Finite Group Databasebeta

← #26226839

Subgroups — #26226839

Order 12032.

902 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689179C211yesmaximal, characteristic
2#384×D47C211yesmaximal, characteristic
3#20245784C211yesmaximal, characteristic
4#20244151C211yesmaximal, characteristic, complemented
5#20244147C211yesmaximal, characteristic, complemented
6#20244144C211yesmaximal, characteristic, complemented
7#20243579C212yesmaximal
8#20243577C212yesmaximal
9#20243553C212yesmaximal, complemented
10#20243520C212yesmaximal, complemented
11C47×#1948C211yesmaximal, characteristic, complemented
12C2×C4⋊C4×D47—21no—
13C2×C4⋊C4×D47—21no—
14C2×C4⋊C4×D47C2212yes—
15C2×C42×D47C2211yescharacteristic
16#15689179C418yescomplemented
17C2×#13093958C2212yes—
18C2×#13093957C2212yes—
19C2×#13093955C2212yes—
20C2×C4⋊(C47⋊C8)C2211yescharacteristic
21C2×(C2×D47)⋊C8C2212yes—
22C2×#13093934C2212yes—
23#15688993C2211yescharacteristic
24#15688992—21no—
25#15688992—21no—
26#15688992C2212yes—
27#15688990—21no—
28#15688988C2211yescharacteristic
29#15688988C2211yescharacteristic
30#15688985—21no—
31#15688980C2211yescharacteristic
32#15688980C2211yescharacteristic
33C4×C4⋊Dic47C2211yescharacteristic
34C4×(C2×D47)⋊C4—21no—
35#15688969—21no—
36#15688886C2212yes—
37#15688885C2212yes—
38#15688885C2212yes—
39C2×#106×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—
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×Dic47D411yescharacteristic
65C2×C4×Dic47D411yescharacteristic
66C2×C4×Dic47C2311yescharacteristic
67C42×D47C2×C414yes—
68C42×D47Q814yes—
69#13093958—42no—
70#13093957—42no—
71#13093955—42no—
72C4⋊(C47⋊C8)C2×C414yescomplemented
73(C2×D47)⋊C8C2×C418yes—
74#13093934C2×C418yescomplemented
75C47⋊#95—21no—
76C47⋊#95—21no—
77C47⋊#95—21no—
78C47⋊#95—21no—
79C47⋊#95—21no—
80C47⋊#95—21no—
81C47⋊#95—21no—
82C22×C8×C47C2312yes—
83C2×C4⋊C4×C47—21no—
84C2×C4⋊C4×C47—21no—
85C2×C4⋊C4×C47C2311yescharacteristic
86C2×C4⋊C4×C47C2311yescharacteristic
87C2×C4⋊C4×C47C2312yes—
88C2×C42×C47—21no—
89C2×C42×C47C2311yescharacteristic
90#106×C47—42no—
91C4⋊C8×C47C2×C414yescomplemented
92#95×C47—21no—
93C23×D47C2×D411yescharacteristic
94C22×C4×C47—21no—
95C22×C4×C47—21no—
96C22×C4×C47—22no—
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—21no—
111C2×C4×D47—21no—
112C2×C4×D47—21no—
113C2×C4×D47—21no—
114C2×C4×D47—44no—
115C2×C4×D47—44no—
116C2×C4×D47—44no—
117C2×C4×D47C4⋊C414yes—
118C2×C4×D47C4⋊C414yes—
119C2×C4×D47C22×C412yes—
120C2×C4×D47C2×Q812yes—
121C2×C8×C47—22no—
122C2×C8×C47—22no—
123C2×C8×C47C22×C414yes—
124C4⋊C4×C47—21no—
125C4⋊C4×C47—22no—
126C4⋊C4×C47—21no—
127C4⋊C4×C47—22no—
128C4⋊C4×C47—21no—
129C4⋊C4×C47—21no—
130C4⋊C4×C47—42no—
131C4⋊C4×C47—42no—
132C42×C47—42no—
133C42×C47C22×C412yes—
134C42×C47C2×Q812yes—
135(C2×D47)⋊C4—42no—
136(C2×D47)⋊C4—42no—
137C4⋊Dic47—22no—
138C4⋊Dic47—21no—
139C4⋊Dic47—21no—
140C4⋊Dic47—22no—
141C4⋊Dic47—21no—
142C4⋊Dic47—21no—
143C4⋊Dic47—42no—
144C4⋊Dic47—42no—
145Dic47⋊C4—44no—
146Dic47⋊C4—44no—
147Dic47⋊C4—44no—
148C4×Dic47—42no—
149C4×Dic47C4⋊C414yes—
150C4×Dic47C4⋊C414yes—
151C4×Dic47C22×C412yes—
152C4×Dic47C2×Q812yes—
153C2×C47⋊C8—22no—
154C2×C47⋊C8—22no—
155C2×C47⋊C8C22×C414yes—
156C23×C47C22×D411yescharacteristic
157C22×D47—21no—
158C22×D47—21no—
159C22×D47—21no—
160C22×D47—21no—
161C22×D47C2×C4⋊C412yes—
162C22×D47AGL(1,ℤ/8)12yes—
163C22×D47#13712yes—
164C2×C4×C47—21no—
165C2×C4×C47—21no—
166C2×C4×C47—42no—
167C2×C4×C47—41no—
168C2×C4×C47—41no—
169C2×C4×C47—42no—
170C2×C4×C47—42no—
171C2×C4×C47—41no—
172C2×C4×C47—41no—
173C2×C4×C47—41no—
174C2×C4×C47—41no—
175C2×C4×C47C2×C4⋊C412yes—
176C2×C4×C47C2×C4⋊C412yes—
177C2×C4×C47AGL(1,ℤ/8)12yes—
178C2×C4×C47#13712yes—
179C2×C4×C47C23×C411yescharacteristic
180C2×C4×C47C22×Q811yescharacteristic
181C2×C4×C47H(5,𝔽2)11yescharacteristic
182C2×C4×C47H(5,𝔽2)11yescharacteristic
183C2×C4×C472−1+411yescharacteristic
184C2×C4×C472−1+411yescharacteristic
185C2×Dic47—21no—
186C2×Dic47—21no—
187C2×Dic47—21no—
188C2×Dic47—21no—
189C2×Dic47—21no—
190C2×Dic47—21no—
191C2×Dic47—42no—
192C2×Dic47—41no—
193C2×Dic47—41no—
194C2×Dic47—41no—
195C2×Dic47—41no—
196C2×Dic47—42no—
197C2×Dic47—42no—
198C2×Dic47—41no—
199C2×Dic47—41no—
200C2×Dic47C2×C4⋊C412yes—
201C2×Dic47C2×C4⋊C412yes—
202C2×Dic47C2×C4⋊C412yes—
203C4×D47—28no—
204C4×D47—22no—
205C4×D47—22no—
206C4×D47—22no—
207C4×D47—22no—
208C4×D47—28no—
209C4×D47—88no—
210C8×C47—24no—
211C47⋊C8—24no—
212#21987—471nomaximal, Sylow
213C22×C47C22×C4⋊C411yescharacteristic
214C22×C47#53511yescharacteristic
215C22×C47#53611yescharacteristic
216C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
217C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
218C22×C47C2×#13711yescharacteristic
219C22×C47C2×#13711yescharacteristic
220C2×D47—22no—
221C2×D47—22no—
222C2×D47—22no—
223C2×D47—22no—
224C2×D47—22no—
225C2×D47—22no—
226C2×D47#42814yes—
227C4×C47—22no—
228C4×C47—82no—
229C4×C47—81no—
230C4×C47—81no—
231C4×C47#42814yes—
232C4×C47#52011yescharacteristic
233C4×C47#52011yescharacteristic
234C4×C47#54411yescharacteristic
235C4×C47#54411yescharacteristic
236Dic47—22no—
237Dic47—22no—
238Dic47—22no—
239Dic47—82no—
240Dic47—81no—
241Dic47—81no—
242C2×#432—471no—
243C2×#384—471no—
244#2283—471no—
245#1948—471no—
246#1948D4711yescharacteristic, complemented
247#1947—471no—
248#1947—471no—
249#1853—472no—
250#1852—472no—
251#1847—472no—
252#1836—472no—
253C2×C47#231711yescharacteristic
254C2×C47C2×#42811yescharacteristic
255C2×C47C2×#42811yescharacteristic
256C2×C47#321711yescharacteristic
257C2×C47#321811yescharacteristic
258C2×C47#323811yescharacteristic
259C2×C47#323911yescharacteristic
260D47—24no—
261C22×C4⋊C4—472no—
262C22×C4⋊C4—941no—
263C22×C4⋊C4—941no—
264C22×C42—471no—
265#432—478no—
266C2×#107—472no—
267C2×#107—472no—
268C2×#106—472no—
269C2×#106C2×D4712yes—
270C2×C4⋊C8—471no—
271C2×C4⋊C8C2×D4711yescharacteristic
272C2×C22⋊C8—472no—
273C2×#97—472no—
274#384—472no—
275#384—471no—
276#384—941no—
277#384—941no—
278#384C2×D4711yescharacteristic
279#383—941no—
280#382—471no—
281#382—471no—
282#381—941no—
283#380—471no—
284#380—471no—
285C4×C4⋊C4—471no—
286C4×C4⋊C4C2×D4711yescharacteristic
287C4×C22⋊C4—941no—
288#376—941no—
289#340—472no—
290#340—472no—
291#340—472no—
292#340C2×D4712yes—
293C47#2198711yescharacteristic, complemented, Sylow
294C23×C4—471no—
295C23×C4—471no—
296C23×C4—471no—
297C23×C4—942no—
298C22×C8—472no—
299C22×C8C22×D4712yes—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—472no—
303C2×C4⋊C4—471no—
304C2×C4⋊C4—471no—
305C2×C4⋊C4—942no—
306C2×C4⋊C4—942no—
307C2×C4⋊C4—942no—
308C2×C4⋊C4—941no—
309C2×C4⋊C4—941no—
310C2×C4⋊C4—1882no—
311C2×C4⋊C4—1882no—
312C2×C4⋊C4—1884no—
313C2×C4⋊C4—1884no—
314C2×C4⋊C4C22×D4712yes—
315C2×C4⋊C4C22×D4711yescharacteristic
316C2×C4⋊C4C22×D4711yescharacteristic
317C2×C22⋊C4—941no—
318C2×C22⋊C4—941no—
319C2×C42—21no—
320C2×C42—471no—
321C2×C42—471no—
322C2×C42—471no—
323C2×C42—474no—
324C2×C42—474no—
325C2×C42—941no—
326C2×C42C22×D4711yescharacteristic
327#107—1882no—
328#107—1882no—
329#106—42no—
330#106—1882no—
331C4⋊C8—474no—
332C4⋊C8C4×D4714yescomplemented
333C22⋊C8—478no—
334#97—478no—
335#95—21no—
336#95—941no—
337#95—941no—
338#95—941no—
339#95—941no—
340#95—941no—
341#95—941no—
342#95—941no—
343C24—471no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—471no—
348C22×C4—471no—
349C22×C4—471no—
350C22×C4—474no—
351C22×C4—474no—
352C22×C4—472no—
353C22×C4—472no—
354C22×C4—942no—
355C22×C4—942no—
356C22×C4—942no—
357C22×C4—942no—
358C22×C4—941no—
359C22×C4—941no—
360C22×C4—941no—
361C22×C4—941no—
362C22×C4—941no—
363C22×C4—941no—
364C22×C4—942no—
365C22×C4—1884no—
366C22×C4—1884no—
367C22×C4—1884no—
368C22×C4D4×D4711yescharacteristic
369C22×C4D4×D4711yescharacteristic
370C22×C4C23×D4711yescharacteristic
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—474no—
374C2×C8—942no—
375C2×C8—942no—
376C2×C8C2×C4×D4714yes—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—941no—
386C4⋊C4—942no—
387C4⋊C4—942no—
388C4⋊C4—941no—
389C4⋊C4—941no—
390C4⋊C4—941no—
391C4⋊C4—1882no—
392C4⋊C4—1884no—
393C4⋊C4—1884no—
394C4⋊C4—1884no—
395C4⋊C4—1882no—
396C22⋊C4—1882no—
397C22⋊C4—1882no—
398C42—42no—
399C42—474no—
400C42—474no—
401C42—472no—
402C42—472no—
403C42—1882no—
404C42C2×C4×D4712yes—
405C42Q8×D4712yes—
406C23—472no—
407C23—472no—
408C23—472no—
409C23—941no—
410C23—941no—
411C23—941no—
412C23—941no—
413C23C2×D4×D4711yescharacteristic
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—472no—
426C2×C4—472no—
427C2×C4—472no—
428C2×C4—948no—
429C2×C4—941no—
430C2×C4—941no—
431C2×C4—941no—
432C2×C4—941no—
433C2×C4—941no—
434C2×C4—941no—
435C2×C4—942no—
436C2×C4—942no—
437C2×C4—942no—
438C2×C4—942no—
439C2×C4—948no—
440C2×C4—3768no—
441C2×C4—1881no—
442C2×C4—1882no—
443C2×C4—1881no—
444C2×C4—1881no—
445C2×C4—1882no—
446C2×C4—1881no—
447C2×C4—1882no—
448C2×C4—1881no—
449C2×C4—1881no—
450C2×C4C4⋊C4×D4712yes—
451C2×C4C4⋊C4×D4712yes—
452C2×C4#1309405112yes—
453C2×C4#1309405312yes—
454C2×C4C22×C4×D4711yescharacteristic
455C2×C4C2×Q8×D4711yescharacteristic
456C2×C4#1309406911yescharacteristic
457C2×C4#1309407011yescharacteristic
458C2×C4#1309407111yescharacteristic
459C2×C4#1309407211yescharacteristic
460C8—24no—
461C8—944no—
462C22—474no—
463C22—942no—
464C22—942no—
465C22—942no—
466C22—942no—
467C22—942no—
468C22—942no—
469C22C2×C4⋊C4×D4711yescharacteristic
470C22#1568968911yescharacteristic
471C22#1568969611yescharacteristic
472C22AGL(1,ℤ/8)×D4711yescharacteristic
473C22C2×#1309405111yescharacteristic
474C22#137×D4711yescharacteristic
475C22C2×#1309405311yescharacteristic
476C4—22no—
477C4—81no—
478C4—82no—
479C4—81no—
480C4—942no—
481C4—942no—
482C4—942no—
483C4—3761no—
484C4—3762no—
485C4—3761no—
486C4#1568916214yes—
487C4#1568960211yescharacteristic
488C4#1568960511yescharacteristic
489C4#1568975011yescharacteristic
490C4#1568975311yescharacteristic
491C2—944no—
492C2#2024601011yescharacteristic
493C2#428×D4711yescharacteristic
494C2C2×#1568916211yescharacteristic
495C2#2025441011yescharacteristic
496C2#2025441411yescharacteristic
497C2#2025460811yescharacteristic
498C2#2025462211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1948C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1948D47
hypocenterC47#21987
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
4. #20244151C2semidirect
5. #20244147C2semidirect
6. #20244144C2semidirect
9. #20243553C2semidirect
10. #20243520C2semidirect
11. C47×#1948C2semidirect
16. #15689179C4semidirect
72. C4⋊(C47⋊C8)C2×C4semidirect
74. #13093934C2×C4semidirect
91. C4⋊C8×C47C2×C4semidirect
246. #1948D47semidirect
293. C47#21987semidirect
332. C4⋊C8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156891792
#384×D472
#202457842
#202441512
#202441472
#202441442
#202435792
#202435772
#202435532
#202435202
C47×#19482
#2198747