Finite Group Databasebeta

← #26226855

Subgroups — #26226855

Order 12032.

786 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689178C211yesmaximal, characteristic
2#20245793C211yesmaximal, characteristic
3#20245781C211yesmaximal, characteristic
4#20244157C212yesmaximal, complemented
5#20244153C212yesmaximal, complemented
6#20243586C212yesmaximal
7#20243584C212yesmaximal
8#20243557C211yesmaximal, characteristic, complemented
9#20243523C211yesmaximal, characteristic, complemented
10#20243522C211yesmaximal, characteristic, complemented
11C47×#1847C211yesmaximal, characteristic, complemented
12C2×#13093982C2212yes—
13C2×C42×D47C2211yescharacteristic
14#15689178C418yescomplemented
15C2×#13093957C2212yes—
16C2×#13093955C2212yes—
17C2×Dic47⋊C8C2212yes—
18C2×(C2×D47)⋊C8C2212yes—
19C2×#13093935C2211yescharacteristic
20#15688993C2211yescharacteristic
21#15688990—21no—
22#15688989C2211yescharacteristic
23#15688988C2212yes—
24#15688985—21no—
25#15688980—21no—
26#15688980—21no—
27#15688980C2211yescharacteristic
28#15688980C2211yescharacteristic
29C4⋊C4×Dic47—21no—
30C4⋊C4×Dic47—21no—
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×D47D411yescharacteristic
45C22×C4×D47D411yescharacteristic
46C22×C4×D47C2311yescharacteristic
47C22×C47⋊C8C2312yes—
48#13093982—42no—
49#13093982—42no—
50C2×C4⋊Dic47—21no—
51C2×C4⋊Dic47—21no—
52C2×C4⋊Dic47C2311yescharacteristic
53C2×C4⋊Dic47C2311yescharacteristic
54C2×C4⋊Dic47C2312yes—
55C2×(C2×D47)⋊C4—21no—
56C2×(C2×D47)⋊C4—21no—
57C2×(C2×D47)⋊C4—22no—
58C2×C4×Dic47—21no—
59C2×C4×Dic47—21no—
60C2×C4×Dic47—22no—
61C2×C4×Dic47D411yescharacteristic
62C2×C4×Dic47D411yescharacteristic
63C2×C4×Dic47C2311yescharacteristic
64C42×D47C2×C414yes—
65C42×D47Q814yes—
66#13093957—42no—
67#13093955—42no—
68Dic47⋊C8C2×C418yes—
69(C2×D47)⋊C8C2×C418yes—
70#13093935C2×C414yes—
71C47⋊#95—21no—
72C47⋊#95—21no—
73C47⋊#95—21no—
74C47⋊#95—21no—
75C47⋊#95—22no—
76C47⋊#95—22no—
77C47⋊#95—21no—
78C47⋊#95—21no—
79C22×C8×C47C2312yes—
80C2×C4⋊C4×C47—21no—
81C2×C4⋊C4×C47—21no—
82C2×C4⋊C4×C47C2311yescharacteristic
83C2×C4⋊C4×C47C2311yescharacteristic
84C2×C4⋊C4×C47C2312yes—
85C2×C42×C47C2311yescharacteristic
86#107×C47—42no—
87#106×C47—42no—
88#97×C47C2×C414yes—
89#95×C47—21no—
90#95×C47—21no—
91C23×D47C2×D411yescharacteristic
92C22×C4×C47—22no—
93C22×C4×C47—21no—
94C22×C4×C47—21no—
95C22×C4×C47C2×D411yescharacteristic
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2411yescharacteristic
98C22×Dic47—21no—
99C22×Dic47—21no—
100C22×Dic47—22no—
101C22×Dic47C2×D411yescharacteristic
102C22×Dic47C2×D411yescharacteristic
103C22×Dic47C2×D411yescharacteristic
104C2×C4×D47—22no—
105C2×C4×D47—22no—
106C2×C4×D47—41no—
107C2×C4×D47—41no—
108C2×C4×D47—41no—
109C2×C4×D47—41no—
110C2×C4×D47C4⋊C414yes—
111C2×C4×D47C4⋊C414yes—
112C2×C4×D47C22×C412yes—
113C2×C4×D47C2×Q812yes—
114C2×C8×C47—22no—
115C2×C8×C47—22no—
116C2×C8×C47C22×C414yes—
117C4⋊C4×C47—22no—
118C4⋊C4×C47—22no—
119C4⋊C4×C47—22no—
120C4⋊C4×C47—22no—
121C4⋊C4×C47—42no—
122C4⋊C4×C47—42no—
123C42×C47C22×C412yes—
124C42×C47C2×Q812yes—
125(C2×D47)⋊C4—42no—
126(C2×D47)⋊C4—42no—
127(C2×D47)⋊C4—44no—
128C4⋊Dic47—22no—
129C4⋊Dic47—22no—
130C4⋊Dic47—22no—
131C4⋊Dic47—22no—
132C4⋊Dic47—42no—
133C4⋊Dic47—42no—
134C4×Dic47—42no—
135C4×Dic47—42no—
136C4×Dic47—44no—
137C4×Dic47C4⋊C414yes—
138C4×Dic47C4⋊C414yes—
139C4×Dic47C22×C412yes—
140C4×Dic47C2×Q812yes—
141C2×C47⋊C8—22no—
142C2×C47⋊C8—22no—
143C2×C47⋊C8C22×C414yes—
144C23×C47C22×D411yescharacteristic
145C22×D47—21no—
146C22×D47—21no—
147C22×D47—21no—
148C22×D47—21no—
149C22×D47—21no—
150C22×D47—21no—
151C22×D47C2×C4⋊C412yes—
152C2×C4×C47—21no—
153C2×C4×C47—21no—
154C2×C4×C47—21no—
155C2×C4×C47—21no—
156C2×C4×C47—42no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47—41no—
162C2×C4×C47—41no—
163C2×C4×C47—42no—
164C2×C4×C47—42no—
165C2×C4×C47C2×C4⋊C412yes—
166C2×C4×C47C2×C4⋊C412yes—
167C2×C4×C47C23×C411yescharacteristic
168C2×C4×C47C22×Q811yescharacteristic
169C2×C4×C47H(5,𝔽2)12yes—
170C2×C4×C472−1+412yes—
171C2×Dic47—21no—
172C2×Dic47—21no—
173C2×Dic47—21no—
174C2×Dic47—21no—
175C2×Dic47—21no—
176C2×Dic47—21no—
177C2×Dic47—42no—
178C2×Dic47—41no—
179C2×Dic47—41no—
180C2×Dic47—41no—
181C2×Dic47—41no—
182C2×Dic47—42no—
183C2×Dic47—42no—
184C2×Dic47—41no—
185C2×Dic47—41no—
186C2×Dic47C2×C4⋊C412yes—
187C2×Dic47C2×C4⋊C412yes—
188C2×Dic47C2×C4⋊C412yes—
189C4×D47—24no—
190C4×D47—24no—
191C4×D47—44no—
192C4×D47—44no—
193C8×C47—24no—
194C47⋊C8—24no—
195#21989—471nomaximal, Sylow
196C22×C47C22×C4⋊C411yescharacteristic
197C22×C47#53511yescharacteristic
198C22×C47#53611yescharacteristic
199C22×C47#57511yescharacteristic
200C22×C47#57511yescharacteristic
201C22×C47#57511yescharacteristic
202C22×C47#57511yescharacteristic
203C2×D47—22no—
204C2×D47—22no—
205C2×D47—22no—
206C2×D47—41no—
207C2×D47—41no—
208C2×D47—41no—
209C2×D47—41no—
210C4×C47—22no—
211C4×C47—22no—
212C4×C47—82no—
213C4×C47—81no—
214C4×C47—81no—
215C4×C47#52012yes—
216C4×C47#54412yes—
217Dic47—22no—
218Dic47—22no—
219Dic47—22no—
220Dic47—82no—
221Dic47—81no—
222Dic47—81no—
223C2×#432—471no—
224#2285—471no—
225#2283—471no—
226#1949—472no—
227#1949—472no—
228#1854—472no—
229#1854—472no—
230#1847—471no—
231#1847D4711yescharacteristic, complemented
232#1836—471no—
233#1836—471no—
234C2×C47#231711yescharacteristic
235C2×C47#291111yescharacteristic
236C2×C47#291111yescharacteristic
237C2×C47#321911yescharacteristic
238C2×C47#321911yescharacteristic
239C2×C47#324011yescharacteristic
240C2×C47#324011yescharacteristic
241D47—42no—
242C2×#117—472no—
243C22×C42—471no—
244#432—478no—
245C2×#107—472no—
246C2×#107C2×D4712yes—
247C2×#106—472no—
248C2×#106C2×D4712yes—
249C2×C4⋊C8—472no—
250C2×C22⋊C8—472no—
251C2×#97—471no—
252C2×#97C2×D4711yescharacteristic
253#384—471no—
254#384C2×D4711yescharacteristic
255#383—941no—
256#382—472no—
257#382—471no—
258#382C2×D4711yescharacteristic
259#381—941no—
260#380—471no—
261#380—471no—
262#380—941no—
263#380—941no—
264C4×C4⋊C4—471no—
265C4×C4⋊C4—471no—
266C4×C4⋊C4—941no—
267C4×C4⋊C4—941no—
268C4×C22⋊C4—941no—
269#376—941no—
270#340—472no—
271#340—472no—
272#340—472no—
273#340—472no—
274C47#2198911yescharacteristic, complemented, Sylow
275C23×C4—471no—
276C23×C4—471no—
277C23×C4—471no—
278C22×C8—472no—
279C22×C8C22×D4712yes—
280#117—1882no—
281#117—1882no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—472no—
285C2×C4⋊C4—471no—
286C2×C4⋊C4—471no—
287C2×C4⋊C4—941no—
288C2×C4⋊C4—941no—
289C2×C4⋊C4C22×D4712yes—
290C2×C4⋊C4C22×D4711yescharacteristic
291C2×C4⋊C4C22×D4711yescharacteristic
292C2×C22⋊C4—942no—
293C2×C22⋊C4—941no—
294C2×C22⋊C4—941no—
295C2×C42—471no—
296C2×C42—471no—
297C2×C42—471no—
298C2×C42—474no—
299C2×C42—474no—
300C2×C42—942no—
301C2×C42—941no—
302C2×C42—941no—
303C2×C42C22×D4711yescharacteristic
304#107—42no—
305#107—1882no—
306#106—42no—
307#106—1882no—
308C4⋊C8—478no—
309C22⋊C8—478no—
310#97—474no—
311#97C4×D4714yes—
312#95—21no—
313#95—21no—
314#95—941no—
315#95—941no—
316#95—941no—
317#95—941no—
318#95—942no—
319#95—942no—
320#95—941no—
321#95—941no—
322C24—471no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—471no—
327C22×C4—471no—
328C22×C4—471no—
329C22×C4—472no—
330C22×C4—472no—
331C22×C4—474no—
332C22×C4—474no—
333C22×C4—942no—
334C22×C4—942no—
335C22×C4—941no—
336C22×C4—941no—
337C22×C4—942no—
338C22×C4—1881no—
339C22×C4—1881no—
340C22×C4—1881no—
341C22×C4—1881no—
342C22×C4D4×D4711yescharacteristic
343C22×C4D4×D4711yescharacteristic
344C22×C4C23×D4711yescharacteristic
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—474no—
348C2×C8—942no—
349C2×C8—942no—
350C2×C8C2×C4×D4714yes—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—942no—
358C4⋊C4—942no—
359C4⋊C4—942no—
360C4⋊C4—942no—
361C4⋊C4—1882no—
362C4⋊C4—1882no—
363C22⋊C4—1882no—
364C22⋊C4—1882no—
365C22⋊C4—1884no—
366C42—474no—
367C42—474no—
368C42—472no—
369C42—472no—
370C42—1882no—
371C42—1882no—
372C42—1884no—
373C42C2×C4×D4712yes—
374C42Q8×D4712yes—
375C23—472no—
376C23—941no—
377C23—941no—
378C23—941no—
379C23—941no—
380C23—941no—
381C23—941no—
382C23C2×D4×D4711yescharacteristic
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—472no—
397C2×C4—472no—
398C2×C4—472no—
399C2×C4—941no—
400C2×C4—941no—
401C2×C4—941no—
402C2×C4—941no—
403C2×C4—941no—
404C2×C4—941no—
405C2×C4—944no—
406C2×C4—944no—
407C2×C4—1884no—
408C2×C4—1884no—
409C2×C4—1881no—
410C2×C4—1881no—
411C2×C4—1881no—
412C2×C4—1881no—
413C2×C4—1882no—
414C2×C4—1882no—
415C2×C4—1882no—
416C2×C4—1881no—
417C2×C4—1881no—
418C2×C4C4⋊C4×D4712yes—
419C2×C4C4⋊C4×D4712yes—
420C2×C4C22×C4×D4711yescharacteristic
421C2×C4C2×Q8×D4711yescharacteristic
422C2×C4#1309406912yes—
423C2×C4#1309407212yes—
424C8—24no—
425C8—944no—
426C22—942no—
427C22—942no—
428C22—942no—
429C22—1881no—
430C22—1881no—
431C22—1881no—
432C22—1881no—
433C22C2×C4⋊C4×D4711yescharacteristic
434C22#1568968611yescharacteristic
435C22#1568969811yescharacteristic
436C22#1568991911yescharacteristic
437C22#1568991911yescharacteristic
438C22#1568992211yescharacteristic
439C22#1568992211yescharacteristic
440C4—22no—
441C4—22no—
442C4—81no—
443C4—82no—
444C4—81no—
445C4—942no—
446C4—942no—
447C4—942no—
448C4—3761no—
449C4—3761no—
450C4—3762no—
451C4#1568960412yes—
452C4#1568975012yes—
453C2—1882no—
454C2#2024600911yescharacteristic
455C2#2025197511yescharacteristic
456C2#2025197511yescharacteristic
457C2#2025442011yescharacteristic
458C2#2025442211yescharacteristic
459C2#2025462911yescharacteristic
460C2#2025463111yescharacteristic
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#21989
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. #20244157C2semidirect
5. #20244153C2semidirect
8. #20243557C2semidirect
9. #20243523C2semidirect
10. #20243522C2semidirect
11. C47×#1847C2semidirect
14. #15689178C4semidirect
231. #1847D47semidirect
274. C47#21989semidirect

Maximal subgroups

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

subgroupindex
C2×#156891782
#202457932
#202457812
#202441572
#202441532
#202435862
#202435842
#202435572
#202435232
#202435222
C47×#18472
#2198947