Finite Group Databasebeta

← #26182428

Subgroups — #26182428

Order 12032.

952 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689715C211yesmaximal, characteristic
2#20252429C211yesmaximal, characteristic, complemented
3#20241758C212yesmaximal, complemented
4#20241751C212yesmaximal, complemented
5#20241745C212yesmaximal, complemented
6#20241734C212yesmaximal, complemented
7#20241724C212yesmaximal, complemented
8#20241717C212yesmaximal, complemented
9C47×#2973C211yesmaximal, characteristic, complemented
10#15689715—24no—
11#15689715—24no—
12C2×#13094018C2211yescharacteristic
13C2×#13094017C411yescharacteristic
14C2×#13094016C2211yescharacteristic
15C2×#13094014C411yescharacteristic
16C2×#13094008C411yescharacteristic
17C2×#13094007C2211yescharacteristic
18C2×#13094006C411yescharacteristic
19C2×#13094005C411yescharacteristic
20C2×#13094003C2211yescharacteristic
21C2×#13094002C411yescharacteristic
22C2×#13094001C411yescharacteristic
23C2×#13093996C411yescharacteristic
24C2×C4×C22⋊D47C2211yescharacteristic
25C22×Dic47⋊C4C2211yescharacteristic
26#15689201—24no—
27#15689176C2212yes—
28#15689087C2212yes—
29#15689087C2212yes—
30#15688848C2214yes—
31#15688848C2214yes—
32#15688848C2214yes—
33#15688846C2214yes—
34C2×C4×D4×C47C2211yescharacteristic
35C47×#435—24no—
36C47×#432C2212yes—
37C47×#407C2212yes—
38C2×C22⋊C8×C47C2212yes—
39C22×Dic94C2×C411yescharacteristic
40C22×C22⋊D47C2×C411yescharacteristic
41C22×C22⋊D47C2×C411yescharacteristic
42C23×Dic47C2×C411yescharacteristic
43C23×Dic47C2×C411yescharacteristic
44C22×C4×D47C2×C411yescharacteristic
45C2×#10625329—22no—
46C2×#10625329—22no—
47#13094018—22no—
48#13094018—22no—
49#13094017—24no—
50#13094016—22no—
51#13094016—22no—
52#13094014—24no—
53#13094008—24no—
54#13094007—22no—
55#13094007—22no—
56#13094006—24no—
57#13094005—24no—
58#13094003—22no—
59#13094003—22no—
60#13094002—24no—
61#13094001—24no—
62#13093996—24no—
63C4×C22⋊D47—22no—
64C4×C22⋊D47—22no—
65C2×C4⋊Dic47C2×C411yescharacteristic
66C2×C4⋊Dic47C2×C411yescharacteristic
67C2×C4⋊Dic47C2×C411yescharacteristic
68C2×Dic47⋊C4—22no—
69C2×Dic47⋊C4—22no—
70C2×Dic47⋊C4C2×C411yescharacteristic
71C2×Dic47⋊C4C2×C411yescharacteristic
72C2×Dic47⋊C4C2×C411yescharacteristic
73C2×Dic47⋊C4C2×C411yescharacteristic
74C2×Dic47⋊C4C2×C411yescharacteristic
75C2×Dic47⋊C4C2×C411yescharacteristic
76C2×Dic47⋊C4C2×C411yescharacteristic
77C2×Dic47⋊C4C2311yescharacteristic
78C2×Dic47⋊C4C2311yescharacteristic
79C2×Dic47⋊C4C2311yescharacteristic
80C2×C22⋊Dic47C2×C411yescharacteristic
81C2×C22⋊Dic47C2×C411yescharacteristic
82C2×C22⋊Dic47C2×C411yescharacteristic
83C2×C22⋊Dic47C2×C411yescharacteristic
84C2×(C2×D47)⋊C4C2×C411yescharacteristic
85C2×(C2×D47)⋊C4C2×C411yescharacteristic
86C2×(C2×D47)⋊C4C2×C411yescharacteristic
87C2×(C2×D47)⋊C4C2311yescharacteristic
88C2×C4×Dic47C2×C411yescharacteristic
89C4⋊(C47⋊C8)—22no—
90C22⋊(C47⋊C8)—22no—
91C22⋊(C47⋊C8)—22no—
92C22⋊(C47⋊C8)C2314yes—
93C4×C47⋊C8—22no—
94C22×D4×C47C2×C411yescharacteristic
95C23×C4×C47C2311yescharacteristic
96C23×C4×C47C2311yescharacteristic
97C2×M16×C47—22no—
98C22×C8×C47—22no—
99C4×D4×C47D414yes—
100C4×D4×C47D414yes—
101C2×C4⋊C4×C47C2×C411yescharacteristic
102C2×C22⋊C4×C47C2×C411yescharacteristic
103C2×C22⋊C4×C47C2×C411yescharacteristic
104C2×C42×C47C2311yescharacteristic
105C4⋊C8×C47—22no—
106C22⋊C8×C47—22no—
107C22⋊C8×C47—22no—
108C22⋊C8×C47C2314yes—
109#97×C47—22no—
110C24×C47C22×C411yescharacteristic
111C24×C47C22×C411yescharacteristic
112C23×D47C22×C411yescharacteristic
113C2×D4×C47—42no—
114C2×D4×C47C22⋊C414yes—
115C22×C4×C47—81no—
116C22×C4×C47—81no—
117C22×C4×C47C22×C411yescharacteristic
118C22×C4×C47C22×C411yescharacteristic
119C22×C4×C47C22×C411yescharacteristic
120C22×C4×C47C22×C411yescharacteristic
121C22×C4×C47C2×D412yes—
122C22×C4×C47C2×D412yes—
123C22×C4×C47C2×D412yes—
124C22×C4×C47C2×D412yes—
125C22×C4×C47C2411yescharacteristic
126C2×C22⋊D47—22no—
127C2×C22⋊D47—22no—
128C2×C22⋊D47—42no—
129C2×C22⋊D47—81no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—42no—
133C22×Dic47—81no—
134C22×Dic47C22×C411yescharacteristic
135C22×Dic47C22×C411yescharacteristic
136C22×Dic47C22×C411yescharacteristic
137C22×Dic47C22×C411yescharacteristic
138C22×Dic47C22×C411yescharacteristic
139C22×Dic47C22×C411yescharacteristic
140C22×Dic47C22×C411yescharacteristic
141C2×C4×D47—22no—
142C2×C4×D47—81no—
143C2×Dic94—22no—
144C2×Dic94—81no—
145M16×C47—82no—
146C2×C8×C47—24no—
147C2×C8×C47—82no—
148C4⋊C4×C47C22⋊C414yes—
149C22⋊C4×C47C22⋊C414yes—
150C22⋊C4×C47C22⋊C414yes—
151C42×C47C2×D412yes—
152C42×C47C2×D412yes—
153C22⋊Dic47—22no—
154C22⋊Dic47—22no—
155C22⋊Dic47—22no—
156C22⋊Dic47—22no—
157(C2×D47)⋊C4—22no—
158(C2×D47)⋊C4—22no—
159(C2×D47)⋊C4—22no—
160(C2×D47)⋊C4—21no—
161(C2×D47)⋊C4—21no—
162C4⋊Dic47—22no—
163C4⋊Dic47—22no—
164C4⋊Dic47—22no—
165Dic47⋊C4—22no—
166Dic47⋊C4—22no—
167Dic47⋊C4—22no—
168Dic47⋊C4—22no—
169Dic47⋊C4—22no—
170Dic47⋊C4—22no—
171Dic47⋊C4—22no—
172Dic47⋊C4—21no—
173Dic47⋊C4—21no—
174Dic47⋊C4—21no—
175Dic47⋊C4—21no—
176Dic47⋊C4—21no—
177Dic47⋊C4—21no—
178C4×Dic47—22no—
179#10625329—82no—
180#10625329—82no—
181C2×C47⋊C8—24no—
182C23×C47—22no—
183C23×C47—22no—
184C23×C47—41no—
185C23×C47—41no—
186C23×C47—41no—
187C23×C47—41no—
188C23×C47C2×C22⋊C412yes—
189C23×C47C2×C22⋊C412yes—
190C23×C47C23×C411yescharacteristic
191C22×D47—21no—
192C22×D47—22no—
193C22×D47—41no—
194C22×D47—41no—
195D4×C47—44no—
196C2×C4×C47—22no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47—42no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47—41no—
204C2×C4×C47—41no—
205C2×C4×C47C2×C22⋊C412yes—
206C2×C4×C47C2×C22⋊C412yes—
207C2×C4×C47C2×C22⋊C412yes—
208C2×C4×C47C2×C22⋊C412yes—
209C2×C4×C47C22×D411yescharacteristic
210C2×C4×C47C22×D411yescharacteristic
211C22⋊D47—44no—
212C22⋊D47—82no—
213C2×Dic47—21no—
214C2×Dic47—21no—
215C2×Dic47—21no—
216C2×Dic47—21no—
217C2×Dic47—21no—
218C2×Dic47—21no—
219C2×Dic47—21no—
220C2×Dic47—22no—
221C2×Dic47—22no—
222C2×Dic47—22no—
223C2×Dic47—44no—
224C2×Dic47—41no—
225C2×Dic47—41no—
226C2×Dic47—41no—
227C2×Dic47—41no—
228C2×Dic47—82no—
229C4×D47—82no—
230Dic94—82no—
231C8×C47—44no—
232C47⋊C8—44no—
233#17406—471nomaximal, Sylow
234C22×C47—21no—
235C22×C47—21no—
236C22×C47—22no—
237C22×C47—22no—
238C22×C47—22no—
239C22×C47—22no—
240C22×C47—22no—
241C22×C47—22no—
242C22×C47—44no—
243C22×C47—82no—
244C22×C47C22×C22⋊C411yescharacteristic
245C22×C47#56812yes—
246C2×D47—41no—
247C2×D47—41no—
248C2×D47—41no—
249C2×D47—82no—
250C4×C47—22no—
251C4×C47—21no—
252C4×C47—21no—
253C4×C47—22no—
254C4×C47—22no—
255C4×C47—22no—
256Dic47—41no—
257Dic47—41no—
258Dic47—41no—
259Dic47—41no—
260Dic47—41no—
261Dic47—41no—
262Dic47—41no—
263C2×#539—471no—
264#2975—471no—
265#2973D4711yescharacteristic, complemented
266#1473—472no—
267#1472—472no—
268#1470—472no—
269#1468—472no—
270#1465—472no—
271#1463—472no—
272C2×C47—22no—
273C2×C47—42no—
274C2×C47—42no—
275C2×C47#287911yescharacteristic
276C2×C47#288311yescharacteristic
277C2×C47#288911yescharacteristic
278D47—81no—
279#539—944no—
280#539—944no—
281C2×#126—471no—
282C2×#126—471no—
283C2×#125—471no—
284C2×#125—471no—
285C2×#123—471no—
286C2×#123—471no—
287C2×#123—471no—
288C2×#123—471no—
289C2×C22⋊Q8—471no—
290C2×C22⋊Q8—471no—
291C2×C22⋊Q8—471no—
292C2×#121—471no—
293C2×C4×D4—471no—
294C2×C4×D4C2×D4711yescharacteristic
295C22×C4⋊C4—471no—
296#436—944no—
297#435—24no—
298#432C2×D4712yes—
299#431—472no—
300#407—472no—
301#407—472no—
302#407C2×D4712yes—
303C2×C22⋊C8C2×D4712yes—
304#324—474no—
305#324—474no—
306#324—474no—
307#323—474no—
308C47#1740611yescharacteristic, complemented, Sylow
309C22×Q8—471no—
310C22×D4—471no—
311C22×D4—471no—
312C22×D4C4×D4711yescharacteristic
313C23×C4—471no—
314C23×C4—471no—
315C23×C4—471no—
316C23×C4C22×D4711yescharacteristic
317C23×C4C22×D4711yescharacteristic
318C2×M16—22no—
319C2×M16—942no—
320C2×M16—942no—
321C22×C8—22no—
322#126—944no—
323#126—942no—
324#126—942no—
325#125—944no—
326#125—942no—
327#125—942no—
328#123—944no—
329#123—942no—
330#123—942no—
331#123—944no—
332#123—944no—
333C22⋊Q8—942no—
334C22⋊Q8—942no—
335C22⋊Q8—944no—
336C22⋊Q8—944no—
337#121—944no—
338C4×D4—942no—
339C4×D4—942no—
340C4×D4D18814yes—
341C4×D4C22⋊D4714yes—
342C2×C4⋊C4—471no—
343C2×C4⋊C4—471no—
344C2×C4⋊C4—471no—
345C2×C4⋊C4—471no—
346C2×C4⋊C4—471no—
347C2×C4⋊C4—471no—
348C2×C4⋊C4—471no—
349C2×C4⋊C4—471no—
350C2×C4⋊C4—471no—
351C2×C4⋊C4—471no—
352C2×C4⋊C4—471no—
353C2×C4⋊C4—471no—
354C2×C4⋊C4—471no—
355C2×C4⋊C4—942no—
356C2×C4⋊C4—942no—
357C2×C4⋊C4C4×D4711yescharacteristic
358C2×C22⋊C4—471no—
359C2×C22⋊C4—471no—
360C2×C22⋊C4—471no—
361C2×C22⋊C4—471no—
362C2×C22⋊C4—471no—
363C2×C22⋊C4—471no—
364C2×C22⋊C4—471no—
365C2×C22⋊C4—471no—
366C2×C22⋊C4C4×D4711yescharacteristic
367C2×C22⋊C4C4×D4711yescharacteristic
368C2×C42—471no—
369C2×C42C22×D4711yescharacteristic
370C4⋊C8—22no—
371C4⋊C8—942no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374C22⋊C8—474no—
375C22⋊C8—942no—
376C22⋊C8—942no—
377C22⋊C8C22×D4714yes—
378#97—22no—
379C4×C8—942no—
380C24—471no—
381C24C2×C4×D4711yescharacteristic
382C24C2×C4×D4711yescharacteristic
383C2×Q8—942no—
384C2×Q8—3761no—
385C2×D4—42no—
386C2×D4—942no—
387C2×D4—942no—
388C2×D4—3761no—
389C2×D4—1882no—
390C2×D4(C2×D47)⋊C414yes—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—471no—
394C22×C4—471no—
395C22×C4—471no—
396C22×C4—471no—
397C22×C4—471no—
398C22×C4—471no—
399C22×C4—471no—
400C22×C4—942no—
401C22×C4—942no—
402C22×C4—942no—
403C22×C4—3761no—
404C22×C4—3761no—
405C22×C4—1882no—
406C22×C4C2×C4×D4711yescharacteristic
407C22×C4C2×C4×D4711yescharacteristic
408C22×C4C2×C4×D4711yescharacteristic
409C22×C4C2×C4×D4711yescharacteristic
410C22×C4C2×D18812yes—
411C22×C4C2×D18812yes—
412C22×C4C2×C22⋊D4712yes—
413C22×C4C2×C22⋊D4712yes—
414C22×C4C23×D4711yescharacteristic
415M16—82no—
416M16—3762no—
417M16—3762no—
418C2×C8—24no—
419C2×C8—82no—
420C2×C8—944no—
421C4⋊C4—942no—
422C4⋊C4—942no—
423C4⋊C4—942no—
424C4⋊C4—942no—
425C4⋊C4—942no—
426C4⋊C4—942no—
427C4⋊C4—942no—
428C4⋊C4—942no—
429C4⋊C4—942no—
430C4⋊C4—942no—
431C4⋊C4—941no—
432C4⋊C4—941no—
433C4⋊C4—941no—
434C4⋊C4—941no—
435C4⋊C4—941no—
436C4⋊C4—941no—
437C4⋊C4(C2×D47)⋊C414yes—
438C22⋊C4—942no—
439C22⋊C4—942no—
440C22⋊C4—942no—
441C22⋊C4—942no—
442C22⋊C4—942no—
443C22⋊C4—942no—
444C22⋊C4—942no—
445C22⋊C4—941no—
446C22⋊C4—941no—
447C22⋊C4(C2×D47)⋊C414yes—
448C22⋊C4(C2×D47)⋊C414yes—
449C42—942no—
450C42C2×D18812yes—
451C42C2×C22⋊D4712yes—
452C23—22no—
453C23—22no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—942no—
459C23—941no—
460C23—1881no—
461C23—1881no—
462C23C2×(C2×D47)⋊C412yes—
463C23C2×(C2×D47)⋊C412yes—
464C23C22×C4×D4711yescharacteristic
465Q8—3762no—
466D4—44no—
467D4—3762no—
468D4—1884no—
469C2×C4—22no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—941no—
479C2×C4—941no—
480C2×C4—941no—
481C2×C4—941no—
482C2×C4—941no—
483C2×C4—941no—
484C2×C4—941no—
485C2×C4—942no—
486C2×C4—942no—
487C2×C4—942no—
488C2×C4—3762no—
489C2×C4—3762no—
490C2×C4—1884no—
491C2×C4—1881no—
492C2×C4—1881no—
493C2×C4—1881no—
494C2×C4—1881no—
495C2×C4C2×(C2×D47)⋊C412yes—
496C2×C4C2×(C2×D47)⋊C412yes—
497C2×C4C2×(C2×D47)⋊C412yes—
498C2×C4C2×(C2×D47)⋊C412yes—
499C2×C4C22×C22⋊D4711yescharacteristic
500C2×C4C22×D18811yescharacteristic
501C8—44no—
502C8—1884no—
503C22—21no—
504C22—21no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—44no—
512C22—82no—
513C22—3762no—
514C22—1881no—
515C22—1881no—
516C22—1881no—
517C22C22×(C2×D47)⋊C411yescharacteristic
518C22#1568988912yes—
519C4—22no—
520C4—21no—
521C4—21no—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—1881no—
526C4—1881no—
527C4—1881no—
528C4—1881no—
529C4—1881no—
530C4—1881no—
531C4—1881no—
532C2—22no—
533C2—42no—
534C2—42no—
535C2—3761no—
536C2#2025183211yescharacteristic
537C2#2025184811yescharacteristic
538C2#2025187011yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2973C2
socleC22×C47C22×C22⋊C4
prime radicalC2×#13093996C4
cosocleC22×C4×C47C24
hypercenter#2973D47
hypocenterC47#17406
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #20252429C2semidirect
3. #20241758C2semidirect
4. #20241751C2semidirect
5. #20241745C2semidirect
6. #20241734C2semidirect
7. #20241724C2semidirect
8. #20241717C2semidirect
9. C47×#2973C2semidirect
265. #2973D47semidirect
308. C47#17406semidirect

Maximal subgroups

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

subgroupindex
C2×#156897152
#202524292
#202417582
#202417512
#202417452
#202417342
#202417242
#202417172
C47×#29732
#1740647