Finite Group Databasebeta

← #26218340

Subgroups — #26218340

Order 12032.

886 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689148C211yesmaximal, characteristic
2C2×#97×D47C211yesmaximal, characteristic
3#20245868C211yesmaximal, characteristic
4#20243955C211yesmaximal, characteristic, complemented
5#20243954C211yesmaximal, characteristic, complemented
6#20243953C211yesmaximal, characteristic, complemented
7#20243952C211yesmaximal, characteristic, complemented
8#20243951C211yesmaximal, characteristic, complemented
9#20243950C211yesmaximal, characteristic, complemented
10#20243949C211yesmaximal, characteristic, complemented
11#20243083C211yesmaximal, characteristic
12#20243083C211yesmaximal, characteristic
13#20243083C211yesmaximal, characteristic
14#20243080C211yesmaximal, characteristic
15C47×#1917C211yesmaximal, characteristic, complemented
16C22×#10625324C2211yescharacteristic
17C22×#10625324C2211yescharacteristic
18C22×C8×D47C2211yescharacteristic
19C22×C8×D47C2211yescharacteristic
20C2×#13093981C2211yescharacteristic
21C2×#13093981C2211yescharacteristic
22C2×C42×D47C2211yescharacteristic
23#15689148—24no—
24C2×C4⋊(C47⋊C8)C2211yescharacteristic
25C2×Dic47⋊C8C2211yescharacteristic
26C2×Dic47⋊C8C2211yescharacteristic
27C2×#13093935C2211yescharacteristic
28C2×#13093934C2211yescharacteristic
29C2×#13093934C2211yescharacteristic
30#97×D47—24no—
31#15689020C411yescharacteristic
32#15689020C411yescharacteristic
33#15689018C411yescharacteristic
34#15689018C411yescharacteristic
35#15688997C411yescharacteristic
36#15688997C411yescharacteristic
37#15688994C411yescharacteristic
38#15688994C411yescharacteristic
39C4×Dic47⋊C4C2211yescharacteristic
40C4×Dic47⋊C4C2211yescharacteristic
41C4×C4⋊Dic47C2211yescharacteristic
42#15688878C2211yescharacteristic
43#15688878C2211yescharacteristic
44#15688878C2211yescharacteristic
45#15688878C2211yescharacteristic
46#15688877C2211yescharacteristic
47#15688877C2211yescharacteristic
48#15688877C2211yescharacteristic
49#15688877C2211yescharacteristic
50#15688877C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53#15688877C2211yescharacteristic
54C2×C4⋊C8×C47C2211yescharacteristic
55C2×#97×C47C2211yescharacteristic
56C4×C4⋊C4×C47C2211yescharacteristic
57C47×#336C2211yescharacteristic
58C47×#336C2211yescharacteristic
59C47×#336C2211yescharacteristic
60C47×#336C2211yescharacteristic
61C22×C4×D47C2×C411yescharacteristic
62C22×C4×D47C2×C411yescharacteristic
63C22×C4×D47C2311yescharacteristic
64C2×#10625324—22no—
65C2×#10625324—22no—
66C2×#10625324—22no—
67C2×#10625324—22no—
68C2×#10625324—22no—
69C2×#10625324—22no—
70C22×C47⋊C8C2311yescharacteristic
71C22×C47⋊C8C2311yescharacteristic
72C22×C47⋊C8C2311yescharacteristic
73C22×C47⋊C8C2311yescharacteristic
74C2×C8×D47—22no—
75C2×C8×D47—22no—
76C2×C8×D47—22no—
77C2×C8×D47—22no—
78C2×C8×D47D414yes—
79C2×C8×D47Q814yes—
80#13093981—22no—
81#13093981—22no—
82#13093981—22no—
83#13093981—22no—
84C2×C4⋊Dic47C2×C411yescharacteristic
85C2×C4⋊Dic47C2×C411yescharacteristic
86C2×Dic47⋊C4C2×C411yescharacteristic
87C2×Dic47⋊C4C2×C411yescharacteristic
88C2×Dic47⋊C4C2×C411yescharacteristic
89C2×Dic47⋊C4C2×C411yescharacteristic
90C2×Dic47⋊C4C2×C411yescharacteristic
91C2×Dic47⋊C4C2×C411yescharacteristic
92C2×Dic47⋊C4C2×C411yescharacteristic
93C2×Dic47⋊C4C2×C411yescharacteristic
94C2×(C2×D47)⋊C4C2×C411yescharacteristic
95C2×(C2×D47)⋊C4C2×C411yescharacteristic
96C2×(C2×D47)⋊C4C2×C411yescharacteristic
97C2×(C2×D47)⋊C4C2×C411yescharacteristic
98C2×C4×Dic47C2×C411yescharacteristic
99C2×C4×Dic47C2×C411yescharacteristic
100C2×C4×Dic47C2311yescharacteristic
101C2×C4×Dic47C2311yescharacteristic
102C2×C4×Dic47C2311yescharacteristic
103C42×D47—22no—
104C42×D47—22no—
105C4⋊(C47⋊C8)—22no—
106Dic47⋊C8—22no—
107Dic47⋊C8—22no—
108#13093935—22no—
109#13093934—22no—
110#13093934—22no—
111C47⋊#95C2×C411yescharacteristic
112C47⋊#95C2×C411yescharacteristic
113C47⋊#95C2×C411yescharacteristic
114C47⋊#95C2×C411yescharacteristic
115C47⋊#95C2×C411yescharacteristic
116C47⋊#95C2×C411yescharacteristic
117C22×C8×C47C2311yescharacteristic
118C22×C8×C47C2311yescharacteristic
119C22×C8×C47C2311yescharacteristic
120C22×C8×C47C2311yescharacteristic
121C2×C4⋊C4×C47C2×C411yescharacteristic
122C2×C4⋊C4×C47C2×C411yescharacteristic
123C2×C42×C47C2311yescharacteristic
124C2×C42×C47C2311yescharacteristic
125C2×C42×C47C2311yescharacteristic
126C4⋊C8×C47—22no—
127#97×C47—22no—
128#95×C47C2×C411yescharacteristic
129#95×C47C2×C411yescharacteristic
130C23×D47C22×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×C47C2411yescharacteristic
138C22×Dic47C22×C411yescharacteristic
139C22×Dic47C22×C411yescharacteristic
140C22×Dic47C22×C411yescharacteristic
141C22×Dic47C22×C411yescharacteristic
142C22×Dic47C22×C411yescharacteristic
143C22×Dic47C22×C411yescharacteristic
144C22×Dic47C22×C411yescharacteristic
145C2×C4×D47—22no—
146C2×C4×D47—22no—
147C2×C4×D47—41no—
148C2×C4×D47—41no—
149C2×C4×D47—41no—
150C2×C4×D47—41no—
151C2×C4×D47C2×D412yes—
152C2×C4×D47C2×Q812yes—
153C2×C4×D47#4112yes—
154C2×C4×D47#4112yes—
155C2×C4×D47#4112yes—
156C2×C4×D47#4112yes—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47—21no—
164C2×C8×C47—21no—
165C2×C8×C47—21no—
166C2×C8×C47—21no—
167C2×C8×C47C2×D412yes—
168C2×C8×C47C2×Q812yes—
169C4⋊C4×C47—41no—
170C4⋊C4×C47—41no—
171C42×C47—21no—
172C42×C47—21no—
173C42×C47—21no—
174C42×C47—21no—
175C42×C47—21no—
176C42×C47—21no—
177(C2×D47)⋊C4—22no—
178(C2×D47)⋊C4—22no—
179(C2×D47)⋊C4—22no—
180(C2×D47)⋊C4—22no—
181C4⋊Dic47—41no—
182C4⋊Dic47—41no—
183Dic47⋊C4—22no—
184Dic47⋊C4—22no—
185Dic47⋊C4—22no—
186Dic47⋊C4—22no—
187Dic47⋊C4—41no—
188Dic47⋊C4—41no—
189Dic47⋊C4—41no—
190Dic47⋊C4—41no—
191C4×Dic47—21no—
192C4×Dic47—21no—
193C4×Dic47—21no—
194C4×Dic47—21no—
195C4×Dic47—21no—
196C4×Dic47—21no—
197C4×Dic47—22no—
198C4×Dic47—22no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8C2×D412yes—
210C2×C47⋊C8C2×Q812yes—
211#10625324—44no—
212#10625324—44no—
213C8×D47—28no—
214C8×D47—28no—
215C23×C47C23×C411yescharacteristic
216C22×D47—21no—
217C22×D47—21no—
218C22×D47—21no—
219C22×D47—21no—
220C22×D47#11712yes—
221C22×D47C4×D412yes—
222C22×D47C4×Q812yes—
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×C47C22×D411yescharacteristic
242C2×C4×C47C22×Q811yescharacteristic
243C2×C4×C47C2×#4111yescharacteristic
244C2×C4×C47C2×#4111yescharacteristic
245C2×C4×C47C2×#4111yescharacteristic
246C2×C4×C47C2×#4111yescharacteristic
247C2×Dic47—21no—
248C2×Dic47—21no—
249C2×Dic47—21no—
250C2×Dic47—21no—
251C2×Dic47—21no—
252C2×Dic47—21no—
253C2×Dic47—21no—
254C2×Dic47—21no—
255C2×Dic47—21no—
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—21no—
263C2×Dic47—21no—
264C2×Dic47—21no—
265C2×Dic47#11712yes—
266C2×Dic47C4×D412yes—
267C2×Dic47C4×Q812yes—
268C4×D47—44no—
269C4×D47—44no—
270C4×D47C22⋊Q814yes—
271C4×D47#12314yes—
272C4×D47#12514yes—
273C4×D47#12614yes—
274C8×C47—22no—
275C8×C47—22no—
276C8×C47—41no—
277C8×C47—41no—
278C47⋊C8—22no—
279C47⋊C8—22no—
280C47⋊C8—41no—
281C47⋊C8—41no—
282#21021—471nomaximal, Sylow
283C22×C47C2×#11711yescharacteristic
284C22×C47C2×C4×D411yescharacteristic
285C22×C47C2×C4×Q811yescharacteristic
286C22×C47#56811yescharacteristic
287C22×C47#56811yescharacteristic
288C22×C47#56811yescharacteristic
289C22×C47#56811yescharacteristic
290C2×D47—22no—
291C2×D47—22no—
292C2×D47—22no—
293C2×D47—22no—
294C2×D47—22no—
295C2×D47—22no—
296C2×D47#38714yes—
297C4×C47—41no—
298C4×C47—41no—
299C4×C47—41no—
300C4×C47—41no—
301C4×C47—41no—
302C4×C47—41no—
303C4×C47C2×C22⋊Q811yescharacteristic
304C4×C47C2×#12311yescharacteristic
305C4×C47C2×#12511yescharacteristic
306C4×C47C2×#12611yescharacteristic
307Dic47—41no—
308Dic47—41no—
309Dic47—41no—
310Dic47—41no—
311Dic47—41no—
312Dic47—41no—
313Dic47#38714yes—
314C2×#424—471no—
315C22×#97—471no—
316#2296—471no—
317#1917—471no—
318#1917—471no—
319#1917—471no—
320#1917—471no—
321#1917—471no—
322#1917—471no—
323#1917—471no—
324#1917D4711yescharacteristic, complemented
325#1729—471no—
326#1729—471no—
327#1729—471no—
328#1728—471no—
329C2×C47C2×#38711yescharacteristic
330C2×C47#292311yescharacteristic
331C2×C47#292311yescharacteristic
332C2×C47#293011yescharacteristic
333C2×C47#293011yescharacteristic
334C2×C47#296111yescharacteristic
335C2×C47#296111yescharacteristic
336D47—24no—
337C22×M16—471no—
338C22×M16—471no—
339C23×C8—471no—
340C23×C8—471no—
341C2×#117—471no—
342C2×#117—471no—
343C22×C42—471no—
344#424—944no—
345C2×C4⋊C8—471no—
346C2×C4⋊C8—471no—
347C2×C4⋊C8—471no—
348C2×C4⋊C8C2×D4711yescharacteristic
349C2×#97—471no—
350C2×#97—471no—
351C2×#97—471no—
352C2×#97—944no—
353C2×#97C2×D4711yescharacteristic
354#389—471no—
355#389—471no—
356#389—471no—
357#389—471no—
358#385—471no—
359#385—471no—
360#385—471no—
361#385—471no—
362C4×C4⋊C4—471no—
363C4×C4⋊C4—471no—
364C4×C4⋊C4—471no—
365C4×C4⋊C4C2×D4711yescharacteristic
366#336—471no—
367#336—471no—
368#336—471no—
369#336—471no—
370#336—471no—
371#336—471no—
372#336—471no—
373#336—471no—
374#336—471no—
375#336—471no—
376#336—471no—
377#336—471no—
378#336C2×D4711yescharacteristic
379#336C2×D4711yescharacteristic
380#336C2×D4711yescharacteristic
381#336C2×D4711yescharacteristic
382C47#2102111yescharacteristic, complemented, Sylow
383C23×C4—471no—
384C23×C4—471no—
385C23×C4—471no—
386C2×M16—942no—
387C2×M16—942no—
388C2×M16—942no—
389C2×M16—942no—
390C2×M16—942no—
391C2×M16—942no—
392C22×C8—471no—
393C22×C8—471no—
394C22×C8—474no—
395C22×C8—474no—
396C22×C8—471no—
397C22×C8—471no—
398C22×C8—942no—
399C22×C8—942no—
400C22×C8—942no—
401C22×C8—942no—
402C22×C8C22×D4711yescharacteristic
403C22×C8C22×D4711yescharacteristic
404C22×C8C22×D4711yescharacteristic
405C22×C8C22×D4711yescharacteristic
406#117—942no—
407#117—942no—
408#117—942no—
409#117—942no—
410C2×C4⋊C4—471no—
411C2×C4⋊C4—471no—
412C2×C4⋊C4—471no—
413C2×C4⋊C4—471no—
414C2×C4⋊C4—471no—
415C2×C4⋊C4—471no—
416C2×C4⋊C4—471no—
417C2×C4⋊C4—471no—
418C2×C4⋊C4—471no—
419C2×C4⋊C4—471no—
420C2×C4⋊C4C4×D4711yescharacteristic
421C2×C4⋊C4C4×D4711yescharacteristic
422C2×C22⋊C4—471no—
423C2×C22⋊C4—471no—
424C2×C22⋊C4—471no—
425C2×C22⋊C4—471no—
426C2×C42—471no—
427C2×C42—471no—
428C2×C42—471no—
429C2×C42—471no—
430C2×C42—471no—
431C2×C42—942no—
432C2×C42—942no—
433C2×C42C22×D4711yescharacteristic
434C2×C42C22×D4711yescharacteristic
435C2×C42C22×D4711yescharacteristic
436C4⋊C8—22no—
437C4⋊C8—942no—
438C4⋊C8—942no—
439C4⋊C8—942no—
440#97—22no—
441#97—942no—
442#97—942no—
443#97—942no—
444#95—471no—
445#95—471no—
446#95—471no—
447#95—471no—
448#95—471no—
449#95—471no—
450#95C4×D4711yescharacteristic
451#95C4×D4711yescharacteristic
452C24—471no—
453C22×C4—471no—
454C22×C4—472no—
455C22×C4—472no—
456C22×C4—472no—
457C22×C4—472no—
458C22×C4—472no—
459C22×C4—472no—
460C22×C4—471no—
461C22×C4—471no—
462C22×C4—471no—
463C22×C4—471no—
464C22×C4—471no—
465C22×C4—471no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—1881no—
469C22×C4—1881no—
470C22×C4—1881no—
471C22×C4—1881no—
472C22×C4C2×C4×D4711yescharacteristic
473C22×C4C2×C4×D4711yescharacteristic
474C22×C4C2×C4×D4711yescharacteristic
475C22×C4C2×C4×D4711yescharacteristic
476C22×C4C2×C4×D4711yescharacteristic
477C22×C4C2×C4×D4711yescharacteristic
478C22×C4C23×D4711yescharacteristic
479M16—1884no—
480M16—1884no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—472no—
492C2×C8—472no—
493C2×C8—948no—
494C2×C8—948no—
495C2×C8—941no—
496C2×C8—941no—
497C2×C8—941no—
498C2×C8—941no—
499C2×C8—941no—
500C2×C8—941no—
501C2×C8—941no—
502C2×C8—941no—
503C2×C8—941no—
504C2×C8—941no—
505C2×C8D4×D4712yes—
506C2×C8Q8×D4712yes—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—942no—
510C4⋊C4—942no—
511C4⋊C4—942no—
512C4⋊C4—942no—
513C4⋊C4—1881no—
514C4⋊C4—1881no—
515C4⋊C4—1881no—
516C4⋊C4—1881no—
517C4⋊C4—1881no—
518C4⋊C4—1881no—
519C22⋊C4—942no—
520C22⋊C4—942no—
521C22⋊C4—942no—
522C22⋊C4—942no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—942no—
530C42—942no—
531C42—941no—
532C42—941no—
533C42—941no—
534C42—941no—
535C42—941no—
536C42—941no—
537C23—472no—
538C23—472no—
539C23—472no—
540C23—941no—
541C23—941no—
542C23—941no—
543C23—941no—
544C23C22×C4×D4711yescharacteristic
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×C4—21no—
562C2×C4—21no—
563C2×C4—472no—
564C2×C4—472no—
565C2×C4—472no—
566C2×C4—474no—
567C2×C4—474no—
568C2×C4—474no—
569C2×C4—474no—
570C2×C4—941no—
571C2×C4—941no—
572C2×C4—941no—
573C2×C4—941no—
574C2×C4—941no—
575C2×C4—941no—
576C2×C4—941no—
577C2×C4—941no—
578C2×C4—941no—
579C2×C4—941no—
580C2×C4—941no—
581C2×C4—941no—
582C2×C4—941no—
583C2×C4—941no—
584C2×C4—941no—
585C2×C4—941no—
586C2×C4—941no—
587C2×C4—941no—
588C2×C4—1884no—
589C2×C4—1884no—
590C2×C4C2×D4×D4711yescharacteristic
591C2×C4C2×Q8×D4711yescharacteristic
592C2×C4#41×D4711yescharacteristic
593C2×C4#41×D4711yescharacteristic
594C2×C4#41×D4711yescharacteristic
595C2×C4#41×D4711yescharacteristic
596C8—22no—
597C8—22no—
598C8—41no—
599C8—41no—
600C8—942no—
601C8—942no—
602C8—1881no—
603C8—1881no—
604C22—474no—
605C22—942no—
606C22—942no—
607C22—942no—
608C22—942no—
609C22—942no—
610C22—942no—
611C22#117×D4711yescharacteristic
612C22C4×D4×D4711yescharacteristic
613C22C4×Q8×D4711yescharacteristic
614C22#1568988811yescharacteristic
615C22#1568988811yescharacteristic
616C22#1568988911yescharacteristic
617C22#1568988911yescharacteristic
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—474no—
625C4—1881no—
626C4—1881no—
627C4—1881no—
628C4—1881no—
629C4—1881no—
630C4—1881no—
631C4C22⋊Q8×D4711yescharacteristic
632C4#123×D4711yescharacteristic
633C4#125×D4711yescharacteristic
634C4#126×D4711yescharacteristic
635C2—944no—
636C2#387×D4711yescharacteristic
637C2#2025203411yescharacteristic
638C2#2025203611yescharacteristic
639C2#2025208611yescharacteristic
640C2#2025208811yescharacteristic
641C2#2025232411yescharacteristic
642C2#2025232411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1917C2
prime radicalC23×D47C22×C4
cosocleC22×C4×C47C24
hypercenter#1917D47
hypocenterC47#21021
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1917

Complements

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

NH ≅ G/Ntype
4. #20243955C2semidirect
5. #20243954C2semidirect
6. #20243953C2semidirect
7. #20243952C2semidirect
8. #20243951C2semidirect
9. #20243950C2semidirect
10. #20243949C2semidirect
15. C47×#1917C2semidirect
324. #1917D47semidirect
382. C47#21021semidirect

Maximal subgroups

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

subgroupindex
C2×#156891482
C2×#97×D472
#202458682
#202439552
#202439542
#202439532
#202439522
#202439512
#202439502
#202439492
#202430832
#202430802
C47×#19172
#2102147