Finite Group Databasebeta

← #26316301

Subgroups — #26316301

Order 12032.

828 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251738C211yesmaximal, characteristic
2C22⋊Q8×Dic47C211yesmaximal, characteristic
3#20244954C211yesmaximal, characteristic, complemented
4#20244924C211yesmaximal, characteristic, complemented
5#20244715C211yesmaximal, characteristic, complemented
6#20244527C211yesmaximal, characteristic, complemented
7#20243983C211yesmaximal, characteristic, complemented
8#20243979C211yesmaximal, characteristic, complemented
9#20243812C211yesmaximal, characteristic
10#20243644C211yesmaximal, characteristic, complemented
11#20243640C211yesmaximal, characteristic, complemented
12#20243417C211yesmaximal, characteristic
13#20243415C211yesmaximal, characteristic
14C2×#15688877C211yesmaximal, characteristic
15C47×#1892C211yesmaximal, characteristic
16C23×C47⋊C8C2211yescharacteristic
17C2×#13094023C2211yescharacteristic
18C2×#13094001—21no—
19C2×Q8×Dic47C2211yescharacteristic
20C22×C4×Dic47C2211yescharacteristic
21C2×#13093958C2211yescharacteristic
22C2×#13093958C2211yescharacteristic
23C2×#13093957C2211yescharacteristic
24C2×#13093957C2211yescharacteristic
25C2×#13093948C2211yescharacteristic
26C2×#13093948C2211yescharacteristic
27#15689030C2211yescharacteristic
28#15689027—21no—
29#15689022C2211yescharacteristic
30#15689022C2211yescharacteristic
31#15689020—21no—
32#15689019C2211yescharacteristic
33#15689009—21no—
34#15688998—21no—
35#15688992C2211yescharacteristic
36#15688983C2211yescharacteristic
37#15688983C2211yescharacteristic
38#15688981—21no—
39C4⋊C4×Dic47—21no—
40C4⋊C4×Dic47C2211yescharacteristic
41C22⋊C4×Dic47—21no—
42#15688886C2211yescharacteristic
43#15688886C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688877C2211yescharacteristic
49#15688877C2211yescharacteristic
50#15688877C2211yescharacteristic
51#15688877C2211yescharacteristic
52C23×C8×C47C2211yescharacteristic
53C2×C22⋊Q8×C47C2211yescharacteristic
54C2×#103×C47C2211yescharacteristic
55C2×#103×C47C2211yescharacteristic
56C47×#380C2211yescharacteristic
57C47×#340C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C22×Dic94—21no—
60C23×Dic47D411yescharacteristic
61C23×Dic47D411yescharacteristic
62C22×C47⋊C8—22no—
63C22×C47⋊C8—22no—
64C22×C47⋊C8—41no—
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67#13094023—41no—
68#13094023—41no—
69#13094001—44no—
70Q8×Dic47—41no—
71Q8×Dic47—41no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47—21no—
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×C4⋊Dic47C2311yescharacteristic
79C2×Dic47⋊C4—21no—
80C2×Dic47⋊C4—21no—
81C2×Dic47⋊C4—21no—
82C2×Dic47⋊C4—21no—
83C2×C22⋊Dic47—21no—
84C2×C22⋊Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47—41no—
88C2×C4×Dic47—41no—
89C2×C4×Dic47D411yescharacteristic
90C2×C4×Dic47D411yescharacteristic
91C2×C4×Dic47C2311yescharacteristic
92C2×C4×Dic47C2311yescharacteristic
93#13093958—22no—
94#13093958—41no—
95#13093957—22no—
96#13093957—41no—
97#13093948—22no—
98#13093948—41no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C22×Q8×C47C2311yescharacteristic
108C23×C4×C47C2311yescharacteristic
109C22×C8×C47—22no—
110C22×C8×C47—22no—
111C22×C8×C47—41no—
112C22×C8×C47C2311yescharacteristic
113C22×C8×C47C2311yescharacteristic
114C22⋊Q8×C47—22no—
115C22⋊Q8×C47Q814yescomplemented
116C2×C4⋊C4×C47—21no—
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C4⋊C4×C47C2311yescharacteristic
120C2×C22⋊C4×C47—21no—
121C2×C22⋊C4×C47—21no—
122#103×C47—22no—
123#103×C47—41no—
124#95×C47—21no—
125C24×C47C2×D411yescharacteristic
126C2×Q8×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47—21no—
131C2×Q8×C47C2×Q812yes—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×Q812yes—
142C22×C4×C47#4112yes—
143C22×C4×C47C2411yescharacteristic
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—21no—
147C22×Dic47—21no—
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—41no—
151C22×Dic47—41no—
152C22×Dic47—41no—
153C22×Dic47Q1614yes—
154C22×Dic47C2×D411yescharacteristic
155C22×Dic47C2×D411yescharacteristic
156C22×Dic47C2×D411yescharacteristic
157C22×Dic47C2×D411yescharacteristic
158C2×Dic94—42no—
159C2×Dic94—42no—
160C2×Dic94—42no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47—21no—
164C2×C8×C47—21no—
165C2×C8×C47—21no—
166C2×C8×C47—44no—
167C2×C8×C47#4112yes—
168C4⋊C4×C47—24no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47C2×Q812yes—
175C22⋊C4×C47—24no—
176C22⋊C4×C47—42no—
177C22⋊Dic47—42no—
178C22⋊Dic47—42no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—42no—
188C4⋊Dic47—42no—
189C4⋊Dic47—42no—
190Dic47⋊C4—42no—
191Dic47⋊C4—42no—
192Dic47⋊C4—42no—
193Dic47⋊C4—42no—
194C4×Dic47—21no—
195C4×Dic47—21no—
196C4×Dic47—21no—
197C4×Dic47—21no—
198C4×Dic47—41no—
199C4×Dic47—41no—
200C4×Dic47—42no—
201C4×Dic47—42no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—44no—
208C2×C47⋊C8#4112yes—
209C23×C47—21no—
210C23×C47—21no—
211C23×C47—21no—
212C23×C47—21no—
213C23×C47C22⋊Q812yes—
214C23×C47C2×Q1612yes—
215C23×C47AGL(1,ℤ/8)12yes—
216C23×C47C22×D411yescharacteristic
217Q8×C47—22no—
218Q8×C47—22no—
219Q8×C47—41no—
220Q8×C47—41no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47C22⋊Q812yes—
240C2×C4×C47C22×Q811yescharacteristic
241C2×C4×C47C2×#4111yescharacteristic
242C2×C4×C47C2×#4111yescharacteristic
243C2×C4×C47C2×#4111yescharacteristic
244C2×C4×C47H(5,𝔽2)11yescharacteristic
245C2×C4×C47H(5,𝔽2)11yescharacteristic
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—21no—
249C2×Dic47—21no—
250C2×Dic47—21no—
251C2×Dic47—21no—
252C2×Dic47—21no—
253C2×Dic47—21no—
254C2×Dic47—21no—
255C2×Dic47—44no—
256C2×Dic47—44no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47C2×Q1612yes—
270C2×Dic47C2×Q1612yes—
271C2×Dic47#13512yes—
272Dic94—84no—
273C8×C47—22no—
274C8×C47—41no—
275C47⋊C8—22no—
276C47⋊C8—41no—
277#29160—471nomaximal, Sylow
278C22×C47—22no—
279C22×C47—22no—
280C22×C47—22no—
281C22×C47—22no—
282C22×C47—22no—
283C22×C47—22no—
284C22×C47#47714yes—
285C22×C47C2×C22⋊Q811yescharacteristic
286C22×C47#53211yescharacteristic
287C22×C47#53411yescharacteristic
288C22×C47C22×Q1611yescharacteristic
289C22×C47C2×#13511yescharacteristic
290C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
291C22×C47#57511yescharacteristic
292C4×C47—22no—
293C4×C47—42no—
294C4×C47—42no—
295C4×C47—81no—
296C4×C47—81no—
297C4×C47#55111yescharacteristic
298C4×C47#55111yescharacteristic
299C4×C47#55411yescharacteristic
300C4×C47#55411yescharacteristic
301Dic47—22no—
302Dic47—22no—
303Dic47—22no—
304Dic47—41no—
305Dic47—81no—
306Dic47—81no—
307Dic47—81no—
308Dic47—81no—
309#2857—471no—
310C4×C22⋊Q8—471no—
311#2100—471no—
312#2095—471no—
313#2057—471no—
314#2024—471no—
315#1921—471no—
316#1921—471no—
317#1892—471no—
318#1892D4711yescharacteristic
319#1863—471no—
320#1863—471no—
321#1808—471no—
322#1808—471no—
323C2×#336—471no—
324C2×C47—24no—
325C2×C47#256411yescharacteristic
326C2×C47C2×#47711yescharacteristic
327C2×C47#307611yescharacteristic
328C2×C47#317011yescharacteristic
329C2×C47#318011yescharacteristic
330C2×C47#318811yescharacteristic
331C2×C47#318911yescharacteristic
332C23×C8—471no—
333C23×C8C2×D4711yescharacteristic
334C2×C4⋊Q8—471no—
335C2×C22⋊Q8—941no—
336C2×C22⋊Q8C2×D4711yescharacteristic
337C2×C4×Q8—471no—
338C22×C42—471no—
339C2×#107—471no—
340C2×#107—471no—
341C2×#107—471no—
342C2×#107—471no—
343C2×#103—471no—
344C2×#103—471no—
345C2×#103C2×D4711yescharacteristic
346C2×#103C2×D4711yescharacteristic
347#391—471no—
348#391—941no—
349#389—471no—
350#389—471no—
351#389—471no—
352#389—941no—
353#387—941no—
354#385—941no—
355#384—471no—
356#380—471no—
357#380—471no—
358#380—941no—
359#380C2×D4711yescharacteristic
360C4×C4⋊C4—471no—
361C4×C4⋊C4—941no—
362C4×C22⋊C4—941no—
363#340—471no—
364#340—471no—
365#340—471no—
366#340—471no—
367#340—471no—
368#340—471no—
369#340C2×D4711yescharacteristic
370#340C2×D4711yescharacteristic
371#336—471no—
372#336—471no—
373#336—471no—
374#336—471no—
375C47#2916011yescharacteristic, complemented, Sylow
376C22×Q8—941no—
377C22×Q8C22×D4711yescharacteristic
378C23×C4—471no—
379C23×C4—471no—
380C23×C4C22×D4711yescharacteristic
381C22×C8—22no—
382C22×C8—22no—
383C22×C8—41no—
384C22×C8—471no—
385C22×C8—471no—
386C22×C8—942no—
387C22×C8—942no—
388C22×C8—1881no—
389C22×C8C22×D4711yescharacteristic
390C22×C8C22×D4711yescharacteristic
391C4⋊Q8—1881no—
392C4⋊Q8—1881no—
393C22⋊Q8—22no—
394C22⋊Q8—1884no—
395C22⋊Q8Dic9414yescomplemented
396C4×Q8—1881no—
397C4×Q8—1881no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—471no—
400C2×C4⋊C4—471no—
401C2×C4⋊C4—471no—
402C2×C4⋊C4—471no—
403C2×C4⋊C4—941no—
404C2×C4⋊C4—941no—
405C2×C4⋊C4—941no—
406C2×C4⋊C4—941no—
407C2×C4⋊C4—941no—
408C2×C4⋊C4—941no—
409C2×C4⋊C4—941no—
410C2×C4⋊C4C22×D4711yescharacteristic
411C2×C4⋊C4C22×D4711yescharacteristic
412C2×C4⋊C4C22×D4711yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—941no—
416C2×C22⋊C4—941no—
417C2×C42—471no—
418C2×C42—471no—
419C2×C42—471no—
420C2×C42—471no—
421C2×C42—941no—
422C2×C42—941no—
423C2×C42—1881no—
424C2×C42—1881no—
425#107—942no—
426#107—942no—
427#107—1881no—
428#107—1881no—
429#103—22no—
430#103—41no—
431#103—942no—
432#103—1881no—
433#95—21no—
434#95—941no—
435#95—941no—
436#95—941no—
437#95—941no—
438#95—941no—
439#95—941no—
440#95—941no—
441#95—941no—
442C24D4×D4711yescharacteristic
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—1882no—
449C2×Q8—1882no—
450C2×Q8—1882no—
451C2×Q8C2×Dic9412yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—471no—
461C22×C4—471no—
462C22×C4—471no—
463C22×C4—474no—
464C22×C4—471no—
465C22×C4—942no—
466C22×C4—942no—
467C22×C4—941no—
468C22×C4—941no—
469C22×C4—941no—
470C22×C4—941no—
471C22×C4—1881no—
472C22×C4—1881no—
473C22×C4—1881no—
474C22×C4C2×Dic9412yes—
475C22×C4D4×D4711yescharacteristic
476C22×C4#1062535212yes—
477C22×C4C23×D4711yescharacteristic
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—44no—
484C2×C8—472no—
485C2×C8—941no—
486C2×C8—941no—
487C2×C8—941no—
488C2×C8—941no—
489C2×C8—941no—
490C2×C8—1884no—
491C2×C8#1062535212yes—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—941no—
499C4⋊C4—941no—
500C4⋊C4—941no—
501C4⋊C4—941no—
502C4⋊C4—941no—
503C4⋊C4—941no—
504C4⋊C4—941no—
505C4⋊C4—941no—
506C4⋊C4—1882no—
507C4⋊C4—1882no—
508C4⋊C4—1882no—
509C4⋊C4—1882no—
510C4⋊C4—1882no—
511C4⋊C4—1882no—
512C4⋊C4—1882no—
513C4⋊C4C2×Dic9412yes—
514C22⋊C4—24no—
515C22⋊C4—42no—
516C22⋊C4—1882no—
517C22⋊C4—1882no—
518C42—941no—
519C42—941no—
520C42—941no—
521C42—941no—
522C42—1881no—
523C42—1881no—
524C42—1882no—
525C42—1882no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#1309400112yes—
531C23Q16×D4712yes—
532C23#1309405112yes—
533C23C2×D4×D4711yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—41no—
537Q8—41no—
538Q8—3764no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—472no—
558C2×C4—472no—
559C2×C4—472no—
560C2×C4—941no—
561C2×C4—941no—
562C2×C4—941no—
563C2×C4—941no—
564C2×C4—941no—
565C2×C4—941no—
566C2×C4—941no—
567C2×C4—941no—
568C2×C4—941no—
569C2×C4—1884no—
570C2×C4—1884no—
571C2×C4—1881no—
572C2×C4—1881no—
573C2×C4—1881no—
574C2×C4—1881no—
575C2×C4—1881no—
576C2×C4—1881no—
577C2×C4—1881no—
578C2×C4—1881no—
579C2×C4—1881no—
580C2×C4—1881no—
581C2×C4—1881no—
582C2×C4—1881no—
583C2×C4#1309400112yes—
584C2×C4C22×Dic9411yescharacteristic
585C2×C4#41×D4711yescharacteristic
586C2×C4C2×#1062535211yescharacteristic
587C2×C4C2×#1062535211yescharacteristic
588C2×C4#1309406911yescharacteristic
589C2×C4#1309407011yescharacteristic
590C8—22no—
591C8—41no—
592C8—942no—
593C8—1881no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22#1568941614yes—
601C22C2×#1309400111yescharacteristic
602C22#1568966811yescharacteristic
603C22#1568968111yescharacteristic
604C22C2×Q16×D4711yescharacteristic
605C22#135×D4711yescharacteristic
606C22C2×#1309405111yescharacteristic
607C22#1568992011yescharacteristic
608C4—22no—
609C4—42no—
610C4—42no—
611C4—81no—
612C4—81no—
613C4—942no—
614C4—942no—
615C4—942no—
616C4—3761no—
617C4—3761no—
618C4—3761no—
619C4—3761no—
620C4—1881no—
621C4#1568979611yescharacteristic
622C4#1568980411yescharacteristic
623C4#1568983011yescharacteristic
624C4#1568983211yescharacteristic
625C2—24no—
626C2#2024857211yescharacteristic
627C2C2×#1568941611yescharacteristic
628C2#2025317211yescharacteristic
629C2#2025396911yescharacteristic
630C2#2025407911yescharacteristic
631C2#2025415411yescharacteristic
632C2#2025415911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#1892C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenter#1892D47
hypocenterC47#29160
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
3. #20244954C2semidirect
4. #20244924C2semidirect
5. #20244715C2semidirect
6. #20244527C2semidirect
7. #20243983C2semidirect
8. #20243979C2semidirect
10. #20243644C2semidirect
11. #20243640C2semidirect
115. C22⋊Q8×C47Q8semidirect
375. C47#29160semidirect
395. C22⋊Q8Dic94semidirect

Maximal subgroups

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

subgroupindex
#202517382
C22⋊Q8×Dic472
#202449542
#202449242
#202447152
#202445272
#202439832
#202439792
#202438122
#202436442
#202436402
#202434172
#202434152
C2×#156888772
C47×#18922
#2916047