Finite Group Databasebeta

← #26112858

Subgroups — #26112858

Order 12032.

564 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254141C211yesmaximal, characteristic
2#20244224C211yesmaximal, characteristic
3#20241960C211yesmaximal, characteristic, complemented
4C2×#15688856C211yesmaximal, characteristic
5#20241267C211yesmaximal, characteristic, complemented
6#20241236C211yesmaximal, characteristic, complemented
7C47×#2368C211yesmaximal, characteristic, complemented
8C2×#13093998C2211yescharacteristic
9#15689451C412yescomplemented
10#15689441—21no—
11#15689354—22no—
12#15689273C412yescomplemented
13#15689262—21no—
14C2×#13093938—21no—
15C2×#13093938—21no—
16C2×C22⋊(C47⋊C8)C2211yescharacteristic
17C2×C22⋊(C47⋊C8)C2211yescharacteristic
18#15688983C2211yescharacteristic
19#15688860—22no—
20#15688856—21no—
21#15688856—21no—
22C2×C22⋊Q8×C47C2211yescharacteristic
23C47×#391—21no—
24C47×#387—21no—
25C47×#387—21no—
26C47×#385—21no—
27C4×C4⋊C4×C47—21no—
28C4×C22⋊C4×C47—21no—
29C4×C22⋊C4×C47C2211yescharacteristic
30C2×#95×C47C2211yescharacteristic
31C22×D188C2×C411yescharacteristic
32C22×C22⋊D47—21no—
33C2×#10625336—22no—
34C22×C47⋊C8—21no—
35C22×C47⋊C8—21no—
36#13093998—21no—
37#13093998—21no—
38#13093998—21no—
39#13093998—21no—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47C2×C411yescharacteristic
42C2×C22⋊Dic47—21no—
43C2×(C2×D47)⋊C4—21no—
44#13093955—22no—
45#13093948—22no—
46#13093944—22no—
47#13093938—42no—
48#13093938—42no—
49C22⋊(C47⋊C8)—21no—
50C22⋊(C47⋊C8)—21no—
51C22⋊(C47⋊C8)—21no—
52C22⋊(C47⋊C8)C2×C412yes—
53C47⋊#95—21no—
54C22×Q8×C47D411yescharacteristic
55C23×C4×C47—21no—
56C23×C4×C47C2311yescharacteristic
57C22⋊Q8×C47—21no—
58C22⋊Q8×C47—21no—
59C22⋊Q8×C47C2×C412yes—
60C22⋊Q8×C47D412yes—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47D411yescharacteristic
65C2×C22⋊C4×C47—21no—
66C2×C22⋊C4×C47D411yescharacteristic
67C2×C22⋊C4×C47D411yescharacteristic
68C2×C42×C47—21no—
69C2×C42×C47—21no—
70C2×C42×C47—21no—
71#95×C47—21no—
72#95×C47—21no—
73#95×C47—21no—
74#95×C47—21no—
75#95×C47—21no—
76#95×C47—21no—
77#95×C47D411yescharacteristic
78#95×C47D411yescharacteristic
79C24×C47C2×D411yescharacteristic
80C23×D47—21no—
81C2×Q8×C47—21no—
82C2×Q8×C47—81no—
83C2×Q8×C47C22⋊C412yes—
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—21no—
90C22×C4×C47—42no—
91C22×C4×C47—41no—
92C22×C4×C47—41no—
93C22×C4×C47—81no—
94C22×C4×C47—81no—
95C22×C4×C47C22×C411yescharacteristic
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47#4111yescharacteristic
100C22×C4×C47#4111yescharacteristic
101C2×C22⋊D47—41no—
102C2×C22⋊D47—41no—
103C2×C22⋊D47—41no—
104C2×C22⋊D47—41no—
105C2×C22⋊D47—41no—
106C2×C22⋊D47—41no—
107C22×Dic47—21no—
108C22×Dic47—41no—
109C2×D188—21no—
110C2×D188—21no—
111C2×D188—81no—
112C4⋊C4×C47—22no—
113C4⋊C4×C47—22no—
114C4⋊C4×C47—21no—
115C4⋊C4×C47—42no—
116C4⋊C4×C47—81no—
117C4⋊C4×C47C22⋊C412yes—
118C22⋊C4×C47—22no—
119C22⋊C4×C47—22no—
120C22⋊C4×C47—21no—
121C22⋊C4×C47—21no—
122C22⋊C4×C47—21no—
123C22⋊C4×C47SD1612yescomplemented
124C42×C47—42no—
125C42×C47—81no—
126C42×C47—81no—
127C22⋊Dic47—41no—
128C22⋊Dic47—41no—
129#10625336—82no—
130(C2×D47)⋊C4—41no—
131(C2×D47)⋊C4—41no—
132C4⋊Dic47—21no—
133C4⋊Dic47—21no—
134C4⋊Dic47—81no—
135C2×C47⋊C8—22no—
136C2×C47⋊C8—41no—
137C2×C47⋊C8—81no—
138C2×C47⋊C8—81no—
139C23×C47—41no—
140C23×C47—41no—
141C23×C47C4×D411yescharacteristic
142C23×C47#12011yescharacteristic
143C23×C47#12111yescharacteristic
144C23×C47C2×SD1611yescharacteristic
145C23×C47#13511yescharacteristic
146C23×C47AGL(1,ℤ/8)11yescharacteristic
147C23×C47#13711yescharacteristic
148C22×D47—41no—
149C22×D47—41no—
150C22×D47—41no—
151C22×D47—81no—
152C22×D47—81no—
153Q8×C47—42no—
154Q8×C47—81no—
155C2×C4×C47—22no—
156C2×C4×C47—22no—
157C2×C4×C47—22no—
158C2×C4×C47—22no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47—41no—
162C2×C4×C47—41no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
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—41no—
172C2×C4×C47—41no—
173C2×C4×C47—41no—
174C2×C4×C47—41no—
175C2×C4×C47—41no—
176C2×C4×C47—41no—
177C2×C4×C47—41no—
178C2×C4×C47—41no—
179C2×C4×C47—82no—
180C2×C4×C47—82no—
181C2×C4×C47C2×C22⋊C411yescharacteristic
182C2×C4×C47C4×D411yescharacteristic
183C2×C4×C47#12111yescharacteristic
184C2×C4×C47#12311yescharacteristic
185C22⋊D47—81no—
186C22⋊D47—81no—
187C22⋊D47—81no—
188C22⋊D47—81no—
189C2×Dic47—41no—
190C2×Dic47—41no—
191C2×Dic47—41no—
192C2×Dic47—81no—
193C2×Dic47—81no—
194C2×Dic47—81no—
195D188—42no—
196D188—81no—
197C47⋊C8—42no—
198C47⋊C8—81no—
199#9278—471nomaximal, Sylow
200C22×C47—21no—
201C22×C47—21no—
202C22×C47—21no—
203C22×C47—21no—
204C22×C47—21no—
205C22×C47—21no—
206C22×C47—41no—
207C22×C47—41no—
208C22×C47—41no—
209C22×C47—41no—
210C22×C47#38611yescharacteristic
211C22×C47C4×SD1611yescharacteristic
212C22×C47#44011yescharacteristic
213C22×C47#44811yescharacteristic
214C22×C47#44911yescharacteristic
215C22×C47#46011yescharacteristic
216C22×C47#46411yescharacteristic
217C2×D47—81no—
218C2×D47—81no—
219C2×D47—81no—
220C2×D47—161no—
221C2×D47—161no—
222C4×C47—21no—
223C4×C47—21no—
224C4×C47—21no—
225C4×C47—21no—
226C4×C47—42no—
227C4×C47—42no—
228C4×C47—42no—
229C4×C47—42no—
230C4×C47—81no—
231C4×C47—81no—
232Dic47—81no—
233Dic47—161no—
234#3187—471no—
235#2368D4711yescharacteristic, complemented
236#1964—471no—
237#1520—471no—
238C2×#327—471no—
239#1294—471no—
240#1282—471no—
241C2×C47—21no—
242C2×C47—21no—
243C2×C47—21no—
244C2×C47—21no—
245C2×C47—21no—
246C2×C47—21no—
247C2×C47#188011yescharacteristic
248C2×C47#190611yescharacteristic
249C2×C47#191311yescharacteristic
250D47—161no—
251C2×C22⋊Q8C2×D4711yescharacteristic
252C2×#121—471no—
253#483—472no—
254#481—941no—
255#465—942no—
256#450—472no—
257#448—941no—
258C2×#99—941no—
259C2×#99—941no—
260C2×C22⋊C8—471no—
261C2×C22⋊C8—471no—
262#391—21no—
263#387—21no—
264#387—21no—
265#385—21no—
266#380—471no—
267C4×C4⋊C4—21no—
268C4×C22⋊C4—21no—
269C4×C22⋊C4C2×D4711yescharacteristic
270C2×#95C2×D4711yescharacteristic
271#328—942no—
272#327—941no—
273#327—941no—
274C47#927811yescharacteristic, complemented, Sylow
275C22×Q8C22⋊D4711yescharacteristic
276C22×D4—471no—
277C22×D4—941no—
278C23×C4—21no—
279C23×C4C22×D4711yescharacteristic
280C2×SD16—942no—
281C22×C8—941no—
282C22×C8—941no—
283C22⋊Q8—21no—
284C22⋊Q8—21no—
285C22⋊Q8C4×D4712yes—
286C22⋊Q8D18812yes—
287#121—941no—
288#121—941no—
289#121—941no—
290#121—941no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—471no—
295C2×C4⋊C4—941no—
296C2×C4⋊C4C22⋊D4711yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—941no—
299C2×C22⋊C4—941no—
300C2×C22⋊C4C22⋊D4711yescharacteristic
301C2×C22⋊C4C22⋊D4711yescharacteristic
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305#106—942no—
306#103—942no—
307#102—942no—
308#99—1882no—
309#99—1882no—
310C22⋊C8—472no—
311C22⋊C8—941no—
312C22⋊C8—941no—
313C22⋊C8—941no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—941no—
321#95C22⋊D4711yescharacteristic
322#95C22⋊D4711yescharacteristic
323C24—941no—
324C24C2×C22⋊D4711yescharacteristic
325C2×Q8—21no—
326C2×Q8—81no—
327C2×Q8(C2×D47)⋊C412yes—
328C2×D4—941no—
329C2×D4—941no—
330C2×D4—3761no—
331C2×D4—1881no—
332C2×D4—1881no—
333C2×D4—1881no—
334C2×D4—1881no—
335C2×D4—1881no—
336C2×D4—1881no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—81no—
347C22×C4—81no—
348C22×C4—941no—
349C22×C4—1881no—
350C22×C4C2×C4×D4711yescharacteristic
351C22×C4C2×D18811yescharacteristic
352C22×C4#1062535011yescharacteristic
353C22×C4#1062535011yescharacteristic
354C22×C4C2×C22⋊D4711yescharacteristic
355C22×C4C2×C22⋊D4711yescharacteristic
356SD16—3762no—
357C2×C8—942no—
358C2×C8—3761no—
359C2×C8—3761no—
360C2×C8—1881no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—42no—
365C4⋊C4—81no—
366C4⋊C4—941no—
367C4⋊C4—941no—
368C4⋊C4—3761no—
369C4⋊C4(C2×D47)⋊C412yes—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—21no—
373C22⋊C4—21no—
374C22⋊C4—21no—
375C22⋊C4—1881no—
376C22⋊C4—1881no—
377C22⋊C4—1881no—
378C22⋊C4—1881no—
379C22⋊C4#1062533612yescomplemented
380C42—42no—
381C42—81no—
382C42—81no—
383C23—41no—
384C23—41no—
385C23—3761no—
386C23—3761no—
387C23—1881no—
388C23—1881no—
389C23—1881no—
390C23C4×C22⋊D4711yescharacteristic
391C23#1309399311yescharacteristic
392C23#1309399811yescharacteristic
393C23C2×#1062533611yescharacteristic
394C23#1309404411yescharacteristic
395C23#1309404911yescharacteristic
396C23#1309405511yescharacteristic
397Q8—42no—
398Q8—81no—
399D4—3761no—
400D4—3761no—
401D4—3761no—
402D4—3761no—
403D4—3761no—
404D4—1882no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—3761no—
432C2×C4—3761no—
433C2×C4—3761no—
434C2×C4—1881no—
435C2×C4—1881no—
436C2×C4—1881no—
437C2×C4C2×(C2×D47)⋊C411yescharacteristic
438C2×C4C4×C22⋊D4711yescharacteristic
439C2×C4#1309399811yescharacteristic
440C2×C4#1309400711yescharacteristic
441C8—3761no—
442C8—1882no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—3761no—
454C22—3761no—
455C22—3761no—
456C22—7521no—
457C22—7521no—
458C22#1568900211yescharacteristic
459C22C4×#1062533611yescharacteristic
460C22#1568922111yescharacteristic
461C22#1568926211yescharacteristic
462C22#1568926811yescharacteristic
463C22#1568932311yescharacteristic
464C22#1568934511yescharacteristic
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—42no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—81no—
474C4—81no—
475C4—3761no—
476C4—7521no—
477C2—21no—
478C2—21no—
479C2—21no—
480C2—21no—
481C2—21no—
482C2—21no—
483C2—7521no—
484C2#2024374311yescharacteristic
485C2#2024388911yescharacteristic
486C2#2024393011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2368C2
socleC22×C47#386
prime radicalC2×#13093998C22
cosocleC23×C4×C47C23
hypercenter#2368D47
hypocenterC47#9278
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2368

Complements

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

NH ≅ G/Ntype
3. #20241960C2semidirect
5. #20241267C2semidirect
6. #20241236C2semidirect
7. C47×#2368C2semidirect
9. #15689451C4semidirect
12. #15689273C4semidirect
123. C22⋊C4×C47SD16semidirect
235. #2368D47semidirect
274. C47#9278semidirect
379. C22⋊C4#10625336semidirect

Maximal subgroups

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

subgroupindex
#202541412
#202442242
#202419602
C2×#156888562
#202412672
#202412362
C47×#23682
#927847