Finite Group Databasebeta

← #26138813

Subgroups — #26138813

Order 12032.

1862 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093982C211yesmaximal, characteristic
2C2×#15688980C211yesmaximal, characteristic
3C2×C4×(C2×D47)⋊C4C211yesmaximal, characteristic
4#20241636C214yesmaximal
5#20241618C211yesmaximal, characteristic, complemented
6#20241617C211yesmaximal, characteristic, complemented
7#20241617C211yesmaximal, characteristic, complemented
8#20241614C214yesmaximal, complemented
9C47×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D47C2211yescharacteristic
11C2×#13093982—24no—
12C2×#13093982—24no—
13C2×#13093982C418yescomplemented
14C22×C4⋊Dic47C2211yescharacteristic
15C22×C4⋊Dic47C2211yescharacteristic
16C22×(C2×D47)⋊C4C2212yes—
17C22×(C2×D47)⋊C4C2212yes—
18C22×(C2×D47)⋊C4C2212yes—
19C22×C4×Dic47C2211yescharacteristic
20C22×C4×Dic47C2212yes—
21#15688980—24no—
22C4×(C2×D47)⋊C4—24no—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2214yes—
26C2×C47⋊#95C2214yes—
27C2×C47⋊#95C2214yes—
28C22×C4⋊C4×C47C2211yescharacteristic
29C22×C4⋊C4×C47C2211yescharacteristic
30C22×C42×C47C2211yescharacteristic
31C2×#95×C47C2214yes—
32C24×D47C2311yescharacteristic
33C23×Dic47C2312yes—
34C23×Dic47C2312yes—
35C23×Dic47C2312yes—
36C23×Dic47C2311yescharacteristic
37C22×C4×D47—22no—
38C22×C4×D47—22no—
39C22×C4×D47—22no—
40C22×C4×D47—22no—
41C22×C4×D47C2×C414yes—
42C22×C4×D47D414yes—
43C22×C4×D47D414yes—
44#13093982—232no—
45C2×C4⋊Dic47—22no—
46C2×C4⋊Dic47—22no—
47C2×C4⋊Dic47—22no—
48C2×C4⋊Dic47—22no—
49C2×C4⋊Dic47—22no—
50C2×C4⋊Dic47C2×C414yes—
51C2×(C2×D47)⋊C4—24no—
52C2×(C2×D47)⋊C4—24no—
53C2×(C2×D47)⋊C4—24no—
54C2×(C2×D47)⋊C4—24no—
55C2×(C2×D47)⋊C4—24no—
56C2×(C2×D47)⋊C4—24no—
57C2×(C2×D47)⋊C4—44no—
58C2×(C2×D47)⋊C4C2×C418yes—
59C2×C4×Dic47—24no—
60C2×C4×Dic47—24no—
61C2×C4×Dic47—22no—
62C2×C4×Dic47—22no—
63C2×C4×Dic47—22no—
64C2×C4×Dic47C2×C418yes—
65C47⋊#95—28no—
66C47⋊#95—24no—
67C47⋊#95—24no—
68C47⋊#95—28no—
69C47⋊#95—44no—
70C23×C4×C47C2312yes—
71C23×C4×C47C2312yes—
72C23×C4×C47C2312yes—
73C23×C4×C47C2311yescharacteristic
74C2×C4⋊C4×C47—22no—
75C2×C4⋊C4×C47—22no—
76C2×C4⋊C4×C47—22no—
77C2×C4⋊C4×C47—22no—
78C2×C4⋊C4×C47—22no—
79C2×C4⋊C4×C47C2×C414yes—
80C2×C42×C47—22no—
81C2×C42×C47—22no—
82C2×C42×C47—22no—
83#95×C47—44no—
84C24×C47C2411yescharacteristic
85C23×D47—22no—
86C23×D47—22no—
87C23×D47—21no—
88C23×D47—21no—
89C23×D47—21no—
90C23×D47—21no—
91C23×D47—22no—
92C23×D47—22no—
93C23×D47C22×C412yes—
94C23×D47C2×D412yes—
95C23×D47C2×D412yes—
96C22×C4×C47—22no—
97C22×C4×C47—22no—
98C22×C4×C47—22no—
99C22×C4×C47—22no—
100C22×C4×C47—22no—
101C22×C4×C47—22no—
102C22×C4×C47—24no—
103C22×C4×C47—22no—
104C22×C4×C47—24no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—24no—
110C22×C4×C47—24no—
111C22×C4×C47C22×C414yes—
112C22×C4×C47C22×C412yes—
113C22×C4×C47C2×D412yes—
114C22×C4×C47C2×D412yes—
115C22×C4×C47#4114yes—
116C22×C4×C47#4112yes—
117C22×C4×C47#4112yes—
118C22×C4×C47#4112yes—
119C22×C4×C47#4112yes—
120C22×Dic47—21no—
121C22×Dic47—21no—
122C22×Dic47—21no—
123C22×Dic47—21no—
124C22×Dic47—22no—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—24no—
128C22×Dic47—24no—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—24no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—24no—
139C22×Dic47C22×C414yes—
140C22×Dic47C22×C412yes—
141C22×Dic47C2×D412yes—
142C22×Dic47C2×D412yes—
143C22×Dic47#4114yes—
144C2×C4×D47—28no—
145C2×C4×D47—28no—
146C2×C4×D47—28no—
147C2×C4×D47—28no—
148C2×C4×D47—28no—
149C2×C4×D47—28no—
150C2×C4×D47C22⋊C4116yes—
151C4⋊C4×C47—28no—
152C4⋊C4×C47—44no—
153C42×C47—44no—
154(C2×D47)⋊C4—216no—
155(C2×D47)⋊C4—216no—
156(C2×D47)⋊C4—48no—
157C4⋊Dic47—28no—
158C4⋊Dic47—44no—
159C4×Dic47—216no—
160C4×Dic47—44no—
161C23×C47C23×C411yescharacteristic
162C23×C47C22×D411yescharacteristic
163C23×C47C22×D411yescharacteristic
164C23×C47C2×#4112yes—
165C23×C47C2×#4111yescharacteristic
166C23×C47C2×#4111yescharacteristic
167C23×C47C2×#4111yescharacteristic
168C23×C47C2×#4111yescharacteristic
169C23×C47C2×#4112yes—
170C23×C47H(5,𝔽2)12yes—
171C23×C472−1+412yes—
172C22×D47—22no—
173C22×D47—22no—
174C22×D47—24no—
175C22×D47—24no—
176C22×D47—22no—
177C22×D47—22no—
178C22×D47—22no—
179C22×D47—22no—
180C22×D47—24no—
181C22×D47—22no—
182C22×D47—22no—
183C22×D47—22no—
184C22×D47—22no—
185C22×D47—24no—
186C22×D47—42no—
187C22×D47—42no—
188C22×D47—42no—
189C22×D47—42no—
190C22×D47—42no—
191C22×D47—42no—
192C22×D47—42no—
193C22×D47—42no—
194C22×D47C2×C22⋊C414yes—
195C2×C4×C47—24no—
196C2×C4×C47—28no—
197C2×C4×C47—24no—
198C2×C4×C47—24no—
199C2×C4×C47—24no—
200C2×C4×C47—42no—
201C2×C4×C47—42no—
202C2×C4×C47—42no—
203C2×C4×C47—44no—
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—44no—
211C2×C4×C47—44no—
212C2×C4×C47C2×C22⋊C414yes—
213C2×C4×C47#11714yes—
214C2×C4×C47#11714yes—
215C2×C4×C47#12314yes—
216C2×C4×C47#12314yes—
217C2×C4×C47#12314yes—
218C2×C4×C47#12314yes—
219C2×Dic47—22no—
220C2×Dic47—22no—
221C2×Dic47—22no—
222C2×Dic47—22no—
223C2×Dic47—22no—
224C2×Dic47—22no—
225C2×Dic47—24no—
226C2×Dic47—28no—
227C2×Dic47—24no—
228C2×Dic47—24no—
229C2×Dic47—24no—
230C2×Dic47—42no—
231C2×Dic47—44no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—42no—
239C2×Dic47—42no—
240C2×Dic47—44no—
241C2×Dic47—44no—
242C2×Dic47C2×C22⋊C414yes—
243C4×D47—232no—
244#13939—471nomaximal, Sylow
245C22×C47C22×C22⋊C411yescharacteristic
246C22×C47C2×#11711yescharacteristic
247C22×C47C2×#11711yescharacteristic
248C22×C47C4×#4112yes—
249C22×C47#52012yes—
250C22×C47C2×#12311yescharacteristic
251C22×C47C2×#12311yescharacteristic
252C22×C47C2×#12311yescharacteristic
253C22×C47C2×#12311yescharacteristic
254C22×C47#52512yes—
255C22×C47#53211yescharacteristic
256C22×C47#53211yescharacteristic
257C22×C47#53511yescharacteristic
258C22×C47#53611yescharacteristic
259C22×C47#53712yes—
260C22×C47#55212yes—
261C22×C47#55212yes—
262C22×C47#55312yes—
263C22×C47#55312yes—
264C22×C47#55512yes—
265C22×C47#55512yes—
266C22×C47#55612yes—
267C22×C47#55612yes—
268C2×D47—24no—
269C2×D47—24no—
270C2×D47—24no—
271C2×D47—44no—
272C2×D47—44no—
273C2×D47—44no—
274C2×D47—44no—
275C2×D47—42no—
276C2×D47—42no—
277C2×D47—42no—
278C2×D47—42no—
279C4×C47—44no—
280C4×C47—44no—
281C4×C47—82no—
282C4×C47#38118yes—
283Dic47—24no—
284Dic47—44no—
285Dic47—44no—
286Dic47—82no—
287C22×#117—471no—
288C2×#380—471no—
289C2×C4×C22⋊C4—471no—
290#1440—474no—
291C4⋊#95—471no—
292C4⋊#95—471no—
293C4⋊#95—471no—
294C4⋊#95D4711yescharacteristic, complemented
295#1434—474no—
296C2×C47C2×#38111yescharacteristic
297C2×C47#229711yescharacteristic
298C2×C47#231111yescharacteristic
299C2×C47#234612yes—
300C2×C47#234712yes—
301C2×C47#248512yes—
302C2×C47#248512yes—
303C2×C47#249911yescharacteristic
304C2×C47#249911yescharacteristic
305C2×C47#250011yescharacteristic
306C2×C47#250011yescharacteristic
307D47—44no—
308C24×C4—471no—
309C2×#117—478no—
310C2×#117—944no—
311C2×#117—944no—
312C22×C4⋊C4—471no—
313C22×C4⋊C4—471no—
314C22×C4⋊C4C2×D4711yescharacteristic
315C22×C4⋊C4C2×D4711yescharacteristic
316C22×C22⋊C4—472no—
317C22×C22⋊C4—472no—
318C22×C22⋊C4—472no—
319C22×C42—472no—
320C22×C42—471no—
321C22×C42C2×D4711yescharacteristic
322#380—944no—
323C4×C22⋊C4—944no—
324C2×#95—472no—
325C2×#95—472no—
326C2×#95—474no—
327C2×#95—474no—
328C2×#95—474no—
329C2×#95C2×D4714yes—
330C47#1393911yescharacteristic, complemented, Sylow
331C25—471no—
332C23×C4—471no—
333C23×C4—474no—
334C23×C4—474no—
335C23×C4—474no—
336C23×C4—472no—
337C23×C4—472no—
338C23×C4—472no—
339C23×C4—942no—
340C23×C4—942no—
341C23×C4—942no—
342C23×C4—942no—
343C23×C4C22×D4712yes—
344C23×C4C22×D4712yes—
345C23×C4C22×D4712yes—
346C23×C4C22×D4711yescharacteristic
347#117—9432no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—474no—
354C2×C4⋊C4—942no—
355C2×C4⋊C4—942no—
356C2×C4⋊C4—942no—
357C2×C4⋊C4—942no—
358C2×C4⋊C4—942no—
359C2×C4⋊C4C4×D4714yes—
360C2×C22⋊C4—478no—
361C2×C22⋊C4—944no—
362C2×C22⋊C4—944no—
363C2×C22⋊C4—944no—
364C2×C22⋊C4—944no—
365C2×C22⋊C4—944no—
366C2×C22⋊C4—944no—
367C2×C22⋊C4—1884no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—478no—
372C2×C42—944no—
373C2×C42—944no—
374C2×C42—942no—
375C2×C42—942no—
376C2×C42—942no—
377#95—44no—
378#95—948no—
379#95—944no—
380#95—944no—
381#95—948no—
382#95—1884no—
383C24—472no—
384C24—472no—
385C24—472no—
386C24—942no—
387C24—941no—
388C24—941no—
389C24—941no—
390C24—942no—
391C24—941no—
392C24—942no—
393C24—942no—
394C24C23×D4711yescharacteristic
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—472no—
411C22×C4—472no—
412C22×C4—472no—
413C22×C4—4716no—
414C22×C4—474no—
415C22×C4—474no—
416C22×C4—941no—
417C22×C4—941no—
418C22×C4—941no—
419C22×C4—941no—
420C22×C4—948no—
421C22×C4—948no—
422C22×C4—948no—
423C22×C4—948no—
424C22×C4—948no—
425C22×C4—948no—
426C22×C4—942no—
427C22×C4—942no—
428C22×C4—942no—
429C22×C4—944no—
430C22×C4—942no—
431C22×C4—944no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—944no—
439C22×C4—942no—
440C22×C4—944no—
441C22×C4C2×C4×D4714yes—
442C22×C4C2×C4×D4712yes—
443C22×C4#1062535012yes—
444C22×C4#1062535012yes—
445C22×C4D4×D4712yes—
446C22×C4D4×D4712yes—
447C22×C4#1062535212yes—
448C22×C4#1062535414yes—
449C22×C4#1062535412yes—
450C4⋊C4—28no—
451C4⋊C4—44no—
452C4⋊C4—948no—
453C4⋊C4—1884no—
454C22⋊C4—9416no—
455C22⋊C4—9416no—
456C22⋊C4—1888no—
457C42—44no—
458C42—9416no—
459C42—1884no—
460C23—474no—
461C23—942no—
462C23—942no—
463C23—944no—
464C23—944no—
465C23—942no—
466C23—942no—
467C23—942no—
468C23—942no—
469C23—944no—
470C23—942no—
471C23—942no—
472C23—942no—
473C23—942no—
474C23—944no—
475C23—1882no—
476C23—1882no—
477C23—1882no—
478C23—1882no—
479C23—1882no—
480C23—1882no—
481C23—1882no—
482C23—1882no—
483C23C22×C4×D4711yescharacteristic
484C23C2×D4×D4711yescharacteristic
485C23C2×D4×D4711yescharacteristic
486C23#41×D4712yes—
487C23C2×#1062535211yescharacteristic
488C23C2×#1062535011yescharacteristic
489C23C2×#1062535011yescharacteristic
490C23C2×#1062535411yescharacteristic
491C23C2×#1062535412yes—
492C23#1309406912yes—
493C23#1309407112yes—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—474no—
512C2×C4—942no—
513C2×C4—942no—
514C2×C4—942no—
515C2×C4—942no—
516C2×C4—942no—
517C2×C4—942no—
518C2×C4—9432no—
519C2×C4—944no—
520C2×C4—944no—
521C2×C4—944no—
522C2×C4—948no—
523C2×C4—944no—
524C2×C4—1884no—
525C2×C4—1884no—
526C2×C4—1882no—
527C2×C4—1882no—
528C2×C4—1882no—
529C2×C4—1882no—
530C2×C4—1882no—
531C2×C4—1882no—
532C2×C4—1884no—
533C2×C4—1882no—
534C2×C4—1882no—
535C2×C4—1882no—
536C2×C4C22⋊C4×D4714yes—
537C2×C4#1309398114yes—
538C2×C4#1309398214yes—
539C2×C4#1309400614yes—
540C2×C4#1309400614yes—
541C2×C4#1309400914yes—
542C2×C4#1309400914yes—
543C22—944no—
544C22—944no—
545C22—944no—
546C22—1884no—
547C22—1884no—
548C22—1882no—
549C22—1882no—
550C22—1882no—
551C22—1882no—
552C22—1884no—
553C22—1884no—
554C22C2×C22⋊C4×D4711yescharacteristic
555C22C2×#1309398111yescharacteristic
556C22C2×#1309398211yescharacteristic
557C22C4×#1062535412yes—
558C22#1568960212yes—
559C22C2×#1309400611yescharacteristic
560C22C2×#1309400611yescharacteristic
561C22C2×#1309400911yescharacteristic
562C22C2×#1309400911yescharacteristic
563C22#1568963612yes—
564C22#1568967211yescharacteristic
565C22#1568967411yescharacteristic
566C22#1568968611yescharacteristic
567C22#1568969311yescharacteristic
568C22#1568970212yes—
569C22#1568981112yes—
570C22#1568981512yes—
571C22#1568982112yes—
572C22#1568982312yes—
573C22#1568983712yes—
574C22#1568983912yes—
575C22#1568984112yes—
576C22#1568984612yes—
577C4—44no—
578C4—44no—
579C4—82no—
580C4—944no—
581C4—3762no—
582C4—1884no—
583C4—1884no—
584C4#1568898518yes—
585C2—1884no—
586C2C2×#1568898511yescharacteristic
587C2#2024587711yescharacteristic
588C2#2024597811yescharacteristic
589C2#2024627212yes—
590C2#2024627712yes—
591C2#2024768712yes—
592C2#2024768812yes—
593C2#2024784311yescharacteristic
594C2#2024784411yescharacteristic
595C2#2024784711yescharacteristic
596C2#2024784911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×C4⋊#95C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenterC4⋊#95D47
hypocenterC47#13939
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
5. #20241618C2semidirect
6. #20241617C2semidirect
7. #20241617C2semidirect
8. #20241614C2semidirect
9. C47×C4⋊#95C2semidirect
13. C2×#13093982C4semidirect
294. C4⋊#95D47semidirect
330. C47#13939semidirect

Maximal subgroups

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

subgroupindex
C22×#130939822
C2×#156889802
C2×C4×(C2×D47)⋊C42
#202416362
#202416182
#202416172
#202416142
C47×C4⋊#952
#1393947