Finite Group Databasebeta

← #26316304

Subgroups — #26316304

Order 12032.

868 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251698C211yesmaximal, characteristic, complemented
2#20244956C211yesmaximal, characteristic, complemented
3#20244909C211yesmaximal, characteristic, complemented
4#20244726C211yesmaximal, characteristic, complemented
5#20244534C211yesmaximal, characteristic, complemented
6#20243980C211yesmaximal, characteristic, complemented
7#20243966C211yesmaximal, characteristic, complemented
8#20243817C211yesmaximal, characteristic
9#20243817C211yesmaximal, characteristic
10#20243649C211yesmaximal, characteristic, complemented
11#20243635C211yesmaximal, characteristic, complemented
12#20243415C211yesmaximal, characteristic, complemented
13#20243411C211yesmaximal, characteristic, complemented
14C2×#15688878C211yesmaximal, characteristic, complemented
15C4×#121×C47C211yesmaximal, characteristic
16C23×C47⋊C8C2211yescharacteristic
17C23×C47⋊C8C2211yescharacteristic
18C2×#13094016C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093945C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27#15689021C2211yescharacteristic
28#15689021C2211yescharacteristic
29#15689007—21no—
30#15689007—21no—
31#15688996—21no—
32#15688996—21no—
33#15688989C2211yescharacteristic
34#15688983C2211yescharacteristic
35#15688983C2211yescharacteristic
36#15688885C2211yescharacteristic
37#15688885C2211yescharacteristic
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688878C2211yescharacteristic
45#15688878C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×#121×C47C2211yescharacteristic
49C2×C4×D4×C47—21no—
50C2×C4×D4×C47C2211yescharacteristic
51C22×C42×C47C2211yescharacteristic
52C47×#390C2211yescharacteristic
53C47×#389C2211yescharacteristic
54C47×#388—21no—
55C47×#386—21no—
56C47×#380C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C4×C22⋊C4×C47—21no—
59C22×C47⋊C8—22no—
60C22×C47⋊C8—22no—
61C22×C47⋊C8—22no—
62C22×C47⋊C8—22no—
63C22×C47⋊C8—41no—
64C22×C47⋊C8—41no—
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69#13094016—41no—
70#13094016—41no—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×C22⋊Dic47—21no—
80C2×C22⋊Dic47—21no—
81#13093957—22no—
82#13093957—41no—
83#13093955—22no—
84#13093955—41no—
85#13093945—22no—
86#13093945—22no—
87#13093945—41no—
88#13093945—41no—
89C47⋊#95—21no—
90C47⋊#95—21no—
91C47⋊#95—21no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C22×D4×C47—21no—
96C22×D4×C47C2311yescharacteristic
97C23×C4×C47—21no—
98C23×C4×C47D411yescharacteristic
99C23×C4×C47D411yescharacteristic
100C23×C4×C47C2311yescharacteristic
101#121×C47—22no—
102#121×C47Q814yes—
103C4×D4×C47—41no—
104C4×D4×C47—41no—
105C4×D4×C47—44no—
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C22⋊C4×C47—21no—
111C2×C22⋊C4×C47—21no—
112C2×C22⋊C4×C47—21no—
113C2×C22⋊C4×C47—21no—
114C2×C22⋊C4×C47—21no—
115C2×C42×C47—21no—
116C2×C42×C47—41no—
117C2×C42×C47—41no—
118C2×C42×C47D411yescharacteristic
119C2×C42×C47D411yescharacteristic
120C2×C42×C47C2311yescharacteristic
121C2×C42×C47C2311yescharacteristic
122#95×C47—21no—
123#95×C47—21no—
124#95×C47—21no—
125C24×C47—21no—
126C24×C47C2×D411yescharacteristic
127C2×D4×C47—24no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47—21no—
133C2×D4×C47—42no—
134C2×D4×C47—42no—
135C2×D4×C47C2×Q812yes—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—22no—
141C22×C4×C47—22no—
142C22×C4×C47—21no—
143C22×C4×C47—22no—
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47—41no—
147C22×C4×C47—41no—
148C22×C4×C47—41no—
149C22×C4×C47—42no—
150C22×C4×C47—42no—
151C22×C4×C47—42no—
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×Q812yes—
158C22×C4×C47#4112yes—
159C22×C4×C47C2411yescharacteristic
160C22×Dic47—21no—
161C22×Dic47—21no—
162C22×Dic47—21no—
163C22×Dic47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—42no—
170C4⋊C4×C47C2×Q812yes—
171C22⋊C4×C47—24no—
172C22⋊C4×C47—42no—
173C22⋊C4×C47—42no—
174C22⋊C4×C47—42no—
175C22⋊C4×C47—42no—
176C42×C47—21no—
177C42×C47—21no—
178C42×C47—21no—
179C42×C47—21no—
180C42×C47—41no—
181C42×C47—41no—
182C42×C47—42no—
183C22⋊Dic47—42no—
184C22⋊Dic47—42no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193Dic47⋊C4—42no—
194Dic47⋊C4—42no—
195Dic47⋊C4—42no—
196Dic47⋊C4—42no—
197C2×C47⋊C8—21no—
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—44no—
208C2×C47⋊C8—44no—
209C2×C47⋊C8#4112yes—
210C2×C47⋊C8#4112yes—
211C23×C47—21no—
212C23×C47—21no—
213C23×C47—21no—
214C23×C47—21no—
215C23×C47—22no—
216C23×C47—41no—
217C23×C47—41no—
218C23×C47—41no—
219C23×C47—41no—
220C23×C47—41no—
221C23×C47—41no—
222C23×C47C22⋊Q812yes—
223C23×C47#13512yes—
224C23×C47AGL(1,ℤ/8)12yes—
225C23×C47C22×D411yescharacteristic
226D4×C47—22no—
227D4×C47—22no—
228D4×C47—48no—
229D4×C47—41no—
230D4×C47—41no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—22no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—21no—
246C2×C4×C47—21no—
247C2×C4×C47—21no—
248C2×C4×C47—21no—
249C2×C4×C47—44no—
250C2×C4×C47—44no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—41no—
257C2×C4×C47—41no—
258C2×C4×C47—41no—
259C2×C4×C47—84no—
260C2×C4×C47C22⋊Q812yes—
261C2×C4×C47#13512yes—
262C2×C4×C47C22×Q811yescharacteristic
263C2×C4×C47C2×#4111yescharacteristic
264C2×C4×C47C2×#4111yescharacteristic
265C2×C4×C47C2×#4111yescharacteristic
266C2×C4×C47H(5,𝔽2)11yescharacteristic
267C2×C4×C47H(5,𝔽2)11yescharacteristic
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280C47⋊C8—22no—
281C47⋊C8—22no—
282C47⋊C8—41no—
283C47⋊C8—41no—
284#29161—471nomaximal, Sylow
285C22×C47—22no—
286C22×C47—22no—
287C22×C47—22no—
288C22×C47—22no—
289C22×C47—22no—
290C22×C47—22no—
291C22×C47—42no—
292C22×C47—42no—
293C22×C47—42no—
294C22×C47—81no—
295C22×C47—81no—
296C22×C47—81no—
297C22×C47—81no—
298C22×C47#47814yes—
299C22×C47C2×C22⋊Q811yescharacteristic
300C22×C47#53211yescharacteristic
301C22×C47#53411yescharacteristic
302C22×C47C2×#13511yescharacteristic
303C22×C47C2×#13511yescharacteristic
304C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
305C22×C47#57511yescharacteristic
306C4×C47—22no—
307C4×C47—22no—
308C4×C47—42no—
309C4×C47—41no—
310C4×C47—41no—
311C4×C47—41no—
312C4×C47—81no—
313C4×C47—81no—
314C4×C47#55111yescharacteristic
315C4×C47#55111yescharacteristic
316C4×C47#55411yescharacteristic
317C4×C47#55411yescharacteristic
318Dic47—81no—
319Dic47—81no—
320Dic47—81no—
321Dic47—81no—
322#2851—471no—
323C4×#121D4711yescharacteristic
324#2101—471no—
325#2092—471no—
326#2059—471no—
327#2025—471no—
328#1921—471no—
329#1919—471no—
330#1893—471no—
331#1893—471no—
332#1864—471no—
333#1862—471no—
334#1808—471no—
335#1807—471no—
336C2×#336—471no—
337C2×C47—24no—
338C2×C47—82no—
339C2×C47#256411yescharacteristic
340C2×C47C2×#47811yescharacteristic
341C2×C47#307611yescharacteristic
342C2×C47#316811yescharacteristic
343C2×C47#318011yescharacteristic
344C2×C47#318811yescharacteristic
345C2×C47#319111yescharacteristic
346C23×C8—471no—
347C23×C8—471no—
348C2×#125—471no—
349C2×#121C2×D4711yescharacteristic
350C2×C4×D4—21no—
351C2×C4×D4C2×D4711yescharacteristic
352C22×C42C2×D4711yescharacteristic
353C2×#107—471no—
354C2×#107—471no—
355C2×#106—471no—
356C2×#106—471no—
357C2×#102—471no—
358C2×#102—471no—
359C2×#102—471no—
360C2×#102—471no—
361#390C2×D4711yescharacteristic
362#389—471no—
363#389—471no—
364#389C2×D4711yescharacteristic
365#388—21no—
366#387—941no—
367#387—941no—
368#386—21no—
369#385—941no—
370#385—941no—
371#382—471no—
372#380—471no—
373#380—471no—
374#380C2×D4711yescharacteristic
375C4×C4⋊C4C2×D4711yescharacteristic
376C4×C22⋊C4—21no—
377#340—471no—
378#340—471no—
379#340—471no—
380#340—471no—
381#340—471no—
382#340—471no—
383#340—471no—
384#340—471no—
385#336—471no—
386#336—471no—
387#336—471no—
388#336—471no—
389C47#2916111yescharacteristic, complemented, Sylow
390C22×D4—21no—
391C22×D4C22×D4711yescharacteristic
392C23×C4—21no—
393C23×C4C22⋊D4711yescharacteristic
394C23×C4C22⋊D4711yescharacteristic
395C23×C4C22×D4711yescharacteristic
396C22×C8—471no—
397C22×C8—471no—
398C22×C8—471no—
399C22×C8—471no—
400C22×C8—942no—
401C22×C8—942no—
402C22×C8—942no—
403C22×C8—942no—
404C22×C8—1881no—
405C22×C8—1881no—
406#125—1881no—
407#125—1881no—
408#121—22no—
409#121Dic9414yes—
410C4×D4—41no—
411C4×D4—41no—
412C4×D4—44no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—471no—
415C2×C4⋊C4—471no—
416C2×C4⋊C4—471no—
417C2×C4⋊C4—471no—
418C2×C4⋊C4—941no—
419C2×C4⋊C4—941no—
420C2×C4⋊C4—941no—
421C2×C4⋊C4—941no—
422C2×C4⋊C4C22×D4711yescharacteristic
423C2×C4⋊C4C22×D4711yescharacteristic
424C2×C4⋊C4C22×D4711yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—941no—
431C2×C22⋊C4—941no—
432C2×C42—21no—
433C2×C42—41no—
434C2×C42—41no—
435C2×C42C22⋊D4711yescharacteristic
436C2×C42C22⋊D4711yescharacteristic
437C2×C42C22×D4711yescharacteristic
438C2×C42C22×D4711yescharacteristic
439#107—942no—
440#107—1881no—
441#106—942no—
442#106—1881no—
443#102—942no—
444#102—942no—
445#102—1881no—
446#102—1881no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—941no—
451#95—941no—
452#95—941no—
453#95—941no—
454#95—941no—
455#95—941no—
456C24—21no—
457C24C2×C22⋊D4711yescharacteristic
458C2×D4—24no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4C2×Dic9412yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—941no—
484C22×C4—941no—
485C22×C4—941no—
486C22×C4—941no—
487C22×C4C2×Dic9412yes—
488C22×C4#1062535012yes—
489C22×C4C2×C22⋊D4711yescharacteristic
490C22×C4C2×C22⋊D4711yescharacteristic
491C22×C4C2×C22⋊D4711yescharacteristic
492C22×C4C2×C22⋊D4711yescharacteristic
493C22×C4C2×C22⋊D4711yescharacteristic
494C22×C4C23×D4711yescharacteristic
495C2×C8—472no—
496C2×C8—472no—
497C2×C8—941no—
498C2×C8—941no—
499C2×C8—941no—
500C2×C8—941no—
501C2×C8—941no—
502C2×C8—941no—
503C2×C8—941no—
504C2×C8—941no—
505C2×C8—941no—
506C2×C8—941no—
507C2×C8—1884no—
508C2×C8—1884no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
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—1882no—
524C4⋊C4—1882no—
525C4⋊C4—1882no—
526C4⋊C4—1882no—
527C4⋊C4C2×Dic9412yes—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—1882no—
534C22⋊C4—1882no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—41no—
540C42—41no—
541C42—42no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—22no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23#1309400312yes—
554C23#1309404412yes—
555C23#1309405012yes—
556C23C22×C22⋊D4711yescharacteristic
557D4—22no—
558D4—22no—
559D4—48no—
560D4—41no—
561D4—41no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—84no—
591C2×C4—1881no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4—1881no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4—1881no—
599C2×C4—1881no—
600C2×C4—1881no—
601C2×C4—1881no—
602C2×C4—1881no—
603C2×C4#1309400312yes—
604C2×C4#1309404412yes—
605C2×C4C22×Dic9411yescharacteristic
606C2×C4#41×D4711yescharacteristic
607C2×C4#41×D4711yescharacteristic
608C2×C4C2×#1062535011yescharacteristic
609C2×C4#1309406911yescharacteristic
610C2×C4#1309406911yescharacteristic
611C8—942no—
612C8—942no—
613C8—1881no—
614C8—1881no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22#1568942314yes—
629C22C2×#1309400311yescharacteristic
630C22#1568966911yescharacteristic
631C22#1568968211yescharacteristic
632C22C2×#1309404411yescharacteristic
633C22C2×#1309404411yescharacteristic
634C22C2×#1309405011yescharacteristic
635C22#1568992111yescharacteristic
636C4—22no—
637C4—22no—
638C4—41no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—81no—
643C4—81no—
644C4—3761no—
645C4—3761no—
646C4—3761no—
647C4—3761no—
648C4#1568980211yescharacteristic
649C4#1568980211yescharacteristic
650C4#1568983211yescharacteristic
651C4#1568983211yescharacteristic
652C2—24no—
653C2—82no—
654C2#2024856411yescharacteristic
655C2C2×#1568942311yescharacteristic
656C2#2025317611yescharacteristic
657C2#2025395211yescharacteristic
658C2#2025408011yescharacteristic
659C2#2025415311yescharacteristic
660C2#2025417611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC4×#121×C47C2
socleC23×C47C22×D4
prime radicalC2×#121×C47C22
hypercenterC4×#121D47
hypocenterC47#29161
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. #20251698C2semidirect
2. #20244956C2semidirect
3. #20244909C2semidirect
4. #20244726C2semidirect
5. #20244534C2semidirect
6. #20243980C2semidirect
7. #20243966C2semidirect
10. #20243649C2semidirect
11. #20243635C2semidirect
12. #20243415C2semidirect
13. #20243411C2semidirect
14. C2×#15688878C2semidirect
389. C47#29161semidirect

Maximal subgroups

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

subgroupindex
#202516982
#202449562
#202449092
#202447262
#202445342
#202439802
#202439662
#202438172
#202436492
#202436352
#202434152
#202434112
C2×#156888782
C4×#121×C472
#2916147