Finite Group Databasebeta

← #26182979

Subgroups — #26182979

Order 12032.

1026 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689677C211yesmaximal, characteristic
2#431×D47C211yesmaximal, characteristic
3#423×D47C211yesmaximal, characteristic
4#20241841C214yesmaximal, complemented
5#20241818C211yesmaximal, characteristic, complemented
6#20241817C211yesmaximal, characteristic, complemented
7#20241816C211yesmaximal, characteristic, complemented
8#20241789C214yesmaximal, complemented
9C47×#1487C211yesmaximal, characteristic, complemented
10C2×M16×D47—21no—
11#15689677—21no—
12#15689677—21no—
13#15689677—21no—
14#15689677—21no—
15#15689677C412yes—
16#15689677C412yes—
17#15689677C2212yes—
18#15689677C2212yes—
19C2×#13094023C411yescharacteristic
20C2×#13094022C411yescharacteristic
21C2×#13094014—21no—
22C2×#13094004C411yescharacteristic
23C2×#13094002—21no—
24C2×#13094000C411yescharacteristic
25C2×C4×Dic94—21no—
26C2×C4×Dic94C2211yescharacteristic
27C2×#13093981—21no—
28C2×C42×D47C2211yescharacteristic
29#15689176C2211yescharacteristic
30#15689175—21no—
31#15689174C2212yes—
32#15689173—21no—
33#15689147C2211yescharacteristic
34#15689146—21no—
35#15689145—21no—
36#15689145C2212yes—
37C4⋊C8×D47—21no—
38C4⋊C8×D47—21no—
39C4⋊C8×D47C2212yes—
40C22⋊C8×D47—21no—
41C4×C8×D47C2212yes—
42#15688855C2214yescomplemented
43Q8⋊(C47⋊C8)C2214yescomplemented
44#15688853C2214yescomplemented
45C2×C4×Q8×C47C2211yescharacteristic
46C47×#431C2211yescharacteristic
47C47×#423C2211yescharacteristic
48C47×Q8⋊C8C2214yescomplemented
49C22×Dic94—21no—
50C22×Dic94C2×C411yescharacteristic
51C22×C4×D47C2×C411yescharacteristic
52C22×C4×D47D411yescharacteristic
53C22×C4×D47D411yescharacteristic
54C2×#10625329—21no—
55C2×#10625324—22no—
56M16×D47—44no—
57C2×C8×D47—22no—
58C2×C8×D47—22no—
59#13094023—21no—
60#13094023—21no—
61#13094023C2×C412yes—
62#13094023C2×C412yes—
63#13094022—21no—
64#13094022—21no—
65#13094022C2×C412yes—
66#13094022C2×C412yes—
67#13094014—22no—
68#13094014—22no—
69#13094014—22no—
70#13094014—22no—
71#13094004—21no—
72#13094004—21no—
73#13094004C2×C412yes—
74#13094004C2×C412yes—
75#13094002—22no—
76#13094002—22no—
77#13094002—22no—
78#13094002—22no—
79#13094000—21no—
80#13094000—21no—
81#13094000C2×C412yes—
82#13094000C2×C412yes—
83C4×Dic94—21no—
84C4×Dic94—21no—
85C4×Dic94—22no—
86C4×Dic94—22no—
87C4×Dic94—22no—
88C4×Dic94—22no—
89C4×Dic94C2×C412yes—
90C4×Dic94C2312yes—
91#13093981—22no—
92#13093981—22no—
93#13093981—22no—
94#13093981—22no—
95C2×C4⋊Dic47—21no—
96C2×C4⋊Dic47C2×C411yescharacteristic
97C2×Dic47⋊C4—21no—
98C2×Dic47⋊C4—21no—
99C2×Dic47⋊C4—21no—
100C2×Dic47⋊C4—21no—
101C2×Dic47⋊C4—21no—
102C2×(C2×D47)⋊C4—21no—
103C2×(C2×D47)⋊C4—21no—
104C2×C4×Dic47—21no—
105C2×C4×Dic47C2×C411yescharacteristic
106C2×C4×Dic47D411yescharacteristic
107C2×C4×Dic47D411yescharacteristic
108C42×D47C2×C411yescharacteristic
109C42×D47C2×C411yescharacteristic
110C42×D47D411yescharacteristic
111C42×D47D411yescharacteristic
112C42×D47D411yescharacteristic
113C42×D47D411yescharacteristic
114C42×D47C2311yescharacteristic
115C42×D47C2311yescharacteristic
116C4⋊(C47⋊C8)—21no—
117C4⋊(C47⋊C8)—21no—
118C4⋊(C47⋊C8)C2312yes—
119Dic47⋊C8—22no—
120Dic47⋊C8—22no—
121Dic47⋊C8—22no—
122C22⋊(C47⋊C8)—21no—
123(C2×D47)⋊C8—22no—
124C4×C47⋊C8C2312yes—
125C8×Dic47—22no—
126C22×Q8×C47C2×C411yescharacteristic
127C2×M16×C47—21no—
128C4×Q8×C47—21no—
129C4×Q8×C47—21no—
130C4×Q8×C47C2×C412yes—
131C4×Q8×C47C2312yes—
132C2×C4⋊C4×C47—21no—
133C2×C4⋊C4×C47C2×C411yescharacteristic
134C2×C42×C47—21no—
135C2×C42×C47C2311yescharacteristic
136C4⋊C8×C47—21no—
137C4⋊C8×C47—21no—
138C4⋊C8×C47C2312yes—
139C22⋊C8×C47—21no—
140C4×C8×C47C2312yes—
141C23×D47C22⋊C411yescharacteristic
142C2×Q8×C47—21no—
143C2×Q8×C47—42no—
144C2×Q8×C47C22×C412yes—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47C22×C411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47C22⋊C411yescharacteristic
153C22×Dic47C22⋊C411yescharacteristic
154C22×Dic47C22⋊C411yescharacteristic
155C2×C4×D47—42no—
156C2×C4×D47—42no—
157C2×C4×D47C22⋊C411yescharacteristic
158C2×C4×D47C22⋊C411yescharacteristic
159C2×C4×D47C22⋊C411yescharacteristic
160C2×C4×D47C22⋊C411yescharacteristic
161C2×C4×D47C22⋊C411yescharacteristic
162C2×C4×D47C22⋊C411yescharacteristic
163C2×C4×D47SD1614yes—
164C2×C4×D47Q1614yes—
165C2×C4×D47C22×C411yescharacteristic
166C2×C4×D47C22×C411yescharacteristic
167C2×C4×D47C2×D411yescharacteristic
168C2×C4×D47C2×D411yescharacteristic
169C2×C4×D47C2×D411yescharacteristic
170C2×C4×D47C2×D411yescharacteristic
171C2×Dic94—21no—
172C2×Dic94—22no—
173C2×Dic94—22no—
174C2×Dic94—42no—
175C2×Dic94—44no—
176C2×Dic94C22×C412yes—
177M16×C47—42no—
178C2×C8×C47—22no—
179C2×C8×C47—22no—
180C4⋊C4×C47—22no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—22no—
183C4⋊C4×C47C22×C412yes—
184C42×C47—22no—
185C42×C47—22no—
186C42×C47C22×C411yescharacteristic
187C42×C47C2×D411yescharacteristic
188C42×C47C2×D411yescharacteristic
189C42×C47C2411yescharacteristic
190(C2×D47)⋊C4—22no—
191(C2×D47)⋊C4—22no—
192(C2×D47)⋊C4—22no—
193(C2×D47)⋊C4—22no—
194C4⋊Dic47—21no—
195C4⋊Dic47—22no—
196C4⋊Dic47—22no—
197C4⋊Dic47C22×C412yes—
198Dic47⋊C4—22no—
199Dic47⋊C4—22no—
200Dic47⋊C4—22no—
201Dic47⋊C4—22no—
202Dic47⋊C4—22no—
203Dic47⋊C4—22no—
204Dic47⋊C4—22no—
205Dic47⋊C4—22no—
206Dic47⋊C4—22no—
207Dic47⋊C4—22no—
208C4×Dic47—22no—
209C4×Dic47—22no—
210C4×Dic47C22⋊C411yescharacteristic
211C4×Dic47C22⋊C411yescharacteristic
212C4×Dic47C22⋊C411yescharacteristic
213C4×Dic47C22⋊C411yescharacteristic
214C4×Dic47C22⋊C411yescharacteristic
215C4×Dic47C22⋊C411yescharacteristic
216C4×Dic47C22×C411yescharacteristic
217C4×Dic47C22×C411yescharacteristic
218C4×Dic47C2×D411yescharacteristic
219C4×Dic47C2×D411yescharacteristic
220C4×Dic47C2×D411yescharacteristic
221C4×Dic47C2×D411yescharacteristic
222#10625329—42no—
223C2×C47⋊C8—22no—
224C2×C47⋊C8—22no—
225#10625324—44no—
226C8×D47—44no—
227C8×D47—44no—
228C23×C47C2×C22⋊C411yescharacteristic
229C22×D47—41no—
230C22×D47—41no—
231C22×D47#10314yes—
232C22×D47C2×C22⋊C411yescharacteristic
233C22×D47C2×C22⋊C411yescharacteristic
234Q8×C47—24no—
235Q8×C47—42no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—41no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47C2×C22⋊C411yescharacteristic
243C2×C4×C47C2×C22⋊C411yescharacteristic
244C2×C4×C47C2×C22⋊C411yescharacteristic
245C2×C4×C47C2×SD1612yes—
246C2×C4×C47C2×Q1612yes—
247C2×C4×C47C23×C411yescharacteristic
248C2×C4×C47C22×D411yescharacteristic
249C2×C4×C47C22×D411yescharacteristic
250C2×Dic47—22no—
251C2×Dic47—22no—
252C2×Dic47—22no—
253C2×Dic47—41no—
254C2×Dic47—42no—
255C2×Dic47—42no—
256C2×Dic47#10314yes—
257C2×Dic47C2×C22⋊C411yescharacteristic
258C2×Dic47C2×C22⋊C411yescharacteristic
259C2×Dic47C2×C22⋊C411yescharacteristic
260C2×Dic47C2×C22⋊C411yescharacteristic
261C2×Dic47C2×C22⋊C411yescharacteristic
262C2×Dic47C2×C22⋊C411yescharacteristic
263C4×D47—22no—
264C4×D47—22no—
265C4×D47—22no—
266C4×D47—22no—
267C4×D47—41no—
268C4×D47—41no—
269C4×D47—41no—
270C4×D47—41no—
271C4×D47#10314yes—
272C4×D47#10314yes—
273C4×D47C2×SD1612yes—
274C4×D47C2×SD1612yes—
275C4×D47C2×Q1612yes—
276C4×D47C2×Q1612yes—
277Dic94—24no—
278Dic94—44no—
279Dic94—42no—
280Dic94—44no—
281C8×C47—42no—
282C8×C47—42no—
283C47⋊C8—42no—
284C47⋊C8—42no—
285#17486—471nomaximal, Sylow
286C22×C47—21no—
287C22×C47—21no—
288C22×C47C2×#10312yes—
289C22×C47C22×C22⋊C411yescharacteristic
290C2×D47—21no—
291C2×D47—21no—
292C2×D47—21no—
293C2×D47—21no—
294C2×D47—44no—
295C2×D47C2×#10312yes—
296C2×D47C2×#10312yes—
297C4×C47—21no—
298C4×C47—21no—
299C4×C47—21no—
300C4×C47—42no—
301C4×C47—42no—
302C4×C47C2×#10312yes—
303C4×C47#56812yes—
304C4×C47C22×SD1611yescharacteristic
305C4×C47C22×Q1611yescharacteristic
306Dic47—21no—
307Dic47—21no—
308Dic47—21no—
309Dic47—42no—
310Dic47—42no—
311Dic47C2×#10312yes—
312Dic47C2×#10312yes—
313Dic47#42112yes—
314C2×#533—471no—
315C2×#431—471no—
316C2×#423—471no—
317#1493—474no—
318#1487—471no—
319#1487—471no—
320#1487—471no—
321#1487D4711yescharacteristic, complemented
322#1481—474no—
323C2×C47—22no—
324C2×C47#287911yescharacteristic
325C2×C47C22×#10311yescharacteristic
326C2×C47C2×#42111yescharacteristic
327D47—22no—
328D47—22no—
329C22×M16—941no—
330#533—472no—
331#533—472no—
332#533—472no—
333#533—472no—
334#533—941no—
335#533—941no—
336#533—941no—
337#533—941no—
338C2×C4⋊Q8—471no—
339C2×C4⋊Q8—471no—
340C2×#125—941no—
341C2×C22⋊Q8—471no—
342C2×C22⋊Q8—471no—
343C2×C22⋊Q8—941no—
344C2×C4×Q8—471no—
345C2×C4×Q8—941no—
346C2×C4×Q8C2×D4711yescharacteristic
347C2×#117—941no—
348C22×C42—471no—
349#431—471no—
350#431—472no—
351#431—941no—
352#431—941no—
353#431C2×D4711yescharacteristic
354#423—471no—
355#423—472no—
356#423—941no—
357#423—941no—
358#423C2×D4711yescharacteristic
359C2×C4⋊C8—472no—
360C2×C4⋊C8—941no—
361C2×C4⋊C8—941no—
362C2×C22⋊C8—941no—
363C2×C4×C8—472no—
364Q8⋊C8—474no—
365Q8⋊C8—474no—
366Q8⋊C8—474no—
367Q8⋊C8C2×D4714yescomplemented
368C47#1748611yescharacteristic, complemented, Sylow
369C22×Q8—471no—
370C22×Q8—941no—
371C22×Q8C4×D4711yescharacteristic
372C23×C4—471no—
373C23×C4—471no—
374C23×C4—471no—
375C2×M16—21no—
376C2×M16—942no—
377C2×M16—941no—
378C2×M16—1884no—
379C22×C8—942no—
380C22×C8—942no—
381C4⋊Q8—472no—
382C4⋊Q8—472no—
383C4⋊Q8—472no—
384C4⋊Q8—472no—
385C4⋊Q8—941no—
386C4⋊Q8—941no—
387C4⋊Q8—941no—
388C4⋊Q8—941no—
389#125—942no—
390#125—942no—
391#125—942no—
392#125—942no—
393C22⋊Q8—472no—
394C22⋊Q8—472no—
395C22⋊Q8—472no—
396C22⋊Q8—472no—
397C22⋊Q8—941no—
398C22⋊Q8—941no—
399C22⋊Q8—942no—
400C22⋊Q8—942no—
401C22⋊Q8—942no—
402C22⋊Q8—942no—
403C22⋊Q8—941no—
404C22⋊Q8—941no—
405C4×Q8—21no—
406C4×Q8—21no—
407C4×Q8—472no—
408C4×Q8—472no—
409C4×Q8—942no—
410C4×Q8—942no—
411C4×Q8—942no—
412C4×Q8—942no—
413C4×Q8—941no—
414C4×Q8—941no—
415C4×Q8C4×D4712yes—
416C4×Q8C22×D4712yes—
417#117—942no—
418#117—942no—
419#117—942no—
420#117—942no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—471no—
423C2×C4⋊C4—941no—
424C2×C4⋊C4—941no—
425C2×C4⋊C4—941no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4—941no—
428C2×C4⋊C4—941no—
429C2×C4⋊C4C4×D4711yescharacteristic
430C2×C22⋊C4—941no—
431C2×C22⋊C4—941no—
432C2×C42—21no—
433C2×C42—471no—
434C2×C42—471no—
435C2×C42—471no—
436C2×C42—471no—
437C2×C42—471no—
438C2×C42—471no—
439C2×C42—471no—
440C2×C42—471no—
441C2×C42—471no—
442C2×C42—471no—
443C2×C42—471no—
444C2×C42—941no—
445C2×C42C22×D4711yescharacteristic
446C4⋊C8—21no—
447C4⋊C8—21no—
448C4⋊C8—472no—
449C4⋊C8—942no—
450C4⋊C8—942no—
451C4⋊C8—942no—
452C4⋊C8—941no—
453C4⋊C8—941no—
454C4⋊C8C22×D4712yes—
455C22⋊C8—21no—
456C22⋊C8—942no—
457C22⋊C8—941no—
458C4×C8—472no—
459C4×C8—942no—
460C4×C8C22×D4712yes—
461C24—471no—
462C2×Q8—21no—
463C2×Q8—42no—
464C2×Q8—472no—
465C2×Q8—942no—
466C2×Q8—942no—
467C2×Q8—941no—
468C2×Q8—1882no—
469C2×Q8—1884no—
470C2×Q8C2×C4×D4712yes—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—471no—
474C22×C4—471no—
475C22×C4—471no—
476C22×C4—474no—
477C22×C4—474no—
478C22×C4—471no—
479C22×C4—471no—
480C22×C4—471no—
481C22×C4—471no—
482C22×C4—471no—
483C22×C4—471no—
484C22×C4—471no—
485C22×C4—471no—
486C22×C4—471no—
487C22×C4—471no—
488C22×C4—471no—
489C22×C4—471no—
490C22×C4—941no—
491C22×C4—941no—
492C22×C4—1882no—
493C22×C4—1882no—
494C22×C4C2×C4×D4711yescharacteristic
495C22×C4D4×D4711yescharacteristic
496C22×C4D4×D4711yescharacteristic
497M16—42no—
498M16—1884no—
499M16—1882no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—942no—
503C2×C8—942no—
504C2×C8—1884no—
505C2×C8—1884no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—472no—
510C4⋊C4—942no—
511C4⋊C4—942no—
512C4⋊C4—942no—
513C4⋊C4—942no—
514C4⋊C4—942no—
515C4⋊C4—942no—
516C4⋊C4—942no—
517C4⋊C4—942no—
518C4⋊C4—942no—
519C4⋊C4—942no—
520C4⋊C4—942no—
521C4⋊C4—941no—
522C4⋊C4—942no—
523C4⋊C4C2×C4×D4712yes—
524C22⋊C4—942no—
525C22⋊C4—942no—
526C22⋊C4—942no—
527C22⋊C4—942no—
528C42—22no—
529C42—22no—
530C42—471no—
531C42—471no—
532C42—471no—
533C42—471no—
534C42—471no—
535C42—471no—
536C42—471no—
537C42—471no—
538C42—471no—
539C42—471no—
540C42—471no—
541C42—471no—
542C42—942no—
543C42—942no—
544C42C2×C4×D4711yescharacteristic
545C42D4×D4711yescharacteristic
546C42D4×D4711yescharacteristic
547C42C23×D4711yescharacteristic
548C23—471no—
549C23—471no—
550C23—474no—
551C23—1881no—
552C23—1881no—
553C23C22⋊C4×D4711yescharacteristic
554Q8—24no—
555Q8—42no—
556Q8—944no—
557Q8—1882no—
558Q8—1884no—
559Q8—1884no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—471no—
567C2×C4—471no—
568C2×C4—471no—
569C2×C4—474no—
570C2×C4—474no—
571C2×C4—474no—
572C2×C4—472no—
573C2×C4—472no—
574C2×C4—472no—
575C2×C4—472no—
576C2×C4—471no—
577C2×C4—471no—
578C2×C4—471no—
579C2×C4—942no—
580C2×C4—942no—
581C2×C4—942no—
582C2×C4—942no—
583C2×C4—942no—
584C2×C4—942no—
585C2×C4—942no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4—1881no—
589C2×C4—1881no—
590C2×C4—1881no—
591C2×C4—1882no—
592C2×C4—1882no—
593C2×C4C22⋊C4×D4711yescharacteristic
594C2×C4C22⋊C4×D4711yescharacteristic
595C2×C4C22⋊C4×D4711yescharacteristic
596C2×C4SD16×D4712yes—
597C2×C4Q16×D4712yes—
598C2×C4C22×C4×D4711yescharacteristic
599C2×C4C2×D4×D4711yescharacteristic
600C2×C4C2×D4×D4711yescharacteristic
601C8—42no—
602C8—42no—
603C8—1882no—
604C8—1882no—
605C22—21no—
606C22—21no—
607C22—472no—
608C22—472no—
609C22—941no—
610C22—941no—
611C22—941no—
612C22—941no—
613C22—1884no—
614C22#103×D4712yes—
615C22C2×C22⋊C4×D4711yescharacteristic
616C4—21no—
617C4—21no—
618C4—21no—
619C4—42no—
620C4—42no—
621C4—472no—
622C4—472no—
623C4—472no—
624C4—941no—
625C4—941no—
626C4—941no—
627C4—1882no—
628C4—1882no—
629C4#103×D4712yes—
630C4#1568988912yes—
631C4C2×SD16×D4711yescharacteristic
632C4C2×Q16×D4711yescharacteristic
633C2—22no—
634C2—942no—
635C2—942no—
636C2#2025183111yescharacteristic
637C2C2×#103×D4711yescharacteristic
638C2#421×D4711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1487C2
socleC22×C47C22×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC42×C47C24
hypercenter#1487D47
hypocenterC47#17486
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
4. #20241841C2semidirect
5. #20241818C2semidirect
6. #20241817C2semidirect
7. #20241816C2semidirect
8. #20241789C2semidirect
9. C47×#1487C2semidirect
42. #15688855C22semidirect
43. Q8⋊(C47⋊C8)C22semidirect
44. #15688853C22semidirect
48. C47×Q8⋊C8C22semidirect
321. #1487D47semidirect
367. Q8⋊C8C2×D47semidirect
368. C47#17486semidirect

Maximal subgroups

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

subgroupindex
C2×#156896772
#431×D472
#423×D472
#202418412
#202418182
#202418172
#202418162
#202417892
C47×#14872
#1748647