Finite Group Databasebeta

← #26389468

Subgroups — #26389468

Order 12032.

2042 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251694C214yesmaximal, complemented
2#20249360C214yesmaximal, complemented
3#20248614C214yesmaximal, complemented
4#20248609C214yesmaximal, complemented
5#20247111C214yesmaximal, complemented
6#20247109C214yesmaximal, complemented
7#20246002C211yesmaximal, characteristic
8#20246000C211yesmaximal, characteristic
9#20246000C211yesmaximal, characteristic
10C2×#15688981C211yesmaximal, characteristic
11C2×#15688981C211yesmaximal, characteristic
12C2×C22⋊C4×Dic47C211yesmaximal, characteristic
13C47×#2315C211yesmaximal, characteristic
14C24×Dic47C2211yescharacteristic
15C2×#13094014C2214yescomplemented
16C22×Dic47⋊C4C2212yes—
17C22×Dic47⋊C4C2212yes—
18C22×C22⋊Dic47C2211yescharacteristic
19C22×C22⋊Dic47C2211yescharacteristic
20C22×C22⋊Dic47C2211yescharacteristic
21C22×C4×Dic47C2212yes—
22#15689064C2214yescomplemented
23#15689064C2214yescomplemented
24#15689063C2214yescomplemented
25#15689061C2214yescomplemented
26#15689057C2214yescomplemented
27#15689056C2214yescomplemented
28#15689054C2214yescomplemented
29#15689053C2214yescomplemented
30#15689020C2214yescomplemented
31#15689018C2214yescomplemented
32#15689009C2214yescomplemented
33#15689008C2214yescomplemented
34#15689005C2214yescomplemented
35#15689004C2214yescomplemented
36#15688998C2214yes—
37#15688998C2214yes—
38#15688996C2214yes—
39#15688996C2214yes—
40#15688995C2214yes—
41#15688995C2214yes—
42#15688987C2214yescomplemented
43#15688983C2214yes—
44#15688981C2212yes—
45#15688981C2212yes—
46#15688981C2212yes—
47#15688981C2214yes—
48#15688981C2212yes—
49#15688981C2214yes—
50#15688981C2212yes—
51#15688981C2212yes—
52#15688981C2212yes—
53#15688981C2212yes—
54C22⋊C4×Dic47C2212yes—
55C22⋊C4×Dic47C2212yes—
56C22⋊C4×Dic47C2212yes—
57C22⋊C4×Dic47C2212yes—
58C2×C47⋊#95C2212yes—
59C2×C47⋊#95C2212yes—
60C2×C47⋊#95C2212yes—
61C22×C22⋊C4×C47C2211yescharacteristic
62C22×C22⋊C4×C47C2211yescharacteristic
63C22×C22⋊C4×C47C2211yescharacteristic
64C47×#385C2214yes—
65C47×#385C2214yes—
66C47×#380C2214yes—
67C23×Dic47—44no—
68C23×Dic47Q818yes—
69C23×Dic47C2311yescharacteristic
70C23×Dic47C2311yescharacteristic
71C23×Dic47C2311yescharacteristic
72C23×Dic47C2312yes—
73C23×Dic47C2312yes—
74C23×Dic47C2312yes—
75C23×Dic47C2311yescharacteristic
76C23×Dic47C2311yescharacteristic
77C23×Dic47C2311yescharacteristic
78#13094014—48no—
79C2×C4⋊Dic47C2314yes—
80C2×C4⋊Dic47C2314yes—
81C2×Dic47⋊C4—44no—
82C2×Dic47⋊C4—44no—
83C2×Dic47⋊C4C2314yes—
84C2×Dic47⋊C4C2314yes—
85C2×Dic47⋊C4C2314yes—
86C2×Dic47⋊C4C2314yes—
87C2×Dic47⋊C4C2314yes—
88C2×Dic47⋊C4C2314yes—
89C2×Dic47⋊C4C2314yes—
90C2×Dic47⋊C4C2314yes—
91C2×C22⋊Dic47—24no—
92C2×C22⋊Dic47—24no—
93C2×C22⋊Dic47—24no—
94C2×C22⋊Dic47C2312yes—
95C2×C22⋊Dic47C2312yes—
96C2×C22⋊Dic47C2312yes—
97C2×C22⋊Dic47C2312yes—
98C2×C22⋊Dic47C2312yes—
99C2×C22⋊Dic47C2312yes—
100C2×C4×Dic47—44no—
101C2×C4×Dic47C2314yes—
102C2×C4×Dic47C2314yes—
103C47⋊#95C2314yes—
104C47⋊#95C2314yes—
105C47⋊#95C2314yes—
106C47⋊#95C2314yes—
107C47⋊#95C2314yes—
108C47⋊#95C2314yes—
109C47⋊#95C2314yes—
110C47⋊#95C2314yes—
111C47⋊#95C2314yes—
112C47⋊#95C2314yes—
113C47⋊#95C2314yes—
114C47⋊#95C2314yes—
115C25×C47C2311yescharacteristic
116C23×C4×C47C2312yes—
117C23×C4×C47C2312yes—
118C23×C4×C47C2312yes—
119C2×C4⋊C4×C47C2314yes—
120C2×C4⋊C4×C47C2314yes—
121C2×C22⋊C4×C47—24no—
122C2×C22⋊C4×C47—24no—
123C2×C22⋊C4×C47—24no—
124C2×C22⋊C4×C47C2312yes—
125C2×C22⋊C4×C47C2312yes—
126C2×C22⋊C4×C47C2312yes—
127C2×C22⋊C4×C47C2312yes—
128C2×C22⋊C4×C47C2312yes—
129C2×C22⋊C4×C47C2312yes—
130#95×C47C2314yes—
131#95×C47C2314yes—
132C24×C47—42no—
133C24×C47—42no—
134C24×C47C2×Q814yes—
135C24×C47#4114yes—
136C24×C47#4114yes—
137C24×C47C2411yescharacteristic
138C24×C47C2411yescharacteristic
139C24×C47C2411yescharacteristic
140C22×C4×C47—24no—
141C22×C4×C47—24no—
142C22×C4×C47—24no—
143C22×C4×C47—42no—
144C22×C4×C47—42no—
145C22×C4×C47—42no—
146C22×C4×C47—42no—
147C22×C4×C47—42no—
148C22×C4×C47—42no—
149C22×C4×C47C2414yes—
150C22×C4×C47C2414yes—
151C22×C4×C47C2414yes—
152C22×Dic47—24no—
153C22×Dic47—24no—
154C22×Dic47—24no—
155C22×Dic47—24no—
156C22×Dic47—24no—
157C22×Dic47—24no—
158C22×Dic47—416no—
159C22×Dic47—42no—
160C22×Dic47—42no—
161C22×Dic47—42no—
162C22×Dic47—42no—
163C22×Dic47—42no—
164C22×Dic47—42no—
165C22×Dic47—42no—
166C22×Dic47—42no—
167C22×Dic47—42no—
168C22×Dic47C2×Q814yes—
169C22×Dic47C2×Q814yes—
170C22×Dic47C2×Q814yes—
171C22×Dic47C2×Q814yes—
172C22×Dic47C2×Q814yes—
173C22×Dic47C2×Q814yes—
174C22×Dic47C2414yes—
175C22×Dic47C2414yes—
176C22×Dic47C2414yes—
177C22×Dic47C2411yescharacteristic
178C22×Dic47C2411yescharacteristic
179C22×Dic47C2411yescharacteristic
180C22×Dic47C2411yescharacteristic
181C4⋊C4×C47—44no—
182C4⋊C4×C47—44no—
183C22⋊C4×C47—24no—
184C22⋊C4×C47—24no—
185C22⋊C4×C47—24no—
186C22⋊C4×C47—24no—
187C22⋊C4×C47—24no—
188C22⋊C4×C47—24no—
189C22⋊Dic47—24no—
190C22⋊Dic47—24no—
191C22⋊Dic47—24no—
192C22⋊Dic47—24no—
193C22⋊Dic47—24no—
194C22⋊Dic47—24no—
195C4⋊Dic47—44no—
196C4⋊Dic47—44no—
197Dic47⋊C4—28no—
198Dic47⋊C4—28no—
199Dic47⋊C4—44no—
200Dic47⋊C4—44no—
201Dic47⋊C4—44no—
202Dic47⋊C4—44no—
203Dic47⋊C4—44no—
204Dic47⋊C4—44no—
205C4×Dic47—28no—
206C4×Dic47—44no—
207C23×C47—22no—
208C23×C47—22no—
209C23×C47—22no—
210C23×C47—22no—
211C23×C47—22no—
212C23×C47—22no—
213C23×C47—44no—
214C23×C47—44no—
215C23×C47—48no—
216C23×C47—48no—
217C23×C47#125116yescomplemented
218C23×C47C22×Q812yes—
219C23×C47C22×Q812yes—
220C23×C47C22×Q812yes—
221C23×C47C2×#4112yes—
222C23×C47C2×#4112yes—
223C23×C47C2×#4112yes—
224C23×C47C2×#4112yes—
225C23×C47C2×#4112yes—
226C23×C47C2×#4112yes—
227C23×C47C2511yescharacteristic
228C2×C4×C47—24no—
229C2×C4×C47—24no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—24no—
236C2×C4×C47—24no—
237C2×C4×C47—48no—
238C2×C4×C47—84no—
239C2×C4×C47—84no—
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—22no—
243C2×Dic47—28no—
244C2×Dic47—28no—
245C2×Dic47—28no—
246C2×Dic47—24no—
247C2×Dic47—24no—
248C2×Dic47—24no—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—24no—
253C2×Dic47—24no—
254C2×Dic47—24no—
255C2×Dic47—44no—
256C2×Dic47—44no—
257C2×Dic47—44no—
258C2×Dic47—48no—
259C2×Dic47—84no—
260C2×Dic47—84no—
261C2×Dic47C22×Q812yes—
262C2×Dic47C22×Q812yes—
263C2×Dic47C22×Q812yes—
264C2×Dic47C22×Q812yes—
265C2×Dic47H(5,𝔽2)14yes—
266#35020—471nomaximal, Sylow
267C22×C47—22no—
268C22×C47—22no—
269C22×C47—22no—
270C22×C47—22no—
271C22×C47—22no—
272C22×C47—22no—
273C22×C47—24no—
274C22×C47—24no—
275C22×C47—24no—
276C22×C47—24no—
277C22×C47—24no—
278C22×C47—24no—
279C22×C47—416no—
280C22×C47C2×#12514yes—
281C22×C47C2×#12514yes—
282C22×C47C2×#12514yes—
283C22×C47C23×Q811yescharacteristic
284C22×C47C22×#4111yescharacteristic
285C22×C47C22×#4111yescharacteristic
286C22×C47C2×H(5,𝔽2)12yes—
287C22×C47#58512yes—
288C4×C47—44no—
289C4×C47—44no—
290C4×C47—44no—
291Dic47—22no—
292Dic47—22no—
293Dic47—22no—
294Dic47—44no—
295Dic47—44no—
296Dic47—44no—
297Dic47#54314yes—
298#2851—474no—
299#2646—474no—
300#2569—474no—
301#2569—474no—
302#2442—474no—
303#2442—474no—
304#2315—471no—
305#2315—471no—
306#2315—471no—
307#2315D4711yescharacteristic
308C2×#380—471no—
309C2×#380—471no—
310C2×C4×C22⋊C4—471no—
311C2×C47—24no—
312C2×C47—24no—
313C2×C47—24no—
314C2×C47C22×#12511yescharacteristic
315C2×C47C2×#54311yescharacteristic
316C2×C47#345811yescharacteristic
317C2×C47#350512yes—
318C2×C47#350512yes—
319C24×C4—471no—
320C2×#125—474no—
321C22×C4⋊C4—472no—
322C22×C4⋊C4—472no—
323C22×C22⋊C4—471no—
324C22×C22⋊C4—471no—
325C22×C22⋊C4—471no—
326C22×C22⋊C4C2×D4711yescharacteristic
327C22×C22⋊C4C2×D4711yescharacteristic
328C22×C22⋊C4C2×D4711yescharacteristic
329C22×C42—472no—
330#400—474no—
331#400—474no—
332#400—474no—
333#400—474no—
334#398—474no—
335#398—474no—
336#398—474no—
337#398—474no—
338#389—474no—
339#389—474no—
340#387—474no—
341#387—474no—
342#387—474no—
343#387—474no—
344#385—474no—
345#385—474no—
346#385—474no—
347#385—474no—
348#385—474no—
349#385—474no—
350#385C2×D4714yes—
351#385C2×D4714yes—
352#382—474no—
353#380—474no—
354#380—472no—
355#380—472no—
356#380—474no—
357#380—474no—
358#380—472no—
359#380—472no—
360#380—472no—
361#380—472no—
362#380—472no—
363#380—472no—
364#380C2×D4714yes—
365C4×C22⋊C4—472no—
366C4×C22⋊C4—472no—
367C4×C22⋊C4—472no—
368C4×C22⋊C4—472no—
369C2×#95—472no—
370C2×#95—472no—
371C2×#95—472no—
372C47#3502011yescharacteristic, complemented, Sylow
373C25C22×D4711yescharacteristic
374C23×C4—471no—
375C23×C4—471no—
376C23×C4—471no—
377C23×C4—478no—
378C23×C4—471no—
379C23×C4—471no—
380C23×C4—471no—
381C23×C4—472no—
382C23×C4—472no—
383C23×C4—472no—
384C23×C4—1884no—
385C23×C4C22×D4712yes—
386C23×C4C22×D4712yes—
387C23×C4C22×D4712yes—
388#125—1888no—
389C2×C4⋊C4—474no—
390C2×C4⋊C4—474no—
391C2×C4⋊C4—474no—
392C2×C4⋊C4—474no—
393C2×C4⋊C4—474no—
394C2×C4⋊C4—474no—
395C2×C4⋊C4—474no—
396C2×C4⋊C4—474no—
397C2×C4⋊C4—474no—
398C2×C4⋊C4—474no—
399C2×C4⋊C4—1884no—
400C2×C4⋊C4—1884no—
401C2×C4⋊C4C22×D4714yes—
402C2×C4⋊C4C22×D4714yes—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—472no—
407C2×C22⋊C4—472no—
408C2×C22⋊C4—472no—
409C2×C22⋊C4—472no—
410C2×C22⋊C4—472no—
411C2×C22⋊C4—472no—
412C2×C22⋊C4—944no—
413C2×C22⋊C4—944no—
414C2×C22⋊C4—944no—
415C2×C22⋊C4C22×D4712yes—
416C2×C22⋊C4C22×D4712yes—
417C2×C22⋊C4C22×D4712yes—
418C2×C22⋊C4C22×D4712yes—
419C2×C22⋊C4C22×D4712yes—
420C2×C22⋊C4C22×D4712yes—
421C2×C42—474no—
422C2×C42—474no—
423C2×C42—1884no—
424#95—474no—
425#95—474no—
426#95—474no—
427#95—474no—
428#95—474no—
429#95—474no—
430#95—474no—
431#95—474no—
432#95—474no—
433#95—474no—
434#95—474no—
435#95—474no—
436#95C22×D4714yes—
437#95C22×D4714yes—
438C24—42no—
439C24—42no—
440C24#1062535014yes—
441C24#1062535214yes—
442C24Q8×D4714yes—
443C24C23×D4711yescharacteristic
444C24C23×D4711yescharacteristic
445C24C23×D4711yescharacteristic
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—471no—
456C22×C4—474no—
457C22×C4—474no—
458C22×C4—474no—
459C22×C4—471no—
460C22×C4—471no—
461C22×C4—471no—
462C22×C4—474no—
463C22×C4—474no—
464C22×C4—474no—
465C22×C4—474no—
466C22×C4—474no—
467C22×C4—474no—
468C22×C4—944no—
469C22×C4—944no—
470C22×C4—944no—
471C22×C4—944no—
472C22×C4—944no—
473C22×C4—944no—
474C22×C4—18816no—
475C22×C4—1882no—
476C22×C4—1882no—
477C22×C4—1882no—
478C22×C4—1882no—
479C22×C4—1882no—
480C22×C4—1882no—
481C22×C4—1882no—
482C22×C4—1882no—
483C22×C4—1882no—
484C22×C4C23×D4714yes—
485C22×C4C23×D4714yes—
486C22×C4C23×D4714yes—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—948no—
490C4⋊C4—948no—
491C4⋊C4—1884no—
492C4⋊C4—1884no—
493C4⋊C4—1884no—
494C4⋊C4—1884no—
495C4⋊C4—1884no—
496C4⋊C4—1884no—
497C4⋊C4—1884no—
498C4⋊C4—1884no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—944no—
506C22⋊C4—944no—
507C22⋊C4—944no—
508C22⋊C4—944no—
509C22⋊C4—944no—
510C22⋊C4—944no—
511C42—948no—
512C42—1884no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—44no—
520C23—48no—
521C23—48no—
522C23—44no—
523C23#13094014116yescomplemented
524C23C2×Q8×D4712yes—
525C23C2×Q8×D4712yes—
526C23C2×Q8×D4712yes—
527C23C2×#1062535212yes—
528C23C2×#1062535212yes—
529C23C2×#1062535212yes—
530C23C2×#1062535012yes—
531C23C2×#1062535012yes—
532C23C2×#1062535012yes—
533C23C24×D4711yescharacteristic
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—48no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—472no—
547C2×C4—474no—
548C2×C4—472no—
549C2×C4—472no—
550C2×C4—472no—
551C2×C4—948no—
552C2×C4—948no—
553C2×C4—948no—
554C2×C4—942no—
555C2×C4—942no—
556C2×C4—942no—
557C2×C4—944no—
558C2×C4—944no—
559C2×C4—944no—
560C2×C4—944no—
561C2×C4—944no—
562C2×C4—944no—
563C2×C4—944no—
564C2×C4—944no—
565C2×C4—944no—
566C2×C4—3764no—
567C2×C4—3764no—
568C2×C4—1884no—
569C2×C4—1884no—
570C2×C4—1884no—
571C2×C4—1888no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—416no—
585C22C2×#1309401414yes—
586C22C2×#1309401414yes—
587C22C2×#1309401414yes—
588C22C22×Q8×D4711yescharacteristic
589C22C22×#1062535211yescharacteristic
590C22C22×#1062535011yescharacteristic
591C22H(5,𝔽2)×D4712yes—
592C22#1568995812yes—
593C4—44no—
594C4—44no—
595C4—44no—
596C4—474no—
597C4—942no—
598C4—942no—
599C4—942no—
600C4—1884no—
601C4—1884no—
602C4—1884no—
603C2—24no—
604C2—24no—
605C2—24no—
606C2C22×#1309401411yescharacteristic
607C2#543×D4711yescharacteristic
608C2#2025679011yescharacteristic
609C2#2025721812yes—
610C2#2025722512yes—
subgroupG/H
center, FrattiniC23C24×D47
commutator, socle, cosocleC23×C47C25
FittingC47×#2315C2
prime radicalC25×C47C23
hypercenter#2315D47
hypocenterC47#35020
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2315

Complements

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

NH ≅ G/Ntype
1. #20251694C2semidirect
2. #20249360C2semidirect
3. #20248614C2semidirect
4. #20248609C2semidirect
5. #20247111C2semidirect
6. #20247109C2semidirect
15. C2×#13094014C22semidirect
22. #15689064C22semidirect
23. #15689064C22semidirect
24. #15689063C22semidirect
25. #15689061C22semidirect
26. #15689057C22semidirect
27. #15689056C22semidirect
28. #15689054C22semidirect
29. #15689053C22semidirect
30. #15689020C22semidirect
31. #15689018C22semidirect
32. #15689009C22semidirect
33. #15689008C22semidirect
34. #15689005C22semidirect
35. #15689004C22semidirect
42. #15688987C22semidirect
217. C23×C47#125semidirect
372. C47#35020semidirect
523. C23#13094014semidirect

Maximal subgroups

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

subgroupindex
#202516942
#202493602
#202486142
#202486092
#202471112
#202471092
#202460022
#202460002
C2×#156889812
C2×C22⋊C4×Dic472
C47×#23152
#3502047