Finite Group Databasebeta

← #26220887

Subgroups — #26220887

Order 12032.

1134 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689750C211yesmaximal, characteristic
2C2×#15689080C211yesmaximal, characteristic
3#20246013C211yesmaximal, characteristic
4#20244359C211yesmaximal, characteristic, complemented
5#20244357C211yesmaximal, characteristic, complemented
6#20244355C211yesmaximal, characteristic, complemented
7#20244352C211yesmaximal, characteristic, complemented
8#20244351C211yesmaximal, characteristic, complemented
9#20244350C211yesmaximal, characteristic, complemented
10#20244347C211yesmaximal, characteristic, complemented
11#20243106C211yesmaximal, characteristic
12#20243106C211yesmaximal, characteristic
13#20243103C211yesmaximal, characteristic
14#340×D47C211yesmaximal, characteristic
15C47×#1993C211yesmaximal, characteristic, complemented
16C22×#10625324C2211yescharacteristic
17C22×C8×D47C2211yescharacteristic
18#15689750—24no—
19#15689750C418yescomplemented
20C2×#13094023C2211yescharacteristic
21C2×#13094022—21no—
22C2×#13094016C2211yescharacteristic
23C2×#13094015C2211yescharacteristic
24C2×#13094014—21no—
25C2×#13094002—21no—
26C2×#13094000—21no—
27C2×#13093982C2211yescharacteristic
28C2×#13093982C2211yescharacteristic
29C2×#13093981C2211yescharacteristic
30C2×C4⋊C4×D47C2211yescharacteristic
31C2×C4⋊C4×D47C2211yescharacteristic
32#15689080—24no—
33C2×#13093934C2211yescharacteristic
34C2×C4×C47⋊C8C2211yescharacteristic
35C2×C8×Dic47C2211yescharacteristic
36#15689020—21no—
37#15689020—21no—
38#15688997—21no—
39#15688997—21no—
40#15688993C2211yescharacteristic
41#15688992C2211yescharacteristic
42#15688988C2211yescharacteristic
43#15688886C2211yescharacteristic
44#15688886C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53#15688885C2211yescharacteristic
54#15688885C2211yescharacteristic
55C2×C4⋊Q8×C47C2211yescharacteristic
56C2×#97×C47C2211yescharacteristic
57C47×#382C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C47×#340C2211yescharacteristic
61C47×#340C2211yescharacteristic
62C22×Dic94—21no—
63C22×C4×D47—21no—
64C22×C4×D47—21no—
65C22×C4×D47C2311yescharacteristic
66C2×#10625324—22no—
67C2×#10625324—22no—
68C2×#10625324Q814yescomplemented
69C22×C47⋊C8C2311yescharacteristic
70C22×C47⋊C8C2311yescharacteristic
71C2×C8×D47—22no—
72C2×C8×D47—22no—
73C2×C8×D47Q814yescomplemented
74#13094023—22no—
75#13094023C2×C414yescomplemented
76#13094022—28no—
77#13094016—22no—
78#13094016C2×C414yescomplemented
79#13094015—22no—
80#13094015C2×C414yescomplemented
81#13094014—28no—
82#13094002—28no—
83#13094000—28no—
84#13093982—22no—
85#13093982—22no—
86#13093982—22no—
87#13093982C2×C414yes—
88#13093981C2×C414yes—
89#13093981D414yes—
90C2×C4⋊Dic47—21no—
91C2×C4⋊Dic47—21no—
92C2×C4⋊Dic47—21no—
93C2×C4⋊Dic47C2311yescharacteristic
94C2×C4⋊Dic47C2311yescharacteristic
95C2×C4⋊Dic47C2311yescharacteristic
96C2×C4⋊Dic47C2311yescharacteristic
97C2×Dic47⋊C4—21no—
98C2×Dic47⋊C4—21no—
99C2×Dic47⋊C4—21no—
100C2×Dic47⋊C4—21no—
101C2×Dic47⋊C4—21no—
102C2×Dic47⋊C4—21no—
103C2×Dic47⋊C4—21no—
104C2×Dic47⋊C4—21no—
105C2×Dic47⋊C4D411yescharacteristic
106C2×Dic47⋊C4D411yescharacteristic
107C4⋊C4×D47—22no—
108C4⋊C4×D47—22no—
109C4⋊C4×D47—22no—
110C4⋊C4×D47C2×C414yes—
111C2×(C2×D47)⋊C4—21no—
112C2×(C2×D47)⋊C4—21no—
113C2×(C2×D47)⋊C4D411yescharacteristic
114C2×(C2×D47)⋊C4D411yescharacteristic
115C2×C4×Dic47—21no—
116C2×C4×Dic47—21no—
117C2×C4×Dic47C2311yescharacteristic
118#13093934—22no—
119C4×C47⋊C8—22no—
120C8×Dic47—22no—
121C47⋊#95—21no—
122C47⋊#95—21no—
123C22×Q8×C47—21no—
124C22×C8×C47C2311yescharacteristic
125C22×C8×C47C2311yescharacteristic
126C4⋊Q8×C47—22no—
127C4⋊Q8×C47C2×C414yescomplemented
128C2×C4⋊C4×C47—21no—
129C2×C4⋊C4×C47C2311yescharacteristic
130C2×C4⋊C4×C47C2311yescharacteristic
131C2×C4⋊C4×C47C2311yescharacteristic
132C2×C4⋊C4×C47C2311yescharacteristic
133C2×C42×C47C2311yescharacteristic
134#97×C47—22no—
135#95×C47—21no—
136#95×C47—21no—
137C23×D47C2×D411yescharacteristic
138C2×Q8×C47—24no—
139C2×Q8×C47—81no—
140C2×Q8×C47—81no—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47—21no—
144C22×C4×C47—21no—
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2411yescharacteristic
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47C2×D411yescharacteristic
153C22×Dic47C2×D411yescharacteristic
154C22×Dic47C2×D411yescharacteristic
155C2×C4×D47—24no—
156C2×C4×D47—42no—
157C2×C4×D47—81no—
158C2×C4×D47—81no—
159C2×C4×D47—81no—
160C2×C4×D47—81no—
161C2×C4×D47C22×C412yes—
162C2×C4×D47C2×D412yes—
163C2×C4×D47C2×Q812yes—
164C2×C4×D47C2×Q812yes—
165C2×C4×D47#4112yes—
166C2×C4×D47#4112yes—
167C2×Dic94—24no—
168C2×Dic94—81no—
169C2×Dic94—81no—
170C2×C8×C47—21no—
171C2×C8×C47—21no—
172C2×C8×C47—21no—
173C2×C8×C47—21no—
174C2×C8×C47C2×Q812yes—
175C2×C8×C47C2×Q812yes—
176C4⋊C4×C47—24no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47C22×C412yes—
184C4⋊C4×C47C22×C412yes—
185C42×C47C22×C412yes—
186C42×C47C2×D412yes—
187(C2×D47)⋊C4—24no—
188(C2×D47)⋊C4—42no—
189(C2×D47)⋊C4C22⋊C414yes—
190(C2×D47)⋊C4C22⋊C414yes—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47—24no—
194C4⋊Dic47—21no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—81no—
199C4⋊Dic47—81no—
200C4⋊Dic47C22×C412yes—
201C4⋊Dic47C22×C412yes—
202Dic47⋊C4—24no—
203Dic47⋊C4—24no—
204Dic47⋊C4—24no—
205Dic47⋊C4—24no—
206Dic47⋊C4—24no—
207Dic47⋊C4—42no—
208Dic47⋊C4—81no—
209Dic47⋊C4—81no—
210Dic47⋊C4C22⋊C414yes—
211Dic47⋊C4C22⋊C414yes—
212C4×Dic47—24no—
213C4×Dic47—42no—
214C4×Dic47C22×C412yes—
215C4×Dic47C2×D412yes—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—21no—
218C2×C47⋊C8—21no—
219C2×C47⋊C8—21no—
220C2×C47⋊C8C2×Q812yes—
221C2×C47⋊C8C2×Q812yes—
222#10625324—28no—
223C8×D47—28no—
224C23×C47C22×D411yescharacteristic
225C22×D47—21no—
226C22×D47—21no—
227C22×D47—21no—
228C22×D47—21no—
229C22×D47C2×C22⋊C412yes—
230C22×D47#12412yes—
231C22×D47C4⋊Q812yes—
232Q8×C47—84no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
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×C47C2×C22⋊C412yes—
250C2×C4×C47C2×C22⋊C412yes—
251C2×C4×C47C23×C411yescharacteristic
252C2×C4×C47C22×D411yescharacteristic
253C2×C4×C47C22×Q811yescharacteristic
254C2×C4×C47C22×Q811yescharacteristic
255C2×C4×C47C2×#4111yescharacteristic
256C2×C4×C47C2×#4111yescharacteristic
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—22no—
262C2×Dic47—22no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47C2×C22⋊C412yes—
274C2×Dic47C2×C22⋊C412yes—
275C2×Dic47C2×C22⋊C412yes—
276C2×Dic47#12412yes—
277C2×Dic47C4⋊Q812yes—
278C4×D47—84no—
279C4×D47—84no—
280C4×D47C4×Q814yes—
281C4×D47C4×Q814yes—
282C4×D47C22⋊Q814yes—
283C4×D47C22⋊Q814yes—
284Dic94—84no—
285C8×C47—22no—
286C8×C47—22no—
287C47⋊C8—22no—
288C47⋊C8—22no—
289#21326—471nomaximal, Sylow
290C22×C47C22×C22⋊C411yescharacteristic
291C22×C47C2×#12411yescharacteristic
292C22×C47C2×C4⋊Q811yescharacteristic
293C22×C47#57611yescharacteristic
294C22×C47#57711yescharacteristic
295C22×C47#57711yescharacteristic
296C22×C47#57811yescharacteristic
297C2×D47—22no—
298C2×D47—22no—
299C2×D47—22no—
300C2×D47—22no—
301C2×D47—22no—
302C2×D47—22no—
303C2×D47#39114yes—
304C4×C47—22no—
305C4×C47—22no—
306C4×C47—42no—
307C4×C47—42no—
308C4×C47—81no—
309C4×C47—81no—
310C4×C47C2×C4×Q811yescharacteristic
311C4×C47C2×C4×Q811yescharacteristic
312C4×C47C2×C22⋊Q811yescharacteristic
313C4×C47C2×C22⋊Q811yescharacteristic
314Dic47—22no—
315Dic47—22no—
316Dic47—42no—
317Dic47—42no—
318Dic47—81no—
319Dic47—81no—
320Dic47#39114yes—
321C2×#544—471no—
322C2×#405—471no—
323#2317—471no—
324#1993—471no—
325#1993D4711yescharacteristic, complemented
326#1992—471no—
327#1992—471no—
328#1991—471no—
329#1991—471no—
330#1990—471no—
331#1990—471no—
332#1736—471no—
333#1736—471no—
334#1735—471no—
335C2×#340—471no—
336C2×C47C2×#39111yescharacteristic
337C2×C47#289711yescharacteristic
338C2×C47#289811yescharacteristic
339C2×C47#313611yescharacteristic
340C2×C47#313711yescharacteristic
341C2×C47#316311yescharacteristic
342C2×C47#316411yescharacteristic
343D47—24no—
344C22×M16—471no—
345C23×C8—471no—
346#544—478no—
347#544—944no—
348C2×C4⋊Q8—471no—
349C2×C4⋊Q8—941no—
350C2×C4⋊Q8C2×D4711yescharacteristic
351C2×#125—471no—
352C2×#125—471no—
353C2×#125—941no—
354C2×C22⋊Q8—941no—
355C2×C22⋊Q8—941no—
356C2×#117—471no—
357C2×#117—471no—
358C2×#117—471no—
359C22×C4⋊C4—471no—
360C22×C4⋊C4—471no—
361#405—944no—
362C2×#97—471no—
363C2×#97C2×D4711yescharacteristic
364C2×C4×C8—471no—
365C2×C4×C8—471no—
366#389—941no—
367#389—941no—
368#385—941no—
369#385—941no—
370#384—471no—
371#384—471no—
372#382—471no—
373#382C2×D4711yescharacteristic
374#340—471no—
375#340—471no—
376#340—471no—
377#340—471no—
378#340—471no—
379#340—471no—
380#340—471no—
381#340—471no—
382#340—471no—
383#340—471no—
384#340—471no—
385#340—471no—
386#340C2×D4711yescharacteristic
387#340C2×D4711yescharacteristic
388#340C2×D4711yescharacteristic
389#340C2×D4711yescharacteristic
390C47#2132611yescharacteristic, complemented, Sylow
391C22×Q8—21no—
392C22×Q8—941no—
393C23×C4—471no—
394C23×C4—941no—
395C23×C4—941no—
396C2×M16—474no—
397C2×M16—942no—
398C2×M16—942no—
399C22×C8—471no—
400C22×C8—471no—
401C22×C8—474no—
402C22×C8—942no—
403C22×C8—942no—
404C22×C8C22×D4711yescharacteristic
405C22×C8C22×D4711yescharacteristic
406C4⋊Q8—22no—
407C4⋊Q8—474no—
408C4⋊Q8—942no—
409C4⋊Q8—948no—
410C4⋊Q8C4×D4714yescomplemented
411#125—474no—
412#125—474no—
413#125—942no—
414#125—948no—
415#125—942no—
416C22⋊Q8—948no—
417C22⋊Q8—948no—
418#117—474no—
419#117—474no—
420#117—474no—
421#117—942no—
422#117—942no—
423#117—942no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—471no—
426C2×C4⋊C4—471no—
427C2×C4⋊C4—471no—
428C2×C4⋊C4—471no—
429C2×C4⋊C4—474no—
430C2×C4⋊C4—471no—
431C2×C4⋊C4—471no—
432C2×C4⋊C4—942no—
433C2×C4⋊C4—942no—
434C2×C4⋊C4—942no—
435C2×C4⋊C4—941no—
436C2×C4⋊C4—941no—
437C2×C4⋊C4—941no—
438C2×C4⋊C4—941no—
439C2×C4⋊C4—941no—
440C2×C4⋊C4—941no—
441C2×C4⋊C4—941no—
442C2×C4⋊C4—941no—
443C2×C4⋊C4—941no—
444C2×C4⋊C4—941no—
445C2×C4⋊C4—941no—
446C2×C4⋊C4C22×D4711yescharacteristic
447C2×C4⋊C4C22×D4711yescharacteristic
448C2×C4⋊C4C22×D4711yescharacteristic
449C2×C4⋊C4C22×D4711yescharacteristic
450C2×C22⋊C4—471no—
451C2×C22⋊C4—471no—
452C2×C22⋊C4—941no—
453C2×C22⋊C4—941no—
454C2×C42—471no—
455C2×C42—941no—
456C2×C42—941no—
457C2×C42C22×D4711yescharacteristic
458#97—22no—
459#97—942no—
460C4×C8—942no—
461C4×C8—942no—
462#95—21no—
463#95—21no—
464#95—941no—
465#95—941no—
466C24—471no—
467C2×Q8—24no—
468C2×Q8—81no—
469C2×Q8—81no—
470C2×Q8—944no—
471C2×Q8—3761no—
472C2×Q8—3761no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—471no—
478C22×C4—472no—
479C22×C4—472no—
480C22×C4—472no—
481C22×C4—472no—
482C22×C4—472no—
483C22×C4—472no—
484C22×C4—471no—
485C22×C4—471no—
486C22×C4—944no—
487C22×C4—941no—
488C22×C4—941no—
489C22×C4—941no—
490C22×C4—941no—
491C22×C4—3761no—
492C22×C4—3761no—
493C22×C4—3761no—
494C22×C4—3761no—
495C22×C4—1882no—
496C22×C4D4×D4711yescharacteristic
497C22×C4D4×D4711yescharacteristic
498C22×C4C23×D4711yescharacteristic
499M16—948no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—472no—
505C2×C8—472no—
506C2×C8—948no—
507C2×C8—941no—
508C2×C8—941no—
509C2×C8—941no—
510C2×C8—941no—
511C2×C8Q8×D4712yes—
512C2×C8Q8×D4712yes—
513C4⋊C4—24no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—474no—
521C4⋊C4—474no—
522C4⋊C4—472no—
523C4⋊C4—472no—
524C4⋊C4—944no—
525C4⋊C4—944no—
526C4⋊C4—944no—
527C4⋊C4—944no—
528C4⋊C4—944no—
529C4⋊C4—944no—
530C4⋊C4—941no—
531C4⋊C4—941no—
532C4⋊C4—941no—
533C4⋊C4—941no—
534C4⋊C4—941no—
535C4⋊C4—941no—
536C4⋊C4—3761no—
537C4⋊C4—3761no—
538C4⋊C4—3761no—
539C4⋊C4—3761no—
540C4⋊C4—1882no—
541C4⋊C4C2×C4×D4712yes—
542C4⋊C4C2×C4×D4712yes—
543C22⋊C4—474no—
544C22⋊C4—474no—
545C22⋊C4—944no—
546C22⋊C4—1882no—
547C42—472no—
548C42—472no—
549C42—944no—
550C42—1882no—
551C42C2×C4×D4712yes—
552C42D4×D4712yes—
553C23—472no—
554C23—472no—
555C23—472no—
556C23—941no—
557C23—941no—
558C23—941no—
559C23—941no—
560C23C2×D4×D4711yescharacteristic
561Q8—84no—
562Q8—3764no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—472no—
580C2×C4—472no—
581C2×C4—472no—
582C2×C4—474no—
583C2×C4—474no—
584C2×C4—474no—
585C2×C4—474no—
586C2×C4—472no—
587C2×C4—472no—
588C2×C4—941no—
589C2×C4—941no—
590C2×C4—941no—
591C2×C4—941no—
592C2×C4—942no—
593C2×C4—942no—
594C2×C4—3764no—
595C2×C4—3764no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4—1881no—
599C2×C4—1881no—
600C2×C4—1881no—
601C2×C4—1881no—
602C2×C4—1881no—
603C2×C4—1881no—
604C2×C4—1881no—
605C2×C4—1881no—
606C2×C4C22⋊C4×D4712yes—
607C2×C4C22⋊C4×D4712yes—
608C2×C4C22×C4×D4711yescharacteristic
609C2×C4C2×D4×D4711yescharacteristic
610C2×C4C2×Q8×D4711yescharacteristic
611C2×C4C2×Q8×D4711yescharacteristic
612C2×C4#41×D4711yescharacteristic
613C2×C4#41×D4711yescharacteristic
614C8—22no—
615C8—22no—
616C8—942no—
617C8—942no—
618C22—474no—
619C22—942no—
620C22—942no—
621C22—942no—
622C22—942no—
623C22—942no—
624C22—942no—
625C22C2×C22⋊C4×D4711yescharacteristic
626C22#124×D4711yescharacteristic
627C22C4⋊Q8×D4711yescharacteristic
628C22#1568992711yescharacteristic
629C22#1568993211yescharacteristic
630C22#1568993311yescharacteristic
631C22#1568993911yescharacteristic
632C4—22no—
633C4—22no—
634C4—42no—
635C4—42no—
636C4—81no—
637C4—81no—
638C4—474no—
639C4—942no—
640C4—942no—
641C4—3761no—
642C4—3761no—
643C4—1882no—
644C4—1882no—
645C4C4×Q8×D4711yescharacteristic
646C4C4×Q8×D4711yescharacteristic
647C4C22⋊Q8×D4711yescharacteristic
648C4C22⋊Q8×D4711yescharacteristic
649C2—944no—
650C2#391×D4711yescharacteristic
651C2#2025191311yescharacteristic
652C2#2025192011yescharacteristic
653C2#2025373911yescharacteristic
654C2#2025375011yescharacteristic
655C2#2025391511yescharacteristic
656C2#2025392311yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1993C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1993D47
hypocenterC47#21326
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
4. #20244359C2semidirect
5. #20244357C2semidirect
6. #20244355C2semidirect
7. #20244352C2semidirect
8. #20244351C2semidirect
9. #20244350C2semidirect
10. #20244347C2semidirect
15. C47×#1993C2semidirect
19. #15689750C4semidirect
68. C2×#10625324Q8semidirect
73. C2×C8×D47Q8semidirect
75. #13094023C2×C4semidirect
78. #13094016C2×C4semidirect
80. #13094015C2×C4semidirect
127. C4⋊Q8×C47C2×C4semidirect
325. #1993D47semidirect
390. C47#21326semidirect
410. C4⋊Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156897502
C2×#156890802
#202460132
#202443592
#202443572
#202443552
#202443522
#202443512
#202443502
#202443472
#202431062
#202431032
#340×D472
C47×#19932
#2132647