Finite Group Databasebeta

← #26221555

Subgroups — #26221555

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×#15689733C211yesmaximal, characteristic
2C2×#15689078C211yesmaximal, characteristic
3#20245995C211yesmaximal, characteristic
4#20244355C211yesmaximal, characteristic, complemented
5#20244351C211yesmaximal, characteristic, complemented
6#20244206C211yesmaximal, characteristic, complemented
7#20244188C211yesmaximal, characteristic, complemented
8#20243822C211yesmaximal, characteristic
9#20243819C211yesmaximal, characteristic
10#20243811C211yesmaximal, characteristic
11#20243220C211yesmaximal, characteristic, complemented
12#20243217C211yesmaximal, characteristic, complemented
13#103×Dic47C211yesmaximal, characteristic, complemented
14#20243207C211yesmaximal, characteristic, complemented
15C47×#1892C211yesmaximal, characteristic
16C22×#10625329C2211yescharacteristic
17#15689733—24no—
18#15689733C418yescomplemented
19C2×#13094022C2211yescharacteristic
20C2×#13094017—21no—
21C2×#13094017—21no—
22C2×#13094015—21no—
23C2×#13094015C2211yescharacteristic
24C2×#13094014—21no—
25C2×#13094003C2211yescharacteristic
26C2×Q8×Dic47C2211yescharacteristic
27C2×#13093989C2211yescharacteristic
28C2×#13093983C2211yescharacteristic
29C2×#13093949C2211yescharacteristic
30C2×#13093949C2211yescharacteristic
31C2×#13093948C2211yescharacteristic
32C2×#13093948C2211yescharacteristic
33C2×#13093947C2211yescharacteristic
34C2×#13093947C2211yescharacteristic
35#15689078—24no—
36C2×#13093934C2211yescharacteristic
37C2×#13093934C2211yescharacteristic
38C2×C8×Dic47C2211yescharacteristic
39C2×C8×Dic47C2211yescharacteristic
40#15689017—21no—
41#15689011—21no—
42#15689008—21no—
43#15689006—21no—
44#15688992C2211yescharacteristic
45#15688988C2211yescharacteristic
46#15688983C2211yescharacteristic
47C4⋊C4×Dic47C2211yescharacteristic
48C4⋊C4×Dic47C2211yescharacteristic
49#15688886C2211yescharacteristic
50#15688886C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53#15688885C2211yescharacteristic
54#15688885C2211yescharacteristic
55C23×C8×C47C2211yescharacteristic
56C2×C22⋊Q8×C47C2211yescharacteristic
57C2×#103×C47C2211yescharacteristic
58C2×#103×C47C2211yescharacteristic
59C47×#380C2211yescharacteristic
60C47×#340C2211yescharacteristic
61C47×#340C2211yescharacteristic
62C22×Dic94C2311yescharacteristic
63C2×#10625329—22no—
64C2×#10625329—22no—
65C2×#10625329—22no—
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68#13094022—22no—
69#13094022C2×C414yes—
70#13094017—28no—
71#13094017—28no—
72#13094015—28no—
73#13094015—22no—
74#13094015C2×C414yes—
75#13094014—28no—
76#13094003—22no—
77#13094003C2×C414yes—
78Q8×Dic47—22no—
79Q8×Dic47C2×C414yes—
80#13093989—22no—
81#13093989C2×C414yes—
82#13093983C2×C414yes—
83#13093983D414yes—
84C2×C4⋊Dic47—21no—
85C2×C4⋊Dic47—21no—
86C2×C4⋊Dic47D411yescharacteristic
87C2×C4⋊Dic47D411yescharacteristic
88C2×C4⋊Dic47C2311yescharacteristic
89C2×C4⋊Dic47C2311yescharacteristic
90C2×C4⋊Dic47C2311yescharacteristic
91C2×Dic47⋊C4—21no—
92C2×Dic47⋊C4—21no—
93C2×Dic47⋊C4—21no—
94C2×Dic47⋊C4—21no—
95C2×Dic47⋊C4—21no—
96C2×Dic47⋊C4—21no—
97C2×Dic47⋊C4—21no—
98C2×C22⋊Dic47—21no—
99C2×C22⋊Dic47—21no—
100C2×C22⋊Dic47D411yescharacteristic
101C2×C22⋊Dic47D411yescharacteristic
102C2×C4×Dic47—21no—
103C2×C4×Dic47—21no—
104C2×C4×Dic47—21no—
105C2×C4×Dic47—21no—
106C2×C4×Dic47C2311yescharacteristic
107C2×C4×Dic47C2311yescharacteristic
108#13093949—22no—
109#13093949—22no—
110#13093948—22no—
111#13093948—22no—
112#13093947—22no—
113#13093947—22no—
114#13093934—22no—
115#13093934—22no—
116C8×Dic47—22no—
117C8×Dic47—22no—
118C47⋊#95—21no—
119C47⋊#95—21no—
120C47⋊#95—21no—
121C47⋊#95—21no—
122C47⋊#95—21no—
123C22×Q8×C47C2311yescharacteristic
124C23×C4×C47C2311yescharacteristic
125C22×C8×C47—22no—
126C22×C8×C47—22no—
127C22×C8×C47—22no—
128C22×C8×C47C2311yescharacteristic
129C22×C8×C47C2311yescharacteristic
130C22⋊Q8×C47—22no—
131C22⋊Q8×C47C2×C414yes—
132C2×C4⋊C4×C47—21no—
133C2×C4⋊C4×C47C2311yescharacteristic
134C2×C4⋊C4×C47C2311yescharacteristic
135C2×C4⋊C4×C47C2311yescharacteristic
136C2×C22⋊C4×C47—21no—
137C2×C22⋊C4×C47—21no—
138#103×C47—22no—
139#103×C47—22no—
140#95×C47—21no—
141C24×C47C2×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×C47—21no—
153C22×C4×C47—21no—
154C22×C4×C47—21no—
155C22×C4×C47—21no—
156C22×C4×C47C22×C412yes—
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2×D412yes—
159C22×C4×C47C2411yescharacteristic
160C22×Dic47—21no—
161C22×Dic47—21no—
162C22×Dic47—21no—
163C22×Dic47—21no—
164C22×Dic47C2×D411yescharacteristic
165C22×Dic47C2×D411yescharacteristic
166C22×Dic47C2×D411yescharacteristic
167C22×Dic47C2×D411yescharacteristic
168C2×Dic94—21no—
169C2×Dic94—21no—
170C2×Dic94—21no—
171C2×Dic94—21no—
172C2×Dic94—21no—
173C2×Dic94C22×C412yes—
174C2×C8×C47—28no—
175C2×C8×C47—21no—
176C2×C8×C47—21no—
177C2×C8×C47—21no—
178C2×C8×C47—21no—
179C2×C8×C47#4112yes—
180C2×C8×C47#4112yes—
181C4⋊C4×C47—24no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47—21no—
184C4⋊C4×C47—21no—
185C4⋊C4×C47—21no—
186C4⋊C4×C47—21no—
187C4⋊C4×C47C22×C412yes—
188C22⋊C4×C47—24no—
189C22⋊C4×C47—81no—
190C22⋊Dic47—24no—
191C22⋊Dic47—81no—
192C22⋊Dic47C22⋊C414yes—
193C22⋊Dic47C22⋊C414yes—
194C4⋊Dic47—21no—
195C4⋊Dic47—24no—
196C4⋊Dic47—24no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47—21no—
200C4⋊Dic47—21no—
201C4⋊Dic47C22⋊C414yes—
202C4⋊Dic47C22⋊C414yes—
203C4⋊Dic47C22×C412yes—
204Dic47⋊C4—24no—
205Dic47⋊C4—24no—
206Dic47⋊C4—24no—
207Dic47⋊C4—24no—
208Dic47⋊C4—24no—
209Dic47⋊C4—81no—
210Dic47⋊C4—81no—
211C4×Dic47—24no—
212C4×Dic47—24no—
213C4×Dic47—81no—
214C4×Dic47—81no—
215C4×Dic47C22×C412yes—
216C4×Dic47C22×C412yes—
217C4×Dic47C2×D412yes—
218C4×Dic47C2×D412yes—
219#10625329—28no—
220C2×C47⋊C8—21no—
221C2×C47⋊C8—21no—
222C2×C47⋊C8—21no—
223C2×C47⋊C8—21no—
224C2×C47⋊C8#4112yes—
225C2×C47⋊C8#4112yes—
226C23×C47—21no—
227C23×C47—21no—
228C23×C47—21no—
229C23×C47—21no—
230C23×C47—21no—
231C23×C47—21no—
232C23×C47C2×C22⋊C412yes—
233C23×C47C22×D411yescharacteristic
234Q8×C47—22no—
235Q8×C47—22no—
236Q8×C47—41no—
237Q8×C47—41no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—22no—
244C2×C4×C47—22no—
245C2×C4×C47—22no—
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×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47C2×C22⋊C412yes—
257C2×C4×C47C23×C411yescharacteristic
258C2×C4×C47C22×D411yescharacteristic
259C2×C4×C47C2×#4111yescharacteristic
260C2×C4×C47C2×#4111yescharacteristic
261C2×C4×C47C2×#4111yescharacteristic
262C2×C4×C47C2×#4111yescharacteristic
263C2×Dic47—21no—
264C2×Dic47—21no—
265C2×Dic47—21no—
266C2×Dic47—21no—
267C2×Dic47—21no—
268C2×Dic47—21no—
269C2×Dic47—21no—
270C2×Dic47—21no—
271C2×Dic47—22no—
272C2×Dic47—22no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280C2×Dic47—41no—
281C2×Dic47—41no—
282C2×Dic47—41no—
283C2×Dic47C2×C22⋊C412yes—
284C2×Dic47C2×C22⋊C412yes—
285C2×Dic47C2×C22⋊C412yes—
286C2×Dic47C2×C22⋊C412yes—
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×C47—22no—
297C22×C47—22no—
298C22×C47—22no—
299C22×C47—41no—
300C22×C47—41no—
301C22×C47—41no—
302C22×C47—41no—
303C22×C47C22×C22⋊C411yescharacteristic
304C22×C47#53211yescharacteristic
305C22×C47#53211yescharacteristic
306C22×C47#57711yescharacteristic
307C22×C47#57711yescharacteristic
308C22×C47#57811yescharacteristic
309C22×C47#57811yescharacteristic
310C4×C47—22no—
311C4×C47—42no—
312C4×C47—42no—
313C4×C47—81no—
314C4×C47—81no—
315C4×C47C4×#4111yescharacteristic
316C4×C47C4×#4111yescharacteristic
317C4×C47#52511yescharacteristic
318C4×C47#52511yescharacteristic
319Dic47—22no—
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#1892D4711yescharacteristic
338#1763—471no—
339#1763—471no—
340C4×#103—471no—
341C4×#103—471no—
342C2×C47—42no—
343C2×C47#229711yescharacteristic
344C2×C47#289811yescharacteristic
345C2×C47#289811yescharacteristic
346C2×C47#317411yescharacteristic
347C2×C47#317711yescharacteristic
348C2×C47#317811yescharacteristic
349C2×C47#317811yescharacteristic
350C22×M16—471no—
351C23×C8C2×D4711yescharacteristic
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⋊Q8C2×D4711yescharacteristic
362C2×C4×Q8—471no—
363C2×C4×Q8—471no—
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×#97—471no—
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#380C2×D4711yescharacteristic
386C4×C4⋊C4—471no—
387C4×C4⋊C4—471no—
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×C4C22×D4711yescharacteristic
400C2×M16—942no—
401C2×M16—942no—
402C2×M16—942no—
403C22×C8—22no—
404C22×C8—22no—
405C22×C8—22no—
406C22×C8—471no—
407C22×C8—471no—
408C22×C8C22×D4711yescharacteristic
409C22×C8C22×D4711yescharacteristic
410C4⋊Q8—474no—
411C4⋊Q8—942no—
412#126—948no—
413#126—948no—
414#125—474no—
415#125—942no—
416#125—948no—
417#125—948no—
418C22⋊Q8—22no—
419C22⋊Q8—474no—
420C22⋊Q8—942no—
421C22⋊Q8C4×D4714yes—
422C4×Q8—474no—
423C4×Q8—474no—
424C4×Q8—942no—
425C4×Q8—942no—
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—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—471no—
449C2×C22⋊C4—471no—
450C2×C22⋊C4—941no—
451C2×C22⋊C4—941no—
452C2×C42—471no—
453C2×C42—471no—
454C2×C42—941no—
455C2×C42—941no—
456C2×C42—941no—
457C2×C42—941no—
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—942no—
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—
476C24D4×D4711yescharacteristic
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—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—471no—
498C22×C4—471no—
499C22×C4—471no—
500C22×C4—471no—
501C22×C4—941no—
502C22×C4—941no—
503C22×C4—941no—
504C22×C4—941no—
505C22×C4C2×C4×D4712yes—
506C22×C4D4×D4711yescharacteristic
507C22×C4D4×D4712yes—
508C22×C4C23×D4711yescharacteristic
509M16—948no—
510C2×C8—28no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—472no—
516C2×C8—472no—
517C2×C8—941no—
518C2×C8—941no—
519C2×C8—941no—
520C2×C8—941no—
521C2×C8#1062535212yes—
522C2×C8#1062535212yes—
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—24no—
548C22⋊C4—81no—
549C22⋊C4—474no—
550C22⋊C4—474no—
551C22⋊C4—944no—
552C22⋊C4—3761no—
553C42—472no—
554C42—472no—
555C42—472no—
556C42—472no—
557C42—944no—
558C42—944no—
559C42—3761no—
560C42—3761no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23C22⋊C4×D4712yes—
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—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
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—41no—
594C2×C4—41no—
595C2×C4—472no—
596C2×C4—472no—
597C2×C4—472no—
598C2×C4—472no—
599C2×C4—941no—
600C2×C4—941no—
601C2×C4—941no—
602C2×C4—941no—
603C2×C4—941no—
604C2×C4—941no—
605C2×C4—941no—
606C2×C4—941no—
607C2×C4—942no—
608C2×C4—942no—
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×C4—1881no—
619C2×C4C22⋊C4×D4712yes—
620C2×C4C22×C4×D4711yescharacteristic
621C2×C4C2×D4×D4711yescharacteristic
622C2×C4#41×D4711yescharacteristic
623C2×C4#41×D4711yescharacteristic
624C2×C4C2×#1062535211yescharacteristic
625C2×C4C2×#1062535211yescharacteristic
626C8—22no—
627C8—22no—
628C8—942no—
629C8—942no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C2×C22⋊C4×D4711yescharacteristic
638C22#1568966811yescharacteristic
639C22#1568966811yescharacteristic
640C22#1568992911yescharacteristic
641C22#1568993011yescharacteristic
642C22#1568993611yescharacteristic
643C22#1568993711yescharacteristic
644C4—22no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4—942no—
650C4—942no—
651C4—942no—
652C4—942no—
653C4—3761no—
654C4—3761no—
655C4—1882no—
656C4—1882no—
657C4C4×#1062535211yescharacteristic
658C4C4×#1062535211yescharacteristic
659C4#1568962911yescharacteristic
660C4#1568963111yescharacteristic
661C2—42no—
662C2#2024587311yescharacteristic
663C2#2025191611yescharacteristic
664C2#2025191711yescharacteristic
665C2#2025400911yescharacteristic
666C2#2025404011yescharacteristic
667C2#2025405111yescharacteristic
668C2#2025405911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1892C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#1892D47
hypocenterC47#21413
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
4. #20244355C2semidirect
5. #20244351C2semidirect
6. #20244206C2semidirect
7. #20244188C2semidirect
11. #20243220C2semidirect
12. #20243217C2semidirect
13. #103×Dic47C2semidirect
14. #20243207C2semidirect
18. #15689733C4semidirect
396. C47#21413semidirect

Maximal subgroups

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

subgroupindex
C2×#156897332
C2×#156890782
#202459952
#202443552
#202443512
#202442062
#202441882
#202438222
#202438192
#202438112
#202432202
#202432172
#103×Dic472
#202432072
C47×#18922
#2141347