Finite Group Databasebeta

← #26238307

Subgroups — #26238307

Order 12032.

850 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
1C2×#15689401C211yesmaximal, characteristic
2#20246475C211yesmaximal, characteristic, complemented
3#20246293C211yesmaximal, characteristic
4#20244359C211yesmaximal, characteristic, complemented
5#20244155C211yesmaximal, characteristic, complemented
6#20244126C211yesmaximal, characteristic, complemented
7#20244120C211yesmaximal, characteristic, complemented
8#20243979C211yesmaximal, characteristic, complemented
9#20243959C211yesmaximal, characteristic, complemented
10#20243651C211yesmaximal, characteristic
11#20243635C211yesmaximal, characteristic
12#20243264C211yesmaximal, characteristic, complemented
13#20243215C211yesmaximal, characteristic
14#102×Dic47C211yesmaximal, characteristic
15C47×#1762C211yesmaximal, characteristic
16C2×#13094023C2211yescharacteristic
17C2×D4×Dic47C2211yescharacteristic
18C2×D4×Dic47C2211yescharacteristic
19#15689401C418yescomplemented
20C2×#13093958C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093957C2211yescharacteristic
23C2×#13093956C2211yescharacteristic
24C2×Dic47⋊C8C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×#13093934C2211yescharacteristic
28C2×C8×Dic47C2211yescharacteristic
29#15689029—21no—
30#15689027—21no—
31#15689022C2211yescharacteristic
32#15689021C2211yescharacteristic
33#15689021C2211yescharacteristic
34#15689020—21no—
35#15689019C2211yescharacteristic
36#15688996—21no—
37#15688995—21no—
38#15688988C2211yescharacteristic
39#15688981—21no—
40C4⋊C4×Dic47C2211yescharacteristic
41C4⋊C4×Dic47C2211yescharacteristic
42C4×Dic47⋊C4—21no—
43C4×Dic47⋊C4C2211yescharacteristic
44C2×C47⋊#95—21no—
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688878C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C2×#102×C47C2211yescharacteristic
55C2×#102×C47C2211yescharacteristic
56C2×#97×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×Dic94—21no—
61C23×Dic47—21no—
62C23×Dic47—21no—
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65#13094023—22no—
66#13094023C2×C414yescomplemented
67D4×Dic47—22no—
68D4×Dic47—22no—
69D4×Dic47—41no—
70D4×Dic47C2×C414yes—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×Dic47⋊C4D411yescharacteristic
81C2×Dic47⋊C4D411yescharacteristic
82C2×C22⋊Dic47—21no—
83C2×C22⋊Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47C2311yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88#13093958C2×C414yescomplemented
89#13093957—41no—
90#13093957C2×C414yescomplemented
91#13093956—41no—
92Dic47⋊C8C2×C414yescomplemented
93#13093945—22no—
94#13093945C2×C414yes—
95#13093934—22no—
96C8×Dic47—41no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95D411yescharacteristic
106C47⋊#95D411yescharacteristic
107C22×D4×C47C2311yescharacteristic
108C23×C4×C47—21no—
109C22×C8×C47C2311yescharacteristic
110C22×C8×C47C2311yescharacteristic
111C4×D4×C47—22no—
112C4×D4×C47—41no—
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C22⋊C4×C47—21no—
117C2×C42×C47—21no—
118C2×C42×C47C2311yescharacteristic
119#102×C47C2×C414yes—
120#102×C47Q814yescomplemented
121#97×C47—41no—
122#95×C47—21no—
123C24×C47—21no—
124C2×D4×C47—21no—
125C2×D4×C47—21no—
126C2×D4×C47—21no—
127C2×D4×C47C22×C412yes—
128C2×D4×C47C2×Q812yes—
129C2×D4×C47#4112yes—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—42no—
134C22×C4×C47—42no—
135C22×C4×C47—81no—
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2411yescharacteristic
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—24no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—42no—
145C22×Dic47—42no—
146C22×Dic47—81no—
147C22×Dic47—81no—
148C22×Dic47—81no—
149C22×Dic47C2×D411yescharacteristic
150C22×Dic47C2×D411yescharacteristic
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153C2×Dic94—24no—
154C2×Dic94—81no—
155C2×Dic94—81no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47C22×C412yes—
161C2×C8×C47C2×Q812yes—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47C22×C412yes—
167C4⋊C4×C47C2×Q812yes—
168C22⋊C4×C47—42no—
169C42×C47—21no—
170C42×C47—21no—
171C42×C47—81no—
172C22⋊Dic47—24no—
173C22⋊Dic47—42no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—81no—
181C4⋊Dic47C22×C412yes—
182C4⋊Dic47C22×C412yes—
183Dic47⋊C4—24no—
184Dic47⋊C4—22no—
185Dic47⋊C4—41no—
186Dic47⋊C4—81no—
187Dic47⋊C4—81no—
188Dic47⋊C4—81no—
189C4×Dic47—21no—
190C4×Dic47—21no—
191C4×Dic47—42no—
192C4×Dic47—81no—
193C4×Dic47C22×C412yes—
194C4×Dic47#4112yes—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8C22×C412yes—
201C23×C47—22no—
202C23×C47—22no—
203C23×C47—22no—
204C23×C47—41no—
205C23×C47—41no—
206C23×C47—41no—
207C23×C47—41no—
208C23×C47C22×D411yescharacteristic
209D4×C47—22no—
210D4×C47—22no—
211D4×C47—22no—
212D4×C47C4×Q814yes—
213C2×C4×C47—22no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—22no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—84no—
229C2×C4×C47C23×C411yescharacteristic
230C2×C4×C47C22×Q811yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×C4×C47H(5,𝔽2)11yescharacteristic
234C2×C4×C472−1+411yescharacteristic
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—21no—
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—22no—
242C2×Dic47—22no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—84no—
257C2×Dic47—84no—
258C2×Dic47C4×D412yes—
259C2×Dic47C4×D412yes—
260C2×Dic47#13712yes—
261Dic94—84no—
262C8×C47—22no—
263C8×C47—22no—
264C47⋊C8—22no—
265C47⋊C8—41no—
266#22979—471nomaximal, Sylow
267C22×C47—24no—
268C22×C47—42no—
269C22×C47—42no—
270C22×C47—42no—
271C22×C47—42no—
272C22×C47—42no—
273C22×C47—42no—
274C22×C47C2×C4×D411yescharacteristic
275C22×C47#54611yescharacteristic
276C22×C47D4×Q811yescharacteristic
277C22×C47C2×#13711yescharacteristic
278C22×C47#57511yescharacteristic
279C22×C47#57611yescharacteristic
280C22×C47#57611yescharacteristic
281C4×C47—42no—
282C4×C47—42no—
283C4×C47—41no—
284C4×C47—41no—
285C4×C47—81no—
286C4×C47C2×C4×Q811yescharacteristic
287C4×C47#52011yescharacteristic
288C4×C47#55211yescharacteristic
289C4×C47#55411yescharacteristic
290Dic47—22no—
291Dic47—22no—
292Dic47—22no—
293Dic47—42no—
294Dic47—42no—
295Dic47—41no—
296Dic47—81no—
297Dic47—81no—
298C2×#474—471no—
299#2369—471no—
300#2349—471no—
301#1993—471no—
302#1949—471no—
303#1943—471no—
304#1942—471no—
305#1921—471no—
306#1918—471no—
307#1864—471no—
308#1862—471no—
309#1771—471no—
310#1762—471no—
311#1762D4711yescharacteristic
312C4×#102—471no—
313C2×C47—44no—
314C2×C47#236011yescharacteristic
315C2×C47C4×#13711yescharacteristic
316C2×C47#294811yescharacteristic
317C2×C47#331411yescharacteristic
318C2×C47#331911yescharacteristic
319C2×C47#338511yescharacteristic
320C2×C47#338911yescharacteristic
321C2×C4⋊Q8—471no—
322C2×C4×D4—471no—
323C2×C4×D4—471no—
324C2×C4×D4C2×D4711yescharacteristic
325#474—478no—
326C2×#107—471no—
327C2×#107—471no—
328C2×#107—471no—
329C2×#106—471no—
330C2×C4⋊C8—471no—
331C2×#102—471no—
332C2×#102—471no—
333C2×#102C2×D4711yescharacteristic
334C2×#102C2×D4711yescharacteristic
335C2×#97—471no—
336C2×#97C2×D4711yescharacteristic
337C2×C4×C8—471no—
338#391—941no—
339#391—941no—
340#389—471no—
341#389—471no—
342#389—471no—
343#389—471no—
344#389—941no—
345#385—941no—
346#385—941no—
347#382—471no—
348#380—941no—
349C4×C4⋊C4—471no—
350C4×C4⋊C4—471no—
351C4×C4⋊C4—471no—
352C4×C4⋊C4—941no—
353C4×C4⋊C4C2×D4711yescharacteristic
354C2×#95—941no—
355#340—471no—
356#340—471no—
357#340—471no—
358#340—471no—
359#340—471no—
360#340—471no—
361#340C2×D4711yescharacteristic
362#340C2×D4711yescharacteristic
363#336—471no—
364#336—471no—
365C47#2297911yescharacteristic, complemented, Sylow
366C22×Q8—941no—
367C22×D4C22×D4711yescharacteristic
368C23×C4—21no—
369C23×C4—941no—
370C23×C4—941no—
371C22×C8—471no—
372C22×C8—471no—
373C22×C8C22×D4711yescharacteristic
374C22×C8C22×D4711yescharacteristic
375C4⋊Q8—474no—
376C4⋊Q8—942no—
377C4×D4—22no—
378C4×D4—41no—
379C4×D4—474no—
380C4×D4—942no—
381C4×D4—942no—
382C4×D4—1881no—
383C2×C4⋊C4—471no—
384C2×C4⋊C4—471no—
385C2×C4⋊C4—471no—
386C2×C4⋊C4—471no—
387C2×C4⋊C4—471no—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4C22×D4711yescharacteristic
395C2×C4⋊C4C22×D4711yescharacteristic
396C2×C4⋊C4C22×D4711yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—941no—
399C2×C22⋊C4—941no—
400C2×C42—21no—
401C2×C42—471no—
402C2×C42—471no—
403C2×C42—941no—
404C2×C42—941no—
405C2×C42C22×D4711yescharacteristic
406#107—474no—
407#107—474no—
408#107—1881no—
409#106—1881no—
410C4⋊C8—474no—
411#102—474no—
412#102—942no—
413#102Dic9414yescomplemented
414#102C4×D4714yes—
415#97—41no—
416#97—942no—
417C4×C8—1881no—
418#95—21no—
419#95—471no—
420#95—471no—
421#95—941no—
422#95—941no—
423#95—941no—
424#95—941no—
425#95—941no—
426#95—941no—
427#95—941no—
428#95—941no—
429C24—21no—
430C2×Q8—944no—
431C2×Q8—3761no—
432C2×Q8—3761no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4C2×Dic9412yes—
437C2×D4C2×C4×D4712yes—
438C2×D4#1062535012yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—81no—
445C22×C4—471no—
446C22×C4—471no—
447C22×C4—471no—
448C22×C4—471no—
449C22×C4—944no—
450C22×C4—941no—
451C22×C4—941no—
452C22×C4—941no—
453C22×C4—941no—
454C22×C4—3761no—
455C22×C4—3761no—
456C22×C4—3761no—
457C22×C4—1882no—
458C22×C4—1882no—
459C22×C4D4×D4711yescharacteristic
460C22×C4D4×D4711yescharacteristic
461C22×C4C23×D4711yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—472no—
467C2×C8—941no—
468C2×C8—941no—
469C2×C8—941no—
470C2×C8—941no—
471C2×C8—941no—
472C2×C8C2×Dic9412yes—
473C2×C8C2×C4×D4712yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—472no—
479C4⋊C4—472no—
480C4⋊C4—942no—
481C4⋊C4—944no—
482C4⋊C4—941no—
483C4⋊C4—941no—
484C4⋊C4—941no—
485C4⋊C4—941no—
486C4⋊C4—941no—
487C4⋊C4—941no—
488C4⋊C4—3761no—
489C4⋊C4—3761no—
490C4⋊C4—3761no—
491C4⋊C4—3761no—
492C4⋊C4—1881no—
493C4⋊C4C2×Dic9412yes—
494C4⋊C4C2×C4×D4712yes—
495C22⋊C4—42no—
496C22⋊C4—944no—
497C22⋊C4—1882no—
498C42—21no—
499C42—21no—
500C42—81no—
501C42—472no—
502C42—472no—
503C42—941no—
504C42—941no—
505C42—3761no—
506C42—1882no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C2×D4×D4711yescharacteristic
515Q8—3764no—
516D4—22no—
517D4—22no—
518D4—22no—
519D4C4×Dic9414yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
536C2×C4—472no—
537C2×C4—472no—
538C2×C4—472no—
539C2×C4—941no—
540C2×C4—941no—
541C2×C4—941no—
542C2×C4—941no—
543C2×C4—941no—
544C2×C4—941no—
545C2×C4—941no—
546C2×C4—942no—
547C2×C4—942no—
548C2×C4—941no—
549C2×C4—941no—
550C2×C4—3764no—
551C2×C4—3764no—
552C2×C4—1881no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4—1881no—
556C2×C4—1881no—
557C2×C4—1881no—
558C2×C4—1881no—
559C2×C4—1881no—
560C2×C4—1881no—
561C2×C4—1881no—
562C2×C4C22×C4×D4711yescharacteristic
563C2×C4C22×Dic9411yescharacteristic
564C2×C4#41×D4711yescharacteristic
565C2×C4C2×#1062535011yescharacteristic
566C2×C4#1309406911yescharacteristic
567C2×C4#1309407111yescharacteristic
568C8—22no—
569C8—22no—
570C8—942no—
571C8—1881no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C4×D4×D4711yescharacteristic
580C22#1568976311yescharacteristic
581C22D4×Dic9411yescharacteristic
582C22#137×D4711yescharacteristic
583C22#1568991911yescharacteristic
584C22#1568992711yescharacteristic
585C22#1568992711yescharacteristic
586C4—41no—
587C4—42no—
588C4—42no—
589C4—41no—
590C4—81no—
591C4—942no—
592C4—942no—
593C4—942no—
594C4—3761no—
595C4—3761no—
596C4—1881no—
597C4—1882no—
598C4—1882no—
599C4C2×C4×Dic9411yescharacteristic
600C4#1568960211yescharacteristic
601C4#1568980611yescharacteristic
602C4#1568983211yescharacteristic
603C2—44no—
604C2#2024639411yescharacteristic
605C2#2025221011yescharacteristic
606C2#2025223711yescharacteristic
607C2#2025547011yescharacteristic
608C2#2025554511yescharacteristic
609C2#2025629911yescharacteristic
610C2#2025634711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1762C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1762D47
hypocenterC47#22979
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. #20246475C2semidirect
4. #20244359C2semidirect
5. #20244155C2semidirect
6. #20244126C2semidirect
7. #20244120C2semidirect
8. #20243979C2semidirect
9. #20243959C2semidirect
12. #20243264C2semidirect
19. #15689401C4semidirect
66. #13094023C2×C4semidirect
88. #13093958C2×C4semidirect
90. #13093957C2×C4semidirect
92. Dic47⋊C8C2×C4semidirect
120. #102×C47Q8semidirect
365. C47#22979semidirect
413. #102Dic94semidirect

Maximal subgroups

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

subgroupindex
C2×#156894012
#202464752
#202462932
#202443592
#202441552
#202441262
#202441202
#202439792
#202439592
#202436512
#202436352
#202432642
#202432152
#102×Dic472
C47×#17622
#2297947