Finite Group Databasebeta

← #26226799

Subgroups — #26226799

Order 12032.

966 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689175C211yesmaximal, characteristic
2#20245792C211yesmaximal, characteristic
3#384×D47C211yesmaximal, characteristic
4#20244147C211yesmaximal, characteristic, complemented
5#20244146C211yesmaximal, characteristic, complemented
6#20244144C211yesmaximal, characteristic, complemented
7#20243577C212yesmaximal
8#20243574C212yesmaximal
9#20243546C212yesmaximal, complemented
10#20243514C212yesmaximal, complemented
11C47×#1947C211yesmaximal, characteristic, complemented
12C2×#13093981—21no—
13C2×C4⋊C4×D47—21no—
14C2×C4⋊C4×D47—21no—
15C2×C4⋊C4×D47C2212yes—
16C2×C42×D47C2211yescharacteristic
17#15689175C418yescomplemented
18C2×#13093958C2212yes—
19C2×#13093957C2212yes—
20C2×#13093957C2212yes—
21C2×C4⋊(C47⋊C8)C2211yescharacteristic
22C2×(C2×D47)⋊C8C2212yes—
23C2×C8×Dic47C2212yes—
24#15688993C2211yescharacteristic
25#15688992—21no—
26#15688992—21no—
27#15688992C2212yes—
28#15688992C2211yescharacteristic
29#15688992C2211yescharacteristic
30#15688987—21no—
31#15688982—21no—
32#15688980C2211yescharacteristic
33#15688980C2211yescharacteristic
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×D47—22no—
45C22×C4×D47D411yescharacteristic
46C22×C4×D47D411yescharacteristic
47C22×C4×D47C2311yescharacteristic
48C22×C47⋊C8C2312yes—
49#13093981—44no—
50C2×C4⋊Dic47—21no—
51C2×C4⋊Dic47—21no—
52C2×C4⋊Dic47C2311yescharacteristic
53C2×C4⋊Dic47C2311yescharacteristic
54C2×C4⋊Dic47C2312yes—
55C2×Dic47⋊C4—21no—
56C2×Dic47⋊C4—21no—
57C2×Dic47⋊C4—21no—
58C2×Dic47⋊C4—21no—
59C2×Dic47⋊C4—22no—
60C2×Dic47⋊C4—22no—
61C2×Dic47⋊C4—22no—
62C4⋊C4×D47—42no—
63C4⋊C4×D47—42no—
64C4⋊C4×D47—44no—
65C4⋊C4×D47—44no—
66C2×(C2×D47)⋊C4—21no—
67C2×(C2×D47)⋊C4—21no—
68C2×C4×Dic47—21no—
69C2×C4×Dic47D411yescharacteristic
70C2×C4×Dic47D411yescharacteristic
71C2×C4×Dic47C2311yescharacteristic
72C42×D47C2×C414yes—
73C42×D47Q814yes—
74#13093958—42no—
75#13093957—42no—
76#13093957—42no—
77C4⋊(C47⋊C8)C2×C414yescomplemented
78(C2×D47)⋊C8C2×C418yes—
79C8×Dic47C2×C418yescomplemented
80C47⋊#95—21no—
81C47⋊#95—21no—
82C47⋊#95—21no—
83C22×C8×C47C2312yes—
84C2×C4⋊C4×C47—21no—
85C2×C4⋊C4×C47—21no—
86C2×C4⋊C4×C47C2311yescharacteristic
87C2×C4⋊C4×C47C2311yescharacteristic
88C2×C4⋊C4×C47C2312yes—
89C2×C42×C47—21no—
90C2×C42×C47C2311yescharacteristic
91#107×C47—42no—
92C4⋊C8×C47C2×C414yescomplemented
93#95×C47—21no—
94C23×D47C2×D411yescharacteristic
95C22×C4×C47—21no—
96C22×C4×C47—21no—
97C22×C4×C47—22no—
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47C2×D411yescharacteristic
100C22×C4×C47C2411yescharacteristic
101C22×Dic47—21no—
102C22×Dic47—21no—
103C22×Dic47—22no—
104C22×Dic47C2×D411yescharacteristic
105C22×Dic47C2×D411yescharacteristic
106C22×Dic47C2×D411yescharacteristic
107C2×C4×D47—21no—
108C2×C4×D47—21no—
109C2×C4×D47—21no—
110C2×C4×D47—21no—
111C2×C4×D47—22no—
112C2×C4×D47—22no—
113C2×C4×D47—44no—
114C2×C4×D47—44no—
115C2×C4×D47—44no—
116C2×C4×D47C4⋊C414yes—
117C2×C4×D47C4⋊C414yes—
118C2×C4×D47D814yes—
119C2×C4×D47Q1614yes—
120C2×C4×D47C22×C412yes—
121C2×C4×D47C2×Q812yes—
122C2×C8×C47—22no—
123C2×C8×C47—22no—
124C2×C8×C47C22×C414yes—
125C4⋊C4×C47—21no—
126C4⋊C4×C47—22no—
127C4⋊C4×C47—21no—
128C4⋊C4×C47—22no—
129C4⋊C4×C47—21no—
130C4⋊C4×C47—21no—
131C4⋊C4×C47—42no—
132C4⋊C4×C47—42no—
133C42×C47—42no—
134C42×C47C22×C412yes—
135C42×C47C2×Q812yes—
136(C2×D47)⋊C4—42no—
137(C2×D47)⋊C4—42no—
138C4⋊Dic47—21no—
139C4⋊Dic47—22no—
140C4⋊Dic47—21no—
141C4⋊Dic47—22no—
142C4⋊Dic47—21no—
143C4⋊Dic47—21no—
144C4⋊Dic47—42no—
145C4⋊Dic47—42no—
146Dic47⋊C4—44no—
147Dic47⋊C4—44no—
148Dic47⋊C4—42no—
149Dic47⋊C4—42no—
150Dic47⋊C4—44no—
151Dic47⋊C4—42no—
152Dic47⋊C4—42no—
153C4×Dic47—42no—
154C4×Dic47C4⋊C414yes—
155C4×Dic47C4⋊C414yes—
156C4×Dic47C22×C412yes—
157C4×Dic47C2×Q812yes—
158C2×C47⋊C8—22no—
159C2×C47⋊C8—22no—
160C2×C47⋊C8C22×C414yes—
161C23×C47C22×D411yescharacteristic
162C22×D47—21no—
163C22×D47—21no—
164C22×D47—21no—
165C22×D47—21no—
166C22×D47C2×C4⋊C412yes—
167C22×D47C2×D812yes—
168C22×D47C2×Q1612yes—
169C2×C4×C47—21no—
170C2×C4×C47—21no—
171C2×C4×C47—42no—
172C2×C4×C47—41no—
173C2×C4×C47—41no—
174C2×C4×C47—41no—
175C2×C4×C47—41no—
176C2×C4×C47—42no—
177C2×C4×C47—42no—
178C2×C4×C47—41no—
179C2×C4×C47—41no—
180C2×C4×C47C2×C4⋊C412yes—
181C2×C4×C47C2×C4⋊C412yes—
182C2×C4×C47C2×D812yes—
183C2×C4×C47C2×Q1612yes—
184C2×C4×C47C23×C411yescharacteristic
185C2×C4×C47C22×Q811yescharacteristic
186C2×C4×C47H(5,𝔽2)11yescharacteristic
187C2×C4×C47H(5,𝔽2)11yescharacteristic
188C2×C4×C472−1+411yescharacteristic
189C2×C4×C472−1+411yescharacteristic
190C2×Dic47—21no—
191C2×Dic47—21no—
192C2×Dic47—42no—
193C2×Dic47—41no—
194C2×Dic47—41no—
195C2×Dic47—41no—
196C2×Dic47—41no—
197C2×Dic47—42no—
198C2×Dic47—42no—
199C2×Dic47—41no—
200C2×Dic47—41no—
201C2×Dic47C2×C4⋊C412yes—
202C2×Dic47C2×C4⋊C412yes—
203C2×Dic47C2×C4⋊C412yes—
204C2×Dic47C2×D812yes—
205C2×Dic47C2×Q1612yes—
206C2×Dic47AGL(1,ℤ/8)12yes—
207C2×Dic47#13712yes—
208C4×D47—22no—
209C4×D47—22no—
210C4×D47—22no—
211C4×D47—22no—
212C4×D47—28no—
213C4×D47—88no—
214C4×D47#107116yescomplemented
215C8×C47—24no—
216C47⋊C8—24no—
217#21982—471nomaximal, Sylow
218C22×C47C22×C4⋊C411yescharacteristic
219C22×C47#53511yescharacteristic
220C22×C47#53611yescharacteristic
221C22×C47C22×D811yescharacteristic
222C22×C47C22×Q1611yescharacteristic
223C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
224C22×C47C2×#13711yescharacteristic
225C2×D47—22no—
226C2×D47—22no—
227C2×D47—22no—
228C2×D47—22no—
229C2×D47—22no—
230C2×D47—22no—
231C2×D47C2×#10714yes—
232C4×C47—22no—
233C4×C47—82no—
234C4×C47—81no—
235C4×C47—81no—
236C4×C47C2×#10714yes—
237C4×C47#52011yescharacteristic
238C4×C47#52011yescharacteristic
239C4×C47#54411yescharacteristic
240C4×C47#54411yescharacteristic
241Dic47—22no—
242Dic47—82no—
243Dic47—81no—
244Dic47—81no—
245Dic47C2×#10714yes—
246Dic47#42814yes—
247C2×#431—471no—
248#2285—471no—
249C2×#384—471no—
250#1947—471no—
251#1947—471no—
252#1947—471no—
253#1947D4711yescharacteristic, complemented
254#1852—472no—
255#1852—472no—
256#1845—472no—
257#1834—472no—
258C2×C47#231711yescharacteristic
259C2×C47C22×#10711yescharacteristic
260C2×C47C2×#42811yescharacteristic
261C2×C47#320511yescharacteristic
262C2×C47#320911yescharacteristic
263C2×C47#322611yescharacteristic
264C2×C47#323011yescharacteristic
265D47—24no—
266C2×#117—941no—
267C22×C4⋊C4—472no—
268C22×C4⋊C4—941no—
269C22×C4⋊C4—941no—
270C22×C42—471no—
271#431—478no—
272C2×#107—472no—
273C2×#107—472no—
274C2×#107—472no—
275C2×#107C2×D4712yes—
276C2×C4⋊C8—471no—
277C2×C4⋊C8C2×D4711yescharacteristic
278C2×C22⋊C8—472no—
279C2×C4×C8—472no—
280#384—471no—
281#384—472no—
282#384—471no—
283#384—471no—
284#384—941no—
285#384—941no—
286#384C2×D4711yescharacteristic
287#382—941no—
288#380—471no—
289#380—471no—
290#380—941no—
291C4×C4⋊C4—471no—
292C4×C4⋊C4—941no—
293C4×C4⋊C4C2×D4711yescharacteristic
294#340—472no—
295#340—472no—
296#340—472no—
297#340C2×D4712yes—
298C47#2198211yescharacteristic, complemented, Sylow
299C23×C4—471no—
300C23×C4—471no—
301C23×C4—471no—
302C23×C4—942no—
303C22×C8—472no—
304C22×C8C22×D4712yes—
305#117—1884no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—472no—
309C2×C4⋊C4—471no—
310C2×C4⋊C4—471no—
311C2×C4⋊C4—942no—
312C2×C4⋊C4—942no—
313C2×C4⋊C4—942no—
314C2×C4⋊C4—941no—
315C2×C4⋊C4—941no—
316C2×C4⋊C4—941no—
317C2×C4⋊C4—941no—
318C2×C4⋊C4—941no—
319C2×C4⋊C4—941no—
320C2×C4⋊C4—1882no—
321C2×C4⋊C4—1882no—
322C2×C4⋊C4—1884no—
323C2×C4⋊C4—1884no—
324C2×C4⋊C4C22×D4712yes—
325C2×C4⋊C4C22×D4711yescharacteristic
326C2×C4⋊C4C22×D4711yescharacteristic
327C2×C22⋊C4—941no—
328C2×C22⋊C4—941no—
329C2×C42—21no—
330C2×C42—471no—
331C2×C42—471no—
332C2×C42—471no—
333C2×C42—474no—
334C2×C42—474no—
335C2×C42—941no—
336C2×C42C22×D4711yescharacteristic
337#107—42no—
338#107—1882no—
339#107—1882no—
340#107—1882no—
341C4⋊C8—474no—
342C4⋊C8C4×D4714yescomplemented
343C22⋊C8—478no—
344C4×C8—478no—
345#95—21no—
346#95—941no—
347#95—941no—
348#95—941no—
349C24—471no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—471no—
354C22×C4—471no—
355C22×C4—474no—
356C22×C4—474no—
357C22×C4—474no—
358C22×C4—471no—
359C22×C4—474no—
360C22×C4—472no—
361C22×C4—472no—
362C22×C4—942no—
363C22×C4—942no—
364C22×C4—941no—
365C22×C4—941no—
366C22×C4—941no—
367C22×C4—941no—
368C22×C4—941no—
369C22×C4—941no—
370C22×C4—942no—
371C22×C4—1884no—
372C22×C4—1884no—
373C22×C4—1884no—
374C22×C4D4×D4711yescharacteristic
375C22×C4D4×D4711yescharacteristic
376C22×C4C23×D4711yescharacteristic
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—474no—
380C2×C8—942no—
381C2×C8—942no—
382C2×C8C2×C4×D4714yes—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—941no—
392C4⋊C4—942no—
393C4⋊C4—941no—
394C4⋊C4—942no—
395C4⋊C4—941no—
396C4⋊C4—941no—
397C4⋊C4—1882no—
398C4⋊C4—1882no—
399C4⋊C4—1882no—
400C4⋊C4—1884no—
401C4⋊C4—1884no—
402C4⋊C4—1882no—
403C4⋊C4—1882no—
404C4⋊C4—1884no—
405C4⋊C4—1882no—
406C22⋊C4—1882no—
407C22⋊C4—1882no—
408C42—42no—
409C42—474no—
410C42—474no—
411C42—472no—
412C42—472no—
413C42—1882no—
414C42C2×C4×D4712yes—
415C42Q8×D4712yes—
416C23—472no—
417C23—472no—
418C23—472no—
419C23—941no—
420C23—941no—
421C23—941no—
422C23—941no—
423C23C2×D4×D4711yescharacteristic
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—472no—
436C2×C4—472no—
437C2×C4—472no—
438C2×C4—472no—
439C2×C4—472no—
440C2×C4—472no—
441C2×C4—4716no—
442C2×C4—472no—
443C2×C4—948no—
444C2×C4—941no—
445C2×C4—941no—
446C2×C4—942no—
447C2×C4—942no—
448C2×C4—942no—
449C2×C4—942no—
450C2×C4—3768no—
451C2×C4—1881no—
452C2×C4—1881no—
453C2×C4—1882no—
454C2×C4—1882no—
455C2×C4—1881no—
456C2×C4—1881no—
457C2×C4—1882no—
458C2×C4—1881no—
459C2×C4—1881no—
460C2×C4C4⋊C4×D4712yes—
461C2×C4C4⋊C4×D4712yes—
462C2×C4D8×D4712yes—
463C2×C4Q16×D4712yes—
464C2×C4C22×C4×D4711yescharacteristic
465C2×C4C2×Q8×D4711yescharacteristic
466C2×C4#1309406911yescharacteristic
467C2×C4#1309407011yescharacteristic
468C2×C4#1309407111yescharacteristic
469C2×C4#1309407211yescharacteristic
470C8—24no—
471C8—944no—
472C22—474no—
473C22—942no—
474C22—942no—
475C22—942no—
476C22—942no—
477C22—942no—
478C22—942no—
479C22C2×C4⋊C4×D4711yescharacteristic
480C22#1568968911yescharacteristic
481C22#1568969611yescharacteristic
482C22C2×D8×D4711yescharacteristic
483C22C2×Q16×D4711yescharacteristic
484C22AGL(1,ℤ/8)×D4711yescharacteristic
485C22#137×D4711yescharacteristic
486C4—22no—
487C4—81no—
488C4—82no—
489C4—81no—
490C4—474no—
491C4—474no—
492C4—942no—
493C4—3761no—
494C4—3762no—
495C4—3761no—
496C4#107×D4714yes—
497C4#1568960211yescharacteristic
498C4#1568960511yescharacteristic
499C4#1568975011yescharacteristic
500C4#1568975311yescharacteristic
501C2—944no—
502C2#2024601011yescharacteristic
503C2C2×#107×D4711yescharacteristic
504C2#428×D4711yescharacteristic
505C2#2025430511yescharacteristic
506C2#2025433511yescharacteristic
507C2#2025450811yescharacteristic
508C2#2025453811yescharacteristic
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#21982
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. #20244147C2semidirect
5. #20244146C2semidirect
6. #20244144C2semidirect
9. #20243546C2semidirect
10. #20243514C2semidirect
11. C47×#1947C2semidirect
17. #15689175C4semidirect
77. C4⋊(C47⋊C8)C2×C4semidirect
79. C8×Dic47C2×C4semidirect
92. C4⋊C8×C47C2×C4semidirect
214. C4×D47#107semidirect
253. #1947D47semidirect
298. C47#21982semidirect
342. C4⋊C8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156891752
#202457922
#384×D472
#202441472
#202441462
#202441442
#202435772
#202435742
#202435462
#202435142
C47×#19472
#2198247