Finite Group Databasebeta

← #27085955

Subgroups — #27085955

Order 12032.

1834 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257204C211yesmaximal, characteristic
2#20257200C211yesmaximal, characteristic
3#20253934C211yesmaximal, characteristic, complemented
4#20253931C212yesmaximal, complemented
5#20253929C212yesmaximal, complemented
6#20253918C214yesmaximal, complemented
7#20253915C218yesmaximal, complemented
8#20253899C212yesmaximal, complemented
9#20253898C211yesmaximal, characteristic, complemented
10C2×#15689519C211yesmaximal, characteristic, complemented
11#498×D47C212yesmaximal, complemented
12#20251982C214yesmaximal
13C4×#13094032C211yesmaximal, characteristic
14C47×#3165C211yesmaximal, characteristic, complemented
15C2×#13094032C2212yes—
16Q8×Dic94—21no—
17#15689853—21no—
18#15689851—21no—
19#15689848—21no—
20#15689834—21no—
21#15689833—21no—
22#15689831—22no—
23#15689828—22no—
24Q8×D188—21no—
25#15689788—21no—
26Q8×C22⋊D47—22no—
27#15689786—22no—
28#15689754C2212yes—
29#15689753C2214yes—
30#15689751C2212yes—
31#15689750C2214yes—
32#15689680C2211yescharacteristic
33#15689678C2212yes—
34#15689678C2212yes—
35C2×#13094024C2211yescharacteristic
36C4⋊Q8×D47C2212yes—
37C4⋊Q8×D47C2212yes—
38#15689604C2212yes—
39#15689603C2212yes—
40C4×#10625350C2211yescharacteristic
41#15689532C2214yes—
42#15689531C2218yes—
43#15689530C2218yes—
44#15689528C2212yes—
45#15689526C2211yescharacteristic
46#15689526C2211yescharacteristic
47#15689524C2218yes—
48#15689522C2214yes—
49#15689521C2212yes—
50#15689520C2218yes—
51#15689519C2211yescharacteristic
52#15689519C2212yes—
53#15689519C2214yes—
54#15689519C2212yes—
55#15689519C2211yescharacteristic
56#15689165C2214yes—
57#15689164C2218yes—
58#15689162C2218yes—
59#15689160C2218yes—
60#15689159C2214yes—
61C2×#13093955C2214yes—
62#106×D47C2218yes—
63#15689081C2212yes—
64#15689080C2214yes—
65C4×#10625324C2212yes—
66C2×C4×C47⋊C8C2211yescharacteristic
67C4×C8×D47C2212yes—
68C47×#533C2211yescharacteristic
69C2×C4⋊Q8×C47C2211yescharacteristic
70C47×#501C2214yes—
71C47×#500C2212yes—
72C47×#498C2212yes—
73C47×#428C2214yes—
74C4×M16×C47C2211yescharacteristic
75C2×#10625350C2311yescharacteristic
76C22×Dic94—21no—
77C2×Q8×D47—22no—
78C2×Q8×D47—22no—
79#13094032Q8116yescomplemented
80C2×#10625324C2314yes—
81C22×C47⋊C8C2312yes—
82C2×C8×D47C2314yes—
83#13094024—21no—
84#13094024—21no—
85#13094024—22no—
86#13094024C2312yes—
87#13094024C2312yes—
88#13094024C2311yescharacteristic
89#13094024C2311yescharacteristic
90#13094023—22no—
91#13094023—24no—
92#13094023—22no—
93#13094023—22no—
94#13094023C2314yes—
95#13094023C2314yes—
96#13094022—24no—
97#13094022—22no—
98#13094022—22no—
99#13094016—22no—
100#13094016C2312yes—
101#13094015—22no—
102#13094015C2314yes—
103#13094015C2314yes—
104#13094014—24no—
105#13094014—24no—
106#13094004—22no—
107#13094004—22no—
108#13094004—24no—
109#13094003—21no—
110#13094003—21no—
111#13094003—24no—
112#13094002—24no—
113#13094002—24no—
114#13094001—24no—
115#13094000—22no—
116#13094000—22no—
117#13094000—24no—
118Q8×Dic47—22no—
119Q8×Dic47—22no—
120C4×Dic94—22no—
121C4×Dic94—21no—
122C4×Dic94—21no—
123C4×Dic94D411yescharacteristic
124C4×Dic94D411yescharacteristic
125#13093989—22no—
126#13093989—22no—
127#13093989—22no—
128C4×D188—22no—
129C4×D188D411yescharacteristic
130C4×D188D411yescharacteristic
131#13093986—22no—
132C4×C22⋊D47—24no—
133C4×C22⋊D47D412yes—
134C4×C22⋊D47D412yes—
135(C22⋊D47)⋊C4—24no—
136#13093983C2312yes—
137#13093983C2311yescharacteristic
138#13093982C2314yes—
139#13093982C2314yes—
140#13093981C2312yes—
141C2×C4⋊Dic47—21no—
142C2×C4⋊Dic47C2312yes—
143C4⋊C4×D47—22no—
144C4⋊C4×D47—22no—
145C4⋊C4×D47C2314yes—
146C4⋊C4×D47C2314yes—
147C42×D47C2312yes—
148#13093958C2318yes—
149#13093957C2314yes—
150#13093957C2314yes—
151#13093956C2318yes—
152#13093955C2318yes—
153#13093955C2318yes—
154#13093955C2314yes—
155#13093955C2314yes—
156#13093934C2314yes—
157C4×C47⋊C8C2311yescharacteristic
158C4×C47⋊C8C2312yes—
159C4×C47⋊C8C2311yescharacteristic
160C8×Dic47C2314yes—
161C22×Q8×C47—21no—
162C2×M16×C47C2312yes—
163C4⋊Q8×C47—21no—
164C4⋊Q8×C47—21no—
165C4⋊Q8×C47C2312yes—
166C4⋊Q8×C47C2312yes—
167C4⋊Q8×C47C2311yescharacteristic
168C4⋊Q8×C47C2311yescharacteristic
169#125×C47C2312yes—
170C22⋊Q8×C47—21no—
171C22⋊Q8×C47—21no—
172C4×Q8×C47—21no—
173C4×Q8×C47—21no—
174#117×C47C2312yes—
175C2×C4⋊C4×C47—21no—
176C2×C4⋊C4×C47C2312yes—
177C2×C42×C47C2311yescharacteristic
178#107×C47C2318yes—
179#106×C47C2318yes—
180#97×C47C2312yes—
181C4×C8×C47C2312yes—
182C2×Q8×C47—21no—
183C2×Q8×C47—21no—
184C2×Q8×C47—22no—
185C2×Q8×C47—22no—
186C2×Q8×C47—82no—
187C22×C4×C47—22no—
188C22×C4×C47C2×D411yescharacteristic
189C22×C4×C47C2×D411yescharacteristic
190C22×C4×C47C2411yescharacteristic
191C2×C22⋊D47C2×D412yes—
192C22×Dic47—22no—
193Q8×D47—84no—
194Q8×D47—84no—
195#10625350C2×Q818yes—
196C2×D188C2×D411yescharacteristic
197C2×C4×D47—24no—
198C2×C4×D47—24no—
199C2×C4×D47C2×D412yes—
200C2×C4×D47C2×D412yes—
201C2×C4×D47C2412yes—
202C2×Dic94—21no—
203C2×Dic94—22no—
204C2×Dic94—21no—
205C2×Dic94—22no—
206C2×Dic94—24no—
207C2×Dic94—24no—
208C2×Dic94—82no—
209C2×Dic94C2×D411yescharacteristic
210M16×C47C2×Q818yescomplemented
211C2×C8×C47C2414yes—
212C4⋊C4×C47—22no—
213C4⋊C4×C47—22no—
214C4⋊C4×C47—22no—
215C4⋊C4×C47—22no—
216C4⋊C4×C47—22no—
217C4⋊C4×C47—21no—
218C4⋊C4×C47—21no—
219C4⋊C4×C47C2412yes—
220C4⋊C4×C47C2414yes—
221C4⋊C4×C47C2412yes—
222C22⋊C4×C47—22no—
223C42×C47—22no—
224C42×C47C2×D411yescharacteristic
225C42×C47C2×D411yescharacteristic
226C42×C47C2411yescharacteristic
227C42×C47C2411yescharacteristic
228C22⋊Dic47—22no—
229C22⋊Dic47C2×D412yes—
230(C2×D47)⋊C4—24no—
231(C2×D47)⋊C4—24no—
232(C2×D47)⋊C4C2×D412yes—
233(C2×D47)⋊C4C2×D412yes—
234C4⋊Dic47—21no—
235C4⋊Dic47—22no—
236C4⋊Dic47—22no—
237C4⋊Dic47—22no—
238C4⋊Dic47—21no—
239C4⋊Dic47—24no—
240C4⋊Dic47—24no—
241C4⋊Dic47C2×D411yescharacteristic
242C4⋊Dic47C2×D411yescharacteristic
243C4⋊Dic47C2414yes—
244C4⋊Dic47C2412yes—
245C4⋊Dic47C2412yes—
246Dic47⋊C4—22no—
247Dic47⋊C4—22no—
248Dic47⋊C4—24no—
249Dic47⋊C4—24no—
250Dic47⋊C4—24no—
251Dic47⋊C4—24no—
252Dic47⋊C4—24no—
253Dic47⋊C4—24no—
254Dic47⋊C4C2×D412yes—
255Dic47⋊C4C2×D412yes—
256C4×Dic47—22no—
257C4×Dic47—24no—
258C4×Dic47—24no—
259C4×Dic47C2×D412yes—
260C4×Dic47C2×D412yes—
261C4×Dic47C2412yes—
262C2×C47⋊C8C2×Q818yescomplemented
263C2×C47⋊C8C2412yes—
264C2×C47⋊C8C2412yes—
265#10625324C2×Q8116yescomplemented
266C8×D47C2×Q8116yescomplemented
267C23×C47C22×D411yescharacteristic
268C22×D47C22×D412yes—
269Q8×C47—42no—
270Q8×C47—44no—
271Q8×C47—42no—
272Q8×C47—82no—
273C2×C4×C47—22no—
274C2×C4×C47—22no—
275C2×C4×C47—22no—
276C2×C4×C47—24no—
277C2×C4×C47—22no—
278C2×C4×C47—82no—
279C2×C4×C47C22×D411yescharacteristic
280C2×C4×C47C22×D411yescharacteristic
281C2×C4×C47C22×D411yescharacteristic
282C2×C4×C47C22×D411yescharacteristic
283C2×C4×C47C22×D411yescharacteristic
284C2×C4×C47C22×Q814yes—
285C2×C4×C47C2511yescharacteristic
286C22⋊D47C4⋊Q818yes—
287C2×Dic47—22no—
288C2×Dic47—22no—
289C2×Dic47—24no—
290C2×Dic47—82no—
291C2×Dic47C22×D412yes—
292C2×Dic47C22×D412yes—
293C2×Dic47C22×D412yes—
294D188C4⋊Q814yes—
295C4×D47—24no—
296C4×D47—24no—
297C4×D47—84no—
298C4×D47—84no—
299C4×D47C22×Q818yes—
300Dic94—42no—
301Dic94—44no—
302Dic94—42no—
303Dic94—84no—
304Dic94—84no—
305Dic94—82no—
306Dic94C4⋊Q814yes—
307C8×C47C22×Q818yes—
308C47⋊C8C22×Q814yes—
309C47⋊C8C22×Q814yes—
310#60665—471nomaximal, Sylow
311C22×C47—21no—
312C22×C47—21no—
313C22×C47C2×C4⋊Q812yes—
314C22×C47C23×D411yescharacteristic
315C2×D47—22no—
316C2×D47—22no—
317C2×D47C2×C4⋊Q814yes—
318C4×C47—21no—
319C4×C47—21no—
320C4×C47—42no—
321C4×C47—44no—
322C4×C47—42no—
323C4×C47C2×C4⋊Q812yes—
324C4×C47#57712yes—
325C4×C47#57712yes—
326C4×C47C23×Q812yes—
327Dic47—22no—
328Dic47—22no—
329Dic47—42no—
330Dic47—44no—
331Dic47—42no—
332Dic47C2×C4⋊Q814yes—
333#3503—471no—
334#3503—471no—
335#3165—472no—
336#3165—472no—
337#3165—471no—
338#3165D4711yescharacteristic, complemented
339#3163—478no—
340#3163—474no—
341#3160—472no—
342#3160—471no—
343C2×#498—472no—
344C2×#498—471no—
345#2912—474no—
346C4×#131—471no—
347C2×C47—22no—
348C2×C47C22×C4⋊Q811yescharacteristic
349C2×C47C2×#57711yescharacteristic
350C2×C47C2×#57711yescharacteristic
351D47—24no—
352C2×#131—472no—
353Q82—941no—
354Q82—941no—
355#557—941no—
356#557—941no—
357#554—942no—
358#554—941no—
359#554—942no—
360#554—941no—
361D4×Q8—941no—
362D4×Q8—942no—
363D4×Q8—941no—
364D4×Q8—942no—
365#544—474no—
366#544—474no—
367#544—472no—
368#544—472no—
369#533—472no—
370#533—472no—
371#533—471no—
372#533C2×D4711yescharacteristic
373C2×C4⋊Q8—472no—
374C2×C4⋊Q8—472no—
375C2×C4⋊Q8—471no—
376C2×C4⋊Q8C2×D4711yescharacteristic
377#520—472no—
378#520—472no—
379C4×#41—471no—
380#501—478no—
381#501—478no—
382#501—474no—
383#501C2×D4714yes—
384#500—471no—
385#500—471no—
386#500—472no—
387#500C2×D4712yes—
388#499—474no—
389#499—478no—
390#498—471no—
391#498—472no—
392#498—474no—
393#498—472no—
394#498—471no—
395#498—478no—
396#498—472no—
397#498C2×D4712yes—
398#428—474no—
399#428—478no—
400#428—478no—
401#428C2×D4714yes—
402#427—474no—
403#427—478no—
404C2×#106—474no—
405C2×#106—478no—
406#405—472no—
407#405—474no—
408C4×M16—472no—
409C4×M16C2×D4711yescharacteristic
410C2×C4×C8—471no—
411C2×C4×C8—472no—
412C47#6066511yescharacteristic, complemented, Sylow
413C2×#41—471no—
414C22×Q8—21no—
415C22×Q8—942no—
416C22×Q8—942no—
417C22×Q8—941no—
418#131—4716no—
419C2×M16—474no—
420C2×M16C22×D4712yes—
421C22×C8—472no—
422C22×C8—474no—
423C4⋊Q8—21no—
424C4⋊Q8—21no—
425C4⋊Q8—474no—
426C4⋊Q8—474no—
427C4⋊Q8—471no—
428C4⋊Q8—472no—
429C4⋊Q8—472no—
430C4⋊Q8—471no—
431C4⋊Q8—942no—
432C4⋊Q8—942no—
433C4⋊Q8—942no—
434C4⋊Q8—942no—
435C4⋊Q8—942no—
436C4⋊Q8—944no—
437C4⋊Q8—942no—
438C4⋊Q8—944no—
439C4⋊Q8—941no—
440C4⋊Q8—941no—
441C4⋊Q8C22×D4711yescharacteristic
442C4⋊Q8C22×D4712yes—
443C4⋊Q8C22×D4712yes—
444C4⋊Q8C22×D4711yescharacteristic
445#125—474no—
446#125—474no—
447#125—472no—
448#125—942no—
449#125—944no—
450#125—944no—
451#125—942no—
452#125C22×D4712yes—
453C22⋊Q8—21no—
454C22⋊Q8—21no—
455C22⋊Q8—944no—
456C22⋊Q8—944no—
457C22⋊Q8—944no—
458C22⋊Q8—944no—
459C22⋊Q8—944no—
460C22⋊Q8—944no—
461C22⋊Q8—942no—
462C22⋊Q8—942no—
463C22⋊Q8—942no—
464C22⋊Q8—942no—
465C22⋊Q8—941no—
466C22⋊Q8—941no—
467C4×Q8—21no—
468C4×Q8—21no—
469C4×Q8—471no—
470C4×Q8—471no—
471C4×Q8—942no—
472C4×Q8—942no—
473C4×Q8—942no—
474C4×Q8—942no—
475C4×Q8—942no—
476C4×Q8—942no—
477C4×Q8—941no—
478C4×Q8—941no—
479C4×D4—472no—
480C4×D4—472no—
481C4×D4—471no—
482C4×D4—471no—
483C4×D4—942no—
484C4×D4—944no—
485C4×D4—942no—
486C4×D4—944no—
487#117—471no—
488#117—472no—
489#117—472no—
490#117—474no—
491#117—474no—
492#117C22×D4712yes—
493C2×C4⋊C4—21no—
494C2×C4⋊C4—472no—
495C2×C4⋊C4—474no—
496C2×C4⋊C4—474no—
497C2×C4⋊C4—942no—
498C2×C4⋊C4—942no—
499C2×C4⋊C4—941no—
500C2×C4⋊C4C22×D4712yes—
501C2×C42—472no—
502C2×C42C22×D4711yescharacteristic
503#107—478no—
504#107—474no—
505#107—474no—
506#107C22×D4718yes—
507#106—478no—
508#106—478no—
509#106—474no—
510#106—478no—
511#106—474no—
512#106C22×D4718yes—
513#97—474no—
514#97C22×D4712yes—
515C4×C8—474no—
516C4×C8—471no—
517C4×C8—471no—
518C4×C8—472no—
519C4×C8C22×D4712yes—
520#41—478no—
521C2×Q8—21no—
522C2×Q8—21no—
523C2×Q8—22no—
524C2×Q8—22no—
525C2×Q8—82no—
526C2×Q8—471no—
527C2×Q8—944no—
528C2×Q8—944no—
529C2×Q8—942no—
530C2×Q8—941no—
531C2×Q8—942no—
532C2×Q8—941no—
533C2×Q8—3762no—
534C2×Q8—3764no—
535C2×Q8—3764no—
536C2×D4—472no—
537C2×D4—471no—
538C22×C4—22no—
539C22×C4—472no—
540C22×C4—472no—
541C22×C4—472no—
542C22×C4—944no—
543C22×C4—944no—
544C22×C4—942no—
545C22×C4D4×D4711yescharacteristic
546C22×C4D4×D4711yescharacteristic
547C22×C4C23×D4711yescharacteristic
548M16—4716no—
549M16Q8×D4718yescomplemented
550C2×C8—478no—
551C2×C8—472no—
552C2×C8—472no—
553C2×C8—4716no—
554C2×C8C23×D4714yes—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—21no—
558C4⋊C4—22no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—472no—
563C4⋊C4—472no—
564C4⋊C4—471no—
565C4⋊C4—471no—
566C4⋊C4—474no—
567C4⋊C4—472no—
568C4⋊C4—472no—
569C4⋊C4—944no—
570C4⋊C4—944no—
571C4⋊C4—944no—
572C4⋊C4—944no—
573C4⋊C4—944no—
574C4⋊C4—944no—
575C4⋊C4—944no—
576C4⋊C4—944no—
577C4⋊C4—942no—
578C4⋊C4—942no—
579C4⋊C4—942no—
580C4⋊C4—941no—
581C4⋊C4—942no—
582C4⋊C4—942no—
583C4⋊C4—941no—
584C4⋊C4C23×D4714yes—
585C4⋊C4C23×D4712yes—
586C4⋊C4C23×D4712yes—
587C22⋊C4—22no—
588C22⋊C4—472no—
589C22⋊C4—472no—
590C22⋊C4—472no—
591C22⋊C4—944no—
592C22⋊C4—944no—
593C22⋊C4—942no—
594C42—22no—
595C42—472no—
596C42—472no—
597C42—472no—
598C42—944no—
599C42—944no—
600C42—942no—
601C42D4×D4711yescharacteristic
602C42D4×D4711yescharacteristic
603C42C23×D4711yescharacteristic
604C42C23×D4711yescharacteristic
605C23—472no—
606C23C2×D4×D4711yescharacteristic
607Q8—42no—
608Q8—42no—
609Q8—44no—
610Q8—82no—
611Q8—474no—
612Q8—3762no—
613Q8—3764no—
614Q8—3764no—
615Q8—1882no—
616Q8—1884no—
617Q8—1882no—
618D4—478no—
619D4—474no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—82no—
626C2×C4—472no—
627C2×C4—478no—
628C2×C4—472no—
629C2×C4—472no—
630C2×C4—944no—
631C2×C4—944no—
632C2×C4—942no—
633C2×C4—944no—
634C2×C4—942no—
635C2×C4—3764no—
636C2×C4—3764no—
637C2×C4—3762no—
638C2×C4C2×D4×D4711yescharacteristic
639C2×C4C2×D4×D4711yescharacteristic
640C2×C4C2×D4×D4711yescharacteristic
641C2×C4C2×D4×D4711yescharacteristic
642C2×C4C2×D4×D4711yescharacteristic
643C2×C4C2×Q8×D4714yes—
644C2×C4C24×D4711yescharacteristic
645C8—474no—
646C8—474no—
647C8C2×Q8×D4718yes—
648C22—21no—
649C22—21no—
650C22—474no—
651C22—942no—
652C22—942no—
653C22C4⋊Q8×D4712yes—
654C22C22×D4×D4711yescharacteristic
655C4—21no—
656C4—21no—
657C4—42no—
658C4—44no—
659C4—42no—
660C4—474no—
661C4—942no—
662C4—942no—
663C4—1884no—
664C4—1882no—
665C4—1882no—
666C4C4⋊Q8×D4712yes—
667C4#1568993212yes—
668C4#1568993312yes—
669C4C22×Q8×D4712yes—
670C2—22no—
671C2—944no—
672C2C2×C4⋊Q8×D4711yescharacteristic
673C2C2×#1568993211yescharacteristic
674C2C2×#1568993311yescharacteristic
subgroupG/H
centerC22C22×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC47×#3165C2
socleC22×C47C23×D4
prime radicalC2×#10625350C23
hypercenter#3165D47
hypocenterC47#60665
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
3. #20253934C2semidirect
4. #20253931C2semidirect
5. #20253929C2semidirect
6. #20253918C2semidirect
7. #20253915C2semidirect
8. #20253899C2semidirect
9. #20253898C2semidirect
10. C2×#15689519C2semidirect
11. #498×D47C2semidirect
14. C47×#3165C2semidirect
79. #13094032Q8semidirect
210. M16×C47C2×Q8semidirect
262. C2×C47⋊C8C2×Q8semidirect
265. #10625324C2×Q8semidirect
266. C8×D47C2×Q8semidirect
338. #3165D47semidirect
412. C47#60665semidirect
549. M16Q8×D47semidirect

Maximal subgroups

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

subgroupindex
#202572042
#202572002
#202539342
#202539312
#202539292
#202539182
#202539152
#202538992
#202538982
C2×#156895192
#498×D472
#202519822
C4×#130940322
C47×#31652
#6066547