Finite Group Databasebeta

← #26187512

Subgroups — #26187512

Order 12032.

642 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253437C211yesmaximal, characteristic
2#20252550C211yesmaximal, characteristic, complemented
3#20252422C211yesmaximal, characteristic, complemented
4#20242303C211yesmaximal, characteristic, complemented
5#20242292C211yesmaximal, characteristic, complemented
6#20242233C211yesmaximal, characteristic, complemented
7#20242226C211yesmaximal, characteristic, complemented
8#20242219C211yesmaximal, characteristic, complemented
9#20242219C211yesmaximal, characteristic, complemented
10#20242213C211yesmaximal, characteristic, complemented
11#20242202C211yesmaximal, characteristic, complemented
12#20242173C211yesmaximal, characteristic, complemented
13#20241849C211yesmaximal, characteristic, complemented
14C2×#15688852C211yesmaximal, characteristic, complemented
15C47×#1452C211yesmaximal, characteristic, complemented
16#15689657C2211yescharacteristic
17C2×C4×D188C2211yescharacteristic
18#15689459C411yescharacteristic
19#15689442C411yescharacteristic
20#15689424C411yescharacteristic
21#15689424C411yescharacteristic
22#15689339C411yescharacteristic
23#15689339C411yescharacteristic
24#15689276C411yescharacteristic
25#15689259C411yescharacteristic
26#15689253C2211yescharacteristic, complemented
27C8×Dic94C2211yescharacteristic, complemented
28C4×#10625325C2211yescharacteristic
29C4×#10625325C2211yescharacteristic
30C4×D376C2211yescharacteristic
31C4×D376C2211yescharacteristic
32#15689202C2211yescharacteristic
33#15689202C2211yescharacteristic
34#15689202C2211yescharacteristic, complemented
35#15689193—21no—
36#15689193—21no—
37#15689191—21no—
38C8×D188C2211yescharacteristic
39C8×D188C2211yescharacteristic
40C8×D188C2211yescharacteristic, complemented
41(C22⋊D47)⋊C8—21no—
42#15689183—21no—
43#15689180C2211yescharacteristic
44#15689148—21no—
45C2×C4⋊(C47⋊C8)C2211yescharacteristic
46#15689085—21no—
47C22⋊C8×D47—21no—
48#15688876C2211yescharacteristic, complemented
49#15688876C2211yescharacteristic, complemented
50#15688874C2211yescharacteristic, complemented
51#15688874C2211yescharacteristic, complemented
52#15688855C2211yescharacteristic, complemented
53#15688855C2211yescharacteristic, complemented
54#15688852C2211yescharacteristic
55#15688852C2211yescharacteristic
56#15688852C2211yescharacteristic
57#15688852C2211yescharacteristic
58#15688852C2211yescharacteristic, complemented
59#15688852C2211yescharacteristic, complemented
60C47×#431C2211yescharacteristic
61C47×#431C2211yescharacteristic
62C2×C4×C8×C47C2211yescharacteristic
63C47×#322C2211yescharacteristic, complemented
64C47×#322C2211yescharacteristic, complemented
65C47×#322C2211yescharacteristic, complemented
66C47×#322C2211yescharacteristic, complemented
67C22×D188C2×C411yescharacteristic
68C22×C4×D47—21no—
69C2×#10625325C2×C411yescharacteristic
70C2×#10625325C2×C411yescharacteristic
71C2×D376C2×C411yescharacteristic
72C2×D376C2×C411yescharacteristic
73C2×#10625324—21no—
74C2×#10625324—21no—
75C2×#10625324—21no—
76C22×C47⋊C8—21no—
77C2×C8×D47—21no—
78C2×C8×D47—21no—
79C2×C8×D47—21no—
80#13094018—21no—
81#13094016C2×C411yescharacteristic
82#13094012C2×C411yescharacteristic
83#13094007—21no—
84#13094003C2×C411yescharacteristic
85#13093998C2×C411yescharacteristic
86C4×Dic94C2311yescharacteristic
87C4×D188—21no—
88C4×D188—21no—
89C4×D188C2×C411yescharacteristic
90C4×D188C2×C411yescharacteristic
91C4×D188C2311yescharacteristic
92C4×D188C2311yescharacteristic
93C4×D188C2311yescharacteristic
94C4×C22⋊D47—21no—
95#13093981—21no—
96C2×C4⋊Dic47C2×C411yescharacteristic
97C2×(C2×D47)⋊C4—21no—
98#13093958C2×C411yescharacteristic
99#13093958C2×C411yescharacteristic
100#13093956C2×C411yescharacteristic
101#13093956C2×C411yescharacteristic
102C4⋊(C47⋊C8)—21no—
103C4⋊(C47⋊C8)C2311yescharacteristic
104C4⋊(C47⋊C8)C2311yescharacteristic
105C4⋊(C47⋊C8)C2311yescharacteristic
106C4⋊(C47⋊C8)C2311yescharacteristic
107Dic47⋊C8—21no—
108Dic47⋊C8—21no—
109#13093949C2×C411yescharacteristic
110#13093949C2×C411yescharacteristic
111#13093946C2×C411yescharacteristic
112#13093946C2×C411yescharacteristic
113#13093946C2×C411yescharacteristic
114#13093946C2×C411yescharacteristic
115#13093946C2×C411yescharacteristic
116#13093946C2×C411yescharacteristic
117C22⋊(C47⋊C8)—21no—
118(C2×D47)⋊C8—21no—
119(C2×D47)⋊C8—21no—
120(C2×D47)⋊C8—21no—
121(C2×D47)⋊C8—21no—
122(C2×D47)⋊C8—21no—
123(C2×D47)⋊C8—21no—
124#13093935—21no—
125#13093934—21no—
126C8×Dic47—21no—
127C22×C8×C47—21no—
128C2×C42×C47C2311yescharacteristic
129C4⋊C8×C47C2×C411yescharacteristic
130C4⋊C8×C47C2×C411yescharacteristic
131C4⋊C8×C47D411yescharacteristic
132C4⋊C8×C47D411yescharacteristic
133C22⋊C8×C47C2×C411yescharacteristic
134C22⋊C8×C47C2×C411yescharacteristic
135C22⋊C8×C47D411yescharacteristic
136C22⋊C8×C47D411yescharacteristic
137C4×C8×C47—21no—
138C4×C8×C47C2311yescharacteristic
139C4×C8×C47C2311yescharacteristic
140C4×C8×C47C2311yescharacteristic
141C4×C8×C47C2311yescharacteristic
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C23×D47—21no—
145C22×C4×C47C22×C411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47#4111yescharacteristic
148C2×C22⋊D47—21no—
149C22×Dic47—21no—
150C2×D188—22no—
151C2×D188—21no—
152C2×D188—41no—
153C2×D188C22×C411yescharacteristic
154C2×D188C22×C411yescharacteristic
155C2×D188C22×C411yescharacteristic
156C2×C4×D47—21no—
157C2×C4×D47—21no—
158C2×C4×D47—21no—
159C2×C4×D47—21no—
160C2×C4×D47—21no—
161C2×C4×D47—81no—
162C2×C4×D47—81no—
163C2×Dic94C22×C411yescharacteristic
164C2×C8×C47—21no—
165C2×C8×C47—21no—
166C2×C8×C47—42no—
167C2×C8×C47C22×C411yescharacteristic
168C2×C8×C47C22×C411yescharacteristic
169C2×C8×C47C22×C411yescharacteristic
170C2×C8×C47C22×C411yescharacteristic
171C2×C8×C47C2×D411yescharacteristic
172C2×C8×C47C2×D411yescharacteristic
173C2×C8×C47C2×D411yescharacteristic
174C2×C8×C47C2×D411yescharacteristic
175C42×C47C22×C411yescharacteristic
176C42×C47C2×D411yescharacteristic
177C42×C47#4111yescharacteristic
178C42×C47C2411yescharacteristic
179C22⋊Dic47—21no—
180(C2×D47)⋊C4—21no—
181(C2×D47)⋊C4—21no—
182(C2×D47)⋊C4—21no—
183(C2×D47)⋊C4—21no—
184(C2×D47)⋊C4—21no—
185(C2×D47)⋊C4—21no—
186(C2×D47)⋊C4—21no—
187(C2×D47)⋊C4—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47C22×C411yescharacteristic
190C4⋊Dic47C22×C411yescharacteristic
191C4⋊Dic47C22×C411yescharacteristic
192C4⋊Dic47C22×C411yescharacteristic
193Dic47⋊C4—21no—
194Dic47⋊C4—21no—
195Dic47⋊C4—21no—
196Dic47⋊C4—21no—
197C4×Dic47—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—81no—
203D376—22no—
204D376—41no—
205#10625325—22no—
206#10625325—41no—
207#10625324—81no—
208#10625324—81no—
209#10625324—81no—
210C8×D47—81no—
211C8×D47—81no—
212C8×D47—81no—
213C23×C47C4×D411yescharacteristic
214C22×D47—21no—
215C22×D47—21no—
216C22×D47—21no—
217C22×D47—42no—
218C22×D47—42no—
219C22×D47—81no—
220C2×C4×C47—22no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47C4×D411yescharacteristic
226C2×C4×C47C4×D411yescharacteristic
227C2×C4×C47C4×D411yescharacteristic
228C2×C4×C47C23×C411yescharacteristic
229C2×C4×C47C22×D411yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C22⋊D47—42no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—81no—
238D188—21no—
239D188—21no—
240D188—21no—
241D188—21no—
242D188—21no—
243D188—42no—
244D188#13112yes—
245C4×D47—41no—
246C4×D47—41no—
247C4×D47—41no—
248C4×D47—41no—
249C4×D47—41no—
250C4×D47—41no—
251C4×D47—81no—
252C4×D47—81no—
253Dic94—21no—
254Dic94—21no—
255C8×C47—21no—
256C8×C47—21no—
257C8×C47—21no—
258C8×C47—21no—
259C8×C47—21no—
260C8×C47—21no—
261C8×C47—21no—
262C8×C47—21no—
263C8×C47—41no—
264C8×C47—41no—
265C47⋊C8—41no—
266C47⋊C8—41no—
267C47⋊C8—41no—
268C47⋊C8—41no—
269#18020—471nomaximal, Sylow
270C22×C47—21no—
271C22×C47—21no—
272C22×C47#44412yes—
273C22×C47C2×C4×D411yescharacteristic
274C2×D47—22no—
275C2×D47—41no—
276C2×D47—41no—
277C2×D47—41no—
278C2×D47—41no—
279C2×D47—41no—
280C2×D47—41no—
281C2×D47—41no—
282C2×D47—41no—
283C2×D47—84no—
284C4×C47—21no—
285C4×C47—21no—
286C4×C47—21no—
287C4×C47—21no—
288C4×C47C2×#13111yescharacteristic
289C4×C47#56811yescharacteristic
290C4×C47#57611yescharacteristic
291C4×C47#57711yescharacteristic
292Dic47—41no—
293Dic47—41no—
294Dic47—41no—
295Dic47—41no—
296Dic47—41no—
297#3105—471no—
298#2994—471no—
299#2974—471no—
300#1594—471no—
301#1592—471no—
302#1583—471no—
303#1582—471no—
304#1581—471no—
305#1581—471no—
306#1580—471no—
307SD16⋊C8—471no—
308#1573—471no—
309#1494—471no—
310C2×D4⋊C8—471no—
311#1452D4711yescharacteristic, complemented
312C2×C47—22no—
313C2×C47#293311yescharacteristic
314C2×C47#294611yescharacteristic
315C2×C47C2×#44411yescharacteristic
316D47—42no—
317D47—81no—
318D47—81no—
319#529—471no—
320C2×C4×D4—471no—
321#484—471no—
322#481—471no—
323#478—471no—
324#478—471no—
325#463—471no—
326#463—471no—
327#450—471no—
328#447—471no—
329#446—471no—
330C8×Q8—471no—
331C4×SD16—471no—
332C4×SD16—471no—
333C4×D8—471no—
334C4×D8—471no—
335#436—471no—
336#436—471no—
337#436—471no—
338#435—941no—
339#435—941no—
340#435—941no—
341C8×D4—471no—
342C8×D4—471no—
343C8×D4—471no—
344C8×D4—941no—
345#433—941no—
346#432—471no—
347#431C2×D4711yescharacteristic
348#431C2×D4711yescharacteristic
349#424—941no—
350C2×C4⋊C8—471no—
351#407—941no—
352C2×C22⋊C8—941no—
353C2×C4×C8C2×D4711yescharacteristic
354#335—471no—
355#335—471no—
356#334—471no—
357#334—471no—
358Q8⋊C8—471no—
359Q8⋊C8—471no—
360D4⋊C8—471no—
361D4⋊C8—471no—
362D4⋊C8—471no—
363D4⋊C8—471no—
364D4⋊C8—471no—
365D4⋊C8—471no—
366#322C2×D4711yescharacteristic, complemented
367#322C2×D4711yescharacteristic, complemented
368#322C2×D4711yescharacteristic, complemented
369#322C2×D4711yescharacteristic, complemented
370C47#1802011yescharacteristic, complemented, Sylow
371C22×D4—471no—
372C23×C4—941no—
373C2×SD16—471no—
374C2×SD16—471no—
375C2×D8—471no—
376C2×D8—471no—
377C2×M16—941no—
378C2×M16—941no—
379C2×M16—941no—
380C22×C8—21no—
381C22×C8—941no—
382C22×C8—941no—
383C22×C8—941no—
384C22×C8—941no—
385#126—941no—
386#125—471no—
387#124—471no—
388#123—941no—
389C22⋊Q8—471no—
390#121—471no—
391C4×Q8—471no—
392C4×D4—471no—
393C4×D4—471no—
394C4×D4—471no—
395C4×D4—471no—
396C4×D4—471no—
397C4×D4—941no—
398C4×D4—941no—
399C4×D4—941no—
400#117—941no—
401C2×C4⋊C4—471no—
402C2×C22⋊C4—941no—
403C2×C42C22×D4711yescharacteristic
404#107—471no—
405#107—471no—
406#106—471no—
407#106—471no—
408C4⋊C8—471no—
409C4⋊C8—471no—
410C4⋊C8—471no—
411C4⋊C8—471no—
412C4⋊C8—941no—
413C4⋊C8—941no—
414C4⋊C8—941no—
415C4⋊C8C4×D4711yescharacteristic
416C4⋊C8C4×D4711yescharacteristic
417C4⋊C8D18811yescharacteristic
418C4⋊C8D18811yescharacteristic
419#103—471no—
420#103—471no—
421#102—471no—
422#102—471no—
423#102—471no—
424#102—471no—
425#102—471no—
426#102—471no—
427C22⋊C8—941no—
428C22⋊C8—941no—
429C22⋊C8—941no—
430C22⋊C8—941no—
431C22⋊C8—941no—
432C22⋊C8—941no—
433C22⋊C8—941no—
434C22⋊C8C4×D4711yescharacteristic
435C22⋊C8C4×D4711yescharacteristic
436C22⋊C8D18811yescharacteristic
437C22⋊C8D18811yescharacteristic
438#97—941no—
439#97—941no—
440C4×C8—21no—
441C4×C8—941no—
442C4×C8C22×D4711yescharacteristic
443C4×C8C22×D4711yescharacteristic
444C4×C8C22×D4711yescharacteristic
445C4×C8C22×D4711yescharacteristic
446C4×C8C22×D4711yescharacteristic
447C4×C8C22×D4711yescharacteristic
448C24—941no—
449C2×Q8—471no—
450C2×D4—471no—
451C2×D4—471no—
452C2×D4—471no—
453C2×D4—942no—
454C2×D4—941no—
455C2×D4—941no—
456C2×D4—1881no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4—941no—
461C22×C4—941no—
462C22×C4—941no—
463C22×C4—3761no—
464C22×C4—3761no—
465C22×C4C2×C4×D4711yescharacteristic
466C22×C4C2×D18811yescharacteristic
467C22×C4#1062535011yescharacteristic
468SD16—942no—
469SD16—1881no—
470D8—942no—
471D8—1881no—
472M16—3761no—
473M16—3761no—
474M16—3761no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—42no—
478C2×C8—941no—
479C2×C8—941no—
480C2×C8—941no—
481C2×C8—941no—
482C2×C8—3761no—
483C2×C8—3761no—
484C2×C8—3761no—
485C2×C8—3761no—
486C2×C8C2×C4×D4711yescharacteristic
487C2×C8C2×C4×D4711yescharacteristic
488C2×C8C2×C4×D4711yescharacteristic
489C2×C8C2×C4×D4711yescharacteristic
490C2×C8C2×D18811yescharacteristic
491C2×C8C2×D18811yescharacteristic
492C2×C8C2×D18811yescharacteristic
493C2×C8C2×D18811yescharacteristic
494C4⋊C4—471no—
495C4⋊C4—471no—
496C4⋊C4—471no—
497C4⋊C4—471no—
498C4⋊C4—941no—
499C4⋊C4—941no—
500C4⋊C4—941no—
501C4⋊C4—941no—
502C4⋊C4—941no—
503C22⋊C4—941no—
504C22⋊C4—941no—
505C22⋊C4—941no—
506C22⋊C4—941no—
507C22⋊C4—941no—
508C22⋊C4—941no—
509C22⋊C4—941no—
510C22⋊C4—941no—
511C22⋊C4—941no—
512C42—941no—
513C42C2×C4×D4711yescharacteristic
514C42C2×D18811yescharacteristic
515C42#1062535011yescharacteristic
516C42C23×D4711yescharacteristic
517C23—941no—
518C23—941no—
519C23—941no—
520C23—3761no—
521C23—1882no—
522C23—1882no—
523C23C4×D18811yescharacteristic
524Q8—941no—
525Q8—941no—
526D4—472no—
527D4—941no—
528D4—941no—
529D4—941no—
530D4—941no—
531D4—941no—
532D4—1882no—
533D4—1882no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—941no—
540C2×C4—941no—
541C2×C4—941no—
542C2×C4—941no—
543C2×C4—941no—
544C2×C4—3761no—
545C2×C4—3761no—
546C2×C4—3761no—
547C2×C4—1881no—
548C2×C4—1881no—
549C2×C4—1881no—
550C2×C4—1881no—
551C2×C4—1881no—
552C2×C4—1881no—
553C2×C4C4×D18811yescharacteristic
554C2×C4C4×D18811yescharacteristic
555C2×C4C4×D18811yescharacteristic
556C2×C4C22×C4×D4711yescharacteristic
557C2×C4C22×D18811yescharacteristic
558C2×C4C2×#1062535011yescharacteristic
559C8—21no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—21no—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—41no—
568C8—41no—
569C8—1881no—
570C8—1881no—
571C8—1881no—
572C8—1881no—
573C22—21no—
574C22—21no—
575C22—942no—
576C22—3764no—
577C22—1881no—
578C22—1881no—
579C22—1881no—
580C22—1881no—
581C22—1881no—
582C22—1881no—
583C22—1881no—
584C22—1881no—
585C22#1568924612yes—
586C22C2×C4×D18811yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—1881no—
592C4—1881no—
593C4—1881no—
594C4—1881no—
595C4—1881no—
596C4#131×D4711yescharacteristic
597C4#1568988811yescharacteristic
598C4#1568992811yescharacteristic
599C4#1568993511yescharacteristic
600C2—22no—
601C2—3761no—
602C2—3761no—
603C2—1882no—
604C2#2025211611yescharacteristic
605C2#2025222211yescharacteristic
606C2C2×#1568924611yescharacteristic
subgroupG/H
centerC2×C4C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1452C2
socleC22×C47C2×C4×D4
prime radical#15689259C4
cosocleC42×C47C24
hypercenter#1452D47
hypocenterC47#18020
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1452

Complements

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

NH ≅ G/Ntype
2. #20252550C2semidirect
3. #20252422C2semidirect
4. #20242303C2semidirect
5. #20242292C2semidirect
6. #20242233C2semidirect
7. #20242226C2semidirect
8. #20242219C2semidirect
9. #20242219C2semidirect
10. #20242213C2semidirect
11. #20242202C2semidirect
12. #20242173C2semidirect
13. #20241849C2semidirect
14. C2×#15688852C2semidirect
15. C47×#1452C2semidirect
26. #15689253C22semidirect
27. C8×Dic94C22semidirect
34. #15689202C22semidirect
40. C8×D188C22semidirect
48. #15688876C22semidirect
49. #15688876C22semidirect
50. #15688874C22semidirect
51. #15688874C22semidirect
52. #15688855C22semidirect
53. #15688855C22semidirect
58. #15688852C22semidirect
59. #15688852C22semidirect
63. C47×#322C22semidirect
64. C47×#322C22semidirect
65. C47×#322C22semidirect
66. C47×#322C22semidirect
311. #1452D47semidirect
366. #322C2×D47semidirect
367. #322C2×D47semidirect
368. #322C2×D47semidirect
369. #322C2×D47semidirect
370. C47#18020semidirect

Maximal subgroups

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

subgroupindex
#202534372
#202525502
#202524222
#202423032
#202422922
#202422332
#202422262
#202422192
#202422132
#202422022
#202421732
#202418492
C2×#156888522
C47×#14522
#1802047