Finite Group Databasebeta

← #26218090

Subgroups — #26218090

Order 12032.

1718 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689699C211yesmaximal, characteristic
2C2×#15689600C211yesmaximal, characteristic, complemented
3C2×#15689096C211yesmaximal, characteristic
4C2×#15689096C211yesmaximal, characteristic
5#20245972C211yesmaximal, characteristic, complemented
6#20243845C218yesmaximal, complemented
7#20243121C211yesmaximal, characteristic, complemented
8C47×#1740C211yesmaximal, characteristic, complemented
9C22×#10625350C2211yescharacteristic
10C23×Dic94C2211yescharacteristic
11#15689699—22no—
12#15689699—22no—
13#15689699—22no—
14#15689699—22no—
15C2×#13094024C412yescomplemented
16C2×#13094012C412yescomplemented
17C2×#13094008C2214yes—
18C2×#13094004C414yescomplemented
19#15689600—24no—
20#15689600—24no—
21C2×C4×Dic94—22no—
22C2×#13093989—22no—
23C2×#13093983C2211yescharacteristic
24C2×#13093983C2211yescharacteristic
25C2×#13093980C2214yes—
26#15689096—24no—
27#15689096—24no—
28C2×#13093938C2214yes—
29C2×#13093938C2214yes—
30#15689029—22no—
31#15689028—22no—
32#15688984C2214yes—
33#15688890C2214yes—
34C2×#117×C47C2211yescharacteristic
35C2×#117×C47C2211yescharacteristic
36C2×#117×C47C2211yescharacteristic
37C47×#342C2214yes—
38C2×#10625350—22no—
39C2×#10625350—22no—
40C2×#10625350—21no—
41C2×#10625350—21no—
42C2×#10625350—21no—
43C2×#10625350—21no—
44C2×#10625350—22no—
45C2×#10625350—22no—
46C22×Dic94—22no—
47C22×Dic94—22no—
48C22×Dic94—22no—
49C22×Dic94—22no—
50C22×Dic94C2×C411yescharacteristic
51C22×Dic94C2×C412yes—
52C22×Dic94C2×C412yes—
53C22×Dic94D412yes—
54C22×Dic94D412yes—
55C22×Dic94D412yes—
56C22×Dic94D412yes—
57C22×D188C2×C411yescharacteristic
58C22×C22⋊D47C2312yes—
59C23×Dic47C2312yes—
60C22×C4×D47C2×C412yes—
61#13094024—24no—
62#13094024—24no—
63#13094012—24no—
64#13094012—24no—
65#13094008—28no—
66#13094008—28no—
67#13094004—28no—
68#13094004—28no—
69C4×Dic94—216no—
70#13093989—216no—
71#13093983—22no—
72#13093983—22no—
73#13093983—22no—
74#13093983D414yes—
75#13093980—28no—
76#13093980—28no—
77C2×C4⋊Dic47—21no—
78C2×C4⋊Dic47—21no—
79C2×C4⋊Dic47C2×C418yes—
80C2×Dic47⋊C4—24no—
81C2×C22⋊Dic47C2312yes—
82C2×C22⋊Dic47C2312yes—
83C2×(C2×D47)⋊C4C2×C418yes—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—24no—
86C2×C4×Dic47—21no—
87#13093938—28no—
88#13093938—28no—
89C47⋊#95—28no—
90C23×C4×C47C2311yescharacteristic
91#117×C47—22no—
92#117×C47—21no—
93#117×C47—21no—
94#117×C47—21no—
95#117×C47—21no—
96#117×C47—22no—
97#117×C47—22no—
98#117×C47D414yes—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47C2×C412yes—
102C2×C22⋊C4×C47C2312yes—
103C2×C22⋊C4×C47C2312yes—
104C2×C22⋊C4×C47C2312yes—
105C2×C42×C47—21no—
106C2×C42×C47—21no—
107C2×C42×C47C2×C412yes—
108C24×C47C2411yescharacteristic
109C23×D47C22×C412yes—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47C22×C414yes—
113C22×C4×C47C22×C411yescharacteristic
114C22×C4×C47C22×C411yescharacteristic
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2×D411yescharacteristic
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2×D411yescharacteristic
119C22×C4×C47C2×D412yes—
120C22×C4×C47C2×D412yes—
121C22×C4×C47#4112yes—
122C22×C4×C47#4112yes—
123C2×C22⋊D47—22no—
124C2×C22⋊D47—22no—
125C2×C22⋊D47—22no—
126C2×C22⋊D47—22no—
127C2×C22⋊D47—22no—
128C2×C22⋊D47—22no—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47C22×C412yes—
136C22×Dic47C22×C414yes—
137C22×Dic47C2×D414yes—
138C22×Dic47C2×D414yes—
139#10625350—42no—
140#10625350—42no—
141#10625350—42no—
142#10625350—42no—
143#10625350—42no—
144#10625350—42no—
145#10625350—42no—
146#10625350—42no—
147C2×D188—21no—
148C2×D188—21no—
149C2×D188—22no—
150C2×D188—22no—
151C2×C4×D47—22no—
152C2×C4×D47—22no—
153C2×C4×D47—24no—
154C2×C4×D47—24no—
155C2×Dic94—24no—
156C2×Dic94—24no—
157C2×Dic94—24no—
158C2×Dic94—24no—
159C2×Dic94—24no—
160C2×Dic94—24no—
161C2×Dic94—24no—
162C2×Dic94—24no—
163C2×Dic94—24no—
164C2×Dic94—24no—
165C2×Dic94—24no—
166C2×Dic94—24no—
167C2×Dic94—21no—
168C2×Dic94—21no—
169C2×Dic94—22no—
170C2×Dic94—22no—
171C2×Dic94C22⋊C414yes—
172C2×Dic94C22⋊C414yes—
173C2×Dic94C22⋊C414yes—
174C2×Dic94C22⋊C414yes—
175C4⋊C4×C47—22no—
176C4⋊C4×C47—22no—
177C4⋊C4×C47—24no—
178C4⋊C4×C47—42no—
179C22⋊C4×C47—22no—
180C22⋊C4×C47—22no—
181C22⋊C4×C47—22no—
182C22⋊C4×C47—22no—
183C22⋊C4×C47—22no—
184C22⋊C4×C47C2×D414yes—
185C42×C47—22no—
186C42×C47—22no—
187C42×C47—24no—
188C42×C47—42no—
189C22⋊Dic47—22no—
190C22⋊Dic47—22no—
191C22⋊Dic47—22no—
192C22⋊Dic47C2×D414yes—
193(C2×D47)⋊C4—216no—
194C4⋊Dic47—24no—
195C4⋊Dic47—216no—
196C4⋊Dic47—42no—
197Dic47⋊C4—216no—
198C4×Dic47—216no—
199C4×Dic47—24no—
200C4×Dic47—42no—
201C23×C47—42no—
202C23×C47C23×C411yescharacteristic
203C23×C47C22×D411yescharacteristic
204C23×C47C22×D411yescharacteristic
205C23×C47C22×D411yescharacteristic
206C23×C47C22×D411yescharacteristic
207C23×C47C2×#4111yescharacteristic
208C23×C47C2×#4111yescharacteristic
209C22×D47—22no—
210C22×D47—22no—
211C22×D47—22no—
212C22×D47—44no—
213C2×C4×C47—24no—
214C2×C4×C47—24no—
215C2×C4×C47—44no—
216C2×C4×C47—44no—
217C2×C4×C47—44no—
218C2×C4×C47—42no—
219C2×C4×C47C2×C22⋊C411yescharacteristic
220C2×C4×C47C2×C22⋊C411yescharacteristic
221C2×C4×C47C2×C22⋊C411yescharacteristic
222C2×C4×C47C2×C22⋊C411yescharacteristic
223C2×C4×C47C4×D412yes—
224C2×C4×C47C4×D412yes—
225C2×C4×C47C4×D412yes—
226C2×C4×C47C4×D412yes—
227C2×C4×C47#12012yes—
228C2×C4×C47#12012yes—
229C2×C4×C47#12112yes—
230C2×C4×C47#12112yes—
231C2×C4×C47#12112yes—
232C2×C4×C47#12112yes—
233C2×C4×C47#12312yes—
234C2×C4×C47#12312yes—
235C22⋊D47—42no—
236C22⋊D47—42no—
237C22⋊D47—42no—
238C22⋊D47—42no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—24no—
244C2×Dic47—24no—
245C2×Dic47—22no—
246C2×Dic47—22no—
247C2×Dic47—22no—
248C2×Dic47—24no—
249C2×Dic47—44no—
250C2×Dic47—44no—
251C2×Dic47—42no—
252C2×Dic47C2×C22⋊C418yes—
253D188—42no—
254D188—42no—
255C4×D47—44no—
256C4×D47—44no—
257Dic94—28no—
258Dic94—28no—
259Dic94—28no—
260Dic94—28no—
261Dic94—42no—
262Dic94—42no—
263#20986—471nomaximal, Sylow
264C22×C47—22no—
265C22×C47—22no—
266C22×C47—22no—
267C22×C47—41no—
268C22×C47—41no—
269C22×C47—41no—
270C22×C47—41no—
271C22×C47C22×C22⋊C411yescharacteristic
272C22×C47C2×C4×D411yescharacteristic
273C22×C47C2×C4×D411yescharacteristic
274C22×C47C2×#12011yescharacteristic
275C22×C47C2×#12111yescharacteristic
276C22×C47C2×#12111yescharacteristic
277C22×C47C2×#12311yescharacteristic
278C2×D47—44no—
279C2×D47—42no—
280C2×D47—82no—
281C2×D47—82no—
282C4×C47—44no—
283C4×C47—44no—
284C4×C47—82no—
285C4×C47#38612yes—
286C4×C47#38612yes—
287C4×C47#38612yes—
288C4×C47#38612yes—
289Dic47—28no—
290Dic47—44no—
291Dic47—42no—
292Dic47—82no—
293C2×#536—471no—
294C2×#519—471no—
295C2×#410—471no—
296C2×#410—471no—
297#2310—471no—
298#1899—478no—
299#1740—471no—
300#1740D4711yescharacteristic, complemented
301C2×C47—21no—
302C2×C47—21no—
303C2×C47—21no—
304C2×C47—21no—
305C2×C47—21no—
306C2×C47—21no—
307C2×C47C2×#38611yescharacteristic
308C2×C47#288412yes—
309D47—82no—
310C22×#41—471no—
311C23×Q8—471no—
312#536—942no—
313#536—942no—
314#536—942no—
315#536—942no—
316C2×C4⋊Q8—472no—
317C2×#124—472no—
318C2×#123—474no—
319C2×C22⋊Q8—474no—
320#519—944no—
321#519—944no—
322C2×C4×Q8—942no—
323C2×C4×Q8—942no—
324C2×#117—471no—
325C2×#117—471no—
326C2×#117—474no—
327C2×#117C2×D4711yescharacteristic
328C2×#117C2×D4711yescharacteristic
329C2×#117C2×D4711yescharacteristic
330#410—944no—
331#410—944no—
332C2×#99—474no—
333C2×#99—474no—
334#391—942no—
335#391—942no—
336#381—474no—
337#342—474no—
338#342C2×D4714yes—
339C47#2098611yescharacteristic, complemented, Sylow
340C2×#41—942no—
341C2×#41—942no—
342C2×#41—941no—
343C2×#41—941no—
344C2×#41—941no—
345C2×#41—941no—
346C2×#41—942no—
347C2×#41—942no—
348C22×Q8—472no—
349C22×Q8—472no—
350C22×Q8—471no—
351C22×Q8—472no—
352C22×Q8—472no—
353C22×Q8—472no—
354C22×Q8—472no—
355C22×Q8—942no—
356C22×Q8—942no—
357C22×Q8—942no—
358C22×Q8—942no—
359C22×D4—472no—
360C22×D4—471no—
361C23×C4—472no—
362C23×C4—472no—
363C23×C4C22×D4711yescharacteristic
364C4⋊Q8—944no—
365C4⋊Q8—944no—
366#124—944no—
367#124—944no—
368#123—948no—
369#123—948no—
370C22⋊Q8—948no—
371C22⋊Q8—948no—
372C4×Q8—9416no—
373C4×Q8—9416no—
374#117—22no—
375#117—21no—
376#117—21no—
377#117—21no—
378#117—21no—
379#117—22no—
380#117—22no—
381#117—474no—
382#117—948no—
383#117—942no—
384#117—948no—
385#117—942no—
386#117—942no—
387#117D18814yes—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—478no—
391C2×C4⋊C4—944no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4C4×D4712yes—
395C2×C22⋊C4—472no—
396C2×C22⋊C4—478no—
397C2×C22⋊C4—472no—
398C2×C22⋊C4C22×D4712yes—
399C2×C22⋊C4C22×D4712yes—
400C2×C22⋊C4C22×D4712yes—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—944no—
404C2×C42—941no—
405C2×C42—941no—
406C2×C42C4×D4712yes—
407#99—948no—
408#99—948no—
409#95—948no—
410C24—472no—
411C24C23×D4711yescharacteristic
412#41—1882no—
413#41—1882no—
414#41—1882no—
415#41—1882no—
416#41—1882no—
417#41—1882no—
418#41—1882no—
419#41—1882no—
420C2×Q8—474no—
421C2×Q8—474no—
422C2×Q8—474no—
423C2×Q8—474no—
424C2×Q8—941no—
425C2×Q8—941no—
426C2×Q8—944no—
427C2×Q8—944no—
428C2×Q8—942no—
429C2×Q8—944no—
430C2×Q8—944no—
431C2×Q8—942no—
432C2×Q8—944no—
433C2×Q8—944no—
434C2×Q8—944no—
435C2×Q8—944no—
436C2×Q8—944no—
437C2×Q8—944no—
438C2×Q8—944no—
439C2×Q8—944no—
440C2×D4—942no—
441C2×D4—942no—
442C2×D4—942no—
443C2×D4—942no—
444C2×D4—942no—
445C2×D4—942no—
446C2×D4—941no—
447C2×D4—941no—
448C2×D4—942no—
449C2×D4—942no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—474no—
453C22×C4—474no—
454C22×C4—474no—
455C22×C4—472no—
456C22×C4—942no—
457C22×C4—942no—
458C22×C4—942no—
459C22×C4—942no—
460C22×C4—942no—
461C22×C4—942no—
462C22×C4—942no—
463C22×C4—944no—
464C22×C4—944no—
465C22×C4—942no—
466C22×C4C2×C4×D4714yes—
467C22×C4C2×C4×D4711yescharacteristic
468C22×C4C2×C4×D4711yescharacteristic
469C22×C4C2×D18812yes—
470C22×C4#1062535012yes—
471C22×C4D4×D4711yescharacteristic
472C22×C4D4×D4711yescharacteristic
473C22×C4D4×D4711yescharacteristic
474C22×C4D4×D4711yescharacteristic
475C22×C4D4×D4712yes—
476C22×C4#1062535412yes—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—42no—
481C4⋊C4—9416no—
482C4⋊C4—9416no—
483C4⋊C4—944no—
484C4⋊C4—1882no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—474no—
491C22⋊C4—9416no—
492C22⋊C4—942no—
493C22⋊C4—942no—
494C22⋊C4—942no—
495C22⋊C4C2×D18814yes—
496C42—22no—
497C42—22no—
498C42—24no—
499C42—42no—
500C42—9416no—
501C42—944no—
502C42—1882no—
503C23—42no—
504C23—942no—
505C23—942no—
506C23—942no—
507C23—1884no—
508C23C22×C4×D4711yescharacteristic
509C23C2×D4×D4711yescharacteristic
510C23C2×D4×D4711yescharacteristic
511C23C2×D4×D4711yescharacteristic
512C23C22×D18811yescharacteristic
513C23C2×#1062535011yescharacteristic
514C23C2×#1062535411yescharacteristic
515Q8—948no—
516Q8—948no—
517Q8—948no—
518Q8—948no—
519Q8—1882no—
520Q8—1882no—
521D4—1882no—
522D4—1882no—
523D4—1882no—
524D4—1882no—
525D4—1882no—
526D4—1882no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—478no—
534C2×C4—944no—
535C2×C4—944no—
536C2×C4—944no—
537C2×C4—944no—
538C2×C4—944no—
539C2×C4—944no—
540C2×C4—942no—
541C2×C4—942no—
542C2×C4—942no—
543C2×C4—944no—
544C2×C4—1884no—
545C2×C4—1882no—
546C2×C4—1884no—
547C2×C4—1884no—
548C2×C4—1884no—
549C2×C4C22⋊C4×D4711yescharacteristic
550C2×C4C22⋊C4×D4711yescharacteristic
551C2×C4C22⋊C4×D4711yescharacteristic
552C2×C4C22⋊C4×D4711yescharacteristic
553C2×C4#1309398612yes—
554C2×C4#1309398612yes—
555C2×C4C4×D18812yes—
556C2×C4C4×D18812yes—
557C2×C4#1309399412yes—
558C2×C4#1309399412yes—
559C2×C4#1309399712yes—
560C2×C4#1309399712yes—
561C2×C4#1309399912yes—
562C2×C4#1309399912yes—
563C2×C4#1309400912yes—
564C2×C4#1309400912yes—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—3762no—
573C22—3762no—
574C22—1882no—
575C22—1884no—
576C22C2×C22⋊C4×D4711yescharacteristic
577C22C2×#1309398611yescharacteristic
578C22C2×C4×D18811yescharacteristic
579C22C2×#1309399411yescharacteristic
580C22C2×#1309399711yescharacteristic
581C22C2×#1309399911yescharacteristic
582C22C2×#1309400911yescharacteristic
583C4—44no—
584C4—44no—
585C4—82no—
586C4—948no—
587C4—3762no—
588C4—1884no—
589C4—1882no—
590C4#1568900112yes—
591C4#1568900112yes—
592C4#1568900112yes—
593C4#1568900112yes—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—3762no—
601C2C2×#1568900111yescharacteristic
602C2#2025185012yes—
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC47×#1740C2
socleC22×C47C22×C22⋊C4
prime radicalC22×#10625350C22
cosocleC24×C47C24
hypercenter#1740D47
hypocenterC47#20986
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
2. C2×#15689600C2semidirect
5. #20245972C2semidirect
6. #20243845C2semidirect
7. #20243121C2semidirect
8. C47×#1740C2semidirect
15. C2×#13094024C4semidirect
16. C2×#13094012C4semidirect
18. C2×#13094004C4semidirect
300. #1740D47semidirect
339. C47#20986semidirect

Maximal subgroups

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

subgroupindex
C2×#156896992
C2×#156896002
C2×#156890962
#202459722
#202438452
#202431212
C47×#17402
#2098647