Finite Group Databasebeta

← #26183606

Subgroups — #26183606

Order 12032.

1046 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
1C2×#15689176C211yesmaximal, characteristic, complemented
2#20245786C211yesmaximal, characteristic, complemented
3#20241965C214yesmaximal, complemented
4#20241960C212yesmaximal, complemented
5#20241956C212yesmaximal
6#20241874C214yesmaximal, complemented
7C2×C47×#529C211yesmaximal, characteristic
8#15689176—21no—
9#15689176—21no—
10#15689176C2212yescomplemented
11#15689176C2212yescomplemented
12C2×C4⋊(C47⋊C8)—21no—
13C2×C22⋊(C47⋊C8)C2212yes—
14C2×C4×C47⋊C8—21no—
15#15688991C2212yescomplemented
16#15688989—21no—
17#15688986—21no—
18#15688983C2212yes—
19C4×C4⋊Dic47—21no—
20C4×C22⋊Dic47—21no—
21#15688969C2212yescomplemented
22#15688860C2218yescomplemented
23#15688857C2218yes—
24C47×#529—21no—
25C47×#529—21no—
26C47×#529—21no—
27C47×#529—21no—
28C47×#529C412yescomplemented
29C47×#529C412yescomplemented
30C47×#529C2212yes—
31C47×#529C2212yes—
32C2×#126×C47—21no—
33C2×#125×C47C411yescharacteristic, complemented
34C2×#124×C47C411yescharacteristic, complemented
35C2×#123×C47—21no—
36C2×C22⋊Q8×C47C2211yescharacteristic
37C2×#121×C47C2211yescharacteristic
38C2×C4×Q8×C47C411yescharacteristic, complemented
39C2×C4×D4×C47—21no—
40C2×C4×D4×C47C411yescharacteristic, complemented
41C2×#117×C47—21no—
42C22×C42×C47C2211yescharacteristic
43C22×C47⋊C8—22no—
44C2×C4⋊Dic47—22no—
45C2×C22⋊Dic47—22no—
46C2×C4×Dic47—22no—
47C4⋊(C47⋊C8)—22no—
48C4⋊(C47⋊C8)—22no—
49C22⋊(C47⋊C8)—22no—
50C22⋊(C47⋊C8)C2314yes—
51C4×C47⋊C8—22no—
52C4×C47⋊C8—22no—
53C47⋊#95—22no—
54C47⋊#95—22no—
55C47⋊#95—22no—
56C47⋊#95C2314yes—
57C22×Q8×C47C2×C411yescharacteristic
58C22×D4×C47—21no—
59C22×D4×C47C2×C411yescharacteristic
60C23×C4×C47—21no—
61C23×C4×C47D411yescharacteristic
62C23×C4×C47D411yescharacteristic
63C23×C4×C47C2311yescharacteristic
64#126×C47—22no—
65#126×C47—22no—
66#126×C47—22no—
67#126×C47—22no—
68#125×C47—21no—
69#125×C47—21no—
70#125×C47C2×C412yes—
71#125×C47C2×C412yes—
72#124×C47—21no—
73#124×C47—21no—
74#124×C47C2×C412yes—
75#124×C47C2×C412yes—
76#123×C47—22no—
77#123×C47—22no—
78#123×C47—22no—
79#123×C47—22no—
80C22⋊Q8×C47—21no—
81C22⋊Q8×C47—21no—
82C22⋊Q8×C47C2×C412yes—
83C22⋊Q8×C47C2312yes—
84#121×C47—21no—
85#121×C47—21no—
86#121×C47C2×C412yes—
87#121×C47C2312yes—
88C4×Q8×C47—21no—
89C4×Q8×C47—21no—
90C4×Q8×C47C2×C412yes—
91C4×Q8×C47C2×C412yes—
92C4×D4×C47—22no—
93C4×D4×C47—22no—
94C4×D4×C47—22no—
95C4×D4×C47—22no—
96C4×D4×C47—21no—
97C4×D4×C47—21no—
98C4×D4×C47C2×C412yes—
99C4×D4×C47C2×C412yes—
100#117×C47—22no—
101#117×C47—22no—
102#117×C47—22no—
103#117×C47—22no—
104C2×C4⋊C4×C47—21no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47C2×C411yescharacteristic
109C2×C4⋊C4×C47C2×C411yescharacteristic
110C2×C22⋊C4×C47—21no—
111C2×C22⋊C4×C47—21no—
112C2×C22⋊C4×C47—21no—
113C2×C22⋊C4×C47—21no—
114C2×C22⋊C4×C47—21no—
115C2×C42×C47—21no—
116C2×C42×C47C2×C411yescharacteristic
117C2×C42×C47C2×C411yescharacteristic
118C2×C42×C47C2×C411yescharacteristic
119C2×C42×C47C2×C411yescharacteristic
120C2×C42×C47D411yescharacteristic
121C2×C42×C47D411yescharacteristic
122C2×C42×C47D411yescharacteristic
123C2×C42×C47D411yescharacteristic
124C2×C42×C47D411yescharacteristic
125C2×C42×C47D411yescharacteristic
126C2×C42×C47C2311yescharacteristic
127C2×C42×C47C2311yescharacteristic
128C24×C47—21no—
129C24×C47C2×D411yescharacteristic
130C2×Q8×C47—21no—
131C2×Q8×C47—42no—
132C2×Q8×C47C22×C412yes—
133C2×D4×C47—22no—
134C2×D4×C47—22no—
135C2×D4×C47—21no—
136C2×D4×C47—42no—
137C2×D4×C47—44no—
138C2×D4×C47C22×C412yes—
139C22×C4×C47—24no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47—22no—
144C22×C4×C47—22no—
145C22×C4×C47—42no—
146C22×C4×C47—42no—
147C22×C4×C47—44no—
148C22×C4×C47C22⋊C411yescharacteristic
149C22×C4×C47C22⋊C411yescharacteristic
150C22×C4×C47C22⋊C411yescharacteristic
151C22×C4×C47C22⋊C411yescharacteristic
152C22×C4×C47C22⋊C411yescharacteristic
153C22×C4×C47C22⋊C411yescharacteristic
154C22×C4×C47C22⋊C411yescharacteristic
155C22×C4×C47C22⋊C411yescharacteristic
156C22×C4×C47C22×C411yescharacteristic
157C22×C4×C47C22×C411yescharacteristic
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2×D411yescharacteristic
161C22×C4×C47C2×D411yescharacteristic
162C22×C4×C47C2×D411yescharacteristic
163C22×C4×C47C2×D411yescharacteristic
164C22×C4×C47C2×D411yescharacteristic
165C22×C4×C47C2411yescharacteristic
166C22×Dic47—24no—
167C4⋊C4×C47—22no—
168C4⋊C4×C47—22no—
169C4⋊C4×C47—22no—
170C4⋊C4×C47—22no—
171C4⋊C4×C47—22no—
172C4⋊C4×C47—22no—
173C4⋊C4×C47—22no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—22no—
177C4⋊C4×C47C22×C412yes—
178C4⋊C4×C47C22×C412yes—
179C22⋊C4×C47—22no—
180C22⋊C4×C47—22no—
181C22⋊C4×C47—22no—
182C22⋊C4×C47—22no—
183C22⋊C4×C47—22no—
184C22⋊C4×C47—22no—
185C22⋊C4×C47—22no—
186C22⋊C4×C47—22no—
187C22⋊C4×C47—22no—
188C22⋊C4×C47—22no—
189C42×C47—22no—
190C42×C47—22no—
191C42×C47C22⋊C411yescharacteristic
192C42×C47C22⋊C411yescharacteristic
193C42×C47C22⋊C411yescharacteristic
194C42×C47C22⋊C411yescharacteristic
195C42×C47C22⋊C411yescharacteristic
196C42×C47C22⋊C411yescharacteristic
197C42×C47C22⋊C411yescharacteristic
198C42×C47C22⋊C411yescharacteristic
199C42×C47C22×C411yescharacteristic
200C42×C47C22×C411yescharacteristic
201C42×C47C22×C411yescharacteristic
202C42×C47C22×C411yescharacteristic
203C42×C47C2×D411yescharacteristic
204C42×C47C2×D411yescharacteristic
205C42×C47C2×D411yescharacteristic
206C42×C47C2×D411yescharacteristic
207C22⋊Dic47—44no—
208C4⋊Dic47—44no—
209C4×Dic47—44no—
210C2×C47⋊C8—24no—
211C2×C47⋊C8—44no—
212C23×C47—22no—
213C23×C47—41no—
214C23×C47—41no—
215C23×C47—42no—
216C23×C47—42no—
217C23×C47—42no—
218C23×C47C2×C22⋊C411yescharacteristic
219C23×C47C2×C22⋊C411yescharacteristic
220C23×C47#13514yes—
221C23×C47C22×D411yescharacteristic
222Q8×C47—42no—
223Q8×C47—42no—
224D4×C47—42no—
225D4×C47—44no—
226D4×C47—44no—
227D4×C47—42no—
228C2×C4×C47—22no—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—22no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—44no—
250C2×C4×C47—42no—
251C2×C4×C47—42no—
252C2×C4×C47—42no—
253C2×C4×C47—44no—
254C2×C4×C47C2×C22⋊C411yescharacteristic
255C2×C4×C47C2×C22⋊C411yescharacteristic
256C2×C4×C47C2×C22⋊C411yescharacteristic
257C2×C4×C47C2×C22⋊C411yescharacteristic
258C2×C4×C47C2×C22⋊C411yescharacteristic
259C2×C4×C47C2×C22⋊C411yescharacteristic
260C2×C4×C47C2×C22⋊C411yescharacteristic
261C2×C4×C47C2×C22⋊C411yescharacteristic
262C2×C4×C47C2×C22⋊C411yescharacteristic
263C2×C4×C47C2×C22⋊C411yescharacteristic
264C2×C4×C47C23×C411yescharacteristic
265C2×C4×C47C22×D411yescharacteristic
266C2×Dic47—44no—
267C2×Dic47—44no—
268C2×Dic47—44no—
269C47⋊C8—44no—
270#17567—471nomaximal, Sylow
271C22×C47—21no—
272C22×C47—21no—
273C22×C47—21no—
274C22×C47—21no—
275C22×C47—21no—
276C22×C47—21no—
277C22×C47—44no—
278C22×C47—42no—
279C22×C47—42no—
280C22×C47—42no—
281C22×C47—84no—
282C22×C47#41612yes—
283C22×C47#41612yes—
284C22×C47C22×C22⋊C411yescharacteristic
285C22×C47C2×#13512yes—
286C4×C47—21no—
287C4×C47—21no—
288C4×C47—21no—
289C4×C47—21no—
290C4×C47—21no—
291C4×C47—21no—
292C4×C47—21no—
293C4×C47—21no—
294C4×C47—21no—
295C4×C47—21no—
296C4×C47—42no—
297C4×C47—42no—
298C4×C47—42no—
299C4×C47#41012yes—
300C4×C47#42112yes—
301Dic47—84no—
302C2×#529D4711yescharacteristic
303C2×#431—471no—
304#2283—471no—
305#1521—474no—
306#1520—472no—
307#1519—472no—
308#1500—474no—
309C2×C47—22no—
310C2×C47—22no—
311C2×C47—22no—
312C2×C47—82no—
313C2×C47C2×#41011yescharacteristic
314C2×C47C2×#41611yescharacteristic
315C2×C47C2×#42111yescharacteristic
316#529—21no—
317#529—21no—
318#529—21no—
319#529—21no—
320#529Dic4712yescomplemented
321#529Dic4712yescomplemented
322#529C2×D4712yes—
323#529C2×D4712yes—
324C2×#126—21no—
325C2×#125Dic4711yescharacteristic, complemented
326C2×#124Dic4711yescharacteristic, complemented
327C2×#123—21no—
328C2×C22⋊Q8C2×D4711yescharacteristic
329C2×#121C2×D4711yescharacteristic
330C2×C4×Q8Dic4711yescharacteristic, complemented
331C2×C4×D4—21no—
332C2×C4×D4Dic4711yescharacteristic, complemented
333C2×#117—21no—
334C22×C42C2×D4711yescharacteristic
335#431—472no—
336#431—472no—
337#431—941no—
338#431—941no—
339C2×C4⋊C8—941no—
340C2×C22⋊C8—472no—
341C2×C4×C8—941no—
342#383—472no—
343#382—941no—
344#381—941no—
345#380—472no—
346C4×C4⋊C4—941no—
347C4×C22⋊C4—941no—
348#376—472no—
349#328—478no—
350#327—478no—
351C47#1756711yescharacteristic, complemented, Sylow
352C22×Q8C2×Dic4711yescharacteristic
353C22×D4—21no—
354C22×D4C2×Dic4711yescharacteristic
355C23×C4—21no—
356C23×C4C22⋊D4711yescharacteristic
357C23×C4C22⋊D4711yescharacteristic
358C23×C4C22×D4711yescharacteristic
359C22×C8—942no—
360#126—22no—
361#126—22no—
362#126—22no—
363#126—22no—
364#125—21no—
365#125—21no—
366#125C2×Dic4712yes—
367#125C2×Dic4712yes—
368#124—21no—
369#124—21no—
370#124C2×Dic4712yes—
371#124C2×Dic4712yes—
372#123—22no—
373#123—22no—
374#123—22no—
375#123—22no—
376C22⋊Q8—21no—
377C22⋊Q8—21no—
378C22⋊Q8C2×Dic4712yes—
379C22⋊Q8C22×D4712yes—
380#121—21no—
381#121—21no—
382#121C2×Dic4712yes—
383#121C22×D4712yes—
384C4×Q8—21no—
385C4×Q8—21no—
386C4×Q8C2×Dic4712yes—
387C4×Q8C2×Dic4712yes—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—21no—
393C4×D4—21no—
394C4×D4C2×Dic4712yes—
395C4×D4C2×Dic4712yes—
396#117—22no—
397#117—22no—
398#117—22no—
399#117—22no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—942no—
405C2×C4⋊C4C2×Dic4711yescharacteristic
406C2×C4⋊C4C2×Dic4711yescharacteristic
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—942no—
413C2×C42—21no—
414C2×C42—942no—
415C2×C42C2×Dic4711yescharacteristic
416C2×C42C2×Dic4711yescharacteristic
417C2×C42C2×Dic4711yescharacteristic
418C2×C42C2×Dic4711yescharacteristic
419C2×C42C22⋊D4711yescharacteristic
420C2×C42C22⋊D4711yescharacteristic
421C2×C42C22⋊D4711yescharacteristic
422C2×C42C22⋊D4711yescharacteristic
423C2×C42C22⋊D4711yescharacteristic
424C2×C42C22⋊D4711yescharacteristic
425C2×C42C22×D4711yescharacteristic
426C2×C42C22×D4711yescharacteristic
427C4⋊C8—942no—
428C4⋊C8—942no—
429C22⋊C8—474no—
430C22⋊C8—942no—
431C4×C8—942no—
432C4×C8—942no—
433#95—474no—
434#95—942no—
435#95—942no—
436#95—942no—
437C24—21no—
438C24C2×C22⋊D4711yescharacteristic
439C2×Q8—21no—
440C2×Q8—42no—
441C2×Q8C22×Dic4712yes—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—21no—
445C2×D4—42no—
446C2×D4—44no—
447C2×D4C22×Dic4712yes—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—944no—
458C22×C4C22⋊Dic4711yescharacteristic
459C22×C4C22⋊Dic4711yescharacteristic
460C22×C4C22⋊Dic4711yescharacteristic
461C22×C4C22⋊Dic4711yescharacteristic
462C22×C4C22⋊Dic4711yescharacteristic
463C22×C4C22⋊Dic4711yescharacteristic
464C22×C4C22⋊Dic4711yescharacteristic
465C22×C4C22⋊Dic4711yescharacteristic
466C22×C4C22×Dic4711yescharacteristic
467C22×C4C22×Dic4711yescharacteristic
468C22×C4C2×C22⋊D4711yescharacteristic
469C22×C4C2×C22⋊D4711yescharacteristic
470C22×C4C2×C22⋊D4711yescharacteristic
471C22×C4C2×C22⋊D4711yescharacteristic
472C22×C4C2×C22⋊D4711yescharacteristic
473C22×C4C2×C22⋊D4711yescharacteristic
474C22×C4C2×C22⋊D4711yescharacteristic
475C22×C4C23×D4711yescharacteristic
476C2×C8—944no—
477C2×C8—1884no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—1884no—
489C4⋊C4C22×Dic4712yes—
490C4⋊C4C22×Dic4712yes—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—1884no—
502C42—22no—
503C42—22no—
504C42—1884no—
505C42C22⋊Dic4711yescharacteristic
506C42C22⋊Dic4711yescharacteristic
507C42C22⋊Dic4711yescharacteristic
508C42C22⋊Dic4711yescharacteristic
509C42C22⋊Dic4711yescharacteristic
510C42C22⋊Dic4711yescharacteristic
511C42C22⋊Dic4711yescharacteristic
512C42C22⋊Dic4711yescharacteristic
513C42C22×Dic4711yescharacteristic
514C42C22×Dic4711yescharacteristic
515C42C22×Dic4711yescharacteristic
516C42C22×Dic4711yescharacteristic
517C42C2×C22⋊D4711yescharacteristic
518C42C2×C22⋊D4711yescharacteristic
519C42C2×C22⋊D4711yescharacteristic
520C42C2×C22⋊D4711yescharacteristic
521C23—22no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23C2×C22⋊Dic4711yescharacteristic
528C23C2×C22⋊Dic4711yescharacteristic
529C23#1309404414yes—
530C23C22×C22⋊D4711yescharacteristic
531Q8—42no—
532Q8—42no—
533D4—42no—
534D4—44no—
535D4—44no—
536D4—42no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—1884no—
564C2×C4—1884no—
565C2×C4—1884no—
566C2×C4C2×C22⋊Dic4711yescharacteristic
567C2×C4C2×C22⋊Dic4711yescharacteristic
568C2×C4C2×C22⋊Dic4711yescharacteristic
569C2×C4C2×C22⋊Dic4711yescharacteristic
570C2×C4C2×C22⋊Dic4711yescharacteristic
571C2×C4C2×C22⋊Dic4711yescharacteristic
572C2×C4C2×C22⋊Dic4711yescharacteristic
573C2×C4C2×C22⋊Dic4711yescharacteristic
574C2×C4C2×C22⋊Dic4711yescharacteristic
575C2×C4C2×C22⋊Dic4711yescharacteristic
576C2×C4C23×Dic4711yescharacteristic
577C2×C4C22×C22⋊D4711yescharacteristic
578C8—1884no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—44no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—84no—
590C22#1568911912yes—
591C22#1568911912yes—
592C22C22×C22⋊Dic4711yescharacteristic
593C22C2×#1309404412yes—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—3764no—
608C4#1568909712yes—
609C4#1568914012yes—
610C2—22no—
611C2—22no—
612C2—22no—
613C2—82no—
614C2C2×#1568909711yescharacteristic
615C2C2×#1568911911yescharacteristic
616C2C2×#1568914011yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊Dic47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#529C2
socleC22×C47C22×C22⋊C4
prime radicalC2×#121×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#529D47
hypocenterC47#17567
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
1. C2×#15689176C2semidirect
2. #20245786C2semidirect
3. #20241965C2semidirect
4. #20241960C2semidirect
6. #20241874C2semidirect
10. #15689176C22semidirect
11. #15689176C22semidirect
15. #15688991C22semidirect
21. #15688969C22semidirect
22. #15688860C22semidirect
28. C47×#529C4semidirect
29. C47×#529C4semidirect
33. C2×#125×C47C4semidirect
34. C2×#124×C47C4semidirect
38. C2×C4×Q8×C47C4semidirect
40. C2×C4×D4×C47C4semidirect
320. #529Dic47semidirect
321. #529Dic47semidirect
325. C2×#125Dic47semidirect
326. C2×#124Dic47semidirect
330. C2×C4×Q8Dic47semidirect
332. C2×C4×D4Dic47semidirect
351. C47#17567semidirect

Maximal subgroups

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

subgroupindex
C2×#156891762
#202457862
#202419652
#202419602
#202419562
#202418742
C2×C47×#5292
#1756747