Finite Group Databasebeta

← #26218296

Subgroups — #26218296

Order 12032.

1748 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256591C211yesmaximal, characteristic, complemented
2C2×#15689097C211yesmaximal, characteristic, complemented
3C2×#15689097C212yesmaximal, complemented
4#20243931C218yesmaximal, complemented
5#20243120C212yesmaximal, complemented
6C47×#3449C211yesmaximal, characteristic
7#15689606—24no—
8#15689601—21no—
9#15689598—21no—
10#15689598C2214yes—
11#41×Dic47—24no—
12C2×#13093983—22no—
13C2×#13093983C2212yes—
14C2×#13093983C2212yes—
15#15689097—24no—
16#15689097—24no—
17#15689097—22no—
18#15689097—22no—
19C2×#13093939C2218yes—
20C2×#13093939C2214yes—
21#15688889C2218yes—
22C22×#41×C47C2211yescharacteristic
23C22×#41×C47C2211yescharacteristic
24C47×#537—22no—
25C47×#536—21no—
26C47×#536C412yescomplemented
27C47×#535—21no—
28C47×#535C412yescomplemented
29C47×#534C2214yes—
30C47×#532—22no—
31C47×#525—22no—
32C47×#525C414yescomplemented
33C2×#117×C47C2211yescharacteristic
34Q8×Dic47—28no—
35D4×Dic47—28no—
36D4×Dic47—216no—
37#13093983—24no—
38#13093983—24no—
39#13093983—22no—
40#13093983—24no—
41#13093983—22no—
42#13093983—22no—
43#13093983—22no—
44#13093983—22no—
45#13093983—22no—
46#13093983D414yescomplemented
47#13093983D414yescomplemented
48C2×C4⋊Dic47—24no—
49C2×C4⋊Dic47—22no—
50C2×C4⋊Dic47—22no—
51C2×C22⋊Dic47—24no—
52C2×C22⋊Dic47C2314yes—
53C2×C22⋊Dic47C2314yes—
54C2×C4×Dic47—24no—
55C2×C4×Dic47—22no—
56C2×C4×Dic47—22no—
57#13093939—28no—
58#13093939—216no—
59C2×#41×C47—22no—
60C2×#41×C47—22no—
61C2×#41×C47—21no—
62C2×#41×C47—21no—
63C2×#41×C47—21no—
64C2×#41×C47—21no—
65C2×#41×C47—22no—
66C2×#41×C47—22no—
67C2×#41×C47—22no—
68C2×#41×C47—22no—
69C2×#41×C47—22no—
70C2×#41×C47—22no—
71C2×#41×C47C2×C412yes—
72C2×#41×C47C2×C412yes—
73C2×#41×C47D412yes—
74C2×#41×C47D412yes—
75C22×Q8×C47C2×C411yescharacteristic, complemented
76C22×Q8×C47D411yescharacteristic
77C22×D4×C47C2×C411yescharacteristic, complemented
78C22×D4×C47D411yescharacteristic
79C22×D4×C47C2312yes—
80C22×D4×C47C2312yes—
81C23×C4×C47C2×C412yescomplemented
82C23×C4×C47D412yes—
83C23×C4×C47C2311yescharacteristic
84C4⋊Q8×C47—22no—
85#127×C47—22no—
86#124×C47—22no—
87#124×C47—22no—
88#123×C47—24no—
89#123×C47C2×C418yescomplemented
90C22⋊Q8×C47—24no—
91C22⋊Q8×C47—24no—
92#121×C47—24no—
93#121×C47—28no—
94#121×C47—24no—
95#120×C47—24no—
96#120×C47C2×C418yescomplemented
97C4×D4×C47—28no—
98#117×C47—21no—
99#117×C47—21no—
100#117×C47C2×C412yescomplemented
101#117×C47C2×C412yescomplemented
102C2×C4⋊C4×C47—21no—
103C2×C22⋊C4×C47C2312yes—
104C2×C42×C47—21no—
105C24×C47C22×C412yes—
106C24×C47C2×D412yes—
107C24×C47C2411yescharacteristic
108#41×C47—24no—
109#41×C47—24no—
110#41×C47—24no—
111#41×C47—24no—
112#41×C47—24no—
113#41×C47—24no—
114#41×C47—24no—
115#41×C47—24no—
116#41×C47—41no—
117#41×C47—42no—
118#41×C47—41no—
119#41×C47—42no—
120#41×C47—42no—
121#41×C47—42no—
122#41×C47—42no—
123#41×C47—41no—
124#41×C47—41no—
125#41×C47—42no—
126C2×Q8×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—22no—
129C2×Q8×C47—22no—
130C2×Q8×C47—22no—
131C2×Q8×C47—22no—
132C2×Q8×C47C22⋊C412yes—
133C2×Q8×C47C22⋊C412yes—
134C2×D4×C47—24no—
135C2×D4×C47—24no—
136C2×D4×C47—22no—
137C2×D4×C47—22no—
138C2×D4×C47—24no—
139C2×D4×C47—24no—
140C2×D4×C47—28no—
141C2×D4×C47—21no—
142C2×D4×C47—21no—
143C2×D4×C47—22no—
144C2×D4×C47—22no—
145C2×D4×C47—22no—
146C2×D4×C47—22no—
147C2×D4×C47C22⋊C412yes—
148C2×D4×C47C22⋊C412yes—
149C2×D4×C47C22×C414yes—
150C2×D4×C47C2×D414yes—
151C22×C4×C47—22no—
152C22×C4×C47—22no—
153C22×C4×C47—22no—
154C22×C4×C47—22no—
155C22×C4×C47—22no—
156C22×C4×C47—22no—
157C22×C4×C47—22no—
158C22×C4×C47—24no—
159C22×C4×C47—24no—
160C22×C4×C47C22⋊C414yes—
161C22×C4×C47C22⋊C414yes—
162C22×C4×C47C22×C412yes—
163C22×C4×C47C22×C411yescharacteristic
164C22×C4×C47C22×C411yescharacteristic
165C22×C4×C47C2×D412yes—
166C22×C4×C47C2×D411yescharacteristic
167C22×C4×C47C2×D411yescharacteristic
168C22×C4×C47C2×D411yescharacteristic
169C22×C4×C47C2×D411yescharacteristic
170C22×C4×C47C2×D412yes—
171C22×C4×C47C2×D412yes—
172C22×C4×C47#4112yes—
173C22×C4×C47#4112yes—
174C22×Dic47—28no—
175C22×Dic47—24no—
176C22×Dic47—24no—
177C4⋊C4×C47—22no—
178C4⋊C4×C47—22no—
179C4⋊C4×C47—28no—
180C22⋊C4×C47—28no—
181C22⋊C4×C47—22no—
182C22⋊C4×C47—28no—
183C22⋊C4×C47C22×C414yes—
184C42×C47—22no—
185C42×C47—22no—
186C22⋊Dic47—28no—
187C22⋊Dic47—24no—
188C22⋊Dic47—24no—
189C22⋊Dic47—24no—
190C22⋊Dic47C2×D418yes—
191C4⋊Dic47—28no—
192C4⋊Dic47—24no—
193C4⋊Dic47—24no—
194C4⋊Dic47—42no—
195C4⋊Dic47—42no—
196C4×Dic47—28no—
197C4×Dic47—24no—
198C4×Dic47—24no—
199C4×Dic47—42no—
200C4×Dic47—42no—
201C23×C47—22no—
202C23×C47—22no—
203C23×C47—22no—
204C23×C47—24no—
205C23×C47—81no—
206C23×C47—82no—
207C23×C47—82no—
208C23×C47C2×C22⋊C414yes—
209C23×C47C23×C411yescharacteristic
210C23×C47C22×D412yes—
211C23×C47C22×D411yescharacteristic
212C23×C47C22×D411yescharacteristic
213C23×C47C2×#4112yes—
214Q8×C47—24no—
215Q8×C47—24no—
216Q8×C47—41no—
217Q8×C47—41no—
218Q8×C47—42no—
219D4×C47—28no—
220D4×C47—28no—
221D4×C47—24no—
222D4×C47—24no—
223D4×C47—42no—
224D4×C47—44no—
225D4×C47—42no—
226D4×C47—41no—
227D4×C47—41no—
228D4×C47—42no—
229D4×C47—88no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—24no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—24no—
236C2×C4×C47—24no—
237C2×C4×C47—22no—
238C2×C4×C47—24no—
239C2×C4×C47—42no—
240C2×C4×C47—44no—
241C2×C4×C47—42no—
242C2×C4×C47—82no—
243C2×C4×C47C2×C22⋊C414yes—
244C2×C4×C47C2×C22⋊C411yescharacteristic
245C2×C4×C47C2×C22⋊C411yescharacteristic
246C2×C4×C47C2×C22⋊C411yescharacteristic
247C2×C4×C47C2×C22⋊C411yescharacteristic
248C2×C4×C47C4×D412yes—
249C2×C4×C47C4×D412yes—
250C2×C4×C47C4×D412yes—
251C2×C4×C47C4×D412yes—
252C2×C4×C47#12011yescharacteristic
253C2×C4×C47#12012yes—
254C2×C4×C47#12011yescharacteristic
255C2×C4×C47#12112yes—
256C2×C4×C47#12112yes—
257C2×C4×C47#12112yes—
258C2×C4×C47#12112yes—
259C2×C4×C47#12311yescharacteristic
260C2×C4×C47#12311yescharacteristic
261C2×C4×C47#12312yes—
262C2×Dic47—28no—
263C2×Dic47—44no—
264C2×Dic47—84no—
265C2×Dic47—88no—
266C2×Dic47—84no—
267#21014—471nomaximal, Sylow
268C22×C47—24no—
269C22×C47—42no—
270C22×C47—42no—
271C22×C47—41no—
272C22×C47—41no—
273C22×C47—41no—
274C22×C47—44no—
275C22×C47—44no—
276C22×C47—44no—
277C22×C47—42no—
278C22×C47—82no—
279C22×C47—84no—
280C22×C47C22×C22⋊C411yescharacteristic
281C22×C47C2×C4×D412yes—
282C22×C47C2×#12011yescharacteristic
283C22×C47C2×#12112yes—
284C22×C47C2×#12311yescharacteristic
285C4×C47—24no—
286C4×C47—44no—
287C4×C47—42no—
288C4×C47#38611yescharacteristic
289C4×C47#38611yescharacteristic
290C4×C47#38611yescharacteristic
291C4×C47#38612yes—
292C4×C47#38612yes—
293C4×C47#38611yescharacteristic
294Dic47—48no—
295Dic47—84no—
296#3449D4711yescharacteristic
297#3428—471no—
298C2×#410—472no—
299C2×#410—471no—
300#1913—478no—
301#1740—472no—
302C2×C47—21no—
303C2×C47—22no—
304C2×C47—21no—
305C2×C47—22no—
306C2×C47—21no—
307C2×C47—44no—
308C2×C47—82no—
309C2×C47C2×#38611yescharacteristic
310C22×#41C2×D4711yescharacteristic
311C22×#41C2×D4711yescharacteristic
312#537—22no—
313#536—21no—
314#536Dic4712yescomplemented
315#535—21no—
316#535Dic4712yescomplemented
317#534C2×D4714yes—
318#532—22no—
319#525—22no—
320#525Dic4714yescomplemented
321#520—944no—
322#519—941no—
323#518—474no—
324#518—941no—
325C4×#41—944no—
326C2×#117—472no—
327C2×#117—472no—
328C2×#117—942no—
329C2×#117C2×D4711yescharacteristic
330#410—944no—
331#410—944no—
332#410—942no—
333#410—942no—
334C2×#99—478no—
335C2×#99—474no—
336#342—478no—
337C47#2101411yescharacteristic, complemented, Sylow
338C2×#41—22no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41—22no—
346C2×#41—21no—
347C2×#41—21no—
348C2×#41—21no—
349C2×#41—21no—
350C2×#41C2×Dic4712yes—
351C2×#41C2×Dic4712yes—
352C2×#41C22⋊D4712yes—
353C2×#41C22⋊D4712yes—
354C22×Q8C2×Dic4711yescharacteristic, complemented
355C22×Q8C22⋊D4711yescharacteristic
356C22×D4C2×Dic4711yescharacteristic, complemented
357C22×D4C22⋊D4711yescharacteristic
358C22×D4C22×D4712yes—
359C22×D4C22×D4712yes—
360C23×C4C2×Dic4712yescomplemented
361C23×C4C22⋊D4712yes—
362C23×C4C22×D4711yescharacteristic
363C4⋊Q8—22no—
364#127—22no—
365#124—22no—
366#124—22no—
367#123—24no—
368#123C2×Dic4718yescomplemented
369C22⋊Q8—24no—
370C22⋊Q8—24no—
371#121—24no—
372#121—28no—
373#121—24no—
374#120—24no—
375#120C2×Dic4718yescomplemented
376C4×Q8—948no—
377C4×D4—28no—
378C4×D4—9416no—
379C4×D4—948no—
380#117—21no—
381#117—21no—
382#117—474no—
383#117—474no—
384#117—944no—
385#117—944no—
386#117—942no—
387#117—942no—
388#117—944no—
389#117—942no—
390#117—942no—
391#117—942no—
392#117—942no—
393#117C2×Dic4712yescomplemented
394#117C2×Dic4712yescomplemented
395C2×C4⋊C4—21no—
396C2×C4⋊C4—944no—
397C2×C4⋊C4—942no—
398C2×C4⋊C4—942no—
399C2×C22⋊C4—474no—
400C2×C22⋊C4—474no—
401C2×C22⋊C4—944no—
402C2×C22⋊C4C22×D4712yes—
403C2×C42—21no—
404C2×C42—944no—
405C2×C42—942no—
406C2×C42—942no—
407#99—9416no—
408#99—948no—
409C24C22×Dic4712yes—
410C24C2×C22⋊D4712yes—
411C24C23×D4711yescharacteristic
412#41—24no—
413#41—24no—
414#41—24no—
415#41—24no—
416#41—24no—
417#41—24no—
418#41—24no—
419#41—24no—
420#41—42no—
421#41—41no—
422#41—41no—
423#41—42no—
424#41—42no—
425#41—42no—
426#41—42no—
427#41—41no—
428#41—42no—
429#41—41no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8C22⋊Dic4712yes—
437C2×Q8C22⋊Dic4712yes—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—24no—
444C2×D4—28no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4C22⋊Dic4712yes—
452C2×D4C22⋊Dic4712yes—
453C2×D4C22×Dic4714yes—
454C2×D4C2×C22⋊D4714yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—948no—
465C22×C4—944no—
466C22×C4—944no—
467C22×C4C22⋊Dic4714yes—
468C22×C4C22⋊Dic4714yes—
469C22×C4D4×D4712yes—
470C22×C4D4×D4712yes—
471C22×C4#1062535212yes—
472C22×C4#1062535212yes—
473C22×C4C22×Dic4712yes—
474C22×C4C22×Dic4711yescharacteristic
475C22×C4C22×Dic4711yescharacteristic
476C22×C4C2×C22⋊D4712yes—
477C22×C4C2×C22⋊D4711yescharacteristic
478C22×C4C2×C22⋊D4711yescharacteristic
479C22×C4C2×C22⋊D4711yescharacteristic
480C22×C4C2×C22⋊D4711yescharacteristic
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—28no—
484C4⋊C4—948no—
485C4⋊C4—944no—
486C4⋊C4—944no—
487C4⋊C4—1882no—
488C4⋊C4—1882no—
489C22⋊C4—28no—
490C22⋊C4—22no—
491C22⋊C4—28no—
492C22⋊C4—478no—
493C22⋊C4—948no—
494C22⋊C4—944no—
495C22⋊C4—944no—
496C22⋊C4—944no—
497C22⋊C4C22×Dic4714yes—
498C42—22no—
499C42—22no—
500C42—948no—
501C42—944no—
502C42—944no—
503C42—1882no—
504C42—1882no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—81no—
510C23—82no—
511C23—82no—
512C23C2×C22⋊Dic4714yes—
513C23C23×Dic4711yescharacteristic
514C23C2×D4×D4712yes—
515C23C22×C22⋊D4711yescharacteristic
516C23C22×C22⋊D4711yescharacteristic
517C23C2×#1062535212yes—
518Q8—24no—
519Q8—24no—
520Q8—41no—
521Q8—41no—
522Q8—42no—
523D4—28no—
524D4—28no—
525D4—24no—
526D4—24no—
527D4—42no—
528D4—44no—
529D4—42no—
530D4—41no—
531D4—41no—
532D4—42no—
533D4—88no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—82no—
547C2×C4—948no—
548C2×C4—3764no—
549C2×C4—3764no—
550C2×C4—3768no—
551C2×C4—1884no—
552C2×C4C2×C22⋊Dic4714yes—
553C2×C4C2×C22⋊Dic4711yescharacteristic
554C2×C4C2×C22⋊Dic4711yescharacteristic
555C2×C4C2×C22⋊Dic4711yescharacteristic
556C2×C4C2×C22⋊Dic4711yescharacteristic
557C2×C4D4×Dic4712yes—
558C2×C4D4×Dic4712yes—
559C2×C4D4×Dic4712yes—
560C2×C4D4×Dic4712yes—
561C2×C4#1309399212yes—
562C2×C4#1309399211yescharacteristic
563C2×C4#1309399211yescharacteristic
564C2×C4#1309399612yes—
565C2×C4#1309399612yes—
566C2×C4#1309399612yes—
567C2×C4#1309399612yes—
568C2×C4#1309400511yescharacteristic
569C2×C4#1309400512yes—
570C2×C4#1309400511yescharacteristic
571C22—24no—
572C22—41no—
573C22—42no—
574C22—42no—
575C22—41no—
576C22—41no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—42no—
581C22—82no—
582C22—84no—
583C22C22×C22⋊Dic4711yescharacteristic
584C22C2×D4×Dic4712yes—
585C22C2×#1309399211yescharacteristic
586C22C2×#1309399612yes—
587C22C2×#1309400511yescharacteristic
588C4—24no—
589C4—44no—
590C4—42no—
591C4—3764no—
592C4—1888no—
593C4#1568900311yescharacteristic
594C4#1568900311yescharacteristic
595C4#1568900311yescharacteristic
596C4#1568900312yes—
597C4#1568900312yes—
598C4#1568900311yescharacteristic
599C2—21no—
600C2—22no—
601C2—21no—
602C2—22no—
603C2—21no—
604C2—44no—
605C2—82no—
606C2C2×#1568900311yescharacteristic
subgroupG/H
centerC4#15689003
commutatorC23×C47C23×C4
FrattiniC24C23×D47
Fitting, prime radicalC47×#3449C2
socleC2×C47C2×#386
cosocleC24×C47C24
hypercenter#3449D47
hypocenterC47#21014
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C23×C4 ◁ #3449

Complements

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

NH ≅ G/Ntype
1. #20256591C2semidirect
2. C2×#15689097C2semidirect
3. C2×#15689097C2semidirect
4. #20243931C2semidirect
5. #20243120C2semidirect
26. C47×#536C4semidirect
28. C47×#535C4semidirect
32. C47×#525C4semidirect
46. #13093983D4semidirect
47. #13093983D4semidirect
75. C22×Q8×C47C2×C4semidirect
77. C22×D4×C47C2×C4semidirect
81. C23×C4×C47C2×C4semidirect
89. #123×C47C2×C4semidirect
96. #120×C47C2×C4semidirect
100. #117×C47C2×C4semidirect
101. #117×C47C2×C4semidirect
314. #536Dic47semidirect
316. #535Dic47semidirect
320. #525Dic47semidirect
337. C47#21014semidirect
354. C22×Q8C2×Dic47semidirect
356. C22×D4C2×Dic47semidirect
360. C23×C4C2×Dic47semidirect
368. #123C2×Dic47semidirect
375. #120C2×Dic47semidirect
393. #117C2×Dic47semidirect
394. #117C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202565912
C2×#156890972
#202439312
#202431202
C47×#34492
#2101447