Finite Group Databasebeta

← #26220809

Subgroups — #26220809

Order 12032.

1174 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D47C211yesmaximal, characteristic
2C2×#15689145C211yesmaximal, characteristic
3C4×#13093982C211yesmaximal, characteristic
4#20244346C211yesmaximal, characteristic, complemented
5#20244345C211yesmaximal, characteristic, complemented
6#20244344C212yesmaximal, complemented
7#20244343C211yesmaximal, characteristic, complemented
8#20244342C212yesmaximal, complemented
9#20243106C212yesmaximal
10#20243106C212yesmaximal
11C47×#1989C211yesmaximal, characteristic, complemented
12C22×#10625324C2212yes—
13C2×#13094016C2211yescharacteristic
14C2×#13094015C2211yescharacteristic
15C2×#13094015C2211yescharacteristic
16C2×#13094014—21no—
17C2×#13094014—21no—
18#125×D47—24no—
19#125×D47C418yescomplemented
20C2×#13093982C2212yes—
21C2×C4⋊C4×D47—21no—
22C2×C4⋊C4×D47—21no—
23C2×C4⋊C4×D47C2211yescharacteristic
24C2×C4⋊C4×D47C2211yescharacteristic
25C2×C42×D47C2211yescharacteristic
26#15689145—24no—
27C2×C4⋊(C47⋊C8)C2211yescharacteristic
28C2×Dic47⋊C8C2212yes—
29C4⋊C4×Dic47C2212yes—
30C4×C4⋊Dic47C2211yescharacteristic
31C4×(C2×D47)⋊C4—21no—
32#15688970—21no—
33#15688968—21no—
34C42×Dic47—21no—
35#15688886C2212yes—
36#15688886C2212yes—
37#15688885C2212yes—
38#15688885C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×#125×C47C2211yescharacteristic
42C2×C4⋊C8×C47C2211yescharacteristic
43C4×C4⋊C4×C47C2211yescharacteristic
44C47×#340C2212yes—
45C47×#340C2212yes—
46C22×C4×D47—21no—
47C22×C4×D47—21no—
48C22×C4×D47D411yescharacteristic
49C22×C4×D47D411yescharacteristic
50C22×C4×D47C2311yescharacteristic
51C2×#10625324—24no—
52C2×#10625324—28no—
53C22×C47⋊C8C2312yes—
54#13094016—22no—
55#13094016C2×C414yes—
56#13094015—22no—
57#13094015—22no—
58#13094015C2×C414yes—
59#13094015C2×C414yes—
60#13094014—28no—
61#13094014—28no—
62#13093982—24no—
63#13093982Q818yes—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47C2311yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2312yes—
69C2×Dic47⋊C4—21no—
70C2×Dic47⋊C4—21no—
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C4⋊C4×D47—28no—
76C4⋊C4×D47—28no—
77C4⋊C4×D47—22no—
78C4⋊C4×D47—22no—
79C4⋊C4×D47C2×C414yes—
80C4⋊C4×D47C2×C414yes—
81C2×(C2×D47)⋊C4—22no—
82C2×C4×Dic47—22no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—22no—
85C2×C4×Dic47D411yescharacteristic
86C2×C4×Dic47D411yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88C42×D47C2×C414yes—
89C42×D47D414yes—
90C4⋊(C47⋊C8)—22no—
91Dic47⋊C8—24no—
92C47⋊#95—21no—
93C47⋊#95—22no—
94C22×C8×C47C2312yes—
95#125×C47—22no—
96#125×C47C2×C414yes—
97C2×C4⋊C4×C47—21no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47C2311yescharacteristic
100C2×C4⋊C4×C47C2311yescharacteristic
101C2×C4⋊C4×C47C2312yes—
102C2×C42×C47—21no—
103C2×C42×C47C2311yescharacteristic
104C4⋊C8×C47—22no—
105#95×C47—21no—
106C23×D47C2×D411yescharacteristic
107C22×C4×C47—22no—
108C22×C4×C47—21no—
109C22×C4×C47—21no—
110C22×C4×C47C2×D411yescharacteristic
111C22×C4×C47C2×D411yescharacteristic
112C22×C4×C47C2411yescharacteristic
113C22×Dic47—22no—
114C22×Dic47—21no—
115C22×Dic47—21no—
116C22×Dic47C2×D411yescharacteristic
117C22×Dic47C2×D411yescharacteristic
118C22×Dic47C2×D411yescharacteristic
119C2×C4×D47—24no—
120C2×C4×D47—24no—
121C2×C4×D47—42no—
122C2×C4×D47—41no—
123C2×C4×D47—41no—
124C2×C4×D47—42no—
125C2×C4×D47—44no—
126C2×C4×D47—42no—
127C2×C4×D47C22⋊C414yes—
128C2×C4×D47C22⋊C414yes—
129C2×C4×D47C22×C412yes—
130C2×C4×D47C2×D412yes—
131C2×C4×D47C2×Q814yes—
132C2×C4×D47#4114yes—
133C2×C8×C47—22no—
134C2×C8×C47—24no—
135C4⋊C4×C47—24no—
136C4⋊C4×C47—24no—
137C4⋊C4×C47—21no—
138C4⋊C4×C47—21no—
139C4⋊C4×C47—22no—
140C4⋊C4×C47C22×C412yes—
141C4⋊C4×C47C22×C412yes—
142C4⋊C4×C47C2×Q814yes—
143C42×C47—42no—
144C42×C47C22×C412yes—
145C42×C47C2×D412yes—
146(C2×D47)⋊C4—28no—
147C4⋊Dic47—21no—
148C4⋊Dic47—21no—
149C4⋊Dic47—22no—
150C4⋊Dic47—24no—
151C4⋊Dic47—24no—
152C4⋊Dic47C22×C412yes—
153C4⋊Dic47C22×C412yes—
154C4⋊Dic47C2×Q814yes—
155Dic47⋊C4—24no—
156Dic47⋊C4—24no—
157Dic47⋊C4—24no—
158Dic47⋊C4—24no—
159Dic47⋊C4—24no—
160Dic47⋊C4—24no—
161C4×Dic47—28no—
162C4×Dic47—44no—
163C4×Dic47—42no—
164C4×Dic47C22⋊C414yes—
165C4×Dic47C22⋊C414yes—
166C4×Dic47C22×C412yes—
167C4×Dic47C2×D412yes—
168C2×C47⋊C8—22no—
169C2×C47⋊C8—24no—
170#10625324—48no—
171C23×C47C22×D411yescharacteristic
172C22×D47—21no—
173C22×D47—22no—
174C22×D47—21no—
175C22×D47C2×C22⋊C412yes—
176C22×D47C22⋊Q814yes—
177C2×C4×C47—21no—
178C2×C4×C47—21no—
179C2×C4×C47—22no—
180C2×C4×C47—22no—
181C2×C4×C47—22no—
182C2×C4×C47—24no—
183C2×C4×C47—42no—
184C2×C4×C47—42no—
185C2×C4×C47—41no—
186C2×C4×C47—42no—
187C2×C4×C47—41no—
188C2×C4×C47C2×C22⋊C412yes—
189C2×C4×C47C2×C22⋊C412yes—
190C2×C4×C47C23×C411yescharacteristic
191C2×C4×C47C22×D411yescharacteristic
192C2×C4×C47C22×Q812yes—
193C2×C4×C47C2×#4112yes—
194C2×Dic47—22no—
195C2×Dic47—21no—
196C2×Dic47—21no—
197C2×Dic47—22no—
198C2×Dic47—22no—
199C2×Dic47—24no—
200C2×Dic47—42no—
201C2×Dic47—42no—
202C2×Dic47—41no—
203C2×Dic47—42no—
204C2×Dic47—41no—
205C2×Dic47C2×C22⋊C412yes—
206C2×Dic47C2×C22⋊C412yes—
207C2×Dic47C2×C22⋊C412yes—
208C2×Dic47C22⋊Q814yes—
209C4×D47—44no—
210C4×D47—44no—
211C4×D47—48no—
212C4×D47—48no—
213C4×D47C4×Q818yes—
214C4×D47#12414yes—
215C4×D47C4⋊Q814yes—
216C8×C47—42no—
217C47⋊C8—42no—
218#21314—471nomaximal, Sylow
219C22×C47C22×C22⋊C411yescharacteristic
220C22×C47C2×C22⋊Q812yes—
221C22×C47#57511yescharacteristic
222C22×C47#57512yes—
223C22×C47#57511yescharacteristic
224C2×D47—24no—
225C2×D47—24no—
226C2×D47—22no—
227C2×D47—22no—
228C2×D47#39114yes—
229C4×C47—22no—
230C4×C47—22no—
231C4×C47—44no—
232C4×C47—42no—
233C4×C47—42no—
234C4×C47C2×C4×Q812yes—
235C4×C47C2×#12411yescharacteristic
236C4×C47C2×C4⋊Q811yescharacteristic
237Dic47—22no—
238Dic47—22no—
239Dic47—44no—
240Dic47—42no—
241Dic47—42no—
242Dic47#39114yes—
243C22×#125—471no—
244C2×#423—471no—
245C4×#117—471no—
246#1989—471no—
247#1989—471no—
248#1989—472no—
249#1989—472no—
250#1989—471no—
251#1989D4711yescharacteristic, complemented
252#1736—472no—
253#1736—472no—
254C2×C47C2×#39111yescharacteristic
255C2×C47#289611yescharacteristic
256C2×C47#289611yescharacteristic
257C2×C47#308012yes—
258C2×C47#308012yes—
259D47—24no—
260C22×M16—472no—
261C2×#125—478no—
262C2×#125—471no—
263C2×#125—471no—
264C2×#125—471no—
265C2×#125—941no—
266C2×#125—941no—
267C2×#125—944no—
268C2×#125C2×D4711yescharacteristic
269C2×#117—472no—
270C22×C4⋊C4—471no—
271C22×C4⋊C4—471no—
272C22×C4⋊C4—941no—
273C22×C4⋊C4—941no—
274C22×C42—471no—
275#423—944no—
276C2×C4⋊C8—472no—
277C2×C4⋊C8—471no—
278C2×C4⋊C8C2×D4711yescharacteristic
279C4×C4⋊C4—472no—
280C4×C4⋊C4—471no—
281C4×C4⋊C4C2×D4711yescharacteristic
282C4×C22⋊C4—941no—
283C4×C22⋊C4—941no—
284#376—941no—
285C43—941no—
286#340—472no—
287#340—472no—
288#340—472no—
289#340—472no—
290#340—472no—
291#340—472no—
292#340C2×D4712yes—
293#340C2×D4712yes—
294C47#2131411yescharacteristic, complemented, Sylow
295C23×C4—471no—
296C23×C4—471no—
297C23×C4—471no—
298C23×C4—941no—
299C23×C4—941no—
300C2×M16—948no—
301C2×M16—944no—
302C22×C8—472no—
303C22×C8C22×D4712yes—
304#125—22no—
305#125—474no—
306#125—474no—
307#125—474no—
308#125—948no—
309#125—948no—
310#125—942no—
311#125—942no—
312#125—942no—
313#125C4×D4714yes—
314#117—478no—
315#117—944no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—471no—
319C2×C4⋊C4—471no—
320C2×C4⋊C4—472no—
321C2×C4⋊C4—474no—
322C2×C4⋊C4—474no—
323C2×C4⋊C4—942no—
324C2×C4⋊C4—942no—
325C2×C4⋊C4—948no—
326C2×C4⋊C4—948no—
327C2×C4⋊C4—941no—
328C2×C4⋊C4—941no—
329C2×C4⋊C4—941no—
330C2×C4⋊C4—941no—
331C2×C4⋊C4—941no—
332C2×C4⋊C4—941no—
333C2×C4⋊C4—941no—
334C2×C4⋊C4—941no—
335C2×C4⋊C4C22×D4711yescharacteristic
336C2×C4⋊C4C22×D4711yescharacteristic
337C2×C4⋊C4C22×D4712yes—
338C2×C22⋊C4—942no—
339C2×C42—21no—
340C2×C42—471no—
341C2×C42—471no—
342C2×C42—471no—
343C2×C42—474no—
344C2×C42—474no—
345C2×C42—942no—
346C2×C42—942no—
347C2×C42—941no—
348C2×C42C22×D4711yescharacteristic
349C4⋊C8—22no—
350C4⋊C8—944no—
351C4⋊C8—942no—
352#95—21no—
353#95—942no—
354#95—941no—
355C24—471no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—471no—
360C22×C4—472no—
361C22×C4—472no—
362C22×C4—474no—
363C22×C4—474no—
364C22×C4—471no—
365C22×C4—471no—
366C22×C4—474no—
367C22×C4—474no—
368C22×C4—944no—
369C22×C4—944no—
370C22×C4—942no—
371C22×C4—941no—
372C22×C4—941no—
373C22×C4—1881no—
374C22×C4—1882no—
375C22×C4—1881no—
376C22×C4—1882no—
377C22×C4—1884no—
378C22×C4—1882no—
379C22×C4D4×D4711yescharacteristic
380C22×C4D4×D4711yescharacteristic
381C22×C4C23×D4711yescharacteristic
382M16—1888no—
383C2×C8—22no—
384C2×C8—24no—
385C2×C8—942no—
386C2×C8—944no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—22no—
392C4⋊C4—472no—
393C4⋊C4—472no—
394C4⋊C4—474no—
395C4⋊C4—944no—
396C4⋊C4—944no—
397C4⋊C4—944no—
398C4⋊C4—944no—
399C4⋊C4—944no—
400C4⋊C4—944no—
401C4⋊C4—944no—
402C4⋊C4—944no—
403C4⋊C4—941no—
404C4⋊C4—941no—
405C4⋊C4—942no—
406C4⋊C4C2×C4×D4712yes—
407C4⋊C4C2×C4×D4712yes—
408C4⋊C4Q8×D4714yes—
409C22⋊C4—948no—
410C42—42no—
411C42—474no—
412C42—474no—
413C42—472no—
414C42—472no—
415C42—948no—
416C42—1882no—
417C42—1884no—
418C42C2×C4×D4712yes—
419C42D4×D4712yes—
420C23—472no—
421C23—474no—
422C23—941no—
423C23—942no—
424C23—941no—
425C23C2×D4×D4711yescharacteristic
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—472no—
438C2×C4—474no—
439C2×C4—478no—
440C2×C4—474no—
441C2×C4—474no—
442C2×C4—472no—
443C2×C4—472no—
444C2×C4—941no—
445C2×C4—942no—
446C2×C4—941no—
447C2×C4—942no—
448C2×C4—942no—
449C2×C4—944no—
450C2×C4—1884no—
451C2×C4—1884no—
452C2×C4—1888no—
453C2×C4—1888no—
454C2×C4—1881no—
455C2×C4—1882no—
456C2×C4—1882no—
457C2×C4—1882no—
458C2×C4—1881no—
459C2×C4C22⋊C4×D4712yes—
460C2×C4C22⋊C4×D4712yes—
461C2×C4C22×C4×D4711yescharacteristic
462C2×C4C2×D4×D4711yescharacteristic
463C2×C4C2×Q8×D4712yes—
464C2×C4#41×D4712yes—
465C8—42no—
466C8—1882no—
467C22—474no—
468C22—942no—
469C22—944no—
470C22—944no—
471C22—942no—
472C22C2×C22⋊C4×D4711yescharacteristic
473C22C22⋊Q8×D4712yes—
474C22#1568991911yescharacteristic
475C22#1568992012yes—
476C22#1568992211yescharacteristic
477C4—22no—
478C4—22no—
479C4—42no—
480C4—44no—
481C4—42no—
482C4—474no—
483C4—942no—
484C4—942no—
485C4—1882no—
486C4—1884no—
487C4—1882no—
488C4C4×Q8×D4712yes—
489C4#124×D4711yescharacteristic
490C4C4⋊Q8×D4711yescharacteristic
491C2—944no—
492C2#391×D4711yescharacteristic
493C2#2025190311yescharacteristic
494C2#2025190411yescharacteristic
495C2#2025320512yes—
496C2#2025320712yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1989C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1989D47
hypocenterC47#21314
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
4. #20244346C2semidirect
5. #20244345C2semidirect
6. #20244344C2semidirect
7. #20244343C2semidirect
8. #20244342C2semidirect
11. C47×#1989C2semidirect
19. #125×D47C4semidirect
251. #1989D47semidirect
294. C47#21314semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D472
C2×#156891452
C4×#130939822
#202443462
#202443452
#202443442
#202443432
#202443422
#202431062
C47×#19892
#2131447