Finite Group Databasebeta

← #26250248

Subgroups — #26250248

Order 12032.

1090 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689846C211yesmaximal, characteristic
2C2×#15689408C211yesmaximal, characteristic, complemented
3#20248306C211yesmaximal, characteristic
4#20244582C212yesmaximal, complemented
5#20244395C212yesmaximal, complemented
6#20244349C211yesmaximal, characteristic, complemented
7#20244182C211yesmaximal, characteristic
8#20243697C212yesmaximal, complemented
9#20243584C212yesmaximal, complemented
10#20243393C211yesmaximal, characteristic, complemented
11C2×C4×Q16×C47C211yesmaximal, characteristic, complemented
12#15689846—24no—
13#15689846—24no—
14C2×#13094024C2211yescharacteristic
15C2×#13094019—22no—
16C2×#13094012—21no—
17C2×#13094000C2212yes—
18C2×#13094000C2212yes—
19C2×C4×Dic94—21no—
20C2×C4×D188C2211yescharacteristic
21C2×#13093982C2212yes—
22#15689408—24no—
23C2×#13093955C2212yes—
24C2×C4⋊(C47⋊C8)C2211yescharacteristic
25C2×#13093948C2212yes—
26C2×#13093948C2212yes—
27C2×(C2×D47)⋊C8C2212yes—
28C2×(C2×D47)⋊C8C2212yes—
29#15689045—22no—
30#15689025—21no—
31#15689021—21no—
32#15688993C2211yescharacteristic
33#15688980C2212yes—
34#15688886C2212yes—
35#15688886C2212yes—
36#15688885C2212yes—
37C22×Q16×C47C2211yescharacteristic
38C2×C4×Q8×C47C2211yescharacteristic
39C2×C4×Q8×C47C2211yescharacteristic
40C4×Q16×C47—24no—
41C2×#107×C47C2211yescharacteristic
42C2×#103×C47C2211yescharacteristic
43C2×#103×C47C2211yescharacteristic
44C2×C4×C8×C47C2211yescharacteristic
45C22×Dic94—21no—
46C22×D188D411yescharacteristic
47C22×C4×D47C2312yes—
48C22×C47⋊C8C2312yes—
49#13094024—22no—
50#13094024D414yes—
51#13094019—216no—
52#13094012—28no—
53#13094000—24no—
54#13094000—24no—
55#13094000—24no—
56#13094000—42no—
57C4×Dic94—28no—
58C4×D188—22no—
59C4×D188—22no—
60#13093982—24no—
61#13093982—24no—
62C2×C4⋊Dic47—21no—
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47D411yescharacteristic
66C2×C4⋊Dic47C2312yes—
67C2×C4⋊Dic47C2312yes—
68C2×Dic47⋊C4—22no—
69C2×Dic47⋊C4—22no—
70C2×(C2×D47)⋊C4—22no—
71C2×(C2×D47)⋊C4—22no—
72C2×(C2×D47)⋊C4—22no—
73C2×(C2×D47)⋊C4—22no—
74C2×(C2×D47)⋊C4D412yes—
75C2×C4×Dic47—22no—
76#13093955—24no—
77C4⋊(C47⋊C8)—22no—
78#13093948—24no—
79#13093948—42no—
80(C2×D47)⋊C8—24no—
81(C2×D47)⋊C8—42no—
82C47⋊#95—22no—
83C22×Q8×C47C2311yescharacteristic
84C22×Q8×C47C2311yescharacteristic
85C2×Q16×C47—22no—
86C2×Q16×C47—22no—
87C2×Q16×C47—22no—
88C22×C8×C47C2311yescharacteristic
89C22×C8×C47C2311yescharacteristic
90C4×Q8×C47—22no—
91C4×Q8×C47—22no—
92C4×Q8×C47—22no—
93C4×Q8×C47D414yes—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47C2311yescharacteristic
97C2×C4⋊C4×C47C2311yescharacteristic
98C2×C42×C47—21no—
99C2×C42×C47—21no—
100C2×C42×C47C2311yescharacteristic
101#107×C47—22no—
102#103×C47—22no—
103#103×C47—22no—
104C4×C8×C47—22no—
105C23×D47C2×D412yes—
106C2×Q8×C47—21no—
107C2×Q8×C47—21no—
108C2×Q8×C47—21no—
109C2×Q8×C47—21no—
110C2×Q8×C47—21no—
111C2×Q8×C47—21no—
112C2×Q8×C47—21no—
113C2×Q8×C47—21no—
114C2×Q8×C47—21no—
115C2×Q8×C47—21no—
116C2×Q8×C47C2×D412yes—
117C2×Q8×C47#4112yes—
118C22×C4×C47—21no—
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2411yescharacteristic
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47C2×D412yes—
128C2×D188—22no—
129C2×D188—22no—
130C2×D188—22no—
131C2×C4×D47—22no—
132C2×C4×D47—22no—
133C2×C4×D47—22no—
134C2×C4×D47—22no—
135C2×C4×D47—22no—
136C2×C4×D47—22no—
137C2×Dic94—24no—
138C2×Dic94—42no—
139C2×Dic94—42no—
140Q16×C47—28no—
141C2×C8×C47—21no—
142C2×C8×C47—21no—
143C2×C8×C47—21no—
144C2×C8×C47—21no—
145C2×C8×C47—21no—
146C2×C8×C47#4112yes—
147C4⋊C4×C47—24no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—42no—
152C4⋊C4×C47C2×D412yes—
153C42×C47—24no—
154C42×C47—42no—
155C42×C47C2×D412yes—
156C42×C47C2×D412yes—
157(C2×D47)⋊C4—28no—
158(C2×D47)⋊C4—28no—
159(C2×D47)⋊C4—24no—
160(C2×D47)⋊C4—44no—
161(C2×D47)⋊C4—82no—
162C4⋊Dic47—24no—
163C4⋊Dic47—22no—
164C4⋊Dic47—22no—
165C4⋊Dic47—22no—
166C4⋊Dic47—22no—
167C4⋊Dic47—42no—
168C4⋊Dic47—81no—
169C4⋊Dic47C2×D414yes—
170Dic47⋊C4—28no—
171Dic47⋊C4—44no—
172C4×Dic47—28no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—22no—
176C23×C47C22×D411yescharacteristic
177C22×D47—22no—
178C22×D47—22no—
179C22×D47—22no—
180C22×D47—22no—
181C22×D47—22no—
182C22×D47—22no—
183C22×D47—22no—
184Q8×C47—22no—
185Q8×C47—22no—
186Q8×C47—22no—
187Q8×C47—22no—
188Q8×C47—41no—
189Q8×C47—41no—
190Q8×C47—41no—
191Q8×C47—41no—
192C2×C4×C47—21no—
193C2×C4×C47—21no—
194C2×C4×C47—22no—
195C2×C4×C47—22no—
196C2×C4×C47—22no—
197C2×C4×C47—41no—
198C2×C4×C47—41no—
199C2×C4×C47—41no—
200C2×C4×C47—41no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47—41no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47#12012yes—
207C2×C4×C47#12012yes—
208C2×C4×C47#13512yes—
209C2×C4×C47#13712yes—
210C2×C4×C47C22×D411yescharacteristic
211C2×C4×C47C22×D411yescharacteristic
212C2×C4×C47C2×#4111yescharacteristic
213C2×C4×C47C2×#4111yescharacteristic
214C2×C4×C47H(5,𝔽2)11yescharacteristic
215C2×C4×C472−1+411yescharacteristic
216C2×Dic47—22no—
217C2×Dic47—24no—
218C2×Dic47—22no—
219C2×Dic47—22no—
220C2×Dic47—42no—
221C2×Dic47—42no—
222C2×Dic47—42no—
223C2×Dic47—42no—
224C2×Dic47—42no—
225D188—44no—
226C4×D47—42no—
227C4×D47—42no—
228C4×D47—42no—
229C4×D47—42no—
230Dic94—48no—
231C8×C47—22no—
232C8×C47—41no—
233C47⋊C8—42no—
234#24123—471nomaximal, Sylow
235C22×C47C2×#12011yescharacteristic
236C22×C47#54611yescharacteristic
237C22×C47#54711yescharacteristic
238C22×C47C2×#13511yescharacteristic
239C22×C47C2×#13711yescharacteristic
240C22×C47#57711yescharacteristic
241C22×C47#57811yescharacteristic
242C2×D47—42no—
243C2×D47—42no—
244C2×D47—42no—
245C2×D47—42no—
246C2×D47—42no—
247C2×D47—42no—
248C2×D47—42no—
249C4×C47—22no—
250C4×C47—42no—
251C4×C47—42no—
252C4×C47—42no—
253C4×C47—81no—
254C4×C47#45214yes—
255C4×C47#52511yescharacteristic
256C4×C47#53711yescharacteristic
257C4×C47#54811yescharacteristic
258C4×C47#55011yescharacteristic
259Dic47—42no—
260Dic47—44no—
261Dic47—82no—
262C2×#556—471no—
263C2×#475—471no—
264C2×C4×Q16D4711yescharacteristic, complemented
265#2538—471no—
266#2035—472no—
267#2001—472no—
268#1990—471no—
269#1954—471no—
270#1873—472no—
271#1854—472no—
272#1803—471no—
273C2×C47#240711yescharacteristic
274C2×C47C2×#45211yescharacteristic
275C2×C47#301311yescharacteristic
276C2×C47#330411yescharacteristic
277C2×C47#332811yescharacteristic
278C2×C47#334211yescharacteristic
279C2×C47#335611yescharacteristic
280D47—82no—
281C22×Q16C2×D4711yescharacteristic
282#556—944no—
283#556—944no—
284C2×C4⋊Q8—471no—
285C2×#126—942no—
286C2×#124—941no—
287C2×C22⋊Q8—472no—
288C2×C22⋊Q8—472no—
289C2×C4×Q8—941no—
290C2×C4×Q8C2×D4711yescharacteristic
291C2×C4×Q8C2×D4711yescharacteristic
292C2×C4×D4—471no—
293C2×#117—472no—
294#475—944no—
295C4×Q16—24no—
296C2×#107C2×D4711yescharacteristic
297C2×#106—472no—
298C2×C4⋊C8—471no—
299C2×#103—472no—
300C2×#103—472no—
301C2×#103C2×D4711yescharacteristic
302C2×#103C2×D4711yescharacteristic
303C2×C22⋊C8—472no—
304C2×C22⋊C8—472no—
305C2×C4×C8C2×D4711yescharacteristic
306#396—942no—
307#390—941no—
308#389—941no—
309#384—471no—
310#380—472no—
311#340—472no—
312#340—472no—
313#340—472no—
314C47#2412311yescharacteristic, complemented, Sylow
315C22×Q8—941no—
316C22×Q8C22×D4711yescharacteristic
317C22×Q8C22×D4711yescharacteristic
318C22×D4—471no—
319C23×C4—472no—
320C2×Q16—22no—
321C2×Q16—22no—
322C2×Q16—22no—
323C22×C8—472no—
324C22×C8C22×D4711yescharacteristic
325C22×C8C22×D4711yescharacteristic
326C4⋊Q8—474no—
327C4⋊Q8—942no—
328#126—9416no—
329#124—948no—
330C22⋊Q8—944no—
331C22⋊Q8—944no—
332C22⋊Q8—944no—
333C22⋊Q8—1882no—
334C4×Q8—22no—
335C4×Q8—22no—
336C4×Q8—22no—
337C4×Q8—948no—
338C4×Q8C22⋊D4714yes—
339C4×D4—942no—
340C4×D4—942no—
341#117—944no—
342#117—944no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—472no—
346C2×C4⋊C4—471no—
347C2×C4⋊C4—472no—
348C2×C4⋊C4—942no—
349C2×C4⋊C4—941no—
350C2×C4⋊C4—941no—
351C2×C4⋊C4—942no—
352C2×C4⋊C4—941no—
353C2×C4⋊C4C22×D4711yescharacteristic
354C2×C4⋊C4C22×D4711yescharacteristic
355C2×C22⋊C4—472no—
356C2×C22⋊C4—942no—
357C2×C22⋊C4—942no—
358C2×C22⋊C4—942no—
359C2×C22⋊C4—942no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—942no—
363C2×C42C22×D4711yescharacteristic
364#107—22no—
365#106—944no—
366C4⋊C8—942no—
367#103—22no—
368#103—22no—
369#103—944no—
370#103—1882no—
371C22⋊C8—944no—
372C22⋊C8—1882no—
373C4×C8—22no—
374#95—942no—
375C24—472no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—944no—
387C2×Q8—1882no—
388C2×Q8—1882no—
389C2×Q8#1062535012yes—
390C2×Q8C2×C22⋊D4712yes—
391C2×D4—942no—
392C2×D4—942no—
393C2×D4—942no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—472no—
399C22×C4—942no—
400C22×C4—942no—
401C22×C4—942no—
402C22×C4—942no—
403C22×C4—942no—
404C22×C4—942no—
405C22×C4—942no—
406C22×C4—942no—
407C22×C4D4×D4711yescharacteristic
408C22×C4D4×D4711yescharacteristic
409C22×C4C23×D4711yescharacteristic
410Q16—28no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—942no—
417C2×C8—942no—
418C2×C8—942no—
419C2×C8#1062535012yes—
420C4⋊C4—24no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—42no—
425C4⋊C4—474no—
426C4⋊C4—948no—
427C4⋊C4—942no—
428C4⋊C4—944no—
429C4⋊C4—942no—
430C4⋊C4—942no—
431C4⋊C4—942no—
432C4⋊C4—3761no—
433C4⋊C4—1884no—
434C4⋊C4—1882no—
435C4⋊C4C2×C22⋊D4712yes—
436C22⋊C4—948no—
437C22⋊C4—948no—
438C22⋊C4—944no—
439C22⋊C4—3762no—
440C22⋊C4—1884no—
441C42—24no—
442C42—42no—
443C42—948no—
444C42D4×D4712yes—
445C42C2×C22⋊D4712yes—
446C23—942no—
447C23—942no—
448C23—942no—
449C23—942no—
450C23—942no—
451C23—942no—
452C23—942no—
453C23C2×D4×D4711yescharacteristic
454Q8—22no—
455Q8—22no—
456Q8—22no—
457Q8—22no—
458Q8—41no—
459Q8—41no—
460Q8—41no—
461Q8—41no—
462Q8—1888no—
463D4—1884no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—942no—
479C2×C4—942no—
480C2×C4—942no—
481C2×C4—944no—
482C2×C4—1882no—
483C2×C4—1882no—
484C2×C4—1882no—
485C2×C4—1882no—
486C2×C4—1882no—
487C2×C4—1882no—
488C2×C4—1882no—
489C2×C4—1882no—
490C2×C4—1882no—
491C2×C4#1309399212yes—
492C2×C4#1309399212yes—
493C2×C4#1309404712yes—
494C2×C4#1309405612yes—
495C2×C4C2×D4×D4711yescharacteristic
496C2×C4C22×C22⋊D4711yescharacteristic
497C2×C4C2×#1062535011yescharacteristic
498C2×C4C2×#1062535011yescharacteristic
499C2×C4#1309407011yescharacteristic
500C2×C4#1309407111yescharacteristic
501C8—22no—
502C8—41no—
503C8—1882no—
504C22—1882no—
505C22—1882no—
506C22—1882no—
507C22—1882no—
508C22—1882no—
509C22—1882no—
510C22—1882no—
511C22C2×#1309399211yescharacteristic
512C22#1568976911yescharacteristic
513C22#1568977811yescharacteristic
514C22C2×#1309404711yescharacteristic
515C22C2×#1309405611yescharacteristic
516C22#1568993011yescharacteristic
517C22#1568993711yescharacteristic
518C4—22no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—81no—
523C4—3762no—
524C4—1884no—
525C4—1882no—
526C4#1568928414yes—
527C4#1568963411yescharacteristic
528C4#1568970111yescharacteristic
529C4#1568978211yescharacteristic
530C4#1568979411yescharacteristic
531C2—3762no—
532C2#2024673311yescharacteristic
533C2C2×#1568928411yescharacteristic
534C2#2025269911yescharacteristic
535C2#2025532911yescharacteristic
536C2#2025567311yescharacteristic
537C2#2025582011yescharacteristic
538C2#2025600511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C4×Q16×C47C2
socleC23×C47C22×D4
prime radicalC22×D188D4
hypercenterC2×C4×Q16D47
hypocenterC47#24123
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#15689408C2semidirect
4. #20244582C2semidirect
5. #20244395C2semidirect
6. #20244349C2semidirect
8. #20243697C2semidirect
9. #20243584C2semidirect
10. #20243393C2semidirect
11. C2×C4×Q16×C47C2semidirect
264. C2×C4×Q16D47semidirect
314. C47#24123semidirect

Maximal subgroups

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

subgroupindex
C2×#156898462
C2×#156894082
#202483062
#202445822
#202443952
#202443492
#202441822
#202436972
#202435842
#202433932
C2×C4×Q16×C472
#2412347