Finite Group Databasebeta

← #26225823

Subgroups — #26225823

Order 12032.

970 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689751C211yesmaximal, characteristic
2C2×#15689184C211yesmaximal, characteristic
3#20244360C211yesmaximal, characteristic, complemented
4#20244356C211yesmaximal, characteristic, complemented
5#20244352C211yesmaximal, characteristic, complemented
6#20244348C211yesmaximal, characteristic, complemented
7#20244345C211yesmaximal, characteristic, complemented
8#20244345C211yesmaximal, characteristic, complemented
9#20244340C211yesmaximal, characteristic, complemented
10#20244336C211yesmaximal, characteristic, complemented
11#20243480C211yesmaximal, characteristic
12#20243480C211yesmaximal, characteristic
13#20243476C211yesmaximal, characteristic
14#20243472C211yesmaximal, characteristic
15C47×#2314C211yesmaximal, characteristic
16#15689751—24no—
17#15689751C418yescomplemented
18C2×#13094024C2211yescharacteristic
19C2×#13094024C2211yescharacteristic
20C2×#13094023—21no—
21C2×#13094016C2211yescharacteristic
22C2×#13094016C2211yescharacteristic
23C2×#13094015—21no—
24C2×#13094001—21no—
25C2×#13094001—21no—
26C2×#13093983C2211yescharacteristic
27C22×C4⋊Dic47C2211yescharacteristic
28#15689184—24no—
29C2×C4⋊(C47⋊C8)C2211yescharacteristic
30C2×C4⋊(C47⋊C8)C2211yescharacteristic
31C2×C22⋊(C47⋊C8)C2211yescharacteristic
32C2×C22⋊(C47⋊C8)C2211yescharacteristic
33C2×#13093935C2211yescharacteristic
34C2×C4×C47⋊C8C2211yescharacteristic
35#15688885C2211yescharacteristic
36#15688885C2211yescharacteristic
37#15688885C2211yescharacteristic
38#15688885C2211yescharacteristic
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×#117×C47C2211yescharacteristic
52C47×#388—21no—
53C47×#388—21no—
54C47×#387—21no—
55C47×#387—21no—
56C47×#384C2211yescharacteristic
57C47×#382C2211yescharacteristic
58C47×#380C2211yescharacteristic
59C47×#380C2211yescharacteristic
60C4×C4⋊C4×C47C2211yescharacteristic
61C4×C4⋊C4×C47C2211yescharacteristic
62C22×Dic94—21no—
63C22×Dic94—21no—
64C23×Dic47—21no—
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69#13094024—22no—
70#13094024—22no—
71#13094024C2×C414yes—
72#13094024C2×C414yes—
73#13094023—28no—
74#13094016—22no—
75#13094016—22no—
76#13094016C2×C414yes—
77#13094016C2×C414yes—
78#13094015—28no—
79#13094001—28no—
80#13094001—28no—
81#13093983—22no—
82#13093983C2×C414yes—
83C2×C4⋊Dic47—22no—
84C2×C4⋊Dic47—21no—
85C2×C4⋊Dic47—21no—
86C2×C4⋊Dic47—21no—
87C2×C4⋊Dic47C2×C414yes—
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—
96C2×C22⋊Dic47—21no—
97C2×C4×Dic47—21no—
98C4⋊(C47⋊C8)—22no—
99C4⋊(C47⋊C8)—22no—
100C22⋊(C47⋊C8)—22no—
101C22⋊(C47⋊C8)—22no—
102#13093935—22no—
103C4×C47⋊C8—22no—
104C23×C4×C47C2311yescharacteristic
105#117×C47C2×C414yes—
106#117×C47D414yes—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47—21no—
109C2×C4⋊C4×C47D411yescharacteristic
110C2×C4⋊C4×C47D411yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C22⋊C4×C47—21no—
116C2×C22⋊C4×C47—21no—
117C2×C22⋊C4×C47D411yescharacteristic
118C2×C22⋊C4×C47D411yescharacteristic
119C2×C42×C47—21no—
120C2×C42×C47—21no—
121C2×C42×C47C2311yescharacteristic
122C2×C42×C47C2311yescharacteristic
123#95×C47—21no—
124#95×C47—21no—
125#95×C47—21no—
126#95×C47—21no—
127#95×C47—21no—
128#95×C47—21no—
129C24×C47C2×D411yescharacteristic
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47C22×C412yes—
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D412yes—
145C22×C4×C47C2411yescharacteristic
146C22×Dic47—21no—
147C22×Dic47—21no—
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—24no—
151C22×Dic47—81no—
152C22×Dic47—81no—
153C2×Dic94—24no—
154C2×Dic94—24no—
155C2×Dic94—81no—
156C2×Dic94—81no—
157C2×Dic94—81no—
158C2×Dic94—81no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—81no—
168C4⋊C4×C47—81no—
169C4⋊C4×C47C22⋊C414yes—
170C4⋊C4×C47C22⋊C414yes—
171C22⋊C4×C47—81no—
172C22⋊C4×C47—81no—
173C22⋊C4×C47C22⋊C414yes—
174C22⋊C4×C47C22⋊C414yes—
175C42×C47—81no—
176C42×C47—81no—
177C42×C47C22×C412yes—
178C42×C47C22×C412yes—
179C42×C47C2×D412yes—
180C42×C47C2×D412yes—
181C22⋊Dic47—24no—
182C4⋊Dic47—24no—
183C4⋊Dic47—24no—
184C4⋊Dic47—24no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47C22×C412yes—
190C4⋊Dic47C22×C412yes—
191C4⋊Dic47C22×C412yes—
192C4⋊Dic47C22×C412yes—
193Dic47⋊C4—24no—
194Dic47⋊C4—24no—
195Dic47⋊C4—24no—
196Dic47⋊C4—24no—
197C4×Dic47—24no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8—21no—
210C23×C47—21no—
211C23×C47—21no—
212C23×C47—21no—
213C23×C47—21no—
214C23×C47—21no—
215C23×C47—21no—
216C23×C47C2×C22⋊C412yes—
217C23×C47C22×D411yescharacteristic
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—22no—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
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—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47C2×C22⋊C412yes—
245C2×C4×C47C2×C22⋊C412yes—
246C2×C4×C47C2×C22⋊C412yes—
247C2×C4×C47C2×C22⋊C412yes—
248C2×C4×C47C2×C22⋊C412yes—
249C2×C4×C47C23×C411yescharacteristic
250C2×C4×C47C22×D411yescharacteristic
251C2×C4×C472−1+411yescharacteristic
252C2×C4×C472−1+411yescharacteristic
253C2×C4×C472−1+411yescharacteristic
254C2×C4×C472−1+411yescharacteristic
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—22no—
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—84no—
268Dic94—84no—
269Dic94—84no—
270C47⋊C8—41no—
271C47⋊C8—41no—
272C47⋊C8—41no—
273C47⋊C8—41no—
274#21843—471nomaximal, Sylow
275C22×C47—22no—
276C22×C47—22no—
277C22×C47—22no—
278C22×C47—41no—
279C22×C47—41no—
280C22×C47—41no—
281C22×C47—41no—
282C22×C47C22×C22⋊C411yescharacteristic
283C22×C47#53611yescharacteristic
284C22×C47#53611yescharacteristic
285C22×C47#57611yescharacteristic
286C22×C47#57711yescharacteristic
287C22×C47#57711yescharacteristic
288C22×C47#57811yescharacteristic
289C4×C47—22no—
290C4×C47—22no—
291C4×C47—22no—
292C4×C47—22no—
293C4×C47—22no—
294C4×C47—81no—
295C4×C47—81no—
296C4×C47—81no—
297C4×C47—81no—
298C4×C47#51911yescharacteristic
299C4×C47#51911yescharacteristic
300C4×C47#53611yescharacteristic
301C4×C47#53611yescharacteristic
302Dic47—42no—
303Dic47—42no—
304Dic47—42no—
305Dic47—42no—
306C2×#544—471no—
307C2×#433—471no—
308#2314D4711yescharacteristic
309#1993—471no—
310#1992—471no—
311#1991—471no—
312#1990—471no—
313#1989—471no—
314#1989—471no—
315#1988—471no—
316#1987—471no—
317#1825—471no—
318#1825—471no—
319#1824—471no—
320#1823—471no—
321C2×C47—42no—
322C2×C47#231011yescharacteristic
323C2×C47#289711yescharacteristic
324C2×C47#289811yescharacteristic
325C2×C47#324111yescharacteristic
326C2×C47#324211yescharacteristic
327C2×C47#324311yescharacteristic
328C2×C47#324411yescharacteristic
329#544—478no—
330#544—944no—
331C2×C4⋊Q8—471no—
332C2×C4⋊Q8—471no—
333C2×C4⋊Q8—941no—
334C2×#125—471no—
335C2×#125—471no—
336C2×#125—941no—
337C2×C22⋊Q8—941no—
338C2×C22⋊Q8—941no—
339C2×#117—471no—
340C2×#117C2×D4711yescharacteristic
341C22×C4⋊C4—471no—
342#433—944no—
343C2×C4⋊C8—471no—
344C2×C4⋊C8—471no—
345C2×C22⋊C8—471no—
346C2×C22⋊C8—471no—
347C2×#97—471no—
348C2×C4×C8—471no—
349#388—21no—
350#388—21no—
351#387—21no—
352#387—21no—
353#384C2×D4711yescharacteristic
354#382C2×D4711yescharacteristic
355#380C2×D4711yescharacteristic
356#380C2×D4711yescharacteristic
357C4×C4⋊C4C2×D4711yescharacteristic
358C4×C4⋊C4C2×D4711yescharacteristic
359#340—471no—
360#340—471no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340—471no—
365#340—471no—
366#340—471no—
367#340—471no—
368#340—471no—
369#340—471no—
370#340—471no—
371#340—471no—
372#340—471no—
373#340—471no—
374#340—471no—
375C47#2184311yescharacteristic, complemented, Sylow
376C22×Q8—941no—
377C22×Q8—941no—
378C23×C4—941no—
379C23×C4C22×D4711yescharacteristic
380C22×C8—471no—
381C22×C8—471no—
382C22×C8—471no—
383C22×C8—471no—
384C4⋊Q8—474no—
385C4⋊Q8—474no—
386C4⋊Q8—948no—
387C4⋊Q8—942no—
388C4⋊Q8—942no—
389#125—474no—
390#125—474no—
391#125—948no—
392#125—942no—
393#125—942no—
394C22⋊Q8—948no—
395C22⋊Q8—948no—
396#117—474no—
397#117—942no—
398#117C4×D4714yes—
399#117D18814yes—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—474no—
403C2×C4⋊C4—471no—
404C2×C4⋊C4—471no—
405C2×C4⋊C4—471no—
406C2×C4⋊C4—471no—
407C2×C4⋊C4—942no—
408C2×C4⋊C4—941no—
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×C4⋊C4C22×D4711yescharacteristic
419C2×C4⋊C4C22×D4711yescharacteristic
420C2×C4⋊C4C22×D4711yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—941no—
424C2×C22⋊C4C22⋊D4711yescharacteristic
425C2×C22⋊C4C22⋊D4711yescharacteristic
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—941no—
429C2×C42C22×D4711yescharacteristic
430C2×C42C22×D4711yescharacteristic
431C4⋊C8—942no—
432C4⋊C8—942no—
433C22⋊C8—942no—
434C22⋊C8—942no—
435#97—942no—
436C4×C8—942no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—21no—
443C24C2×C22⋊D4711yescharacteristic
444C2×Q8—944no—
445C2×Q8—944no—
446C2×Q8—3761no—
447C2×Q8—3761no—
448C2×Q8—3761no—
449C2×Q8—3761no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—944no—
459C22×C4—941no—
460C22×C4—941no—
461C22×C4—941no—
462C22×C4—941no—
463C22×C4—3761no—
464C22×C4—3761no—
465C22×C4C2×C4×D4712yes—
466C22×C4C2×D18812yes—
467C22×C4C2×C22⋊D4711yescharacteristic
468C22×C4C2×C22⋊D4711yescharacteristic
469C22×C4C2×C22⋊D4711yescharacteristic
470C22×C4C2×C22⋊D4711yescharacteristic
471C22×C4C2×C22⋊D4711yescharacteristic
472C22×C4C23×D4711yescharacteristic
473C2×C8—941no—
474C2×C8—941no—
475C2×C8—941no—
476C2×C8—941no—
477C2×C8—941no—
478C2×C8—941no—
479C2×C8—941no—
480C2×C8—941no—
481C2×C8—941no—
482C2×C8—941no—
483C2×C8—941no—
484C2×C8—941no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4—472no—
496C4⋊C4—472no—
497C4⋊C4—472no—
498C4⋊C4—472no—
499C4⋊C4—944no—
500C4⋊C4—944no—
501C4⋊C4—944no—
502C4⋊C4—944no—
503C4⋊C4—944no—
504C4⋊C4—944no—
505C4⋊C4—944no—
506C4⋊C4—941no—
507C4⋊C4—941no—
508C4⋊C4—941no—
509C4⋊C4—941no—
510C4⋊C4(C2×D47)⋊C414yes—
511C4⋊C4(C2×D47)⋊C414yes—
512C22⋊C4—81no—
513C22⋊C4—81no—
514C22⋊C4—944no—
515C22⋊C4(C2×D47)⋊C414yes—
516C22⋊C4(C2×D47)⋊C414yes—
517C42—81no—
518C42—81no—
519C42—944no—
520C42C2×C4×D4712yes—
521C42C2×C4×D4712yes—
522C42C2×D18812yes—
523C42C2×D18812yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23C2×(C2×D47)⋊C412yes—
531C23C22×C22⋊D4711yescharacteristic
532Q8—3764no—
533Q8—3764no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—942no—
561C2×C4—942no—
562C2×C4—942no—
563C2×C4—942no—
564C2×C4—3764no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4—1881no—
571C2×C4—1881no—
572C2×C4—1881no—
573C2×C4C2×(C2×D47)⋊C412yes—
574C2×C4C2×(C2×D47)⋊C412yes—
575C2×C4C2×(C2×D47)⋊C412yes—
576C2×C4C2×(C2×D47)⋊C412yes—
577C2×C4C2×(C2×D47)⋊C412yes—
578C2×C4C22×C4×D4711yescharacteristic
579C2×C4C22×D18811yescharacteristic
580C2×C4#1309407111yescharacteristic
581C2×C4#1309407111yescharacteristic
582C2×C4#1309407111yescharacteristic
583C2×C4#1309407111yescharacteristic
584C8—1881no—
585C8—1881no—
586C8—1881no—
587C8—1881no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C22×(C2×D47)⋊C411yescharacteristic
596C22#1568969411yescharacteristic
597C22#1568969411yescharacteristic
598C22#1568992611yescharacteristic
599C22#1568993111yescharacteristic
600C22#1568993411yescharacteristic
601C22#1568993811yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—81no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—1882no—
612C4—1882no—
613C4—1882no—
614C4—1882no—
615C4#1568960011yescharacteristic
616C4#1568960011yescharacteristic
617C4#1568969911yescharacteristic
618C4#1568969911yescharacteristic
619C2—42no—
620C2#2024597211yescharacteristic
621C2#2025191111yescharacteristic
622C2#2025191811yescharacteristic
623C2#2025463511yescharacteristic
624C2#2025465011yescharacteristic
625C2#2025466511yescharacteristic
626C2#2025468011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2314C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#2314D47
hypocenterC47#21843
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2314

Complements

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

NH ≅ G/Ntype
3. #20244360C2semidirect
4. #20244356C2semidirect
5. #20244352C2semidirect
6. #20244348C2semidirect
7. #20244345C2semidirect
8. #20244345C2semidirect
9. #20244340C2semidirect
10. #20244336C2semidirect
17. #15689751C4semidirect
375. C47#21843semidirect

Maximal subgroups

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

subgroupindex
C2×#156897512
C2×#156891842
#202443602
#202443562
#202443522
#202443482
#202443452
#202443402
#202443362
#202434802
#202434762
#202434722
C47×#23142
#2184347