#2024874911yescharacteristic
592C2#2024895811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1446C2
socle, prime radical, cosocleC24×C47C24
hypercenter#1446D47
hypocenterC47#14174
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
23. #15689042C4semidirect
60. C2×Dic47⋊C4Q8semidirect
248. C2×Dic47C4×Q8semidirect
336. C47#14174semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic472
C2×#156890422
C2×#156890102
C2×#156889742
#202416442
#202416422
#202416232
#202416222
#202416192
#202416132
C47×#14462
#1417447