Finite Group Databasebeta

← #26228521

Subgroups — #26228521

Order 12032.

732 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×#15689423C211yesmaximal, characteristic
2#20246127C211yesmaximal, characteristic, complemented
3#20246031C211yesmaximal, characteristic
4#20244343C211yesmaximal, characteristic, complemented
5#20244144C211yesmaximal, characteristic, complemented
6#20244130C211yesmaximal, characteristic, complemented
7#20244120C211yesmaximal, characteristic, complemented
8#20244082C211yesmaximal, characteristic
9#20244068C211yesmaximal, characteristic
10#20244011C211yesmaximal, characteristic
11#20244011C211yesmaximal, characteristic
12#20243983C211yesmaximal, characteristic, complemented
13#20243969C211yesmaximal, characteristic, complemented
14#20243247C211yesmaximal, characteristic, complemented
15C47×#1802C211yesmaximal, characteristic
16C2×#13094016C2211yescharacteristic
17#15689423C418yescomplemented
18C2×#13093957C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×C4⋊(C47⋊C8)C2211yescharacteristic
23C2×#13093945C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25C2×#13093934C2211yescharacteristic
26C2×C8×Dic47C2211yescharacteristic
27#15689026C2211yescharacteristic
28#15689026C2211yescharacteristic
29#15689022C2211yescharacteristic
30#15689022C2211yescharacteristic
31#15689019C2211yescharacteristic
32#15689019C2211yescharacteristic
33#15689018—21no—
34#15689018—21no—
35#15689016—21no—
36#15689016—21no—
37#15689004—21no—
38#15689004—21no—
39#15688986—21no—
40#15688983—21no—
41C4⋊C4×Dic47C2211yescharacteristic
42C4⋊C4×Dic47C2211yescharacteristic
43C4×C4⋊Dic47C2211yescharacteristic
44#15688969C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C2×C4⋊C8×C47C2211yescharacteristic
55C2×#102×C47C2211yescharacteristic
56C2×#102×C47C2211yescharacteristic
57C47×#384C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62#13094016—22no—
63#13094016C2×C414yes—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47C2311yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×Dic47⋊C4—21no—
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×C22⋊Dic47—21no—
75C2×C22⋊Dic47—21no—
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47—21no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47C2311yescharacteristic
82C2×C4×Dic47C2311yescharacteristic
83#13093957—41no—
84#13093957—41no—
85#13093957C2×C414yes—
86#13093955C2×C414yes—
87C4⋊(C47⋊C8)C2×C414yes—
88#13093945C2×C414yes—
89#13093945C2×C414yes—
90#13093934—41no—
91C8×Dic47—41no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95D411yescharacteristic
100C47⋊#95D411yescharacteristic
101C47⋊#95D411yescharacteristic
102C47⋊#95D411yescharacteristic
103C22×D4×C47C2311yescharacteristic
104C23×C4×C47—21no—
105C22×C8×C47C2311yescharacteristic
106C22×C8×C47C2311yescharacteristic
107C4×D4×C47—22no—
108C4×D4×C47C2×C414yes—
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47—21no—
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C22⋊C4×C47—21no—
115C2×C42×C47C2311yescharacteristic
116C4⋊C8×C47—22no—
117#102×C47—41no—
118#102×C47—41no—
119C24×C47—21no—
120C2×D4×C47—21no—
121C2×D4×C47—21no—
122C2×D4×C47—21no—
123C2×D4×C47—21no—
124C2×D4×C47—21no—
125C2×D4×C47C22×C412yes—
126C22×C4×C47—21no—
127C22×C4×C47—24no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—81no—
131C22×C4×C47—81no—
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2411yescharacteristic
135C22×Dic47—21no—
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47C2×D411yescharacteristic
140C22×Dic47C2×D411yescharacteristic
141C22×Dic47C2×D411yescharacteristic
142C22×Dic47C2×D411yescharacteristic
143C2×C8×C47—21no—
144C2×C8×C47—21no—
145C2×C8×C47—21no—
146C2×C8×C47—21no—
147C2×C8×C47—21no—
148C2×C8×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—81no—
155C4⋊C4×C47—81no—
156C4⋊C4×C47C22×C412yes—
157C22⋊C4×C47—24no—
158C42×C47C22×C412yes—
159C42×C47#4112yes—
160C22⋊Dic47—81no—
161C22⋊Dic47—81no—
162C22⋊Dic47—81no—
163C22⋊Dic47—81no—
164C4⋊Dic47—21no—
165C4⋊Dic47—21no—
166C4⋊Dic47—21no—
167C4⋊Dic47—21no—
168C4⋊Dic47—21no—
169C4⋊Dic47—21no—
170C4⋊Dic47—81no—
171C4⋊Dic47—81no—
172C4⋊Dic47C22×C412yes—
173C4⋊Dic47C22×C412yes—
174Dic47⋊C4—24no—
175Dic47⋊C4—24no—
176Dic47⋊C4—81no—
177Dic47⋊C4—81no—
178C4×Dic47—21no—
179C4×Dic47—21no—
180C4×Dic47—21no—
181C4×Dic47—21no—
182C4×Dic47—81no—
183C4×Dic47—81no—
184C4×Dic47—81no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8C22×C412yes—
190C2×C47⋊C8C22×C412yes—
191C23×C47—22no—
192C23×C47—41no—
193C23×C47—41no—
194C23×C47—41no—
195C23×C47—41no—
196C23×C47—41no—
197C23×C47—41no—
198C23×C47C22×D411yescharacteristic
199D4×C47—22no—
200D4×C47—22no—
201D4×C47—41no—
202D4×C47—41no—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—22no—
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—84no—
217C2×C4×C47C4×D412yes—
218C2×C4×C47C4×D412yes—
219C2×C4×C47C23×C411yescharacteristic
220C2×C4×C47C2×#4111yescharacteristic
221C2×C4×C47H(5,𝔽2)11yescharacteristic
222C2×C4×C472−1+411yescharacteristic
223C2×C4×C472−1+411yescharacteristic
224C2×C4×C472−1+411yescharacteristic
225C2×Dic47—22no—
226C2×Dic47—22no—
227C2×Dic47—21no—
228C2×Dic47—21no—
229C2×Dic47—21no—
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—41no—
248C2×Dic47—41no—
249C8×C47—41no—
250C8×C47—41no—
251C47⋊C8—22no—
252C47⋊C8—22no—
253#22126—471nomaximal, Sylow
254C22×C47—42no—
255C22×C47—42no—
256C22×C47—42no—
257C22×C47—81no—
258C22×C47—81no—
259C22×C47—81no—
260C22×C47—81no—
261C22×C47C2×C4×D411yescharacteristic
262C22×C47#53611yescharacteristic
263C22×C47#53711yescharacteristic
264C22×C47#57611yescharacteristic
265C22×C47#57611yescharacteristic
266C22×C47#57711yescharacteristic
267C22×C47#57811yescharacteristic
268C4×C47—22no—
269C4×C47—22no—
270C4×C47—42no—
271C4×C47—81no—
272C4×C47—81no—
273C4×C47#51911yescharacteristic
274C4×C47#52011yescharacteristic
275C4×C47#54111yescharacteristic
276C4×C47#54211yescharacteristic
277Dic47—42no—
278Dic47—42no—
279Dic47—41no—
280Dic47—41no—
281Dic47—41no—
282Dic47—41no—
283Dic47—81no—
284Dic47—81no—
285C2×#478—471no—
286#2330—471no—
287#2319—471no—
288#1989—471no—
289#1947—471no—
290#1944—471no—
291#1942—471no—
292#1936—471no—
293#1934—471no—
294#1925—471no—
295#1925—471no—
296#1921—471no—
297#1919—471no—
298#1802D4711yescharacteristic
299#1768—471no—
300C2×C47—82no—
301C2×C47#232311yescharacteristic
302C2×C47#294611yescharacteristic
303C2×C47#295111yescharacteristic
304C2×C47#324111yescharacteristic
305C2×C47#324311yescharacteristic
306C2×C47#325811yescharacteristic
307C2×C47#325911yescharacteristic
308C2×#125—471no—
309C2×C4×D4C2×D4711yescharacteristic
310#478—478no—
311C2×#107—471no—
312C2×#107—471no—
313C2×#107—471no—
314C2×#106—471no—
315C2×C4⋊C8—471no—
316C2×C4⋊C8C2×D4711yescharacteristic
317C2×#102—471no—
318C2×#102—471no—
319C2×#102C2×D4711yescharacteristic
320C2×#102C2×D4711yescharacteristic
321C2×#97—471no—
322C2×C4×C8—471no—
323#390—471no—
324#390—471no—
325#389—471no—
326#389—471no—
327#389—471no—
328#389—471no—
329#389—941no—
330#389—941no—
331#388—941no—
332#388—941no—
333#387—941no—
334#387—941no—
335#384C2×D4711yescharacteristic
336#381—941no—
337#380—941no—
338C4×C4⋊C4—471no—
339C4×C4⋊C4—471no—
340C4×C4⋊C4—471no—
341#376—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#2212611yescharacteristic, complemented, Sylow
353C22×D4C22×D4711yescharacteristic
354C23×C4—21no—
355C22×C8—471no—
356C22×C8—471no—
357C22×C8C22×D4711yescharacteristic
358C22×C8C22×D4711yescharacteristic
359#125—474no—
360#125—942no—
361C4×D4—22no—
362C4×D4C4×D4714yes—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—471no—
366C2×C4⋊C4—471no—
367C2×C4⋊C4—471no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—941no—
370C2×C4⋊C4—941no—
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×C22⋊C4—21no—
379C2×C22⋊C4—941no—
380C2×C22⋊C4—941no—
381C2×C22⋊C4—941no—
382C2×C22⋊C4—941no—
383C2×C42—471no—
384C2×C42—471no—
385C2×C42—941no—
386C2×C42—941no—
387C2×C42—941no—
388C2×C42C22×D4711yescharacteristic
389#107—474no—
390#107—1881no—
391#107—1881no—
392#106—474no—
393C4⋊C8—22no—
394C4⋊C8—474no—
395#102—41no—
396#102—41no—
397#102—474no—
398#102—474no—
399#97—1881no—
400C4×C8—1881no—
401#95—471no—
402#95—471no—
403#95—471no—
404#95—471no—
405#95—941no—
406#95—941no—
407#95—941no—
408#95—941no—
409#95—941no—
410#95—941no—
411#95—941no—
412C24—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C2×C4×D4712yes—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—471no—
426C22×C4—471no—
427C22×C4—471no—
428C22×C4—471no—
429C22×C4—941no—
430C22×C4—941no—
431C22×C4—941no—
432C22×C4—941no—
433C22×C4D4×D4711yescharacteristic
434C22×C4D4×D4711yescharacteristic
435C22×C4C23×D4711yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—472no—
443C2×C8—472no—
444C2×C8—941no—
445C2×C8—941no—
446C2×C8—941no—
447C2×C8—941no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—472no—
456C4⋊C4—472no—
457C4⋊C4—944no—
458C4⋊C4—944no—
459C4⋊C4—941no—
460C4⋊C4—941no—
461C4⋊C4—941no—
462C4⋊C4—941no—
463C4⋊C4—941no—
464C4⋊C4—941no—
465C4⋊C4—3761no—
466C4⋊C4—3761no—
467C4⋊C4—3761no—
468C4⋊C4—3761no—
469C4⋊C4C2×C4×D4712yes—
470C22⋊C4—24no—
471C22⋊C4—3761no—
472C22⋊C4—3761no—
473C22⋊C4—3761no—
474C22⋊C4—3761no—
475C42—941no—
476C42—941no—
477C42—941no—
478C42—941no—
479C42—3761no—
480C42—3761no—
481C42—3761no—
482C42C2×C4×D4712yes—
483C42#1062535212yes—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C2×D4×D4711yescharacteristic
492D4—22no—
493D4—22no—
494D4—41no—
495D4—41no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
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—84no—
510C2×C4—942no—
511C2×C4—942no—
512C2×C4—941no—
513C2×C4—941no—
514C2×C4—941no—
515C2×C4—941no—
516C2×C4—941no—
517C2×C4—941no—
518C2×C4—941no—
519C2×C4—941no—
520C2×C4—941no—
521C2×C4—941no—
522C2×C4—941no—
523C2×C4—941no—
524C2×C4—1881no—
525C2×C4—1881no—
526C2×C4—1881no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4(C22⋊D47)⋊C412yes—
535C2×C4(C22⋊D47)⋊C412yes—
536C2×C4C22×C4×D4711yescharacteristic
537C2×C4C2×#1062535211yescharacteristic
538C2×C4#1309406911yescharacteristic
539C2×C4#1309407111yescharacteristic
540C2×C4#1309407111yescharacteristic
541C2×C4#1309407211yescharacteristic
542C8—41no—
543C8—41no—
544C8—942no—
545C8—942no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×(C22⋊D47)⋊C411yescharacteristic
554C22#1568969611yescharacteristic
555C22#1568970211yescharacteristic
556C22#1568992411yescharacteristic
557C22#1568992711yescharacteristic
558C22#1568992911yescharacteristic
559C22#1568993911yescharacteristic
560C4—22no—
561C4—22no—
562C4—42no—
563C4—81no—
564C4—81no—
565C4—3761no—
566C4—3761no—
567C4—1882no—
568C4—1881no—
569C4—1882no—
570C4—1881no—
571C4—1881no—
572C4—1881no—
573C4#1568960011yescharacteristic
574C4#1568960411yescharacteristic
575C4#1568973311yescharacteristic
576C4#1568974011yescharacteristic
577C2—82no—
578C2#2024606611yescharacteristic
579C2#2025221411yescharacteristic
580C2#2025225911yescharacteristic
581C2#2025464011yescharacteristic
582C2#2025466811yescharacteristic
583C2#2025481011yescharacteristic
584C2#2025482511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1802C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1802D47
hypocenterC47#22126
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. #20246127C2semidirect
4. #20244343C2semidirect
5. #20244144C2semidirect
6. #20244130C2semidirect
7. #20244120C2semidirect
12. #20243983C2semidirect
13. #20243969C2semidirect
14. #20243247C2semidirect
17. #15689423C4semidirect
352. C47#22126semidirect

Maximal subgroups

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

subgroupindex
C2×#156894232
#202461272
#202460312
#202443432
#202441442
#202441302
#202441202
#202440822
#202440682
#202440112
#202439832
#202439692
#202432472
C47×#18022
#2212647