Finite Group Databasebeta

← #26221560

Subgroups — #26221560

Order 12032.

1070 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689731C211yesmaximal, characteristic
2C2×#15689080C211yesmaximal, characteristic
3#20245993C211yesmaximal, characteristic
4#20244355C211yesmaximal, characteristic, complemented
5#20244351C211yesmaximal, characteristic, complemented
6#20244206C211yesmaximal, characteristic, complemented
7#20244188C211yesmaximal, characteristic, complemented
8#20243820C211yesmaximal, characteristic
9#20243818C211yesmaximal, characteristic
10#20243810C211yesmaximal, characteristic
11#20243808C211yesmaximal, characteristic
12#20243219C211yesmaximal, characteristic, complemented
13C4×#13093947C211yesmaximal, characteristic, complemented
14C4×#13093948C211yesmaximal, characteristic, complemented
15C47×#1763C211yesmaximal, characteristic, complemented
16C22×#10625324C2211yescharacteristic
17C22×C8×D47C2211yescharacteristic
18#15689731—24no—
19#15689731C418yescomplemented
20C2×#13094022C2211yescharacteristic
21C2×#13094019—21no—
22C2×#13094018—21no—
23C2×#13094016—21no—
24C2×#13094015C2211yescharacteristic
25C2×#13094014—21no—
26C2×#13094004C2211yescharacteristic
27C2×#13094000C2211yescharacteristic
28C2×C4×Dic94C2211yescharacteristic
29C2×#13093981C2211yescharacteristic
30C2×#13093949C2211yescharacteristic
31C2×#13093949C2211yescharacteristic
32C2×#13093948C2211yescharacteristic
33C2×#13093948C2211yescharacteristic
34C2×#13093947C2211yescharacteristic
35C2×#13093947C2211yescharacteristic
36#15689080—24no—
37C2×#13093934C2211yescharacteristic
38C2×C4×C47⋊C8C2211yescharacteristic
39C2×C8×Dic47C2211yescharacteristic
40#15689014—21no—
41#15689012—21no—
42#15689007—21no—
43#15689005—21no—
44#15688992C2211yescharacteristic
45#15688988C2211yescharacteristic
46#15688980C2211yescharacteristic
47#15688980C2211yescharacteristic
48C4×C4⋊Dic47C2211yescharacteristic
49#15688886C2211yescharacteristic
50#15688886C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53#15688885C2211yescharacteristic
54#15688885C2211yescharacteristic
55C2×C4×Q8×C47C2211yescharacteristic
56C2×#103×C47C2211yescharacteristic
57C2×#103×C47C2211yescharacteristic
58C2×#97×C47C2211yescharacteristic
59C4×C4⋊C4×C47C2211yescharacteristic
60C47×#340C2211yescharacteristic
61C47×#340C2211yescharacteristic
62C22×Dic94C2311yescharacteristic
63C22×C4×D47C2311yescharacteristic
64C2×#10625324—22no—
65C2×#10625324—22no—
66C2×#10625324—22no—
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C2×C8×D47—22no—
70C2×C8×D47—22no—
71C2×C8×D47—22no—
72#13094022—22no—
73#13094022C2×C414yes—
74#13094019—28no—
75#13094018—28no—
76#13094016—28no—
77#13094015—22no—
78#13094015C2×C414yes—
79#13094014—28no—
80#13094004—22no—
81#13094004C2×C414yes—
82#13094000—22no—
83#13094000C2×C414yes—
84C4×Dic94—22no—
85C4×Dic94C2×C414yes—
86#13093981C2×C414yes—
87#13093981D414yes—
88C2×C4⋊Dic47—21no—
89C2×C4⋊Dic47—21no—
90C2×C4⋊Dic47C2311yescharacteristic
91C2×C4⋊Dic47C2311yescharacteristic
92C2×C4⋊Dic47C2311yescharacteristic
93C2×Dic47⋊C4—21no—
94C2×Dic47⋊C4—21no—
95C2×Dic47⋊C4—21no—
96C2×Dic47⋊C4—21no—
97C2×Dic47⋊C4—21no—
98C2×Dic47⋊C4—21no—
99C2×Dic47⋊C4—21no—
100C2×Dic47⋊C4D411yescharacteristic
101C2×Dic47⋊C4D411yescharacteristic
102C2×(C2×D47)⋊C4—21no—
103C2×(C2×D47)⋊C4—21no—
104C2×(C2×D47)⋊C4—21no—
105C2×(C2×D47)⋊C4—21no—
106C2×(C2×D47)⋊C4D411yescharacteristic
107C2×(C2×D47)⋊C4D411yescharacteristic
108C2×C4×Dic47—21no—
109C2×C4×Dic47—21no—
110C2×C4×Dic47C2311yescharacteristic
111#13093949—22no—
112#13093949—22no—
113#13093948—22no—
114#13093948—22no—
115#13093947—22no—
116#13093947—22no—
117#13093934—22no—
118C4×C47⋊C8—22no—
119C8×Dic47—22no—
120C47⋊#95—21no—
121C47⋊#95—21no—
122C47⋊#95—21no—
123C47⋊#95—21no—
124C47⋊#95—21no—
125C22×Q8×C47C2311yescharacteristic
126C22×C8×C47C2311yescharacteristic
127C22×C8×C47C2311yescharacteristic
128C4×Q8×C47—22no—
129C4×Q8×C47C2×C414yes—
130C2×C4⋊C4×C47—21no—
131C2×C4⋊C4×C47C2311yescharacteristic
132C2×C4⋊C4×C47C2311yescharacteristic
133C2×C4⋊C4×C47C2311yescharacteristic
134C2×C42×C47—21no—
135C2×C42×C47—21no—
136C2×C42×C47C2311yescharacteristic
137#103×C47—22no—
138#103×C47—22no—
139#97×C47—22no—
140#95×C47—21no—
141C23×D47C2×D411yescharacteristic
142C2×Q8×C47—21no—
143C2×Q8×C47—21no—
144C2×Q8×C47—21no—
145C2×Q8×C47—21no—
146C2×Q8×C47—21no—
147C2×Q8×C47C22×C412yes—
148C22×C4×C47—21no—
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C22×C4×C47—21no—
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2411yescharacteristic
155C22×Dic47—21no—
156C22×Dic47—21no—
157C22×Dic47—21no—
158C22×Dic47—21no—
159C22×Dic47C2×D411yescharacteristic
160C22×Dic47C2×D411yescharacteristic
161C22×Dic47C2×D411yescharacteristic
162C2×C4×D47—21no—
163C2×C4×D47—21no—
164C2×C4×D47—21no—
165C2×C4×D47—21no—
166C2×C4×D47C22×C412yes—
167C2×C4×D47C2×D412yes—
168C2×Dic94—21no—
169C2×Dic94—21no—
170C2×Dic94—21no—
171C2×Dic94—21no—
172C2×Dic94—21no—
173C2×Dic94C22×C412yes—
174C2×C8×C47—21no—
175C2×C8×C47—21no—
176C2×C8×C47—21no—
177C2×C8×C47—21no—
178C2×C8×C47#4112yes—
179C2×C8×C47#4112yes—
180C4⋊C4×C47—24no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47—21no—
184C4⋊C4×C47—21no—
185C4⋊C4×C47—21no—
186C4⋊C4×C47C22×C412yes—
187C42×C47—24no—
188C42×C47—81no—
189C42×C47C22×C412yes—
190C42×C47C2×D412yes—
191(C2×D47)⋊C4—24no—
192(C2×D47)⋊C4—24no—
193(C2×D47)⋊C4—81no—
194(C2×D47)⋊C4—81no—
195(C2×D47)⋊C4C22⋊C414yes—
196(C2×D47)⋊C4C22⋊C414yes—
197C4⋊Dic47—21no—
198C4⋊Dic47—24no—
199C4⋊Dic47—24no—
200C4⋊Dic47—21no—
201C4⋊Dic47—21no—
202C4⋊Dic47—21no—
203C4⋊Dic47—21no—
204C4⋊Dic47C22×C412yes—
205Dic47⋊C4—24no—
206Dic47⋊C4—24no—
207Dic47⋊C4—24no—
208Dic47⋊C4—24no—
209Dic47⋊C4—24no—
210Dic47⋊C4—81no—
211Dic47⋊C4—81no—
212Dic47⋊C4C22⋊C414yes—
213Dic47⋊C4C22⋊C414yes—
214C4×Dic47—24no—
215C4×Dic47—81no—
216C4×Dic47C22×C412yes—
217C4×Dic47C2×D412yes—
218C2×C47⋊C8—21no—
219C2×C47⋊C8—21no—
220C2×C47⋊C8—21no—
221C2×C47⋊C8—21no—
222C2×C47⋊C8#4112yes—
223C2×C47⋊C8#4112yes—
224#10625324—28no—
225C8×D47—28no—
226C23×C47C22×D411yescharacteristic
227C22×D47—21no—
228C22×D47—21no—
229C22×D47—21no—
230C22×D47—21no—
231C22×D47—21no—
232C22×D47—21no—
233C22×D47C2×C22⋊C412yes—
234Q8×C47—22no—
235Q8×C47—22no—
236Q8×C47—41no—
237Q8×C47—41no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—22no—
243C2×C4×C47—22no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47C2×C22⋊C412yes—
255C2×C4×C47C2×C22⋊C412yes—
256C2×C4×C47C23×C411yescharacteristic
257C2×C4×C47C22×D411yescharacteristic
258C2×C4×C47C2×#4111yescharacteristic
259C2×C4×C47C2×#4111yescharacteristic
260C2×C4×C47C2×#4111yescharacteristic
261C2×C4×C47C2×#4111yescharacteristic
262C2×Dic47—21no—
263C2×Dic47—21no—
264C2×Dic47—21no—
265C2×Dic47—21no—
266C2×Dic47—21no—
267C2×Dic47—21no—
268C2×Dic47—22no—
269C2×Dic47—22no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280C2×Dic47C2×C22⋊C412yes—
281C2×Dic47C2×C22⋊C412yes—
282C2×Dic47C2×C22⋊C412yes—
283C4×D47—22no—
284C4×D47—22no—
285C4×D47—22no—
286C4×D47—22no—
287Dic94—22no—
288Dic94—22no—
289Dic94—41no—
290Dic94—41no—
291C8×C47—22no—
292C8×C47—22no—
293C47⋊C8—22no—
294C47⋊C8—22no—
295#21413—471nomaximal, Sylow
296C22×C47C22×C22⋊C411yescharacteristic
297C22×C47#53211yescharacteristic
298C22×C47#53211yescharacteristic
299C22×C47#57711yescharacteristic
300C22×C47#57711yescharacteristic
301C22×C47#57811yescharacteristic
302C22×C47#57811yescharacteristic
303C2×D47—22no—
304C2×D47—22no—
305C2×D47—22no—
306C2×D47—41no—
307C2×D47—41no—
308C2×D47—41no—
309C2×D47—41no—
310C4×C47—22no—
311C4×C47—22no—
312C4×C47—42no—
313C4×C47—42no—
314C4×C47—81no—
315C4×C47—81no—
316C4×C47C4×#4111yescharacteristic
317C4×C47C4×#4111yescharacteristic
318C4×C47#52511yescharacteristic
319C4×C47#52511yescharacteristic
320Dic47—22no—
321Dic47—22no—
322Dic47—22no—
323Dic47—42no—
324Dic47—42no—
325Dic47—81no—
326Dic47—81no—
327C2×#541—471no—
328C2×#405—471no—
329#2314—471no—
330#1992—471no—
331#1991—471no—
332#1960—471no—
333#1956—471no—
334#1894—471no—
335#1894—471no—
336#1892—471no—
337#1892—471no—
338#1763—471no—
339#1763D4711yescharacteristic, complemented
340C4×#103—471no—
341C4×#103—471no—
342C2×C47#229711yescharacteristic
343C2×C47#289811yescharacteristic
344C2×C47#289811yescharacteristic
345C2×C47#317411yescharacteristic
346C2×C47#317711yescharacteristic
347C2×C47#317811yescharacteristic
348C2×C47#317811yescharacteristic
349D47—42no—
350C22×M16—471no—
351C23×C8—471no—
352#541—478no—
353#541—944no—
354C2×C4⋊Q8—471no—
355C2×#126—941no—
356C2×#126—941no—
357C2×#125—471no—
358C2×#125—941no—
359C2×#125—941no—
360C2×C22⋊Q8—471no—
361C2×C22⋊Q8—471no—
362C2×C4×Q8—471no—
363C2×C4×Q8C2×D4711yescharacteristic
364C2×#117—471no—
365C2×#103—471no—
366C2×#103—471no—
367C2×#103—471no—
368C2×#103—471no—
369C2×#103—471no—
370C2×#103—471no—
371C2×#103C2×D4711yescharacteristic
372C2×#103C2×D4711yescharacteristic
373#405—944no—
374C2×#97—471no—
375C2×#97C2×D4711yescharacteristic
376C2×C4×C8—471no—
377C2×C4×C8—471no—
378#388—941no—
379#388—941no—
380#387—941no—
381#387—941no—
382#384—471no—
383#382—471no—
384#380—471no—
385#380—471no—
386C4×C4⋊C4—471no—
387C4×C4⋊C4C2×D4711yescharacteristic
388#340—471no—
389#340—471no—
390#340—471no—
391#340—471no—
392#340—471no—
393#340—471no—
394#340C2×D4711yescharacteristic
395#340C2×D4711yescharacteristic
396C47#2141311yescharacteristic, complemented, Sylow
397C22×Q8—471no—
398C22×Q8C22×D4711yescharacteristic
399C23×C4—471no—
400C2×M16—942no—
401C2×M16—942no—
402C2×M16—942no—
403C22×C8—471no—
404C22×C8—471no—
405C22×C8—942no—
406C22×C8—942no—
407C22×C8—942no—
408C22×C8C22×D4711yescharacteristic
409C22×C8C22×D4711yescharacteristic
410C4⋊Q8—474no—
411C4⋊Q8—942no—
412#126—948no—
413#126—948no—
414#125—474no—
415#125—948no—
416#125—942no—
417#125—948no—
418C22⋊Q8—474no—
419C22⋊Q8—474no—
420C22⋊Q8—942no—
421C22⋊Q8—942no—
422C4×Q8—22no—
423C4×Q8—474no—
424C4×Q8—942no—
425C4×Q8C4×D4714yes—
426#117—474no—
427#117—474no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—471no—
430C2×C4⋊C4—471no—
431C2×C4⋊C4—471no—
432C2×C4⋊C4—471no—
433C2×C4⋊C4—471no—
434C2×C4⋊C4—941no—
435C2×C4⋊C4—941no—
436C2×C4⋊C4—941no—
437C2×C4⋊C4—941no—
438C2×C4⋊C4—941no—
439C2×C4⋊C4—941no—
440C2×C4⋊C4—941no—
441C2×C4⋊C4—941no—
442C2×C4⋊C4—941no—
443C2×C4⋊C4C22×D4711yescharacteristic
444C2×C4⋊C4C22×D4711yescharacteristic
445C2×C4⋊C4C22×D4711yescharacteristic
446C2×C22⋊C4—471no—
447C2×C22⋊C4—471no—
448C2×C22⋊C4—941no—
449C2×C22⋊C4—941no—
450C2×C22⋊C4—941no—
451C2×C22⋊C4—941no—
452C2×C42—21no—
453C2×C42—21no—
454C2×C42—471no—
455C2×C42—941no—
456C2×C42—941no—
457C2×C42C22×D4711yescharacteristic
458#103—22no—
459#103—22no—
460#103—942no—
461#103—942no—
462#103—942no—
463#103—942no—
464#103—942no—
465#103—942no—
466#97—22no—
467#97—942no—
468C4×C8—942no—
469C4×C8—942no—
470#95—21no—
471#95—941no—
472#95—941no—
473#95—941no—
474#95—941no—
475#95—941no—
476C24—471no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—472no—
483C2×Q8—941no—
484C2×Q8—941no—
485C2×Q8—941no—
486C2×Q8—941no—
487C2×Q8—941no—
488C2×Q8C2×C4×D4712yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—471no—
494C22×C4—471no—
495C22×C4—471no—
496C22×C4—472no—
497C22×C4—472no—
498C22×C4—941no—
499C22×C4—941no—
500C22×C4—941no—
501C22×C4—941no—
502C22×C4—941no—
503C22×C4—941no—
504C22×C4—941no—
505C22×C4—941no—
506C22×C4D4×D4711yescharacteristic
507C22×C4D4×D4711yescharacteristic
508C22×C4C23×D4711yescharacteristic
509M16—948no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—472no—
515C2×C8—472no—
516C2×C8—948no—
517C2×C8—941no—
518C2×C8—941no—
519C2×C8—941no—
520C2×C8—941no—
521C2×C8#1062535012yes—
522C2×C8#1062535012yes—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—474no—
530C4⋊C4—474no—
531C4⋊C4—472no—
532C4⋊C4—944no—
533C4⋊C4—944no—
534C4⋊C4—944no—
535C4⋊C4—944no—
536C4⋊C4—944no—
537C4⋊C4—944no—
538C4⋊C4—944no—
539C4⋊C4—941no—
540C4⋊C4—941no—
541C4⋊C4—941no—
542C4⋊C4—941no—
543C4⋊C4—941no—
544C4⋊C4—3761no—
545C4⋊C4—3761no—
546C4⋊C4C2×C4×D4712yes—
547C22⋊C4—474no—
548C22⋊C4—474no—
549C22⋊C4—944no—
550C22⋊C4—944no—
551C22⋊C4—3761no—
552C22⋊C4—3761no—
553C42—24no—
554C42—81no—
555C42—472no—
556C42—472no—
557C42—944no—
558C42—3761no—
559C42C2×C4×D4712yes—
560C42D4×D4712yes—
561C23—472no—
562C23—941no—
563C23—941no—
564C23—941no—
565C23—941no—
566C23—941no—
567C23—941no—
568C23C2×D4×D4711yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573Q8—942no—
574Q8—942no—
575Q8—1881no—
576Q8—1881no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—472no—
594C2×C4—472no—
595C2×C4—472no—
596C2×C4—941no—
597C2×C4—941no—
598C2×C4—941no—
599C2×C4—941no—
600C2×C4—941no—
601C2×C4—941no—
602C2×C4—942no—
603C2×C4—942no—
604C2×C4—942no—
605C2×C4—942no—
606C2×C4—942no—
607C2×C4—942no—
608C2×C4—1881no—
609C2×C4—1881no—
610C2×C4—1881no—
611C2×C4—1881no—
612C2×C4—1881no—
613C2×C4—1881no—
614C2×C4—1881no—
615C2×C4—1881no—
616C2×C4—1881no—
617C2×C4—1881no—
618C2×C4C22⋊C4×D4712yes—
619C2×C4C22⋊C4×D4712yes—
620C2×C4C22×C4×D4711yescharacteristic
621C2×C4C2×D4×D4711yescharacteristic
622C2×C4#41×D4711yescharacteristic
623C2×C4#41×D4711yescharacteristic
624C2×C4C2×#1062535011yescharacteristic
625C2×C4C2×#1062535011yescharacteristic
626C8—22no—
627C8—22no—
628C8—942no—
629C8—942no—
630C22—942no—
631C22—942no—
632C22—942no—
633C22—1881no—
634C22—1881no—
635C22—1881no—
636C22—1881no—
637C22C2×C22⋊C4×D4711yescharacteristic
638C22#1568967011yescharacteristic
639C22#1568967011yescharacteristic
640C22#1568993211yescharacteristic
641C22#1568993311yescharacteristic
642C22#1568993911yescharacteristic
643C22#1568993911yescharacteristic
644C4—22no—
645C4—22no—
646C4—42no—
647C4—42no—
648C4—81no—
649C4—81no—
650C4—942no—
651C4—942no—
652C4—942no—
653C4—3761no—
654C4—3761no—
655C4—1882no—
656C4—1882no—
657C4#1568959111yescharacteristic
658C4C4×#1062535011yescharacteristic
659C4#1568963211yescharacteristic
660C4#1568963211yescharacteristic
661C2—1882no—
662C2#2024587411yescharacteristic
663C2#2025191911yescharacteristic
664C2#2025192011yescharacteristic
665C2#2025401411yescharacteristic
666C2#2025404511yescharacteristic
667C2#2025405411yescharacteristic
668C2#2025406211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1763C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1763D47
hypocenterC47#21413
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
4. #20244355C2semidirect
5. #20244351C2semidirect
6. #20244206C2semidirect
7. #20244188C2semidirect
12. #20243219C2semidirect
13. C4×#13093947C2semidirect
14. C4×#13093948C2semidirect
15. C47×#1763C2semidirect
19. #15689731C4semidirect
339. #1763D47semidirect
396. C47#21413semidirect

Maximal subgroups

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

subgroupindex
C2×#156897312
C2×#156890802
#202459932
#202443552
#202443512
#202442062
#202441882
#202438202
#202438182
#202438102
#202438082
#202432192
C4×#130939472
C4×#130939482
C47×#17632
#2141347