Finite Group Databasebeta

← #26226838

Subgroups — #26226838

Order 12032.

902 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689178C211yesmaximal, characteristic
2#384×D47C211yesmaximal, characteristic
3#20245781C211yesmaximal, characteristic
4#20244148C212yesmaximal, complemented
5#20244143C212yesmaximal, complemented
6#20243579C212yesmaximal
7#20243574C212yesmaximal
8#20243557C211yesmaximal, characteristic, complemented
9#20243523C211yesmaximal, characteristic, complemented
10#20243522C211yesmaximal, characteristic, complemented
11C47×#1847C211yesmaximal, characteristic, complemented
12C2×C4⋊C4×D47—21no—
13C2×C4⋊C4×D47—21no—
14C2×C4⋊C4×D47C2212yes—
15C2×C42×D47C2211yescharacteristic
16#15689178C418yescomplemented
17C2×#13093957C2212yes—
18C2×#13093955C2212yes—
19C2×Dic47⋊C8C2212yes—
20C2×(C2×D47)⋊C8C2212yes—
21C2×#13093935C2211yescharacteristic
22#15688993C2211yescharacteristic
23#15688992—21no—
24#15688992—21no—
25#15688992C2212yes—
26#15688990—21no—
27#15688989C2211yescharacteristic
28#15688985—21no—
29#15688980C2211yescharacteristic
30#15688980C2211yescharacteristic
31C4⋊C4×Dic47C2211yescharacteristic
32C4⋊C4×Dic47C2211yescharacteristic
33#15688970—21no—
34#15688968—21no—
35#15688886C2212yes—
36#15688886C2212yes—
37#15688885C2212yes—
38#15688885C2212yes—
39C2×#107×C47C2212yes—
40C2×#106×C47C2212yes—
41C2×#97×C47C2211yescharacteristic
42C47×#384C2211yescharacteristic
43C47×#382C2211yescharacteristic
44C22×C4×D47—22no—
45C22×C4×D47D411yescharacteristic
46C22×C4×D47D411yescharacteristic
47C22×C4×D47C2311yescharacteristic
48C22×C47⋊C8C2312yes—
49C2×C4⋊Dic47—21no—
50C2×C4⋊Dic47—21no—
51C2×C4⋊Dic47C2311yescharacteristic
52C2×C4⋊Dic47C2311yescharacteristic
53C2×C4⋊Dic47C2312yes—
54C2×Dic47⋊C4—22no—
55C2×Dic47⋊C4—22no—
56C2×Dic47⋊C4—22no—
57C4⋊C4×D47—42no—
58C4⋊C4×D47—42no—
59C4⋊C4×D47—44no—
60C4⋊C4×D47—44no—
61C2×(C2×D47)⋊C4—21no—
62C2×(C2×D47)⋊C4—21no—
63C2×C4×Dic47—21no—
64C2×C4×Dic47—21no—
65C2×C4×Dic47D411yescharacteristic
66C2×C4×Dic47D411yescharacteristic
67C2×C4×Dic47C2311yescharacteristic
68C42×D47C2×C414yes—
69C42×D47Q814yes—
70#13093957—42no—
71#13093955—42no—
72Dic47⋊C8C2×C418yescomplemented
73(C2×D47)⋊C8C2×C418yes—
74#13093935C2×C414yescomplemented
75C47⋊#95—21no—
76C47⋊#95—21no—
77C47⋊#95—21no—
78C47⋊#95—21no—
79C47⋊#95—21no—
80C47⋊#95—21no—
81C22×C8×C47C2312yes—
82C2×C4⋊C4×C47—21no—
83C2×C4⋊C4×C47—21no—
84C2×C4⋊C4×C47C2311yescharacteristic
85C2×C4⋊C4×C47C2311yescharacteristic
86C2×C4⋊C4×C47C2312yes—
87C2×C42×C47C2311yescharacteristic
88#107×C47—42no—
89#106×C47—42no—
90#97×C47C2×C414yescomplemented
91#95×C47—21no—
92#95×C47—21no—
93C23×D47C2×D411yescharacteristic
94C22×C4×C47—22no—
95C22×C4×C47—21no—
96C22×C4×C47—21no—
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47C2411yescharacteristic
100C22×Dic47—21no—
101C22×Dic47—21no—
102C22×Dic47—22no—
103C22×Dic47C2×D411yescharacteristic
104C22×Dic47C2×D411yescharacteristic
105C22×Dic47C2×D411yescharacteristic
106C2×C4×D47—22no—
107C2×C4×D47—22no—
108C2×C4×D47—22no—
109C2×C4×D47—22no—
110C2×C4×D47—22no—
111C2×C4×D47—22no—
112C2×C4×D47—44no—
113C2×C4×D47—44no—
114C2×C4×D47—44no—
115C2×C4×D47C4⋊C414yes—
116C2×C4×D47C4⋊C414yes—
117C2×C4×D47C22×C412yes—
118C2×C4×D47C2×Q812yes—
119C2×C8×C47—22no—
120C2×C8×C47—22no—
121C2×C8×C47C22×C414yes—
122C4⋊C4×C47—22no—
123C4⋊C4×C47—22no—
124C4⋊C4×C47—22no—
125C4⋊C4×C47—22no—
126C4⋊C4×C47—42no—
127C4⋊C4×C47—42no—
128C42×C47C22×C412yes—
129C42×C47C2×Q812yes—
130(C2×D47)⋊C4—42no—
131(C2×D47)⋊C4—42no—
132C4⋊Dic47—22no—
133C4⋊Dic47—22no—
134C4⋊Dic47—22no—
135C4⋊Dic47—22no—
136C4⋊Dic47—42no—
137C4⋊Dic47—42no—
138Dic47⋊C4—44no—
139Dic47⋊C4—44no—
140Dic47⋊C4—44no—
141C4×Dic47—42no—
142C4×Dic47—42no—
143C4×Dic47C4⋊C414yes—
144C4×Dic47C4⋊C414yes—
145C4×Dic47C22×C412yes—
146C4×Dic47C2×Q812yes—
147C2×C47⋊C8—22no—
148C2×C47⋊C8—22no—
149C2×C47⋊C8C22×C414yes—
150C23×C47C22×D411yescharacteristic
151C22×D47—21no—
152C22×D47—21no—
153C22×D47—21no—
154C22×D47—21no—
155C22×D47C2×C4⋊C412yes—
156C22×D47AGL(1,ℤ/8)12yes—
157C22×D47#13712yes—
158C2×C4×C47—21no—
159C2×C4×C47—21no—
160C2×C4×C47—21no—
161C2×C4×C47—21no—
162C2×C4×C47—42no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
166C2×C4×C47—42no—
167C2×C4×C47—42no—
168C2×C4×C47—41no—
169C2×C4×C47—41no—
170C2×C4×C47—41no—
171C2×C4×C47C2×C4⋊C412yes—
172C2×C4×C47C2×C4⋊C412yes—
173C2×C4×C47C23×C411yescharacteristic
174C2×C4×C47C22×Q811yescharacteristic
175C2×C4×C47H(5,𝔽2)12yes—
176C2×C4×C472−1+412yes—
177C2×Dic47—21no—
178C2×Dic47—21no—
179C2×Dic47—21no—
180C2×Dic47—21no—
181C2×Dic47—42no—
182C2×Dic47—41no—
183C2×Dic47—41no—
184C2×Dic47—41no—
185C2×Dic47—42no—
186C2×Dic47—42no—
187C2×Dic47—41no—
188C2×Dic47—41no—
189C2×Dic47—41no—
190C2×Dic47C2×C4⋊C412yes—
191C2×Dic47C2×C4⋊C412yes—
192C2×Dic47C2×C4⋊C412yes—
193C2×Dic47AGL(1,ℤ/8)12yes—
194C2×Dic47#13712yes—
195C4×D47—28no—
196C4×D47—24no—
197C4×D47—24no—
198C4×D47—28no—
199C4×D47—88no—
200C8×C47—24no—
201C47⋊C8—24no—
202#21987—471nomaximal, Sylow
203C22×C47C22×C4⋊C411yescharacteristic
204C22×C47#53511yescharacteristic
205C22×C47#53611yescharacteristic
206C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
207C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
208C22×C47C2×#13711yescharacteristic
209C22×C47C2×#13711yescharacteristic
210C2×D47—22no—
211C2×D47—22no—
212C2×D47—22no—
213C2×D47—22no—
214C2×D47—22no—
215C2×D47—22no—
216C2×D47#42814yes—
217C4×C47—22no—
218C4×C47—22no—
219C4×C47—82no—
220C4×C47—81no—
221C4×C47—81no—
222C4×C47#52012yes—
223C4×C47#54412yes—
224Dic47—22no—
225Dic47—22no—
226Dic47—82no—
227Dic47—81no—
228Dic47—81no—
229Dic47#42814yes—
230C2×#432—471no—
231C2×#384—471no—
232#2283—471no—
233#1948—472no—
234#1947—472no—
235#1853—472no—
236#1852—472no—
237#1847—471no—
238#1847D4711yescharacteristic, complemented
239#1836—471no—
240#1836—471no—
241C2×C47#231711yescharacteristic
242C2×C47C2×#42811yescharacteristic
243C2×C47C2×#42811yescharacteristic
244C2×C47#321711yescharacteristic
245C2×C47#321811yescharacteristic
246C2×C47#323811yescharacteristic
247C2×C47#323911yescharacteristic
248D47—24no—
249C22×C4⋊C4—472no—
250C22×C4⋊C4—941no—
251C22×C4⋊C4—941no—
252C22×C42—471no—
253#432—478no—
254C2×#107—472no—
255C2×#107C2×D4712yes—
256C2×#106—472no—
257C2×#106C2×D4712yes—
258C2×C4⋊C8—472no—
259C2×C22⋊C8—472no—
260C2×#97—471no—
261C2×#97C2×D4711yescharacteristic
262#384—472no—
263#384—471no—
264#384—941no—
265#384—941no—
266#384C2×D4711yescharacteristic
267#383—941no—
268#382—471no—
269#382C2×D4711yescharacteristic
270#381—941no—
271#380—471no—
272#380—471no—
273C4×C4⋊C4—471no—
274C4×C4⋊C4—471no—
275C4×C22⋊C4—941no—
276#376—941no—
277#340—472no—
278#340—472no—
279#340—472no—
280#340—472no—
281C47#2198711yescharacteristic, complemented, Sylow
282C23×C4—471no—
283C23×C4—471no—
284C23×C4—471no—
285C23×C4—942no—
286C22×C8—472no—
287C22×C8C22×D4712yes—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—472no—
291C2×C4⋊C4—471no—
292C2×C4⋊C4—471no—
293C2×C4⋊C4—942no—
294C2×C4⋊C4—942no—
295C2×C4⋊C4—942no—
296C2×C4⋊C4—941no—
297C2×C4⋊C4—941no—
298C2×C4⋊C4—1882no—
299C2×C4⋊C4—1882no—
300C2×C4⋊C4—1884no—
301C2×C4⋊C4—1884no—
302C2×C4⋊C4C22×D4712yes—
303C2×C4⋊C4C22×D4711yescharacteristic
304C2×C4⋊C4C22×D4711yescharacteristic
305C2×C22⋊C4—941no—
306C2×C22⋊C4—941no—
307C2×C42—471no—
308C2×C42—471no—
309C2×C42—471no—
310C2×C42—474no—
311C2×C42—474no—
312C2×C42—941no—
313C2×C42—941no—
314C2×C42C22×D4711yescharacteristic
315#107—42no—
316#107—1882no—
317#106—42no—
318#106—1882no—
319C4⋊C8—478no—
320C22⋊C8—478no—
321#97—474no—
322#97C4×D4714yescomplemented
323#95—21no—
324#95—21no—
325#95—941no—
326#95—941no—
327#95—941no—
328#95—941no—
329#95—941no—
330#95—941no—
331C24—471no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—471no—
336C22×C4—471no—
337C22×C4—471no—
338C22×C4—474no—
339C22×C4—474no—
340C22×C4—472no—
341C22×C4—472no—
342C22×C4—942no—
343C22×C4—942no—
344C22×C4—942no—
345C22×C4—942no—
346C22×C4—942no—
347C22×C4—942no—
348C22×C4—941no—
349C22×C4—941no—
350C22×C4—942no—
351C22×C4—1884no—
352C22×C4—1884no—
353C22×C4—1884no—
354C22×C4D4×D4711yescharacteristic
355C22×C4D4×D4711yescharacteristic
356C22×C4C23×D4711yescharacteristic
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—474no—
360C2×C8—942no—
361C2×C8—942no—
362C2×C8C2×C4×D4714yes—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—942no—
370C4⋊C4—942no—
371C4⋊C4—942no—
372C4⋊C4—942no—
373C4⋊C4—1882no—
374C4⋊C4—1882no—
375C4⋊C4—1884no—
376C4⋊C4—1884no—
377C4⋊C4—1884no—
378C22⋊C4—1882no—
379C22⋊C4—1882no—
380C42—474no—
381C42—474no—
382C42—472no—
383C42—472no—
384C42—1882no—
385C42—1882no—
386C42C2×C4×D4712yes—
387C42Q8×D4712yes—
388C23—472no—
389C23—472no—
390C23—472no—
391C23—941no—
392C23—941no—
393C23—941no—
394C23—941no—
395C23C2×D4×D4711yescharacteristic
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—472no—
410C2×C4—472no—
411C2×C4—472no—
412C2×C4—472no—
413C2×C4—472no—
414C2×C4—948no—
415C2×C4—948no—
416C2×C4—941no—
417C2×C4—941no—
418C2×C4—941no—
419C2×C4—941no—
420C2×C4—944no—
421C2×C4—944no—
422C2×C4—3768no—
423C2×C4—1881no—
424C2×C4—1881no—
425C2×C4—1882no—
426C2×C4—1882no—
427C2×C4—1881no—
428C2×C4—1881no—
429C2×C4—1882no—
430C2×C4—1881no—
431C2×C4—1881no—
432C2×C4C4⋊C4×D4712yes—
433C2×C4C4⋊C4×D4712yes—
434C2×C4C22×C4×D4711yescharacteristic
435C2×C4C2×Q8×D4711yescharacteristic
436C2×C4#1309406912yes—
437C2×C4#1309407212yes—
438C8—24no—
439C8—944no—
440C22—474no—
441C22—942no—
442C22—942no—
443C22—942no—
444C22—942no—
445C22—942no—
446C22—942no—
447C22C2×C4⋊C4×D4711yescharacteristic
448C22#1568968611yescharacteristic
449C22#1568969811yescharacteristic
450C22AGL(1,ℤ/8)×D4711yescharacteristic
451C22AGL(1,ℤ/8)×D4711yescharacteristic
452C22#137×D4711yescharacteristic
453C22#137×D4711yescharacteristic
454C4—22no—
455C4—22no—
456C4—82no—
457C4—81no—
458C4—81no—
459C4—474no—
460C4—942no—
461C4—942no—
462C4—3761no—
463C4—3761no—
464C4—3762no—
465C4#1568960412yes—
466C4#1568975012yes—
467C2—944no—
468C2#2024600911yescharacteristic
469C2#428×D4711yescharacteristic
470C2#428×D4711yescharacteristic
471C2#2025440211yescharacteristic
472C2#2025441111yescharacteristic
473C2#2025461011yescharacteristic
474C2#2025461911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1847C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1847D47
hypocenterC47#21987
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
4. #20244148C2semidirect
5. #20244143C2semidirect
8. #20243557C2semidirect
9. #20243523C2semidirect
10. #20243522C2semidirect
11. C47×#1847C2semidirect
16. #15689178C4semidirect
72. Dic47⋊C8C2×C4semidirect
74. #13093935C2×C4semidirect
90. #97×C47C2×C4semidirect
238. #1847D47semidirect
281. C47#21987semidirect
322. #97C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156891782
#384×D472
#202457812
#202441482
#202441432
#202435792
#202435742
#202435572
#202435232
#202435222
C47×#18472
#2198747