Finite Group Databasebeta

← #26221127

Subgroups — #26221127

Order 12032.

870 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689173C211yesmaximal, characteristic
2#20245783C211yesmaximal, characteristic
3#117×Dic47C211yesmaximal, characteristic
4#20244159C211yesmaximal, characteristic, complemented
5#20244157C211yesmaximal, characteristic, complemented
6#20244155C211yesmaximal, characteristic, complemented
7#20244153C211yesmaximal, characteristic, complemented
8#20243589C211yesmaximal, characteristic
9#20243587C211yesmaximal, characteristic
10#20243585C211yesmaximal, characteristic
11#107×Dic47C211yesmaximal, characteristic, complemented
12#20243280C211yesmaximal, characteristic, complemented
13#106×Dic47C211yesmaximal, characteristic, complemented
14#20243276C211yesmaximal, characteristic, complemented
15C47×#1854C211yesmaximal, characteristic
16C2×#13093983C2211yescharacteristic
17C22×C4×Dic47C2211yescharacteristic
18#15689173C418yescomplemented
19C2×#13093958C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093956C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×Dic47⋊C8C2211yescharacteristic
26C2×Dic47⋊C8C2211yescharacteristic
27C2×C22⋊(C47⋊C8)C2211yescharacteristic
28C2×C8×Dic47C2211yescharacteristic
29C2×C8×Dic47C2211yescharacteristic
30#15688990—21no—
31#15688988C2211yescharacteristic
32#15688988C2211yescharacteristic
33#15688984—21no—
34#15688983C2211yescharacteristic
35C4⋊C4×Dic47C2211yescharacteristic
36C4⋊C4×Dic47C2211yescharacteristic
37C4⋊C4×Dic47C2211yescharacteristic
38C4⋊C4×Dic47C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C4⋊C4×Dic47C2211yescharacteristic
41C22⋊C4×Dic47—21no—
42C22⋊C4×Dic47—21no—
43C22⋊C4×Dic47—21no—
44#15688969—21no—
45#15688968—21no—
46C42×Dic47—21no—
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C2×#117×C47C2211yescharacteristic
54C2×#107×C47C2211yescharacteristic
55C2×#106×C47C2211yescharacteristic
56C2×C22⋊C8×C47C2211yescharacteristic
57C47×#380C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C23×Dic47D411yescharacteristic
61C23×Dic47D411yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64#13093983—22no—
65#13093983—22no—
66C2×C4⋊Dic47C2311yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C22⋊Dic47—21no—
71C2×C22⋊Dic47—21no—
72C2×C4×Dic47—21no—
73C2×C4×Dic47—21no—
74C2×C4×Dic47—21no—
75C2×C4×Dic47—21no—
76C2×C4×Dic47—21no—
77C2×C4×Dic47—21no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47C2×C414yes—
80C2×C4×Dic47D411yescharacteristic
81C2×C4×Dic47D411yescharacteristic
82C2×C4×Dic47Q814yes—
83C2×C4×Dic47C2311yescharacteristic
84C2×C4×Dic47C2311yescharacteristic
85#13093958—22no—
86#13093957—22no—
87#13093957—22no—
88#13093956—22no—
89#13093955—22no—
90#13093955—22no—
91Dic47⋊C8C2×C414yes—
92Dic47⋊C8C2×C414yes—
93C22⋊(C47⋊C8)C2×C414yes—
94C8×Dic47C2×C414yes—
95C8×Dic47C2×C414yes—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C23×C4×C47C2311yescharacteristic
108C22×C8×C47C2311yescharacteristic
109C22×C8×C47C2311yescharacteristic
110#117×C47—22no—
111#117×C47—22no—
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C22⋊C4×C47—21no—
117C2×C22⋊C4×C47—21no—
118C2×C42×C47—21no—
119#107×C47—22no—
120#106×C47—22no—
121C22⋊C8×C47C2×C414yes—
122#95×C47—21no—
123C24×C47C2×D411yescharacteristic
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47C22×C412yes—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×Q812yes—
135C22×C4×C47C2411yescharacteristic
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—41no—
143C22×Dic47—41no—
144C22×Dic47C4⋊C414yes—
145C22×Dic47C4⋊C414yes—
146C22×Dic47C2×D411yescharacteristic
147C22×Dic47C2×D411yescharacteristic
148C22×Dic47C2×D411yescharacteristic
149C22×Dic47C2×D411yescharacteristic
150C2×C8×C47—21no—
151C2×C8×C47—21no—
152C2×C8×C47—21no—
153C2×C8×C47—21no—
154C2×C8×C47C22×C412yes—
155C2×C8×C47C22×C412yes—
156C4⋊C4×C47—21no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47#4112yes—
163C4⋊C4×C47#4112yes—
164C22⋊C4×C47—24no—
165C22⋊C4×C47—42no—
166C42×C47—24no—
167C22⋊Dic47—24no—
168C22⋊Dic47—42no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47#4112yes—
176C4⋊Dic47#4112yes—
177C4×Dic47—24no—
178C4×Dic47—42no—
179C4×Dic47—42no—
180C4×Dic47—42no—
181C4×Dic47—42no—
182C4×Dic47—42no—
183C4×Dic47—42no—
184C4×Dic47C4⋊C414yes—
185C4×Dic47C4⋊C414yes—
186C4×Dic47C22×C412yes—
187C4×Dic47C22×C412yes—
188C4×Dic47C2×Q812yes—
189C4×Dic47C2×Q812yes—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8—21no—
193C2×C47⋊C8—21no—
194C2×C47⋊C8C22×C412yes—
195C2×C47⋊C8C22×C412yes—
196C23×C47—21no—
197C23×C47—21no—
198C23×C47—21no—
199C23×C47—21no—
200C23×C47—21no—
201C23×C47—21no—
202C23×C47C2×C4⋊C412yes—
203C23×C47C22×D411yescharacteristic
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—41no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47C2×C4⋊C412yes—
223C2×C4×C47C23×C411yescharacteristic
224C2×C4×C47C22×Q811yescharacteristic
225C2×C4×C47C2×#4111yescharacteristic
226C2×C4×C47C2×#4111yescharacteristic
227C2×C4×C47C2×#4111yescharacteristic
228C2×C4×C47C2×#4111yescharacteristic
229C2×Dic47—21no—
230C2×Dic47—21no—
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—28no—
236C2×Dic47—22no—
237C2×Dic47—22no—
238C2×Dic47—44no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47C2×C4⋊C412yes—
250C2×Dic47C2×C4⋊C412yes—
251C2×Dic47C2×C4⋊C412yes—
252C2×Dic47C2×C4⋊C412yes—
253C2×Dic47#13512yes—
254C2×Dic47#13512yes—
255C8×C47—22no—
256C8×C47—22no—
257C47⋊C8—22no—
258C47⋊C8—22no—
259#21360—471nomaximal, Sylow
260C22×C47—22no—
261C22×C47—22no—
262C22×C47—22no—
263C22×C47—41no—
264C22×C47—41no—
265C22×C47—41no—
266C22×C47—41no—
267C22×C47C22×C4⋊C411yescharacteristic
268C22×C47#52511yescharacteristic
269C22×C47#52511yescharacteristic
270C22×C47C2×#13511yescharacteristic
271C22×C47C2×#13511yescharacteristic
272C22×C47#57511yescharacteristic
273C22×C47#57511yescharacteristic
274C4×C47—22no—
275C4×C47—42no—
276C4×C47—42no—
277C4×C47—81no—
278C4×C47—81no—
279C4×C47C4×#4111yescharacteristic
280C4×C47C4×#4111yescharacteristic
281C4×C47#53311yescharacteristic
282C4×C47#53311yescharacteristic
283Dic47—22no—
284Dic47—22no—
285Dic47—22no—
286Dic47—42no—
287Dic47—42no—
288Dic47—81no—
289Dic47—81no—
290Dic47#42714yes—
291C2×#431—471no—
292#2283—471no—
293C4×#117—471no—
294#1949—471no—
295#1949—471no—
296#1949—471no—
297#1949—471no—
298#1854—471no—
299#1854—471no—
300#1854—471no—
301#1854D4711yescharacteristic
302C4×#107—471no—
303C4×#107—471no—
304C4×#106—471no—
305C4×#106—471no—
306C2×C47—42no—
307C2×C47#229611yescharacteristic
308C2×C47C2×#42711yescharacteristic
309C2×C47#291111yescharacteristic
310C2×C47#310211yescharacteristic
311C2×C47#310211yescharacteristic
312C2×C47#310211yescharacteristic
313C2×C47#310211yescharacteristic
314C2×#117—471no—
315C2×#117C2×D4711yescharacteristic
316C22×C42—471no—
317#431—478no—
318C2×#107—471no—
319C2×#107—471no—
320C2×#107—471no—
321C2×#107C2×D4711yescharacteristic
322C2×#106—471no—
323C2×#106—471no—
324C2×#106—471no—
325C2×#106C2×D4711yescharacteristic
326C2×C4⋊C8—471no—
327C2×C4⋊C8—471no—
328C2×C22⋊C8—471no—
329C2×C22⋊C8C2×D4711yescharacteristic
330C2×C4×C8—471no—
331C2×C4×C8—471no—
332#383—941no—
333#382—471no—
334#382—471no—
335#381—941no—
336#380—471no—
337#380C2×D4711yescharacteristic
338C4×C4⋊C4—471no—
339C4×C4⋊C4—471no—
340C4×C4⋊C4—471no—
341C4×C4⋊C4—471no—
342C4×C4⋊C4—471no—
343C4×C4⋊C4—471no—
344C4×C22⋊C4—941no—
345C4×C22⋊C4—941no—
346C4×C22⋊C4—941no—
347#376—941no—
348#376—941no—
349C43—941no—
350#340—471no—
351#340—471no—
352#340—471no—
353#340—471no—
354#340—471no—
355#340—471no—
356#340C2×D4711yescharacteristic
357#340C2×D4711yescharacteristic
358C47#2136011yescharacteristic, complemented, Sylow
359C23×C4—471no—
360C23×C4—471no—
361C23×C4C22×D4711yescharacteristic
362C22×C8—471no—
363C22×C8—471no—
364C22×C8C22×D4711yescharacteristic
365C22×C8C22×D4711yescharacteristic
366#117—22no—
367#117—22no—
368#117—942no—
369#117—942no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4C22×D4711yescharacteristic
375C2×C4⋊C4C22×D4711yescharacteristic
376C2×C4⋊C4C22×D4711yescharacteristic
377C2×C4⋊C4C22×D4711yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—941no—
381C2×C22⋊C4—941no—
382C2×C42—21no—
383C2×C42—474no—
384C2×C42—471no—
385C2×C42—471no—
386C2×C42—474no—
387C2×C42—471no—
388C2×C42—471no—
389C2×C42—941no—
390C2×C42—941no—
391C2×C42—941no—
392C2×C42—941no—
393C2×C42—941no—
394C2×C42—941no—
395C2×C42—941no—
396#107—22no—
397#107—942no—
398#107—942no—
399#107—942no—
400#106—22no—
401#106—942no—
402#106—942no—
403#106—942no—
404C4⋊C8—474no—
405C4⋊C8—474no—
406C22⋊C8—474no—
407C22⋊C8C4×D4714yes—
408C4×C8—474no—
409C4×C8—474no—
410#95—21no—
411#95—941no—
412#95—941no—
413#95—941no—
414#95—941no—
415#95—941no—
416#95—941no—
417#95—941no—
418#95—941no—
419#95—941no—
420#95—941no—
421#95—941no—
422C24D4×D4711yescharacteristic
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—471no—
432C22×C4—474no—
433C22×C4—471no—
434C22×C4—471no—
435C22×C4—471no—
436C22×C4—474no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—941no—
440C22×C4—941no—
441C22×C4—941no—
442C22×C4—941no—
443C22×C4—1881no—
444C22×C4—1881no—
445C22×C4C2×C4×D4712yes—
446C22×C4D4×D4711yescharacteristic
447C22×C4Q8×D4712yes—
448C22×C4C23×D4711yescharacteristic
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—472no—
454C2×C8—472no—
455C2×C8—941no—
456C2×C8—941no—
457C2×C8—941no—
458C2×C8—941no—
459C2×C8C2×C4×D4712yes—
460C2×C8C2×C4×D4712yes—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—472no—
468C4⋊C4—472no—
469C4⋊C4—941no—
470C4⋊C4—941no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—941no—
475C4⋊C4#1062535212yes—
476C4⋊C4#1062535212yes—
477C22⋊C4—24no—
478C22⋊C4—42no—
479C22⋊C4—944no—
480C22⋊C4—1882no—
481C42—24no—
482C42—474no—
483C42—474no—
484C42—472no—
485C42—472no—
486C42—472no—
487C42—472no—
488C42—944no—
489C42—1882no—
490C42—1882no—
491C42—1882no—
492C42—1882no—
493C42—1882no—
494C42—1882no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23C4⋊C4×D4712yes—
502C23C2×D4×D4711yescharacteristic
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—472no—
522C2×C4—472no—
523C2×C4—472no—
524C2×C4—472no—
525C2×C4—472no—
526C2×C4—472no—
527C2×C4—948no—
528C2×C4—941no—
529C2×C4—941no—
530C2×C4—941no—
531C2×C4—941no—
532C2×C4—941no—
533C2×C4—941no—
534C2×C4—942no—
535C2×C4—942no—
536C2×C4—1884no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4—1881no—
540C2×C4—1881no—
541C2×C4—1881no—
542C2×C4—1881no—
543C2×C4—1881no—
544C2×C4—1881no—
545C2×C4—1881no—
546C2×C4—1881no—
547C2×C4C4⋊C4×D4712yes—
548C2×C4C22×C4×D4711yescharacteristic
549C2×C4C2×Q8×D4711yescharacteristic
550C2×C4#41×D4711yescharacteristic
551C2×C4#41×D4711yescharacteristic
552C2×C4C2×#1062535211yescharacteristic
553C2×C4C2×#1062535211yescharacteristic
554C8—22no—
555C8—22no—
556C8—942no—
557C8—942no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22C2×C4⋊C4×D4711yescharacteristic
566C22#1568962911yescharacteristic
567C22#1568963111yescharacteristic
568C22#135×D4711yescharacteristic
569C22#135×D4711yescharacteristic
570C22#1568991911yescharacteristic
571C22#1568992211yescharacteristic
572C4—22no—
573C4—42no—
574C4—42no—
575C4—81no—
576C4—81no—
577C4—474no—
578C4—942no—
579C4—942no—
580C4—942no—
581C4—3761no—
582C4—3761no—
583C4—1882no—
584C4—1882no—
585C4C4×#1062535211yescharacteristic
586C4C4×#1062535211yescharacteristic
587C4#1568967511yescharacteristic
588C4#1568967511yescharacteristic
589C2—42no—
590C2#2024586711yescharacteristic
591C2#427×D4711yescharacteristic
592C2#2025197511yescharacteristic
593C2#2025339911yescharacteristic
594C2#2025340111yescharacteristic
595C2#2025340711yescharacteristic
596C2#2025340911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1854C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#1854D47
hypocenterC47#21360
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
4. #20244159C2semidirect
5. #20244157C2semidirect
6. #20244155C2semidirect
7. #20244153C2semidirect
11. #107×Dic47C2semidirect
12. #20243280C2semidirect
13. #106×Dic47C2semidirect
14. #20243276C2semidirect
18. #15689173C4semidirect
358. C47#21360semidirect

Maximal subgroups

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

subgroupindex
C2×#156891732
#202457832
#117×Dic472
#202441592
#202441572
#202441552
#202441532
#202435892
#202435872
#202435852
#107×Dic472
#202432802
#106×Dic472
#202432762
C47×#18542
#2136047