Finite Group Databasebeta

← #26107951

Subgroups — #26107951

Order 12032.

1060 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243376C211yesmaximal, characteristic, complemented
2#20243323C212yesmaximal, complemented
3#20241956C212yesmaximal, complemented
4#20241706C211yesmaximal, characteristic, complemented
5C2×C47×#548C211yesmaximal, characteristic
6C2×#13093948—24no—
7C2×#13093939—21no—
8C2×C22⋊(C47⋊C8)C2212yes—
9#15688983C2212yes—
10#15688890—21no—
11#15688857—24no—
12#15688849—21no—
13#15688847—21no—
14C22×#41×C47C411yescharacteristic, complemented
15C23×Q8×C47C2211yescharacteristic
16C47×#548—24no—
17C47×#548—24no—
18C2×#124×C47—21no—
19C2×#124×C47C411yescharacteristic, complemented
20C2×C22⋊Q8×C47—21no—
21C2×C22⋊Q8×C47C411yescharacteristic, complemented
22C2×#121×C47—21no—
23C2×#121×C47C2211yescharacteristic
24C2×C4×Q8×C47C411yescharacteristic, complemented
25C2×C4×D4×C47—21no—
26C2×C4×D4×C47C2211yescharacteristic
27C22×C47⋊C8—22no—
28C2×C4⋊Dic47—22no—
29C2×C22⋊Dic47—22no—
30#13093948—48no—
31#13093939—42no—
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)—22no—
34C47⋊#95—22no—
35C2×#41×C47—22no—
36C2×#41×C47—22no—
37C2×#41×C47—44no—
38C22×Q8×C47—21no—
39C22×Q8×C47—22no—
40C22×Q8×C47—44no—
41C22×Q8×C47C2×C411yescharacteristic
42C22×Q8×C47C2×C411yescharacteristic
43C22×Q8×C47D414yes—
44C22×D4×C47—21no—
45C22×D4×C47—21no—
46C22×D4×C47C2×C411yescharacteristic
47C22×D4×C47D411yescharacteristic
48C23×C4×C47—21no—
49C23×C4×C47—21no—
50C23×C4×C47C2×C411yescharacteristic
51C23×C4×C47C2311yescharacteristic
52#124×C47—28no—
53#124×C47—22no—
54#124×C47—22no—
55C22⋊Q8×C47—28no—
56C22⋊Q8×C47—22no—
57C22⋊Q8×C47—22no—
58#121×C47—28no—
59#121×C47—22no—
60#121×C47D414yes—
61C4×Q8×C47—22no—
62C4×Q8×C47—22no—
63C4×D4×C47—28no—
64C4×D4×C47—22no—
65C4×D4×C47—22no—
66C2×C4⋊C4×C47—21no—
67C2×C4⋊C4×C47—21no—
68C2×C4⋊C4×C47C2×C411yescharacteristic
69C2×C4⋊C4×C47D411yescharacteristic
70C2×C22⋊C4×C47—21no—
71C2×C22⋊C4×C47—21no—
72C2×C22⋊C4×C47—21no—
73C2×C22⋊C4×C47—21no—
74C2×C22⋊C4×C47D411yescharacteristic
75C2×C22⋊C4×C47D411yescharacteristic
76C2×C42×C47—21no—
77C2×C42×C47C2×C411yescharacteristic
78C24×C47—21no—
79C24×C47C22⋊C411yescharacteristic
80C24×C47C2×D411yescharacteristic
81#41×C47—48no—
82#41×C47—44no—
83#41×C47—44no—
84C2×Q8×C47—24no—
85C2×Q8×C47—21no—
86C2×Q8×C47—21no—
87C2×Q8×C47—24no—
88C2×Q8×C47—28no—
89C2×Q8×C47—42no—
90C2×Q8×C47—44no—
91C2×Q8×C47—44no—
92C2×Q8×C47—48no—
93C2×Q8×C47C22⋊C412yes—
94C2×Q8×C47C22⋊C412yes—
95C2×D4×C47—24no—
96C2×D4×C47—22no—
97C2×D4×C47—24no—
98C2×D4×C47—22no—
99C2×D4×C47—21no—
100C2×D4×C47—21no—
101C2×D4×C47—21no—
102C2×D4×C47—44no—
103C2×D4×C47—42no—
104C2×D4×C47—42no—
105C2×D4×C47—41no—
106C2×D4×C47—41no—
107C2×D4×C47C22⋊C412yes—
108C22×C4×C47—21no—
109C22×C4×C47—24no—
110C22×C4×C47—21no—
111C22×C4×C47—21no—
112C22×C4×C47—24no—
113C22×C4×C47—21no—
114C22×C4×C47—21no—
115C22×C4×C47—42no—
116C22×C4×C47—44no—
117C22×C4×C47—44no—
118C22×C4×C47—42no—
119C22×C4×C47C22⋊C411yescharacteristic
120C22×C4×C47C22⋊C411yescharacteristic
121C22×C4×C47C22⋊C411yescharacteristic
122C22×C4×C47C22×C411yescharacteristic
123C22×C4×C47C2×D412yes—
124C22×C4×C47C2×D412yes—
125C22×C4×C47C2×D411yescharacteristic
126C22×Dic47—42no—
127C4⋊C4×C47—24no—
128C4⋊C4×C47—24no—
129C4⋊C4×C47—21no—
130C4⋊C4×C47—22no—
131C4⋊C4×C47C22⋊C412yes—
132C22⋊C4×C47—24no—
133C22⋊C4×C47—24no—
134C22⋊C4×C47—24no—
135C22⋊C4×C47—22no—
136C22⋊C4×C47—24no—
137C22⋊C4×C47—22no—
138C42×C47—24no—
139C42×C47—21no—
140C42×C47—21no—
141C22⋊Dic47—42no—
142C22⋊Dic47—42no—
143C4⋊Dic47—44no—
144C2×C47⋊C8—44no—
145C2×C47⋊C8—42no—
146C23×C47—21no—
147C23×C47—22no—
148C23×C47—21no—
149C23×C47—21no—
150C23×C47—41no—
151C23×C47—41no—
152C23×C47—42no—
153C23×C47—42no—
154C23×C47—41no—
155C23×C47—41no—
156C23×C47—41no—
157C23×C47—41no—
158C23×C47C2×C22⋊C411yescharacteristic
159C23×C47#12012yes—
160C23×C47#13512yes—
161C23×C47AGL(1,ℤ/8)11yescharacteristic
162C23×C47#13711yescharacteristic
163Q8×C47—24no—
164Q8×C47—22no—
165Q8×C47—24no—
166Q8×C47—48no—
167Q8×C47—42no—
168Q8×C47#9914yescomplemented
169D4×C47—42no—
170D4×C47—48no—
171D4×C47—42no—
172D4×C47—44no—
173D4×C47—42no—
174D4×C47—44no—
175D4×C47—42no—
176C2×C4×C47—24no—
177C2×C4×C47—22no—
178C2×C4×C47—24no—
179C2×C4×C47—22no—
180C2×C4×C47—21no—
181C2×C4×C47—21no—
182C2×C4×C47—22no—
183C2×C4×C47—22no—
184C2×C4×C47—21no—
185C2×C4×C47—21no—
186C2×C4×C47—21no—
187C2×C4×C47—41no—
188C2×C4×C47—42no—
189C2×C4×C47—48no—
190C2×C4×C47—48no—
191C2×C4×C47—42no—
192C2×C4×C47—42no—
193C2×C4×C47—41no—
194C2×C4×C47—41no—
195C2×C4×C47—42no—
196C2×C4×C47—41no—
197C2×C4×C47—41no—
198C2×C4×C47C2×C22⋊C411yescharacteristic
199C2×C4×C47C2×C22⋊C411yescharacteristic
200C2×C4×C47#12012yes—
201C2×Dic47—82no—
202C2×Dic47—82no—
203C2×Dic47—82no—
204C47⋊C8—82no—
205#8377—471nomaximal, Sylow
206C22×C47—22no—
207C22×C47—21no—
208C22×C47—22no—
209C22×C47—21no—
210C22×C47—42no—
211C22×C47—41no—
212C22×C47—41no—
213C22×C47—41no—
214C22×C47—42no—
215C22×C47—41no—
216C22×C47—42no—
217C22×C47—41no—
218C22×C47—42no—
219C22×C47—81no—
220C22×C47—81no—
221C22×C47—82no—
222C22×C47—82no—
223C22×C47#37911yescharacteristic
224C22×C47#41611yescharacteristic
225C22×C47#41711yescharacteristic
226C22×C47#44912yes—
227C22×C47#45212yes—
228C4×C47—22no—
229C4×C47—24no—
230C4×C47—21no—
231C4×C47—41no—
232C4×C47—41no—
233C4×C47—42no—
234C4×C47—41no—
235C4×C47—42no—
236C4×C47C2×#9912yes—
237Dic47—162no—
238C2×#548D4711yescharacteristic
239#1799—471no—
240#1787—472no—
241#1519—472no—
242#1459—471no—
243C2×C47—22no—
244C2×C47—21no—
245C2×C47—21no—
246C2×C47—22no—
247C2×C47—81no—
248C2×C47—82no—
249C2×C47#178111yescharacteristic
250C2×C47#178811yescharacteristic
251C2×C47#179611yescharacteristic
252C22×#41Dic4711yescharacteristic, complemented
253C23×Q8C2×D4711yescharacteristic
254#548—24no—
255#548—24no—
256C2×#124—21no—
257C2×#124Dic4711yescharacteristic, complemented
258C2×C22⋊Q8—21no—
259C2×C22⋊Q8Dic4711yescharacteristic, complemented
260C2×#121—21no—
261C2×#121C2×D4711yescharacteristic
262C2×C4×Q8Dic4711yescharacteristic, complemented
263C2×C4×D4—21no—
264C2×C4×D4C2×D4711yescharacteristic
265C2×#103—944no—
266C2×#99—941no—
267C2×C22⋊C8—472no—
268#380—472no—
269#342—941no—
270#327—944no—
271#324—941no—
272#323—941no—
273C47#837711yescharacteristic, complemented, Sylow
274C2×#41—22no—
275C2×#41—22no—
276C2×#41—44no—
277C22×Q8—21no—
278C22×Q8—22no—
279C22×Q8—44no—
280C22×Q8C2×Dic4711yescharacteristic
281C22×Q8C2×Dic4711yescharacteristic
282C22×Q8C22⋊D4714yes—
283C22×D4—21no—
284C22×D4—21no—
285C22×D4C2×Dic4711yescharacteristic
286C22×D4C22⋊D4711yescharacteristic
287C23×C4—21no—
288C23×C4—21no—
289C23×C4C2×Dic4711yescharacteristic
290C23×C4C22×D4711yescharacteristic
291C22×C8—942no—
292#124—28no—
293#124—22no—
294#124—22no—
295C22⋊Q8—28no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298#121—28no—
299#121—22no—
300#121C22⋊D4714yes—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×D4—28no—
304C4×D4—22no—
305C4×D4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—942no—
309C2×C4⋊C4C2×Dic4711yescharacteristic
310C2×C4⋊C4C22⋊D4711yescharacteristic
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—942no—
316C2×C22⋊C4C22⋊D4711yescharacteristic
317C2×C22⋊C4C22⋊D4711yescharacteristic
318C2×C42—21no—
319C2×C42C2×Dic4711yescharacteristic
320#103—1888no—
321#99—1882no—
322C22⋊C8—942no—
323C22⋊C8—942no—
324#95—942no—
325C24—21no—
326C24C22⋊Dic4711yescharacteristic
327C24C2×C22⋊D4711yescharacteristic
328#41—48no—
329#41—44no—
330#41—44no—
331C2×Q8—24no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—24no—
335C2×Q8—28no—
336C2×Q8—42no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—48no—
340C2×Q8C22⋊Dic4712yes—
341C2×Q8C22⋊Dic4712yes—
342C2×D4—24no—
343C2×D4—22no—
344C2×D4—24no—
345C2×D4—22no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—44no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4C22⋊Dic4712yes—
355C22×C4—21no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—42no—
366C22×C4—1882no—
367C22×C4C22⋊Dic4711yescharacteristic
368C22×C4C22⋊Dic4711yescharacteristic
369C22×C4C22⋊Dic4711yescharacteristic
370C22×C4C22×Dic4711yescharacteristic
371C22×C4C2×C22⋊D4712yes—
372C22×C4C2×C22⋊D4712yes—
373C22×C4C2×C22⋊D4711yescharacteristic
374C2×C8—1884no—
375C2×C8—1882no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—1884no—
381C4⋊C4C22⋊Dic4712yes—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—22no—
386C22⋊C4—24no—
387C22⋊C4—22no—
388C22⋊C4—1882no—
389C22⋊C4—1882no—
390C42—24no—
391C42—21no—
392C42—21no—
393C23—21no—
394C23—22no—
395C23—21no—
396C23—21no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23C2×C22⋊Dic4711yescharacteristic
406C23#1309399312yes—
407C23#1309404412yes—
408C23#1309404911yescharacteristic
409C23#1309405411yescharacteristic
410Q8—24no—
411Q8—22no—
412Q8—24no—
413Q8—48no—
414Q8—42no—
415Q8#1309393914yescomplemented
416D4—42no—
417D4—48no—
418D4—42no—
419D4—44no—
420D4—42no—
421D4—44no—
422D4—42no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—48no—
437C2×C4—48no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—3762no—
446C2×C4—3762no—
447C2×C4—3762no—
448C2×C4C2×C22⋊Dic4711yescharacteristic
449C2×C4C2×C22⋊Dic4711yescharacteristic
450C2×C4#1309399312yes—
451C8—3762no—
452C22—21no—
453C22—22no—
454C22—21no—
455C22—22no—
456C22—42no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—42no—
461C22—41no—
462C22—42no—
463C22—42no—
464C22—41no—
465C22—81no—
466C22—81no—
467C22—82no—
468C22—82no—
469C22#1568897911yescharacteristic
470C22#1568911911yescharacteristic
471C22#1568912411yescharacteristic
472C22#1568926812yes—
473C22#1568928512yes—
474C4—22no—
475C4—24no—
476C4—21no—
477C4—41no—
478C4—42no—
479C4—41no—
480C4—42no—
481C4—41no—
482C4—7522no—
483C4C2×#1309393912yes—
484C2—22no—
485C2—21no—
486C2—21no—
487C2—22no—
488C2—82no—
489C2—81no—
490C2#2024330311yescharacteristic
491C2#2024332811yescharacteristic
492C2#2024336311yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
Fitting, prime radicalC2×C47×#548C2
socleC22×C47#379
cosocleC23×C4×C47C23
hypercenterC2×#548D47
hypocenterC47#8377
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
1. #20243376C2semidirect
2. #20243323C2semidirect
3. #20241956C2semidirect
4. #20241706C2semidirect
14. C22×#41×C47C4semidirect
19. C2×#124×C47C4semidirect
21. C2×C22⋊Q8×C47C4semidirect
24. C2×C4×Q8×C47C4semidirect
168. Q8×C47#99semidirect
252. C22×#41Dic47semidirect
257. C2×#124Dic47semidirect
259. C2×C22⋊Q8Dic47semidirect
262. C2×C4×Q8Dic47semidirect
273. C47#8377semidirect
415. Q8#13093939semidirect

Maximal subgroups

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

subgroupindex
#202433762
#202433232
#202419562
#202417062
C2×C47×#5482
#837747