Finite Group Databasebeta

← #26182503

Subgroups — #26182503

Order 12032.

1290 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251837C212yesmaximal, complemented
2#20241752C212yesmaximal, complemented
3#20241736C214yesmaximal, complemented
4#20241726C211yesmaximal, characteristic, complemented
5#20241718C212yesmaximal, complemented
6#20241706C211yesmaximal, characteristic, complemented
7C2×#15688849C212yesmaximal, complemented
8C2×C47×#547C211yesmaximal, characteristic
9C2×#13094031—22no—
10#15689090—24no—
11#15689090—22no—
12#15689087C2214yes—
13#15689087C2212yes—
14C2×C22⋊(C47⋊C8)C2214yes—
15C2×C22⋊(C47⋊C8)C2212yes—
16#15688849C2211yescharacteristic
17#15688849C2212yes—
18#15688849C2214yes—
19#15688849C2211yescharacteristic
20#15688849C2214yes—
21#15688847C2212yes—
22#15688847C2212yes—
23C22×#41×C47C411yescharacteristic
24C22×#41×C47C2211yescharacteristic
25C47×#547—22no—
26C47×#547—24no—
27C47×#547—22no—
28C2×C4⋊Q8×C47C411yescharacteristic
29C2×#123×C47C412yes—
30C2×#123×C47C2212yes—
31C2×C22⋊Q8×C47C411yescharacteristic
32C2×C22⋊Q8×C47C2211yescharacteristic
33C2×#121×C47C412yes—
34C2×C4×D4×C47C411yescharacteristic
35C2×C4×D4×C47C2211yescharacteristic
36C22×C4⋊C4×C47C2212yes—
37#13094031—48no—
38C2×#10625329—24no—
39C2×#10625329—22no—
40C22×C47⋊C8—24no—
41C22×C47⋊C8—22no—
42C22⋊(C47⋊C8)—24no—
43C22⋊(C47⋊C8)—22no—
44C22⋊(C47⋊C8)—24no—
45C22⋊(C47⋊C8)—22no—
46C22⋊(C47⋊C8)C2312yes—
47C22⋊(C47⋊C8)C2314yes—
48C22⋊(C47⋊C8)C2312yes—
49C2×#41×C47—21no—
50C2×#41×C47—22no—
51C2×#41×C47—21no—
52C2×#41×C47—44no—
53C2×#41×C47—44no—
54C2×#41×C47D412yes—
55C2×#41×C47D414yes—
56C2×#41×C47D412yes—
57C22×Q8×C47C2×C411yescharacteristic
58C22×Q8×C47C2×C411yescharacteristic
59C22×D4×C47C2×C411yescharacteristic
60C22×D4×C47C2×C411yescharacteristic
61C22×D4×C47C2×C412yes—
62C22×D4×C47C2×C412yes—
63C23×C4×C47C2×C412yes—
64C23×C4×C47C2×C412yes—
65C23×C4×C47C2×C411yescharacteristic
66C23×C4×C47C2312yes—
67C23×C4×C47C2311yescharacteristic
68C4⋊Q8×C47—22no—
69C4⋊Q8×C47—22no—
70#123×C47—24no—
71#123×C47—24no—
72#123×C47—24no—
73#123×C47—24no—
74C22⋊Q8×C47—22no—
75C22⋊Q8×C47—22no—
76C22⋊Q8×C47—22no—
77C22⋊Q8×C47—22no—
78#121×C47—24no—
79#121×C47—24no—
80C4×D4×C47—22no—
81C4×D4×C47—21no—
82C4×D4×C47—21no—
83C4×D4×C47—22no—
84C4×D4×C47—22no—
85C2×C4⋊C4×C47—24no—
86C2×C4⋊C4×C47—24no—
87C2×C4⋊C4×C47C2×C412yes—
88C2×C4⋊C4×C47C2×C412yes—
89C2×C4⋊C4×C47C2×C411yescharacteristic
90C2×C4⋊C4×C47C2×C412yes—
91C2×C4⋊C4×C47C2311yescharacteristic
92C2×C4⋊C4×C47C2312yes—
93C2×C22⋊C4×C47C2×C412yes—
94C2×C22⋊C4×C47C2×C411yescharacteristic
95C2×C22⋊C4×C47C2×C412yes—
96C2×C22⋊C4×C47C2×C412yes—
97C2×C22⋊C4×C47C2311yescharacteristic
98C2×C42×C47C2×C411yescharacteristic
99C24×C47C22×C412yes—
100C24×C47C22×C411yescharacteristic
101C24×C47C22×C411yescharacteristic
102#41×C47—28no—
103#41×C47—48no—
104#41×C47—44no—
105#41×C47—48no—
106#41×C47—44no—
107#41×C47—44no—
108C2×Q8×C47—21no—
109C2×Q8×C47—21no—
110C2×Q8×C47—42no—
111C2×Q8×C47—42no—
112C2×Q8×C47C22⋊C412yes—
113C2×Q8×C47C22⋊C412yes—
114C2×D4×C47—24no—
115C2×D4×C47—22no—
116C2×D4×C47—22no—
117C2×D4×C47—22no—
118C2×D4×C47—22no—
119C2×D4×C47—41no—
120C2×D4×C47—44no—
121C2×D4×C47—44no—
122C2×D4×C47C22⋊C412yes—
123C2×D4×C47C22⋊C411yescharacteristic
124C2×D4×C47C22⋊C414yes—
125C2×D4×C47C22⋊C414yes—
126C2×D4×C47C22⋊C411yescharacteristic
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—24no—
134C22×C4×C47—42no—
135C22×C4×C47—42no—
136C22×C4×C47—44no—
137C22×C4×C47—42no—
138C22×C4×C47—44no—
139C22×C4×C47C22×C412yes—
140C22×C4×C47C22×C411yescharacteristic
141C22×C4×C47C22×C411yescharacteristic
142C22×C4×C47C22×C411yescharacteristic
143C22×C4×C47C22×C411yescharacteristic
144C22×C4×C47C22×C412yes—
145C22×C4×C47C22×C412yes—
146C22×C4×C47C2×D414yes—
147C22×C4×C47C2×D412yes—
148C22×C4×C47C2×D414yes—
149C22×C4×C47C2×D412yes—
150C22×C4×C47C2411yescharacteristic
151C4⋊C4×C47—24no—
152C4⋊C4×C47—24no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—24no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47—24no—
158C22⋊C4×C47—22no—
159C22⋊C4×C47—22no—
160C22⋊C4×C47—22no—
161C22⋊C4×C47—24no—
162C22⋊C4×C47—24no—
163C22⋊C4×C47—22no—
164C42×C47—21no—
165C42×C47—21no—
166#10625329—48no—
167#10625329—44no—
168C2×C47⋊C8—22no—
169C2×C47⋊C8—24no—
170C2×C47⋊C8—22no—
171C2×C47⋊C8—48no—
172C2×C47⋊C8—44no—
173C23×C47—22no—
174C23×C47—22no—
175C23×C47—24no—
176C23×C47—21no—
177C23×C47—21no—
178C23×C47—21no—
179C23×C47—21no—
180C23×C47—21no—
181C23×C47—42no—
182C23×C47—41no—
183C23×C47—41no—
184C23×C47—41no—
185C23×C47—41no—
186C23×C47C2×C22⋊C412yes—
187C23×C47C2×C22⋊C412yes—
188C23×C47C2×C22⋊C412yes—
189C23×C47C23×C411yescharacteristic
190Q8×C47—24no—
191Q8×C47—42no—
192Q8×C47—42no—
193Q8×C47—42no—
194D4×C47—24no—
195D4×C47—28no—
196D4×C47—22no—
197D4×C47—44no—
198D4×C47—44no—
199D4×C47—44no—
200D4×C47—44no—
201D4×C47#9914yes—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—28no—
205C2×C4×C47—24no—
206C2×C4×C47—22no—
207C2×C4×C47—21no—
208C2×C4×C47—22no—
209C2×C4×C47—21no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—41no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—44no—
217C2×C4×C47—44no—
218C2×C4×C47—44no—
219C2×C4×C47—42no—
220C2×C4×C47—44no—
221C2×C4×C47—44no—
222C2×C4×C47—42no—
223C2×C4×C47—41no—
224C2×C4×C47—42no—
225C2×C4×C47—41no—
226C2×C4×C47C2×C22⋊C411yescharacteristic
227C2×C4×C47C2×C22⋊C414yes—
228C2×C4×C47C2×C22⋊C411yescharacteristic
229C2×C4×C47#13112yescomplemented
230C2×C4×C47C22×D412yes—
231C47⋊C8—42no—
232C47⋊C8—44no—
233C47⋊C8—42no—
234#17416—471nomaximal, Sylow
235C22×C47—22no—
236C22×C47—21no—
237C22×C47—21no—
238C22×C47—22no—
239C22×C47—22no—
240C22×C47—22no—
241C22×C47—22no—
242C22×C47—22no—
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—44no—
246C22×C47—44no—
247C22×C47—41no—
248C22×C47—41no—
249C22×C47—41no—
250C22×C47—81no—
251C22×C47—81no—
252C22×C47—81no—
253C22×C47—81no—
254C22×C47C2×#9914yes—
255C22×C47C22×C22⋊C411yescharacteristic
256C22×C47C2×#13111yescharacteristic
257C22×C47#56811yescharacteristic
258C4×C47—22no—
259C4×C47—22no—
260C4×C47—21no—
261C4×C47—24no—
262C4×C47—42no—
263C4×C47—41no—
264C4×C47—42no—
265C4×C47—41no—
266C4×C47C2×#9912yes—
267C2×#547D4711yescharacteristic
268#2880—472no—
269#1472—472no—
270#1468—474no—
271#1465—471no—
272#1463—472no—
273#1459—471no—
274C2×#324—472no—
275C2×C47—21no—
276C2×C47—21no—
277C2×C47—24no—
278C2×C47—42no—
279C2×C47—42no—
280C2×C47—81no—
281C2×C47#288011yescharacteristic
282C2×C47C22×#9911yescharacteristic
283C2×C47#288911yescharacteristic
284C22×#41Dic4711yescharacteristic
285C22×#41C2×D4711yescharacteristic
286C2×#131—942no—
287#547—22no—
288#547—24no—
289#547—22no—
290C2×C4⋊Q8Dic4711yescharacteristic
291C2×#123Dic4712yes—
292C2×#123C2×D4712yes—
293C2×C22⋊Q8Dic4711yescharacteristic
294C2×C22⋊Q8C2×D4711yescharacteristic
295C2×#121Dic4712yes—
296C2×C4×D4Dic4711yescharacteristic
297C2×C4×D4C2×D4711yescharacteristic
298C22×C4⋊C4C2×D4712yes—
299#408—942no—
300#408—944no—
301#407—474no—
302#407—472no—
303C2×C22⋊C8—474no—
304C2×C22⋊C8—472no—
305#324—471no—
306#324—472no—
307#324—474no—
308#324—471no—
309#324—474no—
310#323—472no—
311#323—472no—
312C47#1741611yescharacteristic, complemented, Sylow
313C2×#41—21no—
314C2×#41—22no—
315C2×#41—21no—
316C2×#41—44no—
317C2×#41—44no—
318C2×#41C22⋊D4712yes—
319C2×#41C22⋊D4714yes—
320C2×#41C22⋊D4712yes—
321C22×Q8C2×Dic4711yescharacteristic
322C22×Q8C2×Dic4711yescharacteristic
323C22×D4C2×Dic4711yescharacteristic
324C22×D4C2×Dic4711yescharacteristic
325C22×D4C2×Dic4712yes—
326C22×D4C2×Dic4712yes—
327C23×C4C2×Dic4712yes—
328C23×C4C2×Dic4712yes—
329C23×C4C2×Dic4711yescharacteristic
330C23×C4C22×D4712yes—
331C23×C4C22×D4711yescharacteristic
332#131—1888no—
333C2×M16—944no—
334C2×M16—942no—
335C22×C8—944no—
336C22×C8—942no—
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339#123—24no—
340#123—24no—
341#123—24no—
342#123—24no—
343C22⋊Q8—22no—
344C22⋊Q8—22no—
345C22⋊Q8—22no—
346C22⋊Q8—22no—
347#121—24no—
348#121—24no—
349C4×D4—22no—
350C4×D4—21no—
351C4×D4—21no—
352C4×D4—22no—
353C4×D4—22no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4C2×Dic4712yes—
357C2×C4⋊C4C2×Dic4712yes—
358C2×C4⋊C4C2×Dic4711yescharacteristic
359C2×C4⋊C4C2×Dic4712yes—
360C2×C4⋊C4C22×D4711yescharacteristic
361C2×C4⋊C4C22×D4712yes—
362C2×C22⋊C4C2×Dic4712yes—
363C2×C22⋊C4C2×Dic4711yescharacteristic
364C2×C22⋊C4C2×Dic4712yes—
365C2×C22⋊C4C2×Dic4712yes—
366C2×C22⋊C4C22×D4711yescharacteristic
367C2×C42C2×Dic4711yescharacteristic
368C22⋊C8—472no—
369C22⋊C8—472no—
370C22⋊C8—474no—
371C22⋊C8—944no—
372C22⋊C8—942no—
373C22⋊C8—944no—
374C22⋊C8—942no—
375C24C22×Dic4712yes—
376C24C22×Dic4711yescharacteristic
377C24C22×Dic4711yescharacteristic
378#41—28no—
379#41—48no—
380#41—44no—
381#41—48no—
382#41—44no—
383#41—44no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8C22⋊Dic4712yes—
389C2×Q8C22⋊Dic4712yes—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—41no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4C22⋊Dic4712yes—
399C2×D4C22⋊Dic4711yescharacteristic
400C2×D4C22⋊Dic4714yes—
401C2×D4C22⋊Dic4714yes—
402C2×D4C22⋊Dic4711yescharacteristic
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4C22×Dic4712yes—
416C22×C4C22×Dic4711yescharacteristic
417C22×C4C22×Dic4711yescharacteristic
418C22×C4C22×Dic4711yescharacteristic
419C22×C4C22×Dic4711yescharacteristic
420C22×C4C22×Dic4712yes—
421C22×C4C22×Dic4712yes—
422C22×C4C2×C22⋊D4714yes—
423C22×C4C2×C22⋊D4712yes—
424C22×C4C2×C22⋊D4714yes—
425C22×C4C2×C22⋊D4712yes—
426C22×C4C23×D4711yescharacteristic
427M16—1888no—
428M16—1884no—
429C2×C8—942no—
430C2×C8—942no—
431C2×C8—944no—
432C2×C8—1888no—
433C2×C8—1884no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—22no—
447C42—21no—
448C42—21no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23C2×C22⋊Dic4712yes—
463C23C2×C22⋊Dic4712yes—
464C23C2×C22⋊Dic4712yes—
465C23C23×Dic4711yescharacteristic
466Q8—24no—
467Q8—42no—
468Q8—42no—
469Q8—42no—
470D4—24no—
471D4—28no—
472D4—22no—
473D4—44no—
474D4—44no—
475D4—44no—
476D4—44no—
477D4#1309393914yes—
478C2×C4—21no—
479C2×C4—28no—
480C2×C4—21no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4C2×C22⋊Dic4711yescharacteristic
503C2×C4C2×C22⋊Dic4714yes—
504C2×C4C2×C22⋊Dic4711yescharacteristic
505C2×C4#1309403112yescomplemented
506C2×C4C22×C22⋊D4712yes—
507C8—1882no—
508C8—1884no—
509C8—1882no—
510C22—22no—
511C22—21no—
512C22—21no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—44no—
521C22—44no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22C2×#1309393914yes—
530C22C22×C22⋊Dic4711yescharacteristic
531C22C2×#1309403111yescharacteristic
532C22#1568989011yescharacteristic
533C4—22no—
534C4—22no—
535C4—21no—
536C4—24no—
537C4—42no—
538C4—41no—
539C4—42no—
540C4—41no—
541C4C2×#1309393912yes—
542C2—21no—
543C2—21no—
544C2—24no—
545C2—42no—
546C2—42no—
547C2—81no—
548C2#2025183711yescharacteristic
549C2C22×#1309393911yescharacteristic
550C2#2025187111yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#547C2
socleC22×C47C22×C22⋊C4
cosocleC22×C4×C47C24
hypercenterC2×#547D47
hypocenterC47#17416
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×#41 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #20251837C2semidirect
2. #20241752C2semidirect
3. #20241736C2semidirect
4. #20241726C2semidirect
5. #20241718C2semidirect
6. #20241706C2semidirect
7. C2×#15688849C2semidirect
229. C2×C4×C47#131semidirect
312. C47#17416semidirect
505. C2×C4#13094031semidirect

Maximal subgroups

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

subgroupindex
#202518372
#202417522
#202417362
#202417262
#202417182
#202417062
C2×#156888492
C2×C47×#5472
#1741647