Finite Group Databasebeta

← #26143197

Subgroups — #26143197

Order 12032.

1310 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689053C211yesmaximal, characteristic
2C2×#15689021C211yesmaximal, characteristic
3C2×#15689007C212yesmaximal
4#20241655C211yesmaximal, characteristic
5#20241648C212yesmaximal
6#20241644C211yesmaximal, characteristic
7#20241641C211yesmaximal, characteristic
8#20241638C212yesmaximal
9#20241621C212yesmaximal
10#20241615C211yesmaximal, characteristic
11C2×C4×C4⋊C4×C47C211yesmaximal, characteristic
12C22×C4⋊Dic47C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14C22×Dic47⋊C4C2212yes—
15C22×Dic47⋊C4C2212yes—
16#15689053C418yescomplemented
17#15689021—24no—
18#15689007—28no—
19C2×C47⋊#95C2212yes—
20C2×C47⋊#95C2212yes—
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2211yescharacteristic
30C2×C47⋊#95C2212yes—
31C22×C4⋊C4×C47C2211yescharacteristic
32C22×C4⋊C4×C47C2211yescharacteristic
33C22×C42×C47C2211yescharacteristic
34C22×C42×C47C2211yescharacteristic
35C22×C42×C47C2211yescharacteristic
36C4×C4⋊C4×C47—24no—
37C2×#95×C47C2211yescharacteristic
38C2×#95×C47C2211yescharacteristic
39C23×Dic47C2312yes—
40C23×Dic47C2312yes—
41C23×Dic47C2312yes—
42C23×Dic47C2312yes—
43C2×C4⋊Dic47—22no—
44C2×C4⋊Dic47—22no—
45C2×C4⋊Dic47—22no—
46C2×C4⋊Dic47—42no—
47C2×C4⋊Dic47C2×C414yes—
48C2×Dic47⋊C4—24no—
49C2×Dic47⋊C4—24no—
50C2×Dic47⋊C4—24no—
51C2×Dic47⋊C4—24no—
52C2×Dic47⋊C4—24no—
53C2×Dic47⋊C4C2×C418yes—
54C47⋊#95—24no—
55C47⋊#95—24no—
56C47⋊#95—24no—
57C47⋊#95—24no—
58C47⋊#95—24no—
59C47⋊#95—24no—
60C47⋊#95—22no—
61C47⋊#95—42no—
62C47⋊#95—42no—
63C47⋊#95—42no—
64C47⋊#95C2×C414yes—
65C47⋊#95C2×C418yes—
66C23×C4×C47C2311yescharacteristic
67C23×C4×C47C2311yescharacteristic
68C23×C4×C47C2311yescharacteristic
69C23×C4×C47C2311yescharacteristic
70C23×C4×C47C2311yescharacteristic
71C23×C4×C47C2311yescharacteristic
72C23×C4×C47C2311yescharacteristic
73C2×C4⋊C4×C47—22no—
74C2×C4⋊C4×C47—22no—
75C2×C4⋊C4×C47—22no—
76C2×C4⋊C4×C47—22no—
77C2×C4⋊C4×C47—22no—
78C2×C4⋊C4×C47C2×C414yes—
79C2×C42×C47—22no—
80C2×C42×C47—22no—
81C2×C42×C47—22no—
82C2×C42×C47—42no—
83C2×C42×C47—42no—
84C2×C42×C47D414yes—
85C2×C42×C47D414yes—
86#95×C47—22no—
87#95×C47—22no—
88C24×C47C2411yescharacteristic
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—22no—
94C22×C4×C47—22no—
95C22×C4×C47—22no—
96C22×C4×C47—22no—
97C22×C4×C47—21no—
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—21no—
102C22×C4×C47—21no—
103C22×C4×C47—21no—
104C22×C4×C47—21no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47C22×C412yes—
114C22×C4×C47C22×C412yes—
115C22×C4×C47C22×C412yes—
116C22×C4×C47C2×D412yes—
117C22×C4×C47C2×D412yes—
118C22×C4×C47C2×D412yes—
119C22×C4×C47C2×D412yes—
120C22×C4×C47C2×D412yes—
121C22×C4×C47C2×D412yes—
122C22×C4×C47#4112yes—
123C22×C4×C47#4112yes—
124C22×C4×C47#4112yes—
125C22×C4×C47#4112yes—
126C22×Dic47—22no—
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47C22×C414yes—
153C22×Dic47C22×C414yes—
154C4⋊C4×C47—28no—
155C4⋊C4×C47—28no—
156C42×C47—28no—
157C42×C47—44no—
158C42×C47—44no—
159C4⋊Dic47—28no—
160C4⋊Dic47—44no—
161Dic47⋊C4—216no—
162Dic47⋊C4—48no—
163C23×C47C23×C411yescharacteristic
164C23×C47C22×D411yescharacteristic
165C23×C47C22×D411yescharacteristic
166C23×C47C2×#4111yescharacteristic
167C23×C47C2×#4111yescharacteristic
168C23×C47C2×#4111yescharacteristic
169C23×C47C2×#4111yescharacteristic
170C23×C47H(5,𝔽2)12yes—
171C23×C472−1+412yes—
172C23×C472−1+412yes—
173C23×C472−1+412yes—
174C2×C4×C47—22no—
175C2×C4×C47—22no—
176C2×C4×C47—24no—
177C2×C4×C47—22no—
178C2×C4×C47—22no—
179C2×C4×C47—22no—
180C2×C4×C47—22no—
181C2×C4×C47—22no—
182C2×C4×C47—22no—
183C2×C4×C47—22no—
184C2×C4×C47—22no—
185C2×C4×C47—24no—
186C2×C4×C47—24no—
187C2×C4×C47—24no—
188C2×C4×C47—22no—
189C2×C4×C47—24no—
190C2×C4×C47—22no—
191C2×C4×C47—22no—
192C2×C4×C47—22no—
193C2×C4×C47—22no—
194C2×C4×C47—22no—
195C2×C4×C47—24no—
196C2×C4×C47—24no—
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—42no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47C4×D414yes—
209C2×C4×C47C4×D414yes—
210C2×C4×C47#12114yes—
211C2×Dic47—24no—
212C2×Dic47—24no—
213C2×Dic47—24no—
214C2×Dic47—24no—
215C2×Dic47—24no—
216C2×Dic47—24no—
217C2×Dic47—42no—
218C2×Dic47—42no—
219C2×Dic47—42no—
220C2×Dic47—42no—
221C2×Dic47—42no—
222C2×Dic47—42no—
223C2×Dic47—42no—
224C2×Dic47—42no—
225C2×Dic47—42no—
226C2×Dic47—42no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—42no—
239#14314—471nomaximal, Sylow
240C22×C47C2×C4×D411yescharacteristic
241C22×C47C2×C4×D411yescharacteristic
242C22×C47C4×#4111yescharacteristic
243C22×C47#51912yes—
244C22×C47#52012yes—
245C22×C47C2×#12111yescharacteristic
246C22×C47#52511yescharacteristic
247C22×C47#52911yescharacteristic
248C22×C47#53211yescharacteristic
249C22×C47#53411yescharacteristic
250C22×C47#53611yescharacteristic
251C22×C47#53611yescharacteristic
252C22×C47#53611yescharacteristic
253C22×C47#53612yes—
254C22×C47#53712yes—
255C22×C47#53912yes—
256C22×C47#53912yes—
257C22×C47#53912yes—
258C22×C47#54112yes—
259C22×C47#54112yes—
260C22×C47#54112yes—
261C22×C47#54112yes—
262C22×C47#54212yes—
263C4×C47—24no—
264C4×C47—24no—
265C4×C47—24no—
266C4×C47—42no—
267C4×C47—42no—
268C4×C47—42no—
269C4×C47—42no—
270Dic47—44no—
271Dic47—44no—
272Dic47—82no—
273Dic47—82no—
274C2×#398—471no—
275C2×#389—471no—
276C2×#387—472no—
277C2×C4×C4⋊C4D4711yescharacteristic
278#1445—471no—
279#1443—472no—
280#1442—471no—
281#1441—472no—
282#1441—471no—
283#1436—472no—
284#1434—471no—
285C2×C47C4×#12111yescharacteristic
286C2×C47#232211yescharacteristic
287C2×C47#232312yes—
288C2×C47#232711yescharacteristic
289C2×C47#233612yes—
290C2×C47#261811yescharacteristic
291C2×C47#262111yescharacteristic
292C2×C47#263411yescharacteristic
293C2×C47#263811yescharacteristic
294C2×C47#265512yes—
295C2×C47#266412yes—
296C22×C4⋊C4—472no—
297C22×C4⋊C4—471no—
298C22×C4⋊C4—472no—
299C22×C4⋊C4—471no—
300C22×C4⋊C4C2×D4711yescharacteristic
301C22×C4⋊C4C2×D4711yescharacteristic
302C22×C42C2×D4711yescharacteristic
303C22×C42C2×D4711yescharacteristic
304C22×C42C2×D4711yescharacteristic
305#398—478no—
306#389—944no—
307#387—948no—
308C4×C4⋊C4—24no—
309C2×#95—471no—
310C2×#95—472no—
311C2×#95—472no—
312C2×#95—471no—
313C2×#95—472no—
314C2×#95—472no—
315C2×#95—472no—
316C2×#95—472no—
317C2×#95—472no—
318C2×#95—472no—
319C2×#95—472no—
320C2×#95—472no—
321C2×#95C2×D4711yescharacteristic
322C2×#95C2×D4711yescharacteristic
323C47#1431411yescharacteristic, complemented, Sylow
324C23×C4—472no—
325C23×C4—472no—
326C23×C4—472no—
327C23×C4—472no—
328C23×C4C22×D4711yescharacteristic
329C23×C4C22×D4711yescharacteristic
330C23×C4C22×D4711yescharacteristic
331C23×C4C22×D4711yescharacteristic
332C23×C4C22×D4711yescharacteristic
333C23×C4C22×D4711yescharacteristic
334C23×C4C22×D4711yescharacteristic
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—478no—
341C2×C4⋊C4—474no—
342C2×C4⋊C4—944no—
343C2×C4⋊C4—944no—
344C2×C4⋊C4—942no—
345C2×C4⋊C4—942no—
346C2×C4⋊C4—944no—
347C2×C4⋊C4—944no—
348C2×C4⋊C4—944no—
349C2×C4⋊C4—942no—
350C2×C4⋊C4—1882no—
351C2×C4⋊C4C4×D4714yes—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—42no—
356C2×C42—42no—
357C2×C42D18814yes—
358C2×C42C22⋊D4714yes—
359#95—22no—
360#95—22no—
361#95—474no—
362#95—478no—
363#95—944no—
364#95—944no—
365#95—944no—
366#95—944no—
367#95—944no—
368#95—942no—
369#95—944no—
370#95—1882no—
371#95—1882no—
372#95—1882no—
373C24C23×D4711yescharacteristic
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—474no—
399C22×C4—474no—
400C22×C4—942no—
401C22×C4—942no—
402C22×C4—942no—
403C22×C4—942no—
404C22×C4—942no—
405C22×C4—942no—
406C22×C4—942no—
407C22×C4—942no—
408C22×C4—942no—
409C22×C4—942no—
410C22×C4—942no—
411C22×C4—942no—
412C22×C4—942no—
413C22×C4—942no—
414C22×C4—942no—
415C22×C4—942no—
416C22×C4—942no—
417C22×C4—942no—
418C22×C4—942no—
419C22×C4—942no—
420C22×C4—942no—
421C22×C4—942no—
422C22×C4—942no—
423C22×C4—942no—
424C22×C4—942no—
425C22×C4—942no—
426C22×C4C2×C4×D4712yes—
427C22×C4C2×C4×D4712yes—
428C22×C4C2×C4×D4712yes—
429C22×C4C2×D18812yes—
430C22×C4C2×D18812yes—
431C22×C4C2×D18812yes—
432C22×C4#1062535012yes—
433C22×C4#1062535012yes—
434C22×C4#1062535012yes—
435C22×C4#1062535012yes—
436C22×C4C2×C22⋊D4712yes—
437C22×C4C2×C22⋊D4712yes—
438C22×C4C2×C22⋊D4712yes—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—948no—
442C4⋊C4—9416no—
443C4⋊C4—1884no—
444C4⋊C4—1888no—
445C42—28no—
446C42—44no—
447C42—44no—
448C23C22×C4×D4711yescharacteristic
449C23C22×C22⋊D4711yescharacteristic
450C23C22×D18811yescharacteristic
451C23C2×#1062535011yescharacteristic
452C23C2×#1062535011yescharacteristic
453C23C2×#1062535011yescharacteristic
454C23C2×#1062535011yescharacteristic
455C23#1309406912yes—
456C23#1309407112yes—
457C23#1309407112yes—
458C23#1309407212yes—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—944no—
494C2×C4—944no—
495C2×C4—944no—
496C2×C4—944no—
497C2×C4—944no—
498C2×C4—944no—
499C2×C4—1882no—
500C2×C4—1882no—
501C2×C4—1882no—
502C2×C4—1882no—
503C2×C4—1882no—
504C2×C4—1882no—
505C2×C4—1882no—
506C2×C4—1882no—
507C2×C4—1882no—
508C2×C4—1882no—
509C2×C4—1882no—
510C2×C4—1882no—
511C2×C4—1882no—
512C2×C4—1882no—
513C2×C4—1882no—
514C2×C4—1882no—
515C2×C4—1882no—
516C2×C4—1882no—
517C2×C4—1882no—
518C2×C4—1882no—
519C2×C4—1882no—
520C2×C4—1882no—
521C2×C4C4×C22⋊D4714yes—
522C2×C4C4×D18814yes—
523C2×C4#1309399814yes—
524C22C2×C4×C22⋊D4711yescharacteristic
525C22C2×C4×D18811yescharacteristic
526C22C4×#1062535011yescharacteristic
527C22#1568959912yes—
528C22#1568960212yes—
529C22C2×#1309399811yescharacteristic
530C22#1568963411yescharacteristic
531C22#1568965711yescharacteristic
532C22#1568967311yescharacteristic
533C22#1568968211yescharacteristic
534C22#1568969411yescharacteristic
535C22#1568969411yescharacteristic
536C22#1568969711yescharacteristic
537C22#1568969912yes—
538C22#1568970412yes—
539C22#1568971512yes—
540C22#1568971512yes—
541C22#1568972012yes—
542C22#1568973112yes—
543C22#1568973112yes—
544C22#1568973112yes—
545C22#1568973512yes—
546C22#1568973812yes—
547C4—24no—
548C4—24no—
549C4—24no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—3762no—
555C4—3762no—
556C4—1884no—
557C4—1884no—
558C2C4×#1309399811yescharacteristic
559C2#2024605311yescharacteristic
560C2#2024607112yes—
561C2#2024610411yescharacteristic
562C2#2024618712yes—
563C2#2024913011yescharacteristic
564C2#2024915411yescharacteristic
565C2#2024926411yescharacteristic
566C2#2024930511yescharacteristic
567C2#2024947512yes—
568C2#2024954712yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C4×C4⋊C4×C47C2
socle, prime radical, cosocleC24×C47C24
hypercenterC2×C4×C4⋊C4D47
hypocenterC47#14314
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
16. #15689053C4semidirect
323. C47#14314semidirect

Maximal subgroups

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

subgroupindex
C2×#156890532
C2×#156890212
C2×#156890072
#202416552
#202416482
#202416442
#202416412
#202416382
#202416212
#202416152
C2×C4×C4⋊C4×C472
#1431447