Finite Group Databasebeta

← #26225830

Subgroups — #26225830

Order 12032.

970 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689750C211yesmaximal, characteristic
2C2×#15689185C211yesmaximal, characteristic
3#20245998C211yesmaximal, characteristic
4#20244357C211yesmaximal, characteristic, complemented
5#20244352C211yesmaximal, characteristic, complemented
6#20244350C211yesmaximal, characteristic, complemented
7#20244344C211yesmaximal, characteristic, complemented
8#20244342C211yesmaximal, characteristic, complemented
9#20244338C211yesmaximal, characteristic, complemented
10#20244334C211yesmaximal, characteristic, complemented
11#20243482C211yesmaximal, characteristic
12#20243481C211yesmaximal, characteristic
13#20243477C211yesmaximal, characteristic
14#20243473C211yesmaximal, characteristic
15C47×#1993C211yesmaximal, characteristic, complemented
16#15689750—24no—
17#15689750C418yescomplemented
18C2×#13094023C2211yescharacteristic
19C2×#13094022—21no—
20C2×#13094016C2211yescharacteristic
21C2×#13094015C2211yescharacteristic
22C2×#13094014—21no—
23C2×#13094002—21no—
24C2×#13094000—21no—
25C2×#13093982C2211yescharacteristic
26C2×#13093981C2211yescharacteristic
27C2×C4⋊C4×D47C2211yescharacteristic
28#15689185—24no—
29C2×Dic47⋊C8C2211yescharacteristic
30C2×Dic47⋊C8C2211yescharacteristic
31C2×(C2×D47)⋊C8C2211yescharacteristic
32C2×(C2×D47)⋊C8C2211yescharacteristic
33C2×C4×C47⋊C8C2211yescharacteristic
34#15689015—21no—
35#15689015—21no—
36#15689010—21no—
37#15689010—21no—
38#15688993C2211yescharacteristic
39#15688980C2211yescharacteristic
40#15688980C2211yescharacteristic
41C4⋊C4×Dic47C2211yescharacteristic
42C4⋊C4×Dic47C2211yescharacteristic
43#15688886C2211yescharacteristic
44#15688886C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53#15688885C2211yescharacteristic
54#15688885C2211yescharacteristic
55C2×C4⋊Q8×C47C2211yescharacteristic
56C2×#97×C47C2211yescharacteristic
57C47×#382C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C47×#340C2211yescharacteristic
61C47×#340C2211yescharacteristic
62C22×Dic94—21no—
63C22×C4×D47—21no—
64C22×C4×D47C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67#13094023—22no—
68#13094023C2×C414yes—
69#13094022—28no—
70#13094016—22no—
71#13094016C2×C414yes—
72#13094015—22no—
73#13094015C2×C414yes—
74#13094014—28no—
75#13094002—28no—
76#13094000—28no—
77#13093982—22no—
78#13093982C2×C414yes—
79#13093981C2×C414yes—
80#13093981D414yes—
81C2×C4⋊Dic47—21no—
82C2×C4⋊Dic47—21no—
83C2×C4⋊Dic47—21no—
84C2×C4⋊Dic47C2311yescharacteristic
85C2×C4⋊Dic47C2311yescharacteristic
86C2×C4⋊Dic47C2311yescharacteristic
87C2×C4⋊Dic47C2311yescharacteristic
88C2×Dic47⋊C4—21no—
89C2×Dic47⋊C4—21no—
90C2×Dic47⋊C4—21no—
91C2×Dic47⋊C4—21no—
92C2×Dic47⋊C4—21no—
93C2×Dic47⋊C4D411yescharacteristic
94C2×Dic47⋊C4D411yescharacteristic
95C4⋊C4×D47—22no—
96C4⋊C4×D47C2×C414yes—
97C2×(C2×D47)⋊C4—21no—
98C2×(C2×D47)⋊C4—21no—
99C2×(C2×D47)⋊C4—21no—
100C2×(C2×D47)⋊C4D411yescharacteristic
101C2×(C2×D47)⋊C4D411yescharacteristic
102C2×C4×Dic47—21no—
103C2×C4×Dic47—21no—
104C2×C4×Dic47—21no—
105C2×C4×Dic47C2311yescharacteristic
106Dic47⋊C8—22no—
107Dic47⋊C8—22no—
108(C2×D47)⋊C8—22no—
109(C2×D47)⋊C8—22no—
110C4×C47⋊C8—22no—
111C47⋊#95—21no—
112C47⋊#95—21no—
113C47⋊#95—21no—
114C47⋊#95—21no—
115C22×Q8×C47—21no—
116C22×C8×C47C2311yescharacteristic
117C22×C8×C47C2311yescharacteristic
118C4⋊Q8×C47—22no—
119C4⋊Q8×C47C2×C414yes—
120C2×C4⋊C4×C47—21no—
121C2×C4⋊C4×C47C2311yescharacteristic
122C2×C4⋊C4×C47C2311yescharacteristic
123C2×C4⋊C4×C47C2311yescharacteristic
124C2×C4⋊C4×C47C2311yescharacteristic
125C2×C42×C47C2311yescharacteristic
126#97×C47—22no—
127#95×C47—21no—
128#95×C47—21no—
129C23×D47C2×D411yescharacteristic
130C2×Q8×C47—24no—
131C2×Q8×C47—81no—
132C2×Q8×C47—81no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2411yescharacteristic
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47C2×D411yescharacteristic
145C22×Dic47C2×D411yescharacteristic
146C22×Dic47C2×D411yescharacteristic
147C2×C4×D47—21no—
148C2×C4×D47—21no—
149C2×C4×D47—21no—
150C2×C4×D47—21no—
151C2×C4×D47—24no—
152C2×C4×D47—81no—
153C2×C4×D47—81no—
154C2×C4×D47C22×C412yes—
155C2×C4×D47C2×D412yes—
156C2×Dic94—24no—
157C2×Dic94—81no—
158C2×Dic94—81no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47—21no—
164C2×C8×C47—21no—
165C4⋊C4×C47—24no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47C22×C412yes—
173C4⋊C4×C47C22×C412yes—
174C42×C47C22×C412yes—
175C42×C47C2×D412yes—
176(C2×D47)⋊C4—24no—
177(C2×D47)⋊C4—81no—
178(C2×D47)⋊C4—81no—
179(C2×D47)⋊C4C22⋊C414yes—
180(C2×D47)⋊C4C22⋊C414yes—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—24no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47—81no—
189C4⋊Dic47—81no—
190C4⋊Dic47C22×C412yes—
191C4⋊Dic47C22×C412yes—
192Dic47⋊C4—24no—
193Dic47⋊C4—24no—
194Dic47⋊C4—24no—
195Dic47⋊C4—24no—
196Dic47⋊C4—24no—
197Dic47⋊C4C22⋊C414yes—
198Dic47⋊C4C22⋊C414yes—
199C4×Dic47—24no—
200C4×Dic47—81no—
201C4×Dic47—81no—
202C4×Dic47C22×C412yes—
203C4×Dic47C2×D412yes—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8—21no—
210C23×C47C22×D411yescharacteristic
211C22×D47—21no—
212C22×D47—21no—
213C22×D47—21no—
214C22×D47—21no—
215C22×D47—21no—
216C22×D47—21no—
217C22×D47C2×C22⋊C412yes—
218Q8×C47—84no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
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×C47C2×C22⋊C412yes—
236C2×C4×C47C2×C22⋊C412yes—
237C2×C4×C47C23×C411yescharacteristic
238C2×C4×C47C22×D411yescharacteristic
239C2×C4×C472−1+411yescharacteristic
240C2×C4×C472−1+411yescharacteristic
241C2×C4×C472−1+411yescharacteristic
242C2×C4×C472−1+411yescharacteristic
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—21no—
249C2×Dic47—22no—
250C2×Dic47—22no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47C2×C22⋊C412yes—
262C2×Dic47C2×C22⋊C412yes—
263C2×Dic47C2×C22⋊C412yes—
264C4×D47—22no—
265C4×D47—22no—
266C4×D47—22no—
267C4×D47—22no—
268C4×D47—84no—
269Dic94—84no—
270C8×C47—41no—
271C8×C47—41no—
272C47⋊C8—41no—
273C47⋊C8—41no—
274#21843—471nomaximal, Sylow
275C22×C47C22×C22⋊C411yescharacteristic
276C22×C47#53611yescharacteristic
277C22×C47#53611yescharacteristic
278C22×C47#57611yescharacteristic
279C22×C47#57711yescharacteristic
280C22×C47#57711yescharacteristic
281C22×C47#57811yescharacteristic
282C2×D47—22no—
283C2×D47—22no—
284C2×D47—22no—
285C2×D47—41no—
286C2×D47—41no—
287C2×D47—41no—
288C2×D47—41no—
289C4×C47—22no—
290C4×C47—22no—
291C4×C47—42no—
292C4×C47—42no—
293C4×C47—81no—
294C4×C47—81no—
295C4×C47#51911yescharacteristic
296C4×C47#51911yescharacteristic
297C4×C47#53611yescharacteristic
298C4×C47#53611yescharacteristic
299Dic47—22no—
300Dic47—22no—
301Dic47—22no—
302Dic47—42no—
303Dic47—42no—
304Dic47—81no—
305Dic47—81no—
306C2×#544—471no—
307C2×#433—471no—
308#2314—471no—
309#1993D4711yescharacteristic, complemented
310#1992—471no—
311#1991—471no—
312#1990—471no—
313#1989—471no—
314#1989—471no—
315#1988—471no—
316#1987—471no—
317#1825—471no—
318#1825—471no—
319#1824—471no—
320#1823—471no—
321C2×C47#231011yescharacteristic
322C2×C47#289711yescharacteristic
323C2×C47#289811yescharacteristic
324C2×C47#324111yescharacteristic
325C2×C47#324211yescharacteristic
326C2×C47#324311yescharacteristic
327C2×C47#324411yescharacteristic
328D47—42no—
329#544—478no—
330#544—944no—
331C2×C4⋊Q8—471no—
332C2×C4⋊Q8—941no—
333C2×C4⋊Q8C2×D4711yescharacteristic
334C2×#125—471no—
335C2×#125—471no—
336C2×#125—941no—
337C2×C22⋊Q8—941no—
338C2×C22⋊Q8—941no—
339C2×#117—471no—
340C2×#117—471no—
341C22×C4⋊C4—471no—
342#433—944no—
343C2×C4⋊C8—471no—
344C2×C4⋊C8—471no—
345C2×C22⋊C8—471no—
346C2×C22⋊C8—471no—
347C2×#97C2×D4711yescharacteristic
348C2×C4×C8—471no—
349#388—941no—
350#388—941no—
351#387—941no—
352#387—941no—
353#384—471no—
354#382C2×D4711yescharacteristic
355#380—471no—
356#380—471no—
357C4×C4⋊C4—471no—
358C4×C4⋊C4—471no—
359#340—471no—
360#340—471no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340—471no—
365#340—471no—
366#340—471no—
367#340—471no—
368#340—471no—
369#340—471no—
370#340—471no—
371#340C2×D4711yescharacteristic
372#340C2×D4711yescharacteristic
373#340C2×D4711yescharacteristic
374#340C2×D4711yescharacteristic
375C47#2184311yescharacteristic, complemented, Sylow
376C22×Q8—21no—
377C22×Q8—941no—
378C23×C4—471no—
379C23×C4—941no—
380C22×C8—471no—
381C22×C8—471no—
382C22×C8C22×D4711yescharacteristic
383C22×C8C22×D4711yescharacteristic
384C4⋊Q8—22no—
385C4⋊Q8—474no—
386C4⋊Q8—942no—
387C4⋊Q8—948no—
388C4⋊Q8C4×D4714yes—
389#125—474no—
390#125—474no—
391#125—942no—
392#125—948no—
393#125—942no—
394C22⋊Q8—948no—
395C22⋊Q8—948no—
396#117—474no—
397#117—474no—
398#117—474no—
399#117—942no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—471no—
402C2×C4⋊C4—471no—
403C2×C4⋊C4—471no—
404C2×C4⋊C4—471no—
405C2×C4⋊C4—474no—
406C2×C4⋊C4—471no—
407C2×C4⋊C4—471no—
408C2×C4⋊C4—942no—
409C2×C4⋊C4—941no—
410C2×C4⋊C4—941no—
411C2×C4⋊C4—941no—
412C2×C4⋊C4—941no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4—941no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4—941no—
417C2×C4⋊C4C22×D4711yescharacteristic
418C2×C4⋊C4C22×D4711yescharacteristic
419C2×C4⋊C4C22×D4711yescharacteristic
420C2×C4⋊C4C22×D4711yescharacteristic
421C2×C22⋊C4—471no—
422C2×C22⋊C4—471no—
423C2×C22⋊C4—941no—
424C2×C22⋊C4—941no—
425C2×C22⋊C4—941no—
426C2×C42—471no—
427C2×C42—941no—
428C2×C42—941no—
429C2×C42—941no—
430C2×C42C22×D4711yescharacteristic
431C4⋊C8—942no—
432C4⋊C8—942no—
433C22⋊C8—942no—
434C22⋊C8—942no—
435#97—22no—
436C4×C8—942no—
437#95—21no—
438#95—21no—
439#95—941no—
440#95—941no—
441#95—941no—
442#95—941no—
443C24—471no—
444C2×Q8—24no—
445C2×Q8—81no—
446C2×Q8—81no—
447C2×Q8—944no—
448C2×Q8—3761no—
449C2×Q8—3761no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—471no—
455C22×C4—471no—
456C22×C4—471no—
457C22×C4—472no—
458C22×C4—472no—
459C22×C4—944no—
460C22×C4—941no—
461C22×C4—941no—
462C22×C4—941no—
463C22×C4—941no—
464C22×C4—941no—
465C22×C4—941no—
466C22×C4—941no—
467C22×C4—941no—
468C22×C4—3761no—
469C22×C4—3761no—
470C22×C4D4×D4711yescharacteristic
471C22×C4D4×D4711yescharacteristic
472C22×C4C23×D4711yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—941no—
480C2×C8—941no—
481C2×C8—941no—
482C2×C8—941no—
483C2×C8—941no—
484C2×C8—941no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—474no—
493C4⋊C4—474no—
494C4⋊C4—472no—
495C4⋊C4—472no—
496C4⋊C4—944no—
497C4⋊C4—944no—
498C4⋊C4—944no—
499C4⋊C4—944no—
500C4⋊C4—944no—
501C4⋊C4—944no—
502C4⋊C4—941no—
503C4⋊C4—941no—
504C4⋊C4—941no—
505C4⋊C4—941no—
506C4⋊C4—941no—
507C4⋊C4—941no—
508C4⋊C4—3761no—
509C4⋊C4—3761no—
510C4⋊C4C2×C4×D4712yes—
511C4⋊C4C2×C4×D4712yes—
512C22⋊C4—474no—
513C22⋊C4—474no—
514C22⋊C4—944no—
515C22⋊C4—3761no—
516C22⋊C4—3761no—
517C42—472no—
518C42—472no—
519C42—944no—
520C42—3761no—
521C42—3761no—
522C42C2×C4×D4712yes—
523C42D4×D4712yes—
524C23—472no—
525C23—941no—
526C23—941no—
527C23—941no—
528C23—941no—
529C23—941no—
530C23—941no—
531C23C2×D4×D4711yescharacteristic
532Q8—84no—
533Q8—3764no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—472no—
551C2×C4—472no—
552C2×C4—472no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—941no—
556C2×C4—941no—
557C2×C4—941no—
558C2×C4—941no—
559C2×C4—942no—
560C2×C4—942no—
561C2×C4—942no—
562C2×C4—942no—
563C2×C4—942no—
564C2×C4—942no—
565C2×C4—3764no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4—1881no—
571C2×C4—1881no—
572C2×C4—1881no—
573C2×C4—1881no—
574C2×C4—1881no—
575C2×C4—1881no—
576C2×C4C22⋊C4×D4712yes—
577C2×C4C22⋊C4×D4712yes—
578C2×C4C22×C4×D4711yescharacteristic
579C2×C4C2×D4×D4711yescharacteristic
580C2×C4#1309407111yescharacteristic
581C2×C4#1309407111yescharacteristic
582C2×C4#1309407211yescharacteristic
583C2×C4#1309407211yescharacteristic
584C8—41no—
585C8—41no—
586C8—1881no—
587C8—1881no—
588C22—942no—
589C22—942no—
590C22—942no—
591C22—1881no—
592C22—1881no—
593C22—1881no—
594C22—1881no—
595C22C2×C22⋊C4×D4711yescharacteristic
596C22#1568969311yescharacteristic
597C22#1568969811yescharacteristic
598C22#1568992711yescharacteristic
599C22#1568993211yescharacteristic
600C22#1568993311yescharacteristic
601C22#1568993911yescharacteristic
602C4—22no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—81no—
607C4—81no—
608C4—942no—
609C4—942no—
610C4—942no—
611C4—3761no—
612C4—3761no—
613C4—1882no—
614C4—1882no—
615C4#1568959911yescharacteristic
616C4#1568959911yescharacteristic
617C4#1568969611yescharacteristic
618C4#1568969611yescharacteristic
619C2—1882no—
620C2#2024597111yescharacteristic
621C2#2025191311yescharacteristic
622C2#2025192011yescharacteristic
623C2#2025464211yescharacteristic
624C2#2025465711yescharacteristic
625C2#2025466611yescharacteristic
626C2#2025468111yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1993C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1993D47
hypocenterC47#21843
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
4. #20244357C2semidirect
5. #20244352C2semidirect
6. #20244350C2semidirect
7. #20244344C2semidirect
8. #20244342C2semidirect
9. #20244338C2semidirect
10. #20244334C2semidirect
15. C47×#1993C2semidirect
17. #15689750C4semidirect
309. #1993D47semidirect
375. C47#21843semidirect

Maximal subgroups

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

subgroupindex
C2×#156897502
C2×#156891852
#202459982
#202443572
#202443522
#202443502
#202443442
#202443422
#202443382
#202443342
#202434822
#202434812
#202434772
#202434732
C47×#19932
#2184347