Finite Group Databasebeta

← #26370755

Subgroups — #26370755

Order 12032.

738 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689561C211yesmaximal, characteristic, complemented
2#20252969C211yesmaximal, characteristic, complemented
3C2×#15689359C211yesmaximal, characteristic, complemented
4#20245501C212yesmaximal, complemented
5#20245491C212yesmaximal, complemented
6#20245484C212yesmaximal, complemented
7#20245474C212yesmaximal, complemented
8#20245176C211yesmaximal, characteristic, complemented
9#20245105C211yesmaximal, characteristic, complemented
10#20245101C211yesmaximal, characteristic, complemented
11C22×M32×C47C211yesmaximal, characteristic, complemented
12C2×#13094046C2211yescharacteristic
13C22×D376C2211yescharacteristic
14#15689634—21no—
15C2×#13093998—21no—
16#15689561—24no—
17C2×#13093969C2212yescomplemented
18C2×D752C2212yescomplemented
19#15689364C2211yescharacteristic, complemented
20#15689364C2211yescharacteristic, complemented
21#15689359—21no—
22#15689359—21no—
23#15689359C2212yes—
24#15689359C2212yes—
25#15689359C2211yescharacteristic, complemented
26#15689359C2211yescharacteristic, complemented
27#15689352—21no—
28#15689352—21no—
29#15689161C2211yescharacteristic
30C2×#13093958C2211yescharacteristic
31#15689120—21no—
32C2×#13093946—21no—
33#15688951C2212yescomplemented
34#15688949C2212yescomplemented
35#15688928C2212yescomplemented
36#15688925C2212yescomplemented
37#15688925C2212yescomplemented
38#15688925C2212yescomplemented
39C23×C8×C47C2211yescharacteristic
40C2×M32×C47—24no—
41C2×M32×C47C2211yescharacteristic
42C2×M32×C47C2211yescharacteristic
43C2×M32×C47C2212yescomplemented
44C22×C16×C47C2212yescomplemented
45C2×#10625350—21no—
46C22×D188—21no—
47C22×C22⋊D47—41no—
48#13094046—22no—
49#13094046—22no—
50C2×Dic188C2311yescharacteristic
51C2×#10625325—21no—
52C2×D376—22no—
53C2×D376—22no—
54C2×D376—21no—
55C2×D376C2311yescharacteristic
56C2×D376C2312yes—
57#13094007—41no—
58#13094003—21no—
59#13094003—21no—
60#13093998—21no—
61#13093998—21no—
62#13093998—22no—
63#13093998—22no—
64#13093998—41no—
65#13093998—41no—
66#13093993—41no—
67C4×C22⋊D47—41no—
68C4×C22⋊D47—41no—
69#13093983—21no—
70C2×C4⋊Dic47—21no—
71C2×(C2×D47)⋊C4—41no—
72#13093969—24no—
73D752—24no—
74#13093958—21no—
75#13093958C2312yes—
76#13093958C2311yescharacteristic
77#13093958C2311yescharacteristic
78#13093956—21no—
79#13093949—21no—
80#13093949—21no—
81#13093946—21no—
82#13093946—21no—
83#13093946—22no—
84#13093946—22no—
85C23×C4×C47D411yescharacteristic
86C22×C8×C47—22no—
87C22×C8×C47—22no—
88C22×C8×C47D411yescharacteristic
89C22×C8×C47D411yescharacteristic
90C22×C8×C47D411yescharacteristic
91C22×C8×C47C2311yescharacteristic
92C22×C8×C47C2311yescharacteristic
93C22×C8×C47C2311yescharacteristic
94M32×C47—22no—
95M32×C47—24no—
96M32×C47D414yescomplemented
97C2×C16×C47—24no—
98C2×C16×C47C2312yes—
99C2×C16×C47C2312yes—
100C24×C47D811yescharacteristic
101C23×D47—41no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47D811yescharacteristic
105C22×C4×C47D811yescharacteristic
106C22×C4×C47D811yescharacteristic
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D411yescharacteristic
109C22×C4×C47C2×D411yescharacteristic
110C2×C22⋊D47—41no—
111C2×C22⋊D47—41no—
112C2×C22⋊D47—41no—
113C2×C22⋊D47—42no—
114C2×C22⋊D47—42no—
115C2×C22⋊D47—84no—
116C22×Dic47—41no—
117#10625350—42no—
118#10625350—42no—
119C2×D188—22no—
120C2×D188—21no—
121C2×D188—42no—
122C2×D188—42no—
123C2×D188—41no—
124C2×C4×D47—41no—
125C2×Dic94—21no—
126C2×C8×C47—22no—
127C2×C8×C47—22no—
128C2×C8×C47—22no—
129C2×C8×C47—21no—
130C2×C8×C47—21no—
131C2×C8×C47—21no—
132C2×C8×C47—21no—
133C2×C8×C47C2×D411yescharacteristic
134C2×C8×C47C2×D411yescharacteristic
135C2×C8×C47C2×D411yescharacteristic
136C2×C8×C47C2×D412yes—
137C2×C8×C47#4112yes—
138C2×C8×C47C2411yescharacteristic
139C22⋊Dic47—41no—
140C22⋊Dic47—41no—
141C22⋊Dic47—41no—
142(C2×D47)⋊C4—42no—
143(C2×D47)⋊C4—42no—
144(C2×D47)⋊C4—41no—
145(C2×D47)⋊C4—41no—
146C4⋊Dic47—22no—
147C4⋊Dic47—21no—
148C4⋊Dic47—21no—
149C4⋊Dic47—41no—
150Dic47⋊C4—41no—
151Dic47⋊C4—41no—
152C4×Dic47—41no—
153Dic188—21no—
154Dic188—21no—
155D376—21no—
156D376—22no—
157D376—22no—
158D376—21no—
159D376—42no—
160#10625325—42no—
161C16×C47—22no—
162C16×C47C2×D414yescomplemented
163C23×C47—22no—
164C23×C47—22no—
165C23×C47—41no—
166C23×C47C2×D811yescharacteristic
167C23×C47C2×D811yescharacteristic
168C23×C47C2×D811yescharacteristic
169C22×D47—41no—
170C22×D47—42no—
171C22×D47—82no—
172C22×D47—82no—
173C22×D47—82no—
174C2×C4×C47—22no—
175C2×C4×C47—21no—
176C2×C4×C47—21no—
177C2×C4×C47—21no—
178C2×C4×C47—21no—
179C2×C4×C47—22no—
180C2×C4×C47—22no—
181C2×C4×C47#12112yes—
182C2×C4×C47C2×D811yescharacteristic
183C2×C4×C47C2×D811yescharacteristic
184C2×C4×C47C2×D811yescharacteristic
185C2×C4×C47#13512yes—
186C2×C4×C47C22×D411yescharacteristic
187C22⋊D47—82no—
188C22⋊D47—84no—
189C22⋊D47—82no—
190C22⋊D47—82no—
191C22⋊D47—84no—
192C2×Dic47—41no—
193C2×Dic47—41no—
194C2×Dic47—41no—
195C2×Dic47—42no—
196C2×Dic47—82no—
197D188—41no—
198D188—42no—
199D188—42no—
200D188—41no—
201D188—82no—
202C4×D47—82no—
203Dic94—41no—
204Dic94—41no—
205C8×C47—21no—
206C8×C47—21no—
207C8×C47#12112yes—
208C8×C47C22×D411yescharacteristic
209C8×C47C2×#4111yescharacteristic
210#33605—471nomaximal, Sylow
211C22×C47—21no—
212C22×C47—21no—
213C22×C47—21no—
214C22×C47—21no—
215C22×C47—21no—
216C22×C47—21no—
217C22×C47—21no—
218C22×C47—44no—
219C22×C47#46612yes—
220C22×C47#50912yescomplemented
221C22×C47C22×D811yescharacteristic
222C2×D47—82no—
223C2×D47—81no—
224C2×D47—81no—
225C2×D47—82no—
226C2×D47—81no—
227C2×D47—82no—
228C2×D47—164no—
229C4×C47—21no—
230C4×C47—21no—
231C4×C47#46612yes—
232C4×C47C2×#12111yescharacteristic
233C4×C47C2×#13511yescharacteristic
234Dic47—82no—
235Dic47—81no—
236Dic47—81no—
237Dic47—81no—
238C2×#509—471no—
239C22×M32D4711yescharacteristic, complemented
240#3050—471no—
241C2×#466—471no—
242#2220—472no—
243#2218—472no—
244#2217—472no—
245#2215—472no—
246#2159—471no—
247#2140—471no—
248#2139—471no—
249C2×C47—22no—
250C2×C47—22no—
251C2×C47—41no—
252C2×C47C2×#46611yescharacteristic
253C2×C47C2×#50911yescharacteristic
254C2×C47#341411yescharacteristic
255D47—161no—
256D47—162no—
257C2×#135—471no—
258C22×D8—471no—
259C23×C8C2×D4711yescharacteristic
260#525—941no—
261C2×#121—941no—
262#509—944no—
263C2×SD32—472no—
264C2×D16—472no—
265C2×M32—24no—
266C2×M32C2×D4711yescharacteristic
267C2×M32C2×D4711yescharacteristic
268C2×M32C2×D4712yescomplemented
269C22×C16C2×D4712yescomplemented
270#467—471no—
271#467—471no—
272#466—472no—
273#466—472no—
274#466—471no—
275#466—471no—
276#466—941no—
277#466—941no—
278#465—941no—
279#465—941no—
280#427—471no—
281C2×#107—471no—
282#416—941no—
283C2×#102—941no—
284#367—472no—
285#366—472no—
286#358—472no—
287#357—472no—
288#357—472no—
289#357—472no—
290C47#3360511yescharacteristic, complemented, Sylow
291C2×#41—941no—
292C22×D4—941no—
293C22×D4—1881no—
294C23×C4D18811yescharacteristic
295#135—942no—
296#135—942no—
297C2×Q16—471no—
298C2×SD16—941no—
299C2×D8—472no—
300C2×D8—471no—
301C2×D8—941no—
302C2×D8—942no—
303C2×D8—942no—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8D18811yescharacteristic
307C22×C8D18811yescharacteristic
308C22×C8D18811yescharacteristic
309C22×C8C22×D4711yescharacteristic
310C22×C8C22×D4711yescharacteristic
311C22×C8C22×D4711yescharacteristic
312#123—1881no—
313C22⋊Q8—941no—
314C22⋊Q8—941no—
315#121—942no—
316#121—942no—
317#121—941no—
318#121—941no—
319#121—1881no—
320#121—1881no—
321#120—1881no—
322C4×D4—1881no—
323C4×D4—1881no—
324#117—941no—
325C2×C4⋊C4—941no—
326C2×C22⋊C4—1881no—
327SD32—944no—
328D16—944no—
329M32—22no—
330M32—24no—
331M32C22⋊D4714yescomplemented
332C2×C16—24no—
333C2×C16C22×D4712yes—
334C2×C16C22×D4712yes—
335#107—471no—
336#107—471no—
337#107—472no—
338#107—941no—
339#106—941no—
340#103—941no—
341#103—941no—
342#102—942no—
343#102—942no—
344#102—941no—
345#102—941no—
346C24—1881no—
347C24D37611yescharacteristic
348#41—1882no—
349#41—1882no—
350C2×Q8—941no—
351C2×D4—942no—
352C2×D4—941no—
353C2×D4—3764no—
354C2×D4—1881no—
355C2×D4—1881no—
356C2×D4—1881no—
357C2×D4—1882no—
358C2×D4—1882no—
359C2×D4—1882no—
360C2×D4—1881no—
361C2×D4—1882no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—1881no—
365C22×C4—1881no—
366C22×C4D37611yescharacteristic
367C22×C4D37611yescharacteristic
368C22×C4D37611yescharacteristic
369C22×C4C2×D18811yescharacteristic
370C22×C4C2×D18811yescharacteristic
371C22×C4C2×D18811yescharacteristic
372Q16—941no—
373Q16—941no—
374SD16—1882no—
375D8—941no—
376D8—942no—
377D8—942no—
378D8—941no—
379D8—1882no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8C2×D18811yescharacteristic
388C2×C8C2×D18811yescharacteristic
389C2×C8C2×D18811yescharacteristic
390C2×C8#1062535012yes—
391C2×C8C2×C22⋊D4712yes—
392C2×C8C23×D4711yescharacteristic
393C4⋊C4—942no—
394C4⋊C4—941no—
395C4⋊C4—941no—
396C4⋊C4—1881no—
397C4⋊C4—1881no—
398C4⋊C4—1881no—
399C22⋊C4—1881no—
400C22⋊C4—1881no—
401C22⋊C4—1881no—
402C22⋊C4—1882no—
403C22⋊C4—1882no—
404C22⋊C4—1881no—
405C22⋊C4—1881no—
406C42—1881no—
407C16—22no—
408C16C2×C22⋊D4714yescomplemented
409C23—22no—
410C23—22no—
411C23—41no—
412C23—3762no—
413C23—3762no—
414C23—3762no—
415C23—1881no—
416C23—1882no—
417C23C2×D37611yescharacteristic
418C23C2×D37611yescharacteristic
419C23C2×D37611yescharacteristic
420Q8—1881no—
421Q8—1881no—
422D4—3764no—
423D4—3762no—
424D4—3764no—
425D4—3762no—
426D4—3762no—
427D4—3762no—
428D4—1882no—
429D4—1881no—
430D4—1882no—
431D4—1881no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—3762no—
440C2×C4—3762no—
441C2×C4—1881no—
442C2×C4—1881no—
443C2×C4—1882no—
444C2×C4—1881no—
445C2×C4#1309399812yes—
446C2×C4C2×D37611yescharacteristic
447C2×C4C2×D37611yescharacteristic
448C2×C4C2×D37611yescharacteristic
449C2×C4#1309404612yes—
450C2×C4C22×D18811yescharacteristic
451C8—21no—
452C8—21no—
453C8#1309399812yes—
454C8C22×C22⋊D4711yescharacteristic
455C8C2×#1062535011yescharacteristic
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—44no—
464C22—3762no—
465C22—3761no—
466C22—3761no—
467C22—3762no—
468C22—3761no—
469C22—3762no—
470C22—7524no—
471C22#1568935912yes—
472C22#1568956112yescomplemented
473C22C22×D37611yescharacteristic
474C4—21no—
475C4—21no—
476C4—3762no—
477C4—3761no—
478C4—3761no—
479C4—3761no—
480C4#1568935912yes—
481C4C2×#1309399811yescharacteristic
482C4C2×#1309404611yescharacteristic
483C2—22no—
484C2—22no—
485C2—41no—
486C2—7522no—
487C2—7521no—
488C2C2×#1568935911yescharacteristic
489C2C2×#1568956111yescharacteristic
490C2#2025652411yescharacteristic
subgroupG/H
centerC22C22×D376
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC22×M32×C47C2
socleC22×C47C22×D8
hypercenterC22×M32D47
hypocenterC47#33605
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C22×M32

Complements

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

NH ≅ G/Ntype
1. C2×#15689561C2semidirect
2. #20252969C2semidirect
3. C2×#15689359C2semidirect
4. #20245501C2semidirect
5. #20245491C2semidirect
6. #20245484C2semidirect
7. #20245474C2semidirect
8. #20245176C2semidirect
9. #20245105C2semidirect
10. #20245101C2semidirect
11. C22×M32×C47C2semidirect
17. C2×#13093969C22semidirect
18. C2×D752C22semidirect
19. #15689364C22semidirect
20. #15689364C22semidirect
25. #15689359C22semidirect
26. #15689359C22semidirect
33. #15688951C22semidirect
34. #15688949C22semidirect
35. #15688928C22semidirect
36. #15688925C22semidirect
37. #15688925C22semidirect
38. #15688925C22semidirect
43. C2×M32×C47C22semidirect
44. C22×C16×C47C22semidirect
96. M32×C47D4semidirect
162. C16×C47C2×D4semidirect
220. C22×C47#509semidirect
239. C22×M32D47semidirect
268. C2×M32C2×D47semidirect
269. C22×C16C2×D47semidirect
290. C47#33605semidirect
331. M32C22⋊D47semidirect
408. C16C2×C22⋊D47semidirect
472. C22#15689561semidirect

Maximal subgroups

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

subgroupindex
C2×#156895612
#202529692
C2×#156893592
#202455012
#202454912
#202454842
#202454742
#202451762
#202451052
#202451012
C22×M32×C472
#3360547