Finite Group Databasebeta

← #26224365

Subgroups — #26224365

Order 12032.

882 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689486C211yesmaximal, characteristic
2#20246125C211yesmaximal, characteristic, complemented
3#20246110C211yesmaximal, characteristic
4#20244351C211yesmaximal, characteristic, complemented
5#20244192C211yesmaximal, characteristic
6#20244086C211yesmaximal, characteristic, complemented
7#20244082C211yesmaximal, characteristic, complemented
8#20244073C211yesmaximal, characteristic, complemented
9#20244069C211yesmaximal, characteristic, complemented
10#20243520C211yesmaximal, characteristic, complemented
11#20243214C211yesmaximal, characteristic, complemented
12C4×#13093944C211yesmaximal, characteristic, complemented
13C4×#13093945C211yesmaximal, characteristic, complemented
14#20243076C211yesmaximal, characteristic, complemented
15C47×#1762C211yesmaximal, characteristic, complemented
16C2×#13094020C2211yescharacteristic
17C2×#13094015C2211yescharacteristic
18C2×C4×D188C2211yescharacteristic
19#15689486C418yescomplemented
20C2×#13093946C2211yescharacteristic
21C2×#13093946C2211yescharacteristic
22C2×#13093945C2211yescharacteristic
23C2×#13093945C2211yescharacteristic
24C2×#13093944C2211yescharacteristic
25C2×#13093944C2211yescharacteristic
26C2×#13093934C2211yescharacteristic
27C2×#13093934C2211yescharacteristic
28C2×C4×C47⋊C8C2211yescharacteristic
29C2×C8×Dic47C2211yescharacteristic
30C2×C8×Dic47C2211yescharacteristic
31#15689026C2211yescharacteristic
32#15689024C2211yescharacteristic
33#15689022C2211yescharacteristic
34#15689021—21no—
35#15689020—21no—
36#15689019C2211yescharacteristic
37#15689007—21no—
38#15689006—21no—
39#15689002—21no—
40#15689002—21no—
41#15688999—21no—
42#15688999—21no—
43#15688992C2211yescharacteristic
44#15688988C2211yescharacteristic
45C4×C4⋊Dic47C2211yescharacteristic
46#15688969C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C2×#102×C47C2211yescharacteristic
55C2×#102×C47C2211yescharacteristic
56C2×#97×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×D188C2311yescharacteristic
61C22×C22⋊D47—21no—
62C22×C22⋊D47—21no—
63C22×C4×D47—21no—
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66#13094020—22no—
67#13094020C2×C414yes—
68#13094015—22no—
69#13094015C2×C414yescomplemented
70C4×D188—22no—
71C4×D188—22no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×Dic47⋊C4—21no—
81C2×C22⋊Dic47—21no—
82C2×C22⋊Dic47—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47C2311yescharacteristic
88C2×C4×Dic47C2311yescharacteristic
89#13093946—22no—
90#13093946C2×C414yescomplemented
91#13093945—22no—
92#13093945C2×C414yescomplemented
93#13093944—22no—
94#13093944C2×C414yescomplemented
95#13093934—22no—
96#13093934C2×C414yescomplemented
97C4×C47⋊C8—22no—
98C8×Dic47—22no—
99C8×Dic47—22no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95D411yescharacteristic
106C47⋊#95D411yescharacteristic
107C47⋊#95D411yescharacteristic
108C47⋊#95D411yescharacteristic
109C22×D4×C47C2311yescharacteristic
110C23×C4×C47—21no—
111C22×C8×C47C2311yescharacteristic
112C22×C8×C47C2311yescharacteristic
113C4×D4×C47—22no—
114C4×D4×C47—22no—
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C22⋊C4×C47—21no—
119C2×C42×C47—21no—
120C2×C42×C47C2311yescharacteristic
121#102×C47—22no—
122#102×C47C2×C414yescomplemented
123#97×C47—22no—
124#95×C47—21no—
125C24×C47—21no—
126C23×D47—21no—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47C22×C412yes—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—42no—
137C22×C4×C47—81no—
138C22×C4×C47—81no—
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2411yescharacteristic
142C2×C22⋊D47—24no—
143C2×C22⋊D47—24no—
144C2×C22⋊D47—81no—
145C2×C22⋊D47—81no—
146C2×C22⋊D47—81no—
147C2×C22⋊D47—81no—
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153C22×Dic47C2×D411yescharacteristic
154C22×Dic47C2×D411yescharacteristic
155C2×D188—21no—
156C2×D188—21no—
157C2×D188—21no—
158C2×D188—21no—
159C2×D188—21no—
160C2×D188C22×C412yes—
161C2×C4×D47—42no—
162C2×C4×D47—81no—
163C2×C4×D47—81no—
164C2×C8×C47—21no—
165C2×C8×C47—21no—
166C2×C8×C47—21no—
167C2×C8×C47—21no—
168C2×C8×C47C22×C412yes—
169C2×C8×C47#4112yes—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47C22×C412yes—
176C22⋊C4×C47—42no—
177C42×C47—81no—
178C42×C47#4112yes—
179C42×C47#4112yes—
180C22⋊Dic47—81no—
181C22⋊Dic47—81no—
182(C2×D47)⋊C4—42no—
183(C2×D47)⋊C4—81no—
184(C2×D47)⋊C4—81no—
185C4⋊Dic47—21no—
186C4⋊Dic47—24no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—81no—
192C4⋊Dic47C22×C412yes—
193Dic47⋊C4—24no—
194Dic47⋊C4—81no—
195Dic47⋊C4—81no—
196Dic47⋊C4—81no—
197C4×Dic47—21no—
198C4×Dic47—21no—
199C4×Dic47—81no—
200C4×Dic47C22×C412yes—
201C4×Dic47#4112yes—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8C22×C412yes—
207C2×C47⋊C8#4112yes—
208C23×C47—22no—
209C23×C47—41no—
210C23×C47—41no—
211C23×C47—41no—
212C23×C47—41no—
213C23×C47—41no—
214C23×C47—41no—
215C23×C47C22×D411yescharacteristic
216C22×D47—22no—
217C22×D47—41no—
218C22×D47—41no—
219C22×D47—41no—
220C22×D47—41no—
221C22×D47—41no—
222C22×D47—41no—
223D4×C47—22no—
224D4×C47—22no—
225D4×C47—41no—
226D4×C47—41no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—22no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—84no—
241C2×C4×C47#12412yes—
242C2×C4×C47#12412yes—
243C2×C4×C47C23×C411yescharacteristic
244C2×C4×C47C2×#4111yescharacteristic
245C2×C4×C47C2×#4111yescharacteristic
246C2×C4×C47C2×#4111yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C22⋊D47—84no—
250C22⋊D47—84no—
251C2×Dic47—21no—
252C2×Dic47—21no—
253C2×Dic47—21no—
254C2×Dic47—21no—
255C2×Dic47—22no—
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47C4×D412yes—
271C2×Dic47C4×D412yes—
272D188—22no—
273D188—22no—
274D188—41no—
275D188—41no—
276C4×D47—84no—
277C8×C47—22no—
278C8×C47—22no—
279C47⋊C8—22no—
280C47⋊C8—22no—
281#21652—471nomaximal, Sylow
282C22×C47—42no—
283C22×C47—42no—
284C22×C47—42no—
285C22×C47—81no—
286C22×C47—81no—
287C22×C47—81no—
288C22×C47—81no—
289C22×C47C2×C4×D411yescharacteristic
290C22×C47C2×#12411yescharacteristic
291C22×C47#53211yescharacteristic
292C22×C47#57611yescharacteristic
293C22×C47#57611yescharacteristic
294C22×C47#57711yescharacteristic
295C22×C47#57711yescharacteristic
296C2×D47—42no—
297C2×D47—42no—
298C2×D47—42no—
299C2×D47—81no—
300C2×D47—81no—
301C2×D47—81no—
302C2×D47—81no—
303C4×C47—22no—
304C4×C47—22no—
305C4×C47—42no—
306C4×C47—81no—
307C4×C47—81no—
308C4×C47C4×#4111yescharacteristic
309C4×C47C4×#4111yescharacteristic
310C4×C47#52911yescharacteristic
311C4×C47#52911yescharacteristic
312Dic47—22no—
313Dic47—22no—
314Dic47—41no—
315Dic47—41no—
316Dic47—42no—
317Dic47—81no—
318Dic47—81no—
319C2×#490—471no—
320#2330—471no—
321#2328—471no—
322#1991—471no—
323#1957—471no—
324#1936—471no—
325#1936—471no—
326#1934—471no—
327#1934—471no—
328#1836—471no—
329#1762—471no—
330#1762D4711yescharacteristic, complemented
331C4×#102—471no—
332C4×#102—471no—
333#1726—471no—
334C2×C47—82no—
335C2×C47C4×#12411yescharacteristic
336C2×C47#294611yescharacteristic
337C2×C47#294611yescharacteristic
338C2×C47#313611yescharacteristic
339C2×C47#313611yescharacteristic
340C2×C47#317411yescharacteristic
341C2×C47#317611yescharacteristic
342D47—82no—
343C2×#127—471no—
344C2×#125—471no—
345C2×C4×D4—471no—
346C2×C4×D4C2×D4711yescharacteristic
347#490—478no—
348C2×#102—471no—
349C2×#102—471no—
350C2×#102—471no—
351C2×#102—471no—
352C2×#102—471no—
353C2×#102—471no—
354C2×#102C2×D4711yescharacteristic
355C2×#102C2×D4711yescharacteristic
356C2×#97—471no—
357C2×#97—471no—
358C2×#97C2×D4711yescharacteristic
359C2×C4×C8—471no—
360C2×C4×C8—471no—
361C2×C4×C8—471no—
362#390—471no—
363#390—471no—
364#389—471no—
365#389—471no—
366#389—941no—
367#389—941no—
368#387—941no—
369#387—941no—
370#386—941no—
371#386—941no—
372#386—941no—
373#386—941no—
374#384—471no—
375#382—471no—
376C4×C4⋊C4—471no—
377C4×C4⋊C4C2×D4711yescharacteristic
378#376—471no—
379#340—471no—
380#340—471no—
381#340—471no—
382#340—471no—
383#340—471no—
384#340—471no—
385#340C2×D4711yescharacteristic
386#340C2×D4711yescharacteristic
387C47#2165211yescharacteristic, complemented, Sylow
388C22×D4—471no—
389C22×D4—941no—
390C22×D4—941no—
391C22×D4C22×D4711yescharacteristic
392C23×C4—21no—
393C23×C4—941no—
394C22×C8—471no—
395C22×C8—471no—
396C22×C8C22×D4711yescharacteristic
397C22×C8C22×D4711yescharacteristic
398#127—474no—
399#127—942no—
400#125—474no—
401#125—942no—
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—942no—
405C4×D4—942no—
406C2×C4⋊C4—471no—
407C2×C4⋊C4—471no—
408C2×C4⋊C4—471no—
409C2×C4⋊C4—941no—
410C2×C4⋊C4—941no—
411C2×C4⋊C4—941no—
412C2×C4⋊C4—941no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4—941no—
415C2×C4⋊C4C22×D4711yescharacteristic
416C2×C4⋊C4C22×D4711yescharacteristic
417C2×C4⋊C4C22×D4711yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—941no—
420C2×C22⋊C4—941no—
421C2×C22⋊C4—941no—
422C2×C22⋊C4—941no—
423C2×C22⋊C4—941no—
424C2×C42—21no—
425C2×C42—471no—
426C2×C42—471no—
427C2×C42—941no—
428C2×C42C22×D4711yescharacteristic
429#102—22no—
430#102—474no—
431#102—474no—
432#102—474no—
433#102—942no—
434#102—942no—
435#102—942no—
436#102C4×D4714yescomplemented
437#97—22no—
438#97—474no—
439#97—942no—
440C4×C8—942no—
441C4×C8—942no—
442C4×C8—942no—
443#95—21no—
444#95—471no—
445#95—471no—
446#95—471no—
447#95—471no—
448#95—941no—
449#95—941no—
450#95—941no—
451#95—941no—
452#95—941no—
453C24—21no—
454C24—941no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—472no—
461C2×D4—944no—
462C2×D4—944no—
463C2×D4—941no—
464C2×D4—941no—
465C2×D4—941no—
466C2×D4—941no—
467C2×D4—941no—
468C2×D4—3761no—
469C2×D4—3761no—
470C2×D4—3761no—
471C2×D4—3761no—
472C2×D4C2×C4×D4712yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—471no—
480C22×C4—471no—
481C22×C4—471no—
482C22×C4—471no—
483C22×C4—941no—
484C22×C4—941no—
485C22×C4—941no—
486C22×C4—3761no—
487C22×C4—3761no—
488C22×C4—1882no—
489C22×C4D4×D4711yescharacteristic
490C22×C4D4×D4711yescharacteristic
491C22×C4C23×D4711yescharacteristic
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—472no—
497C2×C8—472no—
498C2×C8—941no—
499C2×C8—941no—
500C2×C8—941no—
501C2×C8—941no—
502C2×C8C2×C4×D4712yes—
503C2×C8#1062535012yes—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—472no—
510C4⋊C4—944no—
511C4⋊C4—944no—
512C4⋊C4—941no—
513C4⋊C4—941no—
514C4⋊C4—941no—
515C4⋊C4—941no—
516C4⋊C4—941no—
517C4⋊C4—3761no—
518C4⋊C4—3761no—
519C4⋊C4—3761no—
520C4⋊C4—3761no—
521C4⋊C4C2×C4×D4712yes—
522C22⋊C4—42no—
523C22⋊C4—3761no—
524C22⋊C4—3761no—
525C22⋊C4—3761no—
526C22⋊C4—3761no—
527C22⋊C4—1882no—
528C42—81no—
529C42—472no—
530C42—472no—
531C42—941no—
532C42—941no—
533C42—3761no—
534C42#1062535012yes—
535C42#1062535212yes—
536C23—22no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—942no—
544C23—1881no—
545C23—1881no—
546C23—1881no—
547C23—1881no—
548C23—1881no—
549C23—1881no—
550C23C2×D4×D4711yescharacteristic
551D4—22no—
552D4—22no—
553D4—41no—
554D4—41no—
555D4—942no—
556D4—942no—
557D4—3764no—
558D4—3764no—
559D4—1881no—
560D4—1881no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—84no—
575C2×C4—472no—
576C2×C4—472no—
577C2×C4—941no—
578C2×C4—941no—
579C2×C4—941no—
580C2×C4—941no—
581C2×C4—942no—
582C2×C4—941no—
583C2×C4—941no—
584C2×C4—941no—
585C2×C4—941no—
586C2×C4—941no—
587C2×C4—941no—
588C2×C4—3764no—
589C2×C4—1881no—
590C2×C4—1881no—
591C2×C4—1881no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4—1881no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4#1309401012yes—
598C2×C4#1309401012yes—
599C2×C4C22×C4×D4711yescharacteristic
600C2×C4#41×D4711yescharacteristic
601C2×C4#41×D4711yescharacteristic
602C2×C4C2×#1062535211yescharacteristic
603C2×C4C2×#1062535011yescharacteristic
604C2×C4C2×#1062535011yescharacteristic
605C8—22no—
606C8—22no—
607C8—942no—
608C8—942no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—3761no—
617C22—3761no—
618C22—3761no—
619C22—3761no—
620C22—1882no—
621C22—1882no—
622C22—1882no—
623C22C4×D4×D4711yescharacteristic
624C22C2×#1309401011yescharacteristic
625C22#1568967011yescharacteristic
626C22#1568992711yescharacteristic
627C22#1568992711yescharacteristic
628C22#1568993211yescharacteristic
629C22#1568993311yescharacteristic
630C4—22no—
631C4—22no—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—942no—
636C4—942no—
637C4—3761no—
638C4—3761no—
639C4—1882no—
640C4—1881no—
641C4—1881no—
642C4C4×#1062535211yescharacteristic
643C4C4×#1062535011yescharacteristic
644C4#1568965111yescharacteristic
645C4#1568965311yescharacteristic
646C2—82no—
647C2—3762no—
648C2C4×#1309401011yescharacteristic
649C2#2025221711yescharacteristic
650C2#2025221911yescharacteristic
651C2#2025373811yescharacteristic
652C2#2025374511yescharacteristic
653C2#2025401411yescharacteristic
654C2#2025403611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1762C2
socleC23×C47C22×D4
prime radicalC2×#13094020C22
cosocleC22×C4×C47C24
hypercenter#1762D47
hypocenterC47#21652
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. #20246125C2semidirect
4. #20244351C2semidirect
6. #20244086C2semidirect
7. #20244082C2semidirect
8. #20244073C2semidirect
9. #20244069C2semidirect
10. #20243520C2semidirect
11. #20243214C2semidirect
12. C4×#13093944C2semidirect
13. C4×#13093945C2semidirect
14. #20243076C2semidirect
15. C47×#1762C2semidirect
19. #15689486C4semidirect
69. #13094015C2×C4semidirect
90. #13093946C2×C4semidirect
92. #13093945C2×C4semidirect
94. #13093944C2×C4semidirect
96. #13093934C2×C4semidirect
122. #102×C47C2×C4semidirect
330. #1762D47semidirect
387. C47#21652semidirect
436. #102C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156894862
#202461252
#202461102
#202443512
#202441922
#202440862
#202440822
#202440732
#202440692
#202435202
#202432142
C4×#130939442
C4×#130939452
#202430762
C47×#17622
#2165247