Finite Group Databasebeta

← #27080820

Subgroups — #27080820

Order 12032.

1962 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256683C211yesmaximal, characteristic
2#20252398C218yesmaximal, complemented
3#20252379C218yesmaximal, complemented
4#20252328C211yesmaximal, characteristic, complemented
5#20252325C211yesmaximal, characteristic, complemented
6#20252106C211yesmaximal, characteristic, complemented
7#20252085C212yesmaximal, complemented
8#20252080C212yesmaximal, complemented
9#20252035C212yesmaximal, complemented
10#20252020C211yesmaximal, characteristic, complemented
11#20251814C212yesmaximal, complemented
12C2×#97×D47C211yesmaximal, characteristic, complemented
13C47×#2932C211yesmaximal, characteristic, complemented
14C2×#13094031C2212yes—
15C22×C8×D47C2211yescharacteristic
16C22×C8×D47C2211yescharacteristic
17#15689680C2211yescharacteristic
18#15689678C412yes—
19#15689671C412yes—
20#15689655C2214yes—
21#15689654C414yes—
22#15689652C414yes—
23#15689633C414yes—
24#15689630C2212yes—
25#41×Dic47C2212yes—
26C4×#10625352C2214yes—
27C2×C42×D47C2211yescharacteristic
28#15689252C2218yescomplemented
29#15689251C2218yescomplemented
30#15689197C2218yes—
31#15689195C2218yescomplemented
32#15689192C2218yes—
33C8×C22⋊D47C2218yes—
34(C22⋊D47)⋊C8C2218yes—
35#15689181C2218yescomplemented
36#15689179C2214yes—
37#15689177C2218yescomplemented
38#15689175C2214yes—
39#15689151C2212yes—
40#15689149C2212yes—
41#15689147C2211yescharacteristic
42#15689090C2212yes—
43#15689086C2212yes—
44#15689081C2212yes—
45#15689081C2214yescomplemented
46#15689078C2212yes—
47#15689078C2214yescomplemented
48M16×Dic47C2212yes—
49M16×Dic47C2214yescomplemented
50C2×#13093935C2211yescharacteristic
51C2×#13093934C2211yescharacteristic
52C2×#13093934C2211yescharacteristic
53C2×#13093934C2214yescomplemented
54#97×D47C2214yes—
55#97×D47C2214yes—
56C2×#131×C47C2212yes—
57C47×#532C2211yescharacteristic
58C47×#435C2218yescomplemented
59C47×#423C2211yescharacteristic
60C47×#408C2212yes—
61C2×#97×C47C2211yescharacteristic
62C2×#10625352C2×C414yes—
63C22×C4×D47C2×C412yes—
64C22×C4×D47C2311yescharacteristic
65#13094031—28no—
66C2×#10625329C2312yes—
67C2×#10625329C2314yes—
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2314yes—
70C22×C47⋊C8C2311yescharacteristic
71C2×C8×D47—24no—
72C2×C8×D47—24no—
73C2×C8×D47C2314yes—
74C2×C8×D47C2314yes—
75#13094024C2×C412yes—
76#13094017C2×C418yes—
77#13094016C2×C412yes—
78#13094015C2×C414yes—
79#13094011C2×C414yes—
80#13094006C2×C418yes—
81#13094005C2×C414yes—
82#13094004C2×C414yes—
83#13094003C2×C414yes—
84#13094000C2×C414yes—
85#13093995C2×C414yes—
86#13093995C2×C414yes—
87#13093992C2×C414yes—
88Q8×Dic47C2×C414yes—
89C4×Dic94C2314yes—
90#13093989C2×C414yes—
91D4×Dic47C2×C414yes—
92D4×Dic47C2×C418yes—
93D4×Dic47C2×C414yes—
94C4×C22⋊D47C2318yes—
95(C22⋊D47)⋊C4C2×C418yes—
96#13093983C2312yes—
97#13093983C2314yes—
98#13093982C2×C414yes—
99#13093980C2×C418yes—
100C2×C4×Dic47C2×C412yes—
101C2×C4×Dic47C2311yescharacteristic
102C2×C4×Dic47C2314yes—
103C42×D47C2×C412yes—
104C42×D47C2×C412yes—
105C42×D47C2312yes—
106C42×D47C2312yes—
107C4⋊(C47⋊C8)C2314yes—
108Dic47⋊C8C2318yes—
109Dic47⋊C8C2318yes—
110C22⋊(C47⋊C8)C2318yes—
111(C2×D47)⋊C8C2318yes—
112#13093935C2314yes—
113#13093934C2314yes—
114#13093934C2314yes—
115#13093934C2312yes—
116#13093934C2312yes—
117#13093934C2312yes—
118#13093934C2312yes—
119#13093934C2314yes—
120C8×Dic47C2314yes—
121C8×Dic47C2314yes—
122C8×Dic47C2314yes—
123C2×#41×C47C2312yes—
124#131×C47—28no—
125C2×M16×C47C2312yes—
126C2×M16×C47C2314yes—
127C22×C8×C47C2311yescharacteristic
128C22×C8×C47C2314yes—
129C22×C8×C47C2311yescharacteristic
130C4⋊Q8×C47C2×C411yescharacteristic
131#127×C47C2×C411yescharacteristic
132#124×C47C2×C412yes—
133#121×C47C2×C414yes—
134C4×D4×C47C2314yes—
135C2×C42×C47C2311yescharacteristic
136C4⋊C8×C47C2314yes—
137C22⋊C8×C47C2318yes—
138#97×C47C2314yes—
139C23×D47C22×C411yescharacteristic
140#41×C47—24no—
141#41×C47—24no—
142C2×Q8×C47C22×C412yes—
143C2×D4×C47C22×C414yes—
144C2×D4×C47C22×C412yes—
145C2×D4×C47C22×C414yes—
146C22×C4×C47C22×C412yes—
147C22×C4×C47C2414yes—
148C22×C4×C47C2411yescharacteristic
149C2×C22⋊D47C22×C418yes—
150C22×Dic47C22×C411yescharacteristic
151C22×Dic47C22×C412yes—
152C22×Dic47C22×C418yes—
153#10625352—48no—
154C2×C4×D47—22no—
155C2×C4×D47—22no—
156C2×C4×D47—44no—
157C2×C4×D47C22×C412yes—
158C2×C4×D47C22×C412yes—
159C2×C4×D47C22×C412yes—
160C2×C4×D47C22×C412yes—
161C2×C4×D47C22×C412yes—
162C2×C4×D47C2412yes—
163C2×Dic94C22×C414yes—
164M16×C47—24no—
165M16×C47—28no—
166C2×C8×C47—22no—
167C2×C8×C47—28no—
168C2×C8×C47#4114yescomplemented
169C2×C8×C47C2414yes—
170C2×C8×C47C2412yes—
171C2×C8×C47C2412yes—
172C4⋊C4×C47C22×C414yes—
173C22⋊C4×C47C22×C418yes—
174C42×C47C22×C411yescharacteristic
175C42×C47C22×C411yescharacteristic
176C42×C47C2412yes—
177C22⋊Dic47C22×C418yes—
178C22⋊Dic47C22×C414yes—
179C22⋊Dic47C22×C418yes—
180(C2×D47)⋊C4C22×C418yes—
181C4⋊Dic47C22×C414yes—
182C4⋊Dic47C22×C414yes—
183C4⋊Dic47C22×C418yes—
184Dic47⋊C4C22×C418yes—
185Dic47⋊C4C22×C418yes—
186C4×Dic47C22×C412yes—
187C4×Dic47C22×C412yes—
188C4×Dic47C22×C412yes—
189C4×Dic47C22×C412yes—
190C4×Dic47C22×C412yes—
191C4×Dic47C22×C418yes—
192C4×Dic47C2411yescharacteristic
193C4×Dic47C2414yes—
194C4×Dic47C2411yescharacteristic
195#10625329—24no—
196#10625329—28no—
197C2×C47⋊C8—22no—
198C2×C47⋊C8—28no—
199C2×C47⋊C8#4114yescomplemented
200C2×C47⋊C8C2414yes—
201C2×C47⋊C8C2412yes—
202C2×C47⋊C8C2412yes—
203C8×D47—28no—
204C8×D47—28no—
205C23×C47C23×C414yes—
206C23×C47C23×C411yescharacteristic
207C22×D47—22no—
208C22×D47—24no—
209C22×D47C23×C412yes—
210Q8×C47—24no—
211D4×C47—28no—
212D4×C47—24no—
213D4×C47—44no—
214C2×C4×C47—24no—
215C2×C4×C47—24no—
216C2×C4×C47—42no—
217C2×C4×C47C23×C412yes—
218C2×C4×C47C23×C414yes—
219C2×C4×C47C23×C412yes—
220C2×C4×C47C23×C411yescharacteristic
221C2×C4×C47C2×#4112yes—
222C2×C4×C47C2×#4112yes—
223C2×C4×C47C2511yescharacteristic
224C22⋊D47—216no—
225C2×Dic47—22no—
226C2×Dic47—42no—
227C2×Dic47—48no—
228C2×Dic47C23×C412yes—
229C2×Dic47C23×C412yes—
230C2×Dic47C23×C412yes—
231C2×Dic47C23×C418yes—
232C4×D47—24no—
233C4×D47—24no—
234C4×D47—24no—
235C4×D47—22no—
236C4×D47—22no—
237C4×D47—42no—
238C4×D47—42no—
239Dic94—44no—
240C8×C47—22no—
241C8×C47—24no—
242C8×C47C2×#4114yes—
243C47⋊C8—22no—
244C47⋊C8—24no—
245C47⋊C8C2×#4114yes—
246#60207—471nomaximal, Sylow
247C22×C47—22no—
248C22×C47—24no—
249C22×C47—28no—
250C22×C47C4×#4112yes—
251C22×C47C24×C411yescharacteristic
252C2×D47—28no—
253C2×D47—22no—
254C2×D47—22no—
255C2×D47#56814yes—
256C4×C47—24no—
257C4×C47—22no—
258C4×C47—22no—
259C4×C47C4×#4112yes—
260C4×C47C22×#4111yescharacteristic
261C4×C47C22×#4111yescharacteristic
262Dic47—22no—
263Dic47—28no—
264Dic47—22no—
265Dic47—22no—
266#3444—471no—
267#2970—478no—
268#2968—478no—
269#2961—471no—
270#2961—471no—
271#2932—471no—
272#2932D4711yescharacteristic, complemented
273#2929—472no—
274#2929—472no—
275#2923—472no—
276#2920—471no—
277#2875—472no—
278C22×#97—471no—
279C2×C47—22no—
280C2×C47—44no—
281C2×C47C2×C4×#4111yescharacteristic
282C2×C47C2×#56812yes—
283D47—24no—
284C2×#131—472no—
285C2×#131C2×D4712yes—
286C23×C8—471no—
287C23×C8—471no—
288#533—472no—
289#533—471no—
290#532—472no—
291#532C2×D4711yescharacteristic
292#529—474no—
293#529—474no—
294#529—474no—
295#525—472no—
296#525—474no—
297C4×#41—472no—
298C4×#41—474no—
299C22×C42—471no—
300#446—478no—
301#446—478no—
302#435—478no—
303#435—478no—
304#435—478no—
305#435C2×D4718yescomplemented
306C8×D4—478no—
307C8×D4—478no—
308#433—478no—
309#432—474no—
310#432—478no—
311#431—474no—
312#424—472no—
313#424—472no—
314#423—471no—
315#423C2×D4711yescharacteristic
316#408—472no—
317#408C2×D4712yes—
318#407—472no—
319#405—472no—
320#405—474no—
321#405—472no—
322#405—474no—
323C4×M16—472no—
324C4×M16—474no—
325C2×#97—471no—
326C2×#97—471no—
327C2×#97—474no—
328C2×#97—471no—
329C2×#97—474no—
330C2×#97—474no—
331C2×#97C2×D4711yescharacteristic
332C47#6020711yescharacteristic, complemented, Sylow
333C2×#41—474no—
334C2×#41C22×D4712yes—
335C23×C4—471no—
336C23×C4—472no—
337#131—28no—
338#131—948no—
339C2×M16—472no—
340C2×M16—474no—
341C2×M16C22×D4712yes—
342C2×M16C22×D4714yes—
343C22×C8—471no—
344C22×C8—471no—
345C22×C8—474no—
346C22×C8—474no—
347C22×C8—474no—
348C22×C8—944no—
349C22×C8—944no—
350C22×C8C22×D4711yescharacteristic
351C22×C8C22×D4714yes—
352C22×C8C22×D4711yescharacteristic
353C4⋊Q8—472no—
354C4⋊Q8C4×D4711yescharacteristic
355#127C4×D4711yescharacteristic
356#126—478no—
357#125—474no—
358#125—472no—
359#124—474no—
360#124C4×D4712yes—
361#123—474no—
362#123—478no—
363C22⋊Q8—474no—
364C22⋊Q8—474no—
365C22⋊Q8—474no—
366#121—474no—
367#121—474no—
368#121C4×D4714yes—
369#120—474no—
370C4×Q8—474no—
371C4×Q8—474no—
372C4×Q8—474no—
373C4×D4—478no—
374C4×D4—474no—
375C4×D4—474no—
376C4×D4—478no—
377C4×D4—478no—
378C4×D4C22×D4714yes—
379#117—472no—
380#117—474no—
381#117—474no—
382#117—478no—
383C2×C42—472no—
384C2×C42—474no—
385C2×C42—471no—
386C2×C42—472no—
387C2×C42—472no—
388C2×C42—472no—
389C2×C42—472no—
390C2×C42C22×D4711yescharacteristic
391C4⋊C8—478no—
392C4⋊C8—478no—
393C4⋊C8—474no—
394C4⋊C8C22×D4714yes—
395C22⋊C8—478no—
396C22⋊C8—478no—
397C22⋊C8C22×D4718yes—
398#97—472no—
399#97—472no—
400#97—472no—
401#97—472no—
402#97—474no—
403#97—474no—
404#97—474no—
405#97—474no—
406#97C22×D4714yes—
407C4×C8—474no—
408C4×C8—474no—
409C4×C8—474no—
410C24—471no—
411#41—24no—
412#41—24no—
413#41—1888no—
414C2×Q8—474no—
415C2×Q8C2×C4×D4712yes—
416C2×D4—478no—
417C2×D4C2×C4×D4714yes—
418C2×D4C2×C4×D4712yes—
419C2×D4C2×C4×D4714yes—
420C22×C4—471no—
421C22×C4—472no—
422C22×C4—478no—
423C22×C4—472no—
424C22×C4—472no—
425C22×C4—472no—
426C22×C4—472no—
427C22×C4—472no—
428C22×C4—472no—
429C22×C4—942no—
430C22×C4—942no—
431C22×C4—1884no—
432C22×C4C2×C4×D4712yes—
433C22×C4C23×D4714yes—
434C22×C4C23×D4711yescharacteristic
435M16—24no—
436M16—28no—
437M16—948no—
438M16—944no—
439C2×C8—22no—
440C2×C8—28no—
441C2×C8—474no—
442C2×C8—472no—
443C2×C8—474no—
444C2×C8—472no—
445C2×C8—948no—
446C2×C8—948no—
447C2×C8—942no—
448C2×C8—948no—
449C2×C8#1062535214yescomplemented
450C2×C8C23×D4714yes—
451C2×C8C23×D4712yes—
452C2×C8C23×D4712yes—
453C4⋊C4—478no—
454C4⋊C4—478no—
455C4⋊C4—474no—
456C4⋊C4—474no—
457C4⋊C4—478no—
458C4⋊C4C2×C4×D4714yes—
459C22⋊C4—474no—
460C22⋊C4—478no—
461C22⋊C4—478no—
462C22⋊C4—478no—
463C22⋊C4C2×C4×D4718yes—
464C42—472no—
465C42—472no—
466C42—472no—
467C42—472no—
468C42—472no—
469C42—478no—
470C42—471no—
471C42—474no—
472C42—471no—
473C42C2×C4×D4711yescharacteristic
474C42C2×C4×D4711yescharacteristic
475C42C23×D4712yes—
476C23—472no—
477C23—944no—
478C23—942no—
479C23C22×C4×D4714yes—
480C23C22×C4×D4711yescharacteristic
481Q8—24no—
482Q8—1884no—
483D4—28no—
484D4—24no—
485D4—44no—
486D4—9416no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—42no—
490C2×C4—472no—
491C2×C4—472no—
492C2×C4—472no—
493C2×C4—478no—
494C2×C4—944no—
495C2×C4—944no—
496C2×C4—942no—
497C2×C4—944no—
498C2×C4—942no—
499C2×C4—942no—
500C2×C4—1882no—
501C2×C4—1882no—
502C2×C4—1882no—
503C2×C4—1888no—
504C2×C4C22×C4×D4712yes—
505C2×C4C22×C4×D4712yes—
506C2×C4C22×C4×D4714yes—
507C2×C4C22×C4×D4711yescharacteristic
508C2×C4#41×D4712yes—
509C2×C4C2×#1062535212yes—
510C2×C4C24×D4711yescharacteristic
511C8—22no—
512C8—24no—
513C8—474no—
514C8—944no—
515C8—942no—
516C8C2×#1062535214yes—
517C22—22no—
518C22—28no—
519C22—24no—
520C22—474no—
521C22—948no—
522C22—942no—
523C22—942no—
524C22C4×#1062535212yes—
525C22C23×C4×D4711yescharacteristic
526C4—22no—
527C4—22no—
528C4—24no—
529C4—942no—
530C4—942no—
531C4—948no—
532C4—942no—
533C4C4×#1062535212yes—
534C4C2×#41×D4711yescharacteristic
535C4C22×#1062535211yescharacteristic
536C2—22no—
537C2—44no—
538C2—944no—
539C2C2×C4×#1062535211yescharacteristic
540C2#568×D4712yes—
subgroupG/H
center, FrattiniC2×C4C24×D47
commutator, socleC22×C47C24×C4
FittingC47×#2932C2
prime radical#20256683C2
cosocleC2×C4×C47C25
hypercenter#2932D47
hypocenterC47#60207
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ #2932

Complements

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

NH ≅ G/Ntype
2. #20252398C2semidirect
3. #20252379C2semidirect
4. #20252328C2semidirect
5. #20252325C2semidirect
6. #20252106C2semidirect
7. #20252085C2semidirect
8. #20252080C2semidirect
9. #20252035C2semidirect
10. #20252020C2semidirect
11. #20251814C2semidirect
12. C2×#97×D47C2semidirect
13. C47×#2932C2semidirect
28. #15689252C22semidirect
29. #15689251C22semidirect
31. #15689195C22semidirect
35. #15689181C22semidirect
37. #15689177C22semidirect
45. #15689081C22semidirect
47. #15689078C22semidirect
49. M16×Dic47C22semidirect
53. C2×#13093934C22semidirect
58. C47×#435C22semidirect
168. C2×C8×C47#41semidirect
199. C2×C47⋊C8#41semidirect
272. #2932D47semidirect
305. #435C2×D47semidirect
332. C47#60207semidirect
449. C2×C8#10625352semidirect

Maximal subgroups

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

subgroupindex
#202566832
#202523982
#202523792
#202523282
#202523252
#202521062
#202520852
#202520802
#202520352
#202520202
#202518142
C2×#97×D472
C47×#29322
#6020747