Finite Group Databasebeta

← #26230187

Subgroups — #26230187

Order 12032.

796 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252305C211yesmaximal, characteristic, complemented
2#20252248C211yesmaximal, characteristic, complemented
3#20244293C211yesmaximal, characteristic, complemented
4#20244287C211yesmaximal, characteristic, complemented
5#20243926C211yesmaximal, characteristic, complemented
6#20243921C211yesmaximal, characteristic, complemented
7#20243916C211yesmaximal, characteristic, complemented
8#20243911C211yesmaximal, characteristic, complemented
9#20243884C211yesmaximal, characteristic, complemented
10#20243877C211yesmaximal, characteristic, complemented
11#20243870C211yesmaximal, characteristic, complemented
12#20243863C211yesmaximal, characteristic, complemented
13#20243512C211yesmaximal, characteristic, complemented
14#20243143C211yesmaximal, characteristic, complemented
15C47×#3154C211yesmaximal, characteristic
16#15689890C2211yescharacteristic
17#15689890C2211yescharacteristic
18#15689601C2211yescharacteristic
19#15689598C2211yescharacteristic
20#15689245—22no—
21#15689240—22no—
22#15689224—22no—
23SD16×Dic47—22no—
24C2×#13093960C2211yescharacteristic
25#15689161C2211yescharacteristic
26#15689140C2211yescharacteristic
27#15689140C2211yescharacteristic
28#15689140C2211yescharacteristic
29#15689140C2211yescharacteristic
30#15689132C2211yescharacteristic
31#15689132C2211yescharacteristic
32#15689128C2211yescharacteristic
33#15689128C2211yescharacteristic
34C2×#13093943C2211yescharacteristic
35C2×#13093941C2211yescharacteristic
36#15689097C2211yescharacteristic
37#15689097C2211yescharacteristic
38#15689078C2211yescharacteristic
39#15689078C2211yescharacteristic
40#15688894C2211yescharacteristic
41#15688894C2211yescharacteristic
42C47⋊#338C2211yescharacteristic
43#15688880C2211yescharacteristic
44#15688879C2211yescharacteristic
45#15688879C2211yescharacteristic
46#15688879C2211yescharacteristic
47#15688879C2211yescharacteristic
48C2×#137×C47C2211yescharacteristic
49C2×AGL(1,ℤ/8)×C47C2211yescharacteristic
50C2×#135×C47C2211yescharacteristic
51C22×SD16×C47C2211yescharacteristic
52C47×#536C2211yescharacteristic
53C47×#535C2211yescharacteristic
54C47×#497C412yescomplemented
55C47×#496C412yescomplemented
56C47×#495C412yescomplemented
57C47×#492C412yescomplemented
58C47×#405C2211yescharacteristic
59#13094031—22no—
60#13094031—22no—
61#13094031—41no—
62#13094031—41no—
63C2×#10625329—41no—
64C2×#10625329—41no—
65C2×#10625329C2311yescharacteristic
66C2×#10625329C2311yescharacteristic
67C2×#10625329C2311yescharacteristic
68C2×#10625329C2311yescharacteristic
69Q8×Dic47—22no—
70Q8×Dic47—41no—
71D4×Dic47—22no—
72D4×Dic47—41no—
73#13093983—41no—
74#13093983C2311yescharacteristic
75#13093983C2311yescharacteristic
76#13093983C2311yescharacteristic
77#13093983C2311yescharacteristic
78C2×C22⋊Dic47—41no—
79#13093960—21no—
80#13093960—21no—
81#13093958—21no—
82#13093956—21no—
83#13093951—22no—
84#13093951—22no—
85#13093951—41no—
86#13093951—41no—
87#13093948—22no—
88#13093948—41no—
89#13093945—22no—
90#13093945—41no—
91#13093943—41no—
92#13093941—41no—
93#13093939—41no—
94#13093939—41no—
95#13093934—21no—
96#13093934D412yescomplemented
97C8×Dic47—21no—
98C8×Dic47D412yescomplemented
99C2×#41×C47C2311yescharacteristic
100C2×#41×C47C2311yescharacteristic
101C22×Q8×C47C2311yescharacteristic
102C22×D4×C47C2311yescharacteristic
103#137×C47—41no—
104#137×C47—41no—
105AGL(1,ℤ/8)×C47—41no—
106AGL(1,ℤ/8)×C47—41no—
107#135×C47—21no—
108#135×C47—21no—
109#135×C47—21no—
110#135×C47—21no—
111C2×Q16×C47C2×C411yescharacteristic, complemented
112C2×Q16×C47C2×C412yescomplemented
113C2×SD16×C47—21no—
114C2×SD16×C47—21no—
115C2×SD16×C47—21no—
116C2×SD16×C47—21no—
117C2×SD16×C47C2×C412yes—
118C2×SD16×C47C2×C412yes—
119C2×SD16×C47C2×C412yes—
120C2×SD16×C47C2×C411yescharacteristic, complemented
121C2×SD16×C47C2×C411yescharacteristic, complemented
122C2×SD16×C47C2×C412yescomplemented
123C2×D8×C47C2×C412yes—
124C2×D8×C47C2×C411yescharacteristic, complemented
125C2×M16×C47C2311yescharacteristic
126C22×C8×C47C2311yescharacteristic
127C4⋊Q8×C47C2×C412yescomplemented
128#127×C47C2×C412yes—
129#124×C47C2×C412yes—
130#124×C47C2×C412yescomplemented
131#123×C47—41no—
132C22⋊Q8×C47—41no—
133#121×C47—41no—
134#120×C47—41no—
135#117×C47C2311yescharacteristic
136#97×C47C2×C412yescomplemented
137C4×C8×C47C2×C412yescomplemented
138C24×C47—21no—
139#41×C47—21no—
140#41×C47—21no—
141#41×C47—21no—
142#41×C47—21no—
143#41×C47—21no—
144#41×C47—21no—
145#41×C47—21no—
146#41×C47—21no—
147C2×Q8×C47—21no—
148C2×Q8×C47—21no—
149C2×Q8×C47—21no—
150C2×Q8×C47—21no—
151C2×Q8×C47—21no—
152C2×Q8×C47C22×C411yescharacteristic
153C2×Q8×C47C22×C412yes—
154C2×Q8×C47C22×C411yescharacteristic
155C2×D4×C47—21no—
156C2×D4×C47—21no—
157C2×D4×C47—21no—
158C2×D4×C47—21no—
159C2×D4×C47—21no—
160C2×D4×C47—21no—
161C2×D4×C47—21no—
162C2×D4×C47—42no—
163C2×D4×C47C22×C412yes—
164C2×D4×C47C22×C411yescharacteristic
165C2×D4×C47C22×C411yescharacteristic
166C22×C4×C47—21no—
167C22×C4×C47—21no—
168C22×C4×C47—21no—
169C22×C4×C47C2411yescharacteristic
170C22×Dic47—42no—
171Q16×C47—21no—
172Q16×C47—21no—
173Q16×C47—42no—
174SD16×C47—22no—
175SD16×C47—22no—
176SD16×C47—22no—
177SD16×C47—22no—
178SD16×C47—21no—
179SD16×C47—21no—
180SD16×C47—21no—
181SD16×C47—21no—
182SD16×C47—42no—
183SD16×C47—42no—
184D8×C47—21no—
185D8×C47—21no—
186D8×C47—42no—
187M16×C47—21no—
188M16×C47—21no—
189C2×C8×C47C22×C412yes—
190C2×C8×C47C22×C411yescharacteristic
191C2×C8×C47C22×C411yescharacteristic
192C2×C8×C47C2×D411yescharacteristic
193C2×C8×C47C2×D411yescharacteristic
194C2×C8×C47#4111yescharacteristic
195C2×C8×C47#4111yescharacteristic
196C4⋊C4×C47—21no—
197C4⋊C4×C47—42no—
198C22⋊C4×C47—21no—
199C22⋊C4×C47—42no—
200C22⋊C4×C47—42no—
201C42×C47C22×C412yes—
202C22⋊Dic47—21no—
203C22⋊Dic47—21no—
204C22⋊Dic47—21no—
205C22⋊Dic47—21no—
206C22⋊Dic47—42no—
207C4⋊Dic47—21no—
208C4⋊Dic47—21no—
209C4⋊Dic47—21no—
210C4⋊Dic47—21no—
211C4⋊Dic47—42no—
212C4×Dic47—21no—
213C4×Dic47—21no—
214C4×Dic47—21no—
215C4×Dic47—42no—
216C4×Dic47C2×D412yes—
217#10625329—21no—
218#10625329—21no—
219#10625329—21no—
220#10625329—21no—
221#10625329—21no—
222#10625329—21no—
223#10625329—21no—
224#10625329—21no—
225#10625329—42no—
226#10625329—42no—
227C2×C47⋊C8—21no—
228C2×C47⋊C8—21no—
229C2×C47⋊C8—21no—
230C2×C47⋊C8—42no—
231C2×C47⋊C8—42no—
232C2×C47⋊C8C2×D412yes—
233C23×C47—21no—
234C23×C47—22no—
235C23×C47—21no—
236C23×C47—41no—
237C23×C47—41no—
238C23×C47—161no—
239C23×C47H(5,𝔽2)11yescharacteristic
240Q8×C47—22no—
241Q8×C47—21no—
242Q8×C47—21no—
243Q8×C47—21no—
244Q8×C47—22no—
245Q8×C47—21no—
246Q8×C47—41no—
247Q8×C47—41no—
248D4×C47—21no—
249D4×C47—21no—
250D4×C47—22no—
251D4×C47—21no—
252D4×C47—22no—
253D4×C47—21no—
254D4×C47—41no—
255D4×C47—41no—
256D4×C47—41no—
257D4×C47—41no—
258D4×C47—41no—
259D4×C47—41no—
260D4×C47—162no—
261C2×C4×C47—22no—
262C2×C4×C47—21no—
263C2×C4×C47—22no—
264C2×C4×C47—21no—
265C2×C4×C47—41no—
266C2×C4×C47—41no—
267C2×C4×C47—41no—
268C2×C4×C47—41no—
269C2×C4×C47—41no—
270C2×C4×C47—41no—
271C2×C4×C47C23×C411yescharacteristic
272C2×C4×C47C22×D411yescharacteristic
273C2×C4×C47C2×#4111yescharacteristic
274C2×C4×C47H(5,𝔽2)11yescharacteristic
275C2×C4×C47H(5,𝔽2)11yescharacteristic
276C2×C4×C47H(5,𝔽2)11yescharacteristic
277C2×Dic47—22no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280C2×Dic47—41no—
281C2×Dic47—162no—
282C8×C47—22no—
283C8×C47C4×D411yescharacteristic
284C8×C47C4×D411yescharacteristic
285C8×C47C4×D411yescharacteristic
286C8×C47C4×D411yescharacteristic
287C47⋊C8—22no—
288C47⋊C8—41no—
289C47⋊C8—41no—
290C47⋊C8—41no—
291#22284—471nomaximal, Sylow
292C22×C47—41no—
293C22×C47—41no—
294C22×C47—41no—
295C22×C47—42no—
296C22×C47—81no—
297C22×C47—82no—
298C22×C47—81no—
299C22×C47—82no—
300C22×C47—161no—
301C22×C47#51811yescharacteristic
302C22×C47#53511yescharacteristic
303C22×C47#54011yescharacteristic
304C4×C47—42no—
305C4×C47—41no—
306C4×C47—41no—
307C4×C47—42no—
308C4×C47C2×C4×D411yescharacteristic
309C4×C47#51811yescharacteristic
310C4×C47#53511yescharacteristic
311C4×C47#54011yescharacteristic
312Dic47—42no—
313Dic47—81no—
314Dic47—81no—
315Dic47—81no—
316#3154D4711yescharacteristic
317#2957—471no—
318#2949—471no—
319#1977—471no—
320#1976—471no—
321#1912—471no—
322#1911—471no—
323#1910—471no—
324#1909—471no—
325#1905—471no—
326#1904—471no—
327#1903—471no—
328#1902—471no—
329#1833—471no—
330#1747—471no—
331C2×C47—21no—
332C2×C47—21no—
333C2×C47—21no—
334C2×C47—81no—
335C2×C47—81no—
336C2×C47—82no—
337C2×C47#232811yescharacteristic
338C2×#137C2×D4711yescharacteristic
339C2×AGL(1,ℤ/8)C2×D4711yescharacteristic
340C2×#135C2×D4711yescharacteristic
341C22×SD16C2×D4711yescharacteristic
342#568—471no—
343#568—471no—
344#536C2×D4711yescharacteristic
345#535C2×D4711yescharacteristic
346#519—471no—
347#518—471no—
348#497Dic4712yescomplemented
349#496Dic4712yescomplemented
350#495Dic4712yescomplemented
351#492Dic4712yescomplemented
352#444—942no—
353#443—942no—
354#440—942no—
355C4×SD16—942no—
356C2×#108—471no—
357#427—471no—
358#421—471no—
359#421—471no—
360#421—471no—
361#421—471no—
362#419—471no—
363#419—471no—
364#418—471no—
365#418—471no—
366C2×#101—471no—
367C2×#100—471no—
368#410—471no—
369#410—471no—
370#405—471no—
371#405—471no—
372#405C2×D4711yescharacteristic
373#344—471no—
374#344—471no—
375#338—471no—
376H(3,ℤ/4)—471no—
377H(3,ℤ/4)—471no—
378H(3,ℤ/4)—471no—
379H(3,ℤ/4)—471no—
380H(3,ℤ/4)—471no—
381C47#2228411yescharacteristic, complemented, Sylow
382C2×#41C22×D4711yescharacteristic
383C2×#41C22×D4711yescharacteristic
384C22×Q8C22×D4711yescharacteristic
385C22×D4C22×D4711yescharacteristic
386#137—41no—
387#137—41no—
388AGL(1,ℤ/8)—41no—
389AGL(1,ℤ/8)—41no—
390#135—21no—
391#135—21no—
392#135—21no—
393#135—21no—
394C2×Q16C2×Dic4711yescharacteristic, complemented
395C2×Q16C2×Dic4712yescomplemented
396C2×SD16—21no—
397C2×SD16—21no—
398C2×SD16—21no—
399C2×SD16—21no—
400C2×SD16C2×Dic4712yes—
401C2×SD16C2×Dic4712yes—
402C2×SD16C2×Dic4712yes—
403C2×SD16C2×Dic4711yescharacteristic, complemented
404C2×SD16C2×Dic4711yescharacteristic, complemented
405C2×SD16C2×Dic4712yescomplemented
406C2×D8C2×Dic4712yes—
407C2×D8C2×Dic4711yescharacteristic, complemented
408#131—942no—
409#131—942no—
410#131—1881no—
411#131—1881no—
412C2×M16—471no—
413C2×M16—471no—
414C2×M16—471no—
415C2×M16—471no—
416C2×M16—1881no—
417C2×M16—1881no—
418C2×M16C22×D4711yescharacteristic
419C22×C8C22×D4711yescharacteristic
420C4⋊Q8C2×Dic4712yescomplemented
421#127C2×Dic4712yes—
422#124C2×Dic4712yes—
423#124C2×Dic4712yescomplemented
424#123—41no—
425C22⋊Q8—41no—
426#121—41no—
427#120—41no—
428C4×Q8—942no—
429C4×Q8—1881no—
430C4×D4—942no—
431C4×D4—1881no—
432#117—471no—
433#117—471no—
434#117—471no—
435#117—471no—
436#117—1881no—
437#117C22×D4711yescharacteristic
438C2×C22⋊C4—1881no—
439#108—941no—
440#108—941no—
441#107—941no—
442#106—941no—
443#104—942no—
444#104—942no—
445#104—1881no—
446#104—1881no—
447#103—942no—
448#103—1881no—
449#102—942no—
450#102—1881no—
451#101—1881no—
452#100—1881no—
453#99—1881no—
454#99—1881no—
455#97—472no—
456#97—941no—
457#97C2×Dic4712yescomplemented
458C4×C8—472no—
459C4×C8—941no—
460C4×C8C2×Dic4712yescomplemented
461C24—21no—
462#41—21no—
463#41—21no—
464#41—21no—
465#41—21no—
466#41—21no—
467#41—21no—
468#41—21no—
469#41—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8C22×Dic4711yescharacteristic
476C2×Q8C22×Dic4712yes—
477C2×Q8C22×Dic4711yescharacteristic
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—42no—
486C2×D4C22×Dic4712yes—
487C2×D4C22×Dic4711yescharacteristic
488C2×D4C22×Dic4711yescharacteristic
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—1882no—
493C22×C4C23×D4711yescharacteristic
494Q16—21no—
495Q16—21no—
496Q16—42no—
497SD16—22no—
498SD16—22no—
499SD16—22no—
500SD16—22no—
501SD16—21no—
502SD16—21no—
503SD16—21no—
504SD16—21no—
505SD16—42no—
506SD16—42no—
507D8—21no—
508D8—21no—
509D8—42no—
510M16—21no—
511M16—21no—
512M16—941no—
513M16—941no—
514M16—941no—
515M16—941no—
516M16—941no—
517M16—941no—
518M16—941no—
519M16—941no—
520M16—1882no—
521M16—1882no—
522C2×C8—472no—
523C2×C8—941no—
524C2×C8—941no—
525C2×C8—941no—
526C2×C8—1882no—
527C2×C8—1882no—
528C2×C8D4×D4711yescharacteristic
529C2×C8D4×D4711yescharacteristic
530C2×C8#1062535211yescharacteristic
531C2×C8#1062535211yescharacteristic
532C2×C8C22×Dic4712yes—
533C2×C8C22×Dic4711yescharacteristic
534C2×C8C22×Dic4711yescharacteristic
535C4⋊C4—21no—
536C4⋊C4—42no—
537C4⋊C4—941no—
538C4⋊C4—941no—
539C4⋊C4—941no—
540C4⋊C4—941no—
541C4⋊C4—1882no—
542C22⋊C4—21no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—941no—
546C22⋊C4—941no—
547C22⋊C4—941no—
548C22⋊C4—941no—
549C22⋊C4—1882no—
550C42—472no—
551C42—941no—
552C42—941no—
553C42—941no—
554C42—1882no—
555C42C22×Dic4712yes—
556C23—21no—
557C23—22no—
558C23—21no—
559C23—41no—
560C23—41no—
561C23—161no—
562C23#1309406911yescharacteristic
563Q8—22no—
564Q8—21no—
565Q8—21no—
566Q8—21no—
567Q8—22no—
568Q8—21no—
569Q8—41no—
570Q8—41no—
571D4—21no—
572D4—21no—
573D4—22no—
574D4—21no—
575D4—22no—
576D4—21no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—162no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—942no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4—7522no—
599C2×C4C23×Dic4711yescharacteristic
600C2×C4C2×D4×D4711yescharacteristic
601C2×C4C2×#1062535211yescharacteristic
602C2×C4#1309406911yescharacteristic
603C2×C4#1309406911yescharacteristic
604C2×C4#1309406911yescharacteristic
605C8—22no—
606C8—942no—
607C8—1881no—
608C8—1881no—
609C8—1881no—
610C8D4×Dic4711yescharacteristic
611C8D4×Dic4711yescharacteristic
612C8D4×Dic4711yescharacteristic
613C8D4×Dic4711yescharacteristic
614C22—41no—
615C22—41no—
616C22—41no—
617C22—42no—
618C22—81no—
619C22—82no—
620C22—81no—
621C22—82no—
622C22—161no—
623C22#1568959811yescharacteristic
624C22#1568968611yescharacteristic
625C22#1568972711yescharacteristic
626C4—42no—
627C4—41no—
628C4—41no—
629C4—42no—
630C4—3761no—
631C4—3761no—
632C4—3761no—
633C4—1882no—
634C4C2×D4×Dic4711yescharacteristic
635C4#1568959811yescharacteristic
636C4#1568968611yescharacteristic
637C4#1568972711yescharacteristic
638C2—21no—
639C2—21no—
640C2—21no—
641C2—82no—
642C2—81no—
643C2—81no—
644C2#2024611411yescharacteristic
subgroupG/H
centerC2#20246114
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
Fitting, prime radicalC47×#3154C2
socleC2×C47#2328
cosocleC22×C4×C47C24
hypercenter#3154D47
hypocenterC47#22284
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ #3154

Complements

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

NH ≅ G/Ntype
1. #20252305C2semidirect
2. #20252248C2semidirect
3. #20244293C2semidirect
4. #20244287C2semidirect
5. #20243926C2semidirect
6. #20243921C2semidirect
7. #20243916C2semidirect
8. #20243911C2semidirect
9. #20243884C2semidirect
10. #20243877C2semidirect
11. #20243870C2semidirect
12. #20243863C2semidirect
13. #20243512C2semidirect
14. #20243143C2semidirect
54. C47×#497C4semidirect
55. C47×#496C4semidirect
56. C47×#495C4semidirect
57. C47×#492C4semidirect
96. #13093934D4semidirect
98. C8×Dic47D4semidirect
111. C2×Q16×C47C2×C4semidirect
112. C2×Q16×C47C2×C4semidirect
120. C2×SD16×C47C2×C4semidirect
121. C2×SD16×C47C2×C4semidirect
122. C2×SD16×C47C2×C4semidirect
124. C2×D8×C47C2×C4semidirect
127. C4⋊Q8×C47C2×C4semidirect
130. #124×C47C2×C4semidirect
136. #97×C47C2×C4semidirect
137. C4×C8×C47C2×C4semidirect
348. #497Dic47semidirect
349. #496Dic47semidirect
350. #495Dic47semidirect
351. #492Dic47semidirect
381. C47#22284semidirect
394. C2×Q16C2×Dic47semidirect
395. C2×Q16C2×Dic47semidirect
403. C2×SD16C2×Dic47semidirect
404. C2×SD16C2×Dic47semidirect
405. C2×SD16C2×Dic47semidirect
407. C2×D8C2×Dic47semidirect
420. C4⋊Q8C2×Dic47semidirect
423. #124C2×Dic47semidirect
457. #97C2×Dic47semidirect
460. C4×C8C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202523052
#202522482
#202442932
#202442872
#202439262
#202439212
#202439162
#202439112
#202438842
#202438772
#202438702
#202438632
#202435122
#202431432
C47×#31542
#2228447