Finite Group Databasebeta

← #27080619

Subgroups — #27080619

Order 12032.

2604 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20258159C211yesmaximal, characteristic
2#20258101C211yesmaximal, characteristic, complemented
3#20258100C211yesmaximal, characteristic, complemented
4#20252301C216yesmaximal, complemented
5#20252294C216yesmaximal, complemented
6#20252287C216yesmaximal, complemented
7C2×#15689242C216yesmaximal, complemented
8#20251997C211yesmaximal, characteristic, complemented
9#20251936C211yesmaximal, characteristic, complemented
10#20251816C211yesmaximal, characteristic, complemented
11C47×#2901C211yesmaximal, characteristic, complemented
12#15689959C2211yescharacteristic
13#15689938C414yescomplemented
14#15689934C414yescomplemented
15#15689931C414yescomplemented
16#15689926C414yescomplemented
17#15689921C2216yes—
18C2×#13094044C2216yes—
19#15689890—24no—
20#15689890C2211yescharacteristic
21#15689890C2216yescomplemented
22#15689889C2216yescomplemented
23#15689888—24no—
24C2×#13094032C2216yescomplemented
25C2×#13094031—24no—
26C2×#13094031C2211yescharacteristic
27C2×#13094031C2211yescharacteristic
28C2×#13094031C2216yescomplemented
29#131×D47—24no—
30#15689242C2214yescomplemented
31#15689242C22112yescomplemented
32#15689242C22112yescomplemented
33#15689242C22112yescomplemented
34#15689237C22124yescomplemented
35#15689171C2216yescomplemented
36C2×#13093959C2216yescomplemented
37#15689141C2216yescomplemented
38C2×#13093952C2216yescomplemented
39#15689079C2213yescomplemented
40#15689079C2213yescomplemented
41C4×#10625329C2213yescomplemented
42C2×#13093935C2213yescomplemented
43C47×#585C2211yescharacteristic
44C47×#568C2211yescharacteristic
45C4×#41×C47C2211yescharacteristic
46C47×#421C2216yescomplemented
47C2×#104×C47C2216yescomplemented
48#13094072—21no—
49#13094071C2×C414yes—
50#13094070C2×C414yes—
51#13094069—23no—
52C2×#10625350C2316yes—
53#41×D47—24no—
54#13094055C2×C4112yescomplemented
55#13094054C2×C4112yescomplemented
56#13094050C2×C4112yescomplemented
57#13094049C2×C4112yescomplemented
58#13094044C2×C4112yescomplemented
59#13094044C2×C4112yescomplemented
60#13094044C23112yes—
61#13094044C2314yes—
62C2×#10625337C2×C416yescomplemented
63C2×#10625335C2×C416yescomplemented
64C2×#10625336C2×C416yescomplemented
65C2×#10625334C2×C416yescomplemented
66#13094032—26no—
67#13094032—26no—
68#13094032C23112yes—
69#13094032C2314yes—
70#13094031—24no—
71#13094031—212no—
72#13094031—26no—
73#13094031—26no—
74#13094031C2×C414yes—
75#13094031C2×C414yes—
76#13094031D414yes—
77#13094031D414yes—
78#13094031C23112yes—
79#13094031C2314yes—
80#13094030—212no—
81C2×#10625329—212no—
82C2×#10625329—26no—
83C2×#10625329C2316yes—
84C2×#10625329C2313yes—
85C2×#10625329C2313yes—
86C2×#10625324—26no—
87M16×D47—212no—
88C22×C47⋊C8—26no—
89C22×C47⋊C8C2313yes—
90C22×C47⋊C8C2313yes—
91C2×C8×D47—26no—
92#13093959C23112yes—
93#13093959C2314yes—
94#13093952C23112yes—
95#13093952C2314yes—
96#13093935C2311yescharacteristic
97#13093935C2313yes—
98#13093935C2313yes—
99#13093935C2313yes—
100C4×C47⋊C8C2316yes—
1012−1+4×C47—21no—
1022−1+4×C47C2×C414yes—
103H(5,𝔽2)×C47—23no—
104H(5,𝔽2)×C47C2×C414yes—
105C2×#41×C47—24no—
106C2×#41×C47C2316yes—
107C2×#41×C47C2311yescharacteristic
108#131×C47—24no—
109C2×M16×C47C2316yes—
110C4×Q8×C47—21no—
111C4×D4×C47—23no—
112#117×C47C2313yes—
113C2×C42×C47C2313yes—
114#104×C47C23112yes—
115#104×C47C2314yes—
116#41×C47—24no—
117#41×C47—212no—
118#41×C47—26no—
119#41×C47—26no—
120#41×C47C22×C4112yes—
121#41×C47C22×C414yes—
122#41×C47C2×D414yes—
123#41×C47C2414yes—
124C2×Q8×C47—24no—
125C2×Q8×C47C22×C416yes—
126C2×Q8×C47#4111yescharacteristic
127C2×D4×C47—26no—
128C2×D4×C47—212no—
129C2×D4×C47C22×C416yes—
130C2×D4×C47#4113yes—
131C22×C4×C47—26no—
132C22×C4×C47—23no—
133C22×C4×C47C2413yes—
134C2×C22⋊D47—26no—
135#10625354—24no—
136Q8×D47—24no—
137#10625352—212no—
138D4×D47—212no—
139#10625350—26no—
140#10625350—26no—
141#10625350C22×C4112yes—
142#10625350C2414yes—
143C2×D188C22×C416yes—
144C2×C4×D47—26no—
145C2×Dic94C22×C416yes—
146M16×C47—26no—
147M16×C47C2414yes—
148C2×C8×C47—26no—
149C4⋊C4×C47—23no—
150C22⋊C4×C47—23no—
151C42×C47—23no—
152C42×C47C2411yescharacteristic
153C42×C47C2413yes—
154#10625337C22×C414yes—
155#10625337C22×C4112yes—
156#10625336C22×C414yes—
157#10625336C22×C4112yes—
158#10625335C22×C414yes—
159#10625335C22×C4112yes—
160#10625334C22×C414yes—
161#10625334C22×C4112yes—
162#10625329—26no—
163#10625329—24no—
164#10625329—212no—
165#10625329C22×C4112yes—
166#10625329C2×D4112yes—
167#10625329C2414yes—
168C2×C47⋊C8—26no—
169C2×C47⋊C8—24no—
170C2×C47⋊C8—212no—
171C2×C47⋊C8C22×C416yes—
172C2×C47⋊C8C22×C416yes—
173C2×C47⋊C8C2×D416yes—
174C2×C47⋊C8C2×D416yes—
175C2×C47⋊C8C2413yes—
176C2×C47⋊C8C2411yescharacteristic
177C2×C47⋊C8C2413yes—
178C2×C47⋊C8C2411yescharacteristic
179#10625324—212no—
180#10625324—24no—
181C8×D47—24no—
182C8×D47—212no—
183C23×C47—26no—
184C23×C47C2×#4113yes—
185C22×D47—26no—
186Q8×C47—26no—
187Q8×C47C4×D414yes—
188Q8×C47C23×C414yes—
189D4×C47—212no—
190D4×C47—24no—
191D4×C47—26no—
192D4×C47C4×D4112yes—
193D4×C47C23×C414yes—
194C2×C4×C47—26no—
195C2×C4×C47—212no—
196C2×C4×C47—24no—
197C2×C4×C47—21no—
198C2×C4×C47—23no—
199C2×C4×C47—83no—
200C2×C4×C47C23×C416yes—
201C2×C4×C47C22×D416yes—
202C2×C4×C47C2×#4113yes—
203C2×C4×C47C2511yescharacteristic
204C22⋊D47—212no—
205C22⋊D47—24no—
206C2×Dic47—26no—
207D188—26no—
208D188C23×C414yes—
209C4×D47—212no—
210C4×D47—24no—
211Dic94—26no—
212Dic94C23×C414yes—
213C8×C47—24no—
214C47⋊C8—24no—
215C47⋊C8C23×C411yescharacteristic
216C47⋊C8C23×C411yescharacteristic
217C47⋊C8C23×C416yes—
218C47⋊C8C22×D411yescharacteristic
219C47⋊C8C22×D416yes—
220C47⋊C8C22×D411yescharacteristic
221#60184—471nomaximal, Sylow
222C22×C47—24no—
223C22×C47—43no—
224C22×C47—86no—
225C22×C47C2×C4×D416yes—
226C22×C47C22×#4111yescharacteristic
227C2×D47—24no—
228C2×D47—86no—
229C4×C47—24no—
230C4×C47—43no—
231C4×C47—41no—
232C4×C47C2×C4×D416yes—
233C4×C47C24×C411yescharacteristic
234C4×C47C23×D411yescharacteristic
235Dic47—24no—
236#3584—471no—
237#3573—471no—
238#3573—471no—
239#2957—476no—
240#2956—476no—
241#2955—476no—
242C2×#444—476no—
243#2916—471no—
244#2901—471no—
245#2901D4711yescharacteristic, complemented
246#2875—471no—
247C2×C47—21no—
248C2×C47—26no—
249C2×C47—44no—
250C2×C47C22×C4×D411yescharacteristic
251D47—44no—
252#585—471no—
253#585C2×D4711yescharacteristic
254#578—474no—
255#577—474no—
256#577—474no—
257#576—474no—
258#575—476no—
259C2×#135—476no—
260#568—471no—
261#568—476no—
262#568—476no—
263#568—944no—
264#568—944no—
265#568C2×D4711yescharacteristic
266C2×#131—471no—
267C2×#131—471no—
268C2×#131—476no—
269C2×#131—476no—
270C2×#131—944no—
271C2×#131—944no—
272C4×#41C2×D4711yescharacteristic
273#444—474no—
274#444—4712no—
275#444—4712no—
276#444—4712no—
277#443—4724no—
278#430—476no—
279C2×#108—476no—
280#421—476no—
281#421C2×D4716yescomplemented
282C2×#104—476no—
283C2×#104C2×D4716yescomplemented
284#405—473no—
285#405—473no—
286C4×M16—473no—
287C2×#97—473no—
288C47#6018411yescharacteristic, complemented, Sylow
2892−1+4—21no—
2902−1+4—474no—
2912−1+4—941no—
2922−1+4C4×D4714yes—
293H(5,𝔽2)—23no—
294H(5,𝔽2)—474no—
295H(5,𝔽2)—943no—
296H(5,𝔽2)C4×D4714yes—
297C2×#41—24no—
298C2×#41—476no—
299C2×#41—944no—
300C2×#41C22×D4716yes—
301C2×#41C22×D4711yescharacteristic
302#137—4712no—
303#137—4712no—
304AGL(1,ℤ/8)—4712no—
305AGL(1,ℤ/8)—4712no—
306#135—4712no—
307#135—4712no—
308#135—474no—
309#135—4712no—
310C2×Q16—476no—
311C2×SD16—476no—
312C2×SD16—476no—
313C2×D8—476no—
314#131—24no—
315#131—474no—
316#131—474no—
317#131—474no—
318#131—474no—
319#131—474no—
320#131—4712no—
321#131—474no—
322#131—4712no—
323#131—9412no—
324#131—946no—
325#131—946no—
326#131—946no—
327#131—946no—
328#131—9412no—
329#131—944no—
330C2×M16—473no—
331C2×M16—473no—
332C2×M16—476no—
333C2×M16—946no—
334C2×M16—946no—
335C2×M16—9412no—
336C2×M16—9412no—
337C2×M16C22×D4716yes—
338C22×C8—473no—
339C22×C8—473no—
340C22×C8—946no—
341C22×C8—946no—
342C4×Q8—21no—
343C4×D4—23no—
344#117C22×D4713yes—
345C2×C42C22×D4713yes—
346#108—474no—
347#108—4712no—
348#104—474no—
349#104—4712no—
350#104C22×D4714yes—
351#104C22×D47112yes—
352#97—471no—
353#97—473no—
354#97—473no—
355#97—473no—
356C4×C8—476no—
357#41—24no—
358#41—212no—
359#41—26no—
360#41—26no—
361#41—4712no—
362#41—474no—
363#41—944no—
364#41—9412no—
365#41—946no—
366#41—946no—
367#41C2×C4×D47112yes—
368#41C2×C4×D4714yes—
369#41D4×D4714yes—
370#41C23×D4714yes—
371C2×Q8—24no—
372C2×Q8—476no—
373C2×Q8—944no—
374C2×Q8C2×C4×D4716yes—
375C2×Q8#1062535211yescharacteristic
376C2×D4—26no—
377C2×D4—212no—
378C2×D4—476no—
379C2×D4—946no—
380C2×D4—9412no—
381C2×D4C2×C4×D4716yes—
382C2×D4#1062535213yes—
383C22×C4—26no—
384C22×C4—23no—
385C22×C4—946no—
386C22×C4C23×D4713yes—
387Q16—474no—
388Q16—4712no—
389SD16—474no—
390SD16—4712no—
391SD16—474no—
392SD16—4712no—
393D8—474no—
394D8—4712no—
395M16—26no—
396M16—4712no—
397M16—4712no—
398M16—474no—
399M16—944no—
400M16—9412no—
401M16—944no—
402M16—9412no—
403M16—946no—
404M16C23×D4714yes—
405C2×C8—26no—
406C2×C8—476no—
407C2×C8—476no—
408C2×C8—476no—
409C2×C8—476no—
410C2×C8—473no—
411C2×C8—473no—
412C2×C8—471no—
413C2×C8—471no—
414C2×C8—944no—
415C2×C8—9412no—
416C2×C8—944no—
417C2×C8—9412no—
418C2×C8—946no—
419C4⋊C4—23no—
420C22⋊C4—23no—
421C42—23no—
422C42C23×D4713yes—
423C42C23×D4711yescharacteristic
424C23—26no—
425C23—946no—
426C23C2×#1062535213yes—
427Q8—26no—
428Q8—474no—
429Q8—946no—
430Q8(C22⋊D47)⋊C414yes—
431Q8C22×C4×D4714yes—
432D4—212no—
433D4—24no—
434D4—26no—
435D4—474no—
436D4—9412no—
437D4—944no—
438D4—946no—
439D4(C22⋊D47)⋊C4112yes—
440D4C22×C4×D4714yes—
441C2×C4—26no—
442C2×C4—212no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—23no—
446C2×C4—83no—
447C2×C4—946no—
448C2×C4—9412no—
449C2×C4—944no—
450C2×C4C22×C4×D4716yes—
451C2×C4C2×D4×D4716yes—
452C2×C4C2×#1062535213yes—
453C2×C4C24×D4711yescharacteristic
454C8—24no—
455C8—471no—
456C8—471no—
457C8—476no—
458C8—471no—
459C8—471no—
460C8—476no—
461C8—944no—
462C22—24no—
463C22—43no—
464C22—86no—
465C22—944no—
466C22—3766no—
467C22C2×(C22⋊D47)⋊C416yes—
468C22C22×#1062535211yescharacteristic
469C4—24no—
470C4—41no—
471C4—43no—
472C4—944no—
473C4C2×(C22⋊D47)⋊C416yes—
474C4C23×C4×D4711yescharacteristic
475C4C22×D4×D4711yescharacteristic
476C2—21no—
477C2—26no—
478C2—44no—
479C2—1884no—
480C2C22×(C22⋊D47)⋊C411yescharacteristic
subgroupG/H
centerC4C22×D4×D47
commutatorC4×C47C24×C4
FrattiniC2×C4C24×D47
FittingC47×#2901C2
socleC2×C47C22×C4×D4
prime radical#20258159C2
cosocleC2×C4×C47C25
hypercenter#2901D47
hypocenterC47#60184
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×#41 ◁ #2901

Complements

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

NH ≅ G/Ntype
2. #20258101C2semidirect
3. #20258100C2semidirect
4. #20252301C2semidirect
5. #20252294C2semidirect
6. #20252287C2semidirect
7. C2×#15689242C2semidirect
8. #20251997C2semidirect
9. #20251936C2semidirect
10. #20251816C2semidirect
11. C47×#2901C2semidirect
13. #15689938C4semidirect
14. #15689934C4semidirect
15. #15689931C4semidirect
16. #15689926C4semidirect
21. #15689890C22semidirect
22. #15689889C22semidirect
24. C2×#13094032C22semidirect
28. C2×#13094031C22semidirect
30. #15689242C22semidirect
31. #15689242C22semidirect
32. #15689242C22semidirect
33. #15689242C22semidirect
34. #15689237C22semidirect
35. #15689171C22semidirect
36. C2×#13093959C22semidirect
37. #15689141C22semidirect
38. C2×#13093952C22semidirect
39. #15689079C22semidirect
40. #15689079C22semidirect
41. C4×#10625329C22semidirect
42. C2×#13093935C22semidirect
46. C47×#421C22semidirect
47. C2×#104×C47C22semidirect
54. #13094055C2×C4semidirect
55. #13094054C2×C4semidirect
56. #13094050C2×C4semidirect
57. #13094049C2×C4semidirect
58. #13094044C2×C4semidirect
59. #13094044C2×C4semidirect
62. C2×#10625337C2×C4semidirect
63. C2×#10625335C2×C4semidirect
64. C2×#10625336C2×C4semidirect
65. C2×#10625334C2×C4semidirect
245. #2901D47semidirect
281. #421C2×D47semidirect
283. C2×#104C2×D47semidirect
288. C47#60184semidirect

Maximal subgroups

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

subgroupindex
#202581592
#202581012
#202581002
#202523012
#202522942
#202522872
C2×#156892422
#202519972
#202519362
#202518162
C47×#29012
#6018447