Finite Group Databasebeta

← #27083958

Subgroups — #27083958

Order 12032.

2054 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257214C211yesmaximal, characteristic, complemented
2#20253250C213yesmaximal, complemented
3#20253239C213yesmaximal, complemented
4#20253219C213yesmaximal, complemented
5#20253219C213yesmaximal, complemented
6#20253210C213yesmaximal, complemented
7#20253185C213yesmaximal, complemented
8#20253176C213yesmaximal, complemented
9C2×#15689423C213yesmaximal, complemented
10#20251986C211yesmaximal, characteristic, complemented
11#20251981C211yesmaximal, characteristic, complemented
12#20251956C211yesmaximal, characteristic, complemented
13#20251914C211yesmaximal, characteristic
14#20251914C211yesmaximal, characteristic
15C4×H(5,𝔽2)×C47C211yesmaximal, characteristic
16C2×#13094031C2211yescharacteristic
17C2×#13094031C2211yescharacteristic
18#15689849—21no—
19#15689849—21no—
20#15689832—23no—
21#15689832—23no—
22#15689751C2213yescomplemented
23#15689751C2213yescomplemented
24#15689680C2213yescomplemented
25C2×#13094016C2213yescomplemented
26#15689606C2211yescharacteristic
27#15689606C2211yescharacteristic
28#15689430C2213yescomplemented
29#15689430C2216yescomplemented
30#15689430C2211yescharacteristic, complemented
31#15689423C2213yescomplemented
32#15689423C2216yescomplemented
33#15689423C2216yescomplemented
34#15689423C2216yescomplemented
35#15689423C2216yescomplemented
36#15689423C2213yescomplemented
37#15689418C2213yescomplemented
38#15689413C2216yescomplemented
39#15689413C2213yescomplemented
40#15689408C2213yescomplemented
41#15689403C2216yescomplemented
42#15689403C2213yescomplemented
43#15689166C2213yescomplemented
44#15689166C2213yescomplemented
45#15689163C2213yescomplemented
46#15689163C2213yescomplemented
47#15689159C2213yescomplemented
48#15689159C2213yescomplemented
49C2×#13093957C2213yescomplemented
50C2×#13093955C2213yescomplemented
51#15689150C2213yescomplemented
52#15689150C2213yescomplemented
53#15689147C2213yescomplemented
54C2×C4⋊(C47⋊C8)C2213yescomplemented
55#15689128C2213yes—
56#15689128C2213yes—
57#15689124C2213yes—
58#15689124C2213yes—
59#15689119C2213yes—
60#15689119C2213yes—
61C2×#13093945C2213yes—
62C2×#13093945C2213yes—
63C2×H(5,𝔽2)×C47C2211yescharacteristic
64C47×#520—21no—
65C47×#520C2213yes—
66C47×#520C2211yescharacteristic
67C47×#518—23no—
68C47×#518C2213yes—
69C4×#41×C47—21no—
70C4×#41×C47C2213yes—
71C4×#41×C47C2211yescharacteristic
72C2×C4×D4×C47—23no—
73C2×C4×D4×C47C2213yes—
74#13094031—24no—
75#13094031—24no—
76C2×#10625329C2313yes—
77C2×#10625329C2313yes—
78C22×C47⋊C8C2313yes—
79C22×C47⋊C8C2313yes—
80#13094024C2313yes—
81#13094024C2313yes—
82#13094023—23no—
83#13094023—23no—
84#13094016—23no—
85#13094016—23no—
86#13094016C2316yes—
87#13094016C2313yes—
88#13094016C2311yescharacteristic
89#13094015—23no—
90#13094015—23no—
91#13094003—23no—
92#13094003—23no—
93#13094001—26no—
94#13094001—26no—
95Q8×Dic47—21no—
96Q8×Dic47—21no—
97C4×Dic94—23no—
98C4×Dic94—23no—
99D4×Dic47—23no—
100D4×Dic47—23no—
101#13093983C2313yes—
102#13093983C2313yes—
103C2×C4⋊Dic47C2313yes—
104C2×C4⋊Dic47C2313yes—
105C2×Dic47⋊C4—23no—
106C2×Dic47⋊C4—23no—
107#13093957C2311yescharacteristic
108#13093957C2313yes—
109#13093957C2316yes—
110#13093957C2313yes—
111#13093957C2313yes—
112#13093955C2313yes—
113#13093955C2313yes—
114#13093955C2316yes—
115#13093955C2311yescharacteristic
116#13093955C2313yes—
117C4⋊(C47⋊C8)C2311yescharacteristic
118C4⋊(C47⋊C8)C2313yes—
119C4⋊(C47⋊C8)C2313yes—
120C4⋊(C47⋊C8)C2313yes—
121C4⋊(C47⋊C8)C2316yes—
122#13093948C2311yescharacteristic
123#13093948C2311yescharacteristic
124#13093948C2313yes—
125#13093948C2313yes—
126#13093945C2313yes—
127#13093945C2313yes—
128#13093945C2316yes—
129#13093945C2316yes—
130#13093945C2313yes—
131#13093945C2313yes—
132H(5,𝔽2)×C47—24no—
133H(5,𝔽2)×C47Q818yescomplemented
134C2×#41×C47—21no—
135C2×#41×C47C2313yes—
136C2×#41×C47C2311yescharacteristic
137C22×D4×C47—23no—
138C22×D4×C47C2313yes—
139C23×C4×C47—23no—
140C4×Q8×C47—21no—
141C4×Q8×C47D411yescharacteristic
142C4×Q8×C47D411yescharacteristic
143C4×Q8×C47C2311yescharacteristic
144C4×Q8×C47C2313yes—
145C4×D4×C47—23no—
146C4×D4×C47—23no—
147C4×D4×C47—23no—
148C4×D4×C47—23no—
149C4×D4×C47—23no—
150C4×D4×C47—26no—
151C4×D4×C47—23no—
152C4×D4×C47—23no—
153C4×D4×C47D413yes—
154C4×D4×C47D413yes—
155C4×D4×C47C2313yes—
156C4×D4×C47C2316yes—
157C4×D4×C47C2313yes—
158#117×C47—23no—
159#117×C47—23no—
160#117×C47C2313yes—
161#117×C47C2313yes—
162C2×C4⋊C4×C47—23no—
163C2×C4⋊C4×C47C2313yes—
164C2×C22⋊C4×C47—23no—
165C2×C22⋊C4×C47—26no—
166C2×C42×C47—23no—
167C2×C42×C47C2313yes—
168C24×C47—23no—
169#41×C47—28no—
170#41×C47—26no—
171#41×C47C2×Q8112yes—
172#41×C47C2×Q814yes—
173#41×C47#4114yes—
174C2×Q8×C47C2×D411yescharacteristic
175C2×Q8×C47C2411yescharacteristic
176C2×D4×C47—23no—
177C2×D4×C47—224no—
178C2×D4×C47—23no—
179C2×D4×C47—26no—
180C2×D4×C47C2×D413yes—
181C2×D4×C47C2×Q8112yes—
182C2×D4×C47C2413yes—
183C22×C4×C47—23no—
184C22×C4×C47—23no—
185C22×C4×C47—23no—
186C22×C4×C47—23no—
187C22×C4×C47—26no—
188C22×C4×C47—86no—
189C22×C4×C47C2×D413yes—
190C22×C4×C47C2×D413yes—
191C22×C4×C47C2413yes—
192C22×Dic47—23no—
193C22×Dic47—23no—
194C2×Dic94—23no—
195C2×Dic94—23no—
196C4⋊C4×C47—23no—
197C4⋊C4×C47—23no—
198C4⋊C4×C47—21no—
199C4⋊C4×C47C2×D413yes—
200C4⋊C4×C47C2×D413yes—
201C4⋊C4×C47C2413yes—
202C4⋊C4×C47C2411yescharacteristic
203C22⋊C4×C47—23no—
204C22⋊C4×C47—23no—
205C22⋊C4×C47—23no—
206C22⋊C4×C47—26no—
207C22⋊C4×C47C2×D416yes—
208C42×C47—21no—
209C42×C47—23no—
210C42×C47—23no—
211C42×C47C2×D413yes—
212C42×C47C2×D413yes—
213C42×C47C2411yescharacteristic
214C42×C47C2413yes—
215C22⋊Dic47—23no—
216C22⋊Dic47—23no—
217C4⋊Dic47—23no—
218C4⋊Dic47—23no—
219C4⋊Dic47—23no—
220C4⋊Dic47—23no—
221C4⋊Dic47C2413yes—
222C4⋊Dic47C2411yescharacteristic
223C4⋊Dic47C2413yes—
224C4⋊Dic47C2411yescharacteristic
225Dic47⋊C4—23no—
226Dic47⋊C4—23no—
227Dic47⋊C4—26no—
228Dic47⋊C4—26no—
229Dic47⋊C4—21no—
230Dic47⋊C4—21no—
231C4×Dic47—23no—
232C4×Dic47—23no—
233#10625329—26no—
234#10625329—26no—
235C2×C47⋊C8—26no—
236C2×C47⋊C8—26no—
237C2×C47⋊C8C2411yescharacteristic
238C2×C47⋊C8C2413yes—
239C2×C47⋊C8C2413yes—
240C2×C47⋊C8C2411yescharacteristic
241C23×C47—23no—
242C23×C47—212no—
243C23×C47—83no—
244C23×C47—83no—
245C23×C47C22×D413yes—
246Q8×C47—21no—
247Q8×C47C22⋊Q814yes—
248Q8×C47C22×Q812yes—
249D4×C47—212no—
250D4×C47—212no—
251D4×C47—23no—
252D4×C47C22⋊Q8112yes—
253D4×C47C22×Q816yes—
254C2×C4×C47—21no—
255C2×C4×C47—212no—
256C2×C4×C47—21no—
257C2×C4×C47—23no—
258C2×C4×C47—43no—
259C2×C4×C47—43no—
260C2×C4×C47—46no—
261C2×C4×C47—83no—
262C2×C4×C47—86no—
263C2×C4×C47—83no—
264C2×C4×C47C22×D411yescharacteristic
265C2×C4×C47C22×D411yescharacteristic
266C2×C4×C47C22×D413yes—
267C2×C4×C47C22×D413yes—
268C2×C4×C47C22×D413yes—
269C2×C4×C47C22×Q816yes—
270C2×C4×C47C2×#4116yes—
271C2×C4×C47C2511yescharacteristic
272C2×Dic47—21no—
273C2×Dic47—21no—
274C2×Dic47—23no—
275C2×Dic47—23no—
276C2×Dic47—83no—
277C2×Dic47—83no—
278Dic94—83no—
279Dic94—83no—
280C47⋊C8—21no—
281C47⋊C8—21no—
282C47⋊C8—23no—
283C47⋊C8—23no—
284#60496—471nomaximal, Sylow
285C22×C47—23no—
286C22×C47—23no—
287C22×C47—26no—
288C22×C47—43no—
289C22×C47—43no—
290C22×C47—812no—
291C22×C47C2×C22⋊Q816yes—
292C22×C47C23×D411yescharacteristic
293C4×C47—23no—
294C4×C47—21no—
295C4×C47—23no—
296C4×C47—21no—
297C4×C47—22no—
298C4×C47—41no—
299C4×C47—43no—
300C4×C47C2×C22⋊Q816yes—
301C4×C47C23×Q811yescharacteristic
302C4×C47C22×#4111yescharacteristic
303Dic47—41no—
304Dic47—43no—
305Dic47—41no—
306Dic47—43no—
307#3504—471no—
308C4×H(5,𝔽2)D4711yescharacteristic
309#3084—473no—
310#3083—473no—
311#3081—473no—
312#3081—473no—
313#3080—473no—
314#3077—473no—
315#3076—473no—
316C2×#478—473no—
317#2913—471no—
318#2912—471no—
319#2906—471no—
320#2897—471no—
321#2897—471no—
322C2×C47—26no—
323C2×C47—46no—
324C2×C47C22×C22⋊Q811yescharacteristic
325C2×C47#358411yescharacteristic
326C2×C47#358511yescharacteristic
327C2×H(5,𝔽2)C2×D4711yescharacteristic
328C2×#131—471no—
329C2×#131—471no—
330#557—941no—
331#557—941no—
332#554—943no—
333#554—943no—
334#544—473no—
335#544—473no—
336#533—473no—
337C2×#125—473no—
338#520—21no—
339#520—471no—
340#520—471no—
341#520C2×D4713yes—
342#520C2×D4711yescharacteristic
343#518—23no—
344#518C2×D4713yes—
345C4×#41—21no—
346C4×#41C2×D4713yes—
347C4×#41C2×D4711yescharacteristic
348C2×C4×D4—23no—
349C2×C4×D4C2×D4713yes—
350#479—471no—
351#479—473no—
352#479—476no—
353#478—473no—
354#478—476no—
355#478—476no—
356#478—476no—
357#478—473no—
358#478—476no—
359#477—473no—
360#476—473no—
361#476—476no—
362#475—473no—
363#474—473no—
364#474—476no—
365#428—473no—
366#428—473no—
367#428—473no—
368#428—473no—
369#427—473no—
370#427—473no—
371C2×#107—473no—
372C2×#106—473no—
373#424—473no—
374#424—473no—
375#423—473no—
376C2×C4⋊C8—473no—
377#418—473no—
378#418—473no—
379#417—473no—
380#417—473no—
381#416—473no—
382#416—473no—
383C2×#102—473no—
384C2×#102—473no—
385C47#6049611yescharacteristic, complemented, Sylow
386H(5,𝔽2)—24no—
387H(5,𝔽2)Dic9418yescomplemented
388C2×#41—21no—
389C2×#41C22×D4713yes—
390C2×#41C22×D4711yescharacteristic
391C22×D4—23no—
392C22×D4C22×D4713yes—
393C23×C4—23no—
394#131—944no—
395#131—944no—
396C2×M16—473no—
397C2×M16—473no—
398C22×C8—473no—
399C22×C8—473no—
400C4⋊Q8—473no—
401C4⋊Q8—473no—
402C4⋊Q8—943no—
403C4⋊Q8—943no—
404#125—473no—
405#125—476no—
406#125—471no—
407#125—943no—
408#125—943no—
409#125—943no—
410#125—943no—
411C22⋊Q8—943no—
412C22⋊Q8—943no—
413C22⋊Q8—946no—
414C22⋊Q8—946no—
415C4×Q8—21no—
416C4×Q8—941no—
417C4×Q8—941no—
418C4×Q8—943no—
419C4×Q8—943no—
420C4×Q8C22⋊D4711yescharacteristic
421C4×Q8C22⋊D4711yescharacteristic
422C4×Q8C22×D4711yescharacteristic
423C4×Q8C22×D4713yes—
424C4×D4—23no—
425C4×D4—23no—
426C4×D4—23no—
427C4×D4—23no—
428C4×D4—23no—
429C4×D4—26no—
430C4×D4—23no—
431C4×D4—23no—
432C4×D4—943no—
433C4×D4—943no—
434C4×D4C22⋊D4713yes—
435C4×D4C22⋊D4713yes—
436C4×D4C22×D4713yes—
437C4×D4C22×D4716yes—
438C4×D4C22×D4713yes—
439#117—23no—
440#117—23no—
441#117—473no—
442#117—473no—
443#117C22×D4713yes—
444#117C22×D4713yes—
445C2×C4⋊C4—23no—
446C2×C4⋊C4—473no—
447C2×C4⋊C4—473no—
448C2×C4⋊C4—943no—
449C2×C4⋊C4—943no—
450C2×C4⋊C4C22×D4713yes—
451C2×C22⋊C4—23no—
452C2×C22⋊C4—26no—
453C2×C42—23no—
454C2×C42C22×D4713yes—
455#107—473no—
456#107—476no—
457#107—471no—
458#107—473no—
459#107—473no—
460#106—473no—
461#106—476no—
462#106—471no—
463#106—473no—
464#106—473no—
465C4⋊C8—471no—
466C4⋊C8—473no—
467C4⋊C8—473no—
468C4⋊C8—473no—
469C4⋊C8—476no—
470#103—471no—
471#103—471no—
472#103—473no—
473#103—473no—
474#102—473no—
475#102—473no—
476#102—476no—
477#102—476no—
478#102—473no—
479#102—473no—
480C24—23no—
481#41—28no—
482#41—26no—
483#41C2×Dic94112yes—
484#41C2×Dic9414yes—
485#41#1062535014yes—
486C2×Q8—943no—
487C2×Q8—943no—
488C2×Q8C2×C22⋊D4711yescharacteristic
489C2×Q8C23×D4711yescharacteristic
490C2×D4—23no—
491C2×D4—224no—
492C2×D4—23no—
493C2×D4—26no—
494C2×D4C2×Dic94112yes—
495C2×D4C2×C22⋊D4713yes—
496C2×D4C23×D4713yes—
497C22×C4—23no—
498C22×C4—23no—
499C22×C4—23no—
500C22×C4—23no—
501C22×C4—26no—
502C22×C4—86no—
503C22×C4—943no—
504C22×C4—943no—
505C22×C4C2×C22⋊D4713yes—
506C22×C4C2×C22⋊D4713yes—
507C22×C4C23×D4713yes—
508M16—946no—
509M16—946no—
510C2×C8—471no—
511C2×C8—473no—
512C2×C8—473no—
513C2×C8—471no—
514C2×C8—946no—
515C2×C8—946no—
516C4⋊C4—23no—
517C4⋊C4—23no—
518C4⋊C4—21no—
519C4⋊C4—473no—
520C4⋊C4—471no—
521C4⋊C4—473no—
522C4⋊C4—471no—
523C4⋊C4—943no—
524C4⋊C4—943no—
525C4⋊C4—946no—
526C4⋊C4—946no—
527C4⋊C4—941no—
528C4⋊C4—941no—
529C4⋊C4—943no—
530C4⋊C4—943no—
531C4⋊C4—943no—
532C4⋊C4—943no—
533C4⋊C4C2×C22⋊D4713yes—
534C4⋊C4C2×C22⋊D4713yes—
535C4⋊C4C23×D4713yes—
536C4⋊C4C23×D4711yescharacteristic
537C22⋊C4—23no—
538C22⋊C4—23no—
539C22⋊C4—26no—
540C22⋊C4—23no—
541C22⋊C4—943no—
542C22⋊C4—943no—
543C22⋊C4C2×C22⋊D4716yes—
544C42—21no—
545C42—23no—
546C42—23no—
547C42—943no—
548C42—943no—
549C42C2×C22⋊D4713yes—
550C42C2×C22⋊D4713yes—
551C42C23×D4711yescharacteristic
552C42C23×D4713yes—
553C23—23no—
554C23—212no—
555C23—83no—
556C23—83no—
557C23C22×C22⋊D4713yes—
558Q8—21no—
559Q8—3763no—
560Q8—3763no—
561Q8#1309400314yes—
562Q8C22×Dic9412yes—
563D4—212no—
564D4—212no—
565D4—23no—
566D4#13094003112yes—
567D4C22×Dic9416yes—
568C2×C4—21no—
569C2×C4—212no—
570C2×C4—21no—
571C2×C4—23no—
572C2×C4—43no—
573C2×C4—43no—
574C2×C4—46no—
575C2×C4—83no—
576C2×C4—86no—
577C2×C4—83no—
578C2×C4—941no—
579C2×C4—941no—
580C2×C4—943no—
581C2×C4—943no—
582C2×C4—3763no—
583C2×C4—3763no—
584C2×C4C22×C22⋊D4711yescharacteristic
585C2×C4C22×C22⋊D4713yes—
586C2×C4C22×C22⋊D4711yescharacteristic
587C2×C4C22×C22⋊D4713yes—
588C2×C4C22×C22⋊D4713yes—
589C2×C4C22×Dic9416yes—
590C2×C4C2×#1062535016yes—
591C2×C4C24×D4711yescharacteristic
592C8—941no—
593C8—941no—
594C8—943no—
595C8—943no—
596C22—23no—
597C22—23no—
598C22—26no—
599C22—43no—
600C22—43no—
601C22—812no—
602C22C2×#1309400316yes—
603C22C23×C22⋊D4711yescharacteristic
604C4—22no—
605C4—21no—
606C4—23no—
607C4—21no—
608C4—23no—
609C4—41no—
610C4—43no—
611C4—1883no—
612C4—1881no—
613C4—1883no—
614C4—1881no—
615C4C2×#1309400316yes—
616C4C23×Dic9411yescharacteristic
617C4C22×#1062535011yescharacteristic
618C2—26no—
619C2—46no—
620C2C22×#1309400311yescharacteristic
621C2#2025815911yescharacteristic
622C2#2025816811yescharacteristic
subgroupG/H
centerC22C23×C22⋊D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC4×H(5,𝔽2)×C47C2
socleC22×C47C23×D4
prime radicalC2×H(5,𝔽2)×C47C22
hypercenterC4×H(5,𝔽2)D47
hypocenterC47#60496
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20257214C2semidirect
2. #20253250C2semidirect
3. #20253239C2semidirect
4. #20253219C2semidirect
5. #20253219C2semidirect
6. #20253210C2semidirect
7. #20253185C2semidirect
8. #20253176C2semidirect
9. C2×#15689423C2semidirect
10. #20251986C2semidirect
11. #20251981C2semidirect
12. #20251956C2semidirect
22. #15689751C22semidirect
23. #15689751C22semidirect
24. #15689680C22semidirect
25. C2×#13094016C22semidirect
28. #15689430C22semidirect
29. #15689430C22semidirect
30. #15689430C22semidirect
31. #15689423C22semidirect
32. #15689423C22semidirect
33. #15689423C22semidirect
34. #15689423C22semidirect
35. #15689423C22semidirect
36. #15689423C22semidirect
37. #15689418C22semidirect
38. #15689413C22semidirect
39. #15689413C22semidirect
40. #15689408C22semidirect
41. #15689403C22semidirect
42. #15689403C22semidirect
43. #15689166C22semidirect
44. #15689166C22semidirect
45. #15689163C22semidirect
46. #15689163C22semidirect
47. #15689159C22semidirect
48. #15689159C22semidirect
49. C2×#13093957C22semidirect
50. C2×#13093955C22semidirect
51. #15689150C22semidirect
52. #15689150C22semidirect
53. #15689147C22semidirect
54. C2×C4⋊(C47⋊C8)C22semidirect
133. H(5,𝔽2)×C47Q8semidirect
385. C47#60496semidirect
387. H(5,𝔽2)Dic94semidirect

Maximal subgroups

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

subgroupindex
#202572142
#202532502
#202532392
#202532192
#202532102
#202531852
#202531762
C2×#156894232
#202519862
#202519812
#202519562
#202519142
C4×H(5,𝔽2)×C472
#6049647