Finite Group Databasebeta

← #26184680

Subgroups — #26184680

Order 12032.

972 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D47C211yesmaximal, characteristic
2#20252016C211yesmaximal, characteristic
3#431×D47C211yesmaximal, characteristic
4#20242087C212yesmaximal, complemented
5#20242085C212yesmaximal, complemented
6#20242071C212yesmaximal, complemented
7#20242058C212yesmaximal, complemented
8#20242053C211yesmaximal, characteristic, complemented
9#20242052C211yesmaximal, characteristic, complemented
10#20242052C211yesmaximal, characteristic, complemented
11C47×#1544C211yesmaximal, characteristic, complemented
12#15689680C2211yescharacteristic
13#15689679C412yes—
14#15689678C2211yescharacteristic
15#15689678C2211yescharacteristic
16#15689677—21no—
17#15689677—21no—
18#15689676C412yes—
19#15689675—21no—
20#15689675—21no—
21C4⋊Q8×D47C2211yescharacteristic
22C4⋊Q8×D47C2211yescharacteristic
23#125×D47C412yes—
24C22⋊Q8×D47—21no—
25C22⋊Q8×D47—21no—
26C4×Q8×D47—21no—
27C4×Q8×D47—21no—
28#117×D47C412yes—
29C2×C42×D47C2211yescharacteristic
30#15689179C2212yes—
31#15689177—21no—
32#15689176C2211yescharacteristic
33#15689176C2211yescharacteristic
34#15689175C2212yes—
35#15689174—21no—
36#15689174—21no—
37#15689173—21no—
38#15689145C2212yes—
39C4⋊C8×D47C2212yes—
40#15689085—21no—
41C22⋊C8×D47—21no—
42C4×#10625324—21no—
43C4×C8×D47—21no—
44#15688870C2212yescomplemented
45#15688870C2212yescomplemented
46#15688869C2212yescomplemented
47#15688869C2212yescomplemented
48#15688868C2212yescomplemented
49#15688868C2212yescomplemented
50C47×#533C2211yescharacteristic
51C47×#431C2211yescharacteristic
52C47×#431C2211yescharacteristic
53C47×#332C2212yescomplemented
54C47×#332C2212yescomplemented
55C2×Q8×D47—21no—
56C2×Q8×D47—21no—
57C22×C4×D47C2×C411yescharacteristic
58C22×C4×D47D411yescharacteristic
59C22×C4×D47D411yescharacteristic
60C2×#10625324—22no—
61C2×C8×D47—22no—
62#13094024C2311yescharacteristic
63#13094024C2311yescharacteristic
64#13094023—21no—
65#13094023—21no—
66#13094023C2×C412yes—
67#13094023C2×C412yes—
68#13094022—21no—
69#13094022—21no—
70#13094016C2×C412yes—
71#13094015C2×C412yes—
72#13094015C2×C412yes—
73#13094014—22no—
74#13094014—22no—
75#13094004—21no—
76#13094004—21no—
77#13094003—21no—
78#13094003—21no—
79#13094002—22no—
80#13094002—22no—
81#13094001—22no—
82#13094001—22no—
83#13094000—21no—
84#13094000—21no—
85Q8×Dic47—21no—
86Q8×Dic47—21no—
87C4×Dic94—21no—
88C4×Dic94—21no—
89C4×Dic94—22no—
90C4×Dic94—22no—
91#13093989—22no—
92#13093989—22no—
93#13093989—21no—
94#13093989—21no—
95#13093983C2×C412yes—
96#13093982C2×C412yes—
97#13093982C2×C412yes—
98#13093981—22no—
99#13093980—22no—
100C4⋊C4×D47—22no—
101C4⋊C4×D47—22no—
102C4⋊C4×D47—21no—
103C4⋊C4×D47—21no—
104C4⋊C4×D47C2×C412yes—
105C4⋊C4×D47C2×C412yes—
106C22⋊C4×D47—22no—
107C2×C4×Dic47C2×C411yescharacteristic
108C2×C4×Dic47D411yescharacteristic
109C2×C4×Dic47D411yescharacteristic
110C42×D47—22no—
111C42×D47C2×C411yescharacteristic
112C42×D47C2×C411yescharacteristic
113C42×D47D411yescharacteristic
114C42×D47D411yescharacteristic
115C42×D47D411yescharacteristic
116C42×D47D411yescharacteristic
117C42×D47C2311yescharacteristic
118C42×D47C2311yescharacteristic
119C4⋊(C47⋊C8)C2312yes—
120C4⋊(C47⋊C8)C2312yes—
121Dic47⋊C8—22no—
122Dic47⋊C8—22no—
123C22⋊(C47⋊C8)—21no—
124C22⋊(C47⋊C8)—21no—
125(C2×D47)⋊C8—22no—
126(C2×D47)⋊C8—22no—
127#13093934—22no—
128C4×C47⋊C8—21no—
129C4×C47⋊C8—21no—
130C8×Dic47—22no—
131C4⋊Q8×C47C2311yescharacteristic
132C4⋊Q8×C47C2311yescharacteristic
133#125×C47C2×C412yes—
134C22⋊Q8×C47—21no—
135C22⋊Q8×C47—21no—
136C4×Q8×C47—21no—
137C4×Q8×C47—21no—
138#117×C47C2×C412yes—
139C2×C42×C47C2311yescharacteristic
140C4⋊C8×C47C2312yes—
141C4⋊C8×C47C2312yes—
142C22⋊C8×C47—21no—
143C22⋊C8×C47—21no—
144C4×C8×C47—21no—
145C4×C8×C47—21no—
146C23×D47C22⋊C411yescharacteristic
147C2×Q8×C47—21no—
148C2×Q8×C47—21no—
149C22×C4×C47C22×C411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×Dic47C22⋊C411yescharacteristic
153C22×Dic47C22⋊C411yescharacteristic
154C22×Dic47C22⋊C411yescharacteristic
155Q8×D47—44no—
156Q8×D47—44no—
157C2×C4×D47—24no—
158C2×C4×D47—24no—
159C2×C4×D47—22no—
160C2×C4×D47—22no—
161C2×C4×D47—42no—
162C2×C4×D47C22⋊C411yescharacteristic
163C2×C4×D47C22⋊C411yescharacteristic
164C2×C4×D47C22⋊C411yescharacteristic
165C2×C4×D47C22⋊C411yescharacteristic
166C2×C4×D47C22⋊C411yescharacteristic
167C2×C4×D47C22⋊C411yescharacteristic
168C2×C4×D47C22×C411yescharacteristic
169C2×C4×D47C22×C411yescharacteristic
170C2×C4×D47C2×D411yescharacteristic
171C2×C4×D47C2×D411yescharacteristic
172C2×C4×D47C2×D411yescharacteristic
173C2×C4×D47C2×D411yescharacteristic
174C2×Dic94—21no—
175C2×Dic94—21no—
176C2×Dic94—22no—
177C2×Dic94—22no—
178C2×C8×C47—22no—
179C2×C8×C47—22no—
180C4⋊C4×C47—22no—
181C4⋊C4×C47—22no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47—21no—
184C4⋊C4×C47C22×C412yes—
185C4⋊C4×C47C22×C412yes—
186C22⋊C4×C47—22no—
187C42×C47—22no—
188C42×C47C22×C411yescharacteristic
189C42×C47C2×D411yescharacteristic
190C42×C47C2×D411yescharacteristic
191C42×C47C2411yescharacteristic
192C22⋊Dic47—22no—
193(C2×D47)⋊C4—22no—
194(C2×D47)⋊C4—22no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—22no—
198C4⋊Dic47—22no—
199C4⋊Dic47C22×C412yes—
200C4⋊Dic47C22×C412yes—
201Dic47⋊C4—22no—
202Dic47⋊C4—22no—
203Dic47⋊C4—22no—
204Dic47⋊C4—22no—
205Dic47⋊C4—22no—
206Dic47⋊C4—22no—
207Dic47⋊C4—22no—
208Dic47⋊C4—22no—
209C4×Dic47—22no—
210C4×Dic47—22no—
211C4×Dic47—22no—
212C4×Dic47C22⋊C411yescharacteristic
213C4×Dic47C22⋊C411yescharacteristic
214C4×Dic47C22⋊C411yescharacteristic
215C4×Dic47C22⋊C411yescharacteristic
216C4×Dic47C22⋊C411yescharacteristic
217C4×Dic47C22⋊C411yescharacteristic
218C4×Dic47C22×C411yescharacteristic
219C4×Dic47C22×C411yescharacteristic
220C4×Dic47C2×D411yescharacteristic
221C4×Dic47C2×D411yescharacteristic
222C4×Dic47C2×D411yescharacteristic
223C4×Dic47C2×D411yescharacteristic
224C2×C47⋊C8—22no—
225C2×C47⋊C8—22no—
226#10625324—44no—
227C8×D47—44no—
228C23×C47C2×C22⋊C411yescharacteristic
229C22×D47—22no—
230C22×D47—22no—
231C22×D47—41no—
232C22×D47C2×C22⋊C411yescharacteristic
233C22×D47C2×C22⋊C411yescharacteristic
234Q8×C47—42no—
235Q8×C47—42no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47C2×C22⋊C411yescharacteristic
242C2×C4×C47C2×C22⋊C411yescharacteristic
243C2×C4×C47C2×C22⋊C411yescharacteristic
244C2×C4×C47C23×C411yescharacteristic
245C2×C4×C47C22×D411yescharacteristic
246C2×C4×C47C22×D411yescharacteristic
247C2×Dic47—22no—
248C2×Dic47—22no—
249C2×Dic47—22no—
250C2×Dic47—22no—
251C2×Dic47—41no—
252C2×Dic47C2×C22⋊C411yescharacteristic
253C2×Dic47C2×C22⋊C411yescharacteristic
254C2×Dic47C2×C22⋊C411yescharacteristic
255C2×Dic47C2×C22⋊C411yescharacteristic
256C2×Dic47C2×C22⋊C411yescharacteristic
257C2×Dic47C2×C22⋊C411yescharacteristic
258C4×D47—22no—
259C4×D47—22no—
260C4×D47—22no—
261C4×D47—22no—
262C4×D47—22no—
263C4×D47—22no—
264C4×D47—22no—
265C4×D47—22no—
266C4×D47—41no—
267C4×D47—41no—
268C4×D47—44no—
269C4×D47—44no—
270C4×D47#10114yes—
271C4×D47#13514yes—
272Dic94—44no—
273Dic94—42no—
274Dic94—42no—
275Dic94—44no—
276C8×C47—42no—
277C8×C47—42no—
278C47⋊C8—42no—
279C47⋊C8—42no—
280#17715—471nomaximal, Sylow
281C22×C47—21no—
282C22×C47—21no—
283C22×C47—21no—
284C22×C47C22×C22⋊C411yescharacteristic
285C2×D47—21no—
286C2×D47—21no—
287C2×D47—21no—
288C2×D47—21no—
289C2×D47—44no—
290C2×D47C2×#10112yes—
291C2×D47#41612yes—
292C4×C47—21no—
293C4×C47—21no—
294C4×C47—21no—
295C4×C47—42no—
296C4×C47—42no—
297C4×C47C2×#10112yes—
298C4×C47C2×#13512yes—
299C4×C47#57512yes—
300Dic47—21no—
301Dic47—21no—
302Dic47—21no—
303Dic47—21no—
304Dic47—42no—
305Dic47—42no—
306Dic47C2×#10112yes—
307Dic47#41612yes—
308C2×#533—471no—
309#2920—471no—
310C2×#431—471no—
311#1552—472no—
312#1552—472no—
313#1549—472no—
314#1546—472no—
315#1544—471no—
316#1544—471no—
317#1544—471no—
318#1544D4711yescharacteristic, complemented
319C2×C47—41no—
320C2×C47C22×#10111yescharacteristic
321C2×C47C2×#41611yescharacteristic
322C2×C47#289611yescharacteristic
323D47—22no—
324D47—22no—
325#533—471no—
326#533—471no—
327#533—472no—
328#533—472no—
329#533—471no—
330#533—941no—
331#533—941no—
332#533—941no—
333#533—941no—
334#533C2×D4711yescharacteristic
335C2×C4⋊Q8—471no—
336C2×C4⋊Q8—471no—
337C2×#125—472no—
338C2×C22⋊Q8—941no—
339C2×C22⋊Q8—941no—
340C2×C4×Q8—941no—
341C2×C4×Q8—941no—
342C2×#117—472no—
343C22×C42—471no—
344#432—472no—
345#432—941no—
346#431—471no—
347#431—471no—
348#431—472no—
349#431—941no—
350#431—941no—
351#431—941no—
352#431C2×D4711yescharacteristic
353#431C2×D4711yescharacteristic
354#423—472no—
355C2×C4⋊C8—472no—
356#407—941no—
357C2×C22⋊C8—941no—
358C4×M16—941no—
359C2×C4×C8—941no—
360#332—472no—
361#332—472no—
362#332—472no—
363#332—472no—
364#332—472no—
365#332—472no—
366#332C2×D4712yescomplemented
367#332C2×D4712yescomplemented
368C47#1771511yescharacteristic, complemented, Sylow
369C22×Q8—941no—
370C22×Q8—941no—
371C23×C4—471no—
372C23×C4—471no—
373C23×C4—471no—
374C2×M16—942no—
375C22×C8—942no—
376C4⋊Q8—472no—
377C4⋊Q8—472no—
378C4⋊Q8—471no—
379C4⋊Q8—471no—
380C4⋊Q8—941no—
381C4⋊Q8—941no—
382C4⋊Q8—941no—
383C4⋊Q8—941no—
384C4⋊Q8C22×D4711yescharacteristic
385C4⋊Q8C22×D4711yescharacteristic
386#125—472no—
387#125—472no—
388#125—472no—
389#125—942no—
390#125—942no—
391#125C4×D4712yes—
392C22⋊Q8—21no—
393C22⋊Q8—21no—
394C22⋊Q8—942no—
395C22⋊Q8—942no—
396C22⋊Q8—942no—
397C22⋊Q8—942no—
398C22⋊Q8—941no—
399C22⋊Q8—941no—
400C22⋊Q8—941no—
401C22⋊Q8—941no—
402C22⋊Q8—941no—
403C22⋊Q8—941no—
404C4×Q8—21no—
405C4×Q8—21no—
406C4×Q8—942no—
407C4×Q8—942no—
408C4×Q8—942no—
409C4×Q8—942no—
410C4×Q8—941no—
411C4×Q8—941no—
412C4×Q8—941no—
413C4×Q8—941no—
414C4×Q8—941no—
415C4×Q8—941no—
416#117—472no—
417#117—472no—
418#117—472no—
419#117—942no—
420#117—942no—
421#117C4×D4712yes—
422C2×C4⋊C4—472no—
423C2×C4⋊C4—472no—
424C2×C4⋊C4—942no—
425C2×C4⋊C4—942no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4—941no—
428C2×C22⋊C4—942no—
429C2×C42—471no—
430C2×C42—471no—
431C2×C42—471no—
432C2×C42—471no—
433C2×C42—471no—
434C2×C42—471no—
435C2×C42—471no—
436C2×C42—471no—
437C2×C42—471no—
438C2×C42—471no—
439C2×C42—471no—
440C2×C42—942no—
441C2×C42C22×D4711yescharacteristic
442C4⋊C8—472no—
443C4⋊C8—472no—
444C4⋊C8—942no—
445C4⋊C8—942no—
446C4⋊C8C22×D4712yes—
447C4⋊C8C22×D4712yes—
448C22⋊C8—21no—
449C22⋊C8—21no—
450C22⋊C8—942no—
451C22⋊C8—942no—
452C22⋊C8—941no—
453C22⋊C8—941no—
454#97—942no—
455C4×C8—21no—
456C4×C8—21no—
457C4×C8—942no—
458C4×C8—941no—
459C4×C8—941no—
460C24—471no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—942no—
464C2×Q8—942no—
465C2×Q8—941no—
466C2×Q8—941no—
467C2×Q8—1884no—
468C2×Q8—1884no—
469C22×C4—471no—
470C22×C4—471no—
471C22×C4—471no—
472C22×C4—471no—
473C22×C4—471no—
474C22×C4—471no—
475C22×C4—471no—
476C22×C4—471no—
477C22×C4—471no—
478C22×C4—471no—
479C22×C4—471no—
480C22×C4—471no—
481C22×C4—471no—
482C22×C4—471no—
483C22×C4—471no—
484C22×C4—944no—
485C22×C4—944no—
486C22×C4—942no—
487C22×C4—942no—
488C22×C4—1882no—
489C22×C4C2×C4×D4711yescharacteristic
490C22×C4D4×D4711yescharacteristic
491C22×C4D4×D4711yescharacteristic
492M16—1884no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—942no—
496C2×C8—942no—
497C2×C8—1884no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—472no—
503C4⋊C4—472no—
504C4⋊C4—942no—
505C4⋊C4—942no—
506C4⋊C4—942no—
507C4⋊C4—942no—
508C4⋊C4—942no—
509C4⋊C4—942no—
510C4⋊C4—942no—
511C4⋊C4—942no—
512C4⋊C4—942no—
513C4⋊C4—942no—
514C4⋊C4—941no—
515C4⋊C4—941no—
516C4⋊C4C2×C4×D4712yes—
517C4⋊C4C2×C4×D4712yes—
518C22⋊C4—22no—
519C22⋊C4—942no—
520C22⋊C4—942no—
521C22⋊C4—942no—
522C42—22no—
523C42—471no—
524C42—471no—
525C42—471no—
526C42—471no—
527C42—471no—
528C42—471no—
529C42—471no—
530C42—471no—
531C42—471no—
532C42—471no—
533C42—471no—
534C42—471no—
535C42—942no—
536C42—942no—
537C42—942no—
538C42C2×C4×D4711yescharacteristic
539C42D4×D4711yescharacteristic
540C42D4×D4711yescharacteristic
541C42C23×D4711yescharacteristic
542C23—471no—
543C23—471no—
544C23—942no—
545C23—942no—
546C23—1881no—
547C23C22⋊C4×D4711yescharacteristic
548Q8—42no—
549Q8—42no—
550Q8—1884no—
551Q8—1884no—
552Q8—1882no—
553Q8—1882no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—471no—
560C2×C4—471no—
561C2×C4—474no—
562C2×C4—474no—
563C2×C4—471no—
564C2×C4—471no—
565C2×C4—471no—
566C2×C4—471no—
567C2×C4—942no—
568C2×C4—942no—
569C2×C4—942no—
570C2×C4—942no—
571C2×C4—942no—
572C2×C4—942no—
573C2×C4—942no—
574C2×C4—942no—
575C2×C4—942no—
576C2×C4—942no—
577C2×C4—942no—
578C2×C4—942no—
579C2×C4—1881no—
580C2×C4—1881no—
581C2×C4—1884no—
582C2×C4—1884no—
583C2×C4—1881no—
584C2×C4C22⋊C4×D4711yescharacteristic
585C2×C4C22⋊C4×D4711yescharacteristic
586C2×C4C22⋊C4×D4711yescharacteristic
587C2×C4C22×C4×D4711yescharacteristic
588C2×C4C2×D4×D4711yescharacteristic
589C2×C4C2×D4×D4711yescharacteristic
590C8—42no—
591C8—42no—
592C8—1882no—
593C8—1882no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—472no—
598C22—472no—
599C22—941no—
600C22—941no—
601C22—941no—
602C22—941no—
603C22—1884no—
604C22C2×C22⋊C4×D4711yescharacteristic
605C4—21no—
606C4—21no—
607C4—21no—
608C4—42no—
609C4—42no—
610C4—472no—
611C4—472no—
612C4—941no—
613C4—941no—
614C4—941no—
615C4—941no—
616C4—1882no—
617C4—1882no—
618C4#101×D4712yes—
619C4#135×D4712yes—
620C4#1568992012yes—
621C2—41no—
622C2—942no—
623C2—942no—
624C2C2×#101×D4711yescharacteristic
625C2#416×D4711yescharacteristic
626C2#2025190311yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1544C2
socleC22×C47C22×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC42×C47C24
hypercenter#1544D47
hypocenterC47#17715
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1544

Complements

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

NH ≅ G/Ntype
4. #20242087C2semidirect
5. #20242085C2semidirect
6. #20242071C2semidirect
7. #20242058C2semidirect
8. #20242053C2semidirect
9. #20242052C2semidirect
10. #20242052C2semidirect
11. C47×#1544C2semidirect
44. #15688870C22semidirect
45. #15688870C22semidirect
46. #15688869C22semidirect
47. #15688869C22semidirect
48. #15688868C22semidirect
49. #15688868C22semidirect
53. C47×#332C22semidirect
54. C47×#332C22semidirect
318. #1544D47semidirect
366. #332C2×D47semidirect
367. #332C2×D47semidirect
368. C47#17715semidirect

Maximal subgroups

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

subgroupindex
#533×D472
#202520162
#431×D472
#202420872
#202420852
#202420712
#202420582
#202420532
#202420522
C47×#15442
#1771547