Finite Group Databasebeta

← #26370751

Subgroups — #26370751

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×#15689559C211yesmaximal, characteristic, complemented
2C22×#13093964C211yesmaximal, characteristic, complemented
3#20252969C211yesmaximal, characteristic, complemented
4#20245499C212yesmaximal, complemented
5#20245489C212yesmaximal, complemented
6#20245482C212yesmaximal, complemented
7#20245472C212yesmaximal, complemented
8#20245174C211yesmaximal, characteristic, complemented
9#20245104C211yesmaximal, characteristic, complemented
10#20245098C211yesmaximal, characteristic, complemented
11C2×C47×#466C211yesmaximal, characteristic, complemented
12C2×#13094046C2211yescharacteristic
13#15689634—21no—
14#15689559—24no—
15C2×#13093968C2212yescomplemented
16C2×#13093965C2212yescomplemented
17C2×#13093964—24no—
18C2×#13093964C2211yescharacteristic
19C2×#13093964C2211yescharacteristic
20C2×#13093964C2212yescomplemented
21C22×C47⋊C16C2212yescomplemented
22#15689364C2211yescharacteristic, complemented
23#15689364C2211yescharacteristic, complemented
24#15689359C2211yescharacteristic, complemented
25#15689359C2211yescharacteristic, complemented
26#15689352—21no—
27#15689352—21no—
28#15689161C2211yescharacteristic
29#15689120—21no—
30#15688950C2212yescomplemented
31#15688948C2212yescomplemented
32#15688926C2212yescomplemented
33#15688924C2212yescomplemented
34#15688924C2212yescomplemented
35#15688923C2212yescomplemented
36C22×D8×C47C2211yescharacteristic
37C23×C8×C47C2211yescharacteristic
38C2×#121×C47—21no—
39C47×#466—21no—
40C47×#466—21no—
41C47×#466C2212yes—
42C47×#466C2212yes—
43C2×#107×C47C2211yescharacteristic
44C2×#102×C47—21no—
45C2×#10625350—21no—
46#13094046—22no—
47#13094046—22no—
48C2×Dic188C2311yescharacteristic
49C2×#10625325—21no—
50C2×D376C2311yescharacteristic
51#13094007—41no—
52#13094003—21no—
53#13094003—21no—
54#13093998—21no—
55#13093998—21no—
56#13093993—41no—
57C4×C22⋊D47—41no—
58C4×C22⋊D47—41no—
59#13093983—21no—
60#13093968—24no—
61#13093965—24no—
62#13093964—22no—
63#13093964—24no—
64#13093964D414yescomplemented
65C2×C47⋊C16—24no—
66C2×C47⋊C16C2312yes—
67C2×C47⋊C16C2312yes—
68#13093958C2311yescharacteristic
69#13093958C2311yescharacteristic
70#13093956—21no—
71#13093949—21no—
72#13093949—21no—
73#13093946—21no—
74#13093946—21no—
75C22×D4×C47—21no—
76C22×D4×C47—41no—
77C23×C4×C47D411yescharacteristic
78C2×D8×C47—22no—
79C2×D8×C47—21no—
80C2×D8×C47—22no—
81C2×D8×C47C2312yes—
82C22×C8×C47—22no—
83C22×C8×C47—22no—
84C22×C8×C47D411yescharacteristic
85C22×C8×C47D411yescharacteristic
86C22×C8×C47D411yescharacteristic
87C22×C8×C47C2311yescharacteristic
88C22×C8×C47C2311yescharacteristic
89C22×C8×C47C2311yescharacteristic
90#121×C47—22no—
91#121×C47—22no—
92#121×C47—41no—
93#121×C47—41no—
94C2×C4⋊C4×C47—21no—
95C2×C22⋊C4×C47—41no—
96#107×C47—21no—
97#107×C47C2312yes—
98#102×C47—22no—
99#102×C47—22no—
100C24×C47—41no—
101C24×C47D811yescharacteristic
102C2×D4×C47—22no—
103C2×D4×C47—42no—
104C2×D4×C47—42no—
105C2×D4×C47—42no—
106C2×D4×C47—41no—
107C2×D4×C47—42no—
108C2×D4×C47—84no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—41no—
112C22×C4×C47D811yescharacteristic
113C22×C4×C47D811yescharacteristic
114C22×C4×C47D811yescharacteristic
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2×D411yescharacteristic
117C22×C4×C47C2×D411yescharacteristic
118C2×C22⋊D47—41no—
119C2×C22⋊D47—41no—
120C2×C22⋊D47—41no—
121#10625350—42no—
122#10625350—42no—
123C2×D188—21no—
124C2×C4×D47—41no—
125C2×Dic94—21no—
126D8×C47—22no—
127D8×C47—22no—
128D8×C47—42no—
129C2×C8×C47—22no—
130C2×C8×C47—22no—
131C2×C8×C47—22no—
132C2×C8×C47—21no—
133C2×C8×C47—21no—
134C2×C8×C47—21no—
135C2×C8×C47—21no—
136C2×C8×C47C2×D411yescharacteristic
137C2×C8×C47C2×D411yescharacteristic
138C2×C8×C47C2×D411yescharacteristic
139C2×C8×C47C2×D412yes—
140C2×C8×C47#4112yes—
141C2×C8×C47C2411yescharacteristic
142C4⋊C4×C47—22no—
143C4⋊C4×C47—41no—
144C22⋊C4×C47—42no—
145C22⋊C4×C47—42no—
146C22⋊Dic47—41no—
147C22⋊Dic47—41no—
148C22⋊Dic47—41no—
149(C2×D47)⋊C4—41no—
150(C2×D47)⋊C4—41no—
151C4⋊Dic47—21no—
152C4⋊Dic47—21no—
153Dic47⋊C4—41no—
154Dic47⋊C4—41no—
155C4×Dic47—41no—
156Dic188—21no—
157Dic188—21no—
158D376—21no—
159D376—21no—
160#10625325—42no—
161C47⋊C16—22no—
162C47⋊C16C2×D414yescomplemented
163C23×C47—22no—
164C23×C47—22no—
165C23×C47—41no—
166C23×C47—42no—
167C23×C47—82no—
168C23×C47—82no—
169C23×C47—82no—
170C23×C47C2×D811yescharacteristic
171C23×C47C2×D811yescharacteristic
172C23×C47C2×D811yescharacteristic
173C22×D47—41no—
174D4×C47—42no—
175D4×C47—42no—
176D4×C47—84no—
177D4×C47—84no—
178D4×C47—82no—
179C2×C4×C47—22no—
180C2×C4×C47—21no—
181C2×C4×C47—21no—
182C2×C4×C47—21no—
183C2×C4×C47—21no—
184C2×C4×C47—22no—
185C2×C4×C47—22no—
186C2×C4×C47—42no—
187C2×C4×C47—82no—
188C2×C4×C47#12112yes—
189C2×C4×C47C2×D811yescharacteristic
190C2×C4×C47C2×D811yescharacteristic
191C2×C4×C47C2×D811yescharacteristic
192C2×C4×C47#13512yes—
193C2×C4×C47C22×D411yescharacteristic
194C22⋊D47—82no—
195C22⋊D47—82no—
196C22⋊D47—82no—
197C2×Dic47—41no—
198C2×Dic47—41no—
199C2×Dic47—41no—
200D188—41no—
201D188—41no—
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—82no—
220C22×C47—82no—
221C22×C47—82no—
222C22×C47—164no—
223C22×C47#46612yes—
224C22×C47#50912yescomplemented
225C22×C47C22×D811yescharacteristic
226C2×D47—81no—
227C2×D47—81no—
228C2×D47—81no—
229C4×C47—21no—
230C4×C47—21no—
231C4×C47—82no—
232C4×C47#46612yes—
233C4×C47C2×#12111yescharacteristic
234C4×C47C2×#13511yescharacteristic
235Dic47—81no—
236Dic47—81no—
237Dic47—81no—
238C2×#509—471no—
239C22×M32—471no—
240#3050—471no—
241C2×#466D4711yescharacteristic, complemented
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×C47—162no—
253C2×C47C2×#46611yescharacteristic
254C2×C47C2×#50911yescharacteristic
255C2×C47#341411yescharacteristic
256D47—161no—
257C2×#135—471no—
258C22×D8C2×D4711yescharacteristic
259C23×C8C2×D4711yescharacteristic
260#525—941no—
261C2×#121—21no—
262#509—944no—
263C2×SD32—472no—
264C2×D16—472no—
265C2×M32—471no—
266C2×M32—471no—
267C2×M32—472no—
268C2×M32—944no—
269C22×C16—472no—
270#467—471no—
271#467—471no—
272#466—21no—
273#466—21no—
274#466—471no—
275#466—471no—
276#466C2×D4712yes—
277#466C2×D4712yes—
278#465—941no—
279#465—941no—
280#427—471no—
281C2×#107C2×D4711yescharacteristic
282#416—941no—
283C2×#102—21no—
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—21no—
293C22×D4—41no—
294C23×C4C22⋊D4711yescharacteristic
295#135—942no—
296#135—942no—
297C2×Q16—471no—
298C2×SD16—941no—
299C2×D8—22no—
300C2×D8—22no—
301C2×D8—21no—
302C2×D8—471no—
303C2×D8C22×D4712yes—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8C22⋊D4711yescharacteristic
307C22×C8C22⋊D4711yescharacteristic
308C22×C8C22⋊D4711yescharacteristic
309C22×C8C22×D4711yescharacteristic
310C22×C8C22×D4711yescharacteristic
311C22×C8C22×D4711yescharacteristic
312#123—1881no—
313C22⋊Q8—941no—
314C22⋊Q8—941no—
315#121—22no—
316#121—22no—
317#121—41no—
318#121—41no—
319#121—941no—
320#121—941no—
321#120—1881no—
322C4×D4—1881no—
323C4×D4—1881no—
324#117—941no—
325C2×C4⋊C4—21no—
326C2×C22⋊C4—41no—
327SD32—944no—
328D16—944no—
329M32—474no—
330M32—944no—
331M32—942no—
332C2×C16—472no—
333C2×C16—472no—
334C2×C16—944no—
335#107—21no—
336#107—471no—
337#107—471no—
338#107C22×D4712yes—
339#106—941no—
340#103—941no—
341#103—941no—
342#102—22no—
343#102—22no—
344#102—941no—
345#102—941no—
346C24—41no—
347C24#1062533411yescharacteristic
348#41—1882no—
349#41—1882no—
350C2×Q8—941no—
351C2×D4—22no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4—84no—
358C2×D4—941no—
359C2×D4—1881no—
360C2×D4—1881no—
361C2×D4—1881no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—41no—
365C22×C4—1881no—
366C22×C4#1062533411yescharacteristic
367C22×C4#1062533411yescharacteristic
368C22×C4#1062533411yescharacteristic
369C22×C4C2×C22⋊D4711yescharacteristic
370C22×C4C2×C22⋊D4711yescharacteristic
371C22×C4C2×C22⋊D4711yescharacteristic
372Q16—941no—
373Q16—941no—
374SD16—1882no—
375D8—22no—
376D8—22no—
377D8—42no—
378D8—941no—
379D8—941no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8D4×D4712yes—
388C2×C8#1062535212yes—
389C2×C8C2×C22⋊D4711yescharacteristic
390C2×C8C2×C22⋊D4711yescharacteristic
391C2×C8C2×C22⋊D4711yescharacteristic
392C2×C8C23×D4711yescharacteristic
393C4⋊C4—22no—
394C4⋊C4—41no—
395C4⋊C4—941no—
396C4⋊C4—941no—
397C4⋊C4—1881no—
398C4⋊C4—1881no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—1881no—
402C22⋊C4—1881no—
403C22⋊C4—1881no—
404C22⋊C4—1881no—
405C22⋊C4—1881no—
406C42—1881no—
407C16—474no—
408C16—942no—
409C23—22no—
410C23—22no—
411C23—41no—
412C23—42no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—1881no—
417C23C2×#1062533411yescharacteristic
418C23C2×#1062533411yescharacteristic
419C23C2×#1062533411yescharacteristic
420Q8—1881no—
421Q8—1881no—
422D4—42no—
423D4—42no—
424D4—84no—
425D4—84no—
426D4—82no—
427D4—3762no—
428D4—3762no—
429D4—3762no—
430D4—1881no—
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—42no—
440C2×C4—82no—
441C2×C4—3762no—
442C2×C4—1881no—
443C2×C4—1881no—
444C2×C4—1881no—
445C2×C4#1309399612yes—
446C2×C4C2×#1062533411yescharacteristic
447C2×C4C2×#1062533411yescharacteristic
448C2×C4C2×#1062533411yescharacteristic
449C2×C4#1309404412yes—
450C2×C4C22×C22⋊D4711yescharacteristic
451C8—21no—
452C8—21no—
453C8#1309399612yes—
454C8C2×D4×D4711yescharacteristic
455C8C2×#1062535211yescharacteristic
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—44no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—164no—
468C22—3761no—
469C22—3761no—
470C22—3761no—
471C22#1568935712yes—
472C22#1568955912yescomplemented
473C22C22×#1062533411yescharacteristic
474C4—21no—
475C4—21no—
476C4—82no—
477C4—3761no—
478C4—3761no—
479C4—3761no—
480C4#1568935712yes—
481C4C2×#1309399611yescharacteristic
482C4C2×#1309404411yescharacteristic
483C2—22no—
484C2—22no—
485C2—41no—
486C2—162no—
487C2—7521no—
488C2C2×#1568935711yescharacteristic
489C2C2×#1568955911yescharacteristic
490C2#2025652211yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC2×C47×#466C2
socleC22×C47C22×D8
hypercenterC2×#466D47
hypocenterC47#33605
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C2×#15689559C2semidirect
2. C22×#13093964C2semidirect
3. #20252969C2semidirect
4. #20245499C2semidirect
5. #20245489C2semidirect
6. #20245482C2semidirect
7. #20245472C2semidirect
8. #20245174C2semidirect
9. #20245104C2semidirect
10. #20245098C2semidirect
11. C2×C47×#466C2semidirect
15. C2×#13093968C22semidirect
16. C2×#13093965C22semidirect
20. C2×#13093964C22semidirect
21. C22×C47⋊C16C22semidirect
22. #15689364C22semidirect
23. #15689364C22semidirect
24. #15689359C22semidirect
25. #15689359C22semidirect
30. #15688950C22semidirect
31. #15688948C22semidirect
32. #15688926C22semidirect
33. #15688924C22semidirect
34. #15688924C22semidirect
35. #15688923C22semidirect
64. #13093964D4semidirect
162. C47⋊C16C2×D4semidirect
224. C22×C47#509semidirect
241. C2×#466D47semidirect
290. C47#33605semidirect
472. C22#15689559semidirect

Maximal subgroups

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

subgroupindex
C2×#156895592
C22×#130939642
#202529692
#202454992
#202454892
#202454822
#202454722
#202451742
#202451042
#202450982
C2×C47×#4662
#3360547