Finite Group Databasebeta

← #26224106

Subgroups — #26224106

Order 12032.

750 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
1#20253174C211yesmaximal, characteristic
2#20252320C211yesmaximal, characteristic, complemented
3#20251810C211yesmaximal, characteristic
4#20244330C211yesmaximal, characteristic, complemented
5#20244160C211yesmaximal, characteristic, complemented
6#20243989C211yesmaximal, characteristic, complemented
7#20243985C211yesmaximal, characteristic, complemented
8#20243664C211yesmaximal, characteristic
9#20243660C211yesmaximal, characteristic
10#20243407C211yesmaximal, characteristic
11#20243268C211yesmaximal, characteristic, complemented
12C8×Dic47⋊C4C211yesmaximal, characteristic, complemented
13#20243227C211yesmaximal, characteristic
14#20243225C211yesmaximal, characteristic
15C47×#1772C211yesmaximal, characteristic, complemented
16C2×#13094032C2211yescharacteristic
17C2×#13094030C2211yescharacteristic
18#15689676C2211yescharacteristic
19#15689591C2211yescharacteristic
20#15689433C411yescharacteristic, complemented
21#15689429C411yescharacteristic, complemented
22#15689426C411yescharacteristic, complemented
23#15689422C411yescharacteristic, complemented
24#15689417C411yescharacteristic, complemented
25#15689412C411yescharacteristic, complemented
26#15689407C411yescharacteristic, complemented
27#15689402C411yescharacteristic, complemented
28#15689251—21no—
29#15689251—21no—
30C8×Dic94—21no—
31C8×Dic94—21no—
32C2×#13093959C2211yescharacteristic
33#15689159C2211yescharacteristic
34#15689151C2211yescharacteristic
35#15689150C2211yescharacteristic
36#15689146C2211yescharacteristic
37C2×Dic47⋊C8C2211yescharacteristic
38C2×#13093952C2211yescharacteristic
39C2×#13093950C2211yescharacteristic
40#15689120C2211yescharacteristic
41#15689117C2211yescharacteristic
42#15689080—21no—
43#15689079C2211yescharacteristic
44#15689078C2211yescharacteristic
45#15689077—21no—
46#15689075—21no—
47M16×Dic47C2211yescharacteristic
48C2×C8×Dic47C2211yescharacteristic
49C2×C8×Dic47C2211yescharacteristic
50C4×C8×D47—21no—
51C4×Dic47⋊C4C2211yescharacteristic
52#15688888C2211yescharacteristic
53#15688887C2211yescharacteristic
54#15688884C2211yescharacteristic
55#15688883C2211yescharacteristic
56#15688882C2211yescharacteristic
57#15688882C2211yescharacteristic
58#15688878C2211yescharacteristic
59#15688877C2211yescharacteristic
60C2×#108×C47C2211yescharacteristic
61C47×#427C2211yescharacteristic
62C47×#405C2211yescharacteristic
63C47×#405C2211yescharacteristic
64C2×C4×C8×C47C2211yescharacteristic
65C47×#339C2211yescharacteristic
66C47×#339C2211yescharacteristic
67C2×#10625350C2311yescharacteristic
68#13094032—22no—
69#13094032—22no—
70#13094030—22no—
71#13094030Q814yescomplemented
72C2×#10625329C2311yescharacteristic
73C2×#10625329C2311yescharacteristic
74C2×#10625324—21no—
75C2×#10625324—21no—
76C22×C47⋊C8C2311yescharacteristic
77C22×C47⋊C8C2311yescharacteristic
78C2×C8×D47—21no—
79C2×C8×D47—21no—
80#13094023C2×C411yescharacteristic
81#13094023C2×C411yescharacteristic
82#13094015C2×C411yescharacteristic
83#13094015C2×C411yescharacteristic
84#13094001—21no—
85#13094001—21no—
86C4×Dic94—21no—
87C4×Dic94—21no—
88#13093989C2×C411yescharacteristic
89#13093989C2×C411yescharacteristic
90#13093986C2×C411yescharacteristic
91#13093986C2×C411yescharacteristic
92(C22⋊D47)⋊C4—21no—
93(C22⋊D47)⋊C4—21no—
94#13093983C2311yescharacteristic
95#13093981—21no—
96C2×Dic47⋊C4—21no—
97C2×C4×Dic47C2311yescharacteristic
98C2×C4×Dic47C2311yescharacteristic
99C42×D47—21no—
100#13093959—22no—
101#13093957C2×C411yescharacteristic
102#13093957C2×C411yescharacteristic
103#13093955C2×C411yescharacteristic
104#13093955C2×C411yescharacteristic
105C4⋊(C47⋊C8)—21no—
106C4⋊(C47⋊C8)—21no—
107Dic47⋊C8—21no—
108Dic47⋊C8—21no—
109Dic47⋊C8—21no—
110Dic47⋊C8—21no—
111Dic47⋊C8C2×C411yescharacteristic
112Dic47⋊C8C2×C411yescharacteristic
113Dic47⋊C8C2×C411yescharacteristic
114Dic47⋊C8C2×C411yescharacteristic
115#13093952—22no—
116#13093950—22no—
117#13093949C2×C411yescharacteristic
118#13093949C2×C411yescharacteristic
119#13093947C2×C411yescharacteristic
120#13093947C2×C411yescharacteristic
121#13093946C2×C411yescharacteristic
122#13093946C2×C411yescharacteristic
123#13093944C2×C411yescharacteristic
124#13093944C2×C411yescharacteristic
125#13093935—21no—
126#13093934—21no—
127#13093934—21no—
128C4×C47⋊C8—21no—
129C8×Dic47—21no—
130C8×Dic47—22no—
131C8×Dic47—21no—
132C8×Dic47D411yescharacteristic
133C8×Dic47D411yescharacteristic
134C8×Dic47D411yescharacteristic
135C8×Dic47D411yescharacteristic
136C47⋊#95—21no—
137C2×M16×C47C2311yescharacteristic
138C2×M16×C47C2311yescharacteristic
139C22×C8×C47C2311yescharacteristic
140C22×C8×C47C2311yescharacteristic
141#117×C47C2311yescharacteristic
142#117×C47C2311yescharacteristic
143C2×C42×C47C2311yescharacteristic
144#108×C47—22no—
145#107×C47C2×C411yescharacteristic
146#107×C47C2×C411yescharacteristic
147#106×C47C2×C411yescharacteristic
148#106×C47C2×C411yescharacteristic
149#97×C47—21no—
150#97×C47—21no—
151C4×C8×C47—21no—
152C4×C8×C47—21no—
153C4×C8×C47—22no—
154C22×C4×C47—21no—
155C22×C4×C47C2411yescharacteristic
156C2×C22⋊D47—21no—
157C22×Dic47—21no—
158C22×Dic47#4111yescharacteristic
159C22×Dic47#4111yescharacteristic
160#10625350—21no—
161#10625350—21no—
162#10625350C2×Q812yes—
163#10625350#4112yes—
164C2×D188C22×C411yescharacteristic
165C2×C4×D47—21no—
166C2×C4×D47—21no—
167C2×C4×D47—21no—
168C2×Dic94—21no—
169C2×Dic94—21no—
170C2×Dic94C22×C411yescharacteristic
171M16×C47—21no—
172M16×C47—21no—
173M16×C47—21no—
174M16×C47#4112yes—
175C2×C8×C47—21no—
176C2×C8×C47—21no—
177C2×C8×C47—21no—
178C2×C8×C47—21no—
179C2×C8×C47C22×C411yescharacteristic
180C2×C8×C47C22×C411yescharacteristic
181C2×C8×C47C2×D411yescharacteristic
182C2×C8×C47C2×D411yescharacteristic
183C2×C8×C47C2×Q812yes—
184C2×C8×C47#4112yes—
185C4⋊C4×C47C22×C411yescharacteristic
186C4⋊C4×C47C22×C411yescharacteristic
187C4⋊C4×C47C22×C411yescharacteristic
188C4⋊C4×C47C22×C411yescharacteristic
189C22⋊C4×C47—21no—
190C22⋊C4×C47—21no—
191C42×C47—21no—
192C42×C47—21no—
193C42×C47—21no—
194C42×C47—21no—
195C22⋊Dic47—21no—
196(C2×D47)⋊C4—21no—
197(C2×D47)⋊C4—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47—21no—
200C4⋊Dic47C22×C411yescharacteristic
201C4⋊Dic47C22×C411yescharacteristic
202Dic47⋊C4—24no—
203Dic47⋊C4—21no—
204Dic47⋊C4—21no—
205Dic47⋊C4—21no—
206Dic47⋊C4—21no—
207Dic47⋊C4—21no—
208Dic47⋊C4—21no—
209C4×Dic47—21no—
210C4×Dic47—21no—
211C4×Dic47—21no—
212C4×Dic47—21no—
213C4×Dic47—21no—
214C4×Dic47C22×C411yescharacteristic
215C4×Dic47C22×C411yescharacteristic
216C4×Dic47C2×D411yescharacteristic
217C4×Dic47C2×D411yescharacteristic
218#10625329—21no—
219#10625329—21no—
220#10625329—21no—
221#10625329C2×Q812yes—
222C2×C47⋊C8—21no—
223C2×C47⋊C8—21no—
224C2×C47⋊C8—21no—
225C2×C47⋊C8—21no—
226C2×C47⋊C8—21no—
227C2×C47⋊C8—21no—
228C2×C47⋊C8C22×C411yescharacteristic
229C2×C47⋊C8C22×C411yescharacteristic
230C2×C47⋊C8C2×D411yescharacteristic
231C2×C47⋊C8C2×D411yescharacteristic
232#10625324—24no—
233#10625324—24no—
234C8×D47—24no—
235C8×D47—24no—
236C23×C47C2×#4111yescharacteristic
237C22×D47—21no—
238C2×C4×C47—22no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47C23×C411yescharacteristic
246C2×C4×C47C22×D411yescharacteristic
247C2×C4×C47C22×Q811yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×C4×C47C2×#4111yescharacteristic
250C2×C4×C47C2×#4111yescharacteristic
251C22⋊D47—22no—
252C22⋊D47—22no—
253C2×Dic47—21no—
254C2×Dic47—22no—
255C2×Dic47—22no—
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47C4×D411yescharacteristic
262C2×Dic47C4×D411yescharacteristic
263C2×Dic47C4×D411yescharacteristic
264C2×Dic47C4×D411yescharacteristic
265D188—21no—
266D188C4×Q812yes—
267C4×D47—22no—
268C4×D47—22no—
269C4×D47—42no—
270C4×D47—42no—
271Dic94—21no—
272Dic94—42no—
273Dic94—42no—
274Dic94C4×Q812yes—
275C8×C47—21no—
276C8×C47—21no—
277C8×C47—22no—
278C8×C47—41no—
279C8×C47C22⋊Q812yes—
280C8×C47C22⋊Q812yes—
281C47⋊C8—21no—
282C47⋊C8—21no—
283C47⋊C8—22no—
284C47⋊C8—21no—
285C47⋊C8—21no—
286C47⋊C8—41no—
287#21627—471nomaximal, Sylow
288C22×C47—21no—
289C22×C47—21no—
290C22×C47C2×C4×D411yescharacteristic
291C22×C47#52911yescharacteristic
292C22×C47#53311yescharacteristic
293C2×D47—22no—
294C2×D47—41no—
295C2×D47—41no—
296C4×C47—42no—
297C4×C47—41no—
298C4×C47—41no—
299C4×C47—41no—
300C4×C47—41no—
301C4×C47C2×C4×Q811yescharacteristic
302C4×C47C4×#4111yescharacteristic
303C4×C47C2×C22⋊Q811yescharacteristic
304C4×C47#52511yescharacteristic
305Dic47—21no—
306Dic47—21no—
307Dic47—21no—
308Dic47—22no—
309Dic47—42no—
310Dic47—41no—
311Dic47—41no—
312Dic47#44312yes—
313#3076—471no—
314#2960—471no—
315C4×#131—471no—
316#1986—471no—
317#1950—471no—
318#1922—471no—
319#1922—471no—
320#1866—471no—
321#1866—471no—
322#1806—471no—
323#1772—471no—
324#1772D4711yescharacteristic, complemented
325C8×C4⋊C4—471no—
326#1764—471no—
327#1764—471no—
328C2×C47—21no—
329C2×C47—21no—
330C2×C47C4×C22⋊Q811yescharacteristic
331C2×C47C2×#44311yescharacteristic
332C2×C47#295511yescharacteristic
333D47—42no—
334C2×#131—471no—
335C2×#131—471no—
336#533—471no—
337C4×#41—471no—
338#479—471no—
339#479—471no—
340#478—471no—
341#478—471no—
342#477—471no—
343#476—471no—
344#475—471no—
345#474—471no—
346#446—941no—
347#446—941no—
348C8×Q8—941no—
349C8×Q8—941no—
350C2×#108—471no—
351C2×#108C2×D4711yescharacteristic
352#427—471no—
353#427C2×D4711yescharacteristic
354#424—471no—
355#424—471no—
356#423—471no—
357C2×C4⋊C8—471no—
358C2×#104—471no—
359C2×#104—471no—
360#416—471no—
361#416—471no—
362#405—471no—
363#405—471no—
364#405—941no—
365#405—941no—
366#405C2×D4711yescharacteristic
367#405C2×D4711yescharacteristic
368C4×M16—471no—
369C4×M16—941no—
370C2×C4×C8—471no—
371C2×C4×C8—471no—
372C2×C4×C8—941no—
373C2×C4×C8C2×D4711yescharacteristic
374C4×C4⋊C4—471no—
375#341—471no—
376#341—471no—
377#339—471no—
378#339—471no—
379#339—471no—
380#339—471no—
381#339C2×D4711yescharacteristic
382#339C2×D4711yescharacteristic
383#336—471no—
384#336—471no—
385C47#2162711yescharacteristic, complemented, Sylow
386C2×#41—471no—
387#131—474no—
388#131—942no—
389#131—942no—
390#131—942no—
391C2×M16—471no—
392C2×M16—471no—
393C2×M16—941no—
394C2×M16—941no—
395C2×M16C22×D4711yescharacteristic
396C2×M16C22×D4711yescharacteristic
397C22×C8—471no—
398C22×C8—471no—
399C22×C8—941no—
400C22×C8—941no—
401C22×C8C22×D4711yescharacteristic
402C22×C8C22×D4711yescharacteristic
403C4⋊Q8—471no—
404C4⋊Q8—471no—
405#125—471no—
406#125—471no—
407C22⋊Q8—941no—
408C22⋊Q8—941no—
409C4×Q8—471no—
410C4×Q8—471no—
411C4×Q8—941no—
412C4×Q8—941no—
413C4×D4—471no—
414C4×D4—471no—
415C4×D4—941no—
416C4×D4—941no—
417#117—471no—
418#117—941no—
419#117C22×D4711yescharacteristic
420#117C22×D4711yescharacteristic
421C2×C4⋊C4—941no—
422C2×C42—471no—
423C2×C42—471no—
424C2×C42—941no—
425C2×C42C22×D4711yescharacteristic
426#108—22no—
427#108—942no—
428#107—471no—
429#107—471no—
430#107C4×D4711yescharacteristic
431#107C4×D4711yescharacteristic
432#106—471no—
433#106—471no—
434#106C4×D4711yescharacteristic
435#106C4×D4711yescharacteristic
436C4⋊C8—471no—
437C4⋊C8—471no—
438C4⋊C8—471no—
439C4⋊C8—471no—
440C4⋊C8—941no—
441C4⋊C8—941no—
442C4⋊C8—941no—
443C4⋊C8—941no—
444C4⋊C8—941no—
445C4⋊C8—941no—
446#104—942no—
447#104—942no—
448#103—471no—
449#103—471no—
450#103—471no—
451#103—471no—
452#102—471no—
453#102—471no—
454#102—471no—
455#102—471no—
456#97—21no—
457#97—21no—
458#97—941no—
459#97—941no—
460#97—941no—
461C4×C8—22no—
462C4×C8—21no—
463C4×C8—21no—
464C4×C8—471no—
465C4×C8—471no—
466C4×C8—471no—
467C4×C8—471no—
468C4×C8—941no—
469C4×C8—941no—
470C4×C8—942no—
471C4×C8—941no—
472#95—941no—
473#41—472no—
474#41—472no—
475#41—941no—
476#41—941no—
477C2×Q8—471no—
478C2×Q8—941no—
479C2×Q8—941no—
480C2×D4—471no—
481C2×D4—941no—
482C22×C4—21no—
483C22×C4—471no—
484C22×C4—471no—
485C22×C4—941no—
486C22×C4—941no—
487C22×C4—941no—
488C22×C4—941no—
489C22×C4C23×D4711yescharacteristic
490M16—21no—
491M16—21no—
492M16—21no—
493M16—472no—
494M16—944no—
495M16—941no—
496M16—944no—
497M16—941no—
498M16—941no—
499M16#1062535012yes—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—471no—
505C2×C8—471no—
506C2×C8—471no—
507C2×C8—471no—
508C2×C8—944no—
509C2×C8—941no—
510C2×C8—941no—
511C2×C8—941no—
512C2×C8—941no—
513C2×C8—941no—
514C2×C8—944no—
515C2×C8—941no—
516C2×C8C2×C4×D4711yescharacteristic
517C2×C8C2×C4×D4711yescharacteristic
518C2×C8#1062535012yes—
519C2×C8D4×D4711yescharacteristic
520C2×C8D4×D4711yescharacteristic
521C2×C8Q8×D4712yes—
522C4⋊C4—471no—
523C4⋊C4—471no—
524C4⋊C4—941no—
525C4⋊C4—941no—
526C4⋊C4—941no—
527C4⋊C4—941no—
528C4⋊C4—941no—
529C4⋊C4—941no—
530C4⋊C4—941no—
531C4⋊C4—941no—
532C4⋊C4—944no—
533C4⋊C4C2×C4×D4711yescharacteristic
534C4⋊C4C2×C4×D4711yescharacteristic
535C4⋊C4C2×C4×D4711yescharacteristic
536C4⋊C4C2×C4×D4711yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—941no—
540C22⋊C4—941no—
541C22⋊C4—941no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—471no—
547C42—471no—
548C42—471no—
549C42—471no—
550C42—941no—
551C42—941no—
552C42—941no—
553C42—941no—
554C42—941no—
555C23—941no—
556C23#41×D4711yescharacteristic
557Q8—472no—
558Q8—941no—
559Q8—1882no—
560Q8—1882no—
561D4—472no—
562D4—942no—
563D4—942no—
564D4—941no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—471no—
573C2×C4—471no—
574C2×C4—471no—
575C2×C4—471no—
576C2×C4—942no—
577C2×C4—941no—
578C2×C4—942no—
579C2×C4—942no—
580C2×C4—942no—
581C2×C4—941no—
582C2×C4—941no—
583C2×C4—1882no—
584C2×C4—1882no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4C22×C4×D4711yescharacteristic
589C2×C4C2×D4×D4711yescharacteristic
590C2×C4C2×Q8×D4711yescharacteristic
591C2×C4#41×D4711yescharacteristic
592C2×C4C2×#1062535011yescharacteristic
593C2×C4C2×#1062535011yescharacteristic
594C8—21no—
595C8—21no—
596C8—22no—
597C8—41no—
598C8—941no—
599C8—941no—
600C8—942no—
601C8—941no—
602C8—941no—
603C8—1881no—
604C8#1309400212yes—
605C8#1309400212yes—
606C22—21no—
607C22—21no—
608C22—942no—
609C22—1881no—
610C22—1881no—
611C22C4×D4×D4711yescharacteristic
612C22#1568965111yescharacteristic
613C22#1568967711yescharacteristic
614C4—42no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—472no—
620C4—941no—
621C4—941no—
622C4—941no—
623C4—942no—
624C4—1882no—
625C4—1881no—
626C4—1881no—
627C4C4×Q8×D4711yescharacteristic
628C4C4×#1062535011yescharacteristic
629C4C2×#1309400211yescharacteristic
630C4#1568963211yescharacteristic
631C2—21no—
632C2—21no—
633C2—1882no—
634C2C4×#1309400211yescharacteristic
635C2#443×D4711yescharacteristic
636C2#2025228611yescharacteristic
subgroupG/H
centerC2×C8D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1772C2
socleC22×C47C2×C4×D4
prime radicalC2×#10625350C23
cosocleC22×C4×C47C24
hypercenter#1772D47
hypocenterC47#21627
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ #1772

Complements

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

NH ≅ G/Ntype
2. #20252320C2semidirect
4. #20244330C2semidirect
5. #20244160C2semidirect
6. #20243989C2semidirect
7. #20243985C2semidirect
11. #20243268C2semidirect
12. C8×Dic47⋊C4C2semidirect
15. C47×#1772C2semidirect
20. #15689433C4semidirect
21. #15689429C4semidirect
22. #15689426C4semidirect
23. #15689422C4semidirect
24. #15689417C4semidirect
25. #15689412C4semidirect
26. #15689407C4semidirect
27. #15689402C4semidirect
71. #13094030Q8semidirect
324. #1772D47semidirect
385. C47#21627semidirect

Maximal subgroups

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

subgroupindex
#202531742
#202523202
#202518102
#202443302
#202441602
#202439892
#202439852
#202436642
#202436602
#202434072
#202432682
C8×Dic47⋊C42
#202432272
#202432252
C47×#17722
#2162747