Finite Group Databasebeta

← #27080621

Subgroups — #27080621

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#20258099C211yesmaximal, characteristic, complemented
3#20252303C216yesmaximal, complemented
4#20252295C216yesmaximal, complemented
5#20252289C216yesmaximal, complemented
6C2×#15689243C216yesmaximal, complemented
7#20251997C211yesmaximal, characteristic, complemented
8#20251935C211yesmaximal, characteristic, complemented
9#20251933C211yesmaximal, characteristic, complemented
10#20251814C211yesmaximal, characteristic, complemented
11C47×#3573C211yesmaximal, characteristic, complemented
12#15689959C2211yescharacteristic
13#15689938C414yescomplemented
14#15689934C414yescomplemented
15#15689931C414yescomplemented
16#15689926C414yescomplemented
17#15689921C2216yes—
18C2×#13094044C2216yes—
19#15689890C2211yescharacteristic
20#15689889C2216yescomplemented
21#15689888—24no—
22C2×#13094031C2211yescharacteristic
23C2×#13094030C2216yescomplemented
24#131×D47—24no—
25#41×Dic47C2211yescharacteristic
26#15689243C2214yescomplemented
27#15689243C22112yescomplemented
28#15689243C22112yescomplemented
29#15689243C22112yescomplemented
30#15689238C22124yescomplemented
31#15689171C2216yescomplemented
32C2×#13093959C2216yescomplemented
33#15689140C2216yescomplemented
34#15689139C2216yescomplemented
35C2×#13093951C2216yescomplemented
36C2×#13093950C2216yescomplemented
37#15689081C2213yescomplemented
38#15689078C2213yescomplemented
39M16×Dic47C2213yescomplemented
40C2×#13093934C2213yescomplemented
41C47×#585C2211yescharacteristic
42C47×#568—24no—
43C47×#568C2211yescharacteristic
44C47×#568C2216yescomplemented
45C2×#131×C47—24no—
46C2×#131×C47C2211yescharacteristic
47C2×#131×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—212no—
67#13094031—24no—
68#13094031C2×C414yes—
69#13094031C2×C414yes—
70#13094030—26no—
71#13094030—26no—
72#13094030C23112yes—
73#13094030C2314yes—
74C2×#10625329C2316yes—
75C2×#10625329C2313yes—
76C2×#10625324—26no—
77M16×D47—212no—
78C22×C47⋊C8C2313yes—
79C2×C8×D47—26no—
80Q8×Dic47—21no—
81D4×Dic47—23no—
82#13093983C2313yes—
83C2×C4×Dic47C2313yes—
84#13093959C23112yes—
85#13093959C2314yes—
86#13093951C23112yes—
87#13093951C2314yes—
88#13093950C23112yes—
89#13093950C2314yes—
90#13093934C2311yescharacteristic
91#13093934C2313yes—
92#13093934C2313yes—
93#13093934C2313yes—
94C8×Dic47C2316yes—
952−1+4×C47—21no—
962−1+4×C47C2×C414yes—
97H(5,𝔽2)×C47—23no—
98H(5,𝔽2)×C47C2×C414yes—
99C2×#41×C47—24no—
100C2×#41×C47C2316yes—
101C2×#41×C47C2311yescharacteristic
102#131×C47—24no—
103#131×C47—212no—
104#131×C47—26no—
105#131×C47—26no—
106#131×C47D414yes—
107#131×C47D414yes—
108#131×C47C23112yes—
109#131×C47C2314yes—
110C2×M16×C47—212no—
111C2×M16×C47—26no—
112C2×M16×C47C2316yes—
113C2×M16×C47C2313yes—
114C22×C8×C47—26no—
115C22×C8×C47C2313yes—
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×C47C2413yes—
133C2×C22⋊D47—26no—
134C22×Dic47—23no—
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×C47—24no—
148M16×C47—212no—
149M16×C47C2×D4112yes—
150M16×C47C2414yes—
151C2×C8×C47—26no—
152C2×C8×C47—24no—
153C2×C8×C47—212no—
154C2×C8×C47C2×D416yes—
155C2×C8×C47C2×D416yes—
156C2×C8×C47C2411yescharacteristic
157C2×C8×C47C2413yes—
158C22⋊Dic47—23no—
159#10625337C22×C414yes—
160#10625337C22×C4112yes—
161#10625336C22×C414yes—
162#10625336C22×C4112yes—
163#10625335C22×C414yes—
164#10625335C22×C4112yes—
165#10625334C22×C414yes—
166#10625334C22×C4112yes—
167C4⋊Dic47—23no—
168C4×Dic47—23no—
169C4×Dic47C2413yes—
170C4×Dic47C2411yescharacteristic
171#10625329—26no—
172#10625329C22×C4112yes—
173#10625329C2414yes—
174C2×C47⋊C8—26no—
175C2×C47⋊C8C22×C416yes—
176C2×C47⋊C8C22×C416yes—
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×C47C23×C416yes—
198C2×C4×C47C22×D416yes—
199C2×C4×C47C2×#4113yes—
200C2×C4×C47C2511yescharacteristic
201C22⋊D47—212no—
202C22⋊D47—24no—
203C2×Dic47—26no—
204C2×Dic47—21no—
205C2×Dic47—23no—
206C2×Dic47—83no—
207D188—26no—
208D188C23×C414yes—
209C4×D47—212no—
210C4×D47—24no—
211Dic94—26no—
212Dic94C23×C414yes—
213C8×C47—24no—
214C8×C47C22×D411yescharacteristic
215C8×C47C22×D416yes—
216C8×C47C22×D411yescharacteristic
217C47⋊C8—24no—
218C47⋊C8C23×C411yescharacteristic
219C47⋊C8C23×C411yescharacteristic
220C47⋊C8C23×C416yes—
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×C47C2×C4×D416yes—
231C4×C47C24×C411yescharacteristic
232C4×C47C23×D411yescharacteristic
233Dic47—24no—
234Dic47—43no—
235Dic47—41no—
236#3584—471no—
237#3573—471no—
238#3573D4711yescharacteristic, complemented
239#2957—476no—
240#2956—476no—
241#2955—476no—
242C2×#444—476no—
243#2916—471no—
244#2901—471no—
245#2901—471no—
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—24no—
261#568—471no—
262#568—476no—
263#568—944no—
264#568C2×D4711yescharacteristic
265#568C2×D4716yescomplemented
266C2×#131—24no—
267C2×#131—471no—
268C2×#131—476no—
269C2×#131—944no—
270C2×#131C2×D4711yescharacteristic
271C2×#131C2×D4716yescomplemented
272C4×#41—471no—
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#421—476no—
282C2×#104—476no—
283C2×#104—476no—
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—212no—
316#131—26no—
317#131—26no—
318#131—474no—
319#131—474no—
320#131—474no—
321#131—4712no—
322#131—944no—
323#131—9412no—
324#131—946no—
325#131—946no—
326#131C22⋊D4714yes—
327#131C22⋊D4714yes—
328#131C22×D4714yes—
329#131C22×D47112yes—
330C2×M16—212no—
331C2×M16—26no—
332C2×M16—473no—
333C2×M16—476no—
334C2×M16—9412no—
335C2×M16—946no—
336C2×M16C22×D4716yes—
337C2×M16C22×D4713yes—
338C22×C8—26no—
339C22×C8—473no—
340C22×C8—946no—
341C22×C8C22×D4713yes—
342C4×Q8—941no—
343C4×D4—943no—
344#117—473no—
345C2×C42—473no—
346#108—474no—
347#108—4712no—
348#104—474no—
349#104—4712no—
350#104—474no—
351#104—4712no—
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#41C2×C22⋊D4714yes—
370#41C23×D4714yes—
371C2×Q8—24no—
372C2×Q8—476no—
373C2×Q8—944no—
374C2×Q8C2×C4×D4716yes—
375C2×Q8#1062535011yescharacteristic
376C2×D4—26no—
377C2×D4—212no—
378C2×D4—476no—
379C2×D4—946no—
380C2×D4—9412no—
381C2×D4C2×C4×D4716yes—
382C2×D4#1062535013yes—
383C22×C4—26no—
384C22×C4—946no—
385C22×C4—943no—
386C22×C4C23×D4713yes—
387Q16—474no—
388Q16—4712no—
389SD16—474no—
390SD16—4712no—
391SD16—474no—
392SD16—4712no—
393D8—474no—
394D8—4712no—
395M16—26no—
396M16—212no—
397M16—24no—
398M16—4712no—
399M16—474no—
400M16—946no—
401M16—944no—
402M16—9412no—
403M16C2×C22⋊D47112yes—
404M16C23×D4714yes—
405C2×C8—26no—
406C2×C8—24no—
407C2×C8—212no—
408C2×C8—476no—
409C2×C8—476no—
410C2×C8—473no—
411C2×C8—471no—
412C2×C8—946no—
413C2×C8—944no—
414C2×C8—9412no—
415C2×C8C2×C22⋊D4716yes—
416C2×C8C2×C22⋊D4716yes—
417C2×C8C23×D4713yes—
418C2×C8C23×D4711yescharacteristic
419C4⋊C4—943no—
420C22⋊C4—943no—
421C42—473no—
422C42—471no—
423C42—943no—
424C23—26no—
425C23—946no—
426C23C2×#1062535013yes—
427Q8—26no—
428Q8—474no—
429Q8—946no—
430Q8C4×C22⋊D4714yes—
431Q8C22×C4×D4714yes—
432D4—212no—
433D4—24no—
434D4—26no—
435D4—474no—
436D4—9412no—
437D4—944no—
438D4—946no—
439D4C4×C22⋊D47112yes—
440D4C22×C4×D4714yes—
441C2×C4—26no—
442C2×C4—212no—
443C2×C4—24no—
444C2×C4—946no—
445C2×C4—9412no—
446C2×C4—944no—
447C2×C4—941no—
448C2×C4—943no—
449C2×C4—3763no—
450C2×C4C22×C4×D4716yes—
451C2×C4C22×C22⋊D4716yes—
452C2×C4C2×#1062535013yes—
453C2×C4C24×D4711yescharacteristic
454C8—24no—
455C8—471no—
456C8—471no—
457C8—476no—
458C8—944no—
459C8C22×C22⋊D4711yescharacteristic
460C8C22×C22⋊D4716yes—
461C8C22×C22⋊D4711yescharacteristic
462C22—24no—
463C22—43no—
464C22—86no—
465C22—944no—
466C22—3766no—
467C22C2×C4×C22⋊D4716yes—
468C22C22×#1062535011yescharacteristic
469C4—24no—
470C4—944no—
471C4—1881no—
472C4—1883no—
473C4C2×C4×C22⋊D4716yes—
474C4C23×C4×D4711yescharacteristic
475C4C23×C22⋊D4711yescharacteristic
476C2—21no—
477C2—26no—
478C2—44no—
479C2—1884no—
480C2C22×C4×C22⋊D4711yescharacteristic
subgroupG/H
centerC4C23×C22⋊D47
commutatorC4×C47C24×C4
FrattiniC2×C4C24×D47
FittingC47×#3573C2
socleC2×C47C22×C4×D4
prime radical#20258159C2
cosocleC2×C4×C47C25
hypercenter#3573D47
hypocenterC47#60184
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×#131 ◁ #3573

Complements

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

NH ≅ G/Ntype
2. #20258099C2semidirect
3. #20252303C2semidirect
4. #20252295C2semidirect
5. #20252289C2semidirect
6. C2×#15689243C2semidirect
7. #20251997C2semidirect
8. #20251935C2semidirect
9. #20251933C2semidirect
10. #20251814C2semidirect
11. C47×#3573C2semidirect
13. #15689938C4semidirect
14. #15689934C4semidirect
15. #15689931C4semidirect
16. #15689926C4semidirect
20. #15689889C22semidirect
23. C2×#13094030C22semidirect
26. #15689243C22semidirect
27. #15689243C22semidirect
28. #15689243C22semidirect
29. #15689243C22semidirect
30. #15689238C22semidirect
31. #15689171C22semidirect
32. C2×#13093959C22semidirect
33. #15689140C22semidirect
34. #15689139C22semidirect
35. C2×#13093951C22semidirect
36. C2×#13093950C22semidirect
37. #15689081C22semidirect
38. #15689078C22semidirect
39. M16×Dic47C22semidirect
40. C2×#13093934C22semidirect
44. C47×#568C22semidirect
47. C2×#131×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
238. #3573D47semidirect
265. #568C2×D47semidirect
271. C2×#131C2×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
#202580992
#202523032
#202522952
#202522892
C2×#156892432
#202519972
#202519352
#202519332
#202518142
C47×#35732
#6018447