Finite Group Databasebeta

← #26226849

Subgroups — #26226849

Order 12032.

830 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689179C211yesmaximal, characteristic
2#20245793C211yesmaximal, characteristic
3#20245792C211yesmaximal, characteristic
4#20244152C211yesmaximal, characteristic, complemented
5#20244151C211yesmaximal, characteristic, complemented
6#20244144C211yesmaximal, characteristic, complemented
7#20243588C212yesmaximal
8#20243584C212yesmaximal
9#20243555C212yesmaximal, complemented
10#20243520C212yesmaximal, complemented
11C47×#1947C211yesmaximal, characteristic, complemented
12C2×#13093982C2212yes—
13C2×#13093981—21no—
14C2×C42×D47C2211yescharacteristic
15#15689179C418yescomplemented
16C2×#13093957C2212yes—
17C2×#13093956C2212yes—
18C2×#13093955C2212yes—
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×(C2×D47)⋊C8C2212yes—
21C2×#13093934C2212yes—
22#15688993C2211yescharacteristic
23#15688992C2211yescharacteristic
24#15688992C2211yescharacteristic
25#15688988C2212yes—
26#15688987—21no—
27#15688982—21no—
28#15688980—21no—
29#15688980—21no—
30#15688980C2211yescharacteristic
31#15688980C2211yescharacteristic
32C4⋊C4×Dic47—21no—
33C4⋊C4×Dic47—21no—
34C4×Dic47⋊C4—21no—
35C4×C4⋊Dic47C2211yescharacteristic
36#15688886C2212yes—
37#15688885C2212yes—
38#15688885C2212yes—
39C2×#107×C47C2212yes—
40C2×C4⋊C8×C47C2211yescharacteristic
41C47×#384C2211yescharacteristic
42C4×C4⋊C4×C47C2211yescharacteristic
43C47×#340C2212yes—
44C22×C4×D47D411yescharacteristic
45C22×C4×D47D411yescharacteristic
46C22×C4×D47C2311yescharacteristic
47C22×C47⋊C8C2312yes—
48#13093982—42no—
49#13093982—42no—
50#13093981—44no—
51C2×C4⋊Dic47—21no—
52C2×C4⋊Dic47—21no—
53C2×C4⋊Dic47C2311yescharacteristic
54C2×C4⋊Dic47C2311yescharacteristic
55C2×C4⋊Dic47C2312yes—
56C2×Dic47⋊C4—21no—
57C2×Dic47⋊C4—21no—
58C2×Dic47⋊C4—21no—
59C2×Dic47⋊C4—21no—
60C2×(C2×D47)⋊C4—22no—
61C2×(C2×D47)⋊C4—21no—
62C2×(C2×D47)⋊C4—21no—
63C2×C4×Dic47—21no—
64C2×C4×Dic47—22no—
65C2×C4×Dic47D411yescharacteristic
66C2×C4×Dic47D411yescharacteristic
67C2×C4×Dic47C2311yescharacteristic
68C42×D47C2×C414yes—
69C42×D47Q814yes—
70#13093957—42no—
71#13093956—42no—
72#13093955—42no—
73C4⋊(C47⋊C8)C2×C414yes—
74(C2×D47)⋊C8C2×C418yes—
75#13093934C2×C418yes—
76C47⋊#95—21no—
77C47⋊#95—21no—
78C47⋊#95—21no—
79C47⋊#95—22no—
80C47⋊#95—22no—
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×C47—21no—
88C2×C42×C47C2311yescharacteristic
89#107×C47—42no—
90C4⋊C8×C47C2×C414yes—
91#95×C47—21no—
92C23×D47C2×D411yescharacteristic
93C22×C4×C47—21no—
94C22×C4×C47—21no—
95C22×C4×C47—22no—
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2411yescharacteristic
99C22×Dic47—21no—
100C22×Dic47—21no—
101C22×Dic47—22no—
102C22×Dic47C2×D411yescharacteristic
103C22×Dic47C2×D411yescharacteristic
104C22×Dic47C2×D411yescharacteristic
105C2×C4×D47—22no—
106C2×C4×D47—22no—
107C2×C4×D47—21no—
108C2×C4×D47—21no—
109C2×C4×D47—21no—
110C2×C4×D47—21no—
111C2×C4×D47—41no—
112C2×C4×D47—41no—
113C2×C4×D47C4⋊C414yes—
114C2×C4×D47C4⋊C414yes—
115C2×C4×D47C22×C412yes—
116C2×C4×D47C2×Q812yes—
117C2×C8×C47—22no—
118C2×C8×C47—22no—
119C2×C8×C47C22×C414yes—
120C4⋊C4×C47—21no—
121C4⋊C4×C47—22no—
122C4⋊C4×C47—21no—
123C4⋊C4×C47—22no—
124C4⋊C4×C47—21no—
125C4⋊C4×C47—21no—
126C4⋊C4×C47—42no—
127C4⋊C4×C47—42no—
128C42×C47—42no—
129C42×C47C22×C412yes—
130C42×C47C2×Q812yes—
131(C2×D47)⋊C4—42no—
132(C2×D47)⋊C4—42no—
133(C2×D47)⋊C4—44no—
134C4⋊Dic47—22no—
135C4⋊Dic47—21no—
136C4⋊Dic47—22no—
137C4⋊Dic47—21no—
138C4⋊Dic47—21no—
139C4⋊Dic47—21no—
140C4⋊Dic47—42no—
141C4⋊Dic47—42no—
142Dic47⋊C4—42no—
143Dic47⋊C4—42no—
144Dic47⋊C4—42no—
145Dic47⋊C4—42no—
146C4×Dic47—44no—
147C4×Dic47—42no—
148C4×Dic47C4⋊C414yes—
149C4×Dic47C4⋊C414yes—
150C4×Dic47C22×C412yes—
151C4×Dic47C2×Q812yes—
152C2×C47⋊C8—22no—
153C2×C47⋊C8—22no—
154C2×C47⋊C8C22×C414yes—
155C23×C47C22×D411yescharacteristic
156C22×D47—21no—
157C22×D47—21no—
158C22×D47—21no—
159C22×D47—21no—
160C22×D47—21no—
161C22×D47—21no—
162C22×D47C2×C4⋊C412yes—
163C2×C4×C47—21no—
164C2×C4×C47—21no—
165C2×C4×C47—42no—
166C2×C4×C47—41no—
167C2×C4×C47—41no—
168C2×C4×C47—41no—
169C2×C4×C47—41no—
170C2×C4×C47—42no—
171C2×C4×C47—41no—
172C2×C4×C47—41no—
173C2×C4×C47—42no—
174C2×C4×C47C2×C4⋊C412yes—
175C2×C4×C47C2×C4⋊C412yes—
176C2×C4×C47AGL(1,ℤ/8)12yes—
177C2×C4×C47#13712yes—
178C2×C4×C47C23×C411yescharacteristic
179C2×C4×C47C22×Q811yescharacteristic
180C2×C4×C47H(5,𝔽2)11yescharacteristic
181C2×C4×C47H(5,𝔽2)11yescharacteristic
182C2×C4×C472−1+411yescharacteristic
183C2×C4×C472−1+411yescharacteristic
184C2×Dic47—21no—
185C2×Dic47—21no—
186C2×Dic47—21no—
187C2×Dic47—21no—
188C2×Dic47—21no—
189C2×Dic47—21no—
190C2×Dic47—42no—
191C2×Dic47—41no—
192C2×Dic47—41no—
193C2×Dic47—41no—
194C2×Dic47—41no—
195C2×Dic47—42no—
196C2×Dic47—41no—
197C2×Dic47—41no—
198C2×Dic47—42no—
199C2×Dic47C2×C4⋊C412yes—
200C2×Dic47C2×C4⋊C412yes—
201C2×Dic47C2×C4⋊C412yes—
202C4×D47—22no—
203C4×D47—22no—
204C4×D47—22no—
205C4×D47—22no—
206C4×D47—28no—
207C4×D47—44no—
208C8×C47—24no—
209C47⋊C8—24no—
210#21988—471nomaximal, Sylow
211C22×C47C22×C4⋊C411yescharacteristic
212C22×C47#53511yescharacteristic
213C22×C47#53611yescharacteristic
214C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
215C22×C47C2×#13711yescharacteristic
216C22×C47#57511yescharacteristic
217C22×C47#57511yescharacteristic
218C2×D47—22no—
219C2×D47—22no—
220C2×D47—22no—
221C2×D47—41no—
222C2×D47—41no—
223C2×D47—41no—
224C2×D47—41no—
225C4×C47—22no—
226C4×C47—82no—
227C4×C47—81no—
228C4×C47—81no—
229C4×C47#42814yes—
230C4×C47#52011yescharacteristic
231C4×C47#52011yescharacteristic
232C4×C47#54411yescharacteristic
233C4×C47#54411yescharacteristic
234Dic47—22no—
235Dic47—22no—
236Dic47—22no—
237Dic47—81no—
238Dic47—82no—
239Dic47—81no—
240C2×#432—471no—
241#2285—471no—
242#2285—471no—
243#1948—471no—
244#1948—471no—
245#1947—471no—
246#1947D4711yescharacteristic, complemented
247#1854—472no—
248#1854—472no—
249#1847—472no—
250#1836—472no—
251C2×C47#231711yescharacteristic
252C2×C47C2×#42811yescharacteristic
253C2×C47#291111yescharacteristic
254C2×C47#321411yescharacteristic
255C2×C47#321611yescharacteristic
256C2×C47#323511yescharacteristic
257C2×C47#323711yescharacteristic
258D47—42no—
259C2×#117—472no—
260C2×#117—941no—
261C22×C42—471no—
262#432—478no—
263C2×#107—472no—
264C2×#107C2×D4712yes—
265C2×#106—472no—
266C2×#106—472no—
267C2×C4⋊C8—471no—
268C2×C4⋊C8C2×D4711yescharacteristic
269C2×C22⋊C8—472no—
270C2×#97—472no—
271#384—471no—
272#384—471no—
273#384—471no—
274#384C2×D4711yescharacteristic
275#382—472no—
276#382—941no—
277#380—471no—
278#380—471no—
279#380—941no—
280#380—941no—
281#380—941no—
282C4×C4⋊C4—471no—
283C4×C4⋊C4—941no—
284C4×C4⋊C4—941no—
285C4×C4⋊C4—941no—
286C4×C4⋊C4C2×D4711yescharacteristic
287#340—472no—
288#340—472no—
289#340—472no—
290#340C2×D4712yes—
291C47#2198811yescharacteristic, complemented, Sylow
292C23×C4—471no—
293C23×C4—471no—
294C23×C4—471no—
295C22×C8—472no—
296C22×C8C22×D4712yes—
297#117—1884no—
298#117—1882no—
299#117—1882no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—472no—
303C2×C4⋊C4—471no—
304C2×C4⋊C4—471no—
305C2×C4⋊C4—941no—
306C2×C4⋊C4—941no—
307C2×C4⋊C4—941no—
308C2×C4⋊C4—941no—
309C2×C4⋊C4—941no—
310C2×C4⋊C4—941no—
311C2×C4⋊C4C22×D4712yes—
312C2×C4⋊C4C22×D4711yescharacteristic
313C2×C4⋊C4C22×D4711yescharacteristic
314C2×C22⋊C4—942no—
315C2×C22⋊C4—941no—
316C2×C22⋊C4—941no—
317C2×C42—21no—
318C2×C42—471no—
319C2×C42—471no—
320C2×C42—471no—
321C2×C42—474no—
322C2×C42—474no—
323C2×C42—942no—
324C2×C42—941no—
325C2×C42C22×D4711yescharacteristic
326#107—42no—
327#107—1882no—
328#106—1882no—
329#106—1882no—
330C4⋊C8—474no—
331C4⋊C8C4×D4714yes—
332C22⋊C8—478no—
333#97—478no—
334#95—21no—
335#95—941no—
336#95—941no—
337#95—942no—
338#95—942no—
339#95—941no—
340C24—471no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—471no—
345C22×C4—471no—
346C22×C4—471no—
347C22×C4—474no—
348C22×C4—472no—
349C22×C4—472no—
350C22×C4—474no—
351C22×C4—942no—
352C22×C4—942no—
353C22×C4—941no—
354C22×C4—941no—
355C22×C4—941no—
356C22×C4—941no—
357C22×C4—941no—
358C22×C4—941no—
359C22×C4—942no—
360C22×C4—1881no—
361C22×C4—1881no—
362C22×C4D4×D4711yescharacteristic
363C22×C4D4×D4711yescharacteristic
364C22×C4C23×D4711yescharacteristic
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—474no—
368C2×C8—942no—
369C2×C8—942no—
370C2×C8C2×C4×D4714yes—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—941no—
380C4⋊C4—942no—
381C4⋊C4—942no—
382C4⋊C4—941no—
383C4⋊C4—941no—
384C4⋊C4—941no—
385C4⋊C4—1882no—
386C4⋊C4—1882no—
387C4⋊C4—1882no—
388C4⋊C4—1882no—
389C4⋊C4—1882no—
390C4⋊C4—1882no—
391C22⋊C4—1882no—
392C22⋊C4—1882no—
393C22⋊C4—1884no—
394C42—42no—
395C42—474no—
396C42—474no—
397C42—472no—
398C42—472no—
399C42—1884no—
400C42—1882no—
401C42C2×C4×D4712yes—
402C42Q8×D4712yes—
403C23—472no—
404C23—941no—
405C23—941no—
406C23—941no—
407C23—941no—
408C23—941no—
409C23—941no—
410C23C2×D4×D4711yescharacteristic
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—472no—
423C2×C4—472no—
424C2×C4—472no—
425C2×C4—941no—
426C2×C4—941no—
427C2×C4—941no—
428C2×C4—941no—
429C2×C4—941no—
430C2×C4—941no—
431C2×C4—942no—
432C2×C4—942no—
433C2×C4—942no—
434C2×C4—942no—
435C2×C4—948no—
436C2×C4—1884no—
437C2×C4—1881no—
438C2×C4—1881no—
439C2×C4—1882no—
440C2×C4—1881no—
441C2×C4—1881no—
442C2×C4—1882no—
443C2×C4—1882no—
444C2×C4—1881no—
445C2×C4—1881no—
446C2×C4C4⋊C4×D4712yes—
447C2×C4C4⋊C4×D4712yes—
448C2×C4#1309404812yes—
449C2×C4#1309405612yes—
450C2×C4C22×C4×D4711yescharacteristic
451C2×C4C2×Q8×D4711yescharacteristic
452C2×C4#1309406911yescharacteristic
453C2×C4#1309407011yescharacteristic
454C2×C4#1309407111yescharacteristic
455C2×C4#1309407211yescharacteristic
456C8—24no—
457C8—944no—
458C22—942no—
459C22—942no—
460C22—942no—
461C22—1881no—
462C22—1881no—
463C22—1881no—
464C22—1881no—
465C22C2×C4⋊C4×D4711yescharacteristic
466C22#1568968911yescharacteristic
467C22#1568969611yescharacteristic
468C22C2×#1309404811yescharacteristic
469C22C2×#1309405611yescharacteristic
470C22#1568991911yescharacteristic
471C22#1568992211yescharacteristic
472C4—22no—
473C4—81no—
474C4—81no—
475C4—82no—
476C4—942no—
477C4—942no—
478C4—942no—
479C4—3761no—
480C4—3761no—
481C4—3762no—
482C4#1568916414yes—
483C4#1568960211yescharacteristic
484C4#1568960511yescharacteristic
485C4#1568975011yescharacteristic
486C4#1568975311yescharacteristic
487C2—1882no—
488C2#2024601011yescharacteristic
489C2C2×#1568916411yescharacteristic
490C2#2025197511yescharacteristic
491C2#2025438111yescharacteristic
492C2#2025439811yescharacteristic
493C2#2025458511yescharacteristic
494C2#2025460011yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1947C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1947D47
hypocenterC47#21988
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
4. #20244152C2semidirect
5. #20244151C2semidirect
6. #20244144C2semidirect
9. #20243555C2semidirect
10. #20243520C2semidirect
11. C47×#1947C2semidirect
15. #15689179C4semidirect
246. #1947D47semidirect
291. C47#21988semidirect

Maximal subgroups

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

subgroupindex
C2×#156891792
#202457932
#202457922
#202441522
#202441512
#202441442
#202435882
#202435842
#202435552
#202435202
C47×#19472
#2198847