Finite Group Databasebeta

← #26202122

Subgroups — #26202122

Order 12032.

640 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254513C211yesmaximal, characteristic, complemented
2#20253203C211yesmaximal, characteristic, complemented
3#20252505C211yesmaximal, characteristic, complemented
4#20242926C212yesmaximal, complemented
5#20242919C212yesmaximal, complemented
6#20242892C212yesmaximal, complemented
7#20242885C212yesmaximal, complemented
8#20242162C211yesmaximal, characteristic, complemented
9#20242065C211yesmaximal, characteristic, complemented
10C2×#15688871C211yesmaximal, characteristic, complemented
11C47×#3035C211yesmaximal, characteristic
12C2×#13094024C2211yescharacteristic
13#15689519C2212yes—
14#15689469C2212yes—
15#15689439—22no—
16#15689418C2212yes—
17#15689413—21no—
18#15689413—21no—
19#15689413C2212yes—
20#15689408—21no—
21#15689403—21no—
22#15689254C2212yes—
23#15689201C2212yes—
24#15689195—22no—
25#15689180—21no—
26#15689176C2211yescharacteristic
27#15689166—21no—
28#15689147—21no—
29#15689147C2211yescharacteristic
30C2×C4⋊(C47⋊C8)C2211yescharacteristic
31#15689124—21no—
32#15688875C2212yes—
33#15688875C2212yes—
34#15688871C2212yes—
35#15688871C2212yes—
36#15688869C2212yes—
37#15688867C2212yes—
38C2×#135×C47—21no—
39C47×#532C2211yescharacteristic
40C4×#41×C47C2211yescharacteristic
41C47×#464—21no—
42C47×#463—21no—
43C47×#462C2211yescharacteristic
44C47×#461C2211yescharacteristic
45C47×#460C2211yescharacteristic
46C47×#459C2211yescharacteristic
47C2×C4⋊C8×C47C2211yescharacteristic
48C47×#416—21no—
49C22×Dic94—21no—
50C2×#10625329—21no—
51C2×#10625329—22no—
52C22×C47⋊C8—21no—
53#13094024—21no—
54#13094024—21no—
55#13094024—21no—
56#13094024C2312yes—
57C2×C4⋊Dic47—21no—
58C2×C4⋊Dic47—21no—
59#13093957—22no—
60#13093955—22no—
61#13093955—22no—
62#13093955—22no—
63C4⋊(C47⋊C8)—21no—
64C4⋊(C47⋊C8)—21no—
65C4⋊(C47⋊C8)—22no—
66C4⋊(C47⋊C8)D411yescharacteristic
67C4⋊(C47⋊C8)D411yescharacteristic
68C4⋊(C47⋊C8)C2312yes—
69C4⋊(C47⋊C8)C2312yes—
70#13093948—22no—
71#13093945—22no—
72#13093945—22no—
73#13093945—22no—
74C22⋊(C47⋊C8)—22no—
75C22⋊(C47⋊C8)D412yes—
76#13093935—22no—
77C4×C47⋊C8C2312yes—
78C2×#41×C47—21no—
79C2×#41×C47D411yescharacteristic
80#135×C47—44no—
81C2×Q16×C47—21no—
82C2×SD16×C47—21no—
83C2×SD16×C47—21no—
84C2×D8×C47—21no—
85C22×C8×C47—21no—
86C4⋊Q8×C47C2311yescharacteristic
87#127×C47C2311yescharacteristic
88#124×C47—21no—
89#121×C47—21no—
90#121×C47—21no—
91C4×Q8×C47D411yescharacteristic
92C4×Q8×C47C2311yescharacteristic
93C4×D4×C47—21no—
94C4×D4×C47—21no—
95C4×D4×C47—21no—
96C4×D4×C47—21no—
97C4×D4×C47D411yescharacteristic
98C4×D4×C47C2311yescharacteristic
99#117×C47—21no—
100#117×C47D411yescharacteristic
101C2×C42×C47—21no—
102C2×C42×C47C2311yescharacteristic
103C4⋊C8×C47—21no—
104C4⋊C8×C47C2311yescharacteristic
105C4⋊C8×C47C2311yescharacteristic
106#103×C47—21no—
107#103×C47—21no—
108#102×C47—21no—
109#102×C47—21no—
110#41×C47—22no—
111#41×C47—22no—
112#41×C47—42no—
113#41×C47—42no—
114C2×Q8×C47—21no—
115C2×Q8×C47C2×D411yescharacteristic
116C2×D4×C47—21no—
117C2×D4×C47—21no—
118C2×D4×C47—21no—
119C2×D4×C47—21no—
120C2×D4×C47—41no—
121C2×D4×C47C2×D411yescharacteristic
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—41no—
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47#4111yescharacteristic
129C22×Dic47—41no—
130C2×Dic94—22no—
131C2×Dic94—41no—
132C2×Dic94—82no—
133Q16×C47—42no—
134SD16×C47—42no—
135SD16×C47—42no—
136D8×C47—42no—
137C2×C8×C47—21no—
138C2×C8×C47—21no—
139C2×C8×C47—42no—
140C4⋊C4×C47—21no—
141C4⋊C4×C47—21no—
142C4⋊C4×C47—21no—
143C4⋊C4×C47—21no—
144C4⋊C4×C47C2×D411yescharacteristic
145C4⋊C4×C47C2×D411yescharacteristic
146C22⋊C4×C47—21no—
147C22⋊C4×C47—21no—
148C22⋊C4×C47—21no—
149C22⋊C4×C47—41no—
150C22⋊C4×C47—41no—
151C42×C47—21no—
152C42×C47—21no—
153C42×C47—21no—
154C42×C47C2×D411yescharacteristic
155C42×C47C2×D411yescharacteristic
156C42×C47#4111yescharacteristic
157C42×C47C2411yescharacteristic
158C4⋊Dic47—22no—
159C4⋊Dic47—22no—
160C4⋊Dic47—41no—
161C4⋊Dic47—41no—
162#10625329—42no—
163#10625329—82no—
164C2×C47⋊C8—22no—
165C2×C47⋊C8—22no—
166C2×C47⋊C8—81no—
167C2×C47⋊C8C2×D412yes—
168C2×C47⋊C8C2×D412yes—
169C23×C47—21no—
170C23×C47—41no—
171C23×C47#12111yescharacteristic
172Q8×C47—21no—
173Q8×C47—21no—
174Q8×C47—41no—
175Q8×C47—41no—
176D4×C47—21no—
177D4×C47—21no—
178D4×C47—42no—
179D4×C47—41no—
180D4×C47—41no—
181D4×C47—41no—
182D4×C47—41no—
183D4×C47—81no—
184D4×C47—82no—
185C2×C4×C47—21no—
186C2×C4×C47—21no—
187C2×C4×C47—21no—
188C2×C4×C47—21no—
189C2×C4×C47—21no—
190C2×C4×C47—41no—
191C2×C4×C47—41no—
192C2×C4×C47—41no—
193C2×C4×C47—42no—
194C2×C4×C47—41no—
195C2×C4×C47—81no—
196C2×C4×C47—82no—
197C2×C4×C47#12111yescharacteristic
198C2×C4×C47#12111yescharacteristic
199C2×C4×C47#12111yescharacteristic
200C2×C4×C47AGL(1,ℤ/8)12yes—
201C2×C4×C47#13712yes—
202C2×C4×C47C22×D411yescharacteristic
203C2×C4×C47C22×D411yescharacteristic
204C2×C4×C47C2×#4111yescharacteristic
205C2×Dic47—42no—
206C2×Dic47—82no—
207Dic94—82no—
208Dic94—82no—
209C8×C47—41no—
210C8×C47—41no—
211C47⋊C8—22no—
212C47⋊C8—22no—
213C47⋊C8—42no—
214C47⋊C8—42no—
215#19450—471nomaximal, Sylow
216C22×C47—21no—
217C22×C47—21no—
218C22×C47—41no—
219C22×C47—41no—
220C22×C47—41no—
221C22×C47—81no—
222C22×C47—81no—
223C22×C47—81no—
224C22×C47#46812yes—
225C22×C47C2×#12111yescharacteristic
226C4×C47—21no—
227C4×C47—21no—
228C4×C47—21no—
229C4×C47—41no—
230C4×C47—41no—
231C4×C47—41no—
232C4×C47—81no—
233C4×C47#46812yes—
234C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
235C4×C47C2×#13711yescharacteristic
236C4×C47#57711yescharacteristic
237C4×C47#57811yescharacteristic
238Dic47—82no—
239#3227—471no—
240#3079—471no—
241#3035D4711yescharacteristic
242#2989—471no—
243#1698—472no—
244#1697—472no—
245#1692—472no—
246#1691—472no—
247#1571—471no—
248#1547—471no—
249C2×#333—471no—
250C2×C47—22no—
251C2×C47—81no—
252C2×C47—161no—
253C2×C47#304611yescharacteristic
254C2×C47C2×#46811yescharacteristic
255C2×C47#306811yescharacteristic
256C2×#135—21no—
257#532C2×D4711yescharacteristic
258C2×C4⋊Q8—471no—
259C4×#41C2×D4711yescharacteristic
260#498—472no—
261#486—472no—
262#481—942no—
263#477—472no—
264#476—472no—
265#476—941no—
266#476—941no—
267#475—941no—
268#474—941no—
269#464—21no—
270#463—21no—
271#462C2×D4711yescharacteristic
272#461C2×D4711yescharacteristic
273#460C2×D4711yescharacteristic
274#459C2×D4711yescharacteristic
275#446—472no—
276#436—472no—
277#435—942no—
278#432—941no—
279#431—471no—
280#428—941no—
281#423—471no—
282#423—941no—
283C2×C4⋊C8—471no—
284C2×C4⋊C8C2×D4711yescharacteristic
285#417—941no—
286#416—21no—
287#335—472no—
288#335—472no—
289#333—472no—
290#333—472no—
291#332—472no—
292#331—472no—
293C47#1945011yescharacteristic, complemented, Sylow
294C2×#41—21no—
295C2×#41C22⋊D4711yescharacteristic
296C22×Q8—941no—
297#135—44no—
298C2×Q16—21no—
299C2×SD16—21no—
300C2×SD16—21no—
301C2×D8—21no—
302C2×M16—942no—
303C2×M16—941no—
304C22×C8—21no—
305C22×C8—941no—
306C4⋊Q8—472no—
307C4⋊Q8—941no—
308C4⋊Q8—941no—
309C4⋊Q8—941no—
310C4⋊Q8C22×D4711yescharacteristic
311#127C22×D4711yescharacteristic
312#124—21no—
313#121—21no—
314#121—21no—
315C4×Q8C22⋊D4711yescharacteristic
316C4×Q8C22×D4711yescharacteristic
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4C22⋊D4711yescharacteristic
322C4×D4C22×D4711yescharacteristic
323#117—21no—
324#117C22⋊D4711yescharacteristic
325C2×C4⋊C4—941no—
326C2×C4⋊C4—941no—
327C2×C42—21no—
328C2×C42C22×D4711yescharacteristic
329#107—942no—
330#106—942no—
331#106—942no—
332#106—942no—
333C4⋊C8—21no—
334C4⋊C8—471no—
335C4⋊C8—471no—
336C4⋊C8—472no—
337C4⋊C8—472no—
338C4⋊C8—942no—
339C4⋊C8—941no—
340C4⋊C8—941no—
341C4⋊C8C22×D4711yescharacteristic
342C4⋊C8C22×D4711yescharacteristic
343#103—21no—
344#103—21no—
345#103—942no—
346#102—21no—
347#102—21no—
348#102—942no—
349#102—942no—
350#102—942no—
351C22⋊C8—472no—
352C22⋊C8—942no—
353#97—942no—
354C4×C8—472no—
355#41—22no—
356#41—22no—
357#41—42no—
358#41—42no—
359C2×Q8—21no—
360C2×Q8—942no—
361C2×Q8—3762no—
362C2×Q8—1881no—
363C2×Q8C2×C22⋊D4711yescharacteristic
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4C2×C22⋊D4711yescharacteristic
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—41no—
374C22×C4—1881no—
375C22×C4D4×D4711yescharacteristic
376C22×C4#1062535211yescharacteristic
377C22×C4C2×C22⋊D4711yescharacteristic
378Q16—42no—
379SD16—42no—
380SD16—42no—
381D8—42no—
382M16—3762no—
383M16—1882no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—42no—
387C2×C8—472no—
388C2×C8—472no—
389C2×C8—942no—
390C2×C8—942no—
391C2×C8—3761no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—942no—
397C4⋊C4—942no—
398C4⋊C4—1881no—
399C4⋊C4—1881no—
400C4⋊C4C2×C22⋊D4711yescharacteristic
401C4⋊C4C2×C22⋊D4711yescharacteristic
402C22⋊C4—21no—
403C22⋊C4—21no—
404C22⋊C4—21no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42D4×D4711yescharacteristic
411C42#1062535211yescharacteristic
412C42C2×C22⋊D4711yescharacteristic
413C42C23×D4711yescharacteristic
414C23—21no—
415C23—41no—
416C23#1309399611yescharacteristic
417Q8—21no—
418Q8—21no—
419Q8—41no—
420Q8—41no—
421Q8—3762no—
422Q8—3762no—
423D4—21no—
424D4—21no—
425D4—42no—
426D4—41no—
427D4—41no—
428D4—41no—
429D4—41no—
430D4—81no—
431D4—82no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—3762no—
445C2×C4—1882no—
446C2×C4#1309399611yescharacteristic
447C2×C4#1309399611yescharacteristic
448C2×C4#1309399611yescharacteristic
449C2×C4#1309405012yes—
450C2×C4#1309405412yes—
451C2×C4C2×D4×D4711yescharacteristic
452C2×C4C22×C22⋊D4711yescharacteristic
453C2×C4C2×#1062535211yescharacteristic
454C8—41no—
455C8—41no—
456C8—942no—
457C8—942no—
458C8—1882no—
459C8—1882no—
460C22—21no—
461C22—21no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22#1568936712yes—
469C22C2×#1309399611yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—81no—
477C4—3762no—
478C4#1568936712yes—
479C4C2×#1309405011yescharacteristic
480C4C2×#1309405411yescharacteristic
481C4#1568993011yescharacteristic
482C4#1568993711yescharacteristic
483C2—22no—
484C2—81no—
485C2—161no—
486C2#2025293611yescharacteristic
487C2C2×#1568936711yescharacteristic
488C2#2025311811yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
Fitting, prime radicalC47×#3035C2
socleC22×C47C2×#121
hypercenter#3035D47
hypocenterC47#19450
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #20254513C2semidirect
2. #20253203C2semidirect
3. #20252505C2semidirect
4. #20242926C2semidirect
5. #20242919C2semidirect
6. #20242892C2semidirect
7. #20242885C2semidirect
8. #20242162C2semidirect
9. #20242065C2semidirect
10. C2×#15688871C2semidirect
293. C47#19450semidirect

Maximal subgroups

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

subgroupindex
#202545132
#202532032
#202525052
#202429262
#202429192
#202428922
#202428852
#202421622
#202420652
C2×#156888712
C47×#30352
#1945047