Finite Group Databasebeta

← #26115816

Subgroups — #26115816

Order 12032.

552 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20244374C211yesmaximal, characteristic, complemented
2#20244225C211yesmaximal, characteristic
3#20243417C211yesmaximal, characteristic, complemented
4#20243164C211yesmaximal, characteristic, complemented
5#20241960C211yesmaximal, characteristic, complemented
6#20241956C211yesmaximal, characteristic
7C2×C47×#465C211yesmaximal, characteristic
8C2×#13093958—21no—
9C2×#13093939—22no—
10C2×C22⋊(C47⋊C8)C2211yescharacteristic
11C2×#13093934—21no—
12#15688983C2211yescharacteristic
13#15688983C2211yescharacteristic
14C4×C22⋊Dic47C2211yescharacteristic
15#15688890—22no—
16#15688886—21no—
17#15688877—21no—
18#15688860—22no—
19#15688857—22no—
20C22×SD16×C47C411yescharacteristic, complemented
21C23×C8×C47C2211yescharacteristic
22C2×C22⋊Q8×C47C2211yescharacteristic
23C2×#121×C47C2211yescharacteristic
24C47×#465—21no—
25C47×#465—21no—
26C47×#465—21no—
27C47×#465—21no—
28C2×#106×C47C411yescharacteristic, complemented
29C2×#103×C47C411yescharacteristic, complemented
30C2×#102×C47C411yescharacteristic, complemented
31C22×C47⋊C8—21no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47—21no—
34C2×C22⋊Dic47—21no—
35C2×C22⋊Dic47—21no—
36C2×C22⋊Dic47D412yes—
37C2×C4×Dic47—21no—
38#13093958—42no—
39#13093939—44no—
40C22⋊(C47⋊C8)—21no—
41C22⋊(C47⋊C8)D412yes—
42#13093934—42no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C22×Q8×C47C2×C411yescharacteristic
47C22×D4×C47—21no—
48C22×D4×C47C2×C411yescharacteristic
49C23×C4×C47C2311yescharacteristic
50C2×SD16×C47—21no—
51C2×SD16×C47—21no—
52C2×SD16×C47—42no—
53C22×C8×C47—42no—
54C22×C8×C47C2×C411yescharacteristic
55C22×C8×C47C2×C411yescharacteristic
56C22×C8×C47D411yescharacteristic
57C22×C8×C47D411yescharacteristic
58C22×C8×C47D411yescharacteristic
59C22×C8×C47D411yescharacteristic
60C22⋊Q8×C47—21no—
61C22⋊Q8×C47—21no—
62C22⋊Q8×C47—21no—
63C22⋊Q8×C47—21no—
64#121×C47—21no—
65#121×C47—21no—
66#121×C47—21no—
67#121×C47—21no—
68C2×C4⋊C4×C47—21no—
69C2×C4⋊C4×C47C2×C411yescharacteristic
70C2×C4⋊C4×C47C2×C411yescharacteristic
71C2×C22⋊C4×C47—21no—
72C2×C22⋊C4×C47—21no—
73#106×C47—21no—
74#106×C47—21no—
75#103×C47—21no—
76#103×C47—21no—
77#102×C47—21no—
78#102×C47—21no—
79C24×C47—21no—
80C24×C47C2×D411yescharacteristic
81C2×Q8×C47—21no—
82C2×Q8×C47—21no—
83C2×Q8×C47—41no—
84C2×Q8×C47—41no—
85C2×D4×C47—21no—
86C2×D4×C47—21no—
87C2×D4×C47—41no—
88C2×D4×C47—41no—
89C2×D4×C47—41no—
90C2×D4×C47—41no—
91C2×D4×C47—42no—
92C2×D4×C47—42no—
93C22×C4×C47—21no—
94C22×C4×C47—21no—
95C22×C4×C47—21no—
96C22×C4×C47—41no—
97C22×C4×C47—41no—
98C22×C4×C47C22×C411yescharacteristic
99C22×C4×C47C2×D411yescharacteristic
100C22×C4×C47C2×D411yescharacteristic
101C22×C4×C47C2×D411yescharacteristic
102C22×C4×C47#4111yescharacteristic
103C22×C4×C47#4111yescharacteristic
104C22×Dic47—22no—
105C22×Dic47—41no—
106C22×Dic47—41no—
107SD16×C47—42no—
108SD16×C47—42no—
109C2×C8×C47—22no—
110C2×C8×C47—22no—
111C2×C8×C47—22no—
112C2×C8×C47—41no—
113C2×C8×C47—41no—
114C2×C8×C47—41no—
115C2×C8×C47C22⋊C411yescharacteristic
116C2×C8×C47C22⋊C411yescharacteristic
117C2×C8×C47C22⋊C411yescharacteristic
118C2×C8×C47C22⋊C411yescharacteristic
119C4⋊C4×C47—21no—
120C4⋊C4×C47—21no—
121C4⋊C4×C47—21no—
122C4⋊C4×C47—21no—
123C4⋊C4×C47—41no—
124C4⋊C4×C47—41no—
125C22⋊C4×C47—41no—
126C22⋊C4×C47—41no—
127C22⋊C4×C47—41no—
128C22⋊C4×C47—41no—
129C22⋊Dic47—22no—
130C22⋊Dic47—22no—
131C22⋊Dic47—42no—
132C22⋊Dic47—42no—
133C4⋊Dic47—41no—
134C4⋊Dic47—41no—
135C4⋊Dic47—41no—
136C4⋊Dic47—41no—
137C4×Dic47—41no—
138C4×Dic47—41no—
139C2×C47⋊C8—22no—
140C2×C47⋊C8—41no—
141C2×C47⋊C8—41no—
142C23×C47—41no—
143C23×C47—41no—
144C23×C47—41no—
145C23×C47—42no—
146C23×C47—81no—
147C23×C47—81no—
148C23×C47C4×D411yescharacteristic
149C23×C47#12111yescharacteristic
150C23×C47#12711yescharacteristic
151C23×C47AGL(1,ℤ/8)12yes—
152C23×C47#13712yes—
153Q8×C47—41no—
154Q8×C47—41no—
155Q8×C47—81no—
156D4×C47—41no—
157D4×C47—41no—
158D4×C47—81no—
159D4×C47—82no—
160D4×C47—81no—
161D4×C47—81no—
162C2×C4×C47—21no—
163C2×C4×C47—21no—
164C2×C4×C47—21no—
165C2×C4×C47—21no—
166C2×C4×C47—21no—
167C2×C4×C47—21no—
168C2×C4×C47—41no—
169C2×C4×C47—41no—
170C2×C4×C47—41no—
171C2×C4×C47—41no—
172C2×C4×C47—41no—
173C2×C4×C47—41no—
174C2×C4×C47—42no—
175C2×C4×C47—81no—
176C2×C4×C47—81no—
177C2×C4×C47C2×C22⋊C411yescharacteristic
178C2×C4×C47C4×D411yescharacteristic
179C2×C4×C47#12111yescharacteristic
180C2×C4×C47#12411yescharacteristic
181C2×Dic47—42no—
182C2×Dic47—42no—
183C2×Dic47—42no—
184C2×Dic47—81no—
185C2×Dic47—81no—
186C2×Dic47—81no—
187C2×Dic47—81no—
188C2×Dic47—81no—
189C2×Dic47—81no—
190C8×C47—21no—
191C8×C47—21no—
192C8×C47—21no—
193C8×C47#9912yescomplemented
194C47⋊C8—42no—
195#9786—471nomaximal, Sylow
196C22×C47—21no—
197C22×C47—21no—
198C22×C47—21no—
199C22×C47—21no—
200C22×C47—21no—
201C22×C47—21no—
202C22×C47—42no—
203C22×C47—42no—
204C22×C47—81no—
205C22×C47—81no—
206C22×C47—81no—
207C22×C47—162no—
208C22×C47#39011yescharacteristic
209C22×C47#44012yes—
210C22×C47#46812yes—
211C22×C47#49611yescharacteristic
212C22×C47#49711yescharacteristic
213C4×C47—21no—
214C4×C47—21no—
215C4×C47—21no—
216C4×C47—81no—
217C4×C47—81no—
218C4×C47—81no—
219C4×C47C2×#9911yescharacteristic
220C4×C47#41011yescharacteristic
221Dic47—82no—
222Dic47—161no—
223Dic47—161no—
224C2×#465D4711yescharacteristic
225#1996—471no—
226#1964—471no—
227#1808—471no—
228#1752—471no—
229#1520—471no—
230#1519—471no—
231C2×C47—22no—
232C2×C47—21no—
233C2×C47—22no—
234C2×C47—21no—
235C2×C47—161no—
236C2×C47#196411yescharacteristic
237C2×C47#197311yescharacteristic
238C2×C47#197611yescharacteristic
239C22×SD16Dic4711yescharacteristic, complemented
240C23×C8C2×D4711yescharacteristic
241C2×C22⋊Q8C2×D4711yescharacteristic
242C2×#121C2×D4711yescharacteristic
243#465—21no—
244#465—21no—
245#465—21no—
246#465—21no—
247C2×#107—941no—
248C2×#106Dic4711yescharacteristic, complemented
249C2×#103Dic4711yescharacteristic, complemented
250C2×#102Dic4711yescharacteristic, complemented
251C2×#99—942no—
252C2×C22⋊C8—471no—
253C2×#97—941no—
254#380—471no—
255#380—471no—
256C4×C22⋊C4—471no—
257#342—942no—
258#340—941no—
259#336—941no—
260#328—942no—
261#327—942no—
262C47#978611yescharacteristic, complemented, Sylow
263C22×Q8C2×Dic4711yescharacteristic
264C22×D4—21no—
265C22×D4C2×Dic4711yescharacteristic
266C23×C4C22×D4711yescharacteristic
267C2×SD16—21no—
268C2×SD16—21no—
269C2×SD16—42no—
270C22×C8—42no—
271C22×C8—941no—
272C22×C8C2×Dic4711yescharacteristic
273C22×C8C2×Dic4711yescharacteristic
274C22×C8C22⋊D4711yescharacteristic
275C22×C8C22⋊D4711yescharacteristic
276C22×C8C22⋊D4711yescharacteristic
277C22×C8C22⋊D4711yescharacteristic
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282#121—21no—
283#121—21no—
284#121—21no—
285#121—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—941no—
288C2×C4⋊C4—941no—
289C2×C4⋊C4C2×Dic4711yescharacteristic
290C2×C4⋊C4C2×Dic4711yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—472no—
294C2×C22⋊C4—941no—
295C2×C22⋊C4—941no—
296C2×C42—941no—
297#107—1882no—
298#106—21no—
299#106—21no—
300#103—21no—
301#103—21no—
302#102—21no—
303#102—21no—
304#99—1884no—
305C22⋊C8—472no—
306C22⋊C8—941no—
307#97—1882no—
308#95—941no—
309#95—941no—
310#95—941no—
311C24—21no—
312C24D4×D4711yescharacteristic
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—42no—
324C2×D4—42no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—942no—
331C22×C4—1881no—
332C22×C4—1881no—
333C22×C4D4×D4711yescharacteristic
334C22×C4#1062535211yescharacteristic
335C22×C4#1062535211yescharacteristic
336C22×C4C22×Dic4711yescharacteristic
337C22×C4C2×C22⋊D4711yescharacteristic
338C22×C4C2×C22⋊D4711yescharacteristic
339SD16—42no—
340SD16—42no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—942no—
348C2×C8—1881no—
349C2×C8—1881no—
350C2×C8C22⋊Dic4711yescharacteristic
351C2×C8C22⋊Dic4711yescharacteristic
352C2×C8C22⋊Dic4711yescharacteristic
353C2×C8C22⋊Dic4711yescharacteristic
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—1881no—
361C4⋊C4—1881no—
362C4⋊C4—1881no—
363C4⋊C4—1881no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—942no—
369C22⋊C4—942no—
370C22⋊C4—1882no—
371C22⋊C4—1882no—
372C42—1881no—
373C42—1881no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—81no—
379C23—81no—
380C23D4×Dic4711yescharacteristic
381C23#1309399511yescharacteristic
382C23#1309402011yescharacteristic
383C23#1309405112yes—
384C23#1309405312yes—
385Q8—41no—
386Q8—41no—
387Q8—81no—
388D4—41no—
389D4—41no—
390D4—81no—
391D4—81no—
392D4—81no—
393D4—82no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—3761no—
410C2×C4—3761no—
411C2×C4—3761no—
412C2×C4—3761no—
413C2×C4—3761no—
414C2×C4—3761no—
415C2×C4—1882no—
416C2×C4—1882no—
417C2×C4—1882no—
418C2×C4C2×C22⋊Dic4711yescharacteristic
419C2×C4D4×Dic4711yescharacteristic
420C2×C4#1309399511yescharacteristic
421C2×C4#1309401111yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—1882no—
426C8#1309393912yescomplemented
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—42no—
434C22—42no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—162no—
439C22#1568902611yescharacteristic
440C22#1568922412yes—
441C22#1568936612yes—
442C22#1568950911yescharacteristic
443C22#1568951411yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—81no—
448C4—81no—
449C4—81no—
450C4—3762no—
451C4—7521no—
452C4—7521no—
453C4C2×#1309393911yescharacteristic
454C4#1568909711yescharacteristic
455C2—22no—
456C2—21no—
457C2—22no—
458C2—21no—
459C2—161no—
460C2#2024422511yescharacteristic
461C2#2024427111yescharacteristic
462C2#2024428711yescharacteristic
subgroupG/H
centerC22#15689026
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#465C2
socleC22×C47#390
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenterC2×#465D47
hypocenterC47#9786
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#465

Complements

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

NH ≅ G/Ntype
1. #20244374C2semidirect
3. #20243417C2semidirect
4. #20243164C2semidirect
5. #20241960C2semidirect
20. C22×SD16×C47C4semidirect
28. C2×#106×C47C4semidirect
29. C2×#103×C47C4semidirect
30. C2×#102×C47C4semidirect
193. C8×C47#99semidirect
239. C22×SD16Dic47semidirect
248. C2×#106Dic47semidirect
249. C2×#103Dic47semidirect
250. C2×#102Dic47semidirect
262. C47#9786semidirect
426. C8#13093939semidirect

Maximal subgroups

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

subgroupindex
#202443742
#202442252
#202434172
#202431642
#202419602
#202419562
C2×C47×#4652
#978647