Finite Group Databasebeta

← #26230275

Subgroups — #26230275

Order 12032.

702 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20244329C211yesmaximal, characteristic, complemented
2#20244325C211yesmaximal, characteristic, complemented
3#20244112C211yesmaximal, characteristic, complemented
4#20244107C211yesmaximal, characteristic, complemented
5#20243954C211yesmaximal, characteristic, complemented
6#20243954C211yesmaximal, characteristic, complemented
7#20243947C211yesmaximal, characteristic, complemented
8#20243947C211yesmaximal, characteristic, complemented
9#20243573C211yesmaximal, characteristic
10#20243573C211yesmaximal, characteristic
11#20243372C211yesmaximal, characteristic
12#20243166C211yesmaximal, characteristic
13C22⋊C4×C47⋊C8C211yesmaximal, characteristic
14#20243085C211yesmaximal, characteristic
15C47×#2396C211yesmaximal, characteristic
16C22×#10625329C2211yescharacteristic
17C23×C47⋊C8C2211yescharacteristic
18C2×C4⋊(C47⋊C8)C2211yescharacteristic
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×C4⋊(C47⋊C8)C2211yescharacteristic
21C2×C4⋊(C47⋊C8)C2211yescharacteristic
22C2×C4⋊(C47⋊C8)C2211yescharacteristic
23C2×C4⋊(C47⋊C8)C2211yescharacteristic
24C2×C22⋊(C47⋊C8)C2211yescharacteristic
25C2×C22⋊(C47⋊C8)C2211yescharacteristic
26C2×#13093935C2211yescharacteristic
27C2×#13093935C2211yescharacteristic
28C2×C4×C47⋊C8C2211yescharacteristic
29C2×C4×C47⋊C8C2211yescharacteristic
30#15688878C2211yescharacteristic
31#15688878C2211yescharacteristic
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×C4×Q8×C47C2211yescharacteristic
45C2×#117×C47C2211yescharacteristic
46C47×#400C411yescharacteristic
47C47×#399C411yescharacteristic
48C47×#398C411yescharacteristic
49C47×#398C411yescharacteristic
50C47×#396C411yescharacteristic
51C47×#396C411yescharacteristic
52C47×#395C411yescharacteristic
53C47×#393C411yescharacteristic
54C4×C4⋊C4×C47C2211yescharacteristic
55C4×C4⋊C4×C47C2211yescharacteristic
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C22⋊C4×C47C2211yescharacteristic
58C4×C22⋊C4×C47C2211yescharacteristic
59C2×#10625329—22no—
60C2×#10625329—41no—
61C2×#10625329—41no—
62C22×C47⋊C8—41no—
63C22×C47⋊C8—41no—
64C22×C47⋊C8Q814yescomplemented
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70C22×C47⋊C8C2311yescharacteristic
71C22×C47⋊C8C2311yescharacteristic
72C22×C47⋊C8C2311yescharacteristic
73C4⋊(C47⋊C8)—41no—
74C4⋊(C47⋊C8)—41no—
75C4⋊(C47⋊C8)—41no—
76C4⋊(C47⋊C8)—41no—
77C4⋊(C47⋊C8)—41no—
78C4⋊(C47⋊C8)—41no—
79C22⋊(C47⋊C8)—22no—
80C22⋊(C47⋊C8)—22no—
81#13093935—41no—
82#13093935—41no—
83C4×C47⋊C8—41no—
84C4×C47⋊C8—41no—
85C22×Q8×C47C2×C411yescharacteristic
86C23×C4×C47C2311yescharacteristic
87C4×Q8×C47—41no—
88C4×Q8×C47—41no—
89#117×C47—41no—
90#117×C47—41no—
91C2×C4⋊C4×C47C2×C411yescharacteristic
92C2×C4⋊C4×C47C2×C411yescharacteristic
93C2×C4⋊C4×C47C2×C411yescharacteristic
94C2×C4⋊C4×C47C2×C411yescharacteristic
95C2×C4⋊C4×C47C2×C411yescharacteristic
96C2×C4⋊C4×C47C2×C411yescharacteristic
97C2×C4⋊C4×C47C2×C411yescharacteristic
98C2×C4⋊C4×C47C2×C411yescharacteristic
99C2×C4⋊C4×C47C2×C411yescharacteristic
100C2×C4⋊C4×C47C2×C411yescharacteristic
101C2×C4⋊C4×C47C2×C411yescharacteristic
102C2×C22⋊C4×C47C2×C411yescharacteristic
103C2×C22⋊C4×C47C2×C411yescharacteristic
104C2×C22⋊C4×C47C2×C411yescharacteristic
105C2×C22⋊C4×C47C2×C411yescharacteristic
106C2×C22⋊C4×C47C2×C411yescharacteristic
107C2×C22⋊C4×C47C2×C411yescharacteristic
108C2×C42×C47C2311yescharacteristic
109C2×C42×C47C2311yescharacteristic
110C2×C42×C47C2311yescharacteristic
111C2×C42×C47C2311yescharacteristic
112C2×C42×C47C2311yescharacteristic
113C2×C42×C47C2311yescharacteristic
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
121#95×C47C2×C411yescharacteristic
122#95×C47C2×C411yescharacteristic
123#95×C47C2×C411yescharacteristic
124C24×C47C22×C411yescharacteristic
125C2×Q8×C47—22no—
126C2×Q8×C47—41no—
127C2×Q8×C47—41no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
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×C47C22×C411yescharacteristic
142C22×C4×C47C22×C411yescharacteristic
143C22×C4×C47C22×C411yescharacteristic
144C22×C4×C47C22×C411yescharacteristic
145C22×C4×C47C2×Q812yes—
146C22×C4×C47#4112yes—
147C22×C4×C47C2411yescharacteristic
148C4⋊C4×C47—22no—
149C4⋊C4×C47—22no—
150C4⋊C4×C47—22no—
151C4⋊C4×C47—41no—
152C4⋊C4×C47—41no—
153C4⋊C4×C47—41no—
154C4⋊C4×C47—41no—
155C4⋊C4×C47—41no—
156C4⋊C4×C47—41no—
157C4⋊C4×C47—41no—
158C4⋊C4×C47—41no—
159C22⋊C4×C47—22no—
160C22⋊C4×C47—22no—
161C22⋊C4×C47—41no—
162C22⋊C4×C47—41no—
163C22⋊C4×C47—41no—
164C22⋊C4×C47—41no—
165C42×C47—21no—
166C42×C47—21no—
167C42×C47—21no—
168C42×C47—21no—
169C42×C47—21no—
170C42×C47—21no—
171C42×C47—21no—
172C42×C47—21no—
173C42×C47—21no—
174C42×C47—21no—
175C42×C47—21no—
176C42×C47—21no—
177#10625329—44no—
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—21no—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8—21no—
193C2×C47⋊C8—21no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—44no—
201C2×C47⋊C8C2×Q812yes—
202C2×C47⋊C8C2×Q812yes—
203C23×C47—21no—
204C23×C47—21no—
205C23×C47—21no—
206C23×C47—21no—
207C23×C47—21no—
208C23×C47—21no—
209C23×C47C4×Q812yes—
210C23×C47C23×C411yescharacteristic
211Q8×C47—44no—
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—22no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
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—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—21no—
246C2×C4×C47—21no—
247C2×C4×C47—21no—
248C2×C4×C47—21no—
249C2×C4×C47—21no—
250C2×C4×C47—21no—
251C2×C4×C47—21no—
252C2×C4×C47C4×Q812yes—
253C2×C4×C47#13112yes—
254C2×C4×C47#13112yes—
255C2×C4×C47C22×Q811yescharacteristic
256C2×C4×C47C2×#4111yescharacteristic
257C2×C4×C47H(5,𝔽2)11yescharacteristic
258C2×C4×C47H(5,𝔽2)11yescharacteristic
259C2×C4×C47H(5,𝔽2)11yescharacteristic
260C2×C4×C472−1+411yescharacteristic
261C47⋊C8—22no—
262C47⋊C8—22no—
263C47⋊C8—41no—
264C47⋊C8—41no—
265C47⋊C8—41no—
266C47⋊C8—41no—
267C47⋊C8—41no—
268C47⋊C8—41no—
269#22292—471nomaximal, Sylow
270C22×C47—22no—
271C22×C47—22no—
272C22×C47—22no—
273C22×C47—41no—
274C22×C47—41no—
275C22×C47—41no—
276C22×C47—41no—
277C22×C47C2×C4×Q811yescharacteristic
278C22×C47#51811yescharacteristic
279C22×C47#52011yescharacteristic
280C22×C47C2×#13111yescharacteristic
281C22×C47C2×#13111yescharacteristic
282C22×C47#56811yescharacteristic
283C22×C47#56811yescharacteristic
284C4×C47—22no—
285C4×C47—22no—
286C4×C47—22no—
287C4×C47—41no—
288C4×C47—41no—
289C4×C47—41no—
290C4×C47—41no—
291C4×C47—41no—
292C4×C47—41no—
293C4×C47—41no—
294C4×C47—41no—
295C4×C47—41no—
296C4×C47—41no—
297C4×C47#53811yescharacteristic
298C4×C47#53911yescharacteristic
299C4×C47#54311yescharacteristic
300C4×C47#54411yescharacteristic
301#2396D4711yescharacteristic
302#1985—471no—
303#1984—471no—
304#1940—471no—
305#1939—471no—
306#1917—471no—
307#1917—471no—
308#1916—471no—
309#1916—471no—
310#1851—471no—
311#1851—471no—
312#1798—471no—
313#1752—471no—
314C8×C22⋊C4—471no—
315#1729—471no—
316C2×C47—42no—
317C2×C47#232911yescharacteristic
318C2×C47#296011yescharacteristic
319C2×C47#296111yescharacteristic
320C2×C47#297611yescharacteristic
321C2×C47#298111yescharacteristic
322C2×C47#299211yescharacteristic
323C2×C47#299311yescharacteristic
324C22×M16—471no—
325C23×C8—471no—
326C2×C4×Q8C2×D4711yescharacteristic
327C2×#117C2×D4711yescharacteristic
328C2×C4⋊C8—471no—
329C2×C4⋊C8—471no—
330C2×C4⋊C8—471no—
331C2×C4⋊C8—471no—
332C2×C4⋊C8—471no—
333C2×C4⋊C8—471no—
334C2×C22⋊C8—471no—
335C2×C22⋊C8—471no—
336C2×#97—471no—
337C2×#97—471no—
338C2×C4×C8—471no—
339C2×C4×C8—471no—
340#400Dic4711yescharacteristic
341#399Dic4711yescharacteristic
342#398Dic4711yescharacteristic
343#398Dic4711yescharacteristic
344#396Dic4711yescharacteristic
345#396Dic4711yescharacteristic
346#395Dic4711yescharacteristic
347#393Dic4711yescharacteristic
348C4×C4⋊C4C2×D4711yescharacteristic
349C4×C4⋊C4C2×D4711yescharacteristic
350C4×C4⋊C4C2×D4711yescharacteristic
351C4×C22⋊C4C2×D4711yescharacteristic
352C4×C22⋊C4C2×D4711yescharacteristic
353#336—471no—
354#336—471no—
355#336—471no—
356#336—471no—
357#336—471no—
358#336—471no—
359#336—471no—
360#336—471no—
361#336—471no—
362#336—471no—
363#336—471no—
364#336—471no—
365#336—471no—
366#336—471no—
367C47#2229211yescharacteristic, complemented, Sylow
368C22×Q8C2×Dic4711yescharacteristic
369C23×C4C22×D4711yescharacteristic
370C2×M16—942no—
371C2×M16—1881no—
372C2×M16—1881no—
373C22×C8—471no—
374C22×C8—471no—
375C22×C8—474no—
376C22×C8—471no—
377C22×C8—471no—
378C22×C8—471no—
379C22×C8—471no—
380C22×C8—471no—
381C22×C8—471no—
382C22×C8—1881no—
383C22×C8—1881no—
384C4×Q8—41no—
385C4×Q8—41no—
386#117—41no—
387#117—41no—
388C2×C4⋊C4C2×Dic4711yescharacteristic
389C2×C4⋊C4C2×Dic4711yescharacteristic
390C2×C4⋊C4C2×Dic4711yescharacteristic
391C2×C4⋊C4C2×Dic4711yescharacteristic
392C2×C4⋊C4C2×Dic4711yescharacteristic
393C2×C4⋊C4C2×Dic4711yescharacteristic
394C2×C4⋊C4C2×Dic4711yescharacteristic
395C2×C4⋊C4C2×Dic4711yescharacteristic
396C2×C4⋊C4C2×Dic4711yescharacteristic
397C2×C4⋊C4C2×Dic4711yescharacteristic
398C2×C4⋊C4C2×Dic4711yescharacteristic
399C2×C22⋊C4C2×Dic4711yescharacteristic
400C2×C22⋊C4C2×Dic4711yescharacteristic
401C2×C22⋊C4C2×Dic4711yescharacteristic
402C2×C22⋊C4C2×Dic4711yescharacteristic
403C2×C22⋊C4C2×Dic4711yescharacteristic
404C2×C22⋊C4C2×Dic4711yescharacteristic
405C2×C42C22×D4711yescharacteristic
406C2×C42C22×D4711yescharacteristic
407C2×C42C22×D4711yescharacteristic
408C2×C42C22×D4711yescharacteristic
409C2×C42C22×D4711yescharacteristic
410C2×C42C22×D4711yescharacteristic
411C4⋊C8—1881no—
412C4⋊C8—1881no—
413C4⋊C8—1881no—
414C4⋊C8—1881no—
415C4⋊C8—1881no—
416C4⋊C8—1881no—
417C22⋊C8—942no—
418C22⋊C8—942no—
419#97—1881no—
420#97—1881no—
421C4×C8—1881no—
422C4×C8—1881no—
423#95C2×Dic4711yescharacteristic
424#95C2×Dic4711yescharacteristic
425#95C2×Dic4711yescharacteristic
426#95C2×Dic4711yescharacteristic
427#95C2×Dic4711yescharacteristic
428#95C2×Dic4711yescharacteristic
429#95C2×Dic4711yescharacteristic
430#95C2×Dic4711yescharacteristic
431#95C2×Dic4711yescharacteristic
432#95C2×Dic4711yescharacteristic
433C24C22×Dic4711yescharacteristic
434C2×Q8—22no—
435C2×Q8—41no—
436C2×Q8—41no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4Q8×D4712yes—
442C22×C4#1062535412yes—
443C22×C4C22×Dic4711yescharacteristic
444C22×C4C22×Dic4711yescharacteristic
445C22×C4C22×Dic4711yescharacteristic
446C22×C4C22×Dic4711yescharacteristic
447C22×C4C22×Dic4711yescharacteristic
448C22×C4C22×Dic4711yescharacteristic
449C22×C4C22×Dic4711yescharacteristic
450C22×C4C22×Dic4711yescharacteristic
451C22×C4C22×Dic4711yescharacteristic
452C22×C4C22×Dic4711yescharacteristic
453C22×C4C22×Dic4711yescharacteristic
454C22×C4C22×Dic4711yescharacteristic
455C22×C4C22×Dic4711yescharacteristic
456C22×C4C23×D4711yescharacteristic
457M16—1884no—
458C2×C8—472no—
459C2×C8—472no—
460C2×C8—941no—
461C2×C8—941no—
462C2×C8—941no—
463C2×C8—941no—
464C2×C8—941no—
465C2×C8—941no—
466C2×C8—941no—
467C2×C8—941no—
468C2×C8—941no—
469C2×C8—941no—
470C2×C8—941no—
471C2×C8—941no—
472C2×C8—941no—
473C2×C8—941no—
474C2×C8—941no—
475C2×C8—941no—
476C2×C8—941no—
477C2×C8—941no—
478C2×C8—941no—
479C2×C8—941no—
480C2×C8—941no—
481C2×C8—941no—
482C2×C8—1884no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23Q8×Dic4712yes—
519C23C23×Dic4711yescharacteristic
520Q8—44no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4Q8×Dic4712yes—
562C2×C4#1309403112yes—
563C2×C4#1309403112yes—
564C2×C4C2×Q8×D4711yescharacteristic
565C2×C4C2×#1062535411yescharacteristic
566C2×C4#1309406911yescharacteristic
567C2×C4#1309406911yescharacteristic
568C2×C4#1309407011yescharacteristic
569C2×C4#1309407111yescharacteristic
570C8—942no—
571C8—942no—
572C8—1881no—
573C8—1881no—
574C8—1881no—
575C8—1881no—
576C8—1881no—
577C8—1881no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C2×Q8×Dic4711yescharacteristic
586C22#1568959811yescharacteristic
587C22#1568960611yescharacteristic
588C22C2×#1309403111yescharacteristic
589C22C2×#1309403111yescharacteristic
590C22#1568989011yescharacteristic
591C22#1568989011yescharacteristic
592C4—22no—
593C4—22no—
594C4—22no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4#1568971311yescharacteristic
606C4#1568972211yescharacteristic
607C4#1568974811yescharacteristic
608C4#1568975211yescharacteristic
609C2—42no—
610C2#2024612311yescharacteristic
611C2#2025232311yescharacteristic
612C2#2025232911yescharacteristic
613C2#2025243611yescharacteristic
614C2#2025246411yescharacteristic
615C2#2025252711yescharacteristic
616C2#2025253711yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#2396C2
prime radicalC24×C47C22×C4
cosocleC22×C4×C47C24
hypercenter#2396D47
hypocenterC47#22292
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #2396

Complements

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

NH ≅ G/Ntype
1. #20244329C2semidirect
2. #20244325C2semidirect
3. #20244112C2semidirect
4. #20244107C2semidirect
5. #20243954C2semidirect
6. #20243954C2semidirect
7. #20243947C2semidirect
8. #20243947C2semidirect
64. C22×C47⋊C8Q8semidirect
367. C47#22292semidirect

Maximal subgroups

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

subgroupindex
#202443292
#202443252
#202441122
#202441072
#202439542
#202439472
#202435732
#202433722
#202431662
C22⋊C4×C47⋊C82
#202430852
C47×#23962
#2229247