Finite Group Databasebeta

← #26232951

Subgroups — #26232951

Order 12032.

1066 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689833C211yesmaximal, characteristic
2#20246345C211yesmaximal, characteristic, complemented
3#20244361C211yesmaximal, characteristic, complemented
4#20244357C211yesmaximal, characteristic, complemented
5#20244353C211yesmaximal, characteristic, complemented
6#20244343C211yesmaximal, characteristic, complemented
7#20244343C211yesmaximal, characteristic, complemented
8#20244339C211yesmaximal, characteristic, complemented
9#20243399C211yesmaximal, characteristic
10#20243399C211yesmaximal, characteristic
11#20243389C211yesmaximal, characteristic
12#20243214C211yesmaximal, characteristic
13#20243214C211yesmaximal, characteristic
14C4×#13093946C211yesmaximal, characteristic
15C2×C47×#446C211yesmaximal, characteristic, complemented
16#15689833—24no—
17#15689833C418yescomplemented
18C2×#13094024C2211yescharacteristic
19C2×#13094016C2211yescharacteristic
20C2×#13094016C2211yescharacteristic
21C2×#13094002—21no—
22C2×#13094002—21no—
23C2×#13094000—21no—
24C2×C4×D188C2211yescharacteristic
25C2×C4×D188C2211yescharacteristic
26C2×C4×D188C2211yescharacteristic
27C2×C4⋊C4×D47—21no—
28C2×#13093946C2211yescharacteristic
29C2×#13093946C2211yescharacteristic
30C2×#13093946C2211yescharacteristic
31C2×#13093946C2211yescharacteristic
32#15689030—21no—
33#15689009—21no—
34#15689009—21no—
35#15688993C2211yescharacteristic
36#15688989C2211yescharacteristic
37#15688989C2211yescharacteristic
38C4⋊C4×Dic47—21no—
39C4×C4⋊Dic47C2211yescharacteristic
40C4×C4⋊Dic47C2211yescharacteristic
41C4×C4⋊Dic47C2211yescharacteristic
42#15688886C2211yescharacteristic
43#15688886C2211yescharacteristic
44#15688886C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688886C2211yescharacteristic
50#15688886C2211yescharacteristic
51#15688886C2211yescharacteristic
52#15688886C2211yescharacteristic
53#15688886C2211yescharacteristic
54C2×C4×Q8×C47C2211yescharacteristic
55C47×#446—24no—
56C2×C4⋊C8×C47C2211yescharacteristic
57C2×C4⋊C8×C47C2211yescharacteristic
58C2×C4⋊C8×C47C2211yescharacteristic
59C2×#97×C47C2211yescharacteristic
60C2×#97×C47C2211yescharacteristic
61C2×C4×C8×C47C2211yescharacteristic
62C22×Dic94—21no—
63C22×D188C2311yescharacteristic
64C22×C4×D47—21no—
65C22×C4×D47—21no—
66C22×C4×D47—21no—
67#13094024—22no—
68#13094024C2×C414yes—
69#13094016—22no—
70#13094016—22no—
71#13094016C2×C414yes—
72#13094016C2×C414yes—
73#13094002—28no—
74#13094002—28no—
75#13094000—28no—
76C4×D188—22no—
77C4×D188—22no—
78C4×D188—22no—
79C4×D188C2×C414yes—
80C4×D188C2×C414yes—
81C4×D188C2×C414yes—
82C2×C4⋊Dic47—21no—
83C2×C4⋊Dic47—21no—
84C2×C4⋊Dic47—21no—
85C2×C4⋊Dic47C2311yescharacteristic
86C2×C4⋊Dic47C2311yescharacteristic
87C2×C4⋊Dic47C2311yescharacteristic
88C2×C4⋊Dic47C2311yescharacteristic
89C2×C4⋊Dic47C2311yescharacteristic
90C2×C4⋊Dic47C2311yescharacteristic
91C2×C4⋊Dic47C2311yescharacteristic
92C2×Dic47⋊C4—21no—
93C2×Dic47⋊C4—21no—
94C2×Dic47⋊C4—21no—
95C2×Dic47⋊C4—21no—
96C4⋊C4×D47—28no—
97C2×(C2×D47)⋊C4—21no—
98C2×(C2×D47)⋊C4—21no—
99C2×(C2×D47)⋊C4—21no—
100C2×C4×Dic47—21no—
101C2×C4×Dic47—21no—
102C2×C4×Dic47—21no—
103#13093946—22no—
104#13093946—22no—
105#13093946—22no—
106#13093946Q814yescomplemented
107C47⋊#95—21no—
108C47⋊#95—21no—
109C47⋊#95—21no—
110C47⋊#95—21no—
111C47⋊#95—21no—
112C47⋊#95—21no—
113C47⋊#95—21no—
114C22×Q8×C47D411yescharacteristic
115C22×C8×C47C2311yescharacteristic
116C22×C8×C47C2311yescharacteristic
117C22×C8×C47C2311yescharacteristic
118C22×C8×C47C2311yescharacteristic
119C4×Q8×C47C2×C414yes—
120C4×Q8×C47D414yes—
121C2×C4⋊C4×C47D411yescharacteristic
122C2×C4⋊C4×C47D411yescharacteristic
123C2×C4⋊C4×C47D411yescharacteristic
124C2×C42×C47C2311yescharacteristic
125C2×C42×C47C2311yescharacteristic
126C2×C42×C47C2311yescharacteristic
127C4⋊C8×C47—22no—
128C4⋊C8×C47—22no—
129C4⋊C8×C47—22no—
130#97×C47—22no—
131#97×C47—22no—
132C4×C8×C47—22no—
133C23×D47—21no—
134C2×Q8×C47—41no—
135C2×Q8×C47—41no—
136C2×Q8×C47C22⋊C414yes—
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2411yescharacteristic
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—21no—
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C2×D188—21no—
152C2×D188—21no—
153C2×D188—21no—
154C2×D188C22×C412yes—
155C2×D188C2×Q812yes—
156C2×D188#4112yes—
157C2×C4×D47—24no—
158C2×C4×D47—24no—
159C2×C4×D47—24no—
160C2×C4×D47—42no—
161C2×C4×D47—42no—
162C2×C4×D47—81no—
163C2×C4×D47—81no—
164C2×C4×D47—81no—
165C2×C4×D47—81no—
166C2×Dic94—24no—
167C2×Dic94—81no—
168C2×Dic94—81no—
169C2×C8×C47—21no—
170C2×C8×C47—21no—
171C2×C8×C47—21no—
172C2×C8×C47—21no—
173C2×C8×C47—21no—
174C2×C8×C47—21no—
175C2×C8×C47—21no—
176C2×C8×C47—21no—
177C2×C8×C47—21no—
178C2×C8×C47—21no—
179C2×C8×C47—21no—
180C2×C8×C47C2×Q812yes—
181C4⋊C4×C47C22⋊C414yes—
182C4⋊C4×C47C22⋊C414yes—
183C4⋊C4×C47C22⋊C414yes—
184C42×C47C22×C412yes—
185C42×C47C22×C412yes—
186C42×C47C22×C412yes—
187C42×C47C2×D412yes—
188C42×C47C2×D412yes—
189C42×C47C2×D412yes—
190(C2×D47)⋊C4—24no—
191(C2×D47)⋊C4—24no—
192(C2×D47)⋊C4—24no—
193C4⋊Dic47—24no—
194C4⋊Dic47—21no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47—21no—
200C4⋊Dic47—21no—
201C4⋊Dic47—21no—
202C4⋊Dic47—21no—
203C4⋊Dic47—21no—
204C4⋊Dic47—42no—
205C4⋊Dic47—81no—
206C4⋊Dic47C22×C412yes—
207C4⋊Dic47C22×C412yes—
208C4⋊Dic47C22×C412yes—
209C4⋊Dic47C2×Q812yes—
210Dic47⋊C4—24no—
211Dic47⋊C4—24no—
212Dic47⋊C4—24no—
213Dic47⋊C4—24no—
214C4×Dic47—42no—
215C4×Dic47—81no—
216C4×Dic47—81no—
217C23×C47C22×D411yescharacteristic
218C22×D47—22no—
219C22×D47—22no—
220C22×D47—22no—
221C22×D47—41no—
222C22×D47—41no—
223C22×D47—41no—
224C22×D47—41no—
225Q8×C47—44no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47C2×C22⋊C412yes—
239C2×C4×C47C2×C22⋊C412yes—
240C2×C4×C47C2×C22⋊C412yes—
241C2×C4×C47C2×C22⋊C412yes—
242C2×C4×C47C2×C22⋊C412yes—
243C2×C4×C47C2×C22⋊C412yes—
244C2×C4×C47C23×C411yescharacteristic
245C2×C4×C47C22×D411yescharacteristic
246C2×C4×C47C22×Q811yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C2×C4×C472−1+411yescharacteristic
249C2×C4×C472−1+411yescharacteristic
250C2×Dic47—22no—
251C2×Dic47—22no—
252C2×Dic47—22no—
253C2×Dic47—22no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
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×Dic47—41no—
271D188—22no—
272D188—22no—
273D188—22no—
274D188C4×Q814yes—
275C4×D47—48no—
276C4×D47—84no—
277C4×D47—84no—
278Dic94—84no—
279C8×C47—22no—
280C8×C47—41no—
281C8×C47—41no—
282C8×C47—41no—
283#22526—471nomaximal, Sylow
284C22×C47C22×C22⋊C411yescharacteristic
285C22×C47#54811yescharacteristic
286C22×C47D4×Q811yescharacteristic
287C22×C47#57511yescharacteristic
288C22×C47#57511yescharacteristic
289C22×C47#57611yescharacteristic
290C22×C47#57711yescharacteristic
291C2×D47—24no—
292C2×D47—42no—
293C2×D47—42no—
294C2×D47—42no—
295C2×D47—42no—
296C2×D47—42no—
297C2×D47—42no—
298C4×C47—22no—
299C4×C47—22no—
300C4×C47—22no—
301C4×C47—22no—
302C4×C47—22no—
303C4×C47—22no—
304C4×C47C2×C4×Q811yescharacteristic
305C4×C47#51911yescharacteristic
306C4×C47#54811yescharacteristic
307C4×C47D4×Q811yescharacteristic
308Dic47—42no—
309Dic47—42no—
310Dic47—42no—
311Dic47—42no—
312Dic47—81no—
313Dic47—81no—
314Dic47—81no—
315C2×#554—471no—
316C2×#446D4711yescharacteristic, complemented
317#2355—471no—
318#1993—471no—
319#1992—471no—
320#1991—471no—
321#1989—471no—
322#1989—471no—
323#1988—471no—
324#1804—471no—
325#1804—471no—
326#1802—471no—
327#1762—471no—
328#1762—471no—
329C4×#102—471no—
330C2×C47Q8×C22⋊C411yescharacteristic
331C2×C47#289611yescharacteristic
332C2×C47#289711yescharacteristic
333C2×C47#337011yescharacteristic
334C2×C47#337611yescharacteristic
335C2×C47#338811yescharacteristic
336C2×C47#338911yescharacteristic
337D47—44no—
338#554—478no—
339#554—944no—
340C2×C4⋊Q8—471no—
341C2×#125—471no—
342C2×#125—471no—
343C2×C22⋊Q8—941no—
344C2×C22⋊Q8—941no—
345C2×C22⋊Q8—941no—
346C2×C4×Q8C2×D4711yescharacteristic
347C2×C4×D4—471no—
348C2×C4×D4—471no—
349C2×C4×D4—471no—
350C22×C4⋊C4—941no—
351#446—24no—
352C2×C4⋊C8C2×D4711yescharacteristic
353C2×C4⋊C8C2×D4711yescharacteristic
354C2×C4⋊C8C2×D4711yescharacteristic
355C2×#102—471no—
356C2×#102—471no—
357C2×#102—471no—
358C2×#102—471no—
359C2×#97C2×D4711yescharacteristic
360C2×#97C2×D4711yescharacteristic
361C2×C4×C8C2×D4711yescharacteristic
362#391—941no—
363#387—941no—
364#387—941no—
365#384—471no—
366#382—471no—
367#382—471no—
368C4×C4⋊C4—471no—
369C4×C4⋊C4—471no—
370C4×C4⋊C4—471no—
371C4×C4⋊C4—941no—
372#340—471no—
373#340—471no—
374#340—471no—
375#340—471no—
376#340—471no—
377#340—471no—
378#340—471no—
379#340—471no—
380#340—471no—
381#340—471no—
382#340—471no—
383#340—471no—
384C47#2252611yescharacteristic, complemented, Sylow
385C22×Q8—941no—
386C22×Q8D18811yescharacteristic
387C22×D4—471no—
388C23×C4—941no—
389C23×C4—941no—
390C23×C4—941no—
391C22×C8C22×D4711yescharacteristic
392C22×C8C22×D4711yescharacteristic
393C22×C8C22×D4711yescharacteristic
394C22×C8C22×D4711yescharacteristic
395C4⋊Q8—474no—
396C4⋊Q8—942no—
397#125—474no—
398#125—474no—
399#125—942no—
400#125—942no—
401C22⋊Q8—948no—
402C22⋊Q8—948no—
403C22⋊Q8—948no—
404C4×Q8C4×D4714yes—
405C4×Q8C22⋊D4714yes—
406C4×D4—474no—
407C4×D4—474no—
408C4×D4—474no—
409C4×D4—942no—
410C4×D4—942no—
411C4×D4—942no—
412C2×C4⋊C4—471no—
413C2×C4⋊C4—471no—
414C2×C4⋊C4—471no—
415C2×C4⋊C4—471no—
416C2×C4⋊C4—471no—
417C2×C4⋊C4—471no—
418C2×C4⋊C4—471no—
419C2×C4⋊C4—941no—
420C2×C4⋊C4—941no—
421C2×C4⋊C4—941no—
422C2×C4⋊C4—941no—
423C2×C4⋊C4—941no—
424C2×C4⋊C4—948no—
425C2×C4⋊C4—941no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4D18811yescharacteristic
428C2×C4⋊C4D18811yescharacteristic
429C2×C4⋊C4D18811yescharacteristic
430C2×C22⋊C4—941no—
431C2×C22⋊C4—941no—
432C2×C22⋊C4—941no—
433C2×C42—941no—
434C2×C42—941no—
435C2×C42—941no—
436C2×C42C22×D4711yescharacteristic
437C2×C42C22×D4711yescharacteristic
438C2×C42C22×D4711yescharacteristic
439C4⋊C8—22no—
440C4⋊C8—22no—
441C4⋊C8—22no—
442#102—474no—
443#102—942no—
444#102—942no—
445#102—942no—
446#97—22no—
447#97—22no—
448C4×C8—22no—
449#95—941no—
450#95—941no—
451#95—941no—
452#95—941no—
453#95—941no—
454#95—941no—
455#95—941no—
456C24—941no—
457C2×Q8—41no—
458C2×Q8—41no—
459C2×Q8—944no—
460C2×Q8—3761no—
461C2×Q8—3761no—
462C2×Q8(C2×D47)⋊C414yes—
463C2×D4—472no—
464C2×D4—472no—
465C2×D4—472no—
466C2×D4—941no—
467C2×D4—941no—
468C2×D4—941no—
469C22×C4—941no—
470C22×C4—941no—
471C22×C4—941no—
472C22×C4—941no—
473C22×C4—944no—
474C22×C4—944no—
475C22×C4—944no—
476C22×C4—941no—
477C22×C4—941no—
478C22×C4—941no—
479C22×C4—3761no—
480C22×C4—3761no—
481C22×C4—3761no—
482C22×C4—3761no—
483C22×C4—1882no—
484C22×C4—1882no—
485C22×C4C2×D18811yescharacteristic
486C22×C4C2×D18811yescharacteristic
487C22×C4C2×D18811yescharacteristic
488C22×C4C2×D18811yescharacteristic
489C22×C4C2×D18811yescharacteristic
490C22×C4C2×D18811yescharacteristic
491C22×C4C23×D4711yescharacteristic
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8Q8×D4712yes—
504C4⋊C4—472no—
505C4⋊C4—472no—
506C4⋊C4—472no—
507C4⋊C4—472no—
508C4⋊C4—944no—
509C4⋊C4—944no—
510C4⋊C4—944no—
511C4⋊C4—944no—
512C4⋊C4—944no—
513C4⋊C4—941no—
514C4⋊C4—941no—
515C4⋊C4—941no—
516C4⋊C4—941no—
517C4⋊C4—941no—
518C4⋊C4—941no—
519C4⋊C4—941no—
520C4⋊C4—941no—
521C4⋊C4—941no—
522C4⋊C4—941no—
523C4⋊C4—3761no—
524C4⋊C4—1882no—
525C4⋊C4(C2×D47)⋊C414yes—
526C4⋊C4(C2×D47)⋊C414yes—
527C4⋊C4(C2×D47)⋊C414yes—
528C22⋊C4—944no—
529C22⋊C4—944no—
530C22⋊C4—944no—
531C42—3761no—
532C42—3761no—
533C42—1882no—
534C42C2×C4×D4712yes—
535C42C2×C4×D4712yes—
536C42C2×C4×D4712yes—
537C42C2×C22⋊D4712yes—
538C42C2×C22⋊D4712yes—
539C42C2×C22⋊D4712yes—
540C23—942no—
541C23—942no—
542C23—942no—
543C23—1881no—
544C23—1881no—
545C23—1881no—
546C23—1881no—
547C23C22×D18811yescharacteristic
548Q8—44no—
549Q8—3764no—
550D4—474no—
551D4—942no—
552D4—942no—
553D4—942no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—942no—
567C2×C4—942no—
568C2×C4—942no—
569C2×C4—942no—
570C2×C4—3764no—
571C2×C4—3764no—
572C2×C4—1888no—
573C2×C4—1881no—
574C2×C4—1881no—
575C2×C4—1881no—
576C2×C4—1881no—
577C2×C4—1881no—
578C2×C4—1881no—
579C2×C4—1881no—
580C2×C4—1881no—
581C2×C4—1881no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4—1881no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4—1881no—
589C2×C4—1881no—
590C2×C4C2×(C2×D47)⋊C412yes—
591C2×C4C2×(C2×D47)⋊C412yes—
592C2×C4C2×(C2×D47)⋊C412yes—
593C2×C4C2×(C2×D47)⋊C412yes—
594C2×C4C2×(C2×D47)⋊C412yes—
595C2×C4C2×(C2×D47)⋊C412yes—
596C2×C4C22×C4×D4711yescharacteristic
597C2×C4C22×C22⋊D4711yescharacteristic
598C2×C4C2×Q8×D4711yescharacteristic
599C2×C4#41×D4711yescharacteristic
600C2×C4#1309407111yescharacteristic
601C2×C4#1309407211yescharacteristic
602C8—22no—
603C8—41no—
604C8—41no—
605C8—41no—
606C22—944no—
607C22—1882no—
608C22—1882no—
609C22—1882no—
610C22—1882no—
611C22—1882no—
612C22—1882no—
613C22C22×(C2×D47)⋊C411yescharacteristic
614C22#1568978511yescharacteristic
615C22Q8×D18811yescharacteristic
616C22#1568992311yescharacteristic
617C22#1568992311yescharacteristic
618C22#1568992811yescharacteristic
619C22#1568993511yescharacteristic
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—3761no—
627C4—3761no—
628C4—3761no—
629C4—1882no—
630C4—1882no—
631C4—1882no—
632C4—1882no—
633C4C4×Q8×D4711yescharacteristic
634C4#1568959911yescharacteristic
635C4#1568978011yescharacteristic
636C4Q8×C22⋊D4711yescharacteristic
637C2—1884no—
638C2Q8×(C2×D47)⋊C411yescharacteristic
639C2#2025190811yescharacteristic
640C2#2025191511yescharacteristic
641C2#2025614311yescharacteristic
642C2#2025621711yescharacteristic
643C2#2025633511yescharacteristic
644C2#2025634811yescharacteristic
subgroupG/H
centerC23C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#446C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenterC2×#446D47
hypocenterC47#22526
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
2. #20246345C2semidirect
3. #20244361C2semidirect
4. #20244357C2semidirect
5. #20244353C2semidirect
6. #20244343C2semidirect
7. #20244343C2semidirect
8. #20244339C2semidirect
15. C2×C47×#446C2semidirect
17. #15689833C4semidirect
106. #13093946Q8semidirect
316. C2×#446D47semidirect
384. C47#22526semidirect

Maximal subgroups

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

subgroupindex
C2×#156898332
#202463452
#202443612
#202443572
#202443532
#202443432
#202443392
#202433992
#202433892
#202432142
C4×#130939462
C2×C47×#4462
#2252647