Finite Group Databasebeta

← #26224525

Subgroups — #26224525

Order 12032.

712 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20244003C211yesmaximal, characteristic
2#20244003C211yesmaximal, characteristic
3#20243996C211yesmaximal, characteristic
4#20243954C211yesmaximal, characteristic, complemented
5#20243954C211yesmaximal, characteristic, complemented
6#20243947C211yesmaximal, characteristic, complemented
7#20243531C211yesmaximal, characteristic, complemented
8#20243263C211yesmaximal, characteristic, complemented
9#20243257C211yesmaximal, characteristic, complemented
10C4⋊C4×C47⋊C8C211yesmaximal, characteristic, complemented
11#20243172C211yesmaximal, characteristic
12#20243172C211yesmaximal, characteristic
13#20243172C211yesmaximal, characteristic
14C4×#13093935C211yesmaximal, characteristic, complemented
15C4×#126×C47C211yesmaximal, characteristic
16C22×#10625329C2211yescharacteristic
17C2×C4⋊(C47⋊C8)C2211yescharacteristic
18C2×C4⋊(C47⋊C8)C2211yescharacteristic
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×C22⋊(C47⋊C8)C2211yescharacteristic
21C2×C22⋊(C47⋊C8)C2211yescharacteristic
22C2×C22⋊(C47⋊C8)C2211yescharacteristic
23C2×#13093935C2211yescharacteristic
24C2×#13093935C2211yescharacteristic
25C2×#13093935C2211yescharacteristic
26C2×#13093935C2211yescharacteristic
27C2×#13093935C2211yescharacteristic
28C2×C4×C47⋊C8C2211yescharacteristic
29C2×C4×C47⋊C8C2211yescharacteristic
30C2×C4×C47⋊C8C2211yescharacteristic
31C2×C4×C47⋊C8C2211yescharacteristic
32#15688878C2211yescharacteristic
33#15688878C2211yescharacteristic
34#15688878C2211yescharacteristic
35#15688878C2211yescharacteristic
36#15688878C2211yescharacteristic
37#15688878C2211yescharacteristic
38#15688878C2211yescharacteristic
39#15688878C2211yescharacteristic
40#15688878C2211yescharacteristic
41#15688878C2211yescharacteristic
42#15688878C2211yescharacteristic
43#15688878C2211yescharacteristic
44C2×#126×C47C411yescharacteristic
45C47×#388C411yescharacteristic
46C47×#388C411yescharacteristic
47C47×#388C411yescharacteristic
48C47×#387C411yescharacteristic
49C47×#387C411yescharacteristic
50C47×#387C411yescharacteristic
51C47×#383C411yescharacteristic
52C4×C4⋊C4×C47C2211yescharacteristic
53C4×C4⋊C4×C47C2211yescharacteristic
54C4×C4⋊C4×C47C2211yescharacteristic
55C4×C22⋊C4×C47C2211yescharacteristic
56C4×C22⋊C4×C47C2211yescharacteristic
57C4×C22⋊C4×C47C2211yescharacteristic
58C43×C47C2211yescharacteristic
59C2×#10625329—22no—
60C2×#10625329—22no—
61C2×#10625329—22no—
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70C4⋊(C47⋊C8)—22no—
71C4⋊(C47⋊C8)—22no—
72C4⋊(C47⋊C8)—22no—
73C22⋊(C47⋊C8)—22no—
74C22⋊(C47⋊C8)—22no—
75C22⋊(C47⋊C8)—22no—
76#13093935—22no—
77#13093935—22no—
78#13093935—22no—
79#13093935—22no—
80#13093935—22no—
81C4×C47⋊C8—22no—
82C4×C47⋊C8—22no—
83C4×C47⋊C8—22no—
84C4×C47⋊C8—22no—
85C23×C4×C47C2311yescharacteristic
86#126×C47—24no—
87C2×C4⋊C4×C47C2×C411yescharacteristic
88C2×C4⋊C4×C47C2×C411yescharacteristic
89C2×C4⋊C4×C47C2×C411yescharacteristic
90C2×C4⋊C4×C47C2×C411yescharacteristic
91C2×C4⋊C4×C47C2×C411yescharacteristic
92C2×C4⋊C4×C47C2×C411yescharacteristic
93C2×C22⋊C4×C47C2×C411yescharacteristic
94C2×C22⋊C4×C47C2×C411yescharacteristic
95C2×C22⋊C4×C47C2×C411yescharacteristic
96C2×C22⋊C4×C47C2×C411yescharacteristic
97C2×C22⋊C4×C47C2×C411yescharacteristic
98C2×C22⋊C4×C47C2×C411yescharacteristic
99C2×C42×C47C2×C411yescharacteristic
100C2×C42×C47C2×C411yescharacteristic
101C2×C42×C47C2×C411yescharacteristic
102C2×C42×C47C2×C411yescharacteristic
103C2×C42×C47C2311yescharacteristic
104C2×C42×C47C2311yescharacteristic
105C2×C42×C47C2311yescharacteristic
106C2×C42×C47C2311yescharacteristic
107C2×C42×C47C2311yescharacteristic
108C2×C42×C47C2311yescharacteristic
109#95×C47C2×C411yescharacteristic
110#95×C47C2×C411yescharacteristic
111#95×C47C2×C411yescharacteristic
112#95×C47C2×C411yescharacteristic
113#95×C47C2×C411yescharacteristic
114#95×C47C2×C411yescharacteristic
115#95×C47C2×C411yescharacteristic
116#95×C47C2×C411yescharacteristic
117#95×C47C2×C411yescharacteristic
118#95×C47C2×C411yescharacteristic
119#95×C47C2×C411yescharacteristic
120#95×C47C2×C411yescharacteristic
121C24×C47C22×C411yescharacteristic
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47C22×C411yescharacteristic
129C22×C4×C47C22×C411yescharacteristic
130C22×C4×C47C22×C411yescharacteristic
131C22×C4×C47C22×C411yescharacteristic
132C22×C4×C47C22×C411yescharacteristic
133C22×C4×C47C22×C411yescharacteristic
134C22×C4×C47C22×C411yescharacteristic
135C22×C4×C47C22×C411yescharacteristic
136C22×C4×C47C22×C411yescharacteristic
137C22×C4×C47C22×C411yescharacteristic
138C22×C4×C47C22×C411yescharacteristic
139C22×C4×C47C22×C411yescharacteristic
140C22×C4×C47C22×C411yescharacteristic
141C22×C4×C47C2411yescharacteristic
142C4⋊C4×C47—22no—
143C4⋊C4×C47—22no—
144C4⋊C4×C47—22no—
145C4⋊C4×C47—41no—
146C4⋊C4×C47—41no—
147C4⋊C4×C47M1614yescomplemented
148C22⋊C4×C47—22no—
149C22⋊C4×C47—22no—
150C22⋊C4×C47—22no—
151C22⋊C4×C47—22no—
152C22⋊C4×C47—41no—
153C22⋊C4×C47—41no—
154C42×C47—22no—
155C42×C47—22no—
156C42×C47—21no—
157C42×C47—21no—
158C42×C47—21no—
159C42×C47—21no—
160C42×C47—21no—
161C42×C47—21no—
162C42×C47—21no—
163C42×C47—21no—
164C42×C47—21no—
165C42×C47—21no—
166C42×C47—21no—
167C42×C47—21no—
168C42×C47—41no—
169C42×C47—41no—
170#10625329—44no—
171C2×C47⋊C8—21no—
172C2×C47⋊C8—21no—
173C2×C47⋊C8—21no—
174C2×C47⋊C8—21no—
175C2×C47⋊C8—21no—
176C2×C47⋊C8—21no—
177C2×C47⋊C8—21no—
178C2×C47⋊C8—21no—
179C2×C47⋊C8—21no—
180C2×C47⋊C8—21no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8#4112yes—
190C2×C47⋊C8#4112yes—
191C2×C47⋊C8#4112yes—
192C2×C47⋊C8#4112yes—
193C2×C47⋊C8#4112yes—
194C2×C47⋊C8#4112yes—
195C23×C47—21no—
196C23×C47—21no—
197C23×C47—21no—
198C23×C47—21no—
199C23×C47—21no—
200C23×C47—21no—
201C23×C47—21no—
202C23×C47C23×C411yescharacteristic
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47C2×M1612yes—
243C2×C4×C47C2×M1612yes—
244C2×C4×C47C2×M1612yes—
245C2×C4×C47#13112yes—
246C2×C4×C47C2×#4111yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×C4×C47C2×#4111yescharacteristic
250C2×C4×C47C2×#4111yescharacteristic
251C2×C4×C47C2×#4111yescharacteristic
252C47⋊C8—22no—
253C47⋊C8—22no—
254C47⋊C8—22no—
255C47⋊C8—41no—
256C47⋊C8—41no—
257C47⋊C8—41no—
258C47⋊C8—41no—
259C47⋊C8#12614yescomplemented
260#21667—471nomaximal, Sylow
261C22×C47—41no—
262C22×C47—41no—
263C22×C47—41no—
264C22×C47—41no—
265C22×C47—41no—
266C22×C47—41no—
267C22×C47—41no—
268C22×C47C4×#4111yescharacteristic
269C22×C47C4×#4111yescharacteristic
270C22×C47C4×#4111yescharacteristic
271C22×C47C22×M1611yescharacteristic
272C22×C47C2×#13111yescharacteristic
273C22×C47#56811yescharacteristic
274C22×C47#56811yescharacteristic
275C4×C47—22no—
276C4×C47—22no—
277C4×C47—22no—
278C4×C47—22no—
279C4×C47—41no—
280C4×C47—41no—
281C4×C47—41no—
282C4×C47—41no—
283C4×C47—41no—
284C4×C47—41no—
285C4×C47—41no—
286C4×C47—41no—
287C4×C47—41no—
288C4×C47C2×#12611yescharacteristic
289C4×C47#52911yescharacteristic
290C4×C47#52911yescharacteristic
291C4×C47#52911yescharacteristic
292C4×#126D4711yescharacteristic
293#1924—471no—
294#1924—471no—
295#1923—471no—
296#1917—471no—
297#1917—471no—
298#1916—471no—
299#1839—471no—
300#1771—471no—
301#1770—471no—
302C8×C4⋊C4—471no—
303#1753—471no—
304#1753—471no—
305#1753—471no—
306C4×#97—471no—
307C2×C47—81no—
308C2×C47C4×#12611yescharacteristic
309C2×C47#296511yescharacteristic
310C2×C47#296711yescharacteristic
311C2×C47#296811yescharacteristic
312C2×C47#296911yescharacteristic
313C2×C47#297011yescharacteristic
314C2×C47#297111yescharacteristic
315C22×M16—471no—
316C2×#126Dic4711yescharacteristic
317C2×C4⋊C8—471no—
318C2×C4⋊C8—471no—
319C2×C4⋊C8—471no—
320C2×C22⋊C8—471no—
321C2×C22⋊C8—471no—
322C2×C22⋊C8—471no—
323C2×#97—471no—
324C2×#97—471no—
325C2×#97—471no—
326C2×#97—471no—
327C2×#97—471no—
328C2×C4×C8—471no—
329C2×C4×C8—471no—
330C2×C4×C8—471no—
331C2×C4×C8—471no—
332#388Dic4711yescharacteristic
333#388Dic4711yescharacteristic
334#388Dic4711yescharacteristic
335#387Dic4711yescharacteristic
336#387Dic4711yescharacteristic
337#387Dic4711yescharacteristic
338#383Dic4711yescharacteristic
339C4×C4⋊C4C2×D4711yescharacteristic
340C4×C4⋊C4C2×D4711yescharacteristic
341C4×C4⋊C4C2×D4711yescharacteristic
342C4×C22⋊C4C2×D4711yescharacteristic
343C4×C22⋊C4C2×D4711yescharacteristic
344C4×C22⋊C4C2×D4711yescharacteristic
345C43C2×D4711yescharacteristic
346#336—471no—
347#336—471no—
348#336—471no—
349#336—471no—
350#336—471no—
351#336—471no—
352#336—471no—
353#336—471no—
354#336—471no—
355#336—471no—
356#336—471no—
357#336—471no—
358C47#2166711yescharacteristic, complemented, Sylow
359C23×C4C22×D4711yescharacteristic
360C2×M16—942no—
361C2×M16—942no—
362C2×M16—942no—
363C22×C8—471no—
364C22×C8—471no—
365C22×C8—471no—
366C22×C8—471no—
367C22×C8—471no—
368C22×C8—471no—
369C22×C8—471no—
370C22×C8—471no—
371#126—24no—
372C2×C4⋊C4C2×Dic4711yescharacteristic
373C2×C4⋊C4C2×Dic4711yescharacteristic
374C2×C4⋊C4C2×Dic4711yescharacteristic
375C2×C4⋊C4C2×Dic4711yescharacteristic
376C2×C4⋊C4C2×Dic4711yescharacteristic
377C2×C4⋊C4C2×Dic4711yescharacteristic
378C2×C22⋊C4C2×Dic4711yescharacteristic
379C2×C22⋊C4C2×Dic4711yescharacteristic
380C2×C22⋊C4C2×Dic4711yescharacteristic
381C2×C22⋊C4C2×Dic4711yescharacteristic
382C2×C22⋊C4C2×Dic4711yescharacteristic
383C2×C22⋊C4C2×Dic4711yescharacteristic
384C2×C42C2×Dic4711yescharacteristic
385C2×C42C2×Dic4711yescharacteristic
386C2×C42C2×Dic4711yescharacteristic
387C2×C42C2×Dic4711yescharacteristic
388C2×C42C22×D4711yescharacteristic
389C2×C42C22×D4711yescharacteristic
390C2×C42C22×D4711yescharacteristic
391C2×C42C22×D4711yescharacteristic
392C2×C42C22×D4711yescharacteristic
393C2×C42C22×D4711yescharacteristic
394C4⋊C8—942no—
395C4⋊C8—942no—
396C4⋊C8—942no—
397C22⋊C8—942no—
398C22⋊C8—942no—
399C22⋊C8—942no—
400#97—942no—
401#97—942no—
402#97—942no—
403#97—942no—
404#97—942no—
405C4×C8—942no—
406C4×C8—942no—
407C4×C8—942no—
408C4×C8—942no—
409#95C2×Dic4711yescharacteristic
410#95C2×Dic4711yescharacteristic
411#95C2×Dic4711yescharacteristic
412#95C2×Dic4711yescharacteristic
413#95C2×Dic4711yescharacteristic
414#95C2×Dic4711yescharacteristic
415#95C2×Dic4711yescharacteristic
416#95C2×Dic4711yescharacteristic
417#95C2×Dic4711yescharacteristic
418#95C2×Dic4711yescharacteristic
419#95C2×Dic4711yescharacteristic
420#95C2×Dic4711yescharacteristic
421C24C22×Dic4711yescharacteristic
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4C22×Dic4711yescharacteristic
429C22×C4C22×Dic4711yescharacteristic
430C22×C4C22×Dic4711yescharacteristic
431C22×C4C22×Dic4711yescharacteristic
432C22×C4C22×Dic4711yescharacteristic
433C22×C4C22×Dic4711yescharacteristic
434C22×C4C22×Dic4711yescharacteristic
435C22×C4C22×Dic4711yescharacteristic
436C22×C4C22×Dic4711yescharacteristic
437C22×C4C22×Dic4711yescharacteristic
438C22×C4C22×Dic4711yescharacteristic
439C22×C4C22×Dic4711yescharacteristic
440C22×C4C22×Dic4711yescharacteristic
441C22×C4C23×D4711yescharacteristic
442M16—1884no—
443C2×C8—472no—
444C2×C8—472no—
445C2×C8—472no—
446C2×C8—472no—
447C2×C8—472no—
448C2×C8—472no—
449C2×C8—941no—
450C2×C8—941no—
451C2×C8—941no—
452C2×C8—941no—
453C2×C8—941no—
454C2×C8—941no—
455C2×C8—941no—
456C2×C8—941no—
457C2×C8—941no—
458C2×C8—941no—
459C2×C8—941no—
460C2×C8—941no—
461C2×C8—941no—
462C2×C8—941no—
463C2×C8—941no—
464C2×C8—941no—
465C2×C8—941no—
466C2×C8—941no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4#1062532914yescomplemented
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C42—22no—
480C42—22no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—41no—
494C42—41no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23C23×Dic4711yescharacteristic
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4C2×#1062532912yes—
543C2×C4C2×#1062532912yes—
544C2×C4C2×#1062532912yes—
545C2×C4#1309403112yes—
546C2×C4#41×D4711yescharacteristic
547C2×C4#41×D4711yescharacteristic
548C2×C4#41×D4711yescharacteristic
549C2×C4#41×D4711yescharacteristic
550C2×C4#41×D4711yescharacteristic
551C2×C4#41×D4711yescharacteristic
552C8—474no—
553C8—942no—
554C8—942no—
555C8—942no—
556C8—1881no—
557C8—1881no—
558C8—1881no—
559C8—1881no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#41×Dic4711yescharacteristic
568C22#41×Dic4711yescharacteristic
569C22#41×Dic4711yescharacteristic
570C22C22×#1062532911yescharacteristic
571C22C2×#1309403111yescharacteristic
572C22#1568989011yescharacteristic
573C22#1568989011yescharacteristic
574C4—22no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4#126×D4711yescharacteristic
588C4#1568965611yescharacteristic
589C4#1568965611yescharacteristic
590C4#1568965611yescharacteristic
591C2—81no—
592C2#126×Dic4711yescharacteristic
593C2#2025235511yescharacteristic
594C2#2025237511yescharacteristic
595C2#2025238411yescharacteristic
596C2#2025239311yescharacteristic
597C2#2025240211yescharacteristic
598C2#2025240811yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC4×#126×C47C2
prime radicalC24×C47C22×C4
cosocleC22×C4×C47C24
hypercenterC4×#126D47
hypocenterC47#21667
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C4×#126

Complements

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

NH ≅ G/Ntype
4. #20243954C2semidirect
5. #20243954C2semidirect
6. #20243947C2semidirect
7. #20243531C2semidirect
8. #20243263C2semidirect
9. #20243257C2semidirect
10. C4⋊C4×C47⋊C8C2semidirect
14. C4×#13093935C2semidirect
147. C4⋊C4×C47M16semidirect
259. C47⋊C8#126semidirect
358. C47#21667semidirect
472. C4⋊C4#10625329semidirect

Maximal subgroups

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

subgroupindex
#202440032
#202439962
#202439542
#202439472
#202435312
#202432632
#202432572
C4⋊C4×C47⋊C82
#202431722
C4×#130939352
C4×#126×C472
#2166747