Finite Group Databasebeta

← #26143462

Subgroups — #26143462

Order 12032.

1926 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093981C211yesmaximal, characteristic
2C2×#15688992C211yesmaximal, characteristic, complemented
3C2×#15688988C211yesmaximal, characteristic, complemented
4C2×#15688980C211yesmaximal, characteristic
5C2×#15688980C211yesmaximal, characteristic
6#20241639C214yesmaximal, complemented
7#20241632C214yesmaximal
8#20241618C211yesmaximal, characteristic, complemented
9C47×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D47C2211yescharacteristic
11C2×#13093981—24no—
12C2×#13093981—24no—
13C2×#13093981C418yescomplemented
14C22×C4⋊Dic47C2211yescharacteristic
15C22×C4⋊Dic47C2211yescharacteristic
16C22×Dic47⋊C4C2212yes—
17C22×Dic47⋊C4C2214yes—
18C22×(C2×D47)⋊C4C2212yes—
19C22×(C2×D47)⋊C4C2212yes—
20C22×(C2×D47)⋊C4C2212yes—
21C22×C4×Dic47C2211yescharacteristic
22#15688992—24no—
23#15688988—24no—
24#15688980—24no—
25#15688980—24no—
26C2×C47⋊#95C2214yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2214yes—
30C22×C4⋊C4×C47C2211yescharacteristic
31C22×C4⋊C4×C47C2211yescharacteristic
32C22×C42×C47C2211yescharacteristic
33C2×#95×C47C2214yes—
34C24×D47C2311yescharacteristic
35C23×Dic47C2312yes—
36C23×Dic47C2312yes—
37C23×Dic47C2312yes—
38C23×Dic47C2311yescharacteristic
39C22×C4×D47—22no—
40C22×C4×D47—22no—
41C22×C4×D47—22no—
42C22×C4×D47—22no—
43C22×C4×D47C2×C414yes—
44C22×C4×D47D414yes—
45C22×C4×D47Q814yes—
46#13093981—232no—
47C2×C4⋊Dic47—22no—
48C2×C4⋊Dic47—22no—
49C2×C4⋊Dic47—22no—
50C2×C4⋊Dic47—22no—
51C2×C4⋊Dic47—22no—
52C2×C4⋊Dic47—22no—
53C2×Dic47⋊C4—24no—
54C2×Dic47⋊C4—24no—
55C2×Dic47⋊C4—28no—
56C2×Dic47⋊C4—44no—
57C2×Dic47⋊C4—44no—
58C2×Dic47⋊C4C2×C418yes—
59C2×(C2×D47)⋊C4—24no—
60C2×(C2×D47)⋊C4—24no—
61C2×(C2×D47)⋊C4—24no—
62C2×(C2×D47)⋊C4—24no—
63C2×(C2×D47)⋊C4—44no—
64C2×(C2×D47)⋊C4—44no—
65C2×(C2×D47)⋊C4C2×C418yes—
66C2×C4×Dic47C2×C414yes—
67C2×C4×Dic47D414yes—
68C2×C4×Dic47D414yes—
69C47⋊#95—24no—
70C47⋊#95—24no—
71C47⋊#95—28no—
72C47⋊#95—44no—
73C23×C4×C47C2312yes—
74C23×C4×C47C2312yes—
75C23×C4×C47C2312yes—
76C23×C4×C47C2311yescharacteristic
77C2×C4⋊C4×C47—22no—
78C2×C4⋊C4×C47—22no—
79C2×C4⋊C4×C47—22no—
80C2×C4⋊C4×C47—22no—
81C2×C4⋊C4×C47—22no—
82C2×C4⋊C4×C47—22no—
83C2×C42×C47—22no—
84C2×C42×C47—22no—
85C2×C42×C47C2×C414yes—
86#95×C47—44no—
87C24×C47C2411yescharacteristic
88C23×D47—21no—
89C23×D47—21no—
90C23×D47—21no—
91C23×D47—21no—
92C23×D47—22no—
93C23×D47—22no—
94C23×D47—22no—
95C23×D47—22no—
96C23×D47C22×C412yes—
97C23×D47C2×D412yes—
98C23×D47C2×Q812yes—
99C22×C4×C47—22no—
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—22no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—24no—
112C22×C4×C47—24no—
113C22×C4×C47—24no—
114C22×C4×C47—24no—
115C22×C4×C47C22×C414yes—
116C22×C4×C47C22×C412yes—
117C22×C4×C47C2×D412yes—
118C22×C4×C47C2×D412yes—
119C22×C4×C47C2×D412yes—
120C22×C4×C47C2×Q812yes—
121C22×C4×C47#4112yes—
122C22×C4×C47#4112yes—
123C22×Dic47—21no—
124C22×Dic47—21no—
125C22×Dic47—21no—
126C22×Dic47—21no—
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—24no—
138C22×Dic47—24no—
139C22×Dic47—24no—
140C22×Dic47—24no—
141C22×Dic47C22×C414yes—
142C22×Dic47C22×C412yes—
143C22×Dic47C2×D414yes—
144C22×Dic47C2×D414yes—
145C22×Dic47C2×D412yes—
146C22×Dic47C2×Q812yes—
147C2×C4×D47—28no—
148C2×C4×D47—28no—
149C2×C4×D47—28no—
150C2×C4×D47—28no—
151C2×C4×D47—28no—
152C2×C4×D47—28no—
153C2×C4×D47C4⋊C4116yescomplemented
154C4⋊C4×C47—44no—
155C4⋊C4×C47—44no—
156C42×C47—28no—
157(C2×D47)⋊C4—216no—
158(C2×D47)⋊C4—48no—
159(C2×D47)⋊C4—48no—
160C4⋊Dic47—44no—
161C4⋊Dic47—44no—
162Dic47⋊C4—216no—
163Dic47⋊C4—416no—
164C4×Dic47C22⋊C4116yescomplemented
165C23×C47C23×C411yescharacteristic
166C23×C47C22×D411yescharacteristic
167C23×C47C22×D411yescharacteristic
168C23×C47C22×D411yescharacteristic
169C23×C47C22×Q811yescharacteristic
170C23×C47C2×#4111yescharacteristic
171C23×C47C2×#4111yescharacteristic
172C23×C47H(5,𝔽2)12yes—
173C23×C47H(5,𝔽2)12yes—
174C23×C472−1+412yes—
175C23×C472−1+412yes—
176C22×D47—22no—
177C22×D47—22no—
178C22×D47—24no—
179C22×D47—24no—
180C22×D47—22no—
181C22×D47—22no—
182C22×D47—22no—
183C22×D47—22no—
184C22×D47—22no—
185C22×D47—22no—
186C22×D47—22no—
187C22×D47—22no—
188C22×D47—24no—
189C22×D47—24no—
190C22×D47—42no—
191C22×D47—42no—
192C22×D47—42no—
193C22×D47—42no—
194C22×D47—42no—
195C22×D47—42no—
196C22×D47—42no—
197C22×D47—42no—
198C22×D47C2×C4⋊C414yes—
199C2×C4×C47—24no—
200C2×C4×C47—24no—
201C2×C4×C47—24no—
202C2×C4×C47—42no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—44no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—44no—
214C2×C4×C47—44no—
215C2×C4×C47—44no—
216C2×C4×C47C2×C22⋊C414yes—
217C2×C4×C47C2×C4⋊C414yes—
218C2×C4×C47#11714yes—
219C2×C4×C47#12114yes—
220C2×C4×C47#12114yes—
221C2×C4×C47C22⋊Q814yes—
222C2×C4×C47C22⋊Q814yes—
223C2×Dic47—22no—
224C2×Dic47—22no—
225C2×Dic47—24no—
226C2×Dic47—24no—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—22no—
230C2×Dic47—22no—
231C2×Dic47—24no—
232C2×Dic47—24no—
233C2×Dic47—24no—
234C2×Dic47—24no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—44no—
238C2×Dic47—42no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47—42no—
242C2×Dic47—44no—
243C2×Dic47—44no—
244C2×Dic47—44no—
245C2×Dic47C2×C22⋊C418yes—
246C2×Dic47C2×C4⋊C414yes—
247C4×D47—232no—
248#14340—471nomaximal, Sylow
249C22×C47C22×C22⋊C411yescharacteristic
250C22×C47C22×C4⋊C411yescharacteristic
251C22×C47C2×#11711yescharacteristic
252C22×C47#52012yes—
253C22×C47#52012yes—
254C22×C47C2×#12111yescharacteristic
255C22×C47C2×#12111yescharacteristic
256C22×C47C2×C22⋊Q811yescharacteristic
257C22×C47C2×C22⋊Q811yescharacteristic
258C22×C47#53511yescharacteristic
259C22×C47#53512yes—
260C22×C47#53511yescharacteristic
261C22×C47#53611yescharacteristic
262C22×C47#53612yes—
263C22×C47#53611yescharacteristic
264C22×C47#53712yes—
265C22×C47#53712yes—
266C22×C47#54012yes—
267C22×C47#54112yes—
268C22×C47#54212yes—
269C22×C47#54212yes—
270C22×C47#54412yes—
271C22×C47#54412yes—
272C2×D47—24no—
273C2×D47—24no—
274C2×D47—24no—
275C2×D47—44no—
276C2×D47—44no—
277C2×D47—42no—
278C2×D47—42no—
279C2×D47—42no—
280C2×D47—42no—
281C2×D47—44no—
282C2×D47—44no—
283C4×C47—44no—
284C4×C47—82no—
285C4×C47—82no—
286C4×C47#38018yes—
287Dic47—28no—
288Dic47—24no—
289Dic47—82no—
290Dic47—82no—
291C22×#117—471no—
292C2×#384—471no—
293C2×#382—471no—
294C2×#380—471no—
295C2×#380—471no—
296#1441—474no—
297#1439—474no—
298C4⋊#95—471no—
299C4⋊#95D4711yescharacteristic, complemented
300C2×C47C2×#38011yescharacteristic
301C2×C47#231112yes—
302C2×C47#231412yes—
303C2×C47#231711yescharacteristic
304C2×C47#231711yescharacteristic
305C2×C47#261911yescharacteristic
306C2×C47#262011yescharacteristic
307C2×C47#262011yescharacteristic
308C2×C47#262111yescharacteristic
309C2×C47#262812yes—
310C2×C47#262912yes—
311D47—44no—
312C24×C4—471no—
313C2×#117—478no—
314C2×#117—944no—
315C2×#117—944no—
316C22×C4⋊C4—472no—
317C22×C4⋊C4—471no—
318C22×C4⋊C4—471no—
319C22×C4⋊C4—474no—
320C22×C4⋊C4C2×D4711yescharacteristic
321C22×C4⋊C4C2×D4711yescharacteristic
322C22×C22⋊C4—472no—
323C22×C22⋊C4—472no—
324C22×C22⋊C4—472no—
325C22×C42—471no—
326C22×C42C2×D4711yescharacteristic
327#384—944no—
328#382—944no—
329#380—944no—
330#380—944no—
331C2×#95—474no—
332C2×#95—472no—
333C2×#95—472no—
334C2×#95—474no—
335C2×#95C2×D4714yes—
336C47#1434011yescharacteristic, complemented, Sylow
337C25—471no—
338C23×C4—472no—
339C23×C4—471no—
340C23×C4—474no—
341C23×C4—474no—
342C23×C4—474no—
343C23×C4—472no—
344C23×C4—472no—
345C23×C4—942no—
346C23×C4—942no—
347C23×C4—942no—
348C23×C4—942no—
349C23×C4C22×D4712yes—
350C23×C4C22×D4712yes—
351C23×C4C22×D4712yes—
352C23×C4C22×D4711yescharacteristic
353#117—9432no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—478no—
361C2×C4⋊C4—944no—
362C2×C4⋊C4—944no—
363C2×C4⋊C4—948no—
364C2×C4⋊C4—942no—
365C2×C4⋊C4—942no—
366C2×C4⋊C4—942no—
367C2×C4⋊C4—942no—
368C2×C4⋊C4—942no—
369C2×C4⋊C4—942no—
370C2×C4⋊C4—1884no—
371C2×C4⋊C4—1884no—
372C2×C22⋊C4—478no—
373C2×C22⋊C4—944no—
374C2×C22⋊C4—944no—
375C2×C22⋊C4—944no—
376C2×C22⋊C4—944no—
377C2×C22⋊C4—1884no—
378C2×C22⋊C4—1884no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—474no—
382C2×C42—474no—
383C2×C42—474no—
384C2×C42C4×D4714yes—
385#95—44no—
386#95—944no—
387#95—944no—
388#95—948no—
389#95—1884no—
390C24—472no—
391C24—472no—
392C24—472no—
393C24—941no—
394C24—941no—
395C24—941no—
396C24—941no—
397C24—942no—
398C24—942no—
399C24—942no—
400C24—942no—
401C24C23×D4711yescharacteristic
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—474no—
419C22×C4—472no—
420C22×C4—472no—
421C22×C4—474no—
422C22×C4—474no—
423C22×C4—472no—
424C22×C4—4716no—
425C22×C4—941no—
426C22×C4—941no—
427C22×C4—941no—
428C22×C4—941no—
429C22×C4—942no—
430C22×C4—942no—
431C22×C4—942no—
432C22×C4—942no—
433C22×C4—948no—
434C22×C4—948no—
435C22×C4—948no—
436C22×C4—948no—
437C22×C4—948no—
438C22×C4—948no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—944no—
446C22×C4—944no—
447C22×C4—944no—
448C22×C4—944no—
449C22×C4C2×C4×D4714yes—
450C22×C4C2×C4×D4712yes—
451C22×C4#1062535012yes—
452C22×C4#1062535012yes—
453C22×C4D4×D4712yes—
454C22×C4D4×D4712yes—
455C22×C4D4×D4712yes—
456C22×C4Q8×D4712yes—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—9416no—
460C4⋊C4—18816no—
461C4⋊C4—1884no—
462C4⋊C4—1884no—
463C22⋊C4—9416no—
464C22⋊C4—1888no—
465C22⋊C4—1888no—
466C42—28no—
467C42—4716no—
468C23—474no—
469C23—944no—
470C23—944no—
471C23—942no—
472C23—942no—
473C23—944no—
474C23—944no—
475C23—942no—
476C23—942no—
477C23—942no—
478C23—942no—
479C23—942no—
480C23—942no—
481C23—942no—
482C23—942no—
483C23—1882no—
484C23—1882no—
485C23—1882no—
486C23—1882no—
487C23—1882no—
488C23—1882no—
489C23—1882no—
490C23—1882no—
491C23C22×C4×D4711yescharacteristic
492C23C2×D4×D4711yescharacteristic
493C23C2×D4×D4711yescharacteristic
494C23C2×D4×D4711yescharacteristic
495C23C2×Q8×D4711yescharacteristic
496C23C2×#1062535011yescharacteristic
497C23C2×#1062535011yescharacteristic
498C23#1309406912yes—
499C23#1309407012yes—
500C23#1309407112yes—
501C23#1309407212yes—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—478no—
520C2×C4—474no—
521C2×C4—942no—
522C2×C4—942no—
523C2×C4—944no—
524C2×C4—944no—
525C2×C4—942no—
526C2×C4—942no—
527C2×C4—942no—
528C2×C4—942no—
529C2×C4—944no—
530C2×C4—944no—
531C2×C4—944no—
532C2×C4—944no—
533C2×C4—9432no—
534C2×C4—1884no—
535C2×C4—1882no—
536C2×C4—1882no—
537C2×C4—1882no—
538C2×C4—1884no—
539C2×C4—1882no—
540C2×C4—1882no—
541C2×C4—1882no—
542C2×C4—1884no—
543C2×C4—1884no—
544C2×C4C22⋊C4×D4714yes—
545C2×C4C4⋊C4×D4714yes—
546C2×C4#1309398114yes—
547C2×C4#1309399714yes—
548C2×C4#1309399714yes—
549C2×C4#1309400214yes—
550C2×C4#1309400214yes—
551C22—944no—
552C22—944no—
553C22—944no—
554C22—1882no—
555C22—1882no—
556C22—1882no—
557C22—1882no—
558C22—1884no—
559C22—1884no—
560C22—1884no—
561C22—1884no—
562C22C2×C22⋊C4×D4711yescharacteristic
563C22C2×C4⋊C4×D4711yescharacteristic
564C22C2×#1309398111yescharacteristic
565C22#1568960312yes—
566C22#1568960412yes—
567C22C2×#1309399711yescharacteristic
568C22C2×#1309399711yescharacteristic
569C22C2×#1309400211yescharacteristic
570C22C2×#1309400211yescharacteristic
571C22#1568968611yescharacteristic
572C22#1568968912yes—
573C22#1568969111yescharacteristic
574C22#1568969311yescharacteristic
575C22#1568969612yes—
576C22#1568969811yescharacteristic
577C22#1568970212yes—
578C22#1568970312yes—
579C22#1568972412yes—
580C22#1568973112yes—
581C22#1568973812yes—
582C22#1568974512yes—
583C22#1568975012yes—
584C22#1568975312yes—
585C4—44no—
586C4—82no—
587C4—82no—
588C4—944no—
589C4—948no—
590C4—3762no—
591C4—3762no—
592C4#1568898218yes—
593C2—1884no—
594C2C2×#1568898211yescharacteristic
595C2#2024597712yes—
596C2#2024599312yes—
597C2#2024600911yescharacteristic
598C2#2024601311yescharacteristic
599C2#2024914311yescharacteristic
600C2#2024915111yescharacteristic
601C2#2024915211yescharacteristic
602C2#2024915911yescharacteristic
603C2#2024921512yes—
604C2#2024922312yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×C4⋊#95C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenterC4⋊#95D47
hypocenterC47#14340
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#15688992C2semidirect
3. C2×#15688988C2semidirect
6. #20241639C2semidirect
8. #20241618C2semidirect
9. C47×C4⋊#95C2semidirect
13. C2×#13093981C4semidirect
153. C2×C4×D47C4⋊C4semidirect
164. C4×Dic47C22⋊C4semidirect
299. C4⋊#95D47semidirect
336. C47#14340semidirect

Maximal subgroups

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

subgroupindex
C22×#130939812
C2×#156889922
C2×#156889882
C2×#156889802
#202416392
#202416322
#202416182
C47×C4⋊#952
#1434047