Finite Group Databasebeta

← #26142828

Subgroups — #26142828

Order 12032.

1406 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689018C212yesmaximal
2C2×#15688992C211yesmaximal, characteristic
3C2×#15688987C211yesmaximal, characteristic
4C2×#15688968C211yesmaximal, characteristic
5#20241652C211yesmaximal, characteristic
6#20241651C212yesmaximal
7#20241647C211yesmaximal, characteristic
8#20241646C212yesmaximal
9#20241645C211yesmaximal, characteristic
10Dic47⋊#95C212yesmaximal
11C47×#1444C211yesmaximal, characteristic
12C22×C4⋊Dic47C2211yescharacteristic
13C22×Dic47⋊C4C2212yes—
14C22×Dic47⋊C4C2212yes—
15C22×Dic47⋊C4C2212yes—
16C22×Dic47⋊C4C2212yes—
17C22×C4×Dic47C2211yescharacteristic
18C22×C4×Dic47C2211yescharacteristic
19C22×C4×Dic47C2211yescharacteristic
20#15689018—28no—
21#15688992—24no—
22#15688987C418yescomplemented
23#15688968—24no—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2211yescharacteristic
27C2×C47⋊#95C2211yescharacteristic
28C2×C47⋊#95C2211yescharacteristic
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2211yescharacteristic
34C22×C4⋊C4×C47C2211yescharacteristic
35C22×C4⋊C4×C47C2212yes—
36C2×#95×C47C2211yescharacteristic
37C2×#95×C47C2212yes—
38C2×#95×C47C2211yescharacteristic
39C23×Dic47C2312yes—
40C23×Dic47C2312yes—
41C23×Dic47C2311yescharacteristic
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2311yescharacteristic
44C23×Dic47C2311yescharacteristic
45C2×C4⋊Dic47—22no—
46C2×C4⋊Dic47—42no—
47C2×Dic47⋊C4—24no—
48C2×Dic47⋊C4—24no—
49C2×Dic47⋊C4—24no—
50C2×Dic47⋊C4—24no—
51C2×Dic47⋊C4—24no—
52C2×Dic47⋊C4—24no—
53C2×Dic47⋊C4—24no—
54C2×Dic47⋊C4—42no—
55C2×Dic47⋊C4—42no—
56C2×Dic47⋊C4—42no—
57C2×Dic47⋊C4—42no—
58C2×Dic47⋊C4C2×C418yes—
59C2×C4×Dic47—22no—
60C2×C4×Dic47—22no—
61C2×C4×Dic47—22no—
62C2×C4×Dic47—22no—
63C2×C4×Dic47C2×C414yes—
64C2×C4×Dic47D414yes—
65C2×C4×Dic47D414yes—
66C2×C4×Dic47Q814yes—
67C2×C4×Dic47Q814yes—
68C47⋊#95—22no—
69C47⋊#95—22no—
70C47⋊#95—22no—
71C47⋊#95—24no—
72C47⋊#95—24no—
73C47⋊#95—42no—
74C47⋊#95—42no—
75C47⋊#95—42no—
76C47⋊#95C2×C414yes—
77C47⋊#95C2×C418yes—
78C23×C4×C47C2312yes—
79C23×C4×C47C2312yes—
80C23×C4×C47C2311yescharacteristic
81C23×C4×C47C2311yescharacteristic
82C23×C4×C47C2311yescharacteristic
83C2×C4⋊C4×C47—22no—
84C2×C4⋊C4×C47—24no—
85C2×C4⋊C4×C47—42no—
86C2×C4⋊C4×C47—42no—
87C2×C4⋊C4×C47—42no—
88#95×C47—22no—
89#95×C47—42no—
90#95×C47C2×C414yes—
91C24×C47C2411yescharacteristic
92C22×C4×C47—22no—
93C22×C4×C47—22no—
94C22×C4×C47—22no—
95C22×C4×C47—21no—
96C22×C4×C47—21no—
97C22×C4×C47—21no—
98C22×C4×C47—21no—
99C22×C4×C47—22no—
100C22×C4×C47—22no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47C22×C414yes—
116C22×C4×C47C22×C412yes—
117C22×C4×C47C2×D412yes—
118C22×C4×C47C2×D412yes—
119C22×C4×C47C2×Q812yes—
120C22×C4×C47C2×Q812yes—
121C22×Dic47—22no—
122C22×Dic47—22no—
123C22×Dic47—21no—
124C22×Dic47—21no—
125C22×Dic47—22no—
126C22×Dic47—21no—
127C22×Dic47—21no—
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47C22×C414yes—
145C22×Dic47C22×C412yes—
146C22×Dic47C22×C412yes—
147C22×Dic47C2×D412yes—
148C22×Dic47C2×D412yes—
149C22×Dic47C2×D412yes—
150C22×Dic47C2×D412yes—
151C22×Dic47C2×Q812yes—
152C22×Dic47C2×Q812yes—
153C22×Dic47C2×Q812yes—
154C22×Dic47C2×Q812yes—
155C22×Dic47#4112yes—
156C22×Dic47#4112yes—
157C4⋊C4×C47—44no—
158C4⋊C4×C47—48no—
159C4⋊Dic47—44no—
160Dic47⋊C4—216no—
161Dic47⋊C4—48no—
162Dic47⋊C4—48no—
163Dic47⋊C4—48no—
164C4×Dic47—28no—
165C4×Dic47—28no—
166C4×Dic47C4⋊C4116yescomplemented
167C23×C47C23×C411yescharacteristic
168C23×C47C22×D411yescharacteristic
169C23×C47C22×D411yescharacteristic
170C23×C47C22×Q811yescharacteristic
171C23×C47C22×Q811yescharacteristic
172C23×C47C2×#4111yescharacteristic
173C23×C47C2×#4111yescharacteristic
174C23×C47H(5,𝔽2)12yes—
175C23×C47H(5,𝔽2)12yes—
176C23×C472−1+412yes—
177C23×C472−1+412yes—
178C2×C4×C47—24no—
179C2×C4×C47—22no—
180C2×C4×C47—22no—
181C2×C4×C47—24no—
182C2×C4×C47—24no—
183C2×C4×C47—24no—
184C2×C4×C47—24no—
185C2×C4×C47—22no—
186C2×C4×C47—22no—
187C2×C4×C47—24no—
188C2×C4×C47—24no—
189C2×C4×C47—42no—
190C2×C4×C47—41no—
191C2×C4×C47—41no—
192C2×C4×C47—41no—
193C2×C4×C47—41no—
194C2×C4×C47—42no—
195C2×C4×C47—42no—
196C2×C4×C47—42no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47—42no—
201C2×C4×C47—42no—
202C2×C4×C47—42no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47C2×C4⋊C414yes—
207C2×Dic47—24no—
208C2×Dic47—22no—
209C2×Dic47—22no—
210C2×Dic47—24no—
211C2×Dic47—24no—
212C2×Dic47—24no—
213C2×Dic47—24no—
214C2×Dic47—22no—
215C2×Dic47—22no—
216C2×Dic47—24no—
217C2×Dic47—24no—
218C2×Dic47—42no—
219C2×Dic47—41no—
220C2×Dic47—41no—
221C2×Dic47—42no—
222C2×Dic47—41no—
223C2×Dic47—41no—
224C2×Dic47—42no—
225C2×Dic47—42no—
226C2×Dic47—42no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47C2×C4⋊C414yes—
236C2×Dic47C2×C4⋊C414yes—
237C2×Dic47C4×D414yes—
238C2×Dic47C4×Q814yes—
239C2×Dic47#12114yes—
240C2×Dic47C22⋊Q814yes—
241C2×Dic47#12514yes—
242C2×Dic47C4⋊Q814yes—
243#14277—471nomaximal, Sylow
244C22×C47C22×C4⋊C411yescharacteristic
245C22×C47C2×C4×D411yescharacteristic
246C22×C47C2×C4×Q811yescharacteristic
247C22×C47#52012yes—
248C22×C47#52012yes—
249C22×C47C2×#12111yescharacteristic
250C22×C47C2×C22⋊Q811yescharacteristic
251C22×C47C2×#12511yescharacteristic
252C22×C47C2×C4⋊Q811yescharacteristic
253C22×C47#53511yescharacteristic
254C22×C47#53511yescharacteristic
255C22×C47#53512yes—
256C22×C47#53611yescharacteristic
257C22×C47#53612yes—
258C22×C47#53611yescharacteristic
259C22×C47#54012yes—
260C22×C47#54112yes—
261C22×C47#54212yes—
262C22×C47#54212yes—
263C22×C47#54412yes—
264C22×C47#54412yes—
265C22×C47#54412yes—
266C22×C47#54412yes—
267C4×C47—24no—
268C4×C47—44no—
269C4×C47—42no—
270C4×C47—42no—
271C4×C47—82no—
272Dic47—24no—
273Dic47—44no—
274Dic47—42no—
275Dic47—42no—
276Dic47—82no—
277Dic47#38918yes—
278C2×#389—472no—
279C2×#384—471no—
280C2×#382—471no—
281C2×#376—471no—
282#1444—471no—
283#1444—472no—
284#1444D4711yescharacteristic
285#1442—471no—
286#1442—472no—
287#1442—471no—
288C4⋊#95—472no—
289C2×C47C2×#38911yescharacteristic
290C2×C47#231711yescharacteristic
291C2×C47#231711yescharacteristic
292C2×C47#232612yes—
293C2×C47#233012yes—
294C2×C47#261911yescharacteristic
295C2×C47#262111yescharacteristic
296C2×C47#262811yescharacteristic
297C2×C47#262811yescharacteristic
298C2×C47#264712yes—
299C2×C47#266012yes—
300C22×C4⋊C4—472no—
301C22×C4⋊C4—472no—
302C22×C4⋊C4—472no—
303C22×C4⋊C4—472no—
304C22×C4⋊C4—471no—
305C22×C4⋊C4C2×D4711yescharacteristic
306C22×C4⋊C4C2×D4712yes—
307C22×C42—471no—
308C22×C42—471no—
309C22×C42—471no—
310#389—948no—
311#384—944no—
312#382—478no—
313#376—944no—
314C2×#95—472no—
315C2×#95—471no—
316C2×#95—471no—
317C2×#95—471no—
318C2×#95—472no—
319C2×#95—472no—
320C2×#95—471no—
321C2×#95—472no—
322C2×#95—472no—
323C2×#95—472no—
324C2×#95C2×D4711yescharacteristic
325C2×#95C2×D4712yes—
326C2×#95C2×D4711yescharacteristic
327C47#1427711yescharacteristic, complemented, Sylow
328C23×C4—471no—
329C23×C4—471no—
330C23×C4—472no—
331C23×C4—471no—
332C23×C4—471no—
333C23×C4—472no—
334C23×C4C22×D4712yes—
335C23×C4C22×D4712yes—
336C23×C4C22×D4711yescharacteristic
337C23×C4C22×D4711yescharacteristic
338C23×C4C22×D4711yescharacteristic
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—42no—
342C2×C4⋊C4—42no—
343C2×C4⋊C4—42no—
344C2×C4⋊C4—478no—
345C2×C4⋊C4—944no—
346C2×C4⋊C4—944no—
347C2×C4⋊C4—944no—
348C2×C4⋊C4—944no—
349C2×C4⋊C4—944no—
350C2×C4⋊C4—944no—
351C2×C4⋊C4—944no—
352C2×C4⋊C4—942no—
353C2×C4⋊C4—1882no—
354C2×C4⋊C4—1882no—
355C2×C4⋊C4—1882no—
356C2×C4⋊C4—1882no—
357C2×C4⋊C4—1882no—
358C2×C42—474no—
359C2×C42—474no—
360C2×C42—474no—
361C2×C42—474no—
362C2×C42—474no—
363C2×C42—942no—
364C2×C42—942no—
365C2×C42—942no—
366C2×C42—942no—
367#95—22no—
368#95—42no—
369#95—478no—
370#95—474no—
371#95—942no—
372#95—942no—
373#95—944no—
374#95—944no—
375#95—942no—
376#95—1882no—
377#95—1882no—
378#95—1882no—
379#95C4×D4714yes—
380C24C23×D4711yescharacteristic
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—472no—
405C22×C4—472no—
406C22×C4—472no—
407C22×C4—472no—
408C22×C4—472no—
409C22×C4—472no—
410C22×C4—472no—
411C22×C4—472no—
412C22×C4—472no—
413C22×C4—472no—
414C22×C4—474no—
415C22×C4—472no—
416C22×C4—472no—
417C22×C4—942no—
418C22×C4—942no—
419C22×C4—941no—
420C22×C4—941no—
421C22×C4—942no—
422C22×C4—941no—
423C22×C4—941no—
424C22×C4—942no—
425C22×C4—942no—
426C22×C4—942no—
427C22×C4—942no—
428C22×C4—942no—
429C22×C4—942no—
430C22×C4—942no—
431C22×C4—942no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4C2×C4×D4714yes—
441C22×C4C2×C4×D4712yes—
442C22×C4D4×D4712yes—
443C22×C4D4×D4712yes—
444C22×C4Q8×D4712yes—
445C22×C4Q8×D4712yes—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—9416no—
449C4⋊C4—1888no—
450C4⋊C4—1884no—
451C4⋊C4—1888no—
452C4⋊C4—1888no—
453C42—4716no—
454C42—948no—
455C42—948no—
456C23C22×C4×D4711yescharacteristic
457C23C2×D4×D4711yescharacteristic
458C23C2×D4×D4711yescharacteristic
459C23C2×Q8×D4711yescharacteristic
460C23C2×Q8×D4711yescharacteristic
461C23#41×D4711yescharacteristic
462C23#41×D4711yescharacteristic
463C23#1309406912yes—
464C23#1309406912yes—
465C23#1309407112yes—
466C23#1309407212yes—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—474no—
496C2×C4—474no—
497C2×C4—474no—
498C2×C4—474no—
499C2×C4—474no—
500C2×C4—474no—
501C2×C4—474no—
502C2×C4—474no—
503C2×C4—944no—
504C2×C4—944no—
505C2×C4—942no—
506C2×C4—942no—
507C2×C4—944no—
508C2×C4—944no—
509C2×C4—944no—
510C2×C4—944no—
511C2×C4—942no—
512C2×C4—942no—
513C2×C4—944no—
514C2×C4—1882no—
515C2×C4—1882no—
516C2×C4—1882no—
517C2×C4—1882no—
518C2×C4—1882no—
519C2×C4—1882no—
520C2×C4—1882no—
521C2×C4—1882no—
522C2×C4—1882no—
523C2×C4—1882no—
524C2×C4—1882no—
525C2×C4—1882no—
526C2×C4—1881no—
527C2×C4—1881no—
528C2×C4—1882no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4C4⋊C4×D4714yes—
532C22C2×C4⋊C4×D4711yescharacteristic
533C22C4×D4×D4711yescharacteristic
534C22C4×Q8×D4711yescharacteristic
535C22#1568960212yes—
536C22#1568960412yes—
537C22#121×D4711yescharacteristic
538C22C22⋊Q8×D4711yescharacteristic
539C22#125×D4711yescharacteristic
540C22C4⋊Q8×D4711yescharacteristic
541C22#1568968611yescharacteristic
542C22#1568968812yes—
543C22#1568968811yescharacteristic
544C22#1568969511yescharacteristic
545C22#1568969612yes—
546C22#1568969811yescharacteristic
547C22#1568972312yes—
548C22#1568973212yes—
549C22#1568974112yes—
550C22#1568974312yes—
551C22#1568975012yes—
552C22#1568975012yes—
553C22#1568975212yes—
554C22#1568975212yes—
555C4—24no—
556C4—42no—
557C4—44no—
558C4—42no—
559C4—82no—
560C4—478no—
561C4—944no—
562C4—3762no—
563C4—1882no—
564C4—1884no—
565C4—1882no—
566C2#389×D4711yescharacteristic
567C2#2024600711yescharacteristic
568C2#2024600911yescharacteristic
569C2#2024609412yes—
570C2#2024612612yes—
571C2#2024913711yescharacteristic
572C2#2024915611yescharacteristic
573C2#2024921111yescharacteristic
574C2#2024921411yescharacteristic
575C2#2024936212yes—
576C2#2024950612yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1444C2
socle, prime radical, cosocleC24×C47C24
hypercenter#1444D47
hypocenterC47#14277
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
22. #15688987C4semidirect
166. C4×Dic47C4⋊C4semidirect
327. C47#14277semidirect

Maximal subgroups

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

subgroupindex
C2×#156890182
C2×#156889922
C2×#156889872
C2×#156889682
#202416522
#202416512
#202416472
#202416462
#202416452
Dic47⋊#952
C47×#14442
#1427747