Finite Group Databasebeta

← #26111062

Subgroups — #26111062

Order 12032.

692 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246337C211yesmaximal, characteristic, complemented
2#20244316C211yesmaximal, characteristic, complemented
3#20244195C211yesmaximal, characteristic, complemented
4C2×D4⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
5#20241272C211yesmaximal, characteristic, complemented
6#20241227C211yesmaximal, characteristic, complemented
7C2×C47×#459C211yesmaximal, characteristic
8C2×#13093983—21no—
9C2×C4⋊(C47⋊C8)C2211yescharacteristic
10C2×#13093945—21no—
11C2×#13093945—21no—
12C2×#13093945—22no—
13C2×C4×C47⋊C8C2211yescharacteristic
14#15689026—21no—
15#15689016—22no—
16#15688993C2211yescharacteristic
17#15688983—21no—
18C4⋊C4×Dic47—21no—
19C4×C4⋊Dic47C2211yescharacteristic
20D4⋊(C47⋊C8)—22no—
21C22×D8×C47—21no—
22C2×#127×C47C2211yescharacteristic
23C2×C4×D4×C47C2211yescharacteristic
24C47×#459—22no—
25C47×#459C414yescomplemented
26C2×C4⋊C8×C47C2211yescharacteristic
27C2×#102×C47—21no—
28C22×C47⋊C8—21no—
29C22×C47⋊C8—21no—
30#13093983—42no—
31#13093983—42no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47D411yescharacteristic
35C2×C4⋊Dic47D411yescharacteristic
36C2×C4⋊Dic47D412yes—
37C2×C22⋊Dic47—22no—
38C2×C22⋊Dic47—22no—
39C2×C4×Dic47—22no—
40C2×C4×Dic47—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43#13093945—44no—
44#13093945—42no—
45#13093945—42no—
46C4×C47⋊C8—21no—
47C4×C47⋊C8—21no—
48C47⋊#95—22no—
49C47⋊#95—21no—
50C22×D4×C47—21no—
51C22×D4×C47—21no—
52C22×D4×C47—21no—
53C22×D4×C47D411yescharacteristic
54C23×C4×C47—21no—
55C2×D8×C47—24no—
56C2×D8×C47—42no—
57C2×D8×C47—42no—
58C22×C8×C47—21no—
59#127×C47—21no—
60#127×C47—21no—
61#127×C47—21no—
62#127×C47C2×C412yes—
63C4×D4×C47—21no—
64C4×D4×C47—21no—
65C4×D4×C47C2×C412yes—
66C4×D4×C47D412yes—
67C2×C4⋊C4×C47D411yescharacteristic
68C2×C22⋊C4×C47—21no—
69C2×C42×C47C2311yescharacteristic
70C4⋊C8×C47—21no—
71C4⋊C8×C47C2×C412yes—
72#102×C47—24no—
73C24×C47—21no—
74C24×C47—41no—
75C2×D4×C47—22no—
76C2×D4×C47—22no—
77C2×D4×C47—22no—
78C2×D4×C47—21no—
79C2×D4×C47—21no—
80C2×D4×C47—21no—
81C2×D4×C47—21no—
82C2×D4×C47—21no—
83C2×D4×C47—42no—
84C2×D4×C47—42no—
85C2×D4×C47—41no—
86C2×D4×C47—41no—
87C2×D4×C47—41no—
88C2×D4×C47—41no—
89C2×D4×C47—41no—
90C2×D4×C47—41no—
91C2×D4×C47—41no—
92C2×D4×C47—41no—
93C2×D4×C47—41no—
94C2×D4×C47—41no—
95C2×D4×C47—41no—
96C2×D4×C47C22⋊C412yes—
97C22×C4×C47—21no—
98C22×C4×C47—22no—
99C22×C4×C47—22no—
100C22×C4×C47—41no—
101C22×C4×C47—41no—
102C22×C4×C47—41no—
103C22×C4×C47—41no—
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D411yescharacteristic
107C22×Dic47—21no—
108C22×Dic47—21no—
109C22×Dic47—22no—
110D8×C47—48no—
111C2×C8×C47—22no—
112C2×C8×C47—41no—
113C2×C8×C47—41no—
114C4⋊C4×C47—21no—
115C4⋊C4×C47C22⋊C412yes—
116C22⋊C4×C47—22no—
117C22⋊C4×C47—22no—
118C42×C47C22×C411yescharacteristic
119C42×C47C2×D411yescharacteristic
120C42×C47#4111yescharacteristic
121C42×C47#4111yescharacteristic
122C22⋊Dic47—44no—
123C22⋊Dic47—82no—
124C4⋊Dic47—21no—
125C4⋊Dic47—21no—
126C4⋊Dic47—21no—
127C4⋊Dic47—41no—
128C4⋊Dic47—41no—
129C4⋊Dic47—41no—
130C4⋊Dic47—41no—
131C4⋊Dic47—42no—
132C4⋊Dic47D812yescomplemented
133C4×Dic47—44no—
134C4×Dic47—41no—
135C4×Dic47—41no—
136C2×C47⋊C8—22no—
137C2×C47⋊C8—41no—
138C2×C47⋊C8—41no—
139C2×C47⋊C8—41no—
140C2×C47⋊C8—41no—
141C2×C47⋊C8—41no—
142C23×C47—22no—
143C23×C47—41no—
144C23×C47—41no—
145C23×C47—41no—
146C23×C47—41no—
147C23×C47—41no—
148C23×C47—41no—
149C23×C47—42no—
150C23×C47—81no—
151C23×C47—81no—
152C23×C47—81no—
153C23×C47—81no—
154C23×C47—81no—
155C23×C47—81no—
156C23×C47#12011yescharacteristic
157D4×C47—22no—
158D4×C47—22no—
159D4×C47—44no—
160D4×C47—42no—
161D4×C47—42no—
162D4×C47—42no—
163D4×C47—42no—
164D4×C47—41no—
165D4×C47—41no—
166D4×C47—82no—
167D4×C47—81no—
168D4×C47—81no—
169D4×C47—81no—
170D4×C47—81no—
171C2×C4×C47—21no—
172C2×C4×C47—21no—
173C2×C4×C47—22no—
174C2×C4×C47—41no—
175C2×C4×C47—41no—
176C2×C4×C47—42no—
177C2×C4×C47—42no—
178C2×C4×C47—42no—
179C2×C4×C47—42no—
180C2×C4×C47C2×C22⋊C411yescharacteristic
181C2×C4×C47C4×D411yescharacteristic
182C2×C4×C47C4×D411yescharacteristic
183C2×C4×C47#12111yescharacteristic
184C2×C4×C47#12111yescharacteristic
185C2×C4×C47#12311yescharacteristic
186C2×C4×C47C2×D811yescharacteristic
187C2×C4×C47#13511yescharacteristic
188C2×C4×C47#13511yescharacteristic
189C2×C4×C47#13511yescharacteristic
190C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
191C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
192C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
193C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
194C2×Dic47—22no—
195C2×Dic47—41no—
196C2×Dic47—41no—
197C2×Dic47—42no—
198C2×Dic47—41no—
199C2×Dic47—41no—
200C2×Dic47—42no—
201C2×Dic47—42no—
202C2×Dic47—41no—
203C8×C47—42no—
204C47⋊C8—42no—
205C47⋊C8—81no—
206#8980—471nomaximal, Sylow
207C22×C47—42no—
208C22×C47—42no—
209C22×C47—42no—
210C22×C47—81no—
211C22×C47—81no—
212C22×C47—81no—
213C22×C47—81no—
214C22×C47—82no—
215C22×C47—82no—
216C22×C47—82no—
217C22×C47—161no—
218C22×C47—161no—
219C22×C47—161no—
220C22×C47—161no—
221C22×C47#38611yescharacteristic
222C22×C47#44711yescharacteristic
223C22×C47#44911yescharacteristic
224C22×C47#44911yescharacteristic
225C22×C47#44911yescharacteristic
226C22×C47#45311yescharacteristic
227C22×C47#45411yescharacteristic
228C4×C47—21no—
229C4×C47—21no—
230C4×C47—42no—
231C4×C47#41611yescharacteristic
232C4×C47#41811yescharacteristic
233C4×C47C4×D811yescharacteristic
234C4×C47#44211yescharacteristic
235C4×C47#45911yescharacteristic
236C4×C47#46311yescharacteristic
237C4×C47#48211yescharacteristic
238C4×C47#48211yescharacteristic
239Dic47—42no—
240Dic47—82no—
241Dic47—81no—
242C2×#459D4711yescharacteristic
243#2354—471no—
244#1982—471no—
245#1957—471no—
246C2×D4⋊C8—471no—
247#1296—471no—
248#1279—471no—
249C2×C47—82no—
250C2×C47—162no—
251C2×C47#162111yescharacteristic
252C2×C47#162411yescharacteristic
253C2×C47#162511yescharacteristic
254C2×C47#162811yescharacteristic
255C2×C47#187511yescharacteristic
256C2×C47#187911yescharacteristic
257C2×C47#188311yescharacteristic
258C22×D8—21no—
259C2×#127C2×D4711yescharacteristic
260C2×C4×D4C2×D4711yescharacteristic
261C2×#117—941no—
262#459—22no—
263#459Dic4714yescomplemented
264C2×C4⋊C8—471no—
265C2×C4⋊C8C2×D4711yescharacteristic
266C2×#102—21no—
267C2×#102—942no—
268C2×#102—941no—
269C2×#102—941no—
270C2×C4×C8—471no—
271#390—941no—
272#388—942no—
273#384—471no—
274#380—941no—
275C4×C4⋊C4—471no—
276C4×C4⋊C4—941no—
277D4⋊C8—942no—
278C47#898011yescharacteristic, complemented, Sylow
279C22×D4—21no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C22⋊D4711yescharacteristic
283C23×C4—21no—
284C2×D8—24no—
285C2×D8—42no—
286C2×D8—42no—
287C22×C8—21no—
288C22×C8—941no—
289C22×C8—941no—
290#127—21no—
291#127—21no—
292#127—21no—
293#127C2×Dic4712yes—
294C4×D4—21no—
295C4×D4—21no—
296C4×D4C2×Dic4712yes—
297C4×D4C22⋊D4712yes—
298#117—1882no—
299#117—1882no—
300C2×C4⋊C4—471no—
301C2×C4⋊C4—471no—
302C2×C4⋊C4—472no—
303C2×C4⋊C4—941no—
304C2×C4⋊C4—941no—
305C2×C4⋊C4C22⋊D4711yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—942no—
308C2×C22⋊C4—942no—
309C2×C42—942no—
310C2×C42—941no—
311C2×C42C22×D4711yescharacteristic
312C4⋊C8—21no—
313C4⋊C8—941no—
314C4⋊C8—941no—
315C4⋊C8C2×Dic4712yes—
316#102—24no—
317#102—1882no—
318#102—1882no—
319#102—1884no—
320C4×C8—941no—
321C4×C8—941no—
322#95—942no—
323#95—941no—
324C24—21no—
325C24—41no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4C22⋊Dic4712yes—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—941no—
356C22×C4—941no—
357C22×C4—942no—
358C22×C4D4×D4711yescharacteristic
359C22×C4D4×D4711yescharacteristic
360C22×C4C2×C22⋊D4711yescharacteristic
361D8—48no—
362C2×C8—22no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—942no—
366C2×C8—1881no—
367C2×C8—1881no—
368C2×C8—1881no—
369C2×C8—1881no—
370C2×C8—1881no—
371C4⋊C4—21no—
372C4⋊C4—472no—
373C4⋊C4—941no—
374C4⋊C4—941no—
375C4⋊C4—941no—
376C4⋊C4—1882no—
377C4⋊C4—1881no—
378C4⋊C4—1881no—
379C4⋊C4—1881no—
380C4⋊C4—1881no—
381C4⋊C4C22⋊Dic4712yes—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—3762no—
385C22⋊C4—1884no—
386C42—1884no—
387C42—1881no—
388C42—1881no—
389C42#1062535211yescharacteristic
390C42#1062535211yescharacteristic
391C42C22×Dic4711yescharacteristic
392C42C2×C22⋊D4711yescharacteristic
393C23—22no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—42no—
401C23—81no—
402C23—81no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23#1309399211yescharacteristic
408D4—22no—
409D4—22no—
410D4—44no—
411D4—42no—
412D4—42no—
413D4—42no—
414D4—41no—
415D4—41no—
416D4—42no—
417D4—82no—
418D4—81no—
419D4—81no—
420D4—81no—
421D4—81no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—942no—
432C2×C4—1882no—
433C2×C4—1881no—
434C2×C4—1881no—
435C2×C4—1882no—
436C2×C4—1881no—
437C2×C4—1882no—
438C2×C4—1881no—
439C2×C4—1881no—
440C2×C4C2×C22⋊Dic4711yescharacteristic
441C2×C4D4×Dic4711yescharacteristic
442C2×C4D4×Dic4711yescharacteristic
443C2×C4#1309399611yescharacteristic
444C2×C4#1309399611yescharacteristic
445C2×C4#1309400511yescharacteristic
446C2×C4D8×D4711yescharacteristic
447C2×C4#1309404311yescharacteristic
448C2×C4#1309404411yescharacteristic
449C2×C4#1309404411yescharacteristic
450C2×C4#1309404811yescharacteristic
451C2×C4#1309404811yescharacteristic
452C2×C4#1309405011yescharacteristic
453C2×C4#1309405011yescharacteristic
454C8—42no—
455C8—3761no—
456C8—1882no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—161no—
468C22—161no—
469C22—161no—
470C22—161no—
471C22#1568900311yescharacteristic
472C22#1568925511yescharacteristic
473C22#1568926511yescharacteristic
474C22#1568926611yescharacteristic
475C22#1568926611yescharacteristic
476C22#1568928911yescharacteristic
477C22#1568929411yescharacteristic
478C4—21no—
479C4—21no—
480C4—42no—
481C4—3762no—
482C4—3761no—
483C4—1882no—
484C4#1568911911yescharacteristic
485C4#1568912811yescharacteristic
486C4D8×Dic4711yescharacteristic
487C4#1568923511yescharacteristic
488C4#1568931611yescharacteristic
489C4#1568933611yescharacteristic
490C4#1568944411yescharacteristic
491C4#1568944411yescharacteristic
492C2—82no—
493C2—162no—
494C2#2024245211yescharacteristic
495C2#2024247011yescharacteristic
496C2#2024247611yescharacteristic
497C2#2024249611yescharacteristic
498C2#2024371111yescharacteristic
499C2#2024373811yescharacteristic
500C2#2024376611yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC2×C47×#459C2
socleC23×C47#120
cosocleC2×C42×C47C23
hypercenterC2×#459D47
hypocenterC47#8980
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. #20246337C2semidirect
2. #20244316C2semidirect
3. #20244195C2semidirect
4. C2×D4⋊(C47⋊C8)C2semidirect
5. #20241272C2semidirect
6. #20241227C2semidirect
25. C47×#459C4semidirect
132. C4⋊Dic47D8semidirect
263. #459Dic47semidirect
278. C47#8980semidirect

Maximal subgroups

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

subgroupindex
#202463372
#202443162
#202441952
C2×D4⋊(C47⋊C8)2
#202412722
#202412272
C2×C47×#4592
#898047