Finite Group Databasebeta

← #26219665

Subgroups — #26219665

Order 12032.

1058 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×#15689164C211yesmaximal, characteristic
2C2×#15689162C211yesmaximal, characteristic
3C2×#15689075C211yesmaximal, characteristic
4#20245869C211yesmaximal, characteristic
5#389×D47C211yesmaximal, characteristic
6#20244131C211yesmaximal, characteristic, complemented
7#20244128C211yesmaximal, characteristic, complemented
8#20244123C211yesmaximal, characteristic, complemented
9#20244122C211yesmaximal, characteristic, complemented
10#20244120C211yesmaximal, characteristic, complemented
11#20244115C211yesmaximal, characteristic, complemented
12#20244114C211yesmaximal, characteristic, complemented
13#20243106C211yesmaximal, characteristic
14#20243106C211yesmaximal, characteristic
15C47×#1943C211yesmaximal, characteristic, complemented
16C22×#10625324C2211yescharacteristic
17C22×#10625324C2211yescharacteristic
18C2×#13093982C2211yescharacteristic
19C2×#13093982C2211yescharacteristic
20C2×C4⋊C4×D47—21no—
21C2×C4⋊C4×D47C2211yescharacteristic
22C2×C4⋊C4×D47C2211yescharacteristic
23C2×C42×D47C2211yescharacteristic
24#15689164C418yescomplemented
25#15689162—24no—
26C2×#13093958C2211yescharacteristic
27C2×#13093957C2211yescharacteristic
28C2×#13093957C2211yescharacteristic
29C2×#13093956C2211yescharacteristic
30C2×#13093955C2211yescharacteristic
31C2×#13093955C2211yescharacteristic
32#15689075—24no—
33C2×#13093935C2211yescharacteristic
34C2×C8×Dic47C2211yescharacteristic
35C2×C8×Dic47C2211yescharacteristic
36#15689022C2211yescharacteristic
37#15689022C2211yescharacteristic
38#15689021C2211yescharacteristic
39#15689021C2211yescharacteristic
40#15689020—21no—
41#15689019C2211yescharacteristic
42#15689019C2211yescharacteristic
43#15689018—21no—
44#15688997—21no—
45#15688994—21no—
46C4⋊C4×Dic47—21no—
47#15688974—21no—
48#95×D47—21no—
49#15688886C2211yescharacteristic
50#15688886C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53#15688885C2211yescharacteristic
54#15688885C2211yescharacteristic
55C2×#107×C47C2211yescharacteristic
56C2×#106×C47C2211yescharacteristic
57C2×#97×C47C2211yescharacteristic
58C47×#389C2211yescharacteristic
59C47×#389C2211yescharacteristic
60C47×#340C2211yescharacteristic
61C47×#340C2211yescharacteristic
62C22×C4×D47—21no—
63C22×C4×D47—21no—
64C22×C4×D47D411yescharacteristic
65C22×C4×D47D411yescharacteristic
66C22×C4×D47C2311yescharacteristic
67C2×#10625324—22no—
68C2×#10625324—22no—
69C2×#10625324C2×C414yes—
70C2×#10625324D414yes—
71C2×#10625324Q814yes—
72C2×#10625324Q814yes—
73C22×C47⋊C8C2311yescharacteristic
74C22×C47⋊C8C2311yescharacteristic
75#13093982—22no—
76#13093982—22no—
77#13093982—22no—
78#13093982C2×C414yes—
79C2×C4⋊Dic47—21no—
80C2×C4⋊Dic47—21no—
81C2×C4⋊Dic47C2311yescharacteristic
82C2×C4⋊Dic47C2311yescharacteristic
83C2×C4⋊Dic47C2311yescharacteristic
84C2×C4⋊Dic47C2311yescharacteristic
85C2×Dic47⋊C4—21no—
86C2×Dic47⋊C4—21no—
87C2×Dic47⋊C4—21no—
88C2×Dic47⋊C4—21no—
89C2×Dic47⋊C4—21no—
90C2×Dic47⋊C4—21no—
91C4⋊C4×D47—22no—
92C4⋊C4×D47—22no—
93C4⋊C4×D47—22no—
94C4⋊C4×D47—44no—
95C4⋊C4×D47C2×C414yes—
96C2×(C2×D47)⋊C4—21no—
97C2×(C2×D47)⋊C4—21no—
98C2×C4×Dic47—21no—
99C2×C4×Dic47—21no—
100C2×C4×Dic47D411yescharacteristic
101C2×C4×Dic47D411yescharacteristic
102C2×C4×Dic47C2311yescharacteristic
103C42×D47—22no—
104C42×D47—22no—
105#13093958—22no—
106#13093957—22no—
107#13093957C2×C414yescomplemented
108#13093956C2×C414yescomplemented
109#13093955—22no—
110#13093955C2×C414yescomplemented
111#13093935—22no—
112C8×Dic47—22no—
113C8×Dic47—22no—
114C47⋊#95—21no—
115C47⋊#95—21no—
116C47⋊#95—21no—
117C47⋊#95—21no—
118C47⋊#95—21no—
119C47⋊#95—21no—
120C22×C8×C47C2311yescharacteristic
121C22×C8×C47C2311yescharacteristic
122C2×C4⋊C4×C47—21no—
123C2×C4⋊C4×C47—21no—
124C2×C4⋊C4×C47C2311yescharacteristic
125C2×C4⋊C4×C47C2311yescharacteristic
126C2×C4⋊C4×C47C2311yescharacteristic
127C2×C4⋊C4×C47C2311yescharacteristic
128C2×C42×C47C2311yescharacteristic
129#107×C47C2×C414yescomplemented
130#106×C47—22no—
131#97×C47—22no—
132#95×C47—21no—
133#95×C47—21no—
134C23×D47C2×D411yescharacteristic
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2411yescharacteristic
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47C2×D411yescharacteristic
147C22×Dic47C2×D411yescharacteristic
148C22×Dic47C2×D411yescharacteristic
149C2×C4×D47—24no—
150C2×C4×D47—22no—
151C2×C4×D47—22no—
152C2×C4×D47—22no—
153C2×C4×D47—22no—
154C2×C4×D47—41no—
155C2×C4×D47—41no—
156C2×C4×D47—42no—
157C2×C4×D47—42no—
158C2×C4×D47—42no—
159C2×C4×D47—42no—
160C2×C4×D47—42no—
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×C8×C47—21no—
168C2×C8×C47—21no—
169C2×C8×C47C22×C412yes—
170C2×C8×C47C2×D412yes—
171C2×C8×C47C2×Q812yes—
172C2×C8×C47C2×Q812yes—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—42no—
180C4⋊C4×C47—42no—
181C4⋊C4×C47C22×C412yes—
182C4⋊C4×C47C22×C412yes—
183C42×C47—21no—
184C42×C47—21no—
185(C2×D47)⋊C4—24no—
186(C2×D47)⋊C4—42no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47—42no—
194C4⋊Dic47—42no—
195C4⋊Dic47C22×C412yes—
196C4⋊Dic47C22×C412yes—
197Dic47⋊C4—24no—
198Dic47⋊C4—42no—
199Dic47⋊C4—42no—
200Dic47⋊C4—42no—
201Dic47⋊C4—42no—
202Dic47⋊C4—42no—
203C4×Dic47—21no—
204C4×Dic47—21no—
205C4×Dic47—24no—
206C4×Dic47—22no—
207C4×Dic47—22no—
208C4×Dic47—42no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8—21no—
211C2×C47⋊C8C22×C412yes—
212C2×C47⋊C8C2×D412yes—
213C2×C47⋊C8C2×Q812yes—
214C2×C47⋊C8C2×Q812yes—
215#10625324—28no—
216#10625324C4⋊C4116yescomplemented
217C23×C47C22×D411yescharacteristic
218C22×D47—21no—
219C22×D47—21no—
220C22×D47—21no—
221C22×D47—21no—
222C22×D47C4×D412yes—
223C22×D47#12112yes—
224C22×D47C4⋊Q812yes—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
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—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47C23×C411yescharacteristic
244C2×C4×C47C22×D411yescharacteristic
245C2×C4×C47C22×Q811yescharacteristic
246C2×C4×C47C22×Q811yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×Dic47—21no—
250C2×Dic47—21no—
251C2×Dic47—21no—
252C2×Dic47—21no—
253C2×Dic47—22no—
254C2×Dic47—22no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47C4×D412yes—
266C2×Dic47#12112yes—
267C2×Dic47C4⋊Q812yes—
268C2×Dic47AGL(1,ℤ/8)12yes—
269C2×Dic47#13712yes—
270C4×D47—44no—
271C4×D47—44no—
272C4×D47—48no—
273C4×D47—84no—
274C4×D47C2×C4⋊C414yes—
275C4×D47C4×Q814yes—
276C4×D47C22⋊Q814yes—
277C4×D47#12514yes—
278C8×C47—22no—
279C8×C47C2×C4⋊C414yescomplemented
280C47⋊C8—22no—
281C47⋊C8C2×C4⋊C414yescomplemented
282#21159—471nomaximal, Sylow
283C22×C47C2×C4×D411yescharacteristic
284C22×C47C2×#12111yescharacteristic
285C22×C47C2×C4⋊Q811yescharacteristic
286C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C47C2×#13711yescharacteristic
288C22×C47#57511yescharacteristic
289C22×C47#57511yescharacteristic
290C2×D47—22no—
291C2×D47—22no—
292C2×D47—22no—
293C2×D47—22no—
294C2×D47—22no—
295C2×D47—22no—
296C2×D47#38914yes—
297C4×C47—42no—
298C4×C47—42no—
299C4×C47—41no—
300C4×C47—41no—
301C4×C47—81no—
302C4×C47—81no—
303C4×C47C22×C4⋊C411yescharacteristic
304C4×C47C2×C4×Q811yescharacteristic
305C4×C47C2×C22⋊Q811yescharacteristic
306C4×C47C2×#12511yescharacteristic
307Dic47—22no—
308Dic47—22no—
309Dic47—42no—
310Dic47—42no—
311Dic47—81no—
312Dic47—81no—
313Dic47#38914yes—
314C2×#428—471no—
315C2×#428—471no—
316C2×C4×M16—471no—
317#2296—471no—
318C2×#389—471no—
319#1944—471no—
320#1944—471no—
321#1943—471no—
322#1943D4711yescharacteristic, complemented
323#1942—471no—
324#1942—471no—
325#1941—471no—
326#1941—471no—
327#1736—471no—
328#1736—471no—
329C2×C47C2×#38911yescharacteristic
330C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
331C2×C47C4×#13711yescharacteristic
332C2×C47#305511yescharacteristic
333C2×C47#305611yescharacteristic
334C2×C47#316511yescharacteristic
335C2×C47#316611yescharacteristic
336D47—24no—
337C22×M16—471no—
338C22×M16—471no—
339C2×#117—471no—
340C2×#117—471no—
341C22×C4⋊C4—471no—
342C22×C4⋊C4—471no—
343C22×C4⋊C4—941no—
344C22×C42—471no—
345#428—478no—
346#428—944no—
347C2×#107—471no—
348C2×#107—471no—
349C2×#107—471no—
350C2×#107C2×D4711yescharacteristic
351C2×#106—471no—
352C2×#106—471no—
353C2×#106—471no—
354C2×#106C2×D4711yescharacteristic
355C4×M16—944no—
356C2×#97—471no—
357C2×#97C2×D4711yescharacteristic
358C2×C4×C8—471no—
359C2×C4×C8—471no—
360#389—471no—
361#389—471no—
362#389—471no—
363#389—471no—
364#389—471no—
365#389—471no—
366#389—941no—
367#389—941no—
368#389C2×D4711yescharacteristic
369#389C2×D4711yescharacteristic
370#385—941no—
371#385—941no—
372C4×C4⋊C4—941no—
373C4×C4⋊C4—941no—
374C2×#95—941no—
375#340—471no—
376#340—471no—
377#340—471no—
378#340—471no—
379#340—471no—
380#340—471no—
381#340C2×D4711yescharacteristic
382#340C2×D4711yescharacteristic
383C47#2115911yescharacteristic, complemented, Sylow
384C23×C4—471no—
385C23×C4—471no—
386C23×C4—471no—
387C23×C4—941no—
388C23×C4—941no—
389C2×M16—474no—
390C2×M16—474no—
391C2×M16—474no—
392C2×M16—474no—
393C2×M16—942no—
394C2×M16—942no—
395C22×C8—471no—
396C22×C8—471no—
397C22×C8C22×D4711yescharacteristic
398C22×C8C22×D4711yescharacteristic
399#117—474no—
400#117—942no—
401#117—942no—
402#117—942no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—471no—
406C2×C4⋊C4—471no—
407C2×C4⋊C4—474no—
408C2×C4⋊C4—471no—
409C2×C4⋊C4—471no—
410C2×C4⋊C4—942no—
411C2×C4⋊C4—942no—
412C2×C4⋊C4—942no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4—941no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4—941no—
417C2×C4⋊C4—941no—
418C2×C4⋊C4—941no—
419C2×C4⋊C4—941no—
420C2×C4⋊C4—941no—
421C2×C4⋊C4—1884no—
422C2×C4⋊C4C22×D4711yescharacteristic
423C2×C4⋊C4C22×D4711yescharacteristic
424C2×C4⋊C4C22×D4711yescharacteristic
425C2×C4⋊C4C22×D4711yescharacteristic
426C2×C22⋊C4—941no—
427C2×C22⋊C4—941no—
428C2×C42—471no—
429C2×C42—471no—
430C2×C42—471no—
431C2×C42—942no—
432C2×C42—942no—
433C2×C42—941no—
434C2×C42—941no—
435C2×C42C22×D4711yescharacteristic
436#107—474no—
437#107—942no—
438#107—942no—
439#107C4×D4714yescomplemented
440#106—22no—
441#106—474no—
442#106—474no—
443#106—942no—
444#97—22no—
445#97—942no—
446C4×C8—942no—
447C4×C8—942no—
448#95—21no—
449#95—21no—
450#95—941no—
451#95—941no—
452#95—941no—
453#95—941no—
454#95—941no—
455#95—941no—
456C24—471no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—471no—
462C22×C4—472no—
463C22×C4—472no—
464C22×C4—472no—
465C22×C4—472no—
466C22×C4—472no—
467C22×C4—472no—
468C22×C4—471no—
469C22×C4—471no—
470C22×C4—942no—
471C22×C4—942no—
472C22×C4—942no—
473C22×C4—942no—
474C22×C4—944no—
475C22×C4—941no—
476C22×C4—941no—
477C22×C4—941no—
478C22×C4—941no—
479C22×C4—1881no—
480C22×C4—1881no—
481C22×C4—1882no—
482C22×C4—1882no—
483C22×C4—1882no—
484C22×C4—1882no—
485C22×C4—1882no—
486C22×C4D4×D4711yescharacteristic
487C22×C4D4×D4711yescharacteristic
488C22×C4C23×D4711yescharacteristic
489M16—4716no—
490M16—948no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—472no—
494C2×C8—472no—
495C2×C8—472no—
496C2×C8—472no—
497C2×C8—941no—
498C2×C8—941no—
499C2×C8C2×C4×D4712yes—
500C2×C8D4×D4712yes—
501C2×C8Q8×D4712yes—
502C2×C8Q8×D4712yes—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—472no—
512C4⋊C4—472no—
513C4⋊C4—944no—
514C4⋊C4—941no—
515C4⋊C4—941no—
516C4⋊C4—941no—
517C4⋊C4—941no—
518C4⋊C4—941no—
519C4⋊C4—941no—
520C4⋊C4—1882no—
521C4⋊C4—1882no—
522C4⋊C4—1882no—
523C4⋊C4—1882no—
524C4⋊C4—1882no—
525C4⋊C4—1882no—
526C4⋊C4—1882no—
527C4⋊C4C2×C4×D4712yes—
528C4⋊C4C2×C4×D4712yes—
529C22⋊C4—944no—
530C22⋊C4—1882no—
531C42—21no—
532C42—21no—
533C42—942no—
534C42—942no—
535C42—944no—
536C42—941no—
537C42—941no—
538C42—1882no—
539C23—472no—
540C23—472no—
541C23—472no—
542C23—941no—
543C23—941no—
544C23—941no—
545C23—941no—
546C23C2×D4×D4711yescharacteristic
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—472no—
566C2×C4—472no—
567C2×C4—472no—
568C2×C4—474no—
569C2×C4—474no—
570C2×C4—474no—
571C2×C4—474no—
572C2×C4—472no—
573C2×C4—472no—
574C2×C4—941no—
575C2×C4—941no—
576C2×C4—941no—
577C2×C4—941no—
578C2×C4—942no—
579C2×C4—942no—
580C2×C4—3764no—
581C2×C4—1884no—
582C2×C4—1884no—
583C2×C4—1888no—
584C2×C4—1881no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4—1881no—
589C2×C4—1881no—
590C2×C4—1881no—
591C2×C4—1881no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4C22×C4×D4711yescharacteristic
595C2×C4C2×D4×D4711yescharacteristic
596C2×C4C2×Q8×D4711yescharacteristic
597C2×C4C2×Q8×D4711yescharacteristic
598C2×C4#41×D4711yescharacteristic
599C2×C4#41×D4711yescharacteristic
600C8—22no—
601C8—474no—
602C8—942no—
603C8C4⋊C4×D4714yescomplemented
604C22—474no—
605C22—942no—
606C22—942no—
607C22—942no—
608C22—942no—
609C22—942no—
610C22—942no—
611C22C4×D4×D4711yescharacteristic
612C22#121×D4711yescharacteristic
613C22C4⋊Q8×D4711yescharacteristic
614C22AGL(1,ℤ/8)×D4711yescharacteristic
615C22#137×D4711yescharacteristic
616C22#1568991911yescharacteristic
617C22#1568992211yescharacteristic
618C4—42no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—474no—
625C4—942no—
626C4—942no—
627C4—3761no—
628C4—3761no—
629C4—1882no—
630C4—1882no—
631C4C2×C4⋊C4×D4711yescharacteristic
632C4C4×Q8×D4711yescharacteristic
633C4C22⋊Q8×D4711yescharacteristic
634C4#125×D4711yescharacteristic
635C2—944no—
636C2#389×D4711yescharacteristic
637C2#2025219911yescharacteristic
638C2#2025220411yescharacteristic
639C2#2025300211yescharacteristic
640C2#2025301311yescharacteristic
641C2#2025393311yescharacteristic
642C2#2025393911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1943C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1943D47
hypocenterC47#21159
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1943

Complements

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

NH ≅ G/Ntype
6. #20244131C2semidirect
7. #20244128C2semidirect
8. #20244123C2semidirect
9. #20244122C2semidirect
10. #20244120C2semidirect
11. #20244115C2semidirect
12. #20244114C2semidirect
15. C47×#1943C2semidirect
24. #15689164C4semidirect
107. #13093957C2×C4semidirect
108. #13093956C2×C4semidirect
110. #13093955C2×C4semidirect
129. #107×C47C2×C4semidirect
216. #10625324C4⋊C4semidirect
279. C8×C47C2×C4⋊C4semidirect
281. C47⋊C8C2×C4⋊C4semidirect
322. #1943D47semidirect
383. C47#21159semidirect
439. #107C4×D47semidirect
603. C8C4⋊C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156891642
C2×#156891622
C2×#156890752
#202458692
#389×D472
#202441312
#202441282
#202441232
#202441222
#202441202
#202441152
#202441142
#202431062
C47×#19432
#2115947