Finite Group Databasebeta

← #26112608

Subgroups — #26112608

Order 12032.

478 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252046C211yesmaximal, characteristic, complemented
2#20243487C211yesmaximal, characteristic, complemented
3#20241904C211yesmaximal, characteristic
4#20241748C211yesmaximal, characteristic
5#20241265C211yesmaximal, characteristic, complemented
6#20241235C211yesmaximal, characteristic, complemented
7C2×C47×#482C211yesmaximal, characteristic
8#15689184—22no—
9#15689180—21no—
10#15689176—21no—
11#15689150—21no—
12#15689087C2211yescharacteristic
13#15689087C2211yescharacteristic
14C2×C22⋊(C47⋊C8)C2211yescharacteristic
15#15689079—21no—
16C4×C22⋊Dic47C2211yescharacteristic
17#15688882—21no—
18#15688882—21no—
19#15688857—22no—
20#15688847—21no—
21#15688847—21no—
22C2×#121×C47C2211yescharacteristic
23C2×#117×C47C2211yescharacteristic
24C47×#482—21no—
25C47×#482—21no—
26C47×#482—21no—
27C47×#482—21no—
28C2×#107×C47C411yescharacteristic, complemented
29C2×#106×C47C411yescharacteristic, complemented
30C2×#102×C47C411yescharacteristic, complemented
31C2×#102×C47C411yescharacteristic, complemented
32C2×C22⋊C8×C47C2211yescharacteristic
33C2×#10625329—21no—
34C2×#10625329—21no—
35C22×C47⋊C8—21no—
36C2×C22⋊Dic47—21no—
37C2×C4×Dic47—21no—
38C4⋊(C47⋊C8)—22no—
39C4⋊(C47⋊C8)—22no—
40C22⋊(C47⋊C8)—21no—
41C22⋊(C47⋊C8)—21no—
42C22⋊(C47⋊C8)—21no—
43C22⋊(C47⋊C8)—21no—
44C22⋊(C47⋊C8)D411yescharacteristic
45C22⋊(C47⋊C8)D411yescharacteristic
46C22⋊(C47⋊C8)D412yes—
47#13093935—22no—
48C4×C47⋊C8—22no—
49C47⋊#95—21no—
50C22×D4×C47—21no—
51C22×D4×C47C2×C411yescharacteristic
52C23×C4×C47C2311yescharacteristic
53C22×C8×C47C2×C411yescharacteristic
54C22×C8×C47C2×C411yescharacteristic
55#121×C47—21no—
56#121×C47—21no—
57#121×C47—21no—
58#121×C47—21no—
59#117×C47—21no—
60#117×C47—21no—
61#117×C47D411yescharacteristic
62#117×C47D411yescharacteristic
63#117×C47D411yescharacteristic
64#117×C47D411yescharacteristic
65C2×C4⋊C4×C47C2×C411yescharacteristic
66C2×C4⋊C4×C47C2×C411yescharacteristic
67C2×C4⋊C4×C47C2×C411yescharacteristic
68C2×C22⋊C4×C47—21no—
69C2×C22⋊C4×C47—21no—
70C2×C42×C47—21no—
71#107×C47—21no—
72#107×C47—21no—
73#106×C47—21no—
74#106×C47—21no—
75#102×C47—21no—
76#102×C47—21no—
77#102×C47—21no—
78#102×C47—21no—
79C22⋊C8×C47—21no—
80C22⋊C8×C47—21no—
81C24×C47—21no—
82C24×C47#4111yescharacteristic
83C2×D4×C47—21no—
84C2×D4×C47—21no—
85C2×D4×C47—41no—
86C2×D4×C47—41no—
87C2×D4×C47—41no—
88C2×D4×C47—41no—
89C2×D4×C47—82no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—41no—
94C22×C4×C47—41no—
95C22×C4×C47C22×C411yescharacteristic
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47C2×D411yescharacteristic
100C22×C4×C47#4111yescharacteristic
101C22×Dic47—41no—
102C2×C8×C47—21no—
103C2×C8×C47—21no—
104C2×C8×C47—41no—
105C2×C8×C47—41no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—21no—
110C4⋊C4×C47C22⋊C411yescharacteristic
111C4⋊C4×C47C22⋊C411yescharacteristic
112C4⋊C4×C47C22⋊C411yescharacteristic
113C4⋊C4×C47C22⋊C411yescharacteristic
114C22⋊C4×C47—21no—
115C22⋊C4×C47—21no—
116C22⋊C4×C47—21no—
117C22⋊C4×C47—21no—
118C22⋊C4×C47—41no—
119C22⋊C4×C47—41no—
120C42×C47—21no—
121C42×C47—21no—
122C42×C47—21no—
123C42×C47—21no—
124C22⋊Dic47—22no—
125C22⋊Dic47—41no—
126C4×Dic47—41no—
127C4×Dic47—41no—
128#10625329—41no—
129#10625329—41no—
130#10625329—41no—
131#10625329—41no—
132C2×C47⋊C8—21no—
133C2×C47⋊C8—22no—
134C2×C47⋊C8—21no—
135C2×C47⋊C8—41no—
136C2×C47⋊C8—41no—
137C2×C47⋊C8—41no—
138C23×C47—21no—
139C23×C47—21no—
140C23×C47—41no—
141C23×C47—41no—
142C23×C47—41no—
143C23×C47—41no—
144C23×C47—41no—
145C23×C47—81no—
146C23×C47—81no—
147C23×C47C4×D411yescharacteristic
148C23×C47#12111yescharacteristic
149C23×C47#12311yescharacteristic
150D4×C47—41no—
151D4×C47—41no—
152D4×C47—81no—
153D4×C47—82no—
154D4×C47—82no—
155C2×C4×C47—21no—
156C2×C4×C47—21no—
157C2×C4×C47—21no—
158C2×C4×C47—21no—
159C2×C4×C47—21no—
160C2×C4×C47—21no—
161C2×C4×C47—21no—
162C2×C4×C47—21no—
163C2×C4×C47—21no—
164C2×C4×C47—21no—
165C2×C4×C47—41no—
166C2×C4×C47—41no—
167C2×C4×C47—41no—
168C2×C4×C47—41no—
169C2×C4×C47—41no—
170C2×C4×C47—41no—
171C2×C4×C47—81no—
172C2×C4×C47—81no—
173C2×C4×C47C2×C22⋊C411yescharacteristic
174C2×C4×C47C4×D411yescharacteristic
175C2×C4×C47#12011yescharacteristic
176C2×C4×C47#12111yescharacteristic
177C2×Dic47—42no—
178C2×Dic47—81no—
179C2×Dic47—81no—
180C8×C47—41no—
181C8×C47—41no—
182C47⋊C8—41no—
183C47⋊C8—41no—
184C47⋊C8—42no—
185C47⋊C8—81no—
186#9242—471nomaximal, Sylow
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—21no—
191C22×C47—21no—
192C22×C47—21no—
193C22×C47—21no—
194C22×C47—21no—
195C22×C47—81no—
196C22×C47—81no—
197C22×C47—81no—
198C22×C47—81no—
199C22×C47—81no—
200C22×C47—161no—
201C22×C47—161no—
202C22×C47#38611yescharacteristic
203C22×C47#44311yescharacteristic
204C22×C47#44411yescharacteristic
205C4×C47—21no—
206C4×C47—21no—
207C4×C47—21no—
208C4×C47—41no—
209C4×C47—41no—
210C4×C47—41no—
211C4×C47—41no—
212C4×C47—81no—
213C4×C47#41011yescharacteristic
214C4×C47#41011yescharacteristic
215Dic47—82no—
216C2×#482D4711yescharacteristic
217#2924—471no—
218#1826—471no—
219#1507—471no—
220#1471—471no—
221#1293—471no—
222#1282—471no—
223C2×C47—21no—
224C2×C47—21no—
225C2×C47—41no—
226C2×C47—41no—
227C2×C47—161no—
228C2×C47#187611yescharacteristic
229C2×C47#189911yescharacteristic
230C2×C47#190311yescharacteristic
231C2×#121C2×D4711yescharacteristic
232C2×#117C2×D4711yescharacteristic
233#482—21no—
234#482—21no—
235#482—21no—
236#482—21no—
237#433—942no—
238#432—941no—
239#431—941no—
240C2×#107Dic4711yescharacteristic, complemented
241C2×#106Dic4711yescharacteristic, complemented
242#424—941no—
243C2×#102Dic4711yescharacteristic, complemented
244C2×#102Dic4711yescharacteristic, complemented
245#407—471no—
246#407—471no—
247C2×C22⋊C8—471no—
248C2×C22⋊C8C2×D4711yescharacteristic
249#405—941no—
250C4×C22⋊C4—471no—
251#339—941no—
252#339—941no—
253#327—942no—
254#323—941no—
255#323—941no—
256C47#924211yescharacteristic, complemented, Sylow
257C22×D4—21no—
258C22×D4C2×Dic4711yescharacteristic
259C23×C4C22×D4711yescharacteristic
260C2×M16—941no—
261C2×M16—941no—
262C22×C8—941no—
263C22×C8C2×Dic4711yescharacteristic
264C22×C8C2×Dic4711yescharacteristic
265#121—21no—
266#121—21no—
267#121—21no—
268#121—21no—
269#117—21no—
270#117—21no—
271#117C22⋊D4711yescharacteristic
272#117C22⋊D4711yescharacteristic
273#117C22⋊D4711yescharacteristic
274#117C22⋊D4711yescharacteristic
275C2×C4⋊C4C2×Dic4711yescharacteristic
276C2×C4⋊C4C2×Dic4711yescharacteristic
277C2×C4⋊C4C2×Dic4711yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—941no—
281C2×C42—21no—
282C2×C42—941no—
283#107—21no—
284#107—21no—
285#106—21no—
286#106—21no—
287C4⋊C8—942no—
288C4⋊C8—942no—
289#102—21no—
290#102—21no—
291#102—21no—
292#102—21no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—471no—
296C22⋊C8—471no—
297C22⋊C8—472no—
298C22⋊C8—941no—
299C22⋊C8—941no—
300C22⋊C8—941no—
301C22⋊C8—941no—
302#97—942no—
303C4×C8—942no—
304#95—941no—
305C24—21no—
306C24#1062535211yescharacteristic
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—82no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—1881no—
320C22×C4D4×D4711yescharacteristic
321C22×C4D4×D4711yescharacteristic
322C22×C4#1062535211yescharacteristic
323C22×C4C22×Dic4711yescharacteristic
324C22×C4C2×C22⋊D4711yescharacteristic
325C22×C4C2×C22⋊D4711yescharacteristic
326M16—1881no—
327M16—1881no—
328M16—1881no—
329M16—1881no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—941no—
335C2×C8—942no—
336C2×C8—941no—
337C2×C8—1881no—
338C2×C8—1881no—
339C2×C8—1881no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4C22⋊Dic4711yescharacteristic
345C4⋊C4C22⋊Dic4711yescharacteristic
346C4⋊C4C22⋊Dic4711yescharacteristic
347C4⋊C4C22⋊Dic4711yescharacteristic
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—942no—
355C22⋊C4—1881no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—1881no—
361C42—1881no—
362C23—21no—
363C23—21no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—81no—
370C23—81no—
371C23D4×Dic4711yescharacteristic
372C23#1309399611yescharacteristic
373C23#1309400511yescharacteristic
374D4—41no—
375D4—41no—
376D4—81no—
377D4—82no—
378D4—82no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—3761no—
398C2×C4—3761no—
399C2×C4—1882no—
400C2×C4C2×C22⋊Dic4711yescharacteristic
401C2×C4D4×Dic4711yescharacteristic
402C2×C4#1309399211yescharacteristic
403C2×C4#1309399611yescharacteristic
404C8—41no—
405C8—41no—
406C8—3761no—
407C8—1881no—
408C8—1881no—
409C8—1882no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—161no—
424C22—161no—
425C22#1568900311yescharacteristic
426C22#1568924011yescharacteristic
427C22#1568924511yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—81no—
436C4—3762no—
437C4#1568909711yescharacteristic
438C4#1568909711yescharacteristic
439C2—21no—
440C2—21no—
441C2—41no—
442C2—41no—
443C2—161no—
444C2#2024371711yescharacteristic
445C2#2024384711yescharacteristic
446C2#2024387011yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#482C2
socleC22×C47#386
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenterC2×#482D47
hypocenterC47#9242
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#482

Complements

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

NH ≅ G/Ntype
1. #20252046C2semidirect
2. #20243487C2semidirect
5. #20241265C2semidirect
6. #20241235C2semidirect
28. C2×#107×C47C4semidirect
29. C2×#106×C47C4semidirect
30. C2×#102×C47C4semidirect
31. C2×#102×C47C4semidirect
240. C2×#107Dic47semidirect
241. C2×#106Dic47semidirect
243. C2×#102Dic47semidirect
244. C2×#102Dic47semidirect
256. C47#9242semidirect

Maximal subgroups

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

subgroupindex
#202520462
#202434872
#202419042
#202417482
#202412652
#202412352
C2×C47×#4822
#924247