Finite Group Databasebeta

← #26137665

Subgroups — #26137665

Order 12032.

2782 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093982C211yesmaximal, characteristic
2C22×#13093981C211yesmaximal, characteristic
3C22×C4⋊C4×D47C211yesmaximal, characteristic
4#20241618C211yesmaximal, characteristic, complemented
5#20241617C211yesmaximal, characteristic, complemented
6#20241617C211yesmaximal, characteristic, complemented
7Dic47⋊#95C214yesmaximal, complemented
8(C2×D47)⋊#95C214yesmaximal
9C47×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D47C2212yes—
11C23×C4×D47C2211yescharacteristic
12C2×#13093982—24no—
13C2×#13093982—24no—
14C2×#13093982C418yescomplemented
15C2×#13093981—24no—
16C2×#13093981—24no—
17C2×#13093981C418yescomplemented
18C22×C4⋊Dic47C2211yescharacteristic
19C22×C4⋊Dic47C2211yescharacteristic
20C22×Dic47⋊C4C2212yes—
21C22×Dic47⋊C4C2212yes—
22C2×C4⋊C4×D47—24no—
23C2×C4⋊C4×D47—24no—
24C2×C4⋊C4×D47C418yescomplemented
25C22×(C2×D47)⋊C4C2212yes—
26C22×(C2×D47)⋊C4C2212yes—
27C22×C4×Dic47C2211yescharacteristic
28C22×C4×Dic47C2212yes—
29C2×C47⋊#95C2214yes—
30C2×C47⋊#95C2214yes—
31C2×C47⋊#95C2214yes—
32C22×C4⋊C4×C47C2211yescharacteristic
33C22×C4⋊C4×C47C2211yescharacteristic
34C22×C42×C47C2211yescharacteristic
35C2×#95×C47C2214yes—
36C24×D47C2311yescharacteristic
37C23×Dic47C2312yes—
38C23×Dic47C2312yes—
39C23×Dic47C2312yes—
40C23×Dic47C2311yescharacteristic
41C22×C4×D47—24no—
42C22×C4×D47—24no—
43C22×C4×D47—44no—
44C22×C4×D47—44no—
45C22×C4×D47C2×C418yes—
46C22×C4×D47C2×C414yes—
47C22×C4×D47C2×C414yes—
48C22×C4×D47C2×C414yes—
49C22×C4×D47D414yes—
50C22×C4×D47D414yes—
51C22×C4×D47D414yes—
52C22×C4×D47Q814yes—
53#13093982—232no—
54#13093981—232no—
55C2×C4⋊Dic47—22no—
56C2×C4⋊Dic47—22no—
57C2×C4⋊Dic47—22no—
58C2×C4⋊Dic47—22no—
59C2×C4⋊Dic47C2×C414yes—
60C2×C4⋊Dic47C2×C414yescomplemented
61C2×Dic47⋊C4—24no—
62C2×Dic47⋊C4—24no—
63C2×Dic47⋊C4—24no—
64C2×Dic47⋊C4—24no—
65C2×Dic47⋊C4C2×C418yes—
66C2×Dic47⋊C4C2×C418yescomplemented
67C4⋊C4×D47—232no—
68C2×(C2×D47)⋊C4—24no—
69C2×(C2×D47)⋊C4—24no—
70C2×(C2×D47)⋊C4—24no—
71C2×(C2×D47)⋊C4—24no—
72C2×(C2×D47)⋊C4C2×C418yes—
73C2×(C2×D47)⋊C4C2×C418yes—
74C2×C4×Dic47—22no—
75C2×C4×Dic47—22no—
76C2×C4×Dic47—24no—
77C2×C4×Dic47—24no—
78C2×C4×Dic47C2×C414yescomplemented
79C2×C4×Dic47C2×C418yescomplemented
80C47⋊#95—44no—
81C47⋊#95—44no—
82C47⋊#95—44no—
83C23×C4×C47C2312yes—
84C23×C4×C47C2312yes—
85C23×C4×C47C2312yes—
86C23×C4×C47C2311yescharacteristic
87C2×C4⋊C4×C47—22no—
88C2×C4⋊C4×C47—22no—
89C2×C4⋊C4×C47—22no—
90C2×C4⋊C4×C47—22no—
91C2×C4⋊C4×C47C2×C414yes—
92C2×C4⋊C4×C47C2×C414yescomplemented
93C2×C42×C47—22no—
94C2×C42×C47—22no—
95C2×C42×C47C2×C414yescomplemented
96#95×C47—44no—
97C24×C47C2411yescharacteristic
98C23×D47—22no—
99C23×D47—22no—
100C23×D47—22no—
101C23×D47—22no—
102C23×D47C22×C412yes—
103C23×D47C22×C412yes—
104C23×D47C22×C412yes—
105C23×D47C2×D412yes—
106C23×D47C2×D412yes—
107C23×D47C2×D412yes—
108C23×D47C2×Q812yes—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—24no—
117C22×C4×C47—24no—
118C22×C4×C47—24no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—24no—
122C22×C4×C47—22no—
123C22×C4×C47C22×C414yes—
124C22×C4×C47C22×C414yes—
125C22×C4×C47C22×C414yes—
126C22×C4×C47C22×C412yes—
127C22×C4×C47C22×C412yes—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C2×D412yes—
130C22×C4×C47C2×D412yes—
131C22×C4×C47C2×D412yes—
132C22×C4×C47C2×Q812yes—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—24no—
142C22×Dic47—24no—
143C22×Dic47—24no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—24no—
147C22×Dic47C22×C414yes—
148C22×Dic47C22×C414yes—
149C22×Dic47C22×C414yes—
150C22×Dic47C22×C412yes—
151C22×Dic47C22×C412yes—
152C22×Dic47C22×C412yes—
153C22×Dic47C2×D412yes—
154C22×Dic47C2×D412yes—
155C22×Dic47C2×D412yes—
156C22×Dic47C2×Q812yes—
157C2×C4×D47—216no—
158C2×C4×D47—416no—
159C2×C4×D47—48no—
160C2×C4×D47—416no—
161C2×C4×D47—48no—
162C2×C4×D47C42116yes—
163C2×C4×D47C22⋊C4116yes—
164C2×C4×D47C22⋊C4116yes—
165C2×C4×D47C22⋊C4116yes—
166C2×C4×D47C4⋊C4116yes—
167C2×C4×D47C4⋊C4116yes—
168C2×C4×D47C4⋊C4116yes—
169C4⋊C4×C47—28no—
170C4⋊C4×C47—28no—
171C42×C47—28no—
172(C2×D47)⋊C4—216no—
173(C2×D47)⋊C4—216no—
174C4⋊Dic47—28no—
175C4⋊Dic47—28no—
176Dic47⋊C4—216no—
177Dic47⋊C4—216no—
178C4×Dic47—28no—
179C4×Dic47—216no—
180C23×C47C23×C411yescharacteristic
181C23×C47C23×C411yescharacteristic
182C23×C47C23×C411yescharacteristic
183C23×C47C22×D411yescharacteristic
184C23×C47C22×D411yescharacteristic
185C23×C47C22×D411yescharacteristic
186C23×C47C22×Q811yescharacteristic
187C23×C47H(5,𝔽2)12yes—
188C23×C47H(5,𝔽2)12yes—
189C23×C472−1+412yes—
190C23×C472−1+412yes—
191C22×D47—24no—
192C22×D47—24no—
193C22×D47—24no—
194C22×D47—24no—
195C22×D47—24no—
196C22×D47—24no—
197C22×D47—22no—
198C22×D47—24no—
199C22×D47—22no—
200C22×D47—22no—
201C22×D47—24no—
202C22×D47—22no—
203C22×D47—24no—
204C22×D47—24no—
205C22×D47—24no—
206C22×D47—24no—
207C22×D47C2×C4214yes—
208C22×D47C2×C22⋊C414yes—
209C22×D47C2×C22⋊C414yes—
210C22×D47C2×C22⋊C414yes—
211C22×D47C2×C4⋊C414yes—
212C22×D47C2×C4⋊C414yes—
213C22×D47C2×C4⋊C414yes—
214C2×C4×C47—24no—
215C2×C4×C47—24no—
216C2×C4×C47—24no—
217C2×C4×C47—28no—
218C2×C4×C47—28no—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—42no—
222C2×C4×C47—44no—
223C2×C4×C47—42no—
224C2×C4×C47—42no—
225C2×C4×C47—42no—
226C2×C4×C47—42no—
227C2×C4×C47—42no—
228C2×C4×C47—42no—
229C2×C4×C47—44no—
230C2×C4×C47—42no—
231C2×C4×C47C2×C4214yes—
232C2×C4×C47C2×C22⋊C414yes—
233C2×C4×C47C2×C22⋊C414yes—
234C2×C4×C47C2×C22⋊C414yes—
235C2×C4×C47C2×C4⋊C414yes—
236C2×C4×C47C2×C4⋊C414yes—
237C2×C4×C47C2×C4⋊C414yes—
238C2×Dic47—24no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—28no—
242C2×Dic47—24no—
243C2×Dic47—24no—
244C2×Dic47—28no—
245C2×Dic47—42no—
246C2×Dic47—44no—
247C2×Dic47—42no—
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—42no—
251C2×Dic47—42no—
252C2×Dic47—42no—
253C2×Dic47—42no—
254C2×Dic47—44no—
255C2×Dic47C2×C4214yes—
256C2×Dic47C2×C22⋊C414yes—
257C2×Dic47C2×C22⋊C414yes—
258C2×Dic47C2×C22⋊C414yes—
259C2×Dic47C2×C4⋊C414yes—
260C2×Dic47C2×C4⋊C414yes—
261C2×Dic47C2×C4⋊C414yes—
262C4×D47—432no—
263C4×D47#95164yescomplemented
264#13795—471nomaximal, Sylow
265C22×C47C22×C4211yescharacteristic
266C22×C47C22×C22⋊C411yescharacteristic
267C22×C47C22×C22⋊C411yescharacteristic
268C22×C47C22×C22⋊C411yescharacteristic
269C22×C47C22×C4⋊C411yescharacteristic
270C22×C47C22×C4⋊C411yescharacteristic
271C22×C47C22×C4⋊C411yescharacteristic
272C22×C47#51812yes—
273C22×C47#51912yes—
274C22×C47#52012yes—
275C22×C47#52012yes—
276C22×C47#52012yes—
277C22×C47#52012yes—
278C22×C47#53511yescharacteristic
279C22×C47#53512yes—
280C22×C47#53511yescharacteristic
281C22×C47#53611yescharacteristic
282C22×C47#53612yes—
283C22×C47#53611yescharacteristic
284C22×C47#53712yes—
285C22×C47#53712yes—
286C22×C47#54412yes—
287C22×C47#54412yes—
288C2×D47—24no—
289C2×D47—24no—
290C2×D47—28no—
291C2×D47—24no—
292C2×D47—24no—
293C2×D47—28no—
294C2×D47—24no—
295C2×D47—28no—
296C2×D47—24no—
297C2×D47—28no—
298C2×D47C2×#9518yes—
299C4×C47—44no—
300C4×C47—44no—
301C4×C47—44no—
302C4×C47C2×#9518yes—
303Dic47—44no—
304Dic47—44no—
305Dic47—44no—
306Dic47C2×#9518yes—
307C22×#117—471no—
308C22×#117—471no—
309C23×C4⋊C4—471no—
310C4⋊#95—471no—
311C4⋊#95—474no—
312C4⋊#95—471no—
313C4⋊#95—471no—
314C4⋊#95D4711yescharacteristic, complemented
315C22⋊#95—474no—
316C2×C47C22×#9511yescharacteristic
317C2×C47#227412yes—
318C2×C47#230911yescharacteristic
319C2×C47#231011yescharacteristic
320C2×C47#231112yes—
321C2×C47#231111yescharacteristic
322C2×C47#231111yescharacteristic
323C2×C47#231612yes—
324C2×C47#231711yescharacteristic
325C2×C47#231712yes—
326C2×C47#231711yescharacteristic
327D47—28no—
328C24×C4—471no—
329C24×C4—472no—
330C2×#117—478no—
331C2×#117—478no—
332C2×#117—944no—
333C2×#117—944no—
334C2×#117—944no—
335C2×#117—944no—
336C22×C4⋊C4—472no—
337C22×C4⋊C4—471no—
338C22×C4⋊C4—472no—
339C22×C4⋊C4—471no—
340C22×C4⋊C4—478no—
341C22×C4⋊C4—944no—
342C22×C4⋊C4—944no—
343C22×C4⋊C4C2×D4711yescharacteristic
344C22×C4⋊C4C2×D4711yescharacteristic
345C22×C22⋊C4—472no—
346C22×C22⋊C4—472no—
347C22×C42—472no—
348C22×C42—471no—
349C22×C42C2×D4711yescharacteristic
350C2×#95—474no—
351C2×#95—474no—
352C2×#95—474no—
353C2×#95C2×D4714yes—
354C47#1379511yescharacteristic, complemented, Sylow
355C25—471no—
356C23×C4—471no—
357C23×C4—474no—
358C23×C4—474no—
359C23×C4—474no—
360C23×C4—474no—
361C23×C4—474no—
362C23×C4—474no—
363C23×C4—474no—
364C23×C4—472no—
365C23×C4—472no—
366C23×C4—472no—
367C23×C4—478no—
368C23×C4—944no—
369C23×C4—944no—
370C23×C4—1884no—
371C23×C4—1884no—
372C23×C4C22×D4712yes—
373C23×C4C22×D4712yes—
374C23×C4C22×D4712yes—
375C23×C4C22×D4711yescharacteristic
376#117—9432no—
377#117—9432no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—478no—
383C2×C4⋊C4—474no—
384C2×C4⋊C4—478no—
385C2×C4⋊C4—474no—
386C2×C4⋊C4—9432no—
387C2×C4⋊C4—944no—
388C2×C4⋊C4—944no—
389C2×C4⋊C4—944no—
390C2×C4⋊C4—944no—
391C2×C4⋊C4—942no—
392C2×C4⋊C4—942no—
393C2×C4⋊C4—942no—
394C2×C4⋊C4—942no—
395C2×C4⋊C4C4×D4714yes—
396C2×C4⋊C4C4×D4714yescomplemented
397C2×C22⋊C4—478no—
398C2×C22⋊C4—478no—
399C2×C22⋊C4—944no—
400C2×C22⋊C4—944no—
401C2×C22⋊C4—944no—
402C2×C22⋊C4—944no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42—478no—
406C2×C42—474no—
407C2×C42—944no—
408C2×C42—944no—
409C2×C42—942no—
410C2×C42—942no—
411C2×C42C4×D4714yescomplemented
412#95—44no—
413#95—1884no—
414#95—1884no—
415#95—1884no—
416C24—472no—
417C24—472no—
418C24—472no—
419C24—472no—
420C24—472no—
421C24—472no—
422C24—472no—
423C24—942no—
424C24—942no—
425C24—942no—
426C24—942no—
427C24C23×D4711yescharacteristic
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—472no—
443C22×C4—472no—
444C22×C4—472no—
445C22×C4—472no—
446C22×C4—472no—
447C22×C4—472no—
448C22×C4—472no—
449C22×C4—4716no—
450C22×C4—4716no—
451C22×C4—4716no—
452C22×C4—4716no—
453C22×C4—4716no—
454C22×C4—4716no—
455C22×C4—4716no—
456C22×C4—474no—
457C22×C4—474no—
458C22×C4—474no—
459C22×C4—9416no—
460C22×C4—942no—
461C22×C4—942no—
462C22×C4—942no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—942no—
466C22×C4—942no—
467C22×C4—944no—
468C22×C4—942no—
469C22×C4—944no—
470C22×C4—944no—
471C22×C4—942no—
472C22×C4—942no—
473C22×C4—944no—
474C22×C4—18816no—
475C22×C4—1888no—
476C22×C4—18816no—
477C22×C4—1888no—
478C22×C4C2×C4×D4714yes—
479C22×C4C2×C4×D4714yes—
480C22×C4C2×C4×D4714yes—
481C22×C4C2×C4×D4712yes—
482C22×C4C2×C4×D4712yes—
483C22×C4C2×C4×D4712yes—
484C22×C4D4×D4712yes—
485C22×C4D4×D4712yes—
486C22×C4D4×D4712yes—
487C22×C4Q8×D4712yes—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—9416no—
491C4⋊C4—9416no—
492C4⋊C4—948no—
493C4⋊C4—948no—
494C22⋊C4—9416no—
495C22⋊C4—9416no—
496C42—28no—
497C42—9416no—
498C42—948no—
499C23—474no—
500C23—474no—
501C23—474no—
502C23—474no—
503C23—474no—
504C23—474no—
505C23—474no—
506C23—944no—
507C23—944no—
508C23—944no—
509C23—944no—
510C23—944no—
511C23—944no—
512C23—944no—
513C23—944no—
514C23—942no—
515C23—944no—
516C23—942no—
517C23—942no—
518C23—944no—
519C23—942no—
520C23—944no—
521C23—944no—
522C23C22×C4×D4711yescharacteristic
523C23C22×C4×D4711yescharacteristic
524C23C22×C4×D4711yescharacteristic
525C23C2×D4×D4711yescharacteristic
526C23C2×D4×D4711yescharacteristic
527C23C2×D4×D4711yescharacteristic
528C23C2×Q8×D4711yescharacteristic
529C23#1309406912yes—
530C23#1309407012yes—
531C23#1309407112yes—
532C23#1309407212yes—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—474no—
551C2×C4—474no—
552C2×C4—474no—
553C2×C4—474no—
554C2×C4—474no—
555C2×C4—474no—
556C2×C4—474no—
557C2×C4—4764no—
558C2×C4—944no—
559C2×C4—944no—
560C2×C4—944no—
561C2×C4—948no—
562C2×C4—944no—
563C2×C4—944no—
564C2×C4—948no—
565C2×C4—18832no—
566C2×C4—1882no—
567C2×C4—1882no—
568C2×C4—1884no—
569C2×C4—1882no—
570C2×C4—1882no—
571C2×C4—1882no—
572C2×C4—1882no—
573C2×C4—1882no—
574C2×C4—1882no—
575C2×C4—1884no—
576C2×C4C42×D4714yes—
577C2×C4C22⋊C4×D4714yes—
578C2×C4C22⋊C4×D4714yes—
579C2×C4C22⋊C4×D4714yes—
580C2×C4C4⋊C4×D4714yes—
581C2×C4C4⋊C4×D4714yes—
582C2×C4C4⋊C4×D4714yes—
583C22—478no—
584C22—944no—
585C22—944no—
586C22—948no—
587C22—944no—
588C22—944no—
589C22—948no—
590C22—948no—
591C22—944no—
592C22—948no—
593C22—944no—
594C22C2×C42×D4711yescharacteristic
595C22C2×C22⋊C4×D4711yescharacteristic
596C22C2×C22⋊C4×D4711yescharacteristic
597C22C2×C22⋊C4×D4711yescharacteristic
598C22C2×C4⋊C4×D4711yescharacteristic
599C22C2×C4⋊C4×D4711yescharacteristic
600C22C2×C4⋊C4×D4711yescharacteristic
601C22#1568959512yes—
602C22#1568959912yes—
603C22#1568960212yes—
604C22#1568960312yes—
605C22#1568960412yes—
606C22#1568960512yes—
607C22#1568968611yescharacteristic
608C22#1568968912yes—
609C22#1568969111yescharacteristic
610C22#1568969311yescharacteristic
611C22#1568969612yes—
612C22#1568969811yescharacteristic
613C22#1568970212yes—
614C22#1568970312yes—
615C22#1568975012yes—
616C22#1568975312yes—
617C4—44no—
618C4—44no—
619C4—44no—
620C4—478no—
621C4—1884no—
622C4—1884no—
623C4—1884no—
624C4#95×D4718yes—
625C2—948no—
626C2C2×#95×D4711yescharacteristic
627C2#2024574212yes—
628C2#2024596711yescharacteristic
629C2#2024597111yescharacteristic
630C2#2024597712yes—
631C2#2024597811yescharacteristic
632C2#2024597911yescharacteristic
633C2#2024600312yes—
634C2#2024600911yescharacteristic
635C2#2024601012yes—
636C2#2024601311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC22×C47C22×C42
FittingC47×C4⋊#95C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenterC4⋊#95D47
hypocenterC47#13795
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
4. #20241618C2semidirect
5. #20241617C2semidirect
6. #20241617C2semidirect
7. Dic47⋊#95C2semidirect
9. C47×C4⋊#95C2semidirect
14. C2×#13093982C4semidirect
17. C2×#13093981C4semidirect
24. C2×C4⋊C4×D47C4semidirect
60. C2×C4⋊Dic47C2×C4semidirect
66. C2×Dic47⋊C4C2×C4semidirect
78. C2×C4×Dic47C2×C4semidirect
79. C2×C4×Dic47C2×C4semidirect
92. C2×C4⋊C4×C47C2×C4semidirect
95. C2×C42×C47C2×C4semidirect
263. C4×D47#95semidirect
314. C4⋊#95D47semidirect
354. C47#13795semidirect
396. C2×C4⋊C4C4×D47semidirect
411. C2×C42C4×D47semidirect

Maximal subgroups

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

subgroupindex
C22×#130939822
C22×#130939812
C22×C4⋊C4×D472
#202416182
#202416172
Dic47⋊#952
(C2×D47)⋊#952
C47×C4⋊#952
#1379547