Finite Group Databasebeta

← #26147004

Subgroups — #26147004

Order 12032.

1130 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689022C211yesmaximal, characteristic
2C2×#15689009C211yesmaximal, characteristic
3#20241656C211yesmaximal, characteristic
4#20241653C211yesmaximal, characteristic
5#20241653C212yesmaximal
6#20241650C212yesmaximal
7#20241645C212yesmaximal
8#20241645C212yesmaximal
9#20241641C211yesmaximal, characteristic
10#20241613C211yesmaximal, characteristic
11C2×C47×#400C211yesmaximal, characteristic
12C22×C4⋊Dic47C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14C22×C4⋊Dic47C2212yes—
15C22×C4⋊Dic47C2212yes—
16C22×C4×Dic47C2211yescharacteristic
17#15689022—24no—
18#15689009—24no—
19C2×C47⋊#95C2211yescharacteristic
20C2×C47⋊#95C2211yescharacteristic
21C2×C47⋊#95C2211yescharacteristic
22C2×C47⋊#95C2211yescharacteristic
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C22×C4⋊C4×C47C2211yescharacteristic
33C22×C4⋊C4×C47C2211yescharacteristic
34C47×#400C418yescomplemented
35C2×#95×C47C2211yescharacteristic
36C2×#95×C47C2212yes—
37C2×#95×C47C2212yes—
38C23×Dic47C2311yescharacteristic
39C23×Dic47C2312yes—
40C23×Dic47C2311yescharacteristic
41C23×Dic47C2312yes—
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2311yescharacteristic
44C2×C4⋊Dic47—22no—
45C2×C4⋊Dic47—22no—
46C2×C4⋊Dic47—24no—
47C2×C4⋊Dic47—42no—
48C2×C4⋊Dic47—42no—
49C2×C4⋊Dic47—42no—
50C2×C4⋊Dic47—42no—
51C2×C4⋊Dic47—42no—
52C2×C4⋊Dic47—42no—
53C2×C4⋊Dic47—42no—
54C2×C4×Dic47—42no—
55C2×C4×Dic47D414yes—
56C47⋊#95—22no—
57C47⋊#95—22no—
58C47⋊#95—22no—
59C47⋊#95—22no—
60C47⋊#95—22no—
61C47⋊#95—24no—
62C47⋊#95—42no—
63C47⋊#95—42no—
64C47⋊#95—42no—
65C47⋊#95—42no—
66C47⋊#95—42no—
67C47⋊#95—42no—
68C47⋊#95—42no—
69C23×C4×C47C2311yescharacteristic
70C23×C4×C47C2311yescharacteristic
71C23×C4×C47C2312yes—
72C23×C4×C47C2312yes—
73C23×C4×C47C2311yescharacteristic
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×C4⋊C4×C47C2×C414yes—
80#95×C47C2×C418yes—
81#95×C47C2×C418yes—
82#95×C47C2×C414yes—
83C24×C47C2411yescharacteristic
84C22×C4×C47—22no—
85C22×C4×C47—22no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—22no—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—22no—
92C22×C4×C47—21no—
93C22×C4×C47—22no—
94C22×C4×C47—22no—
95C22×C4×C47—22no—
96C22×C4×C47—22no—
97C22×C4×C47—22no—
98C22×C4×C47—22no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—22no—
102C22×C4×C47—21no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47C22×C412yes—
109C22×C4×C47C22×C412yes—
110C22×C4×C47C22×C414yes—
111C22×C4×C47C22×C414yes—
112C22×C4×C47C22×C412yes—
113C22×C4×C47C2×D412yes—
114C22×C4×C47#4112yes—
115C22×Dic47—21no—
116C22×Dic47—22no—
117C22×Dic47—22no—
118C22×Dic47—22no—
119C22×Dic47—21no—
120C22×Dic47—22no—
121C22×Dic47—21no—
122C22×Dic47—21no—
123C22×Dic47—21no—
124C22×Dic47—21no—
125C22×Dic47—22no—
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—21no—
139C22×Dic47—22no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47C2×D412yes—
148C22×Dic47C2×D412yes—
149C4⋊C4×C47—28no—
150C4⋊C4×C47—28no—
151C4⋊Dic47—44no—
152C4⋊Dic47—48no—
153C4⋊Dic47—44no—
154C4⋊Dic47—48no—
155C4×Dic47—44no—
156C23×C47C23×C411yescharacteristic
157C23×C47C22×D411yescharacteristic
158C23×C47C2×#4111yescharacteristic
159C23×C47H(5,𝔽2)12yes—
160C23×C47H(5,𝔽2)12yes—
161C23×C47H(5,𝔽2)11yescharacteristic
162C23×C472−1+412yes—
163C23×C472−1+411yescharacteristic
164C23×C472−1+412yes—
165C23×C472−1+411yescharacteristic
166C23×C472−1+411yescharacteristic
167C2×C4×C47—24no—
168C2×C4×C47—22no—
169C2×C4×C47—22no—
170C2×C4×C47—24no—
171C2×C4×C47—22no—
172C2×C4×C47—22no—
173C2×C4×C47—24no—
174C2×C4×C47—22no—
175C2×C4×C47—24no—
176C2×C4×C47—22no—
177C2×C4×C47—24no—
178C2×C4×C47—24no—
179C2×C4×C47—24no—
180C2×C4×C47—24no—
181C2×C4×C47—24no—
182C2×C4×C47—41no—
183C2×C4×C47—42no—
184C2×C4×C47—42no—
185C2×C4×C47—41no—
186C2×C4×C47—41no—
187C2×C4×C47—41no—
188C2×C4×C47—42no—
189C2×C4×C47—41no—
190C2×C4×C47—42no—
191C2×C4×C47—42no—
192C2×C4×C47—41no—
193C2×C4×C47—42no—
194C2×C4×C47—42no—
195C2×C4×C47—42no—
196C2×C4×C47—42no—
197C2×C4×C47C4×D414yes—
198C2×Dic47—22no—
199C2×Dic47—22no—
200C2×Dic47—22no—
201C2×Dic47—22no—
202C2×Dic47—22no—
203C2×Dic47—22no—
204C2×Dic47—42no—
205C2×Dic47—42no—
206C2×Dic47—42no—
207C2×Dic47—42no—
208C2×Dic47—41no—
209C2×Dic47—42no—
210C2×Dic47—42no—
211C2×Dic47—41no—
212C2×Dic47—41no—
213C2×Dic47—41no—
214C2×Dic47—41no—
215C2×Dic47—41no—
216C2×Dic47—42no—
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—
232#14629—471nomaximal, Sylow
233C22×C47C2×C4×D411yescharacteristic
234C22×C47#51911yescharacteristic
235C22×C47#52012yes—
236C22×C47#52011yescharacteristic
237C22×C47#52012yes—
238C22×C47#53411yescharacteristic
239C22×C47#53511yescharacteristic
240C22×C47#53611yescharacteristic
241C22×C47#53611yescharacteristic
242C22×C47#53611yescharacteristic
243C22×C47#53711yescharacteristic
244C22×C47#53912yes—
245C22×C47#54111yescharacteristic
246C22×C47#54212yes—
247C22×C47#54211yescharacteristic
248C22×C47#56012yes—
249C22×C47#56212yes—
250C22×C47#56312yes—
251C22×C47#56312yes—
252C22×C47#56312yes—
253C22×C47#56312yes—
254C22×C47#56312yes—
255C22×C47#56412yes—
256C4×C47—24no—
257C4×C47—42no—
258C4×C47—42no—
259C4×C47—44no—
260C4×C47—44no—
261Dic47—42no—
262Dic47—42no—
263Dic47—81no—
264Dic47—82no—
265Dic47—82no—
266Dic47—81no—
267C2×#400D4711yescharacteristic
268C2×#389—471no—
269C2×#387—471no—
270#1445—471no—
271#1444—471no—
272#1444—472no—
273#1443—472no—
274#1442—472no—
275#1442—472no—
276#1441—471no—
277#1434—471no—
278C2×C47#232211yescharacteristic
279C2×C47#232311yescharacteristic
280C2×C47#232611yescharacteristic
281C2×C47#238212yes—
282C2×C47#238312yes—
283C2×C47#281311yescharacteristic
284C2×C47#281711yescharacteristic
285C2×C47#282111yescharacteristic
286C2×C47#282211yescharacteristic
287C2×C47#283812yes—
288C2×C47#283912yes—
289C22×C4⋊C4—471no—
290C22×C4⋊C4—472no—
291C22×C4⋊C4—471no—
292C22×C4⋊C4—472no—
293C22×C4⋊C4C2×D4711yescharacteristic
294C22×C4⋊C4C2×D4711yescharacteristic
295C22×C42—471no—
296#400Dic4718yescomplemented
297#389—944no—
298#387—944no—
299C2×#95—471no—
300C2×#95—471no—
301C2×#95—471no—
302C2×#95—472no—
303C2×#95—471no—
304C2×#95—471no—
305C2×#95—472no—
306C2×#95—472no—
307C2×#95—472no—
308C2×#95—472no—
309C2×#95—472no—
310C2×#95—472no—
311C2×#95—472no—
312C2×#95C2×D4711yescharacteristic
313C2×#95C2×D4712yes—
314C2×#95C2×D4712yes—
315C47#1462911yescharacteristic, complemented, Sylow
316C23×C4—471no—
317C23×C4—471no—
318C23×C4—471no—
319C23×C4—472no—
320C23×C4—471no—
321C23×C4—472no—
322C23×C4C22×D4712yes—
323C23×C4C22×D4711yescharacteristic
324C23×C4C22×D4712yes—
325C23×C4C22×D4711yescharacteristic
326C23×C4C22×D4711yescharacteristic
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—942no—
332C2×C4⋊C4—944no—
333C2×C4⋊C4—942no—
334C2×C4⋊C4—1882no—
335C2×C4⋊C4—1882no—
336C2×C4⋊C4—1882no—
337C2×C4⋊C4—1882no—
338C2×C4⋊C4—1882no—
339C2×C4⋊C4—1882no—
340C2×C4⋊C4—1882no—
341C2×C4⋊C4C2×Dic4714yes—
342C2×C4⋊C4C2×Dic4714yes—
343C2×C42—474no—
344C2×C42—1882no—
345#95—942no—
346#95—942no—
347#95—942no—
348#95—944no—
349#95—942no—
350#95—942no—
351#95—1882no—
352#95—1882no—
353#95—1882no—
354#95—1882no—
355#95—1882no—
356#95—1882no—
357#95—1882no—
358#95C2×Dic4714yes—
359#95C2×Dic4718yes—
360#95C2×Dic4718yes—
361C24C23×D4711yescharacteristic
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—472no—
387C22×C4—472no—
388C22×C4—941no—
389C22×C4—942no—
390C22×C4—942no—
391C22×C4—942no—
392C22×C4—942no—
393C22×C4—941no—
394C22×C4—941no—
395C22×C4—941no—
396C22×C4—941no—
397C22×C4—941no—
398C22×C4—942no—
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×C4—942no—
408C22×C4—941no—
409C22×C4—942no—
410C22×C4—942no—
411C22×C4—942no—
412C22×C4—942no—
413C22×C4—941no—
414C22×C4—942no—
415C22×C4—942no—
416C22×C4—942no—
417C22×C4—941no—
418C22×C4—942no—
419C22×C4—941no—
420C22×C4D4×D4712yes—
421C22×C4#1062535212yes—
422C22×C4C22×Dic4714yes—
423C22×C4C22×Dic4712yes—
424C22×C4C22×Dic4712yes—
425C22×C4C22×Dic4714yes—
426C22×C4C22×Dic4712yes—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—1884no—
430C4⋊C4—1884no—
431C4⋊C4—1888no—
432C4⋊C4—1888no—
433C42—1884no—
434C23C23×Dic4711yescharacteristic
435C23C2×D4×D4711yescharacteristic
436C23C2×#1062535211yescharacteristic
437C23#1309407012yes—
438C23#1309407011yescharacteristic
439C23#1309407012yes—
440C23#1309407111yescharacteristic
441C23#1309407112yes—
442C23#1309407112yes—
443C23#1309407211yescharacteristic
444C23#1309407211yescharacteristic
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—942no—
476C2×C4—942no—
477C2×C4—942no—
478C2×C4—942no—
479C2×C4—942no—
480C2×C4—942no—
481C2×C4—1882no—
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—1882no—
492C2×C4—1882no—
493C2×C4—1882no—
494C2×C4—1882no—
495C2×C4—1882no—
496C2×C4—1881no—
497C2×C4—1882no—
498C2×C4—1881no—
499C2×C4—1881no—
500C2×C4—1881no—
501C2×C4—1881no—
502C2×C4—1881no—
503C2×C4—1882no—
504C2×C4—1882no—
505C2×C4—1882no—
506C2×C4—1882no—
507C2×C4—1882no—
508C2×C4—1882no—
509C2×C4D4×Dic4714yes—
510C22C2×D4×Dic4711yescharacteristic
511C22#1568960111yescharacteristic
512C22#1568960612yes—
513C22#1568960612yes—
514C22#1568960611yescharacteristic
515C22#1568968411yescharacteristic
516C22#1568969111yescharacteristic
517C22#1568969311yescharacteristic
518C22#1568969511yescharacteristic
519C22#1568969611yescharacteristic
520C22#1568970311yescharacteristic
521C22#1568971712yes—
522C22#1568973311yescharacteristic
523C22#1568974212yes—
524C22#1568974611yescharacteristic
525C22#1568986412yes—
526C22#1568987112yes—
527C22#1568987512yes—
528C22#1568987512yes—
529C22#1568987512yes—
530C22#1568987712yes—
531C22#1568987712yes—
532C22C47⋊#56412yes—
533C4—24no—
534C4—42no—
535C4—44no—
536C4—42no—
537C4—44no—
538C4—3761no—
539C4—3762no—
540C4—3761no—
541C4—3762no—
542C4—1882no—
543C4—1882no—
544C2#2024606011yescharacteristic
545C2#2024607211yescharacteristic
546C2#2024610111yescharacteristic
547C2#2024657412yes—
548C2#2024658012yes—
549C2#2025139311yescharacteristic
550C2#2025142411yescharacteristic
551C2#2025145311yescharacteristic
552C2#2025145711yescharacteristic
553C2#2025161712yes—
554C2#2025163212yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#400C2
socle, prime radical, cosocleC24×C47C24
hypercenterC2×#400D47
hypocenterC47#14629
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
34. C47×#400C4semidirect
296. #400Dic47semidirect
315. C47#14629semidirect

Maximal subgroups

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

subgroupindex
C2×#156890222
C2×#156890092
#202416562
#202416532
#202416502
#202416452
#202416412
#202416132
C2×C47×#4002
#1462947