Finite Group Databasebeta

← #27101006

Subgroups — #27101006

Order 12032.

2090 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257189C211yesmaximal, characteristic
2C2×#15689788C211yesmaximal, characteristic
3#20255267C214yesmaximal, complemented
4#20255263C214yesmaximal, complemented
5#20255138C214yesmaximal, complemented
6#20255135C214yesmaximal, complemented
7#20254336C211yesmaximal, characteristic, complemented
8#20254161C214yesmaximal, complemented
9#20253428C211yesmaximal, characteristic, complemented
10#20253423C211yesmaximal, characteristic, complemented
11#451×D47C214yesmaximal, complemented
12#20252411C211yesmaximal, characteristic
13C47×#3209C211yesmaximal, characteristic, complemented
14C22×Q8×D47C2212yes—
15C2×Q16×D47C2218yescomplemented
16#15689834C2212yes—
17#15689833—22no—
18#15689833C2212yes—
19#15689829—22no—
20Q8×D188C2212yes—
21#15689788—22no—
22#15689788—22no—
23#15689788C2214yes—
24#15689788C2214yes—
25#15689788C2212yes—
26#15689751—21no—
27#15689748—22no—
28#15689680C2211yescharacteristic
29C2×#13094023—21no—
30C2×#13094002—22no—
31C2×#13094000C2212yes—
32C22⋊Q8×D47C2214yes—
33#15689597—21no—
34C2×C4×Dic94C2211yescharacteristic
35C2×C4×D188C2211yescharacteristic
36C2×#13093986—21no—
37#15689502C2212yescomplemented
38#15689473C2212yescomplemented
39#15689459C2212yes—
40#15689457C2218yescomplemented
41#15689456C2212yes—
42#15689419C2212yescomplemented
43#15689415C2212yescomplemented
44#15689363C2218yescomplemented
45#15689361C2218yescomplemented
46#15689334C2212yescomplemented
47#15689330C2214yescomplemented
48#15689330C2212yescomplemented
49#15689281C2212yes—
50#15689280C2218yescomplemented
51#15689279C2214yes—
52#15689278C2212yes—
53#15689277C2218yescomplemented
54#15689277C2218yescomplemented
55C4×Dic188C2214yescomplemented
56C4×#10625337C2214yescomplemented
57C8×D188C2214yes—
58#15689188C2214yes—
59#15689176C2211yescharacteristic
60#103×D47C2218yescomplemented
61C22⋊C8×D47C2214yes—
62C47×#533C2211yescharacteristic
63C2×C4⋊Q8×C47C2211yescharacteristic
64C47×#494C2212yescomplemented
65C47×#487C2212yescomplemented
66C47×#462C2214yescomplemented
67C47×#451C2214yes—
68C47×#431C2211yescharacteristic
69C22×Dic94—22no—
70C22×Dic94C2311yescharacteristic
71C2×Q8×D47—24no—
72C2×Q8×D47—48no—
73C2×Q8×D47C2318yes—
74C2×Q8×D47C2314yes—
75C22×D188D411yescharacteristic
76C22×C4×D47—22no—
77C22×C4×D47C2312yes—
78C2×Dic188C2314yes—
79C2×#10625337C2314yes—
80C2×#10625337C2318yes—
81Q16×D47—416no—
82C2×C8×D47C2318yes—
83#13094024—22no—
84#13094024C2311yescharacteristic
85#13094024C2311yescharacteristic
86#13094023—22no—
87#13094023—24no—
88#13094023—24no—
89#13094016—21no—
90#13094016—22no—
91#13094015—22no—
92#13094004—24no—
93#13094004C2314yes—
94#13094003—21no—
95#13094003C2312yes—
96#13094002—24no—
97#13094002—24no—
98#13094002—24no—
99#13094002—28no—
100#13094002—28no—
101#13094001—24no—
102#13094000—24no—
103#13094000—24no—
104#13094000C2318yes—
105#13094000C2314yes—
106C4×Dic94—21no—
107C4×Dic94—21no—
108C4×Dic94—21no—
109C4×Dic94C2312yes—
110C4×Dic94C2312yes—
111C4×Dic94C2312yes—
112C4×D188—22no—
113C4×D188—22no—
114C4×D188D412yes—
115C4×D188D412yes—
116C4×D188C2312yes—
117C4×D188C2312yes—
118#13093986—24no—
119#13093986—24no—
120#13093986—22no—
121#13093986—22no—
122#13093983—21no—
123C2×C4⋊Dic47D411yescharacteristic
124C2×C4⋊Dic47C2311yescharacteristic
125C2×Dic47⋊C4—21no—
126C2×Dic47⋊C4—22no—
127C4⋊C4×D47—24no—
128C4⋊C4×D47C2314yes—
129C2×(C2×D47)⋊C4—22no—
130C2×(C2×D47)⋊C4D412yes—
131C22⋊C4×D47—24no—
132C2×C4×Dic47—21no—
133#13093958C2314yes—
134#13093957C2314yes—
135C4⋊(C47⋊C8)C2312yes—
136#13093949C2314yes—
137#13093949C2314yes—
138#13093948C2314yes—
139#13093948C2318yes—
140#13093947C2314yes—
141C22⋊(C47⋊C8)C2312yes—
142(C2×D47)⋊C8C2318yes—
143C4×C47⋊C8C2312yes—
144C22×Q8×C47C2312yes—
145C2×Q16×C47C2318yes—
146C4⋊Q8×C47—21no—
147C4⋊Q8×C47—21no—
148C4⋊Q8×C47C2312yes—
149C4⋊Q8×C47C2312yes—
150C4⋊Q8×C47C2311yescharacteristic
151C4⋊Q8×C47C2311yescharacteristic
152#125×C47—21no—
153C22⋊Q8×C47—21no—
154C22⋊Q8×C47C2312yes—
155C4×Q8×C47—21no—
156C4×Q8×C47C2312yes—
157#117×C47—21no—
158C2×C4⋊C4×C47—21no—
159C2×C4⋊C4×C47—21no—
160C2×C42×C47C2311yescharacteristic
161C4⋊C8×C47C2312yes—
162#103×C47C2318yes—
163C22⋊C8×C47C2312yes—
164C4×C8×C47C2312yes—
165C23×D47C2×D412yes—
166C2×Q8×C47—22no—
167C2×Q8×C47—21no—
168C2×Q8×C47—44no—
169C2×Q8×C47C2414yes—
170C2×Q8×C47C2412yes—
171C22×C4×C47—22no—
172C22×C4×C47C2×D411yescharacteristic
173C22×C4×C47C2×D411yescharacteristic
174C22×C4×C47C2411yescharacteristic
175C22×Dic47—21no—
176C22×Dic47—21no—
177C22×Dic47C2×D412yes—
178Q8×D47—216no—
179Q8×D47—48no—
180Q8×D47—416no—
181Q8×D47—48no—
182C2×D188Q1618yescomplemented
183C2×D188C2×D412yes—
184C2×D188C2×D412yes—
185C2×C4×D47—28no—
186C2×C4×D47—24no—
187C2×C4×D47—24no—
188C2×C4×D47—44no—
189C2×C4×D47—48no—
190C2×C4×D47C2×D414yes—
191C2×C4×D47C2414yes—
192C2×Dic94—21no—
193C2×Dic94—24no—
194C2×Dic94—28no—
195C2×Dic94—21no—
196C2×Dic94—24no—
197C2×Dic94—48no—
198C2×Dic94C2412yes—
199C2×Dic94C2412yes—
200Q16×C47—48no—
201C2×C8×C47C2414yes—
202C4⋊C4×C47—22no—
203C4⋊C4×C47—22no—
204C4⋊C4×C47—22no—
205C4⋊C4×C47—22no—
206C4⋊C4×C47—21no—
207C4⋊C4×C47—21no—
208C4⋊C4×C47—21no—
209C4⋊C4×C47—22no—
210C4⋊C4×C47—22no—
211C4⋊C4×C47C2412yes—
212C22⋊C4×C47—22no—
213C42×C47—22no—
214C42×C47C2×D411yescharacteristic
215C42×C47C2×D411yescharacteristic
216C42×C47C2411yescharacteristic
217C42×C47C2411yescharacteristic
218C22⋊Dic47—22no—
219#10625337—28no—
220#10625337—48no—
221(C2×D47)⋊C4—24no—
222(C2×D47)⋊C4—24no—
223(C2×D47)⋊C4—28no—
224(C2×D47)⋊C4C2×D414yes—
225(C2×D47)⋊C4C2×D414yes—
226C4⋊Dic47—21no—
227C4⋊Dic47—21no—
228C4⋊Dic47—21no—
229C4⋊Dic47—21no—
230C4⋊Dic47—24no—
231C4⋊Dic47C2×D412yes—
232C4⋊Dic47C2×D412yes—
233C4⋊Dic47C2412yes—
234C4⋊Dic47C2412yes—
235Dic47⋊C4—22no—
236Dic47⋊C4—22no—
237Dic47⋊C4—22no—
238Dic47⋊C4—22no—
239Dic47⋊C4—24no—
240Dic47⋊C4—24no—
241Dic47⋊C4—28no—
242C4×Dic47—22no—
243C4×Dic47—22no—
244C4×Dic47—22no—
245C2×C47⋊C8C2414yes—
246Dic188—28no—
247C8×D47—48no—
248C23×C47C22×D411yescharacteristic
249C22×D47—42no—
250C22×D47—42no—
251C22×D47C2×Q1618yescomplemented
252C22×D47C22×D414yes—
253Q8×C47—24no—
254Q8×C47—28no—
255Q8×C47—44no—
256Q8×C47—81no—
257C2×C4×C47—22no—
258C2×C4×C47—22no—
259C2×C4×C47—22no—
260C2×C4×C47—24no—
261C2×C4×C47—41no—
262C2×C4×C47—44no—
263C2×C4×C47C2×Q1614yescomplemented
264C2×C4×C47C22×D411yescharacteristic
265C2×C4×C47C22×D411yescharacteristic
266C2×C4×C47C22×D411yescharacteristic
267C2×C4×C47C22×D411yescharacteristic
268C2×C4×C47C22×D411yescharacteristic
269C2×C4×C47C2511yescharacteristic
270C2×Dic47—24no—
271C2×Dic47—22no—
272C2×Dic47—22no—
273C2×Dic47—22no—
274C2×Dic47—22no—
275C2×Dic47—42no—
276C2×Dic47—42no—
277C2×Dic47C22×D414yes—
278D188C2×Q1618yescomplemented
279D188C2×Q1618yescomplemented
280C4×D47—28no—
281C4×D47—44no—
282C4×D47—416no—
283C4×D47—84no—
284C4×D47—84no—
285Dic94—24no—
286Dic94—22no—
287Dic94—24no—
288Dic94—416no—
289Dic94—84no—
290Dic94—81no—
291Dic94H(5,𝔽2)14yes—
292C8×C47—24no—
293C47⋊C8—24no—
294#61563—471nomaximal, Sylow
295C22×C47—21no—
296C22×C47—21no—
297C22×C47C22×Q1612yes—
298C22×C47C23×D411yescharacteristic
299C2×D47—24no—
300C2×D47—24no—
301C2×D47—48no—
302C2×D47C22×Q1618yes—
303C4×C47—21no—
304C4×C47—21no—
305C4×C47—21no—
306C4×C47—42no—
307C4×C47—44no—
308C4×C47—42no—
309C4×C47C22×Q1612yes—
310C4×C47C22×Q1612yes—
311C4×C47C2×H(5,𝔽2)12yes—
312Dic47—24no—
313Dic47—24no—
314Dic47—42no—
315Dic47—42no—
316Dic47—42no—
317#3501—471no—
318C2×D4×Q8—471no—
319#3299—474no—
320#3299—474no—
321D4×Q16—474no—
322D4×Q16—474no—
323#3209—471no—
324#3209D4711yescharacteristic, complemented
325#3189—474no—
326#3104—471no—
327#3104—471no—
328C2×#451—474no—
329#2972—471no—
330C2×C47—22no—
331C2×C47#348411yescharacteristic
332C2×C47C23×Q1611yescharacteristic
333C2×C47#358611yescharacteristic
334D47—28no—
335C23×Q8—472no—
336C22×Q16—478no—
337#554—472no—
338#554—472no—
339#554—942no—
340#554—942no—
341D4×Q8—474no—
342D4×Q8—474no—
343D4×Q8—472no—
344D4×Q8—472no—
345D4×Q8—942no—
346D4×Q8—942no—
347#544—941no—
348#543—942no—
349#533—471no—
350#533C2×D4711yescharacteristic
351C2×C4⋊Q8—941no—
352C2×C4⋊Q8C2×D4711yescharacteristic
353C2×C22⋊Q8—472no—
354C2×C22⋊Q8—474no—
355C2×C22⋊Q8—942no—
356#518—941no—
357C2×C4×Q8—471no—
358C2×C4×D4—471no—
359C2×C4×D4—941no—
360#494—472no—
361#494C2×D4712yescomplemented
362#487—472no—
363#487C2×D4712yescomplemented
364#484—472no—
365#484—478no—
366#484—472no—
367#477—472no—
368#477—472no—
369#467—478no—
370#467—478no—
371#462—472no—
372#462—474no—
373#462—472no—
374#462C2×D4714yescomplemented
375#451—474no—
376#451—478no—
377#451—478no—
378#451—472no—
379#451—478no—
380#451—472no—
381#451C2×D4714yes—
382C4×Q16—474no—
383C4×Q16—474no—
384C8×D4—474no—
385C8×D4—474no—
386#431—471no—
387#431C2×D4711yescharacteristic
388C2×#103—478no—
389C2×C22⋊C8—474no—
390C47#6156311yescharacteristic, complemented, Sylow
391C22×Q8—478no—
392C22×Q8—474no—
393C22×Q8—471no—
394C22×Q8—942no—
395C22×Q8—944no—
396C22×Q8—1888no—
397C22×Q8C22×D4712yes—
398C22×D4—471no—
399C23×C4—472no—
400C23×C4—942no—
401C2×Q16—474no—
402C2×Q16—478no—
403C2×Q16—474no—
404C2×Q16—18816no—
405C2×Q16C22×D4718yes—
406C22×C8—478no—
407C4⋊Q8—21no—
408C4⋊Q8—21no—
409C4⋊Q8—471no—
410C4⋊Q8—471no—
411C4⋊Q8—942no—
412C4⋊Q8—944no—
413C4⋊Q8—944no—
414C4⋊Q8—942no—
415C4⋊Q8C22×D4712yes—
416C4⋊Q8C22×D4712yes—
417C4⋊Q8C22×D4711yescharacteristic
418C4⋊Q8C22×D4711yescharacteristic
419#125—21no—
420#125—942no—
421#125—942no—
422#125—941no—
423C22⋊Q8—21no—
424C22⋊Q8—478no—
425C22⋊Q8—474no—
426C22⋊Q8—472no—
427C22⋊Q8—474no—
428C22⋊Q8—944no—
429C22⋊Q8—948no—
430C22⋊Q8—948no—
431C22⋊Q8—944no—
432C22⋊Q8—944no—
433C22⋊Q8—944no—
434C22⋊Q8—944no—
435C22⋊Q8—944no—
436C22⋊Q8—944no—
437C22⋊Q8—941no—
438C22⋊Q8C22×D4712yes—
439C4×Q8—21no—
440C4×Q8—472no—
441C4×Q8—472no—
442C4×Q8—472no—
443C4×Q8—941no—
444C4×Q8—941no—
445C4×Q8—941no—
446C4×Q8C22×D4712yes—
447C4×D4—472no—
448C4×D4—472no—
449C4×D4—472no—
450C4×D4—472no—
451C4×D4—942no—
452C4×D4—942no—
453C4×D4—944no—
454C4×D4—944no—
455C4×D4—942no—
456C4×D4—942no—
457#117—21no—
458#117—941no—
459C2×C4⋊C4—21no—
460C2×C4⋊C4—21no—
461C2×C4⋊C4—471no—
462C2×C4⋊C4—471no—
463C2×C4⋊C4—474no—
464C2×C4⋊C4—944no—
465C2×C4⋊C4—942no—
466C2×C4⋊C4—941no—
467C2×C22⋊C4—472no—
468C2×C22⋊C4—944no—
469C2×C22⋊C4—942no—
470C2×C42—941no—
471C2×C42C22×D4711yescharacteristic
472#107—474no—
473#107—474no—
474C4⋊C8—472no—
475C4⋊C8C22×D4712yes—
476#103—478no—
477#103—474no—
478#103—474no—
479#103—474no—
480#103—474no—
481#103C22×D4718yes—
482C22⋊C8—472no—
483C22⋊C8—478no—
484C22⋊C8C22×D4712yes—
485C4×C8—472no—
486C4×C8C22×D4712yes—
487C24—472no—
488C2×Q8—22no—
489C2×Q8—21no—
490C2×Q8—44no—
491C2×Q8—472no—
492C2×Q8—472no—
493C2×Q8—944no—
494C2×Q8—948no—
495C2×Q8—944no—
496C2×Q8—941no—
497C2×Q8—941no—
498C2×Q8—9416no—
499C2×Q8—1888no—
500C2×Q8—1888no—
501C2×Q8—1888no—
502C2×Q8—18816no—
503C2×Q8C23×D4714yes—
504C2×Q8C23×D4712yes—
505C2×D4—478no—
506C2×D4—472no—
507C2×D4—472no—
508C22×C4—22no—
509C22×C4—472no—
510C22×C4—474no—
511C22×C4—474no—
512C22×C4—944no—
513C22×C4—944no—
514C22×C4—948no—
515C22×C4—941no—
516C22×C4—941no—
517C22×C4—1884no—
518C22×C4—1888no—
519C22×C4D4×D4711yescharacteristic
520C22×C4D4×D4711yescharacteristic
521C22×C4C23×D4711yescharacteristic
522Q16—48no—
523Q16—948no—
524Q16—948no—
525Q16—1888no—
526C2×C8—474no—
527C2×C8—1888no—
528C2×C8C23×D4714yes—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—472no—
539C4⋊C4—472no—
540C4⋊C4—472no—
541C4⋊C4—472no—
542C4⋊C4—944no—
543C4⋊C4—944no—
544C4⋊C4—948no—
545C4⋊C4—944no—
546C4⋊C4—942no—
547C4⋊C4—942no—
548C4⋊C4—942no—
549C4⋊C4—942no—
550C4⋊C4—941no—
551C4⋊C4—941no—
552C4⋊C4—941no—
553C4⋊C4—941no—
554C4⋊C4C23×D4712yes—
555C22⋊C4—22no—
556C22⋊C4—474no—
557C22⋊C4—474no—
558C22⋊C4—944no—
559C22⋊C4—944no—
560C22⋊C4—948no—
561C22⋊C4—942no—
562C42—22no—
563C42—942no—
564C42—942no—
565C42—942no—
566C42D4×D4711yescharacteristic
567C42D4×D4711yescharacteristic
568C42C23×D4711yescharacteristic
569C42C23×D4711yescharacteristic
570C23—474no—
571C23—478no—
572C23—1882no—
573C23—1882no—
574C23C2×D4×D4711yescharacteristic
575Q8—24no—
576Q8—28no—
577Q8—44no—
578Q8—81no—
579Q8—474no—
580Q8—944no—
581Q8—942no—
582Q8—944no—
583Q8—3761no—
584Q8—3764no—
585Q8—18816no—
586D4—478no—
587D4—478no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—474no—
595C2×C4—944no—
596C2×C4—942no—
597C2×C4—948no—
598C2×C4—942no—
599C2×C4—942no—
600C2×C4—942no—
601C2×C4—3764no—
602C2×C4—3764no—
603C2×C4—1884no—
604C2×C4—18816no—
605C2×C4—1882no—
606C2×C4—1882no—
607C2×C4Q16×D4714yescomplemented
608C2×C4C2×D4×D4711yescharacteristic
609C2×C4C2×D4×D4711yescharacteristic
610C2×C4C2×D4×D4711yescharacteristic
611C2×C4C2×D4×D4711yescharacteristic
612C2×C4C2×D4×D4711yescharacteristic
613C2×C4C24×D4711yescharacteristic
614C8—24no—
615C8—944no—
616C22—21no—
617C22—21no—
618C22—478no—
619C22—944no—
620C22—944no—
621C22—1888no—
622C22C2×Q16×D4712yes—
623C22C22×D4×D4711yescharacteristic
624C4—21no—
625C4—21no—
626C4—21no—
627C4—42no—
628C4—44no—
629C4—42no—
630C4—944no—
631C4—944no—
632C4—1882no—
633C4—1882no—
634C4—1882no—
635C4C2×Q16×D4712yes—
636C4C2×Q16×D4712yes—
637C4H(5,𝔽2)×D4712yes—
638C2—22no—
639C2—948no—
640C2#2025702211yescharacteristic
641C2C22×Q16×D4711yescharacteristic
642C2#2025817111yescharacteristic
subgroupG/H
centerC22C22×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC47×#3209C2
socleC22×C47C23×D4
prime radicalC22×D188D4
hypercenter#3209D47
hypocenterC47#61563
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
3. #20255267C2semidirect
4. #20255263C2semidirect
5. #20255138C2semidirect
6. #20255135C2semidirect
7. #20254336C2semidirect
8. #20254161C2semidirect
9. #20253428C2semidirect
10. #20253423C2semidirect
11. #451×D47C2semidirect
13. C47×#3209C2semidirect
15. C2×Q16×D47C22semidirect
37. #15689502C22semidirect
38. #15689473C22semidirect
40. #15689457C22semidirect
42. #15689419C22semidirect
43. #15689415C22semidirect
44. #15689363C22semidirect
45. #15689361C22semidirect
46. #15689334C22semidirect
47. #15689330C22semidirect
48. #15689330C22semidirect
50. #15689280C22semidirect
53. #15689277C22semidirect
54. #15689277C22semidirect
55. C4×Dic188C22semidirect
56. C4×#10625337C22semidirect
60. #103×D47C22semidirect
64. C47×#494C22semidirect
65. C47×#487C22semidirect
66. C47×#462C22semidirect
182. C2×D188Q16semidirect
251. C22×D47C2×Q16semidirect
263. C2×C4×C47C2×Q16semidirect
278. D188C2×Q16semidirect
279. D188C2×Q16semidirect
324. #3209D47semidirect
361. #494C2×D47semidirect
363. #487C2×D47semidirect
374. #462C2×D47semidirect
390. C47#61563semidirect
607. C2×C4Q16×D47semidirect

Maximal subgroups

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

subgroupindex
#202571892
C2×#156897882
#202552672
#202552632
#202551382
#202551352
#202543362
#202541612
#202534282
#202534232
#451×D472
#202524112
C47×#32092
#6156347