Finite Group Databasebeta

← #26279289

Subgroups — #26279289

Order 12032.

864 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689403C211yesmaximal, characteristic
2C2×C4×#10625335C211yesmaximal, characteristic
3#20247227C211yesmaximal, characteristic, complemented
4#20244939C211yesmaximal, characteristic, complemented
5#20244870C211yesmaximal, characteristic, complemented
6#20244780C211yesmaximal, characteristic, complemented
7#20244721C211yesmaximal, characteristic, complemented
8#20244118C211yesmaximal, characteristic, complemented
9#20243947C211yesmaximal, characteristic, complemented
10#20243722C211yesmaximal, characteristic
11#20243722C211yesmaximal, characteristic
12#20243649C211yesmaximal, characteristic
13#20243641C211yesmaximal, characteristic, complemented
14#20243635C211yesmaximal, characteristic
15C47×#2349C211yesmaximal, characteristic
16C22×#10625335C2211yescharacteristic
17C2×#13094024C2211yescharacteristic
18C2×C4×Dic94C2211yescharacteristic
19#15689403—24no—
20C4×#10625335—24no—
21C2×#13093957C2211yescharacteristic
22C2×#13093957C2211yescharacteristic
23C2×#13093957C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×#13093947C2211yescharacteristic
28C2×#13093947C2211yescharacteristic
29C2×#13093947C2211yescharacteristic
30C2×#13093945C2211yescharacteristic
31C2×#13093945C2211yescharacteristic
32C2×#13093945C2211yescharacteristic
33C2×C4×C47⋊C8C2211yescharacteristic
34#15689057—21no—
35#15689043—21no—
36#15689029C2211yescharacteristic
37#15689029C2211yescharacteristic
38#15689027—21no—
39#15689021C2211yescharacteristic
40#15689021C2211yescharacteristic
41#15689009—21no—
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48#15688878C2211yescharacteristic
49#15688878C2211yescharacteristic
50C2×C4×D4×C47C2211yescharacteristic
51C2×C4×D4×C47C2211yescharacteristic
52C2×C4×D4×C47C2211yescharacteristic
53C47×#389C2211yescharacteristic
54C47×#389C2211yescharacteristic
55C47×#385—21no—
56C47×#385—21no—
57C47×#382C2211yescharacteristic
58C47×#380—21no—
59C4×C4⋊C4×C47C2211yescharacteristic
60C2×#95×C47—21no—
61C22×Dic94—21no—
62C22×Dic94C2311yescharacteristic
63C2×#10625335—22no—
64C2×#10625335—22no—
65C2×#10625335Q814yescomplemented
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70#13094024—22no—
71#13094024—41no—
72C4×Dic94—22no—
73C4×Dic94—41no—
74C2×C4⋊Dic47—21no—
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×Dic47⋊C4—21no—
81C2×C4×Dic47—21no—
82#13093957—22no—
83#13093957—22no—
84#13093957—22no—
85#13093955—22no—
86#13093955—41no—
87C4⋊(C47⋊C8)—22no—
88#13093947—22no—
89#13093947—22no—
90#13093947—41no—
91#13093945—22no—
92#13093945—22no—
93#13093945D414yes—
94C4×C47⋊C8—22no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C22×D4×C47C2311yescharacteristic
102C23×C4×C47—21no—
103C23×C4×C47—21no—
104C23×C4×C47—21no—
105C4×D4×C47—22no—
106C4×D4×C47—22no—
107C4×D4×C47—22no—
108C4×D4×C47—22no—
109C4×D4×C47—41no—
110C4×D4×C47—41no—
111C2×C4⋊C4×C47—21no—
112C2×C4⋊C4×C47—21no—
113C2×C4⋊C4×C47D411yescharacteristic
114C2×C4⋊C4×C47D411yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C22⋊C4×C47—21no—
119C2×C22⋊C4×C47—21no—
120C2×C22⋊C4×C47—21no—
121C2×C42×C47C2311yescharacteristic
122C2×C42×C47C2311yescharacteristic
123C2×C42×C47C2311yescharacteristic
124#95×C47—21no—
125#95×C47—21no—
126#95×C47—21no—
127#95×C47—21no—
128#95×C47D411yescharacteristic
129#95×C47D411yescharacteristic
130C24×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47—21no—
133C2×D4×C47—21no—
134C2×D4×C47C2×D412yes—
135C2×D4×C47C2×Q812yes—
136C2×D4×C47#4112yes—
137C22×C4×C47—21no—
138C22×C4×C47—24no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—42no—
142C22×C4×C47—42no—
143C22×C4×C47—42no—
144C22×C4×C47—42no—
145C22×C4×C47—81no—
146C22×C4×C47—81no—
147C22×C4×C47—81no—
148C22×C4×C47—81no—
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2411yescharacteristic
156C22×Dic47—21no—
157C22×Dic47—21no—
158C22×Dic47—21no—
159C22×Dic47—21no—
160C2×Dic94—21no—
161C2×Dic94—21no—
162C2×Dic94—21no—
163C2×Dic94—21no—
164C2×Dic94—21no—
165C2×Dic94—42no—
166C2×Dic94—42no—
167C2×Dic94—81no—
168C2×Dic94C2×Q812yes—
169C4⋊C4×C47—22no—
170C4⋊C4×C47—22no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—81no—
177C4⋊C4×C47—81no—
178C4⋊C4×C47#4112yes—
179C22⋊C4×C47—24no—
180C22⋊C4×C47—42no—
181C22⋊C4×C47—42no—
182C42×C47—21no—
183C42×C47—21no—
184C42×C47—21no—
185C42×C47—21no—
186C42×C47—21no—
187C42×C47—21no—
188#10625335—28no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47—21no—
194C4⋊Dic47—42no—
195C4⋊Dic47C2×D412yes—
196Dic47⋊C4—42no—
197Dic47⋊C4—81no—
198Dic47⋊C4—81no—
199C4×Dic47—42no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8C2×D412yes—
211C2×C47⋊C8C2×Q812yes—
212C23×C47—22no—
213C23×C47—22no—
214C23×C47—22no—
215C23×C47—41no—
216C23×C47—41no—
217C23×C47—41no—
218C23×C47—41no—
219C23×C47C22×D411yescharacteristic
220D4×C47—22no—
221D4×C47—22no—
222D4×C47—22no—
223D4×C47C22⋊Q814yes—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—22no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—84no—
250C2×C4×C47—84no—
251C2×C4×C47—84no—
252C2×C4×C47#13512yes—
253C2×C4×C47#13712yes—
254C2×C4×C47C22×D411yescharacteristic
255C2×C4×C47C22×Q811yescharacteristic
256C2×C4×C47C2×#4111yescharacteristic
257C2×C4×C47C2×#4111yescharacteristic
258C2×C4×C47H(5,𝔽2)11yescharacteristic
259C2×C4×C472−1+411yescharacteristic
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272Dic94—22no—
273Dic94—22no—
274Dic94—41no—
275Dic94—41no—
276Dic94—84no—
277C47⋊C8—22no—
278C47⋊C8—22no—
279C47⋊C8—41no—
280C47⋊C8—41no—
281#26372—471nomaximal, Sylow
282C22×C47—24no—
283C22×C47—42no—
284C22×C47—42no—
285C22×C47—42no—
286C22×C47—42no—
287C22×C47—42no—
288C22×C47—42no—
289C22×C47#52511yescharacteristic
290C22×C47D4211yescharacteristic
291C22×C47D4×Q811yescharacteristic
292C22×C47C2×#13511yescharacteristic
293C22×C47C2×#13711yescharacteristic
294C22×C47#57611yescharacteristic
295C22×C47#57711yescharacteristic
296C4×C47—22no—
297C4×C47—22no—
298C4×C47—41no—
299C4×C47—41no—
300C4×C47—42no—
301C4×C47—41no—
302C4×C47—41no—
303C4×C47—81no—
304C4×C47—81no—
305C4×C47C2×C22⋊Q811yescharacteristic
306C4×C47#53911yescharacteristic
307C4×C47#54711yescharacteristic
308C4×C47#55411yescharacteristic
309Dic47—42no—
310Dic47—42no—
311Dic47—81no—
312Dic47—81no—
313C2×#474—471no—
314C2×C4×SD16—471no—
315#2451—471no—
316#2349D4711yescharacteristic
317#2097—471no—
318#2085—471no—
319#2068—471no—
320#2058—471no—
321#1942—471no—
322#1916—471no—
323#1877—471no—
324#1877—471no—
325#1864—471no—
326#1863—471no—
327#1862—471no—
328C2×C47—44no—
329C2×C47#244911yescharacteristic
330C2×C47#310711yescharacteristic
331C2×C47#311811yescharacteristic
332C2×C47#326611yescharacteristic
333C2×C47#328011yescharacteristic
334C2×C47#338111yescharacteristic
335C2×C47#338511yescharacteristic
336C22×SD16—471no—
337C2×C4⋊Q8—471no—
338C2×C4×Q8—471no—
339C2×C4×D4C2×D4711yescharacteristic
340C2×C4×D4C2×D4711yescharacteristic
341C2×C4×D4C2×D4711yescharacteristic
342#474—944no—
343C4×SD16—944no—
344C2×#107—471no—
345C2×#107—471no—
346C2×#107—471no—
347C2×#106—471no—
348C2×#106—471no—
349C2×C4⋊C8—471no—
350C2×#103—471no—
351C2×#103—471no—
352C2×#103—471no—
353C2×#102—471no—
354C2×#102—471no—
355C2×#102—471no—
356C2×C4×C8—471no—
357#398—941no—
358#395—941no—
359#391—471no—
360#391—471no—
361#391—941no—
362#389—471no—
363#389—471no—
364#389C2×D4711yescharacteristic
365#389C2×D4711yescharacteristic
366#387—941no—
367#385—21no—
368#385—21no—
369#382C2×D4711yescharacteristic
370#380—21no—
371C4×C4⋊C4C2×D4711yescharacteristic
372C2×#95—21no—
373#340—471no—
374#340—471no—
375#340—471no—
376#340—471no—
377#336—471no—
378#336—471no—
379#336—471no—
380#336—471no—
381C47#2637211yescharacteristic, complemented, Sylow
382C22×Q8—471no—
383C22×Q8—941no—
384C22×D4C22×D4711yescharacteristic
385C23×C4—21no—
386C23×C4—21no—
387C23×C4—21no—
388C2×SD16—474no—
389C2×SD16—942no—
390C2×SD16—942no—
391C22×C8—471no—
392C22×C8—471no—
393C22×C8—471no—
394C22×C8—471no—
395C4⋊Q8—942no—
396C4⋊Q8—1881no—
397C4×Q8—942no—
398C4×Q8—1881no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—41no—
404C4×D4—41no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—471no—
408C2×C4⋊C4—471no—
409C2×C4⋊C4—471no—
410C2×C4⋊C4—941no—
411C2×C4⋊C4—941no—
412C2×C4⋊C4—941no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4C22⋊D4711yescharacteristic
415C2×C4⋊C4C22⋊D4711yescharacteristic
416C2×C4⋊C4C22×D4711yescharacteristic
417C2×C4⋊C4C22×D4711yescharacteristic
418C2×C4⋊C4C22×D4711yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C42—941no—
423C2×C42C22×D4711yescharacteristic
424C2×C42C22×D4711yescharacteristic
425C2×C42C22×D4711yescharacteristic
426#107—942no—
427#107—942no—
428#107—942no—
429#106—942no—
430#106—1881no—
431C4⋊C8—942no—
432#103—942no—
433#103—942no—
434#103—1881no—
435#102—474no—
436#102—942no—
437#102—942no—
438C4×C8—942no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—21no—
443#95—941no—
444#95—941no—
445#95—941no—
446#95—941no—
447#95—941no—
448#95—941no—
449#95C22⋊D4711yescharacteristic
450#95C22⋊D4711yescharacteristic
451C24—21no—
452C2×Q8—472no—
453C2×Q8—941no—
454C2×Q8—941no—
455C2×Q8—941no—
456C2×Q8—941no—
457C2×Q8—941no—
458C2×Q8—3761no—
459C2×Q8—1882no—
460C2×Q8—1882no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4#1062535012yes—
465C2×D4D4×D4712yes—
466C2×D4Q8×D4712yes—
467C22×C4—21no—
468C22×C4—24no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—941no—
480C22×C4—941no—
481C22×C4—941no—
482C22×C4—941no—
483C22×C4C2×C22⋊D4711yescharacteristic
484C22×C4C2×C22⋊D4711yescharacteristic
485C22×C4C2×C22⋊D4711yescharacteristic
486C22×C4C2×C22⋊D4711yescharacteristic
487C22×C4C2×C22⋊D4711yescharacteristic
488C22×C4C2×C22⋊D4711yescharacteristic
489C22×C4C23×D4711yescharacteristic
490SD16—948no—
491C2×C8—472no—
492C2×C8—472no—
493C2×C8—941no—
494C2×C8—941no—
495C2×C8—941no—
496C2×C8—941no—
497C2×C8—941no—
498C2×C8—941no—
499C2×C8—941no—
500C2×C8—941no—
501C2×C8—941no—
502C2×C8—941no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—472no—
513C4⋊C4—941no—
514C4⋊C4—941no—
515C4⋊C4—941no—
516C4⋊C4—941no—
517C4⋊C4—941no—
518C4⋊C4—3761no—
519C4⋊C4—3761no—
520C4⋊C4—1882no—
521C4⋊C4—1882no—
522C4⋊C4#1062535012yes—
523C22⋊C4—24no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—1882no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C22×C22⋊D4711yescharacteristic
541Q8—942no—
542Q8—942no—
543Q8—3764no—
544Q8—1881no—
545Q8—1881no—
546D4—22no—
547D4—22no—
548D4—22no—
549D4#1309400214yes—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—942no—
579C2×C4—942no—
580C2×C4—1881no—
581C2×C4—1881no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4—1881no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4—1881no—
589C2×C4—1881no—
590C2×C4#1309404412yes—
591C2×C4#1309405412yes—
592C2×C4C2×D4×D4711yescharacteristic
593C2×C4C2×Q8×D4711yescharacteristic
594C2×C4C2×#1062535011yescharacteristic
595C2×C4C2×#1062535011yescharacteristic
596C2×C4#1309406911yescharacteristic
597C2×C4#1309407211yescharacteristic
598C8—942no—
599C8—942no—
600C8—1881no—
601C8—1881no—
602C22—24no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22#1568963411yescharacteristic
610C22D4×C22⋊D4711yescharacteristic
611C22Q8×C22⋊D4711yescharacteristic
612C22C2×#1309404411yescharacteristic
613C22C2×#1309405411yescharacteristic
614C22#1568992611yescharacteristic
615C22#1568993411yescharacteristic
616C4—22no—
617C4—22no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—42no—
622C4—41no—
623C4—81no—
624C4—81no—
625C4—3761no—
626C4—3761no—
627C4—1882no—
628C4—1882no—
629C4C2×#1309400211yescharacteristic
630C4#1568972011yescharacteristic
631C4#1568977411yescharacteristic
632C4#1568982811yescharacteristic
633C2—44no—
634C2#2024719711yescharacteristic
635C2#2025345911yescharacteristic
636C2#2025357211yescharacteristic
637C2#2025489411yescharacteristic
638C2#2025506911yescharacteristic
639C2#2025626311yescharacteristic
640C2#2025629811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2349C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
hypercenter#2349D47
hypocenterC47#26372
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #20247227C2semidirect
4. #20244939C2semidirect
5. #20244870C2semidirect
6. #20244780C2semidirect
7. #20244721C2semidirect
8. #20244118C2semidirect
9. #20243947C2semidirect
13. #20243641C2semidirect
65. C2×#10625335Q8semidirect
381. C47#26372semidirect

Maximal subgroups

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

subgroupindex
C2×#156894032
C2×C4×#106253352
#202472272
#202449392
#202448702
#202447802
#202447212
#202441182
#202439472
#202437222
#202436492
#202436412
#202436352
C47×#23492
#2637247