Finite Group Databasebeta

← #26234014

Subgroups — #26234014

Order 12032.

782 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689165C211yesmaximal, characteristic
2#20246275C211yesmaximal, characteristic
3#20246267C211yesmaximal, characteristic
4#20244124C211yesmaximal, characteristic, complemented
5#20244119C211yesmaximal, characteristic, complemented
6#20243979C211yesmaximal, characteristic, complemented
7#20243972C211yesmaximal, characteristic, complemented
8#20243965C211yesmaximal, characteristic, complemented
9#20243958C211yesmaximal, characteristic, complemented
10#20243480C211yesmaximal, characteristic
11#20243472C211yesmaximal, characteristic
12#20243287C211yesmaximal, characteristic, complemented
13#106×Dic47C211yesmaximal, characteristic, complemented
14#20243169C211yesmaximal, characteristic
15C47×#1809C211yesmaximal, characteristic
16C2×#13093983C2211yescharacteristic
17C22×C4⋊Dic47C2211yescharacteristic
18#15689165C418yescomplemented
19C2×#13093958C2211yescharacteristic
20C2×#13093958C2211yescharacteristic
21C2×#13093956C2211yescharacteristic
22C2×#13093956C2211yescharacteristic
23C2×C22⋊(C47⋊C8)C2211yescharacteristic
24C2×#13093934C2211yescharacteristic
25C2×C8×Dic47C2211yescharacteristic
26#15689019C2211yescharacteristic
27#15689019C2211yescharacteristic
28#15689019C2211yescharacteristic
29#15689019C2211yescharacteristic
30#15689017—21no—
31#15689017—21no—
32#15689008—21no—
33#15689008—21no—
34#15688987—21no—
35#15688981—21no—
36C4⋊C4×Dic47C2211yescharacteristic
37C4⋊C4×Dic47C2211yescharacteristic
38C4⋊C4×Dic47C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C22⋊C4×Dic47—21no—
41C4×C22⋊Dic47C2211yescharacteristic
42#15688968—21no—
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C22×M16×C47C2211yescharacteristic
54C2×#107×C47C2211yescharacteristic
55C2×#106×C47C2211yescharacteristic
56C47×#380C2211yescharacteristic
57C47×#380C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C23×Dic47—21no—
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63#13093983—22no—
64#13093983C2×C414yes—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47—22no—
67C2×C4⋊Dic47C2×C414yes—
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47D411yescharacteristic
78C2×C22⋊Dic47D411yescharacteristic
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47C2311yescharacteristic
85C2×C4×Dic47C2311yescharacteristic
86#13093958C2×C414yes—
87#13093958C2×C414yes—
88#13093956C2×C414yes—
89#13093956C2×C414yes—
90C22⋊(C47⋊C8)—22no—
91#13093934—22no—
92C8×Dic47—22no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95D411yescharacteristic
102C47⋊#95D411yescharacteristic
103C23×C4×C47C2311yescharacteristic
104C2×M16×C47—22no—
105C2×M16×C47—22no—
106C2×M16×C47C2×C414yes—
107C22×C8×C47C2311yescharacteristic
108C22×C8×C47C2311yescharacteristic
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C22⋊C4×C47—21no—
114C2×C22⋊C4×C47—21no—
115#107×C47—22no—
116#106×C47—22no—
117#95×C47—21no—
118#95×C47—21no—
119C24×C47C2×D411yescharacteristic
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47#4112yes—
131C22×C4×C47C2411yescharacteristic
132C22×Dic47—21no—
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47—24no—
137C22×Dic47—81no—
138C22×Dic47—81no—
139C22×Dic47C2×D411yescharacteristic
140C22×Dic47C2×D411yescharacteristic
141C22×Dic47C2×D411yescharacteristic
142C22×Dic47C2×D411yescharacteristic
143M16×C47—28no—
144C2×C8×C47—21no—
145C2×C8×C47—21no—
146C2×C8×C47C22×C412yes—
147C2×C8×C47C22×C412yes—
148C2×C8×C47#4112yes—
149C2×C8×C47#4112yes—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—21no—
157C4⋊C4×C47—21no—
158C22⋊C4×C47—42no—
159C22⋊C4×C47—81no—
160C22⋊Dic47—24no—
161C22⋊Dic47—22no—
162C22⋊Dic47—41no—
163C4⋊Dic47—24no—
164C4⋊Dic47—21no—
165C4⋊Dic47—21no—
166C4⋊Dic47—21no—
167C4⋊Dic47—21no—
168C4⋊Dic47C22×C412yes—
169C4⋊Dic47C22×C412yes—
170C4⋊Dic47C22×C412yes—
171C4⋊Dic47C22×C412yes—
172Dic47⋊C4—81no—
173Dic47⋊C4—81no—
174Dic47⋊C4—81no—
175Dic47⋊C4—81no—
176C4×Dic47—24no—
177C4×Dic47—21no—
178C4×Dic47—21no—
179C4×Dic47—21no—
180C4×Dic47—21no—
181C4×Dic47—42no—
182C4×Dic47—42no—
183C4×Dic47—81no—
184C4×Dic47—81no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8—21no—
191C23×C47—21no—
192C23×C47—21no—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47—21no—
197C23×C47C4×D412yes—
198C23×C47C22×D411yescharacteristic
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47—22no—
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—22no—
205C2×C4×C47—41no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
211C2×C4×C47—41no—
212C2×C4×C47—41no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47C4×D412yes—
218C2×C4×C47C23×C411yescharacteristic
219C2×C4×C47C2×#4111yescharacteristic
220C2×C4×C47C2×#4111yescharacteristic
221C2×C4×C47C2×#4111yescharacteristic
222C2×C4×C472−1+411yescharacteristic
223C2×C4×C472−1+411yescharacteristic
224C2×Dic47—21no—
225C2×Dic47—21no—
226C2×Dic47—22no—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—22no—
230C2×Dic47—21no—
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—21no—
238C2×Dic47—21no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—84no—
248C2×Dic47#13712yes—
249C8×C47—22no—
250C8×C47#11714yes—
251C47⋊C8—41no—
252C47⋊C8—41no—
253#22621—471nomaximal, Sylow
254C22×C47—22no—
255C22×C47—22no—
256C22×C47—22no—
257C22×C47—41no—
258C22×C47—41no—
259C22×C47—41no—
260C22×C47—41no—
261C22×C47C2×C4×D411yescharacteristic
262C22×C47#54711yescharacteristic
263C22×C47#54711yescharacteristic
264C22×C47C2×#13711yescharacteristic
265C22×C47#57511yescharacteristic
266C22×C47#57611yescharacteristic
267C22×C47#57711yescharacteristic
268C4×C47—22no—
269C4×C47—81no—
270C4×C47—81no—
271C4×C47—81no—
272C4×C47—81no—
273C4×C47C2×#11711yescharacteristic
274C4×C47#51911yescharacteristic
275C4×C47#55211yescharacteristic
276C4×C47#55211yescharacteristic
277Dic47—22no—
278Dic47—41no—
279Dic47—41no—
280Dic47—41no—
281Dic47—42no—
282Dic47—42no—
283Dic47—42no—
284Dic47—42no—
285C2×#428—471no—
286#2346—471no—
287#2345—471no—
288#1943—471no—
289#1942—471no—
290#1921—471no—
291#1920—471no—
292#1919—471no—
293#1918—471no—
294#1825—471no—
295#1823—471no—
296#1809D4711yescharacteristic
297#1776—471no—
298C4×#106—471no—
299#1753—471no—
300C2×C47—42no—
301C2×C47#234411yescharacteristic
302C2×C47C4×#13711yescharacteristic
303C2×C47#294611yescharacteristic
304C2×C47#334411yescharacteristic
305C2×C47#334711yescharacteristic
306C2×C47#335111yescharacteristic
307C2×C47#335711yescharacteristic
308C22×M16C2×D4711yescharacteristic
309C2×#117—471no—
310C22×C4⋊C4—471no—
311#428—478no—
312C2×#107—471no—
313C2×#107—471no—
314C2×#107C2×D4711yescharacteristic
315C2×#106—471no—
316C2×#106—471no—
317C2×#106C2×D4711yescharacteristic
318C2×C22⋊C8—471no—
319C2×#97—471no—
320C2×C4×C8—471no—
321#389—471no—
322#389—471no—
323#389—471no—
324#389—471no—
325#388—941no—
326#388—941no—
327#387—941no—
328#387—941no—
329#382—941no—
330#380—941no—
331#380C2×D4711yescharacteristic
332#380C2×D4711yescharacteristic
333C4×C4⋊C4—471no—
334C4×C4⋊C4—471no—
335C4×C4⋊C4—471no—
336C4×C4⋊C4—471no—
337C4×C22⋊C4—471no—
338C4×C22⋊C4—941no—
339#376—941no—
340#340—471no—
341#340—471no—
342#340—471no—
343#340—471no—
344#340—471no—
345#340—471no—
346#340—471no—
347#340—471no—
348#340C2×D4711yescharacteristic
349#340C2×D4711yescharacteristic
350#336—471no—
351#336—471no—
352C47#2262111yescharacteristic, complemented, Sylow
353C23×C4—941no—
354C23×C4C22×D4711yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16C4×D4714yes—
358C22×C8—471no—
359C22×C8—471no—
360C22×C8C22×D4711yescharacteristic
361C22×C8C22×D4711yescharacteristic
362#117—474no—
363#117—942no—
364C2×C4⋊C4—474no—
365C2×C4⋊C4—471no—
366C2×C4⋊C4—471no—
367C2×C4⋊C4—471no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—941no—
370C2×C4⋊C4—942no—
371C2×C4⋊C4—941no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4C22×D4711yescharacteristic
376C2×C4⋊C4C22×D4711yescharacteristic
377C2×C4⋊C4C22×D4711yescharacteristic
378C2×C4⋊C4C22×D4711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—471no—
382C2×C22⋊C4—471no—
383C2×C22⋊C4—941no—
384C2×C42—471no—
385C2×C42—471no—
386C2×C42—941no—
387C2×C42—941no—
388C2×C42—941no—
389C2×C42—941no—
390C2×C42—941no—
391#107—22no—
392#107—474no—
393#107—474no—
394#106—22no—
395#106—474no—
396#106—474no—
397C22⋊C8—942no—
398#97—942no—
399C4×C8—942no—
400#95—21no—
401#95—21no—
402#95—471no—
403#95—471no—
404#95—941no—
405#95—941no—
406#95—941no—
407#95—941no—
408#95—941no—
409#95—941no—
410#95—941no—
411#95—941no—
412C24D4×D4711yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—471no—
422C22×C4—471no—
423C22×C4—471no—
424C22×C4—471no—
425C22×C4—944no—
426C22×C4—941no—
427C22×C4—941no—
428C22×C4—941no—
429C22×C4—941no—
430C22×C4—3761no—
431C22×C4—3761no—
432C22×C4C2×C4×D4712yes—
433C22×C4D4×D4711yescharacteristic
434C22×C4#1062535412yes—
435C22×C4C23×D4711yescharacteristic
436M16—28no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—941no—
440C2×C8—941no—
441C2×C8—941no—
442C2×C8—941no—
443C2×C8—941no—
444C2×C8—941no—
445C2×C8C2×C4×D4712yes—
446C2×C8C2×C4×D4712yes—
447C2×C8#1062535212yes—
448C2×C8#1062535212yes—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—472no—
458C4⋊C4—472no—
459C4⋊C4—472no—
460C4⋊C4—472no—
461C4⋊C4—944no—
462C4⋊C4—941no—
463C4⋊C4—941no—
464C4⋊C4—941no—
465C4⋊C4—941no—
466C4⋊C4—3761no—
467C4⋊C4—3761no—
468C4⋊C4—3761no—
469C4⋊C4—3761no—
470C22⋊C4—42no—
471C22⋊C4—81no—
472C22⋊C4—942no—
473C22⋊C4—944no—
474C22⋊C4—1881no—
475C42—944no—
476C42—941no—
477C42—941no—
478C42—941no—
479C42—941no—
480C42—3761no—
481C42—3761no—
482C42—1882no—
483C42—1882no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23#1309398612yes—
491C23C2×D4×D4711yescharacteristic
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—472no—
511C2×C4—941no—
512C2×C4—941no—
513C2×C4—941no—
514C2×C4—941no—
515C2×C4—941no—
516C2×C4—942no—
517C2×C4—942no—
518C2×C4—942no—
519C2×C4—942no—
520C2×C4—941no—
521C2×C4—941no—
522C2×C4—941no—
523C2×C4—941no—
524C2×C4—941no—
525C2×C4—941no—
526C2×C4—3764no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4#1309398612yes—
536C2×C4C22×C4×D4711yescharacteristic
537C2×C4C2×#1062535211yescharacteristic
538C2×C4C2×#1062535211yescharacteristic
539C2×C4C2×#1062535411yescharacteristic
540C2×C4#1309407111yescharacteristic
541C2×C4#1309407111yescharacteristic
542C8—22no—
543C8—1881no—
544C8—1881no—
545C8#1309398014yes—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22C2×#1309398611yescharacteristic
554C22#1568977311yescharacteristic
555C22#1568977311yescharacteristic
556C22#137×D4711yescharacteristic
557C22#1568991911yescharacteristic
558C22#1568992711yescharacteristic
559C22#1568993211yescharacteristic
560C4—22no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—942no—
566C4—1881no—
567C4—1882no—
568C4—1882no—
569C4—1882no—
570C4—1882no—
571C4—1881no—
572C4—1881no—
573C4C2×#1309398011yescharacteristic
574C4#1568960011yescharacteristic
575C4#1568981511yescharacteristic
576C4#1568981511yescharacteristic
577C2—42no—
578C2#2024625911yescharacteristic
579C2#2025220611yescharacteristic
580C2#2025221811yescharacteristic
581C2#2025583811yescharacteristic
582C2#2025588311yescharacteristic
583C2#2025594311yescharacteristic
584C2#2025600911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1809C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#1809D47
hypocenterC47#22621
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
4. #20244124C2semidirect
5. #20244119C2semidirect
6. #20243979C2semidirect
7. #20243972C2semidirect
8. #20243965C2semidirect
9. #20243958C2semidirect
12. #20243287C2semidirect
13. #106×Dic47C2semidirect
18. #15689165C4semidirect
352. C47#22621semidirect

Maximal subgroups

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

subgroupindex
C2×#156891652
#202462752
#202462672
#202441242
#202441192
#202439792
#202439722
#202439652
#202439582
#202434802
#202434722
#202432872
#106×Dic472
#202431692
C47×#18092
#2262147