Finite Group Databasebeta

← #26232943

Subgroups — #26232943

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
2C2×#15689254C211yesmaximal, characteristic, complemented
3#20244360C211yesmaximal, characteristic, complemented
4#20244356C211yesmaximal, characteristic, complemented
5#20244352C211yesmaximal, characteristic, complemented
6#20244345C211yesmaximal, characteristic, complemented
7#20244345C211yesmaximal, characteristic, complemented
8#20244340C211yesmaximal, characteristic, complemented
9#20243395C211yesmaximal, characteristic
10#20243395C211yesmaximal, characteristic
11#20243385C211yesmaximal, characteristic
12#20243216C211yesmaximal, characteristic
13#20243216C211yesmaximal, characteristic
14C4×#13093944C211yesmaximal, characteristic
15C47×#2355C211yesmaximal, 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—
28#15689254—24no—
29C2×C4⋊(C47⋊C8)C2211yescharacteristic
30C2×C4⋊(C47⋊C8)C2211yescharacteristic
31C2×C4⋊(C47⋊C8)C2211yescharacteristic
32C2×#13093944C2211yescharacteristic
33C2×#13093944C2211yescharacteristic
34C2×#13093944C2211yescharacteristic
35C2×#13093944C2211yescharacteristic
36C2×#13093935C2211yescharacteristic
37C2×#13093935C2211yescharacteristic
38C2×C4×C47⋊C8C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51C2×C4×Q8×C47C2211yescharacteristic
52C47×#391—21no—
53C47×#387—21no—
54C47×#387—21no—
55C47×#384C2211yescharacteristic
56C47×#382C2211yescharacteristic
57C47×#382C2211yescharacteristic
58C4×C4⋊C4×C47—21no—
59C4×C4⋊C4×C47C2211yescharacteristic
60C4×C4⋊C4×C47C2211yescharacteristic
61C4×C4⋊C4×C47C2211yescharacteristic
62C22×Dic94—21no—
63C22×D188C2311yescharacteristic
64C22×C4×D47—21no—
65C22×C4×D47—21no—
66C22×C4×D47—21no—
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70C22×C47⋊C8C2311yescharacteristic
71#13094024—22no—
72#13094024C2×C414yes—
73#13094016—22no—
74#13094016—22no—
75#13094016C2×C414yes—
76#13094016C2×C414yes—
77#13094002—28no—
78#13094002—28no—
79#13094000—28no—
80C4×D188—22no—
81C4×D188—22no—
82C4×D188—22no—
83C4×D188C2×C414yes—
84C4×D188C2×C414yes—
85C4×D188C2×C414yes—
86C2×C4⋊Dic47—21no—
87C2×C4⋊Dic47C2311yescharacteristic
88C2×C4⋊Dic47C2311yescharacteristic
89C2×C4⋊Dic47C2311yescharacteristic
90C2×Dic47⋊C4—21no—
91C2×Dic47⋊C4—21no—
92C2×Dic47⋊C4—21no—
93C2×Dic47⋊C4—21no—
94C4⋊C4×D47—28no—
95C2×(C2×D47)⋊C4—21no—
96C2×(C2×D47)⋊C4—21no—
97C2×(C2×D47)⋊C4—21no—
98C4⋊(C47⋊C8)—22no—
99C4⋊(C47⋊C8)—22no—
100C4⋊(C47⋊C8)—22no—
101#13093944—22no—
102#13093944—22no—
103#13093944—22no—
104#13093944Q814yescomplemented
105#13093935—22no—
106#13093935—22no—
107C4×C47⋊C8—22no—
108C22×Q8×C47D411yescharacteristic
109C4×Q8×C47C2×C414yes—
110C4×Q8×C47D414yes—
111C2×C4⋊C4×C47—21no—
112C2×C4⋊C4×C47—21no—
113C2×C4⋊C4×C47D411yescharacteristic
114C2×C4⋊C4×C47D411yescharacteristic
115C2×C4⋊C4×C47D411yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C4⋊C4×C47C2311yescharacteristic
120C2×C42×C47—21no—
121C2×C42×C47—21no—
122C2×C42×C47—21no—
123C2×C42×C47C2311yescharacteristic
124C2×C42×C47C2311yescharacteristic
125C2×C42×C47C2311yescharacteristic
126#95×C47—21no—
127#95×C47—21no—
128#95×C47—21no—
129#95×C47—21no—
130#95×C47—21no—
131#95×C47—21no—
132#95×C47—21no—
133C23×D47—21no—
134C2×Q8×C47—41no—
135C2×Q8×C47—41no—
136C2×Q8×C47C22⋊C414yes—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2411yescharacteristic
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—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—42no—
177C4⋊C4×C47—81no—
178C4⋊C4×C47C22⋊C414yes—
179C4⋊C4×C47C22⋊C414yes—
180C4⋊C4×C47C22⋊C414yes—
181C4⋊C4×C47C2×Q812yes—
182C42×C47—42no—
183C42×C47—81no—
184C42×C47—81no—
185C42×C47C22×C412yes—
186C42×C47C22×C412yes—
187C42×C47C22×C412yes—
188C42×C47C2×D412yes—
189C42×C47C2×D412yes—
190C42×C47C2×D412yes—
191(C2×D47)⋊C4—24no—
192(C2×D47)⋊C4—24no—
193(C2×D47)⋊C4—24no—
194C4⋊Dic47—24no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47C22×C412yes—
199C4⋊Dic47C22×C412yes—
200C4⋊Dic47C22×C412yes—
201Dic47⋊C4—24no—
202Dic47⋊C4—24no—
203Dic47⋊C4—24no—
204Dic47⋊C4—24no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8—21no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8C2×Q812yes—
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×C47—22no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47C2×C22⋊C412yes—
251C2×C4×C47C2×C22⋊C412yes—
252C2×C4×C47C2×C22⋊C412yes—
253C2×C4×C47C2×C22⋊C412yes—
254C2×C4×C47C2×C22⋊C412yes—
255C2×C4×C47C2×C22⋊C412yes—
256C2×C4×C47C23×C411yescharacteristic
257C2×C4×C47C22×D411yescharacteristic
258C2×C4×C47C22×Q811yescharacteristic
259C2×C4×C47C2×#4111yescharacteristic
260C2×C4×C472−1+411yescharacteristic
261C2×C4×C472−1+411yescharacteristic
262C2×Dic47—22no—
263C2×Dic47—22no—
264C2×Dic47—22no—
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—
279C47⋊C8—22no—
280C47⋊C8—41no—
281C47⋊C8—41no—
282C47⋊C8—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×C47—42no—
305C4×C47—81no—
306C4×C47—81no—
307C4×C47—81no—
308C4×C47C2×C4×Q811yescharacteristic
309C4×C47#51911yescharacteristic
310C4×C47#54811yescharacteristic
311C4×C47D4×Q811yescharacteristic
312Dic47—42no—
313Dic47—42no—
314Dic47—42no—
315C2×#554—471no—
316C2×#446—471no—
317#2355D4711yescharacteristic, complemented
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—944no—
352C2×C4⋊C8—471no—
353C2×C4⋊C8—471no—
354C2×C4⋊C8—471no—
355C2×#102—471no—
356C2×#102—471no—
357C2×#102—471no—
358C2×#102—471no—
359C2×#97—471no—
360C2×#97—471no—
361C2×C4×C8—471no—
362#391—21no—
363#387—21no—
364#387—21no—
365#384C2×D4711yescharacteristic
366#382C2×D4711yescharacteristic
367#382C2×D4711yescharacteristic
368C4×C4⋊C4—21no—
369C4×C4⋊C4C2×D4711yescharacteristic
370C4×C4⋊C4C2×D4711yescharacteristic
371C4×C4⋊C4C2×D4711yescharacteristic
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×Q8C22⋊D4711yescharacteristic
387C22×D4—471no—
388C23×C4—941no—
389C23×C4—941no—
390C23×C4—941no—
391C22×C8—471no—
392C22×C8—471no—
393C22×C8—471no—
394C22×C8—471no—
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×Q8D18814yes—
406C4×D4—474no—
407C4×D4—474no—
408C4×D4—474no—
409C4×D4—942no—
410C4×D4—942no—
411C4×D4—942no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—471no—
415C2×C4⋊C4—471no—
416C2×C4⋊C4—471no—
417C2×C4⋊C4—941no—
418C2×C4⋊C4—941no—
419C2×C4⋊C4—941no—
420C2×C4⋊C4—941no—
421C2×C4⋊C4—941no—
422C2×C4⋊C4—948no—
423C2×C4⋊C4C22⋊D4711yescharacteristic
424C2×C4⋊C4C22⋊D4711yescharacteristic
425C2×C4⋊C4C22⋊D4711yescharacteristic
426C2×C4⋊C4C22×D4711yescharacteristic
427C2×C4⋊C4C22×D4711yescharacteristic
428C2×C4⋊C4C22×D4711yescharacteristic
429C2×C4⋊C4C22×D4711yescharacteristic
430C2×C22⋊C4—941no—
431C2×C22⋊C4—941no—
432C2×C22⋊C4—941no—
433C2×C42—21no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42C22×D4711yescharacteristic
437C2×C42C22×D4711yescharacteristic
438C2×C42C22×D4711yescharacteristic
439C4⋊C8—942no—
440C4⋊C8—942no—
441C4⋊C8—942no—
442#102—474no—
443#102—942no—
444#102—942no—
445#102—942no—
446#97—942no—
447#97—942no—
448C4×C8—942no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—21no—
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—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—941no—
474C22×C4—941no—
475C22×C4—941no—
476C22×C4—944no—
477C22×C4—944no—
478C22×C4—944no—
479C22×C4—3761no—
480C22×C4—3761no—
481C22×C4—3761no—
482C22×C4—3761no—
483C22×C4—1882no—
484C22×C4—1882no—
485C22×C4C2×C22⋊D4711yescharacteristic
486C22×C4C2×C22⋊D4711yescharacteristic
487C22×C4C2×C22⋊D4711yescharacteristic
488C22×C4C2×C22⋊D4711yescharacteristic
489C22×C4C2×C22⋊D4711yescharacteristic
490C22×C4C2×C22⋊D4711yescharacteristic
491C22×C4C23×D4711yescharacteristic
492C2×C8—472no—
493C2×C8—941no—
494C2×C8—941no—
495C2×C8—941no—
496C2×C8—941no—
497C2×C8—941no—
498C2×C8—941no—
499C2×C8—941no—
500C2×C8—941no—
501C2×C8—941no—
502C2×C8—941no—
503C2×C8—941no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—42no—
512C4⋊C4—81no—
513C4⋊C4—472no—
514C4⋊C4—472no—
515C4⋊C4—472no—
516C4⋊C4—944no—
517C4⋊C4—944no—
518C4⋊C4—944no—
519C4⋊C4—944no—
520C4⋊C4—944no—
521C4⋊C4—941no—
522C4⋊C4—941no—
523C4⋊C4—941no—
524C4⋊C4(C2×D47)⋊C414yes—
525C4⋊C4(C2×D47)⋊C414yes—
526C4⋊C4(C2×D47)⋊C414yes—
527C4⋊C4Q8×D4712yes—
528C22⋊C4—944no—
529C22⋊C4—944no—
530C22⋊C4—944no—
531C42—42no—
532C42—81no—
533C42—81no—
534C42C2×C4×D4712yes—
535C42C2×C4×D4712yes—
536C42C2×C4×D4712yes—
537C42C2×D18812yes—
538C42C2×D18812yes—
539C42C2×D18812yes—
540C23—942no—
541C23—942no—
542C23—942no—
543C23—1881no—
544C23—1881no—
545C23—1881no—
546C23—1881no—
547C23C22×C22⋊D4711yescharacteristic
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—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—942no—
579C2×C4—942no—
580C2×C4—942no—
581C2×C4—3764no—
582C2×C4—3764no—
583C2×C4—1888no—
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×D18811yescharacteristic
598C2×C4C2×Q8×D4711yescharacteristic
599C2×C4#41×D4711yescharacteristic
600C2×C4#1309407111yescharacteristic
601C2×C4#1309407211yescharacteristic
602C8—942no—
603C8—1881no—
604C8—1881no—
605C8—1881no—
606C22—944no—
607C22—1882no—
608C22—1882no—
609C22—1882no—
610C22—1882no—
611C22—1882no—
612C22—1882no—
613C22C22×(C2×D47)⋊C411yescharacteristic
614C22#1568978011yescharacteristic
615C22Q8×C22⋊D4711yescharacteristic
616C22#1568992111yescharacteristic
617C22#1568992111yescharacteristic
618C22#1568992611yescharacteristic
619C22#1568993111yescharacteristic
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—42no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—1882no—
631C4—1882no—
632C4—1882no—
633C4C4×Q8×D4711yescharacteristic
634C4#1568959911yescharacteristic
635C4#1568978511yescharacteristic
636C4Q8×D18811yescharacteristic
637C2—1884no—
638C2Q8×(C2×D47)⋊C411yescharacteristic
639C2#2025190511yescharacteristic
640C2#2025191111yescharacteristic
641C2#2025613511yescharacteristic
642C2#2025620911yescharacteristic
643C2#2025632711yescharacteristic
644C2#2025634211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2355C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenter#2355D47
hypocenterC47#22526
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2355

Complements

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

NH ≅ G/Ntype
2. C2×#15689254C2semidirect
3. #20244360C2semidirect
4. #20244356C2semidirect
5. #20244352C2semidirect
6. #20244345C2semidirect
7. #20244345C2semidirect
8. #20244340C2semidirect
15. C47×#2355C2semidirect
17. #15689833C4semidirect
104. #13093944Q8semidirect
317. #2355D47semidirect
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
C2×#156892542
#202443602
#202443562
#202443522
#202443452
#202443402
#202433952
#202433852
#202432162
C4×#130939442
C47×#23552
#2252647