Finite Group Databasebeta

← #26220797

Subgroups — #26220797

Order 12032.

1250 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689676C211yesmaximal, characteristic
2C2×#15689146C211yesmaximal, characteristic
3C2×C4⋊C4×Dic47C211yesmaximal, characteristic
4#20244344C212yesmaximal, complemented
5#20244342C212yesmaximal, complemented
6#20244338C212yesmaximal, complemented
7#20244334C212yesmaximal, complemented
8#20243109C211yesmaximal, characteristic
9#20243108C211yesmaximal, characteristic
10#20243107C211yesmaximal, characteristic
11C47×#1736C211yesmaximal, characteristic
12C22×#10625329C2211yescharacteristic
13#15689676—24no—
14#15689676C418yescomplemented
15C2×#13094023C2211yescharacteristic
16C2×#13094023C2211yescharacteristic
17C2×#13094015C2211yescharacteristic
18C2×#13094015C2211yescharacteristic
19C2×#13094001—21no—
20C2×#13094001—21no—
21C2×C4×Dic94—21no—
22C2×C4×Dic94—21no—
23C2×#13093983C2211yescharacteristic
24C22×C4⋊Dic47C2211yescharacteristic
25C22×C4×Dic47—21no—
26C22×C4×Dic47C2211yescharacteristic
27#15689146—24no—
28C2×Dic47⋊C8C2212yes—
29C2×Dic47⋊C8C2212yes—
30C4⋊C4×Dic47—21no—
31C4⋊C4×Dic47—21no—
32C4⋊C4×Dic47C2212yes—
33C4⋊C4×Dic47C2212yes—
34C2×C47⋊#95—21no—
35#15688886C2212yes—
36#15688886C2212yes—
37#15688885C2212yes—
38#15688885C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C22×M16×C47C2211yescharacteristic
42C2×#117×C47C2211yescharacteristic
43C22×C4⋊C4×C47C2211yescharacteristic
44C47×#340C2212yes—
45C47×#340C2212yes—
46C22×Dic94—21no—
47C22×Dic94—21no—
48C23×Dic47—21no—
49C23×Dic47D411yescharacteristic
50C23×Dic47D411yescharacteristic
51C2×#10625329—22no—
52C2×#10625329—22no—
53C2×#10625329—22no—
54C22×C47⋊C8C2312yes—
55#13094023—22no—
56#13094023—22no—
57#13094023C2×C414yescomplemented
58#13094023C2×C414yescomplemented
59#13094015—22no—
60#13094015—22no—
61#13094015C2×C414yescomplemented
62#13094015C2×C414yescomplemented
63#13094001—28no—
64#13094001—28no—
65C4×Dic94—28no—
66C4×Dic94—28no—
67#13093983—22no—
68#13093983C2×C414yes—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47—22no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47Q814yescomplemented
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2312yes—
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×Dic47⋊C4—21no—
81C2×C22⋊Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—22no—
84C2×C4×Dic47—22no—
85C2×C4×Dic47—44no—
86C2×C4×Dic47C2×C414yes—
87C2×C4×Dic47D411yescharacteristic
88C2×C4×Dic47D411yescharacteristic
89C2×C4×Dic47D414yes—
90C2×C4×Dic47C2311yescharacteristic
91C2×C4×Dic47C2311yescharacteristic
92Dic47⋊C8—24no—
93Dic47⋊C8—24no—
94C47⋊#95—22no—
95C47⋊#95—22no—
96C23×C4×C47—21no—
97C23×C4×C47C2311yescharacteristic
98C2×M16×C47—22no—
99C2×M16×C47—22no—
100C2×M16×C47—22no—
101C22×C8×C47C2312yes—
102#117×C47—22no—
103#117×C47C2×C414yes—
104C2×C4⋊C4×C47—21no—
105C2×C4⋊C4×C47—22no—
106C2×C4⋊C4×C47Q814yescomplemented
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C4⋊C4×C47C2312yes—
110C2×C22⋊C4×C47—21no—
111C2×C42×C47—21no—
112C24×C47C2×D411yescharacteristic
113C22×C4×C47—24no—
114C22×C4×C47—21no—
115C22×C4×C47—21no—
116C22×C4×C47—22no—
117C22×C4×C47—42no—
118C22×C4×C47—42no—
119C22×C4×C47C22×C412yes—
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D412yes—
122C22×C4×C47C2×Q812yes—
123C22×C4×C47C2×Q812yes—
124C22×C4×C47#4112yes—
125C22×C4×C47#4112yes—
126C22×C4×C47C2411yescharacteristic
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—24no—
130C22×Dic47—22no—
131C22×Dic47—21no—
132C22×Dic47—21no—
133C22×Dic47—41no—
134C22×Dic47—41no—
135C22×Dic47—42no—
136C22×Dic47—42no—
137C22×Dic47C22⋊C414yes—
138C22×Dic47C22⋊C414yes—
139C22×Dic47C2×D411yescharacteristic
140C22×Dic47C2×D411yescharacteristic
141C22×Dic47C2×D411yescharacteristic
142C22×Dic47C2×D411yescharacteristic
143C2×Dic94—24no—
144C2×Dic94—24no—
145C2×Dic94—42no—
146C2×Dic94—42no—
147C2×Dic94—42no—
148C2×Dic94—42no—
149M16×C47—44no—
150C2×C8×C47—22no—
151C2×C8×C47—22no—
152C2×C8×C47—22no—
153C4⋊C4×C47—24no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47C22×C412yes—
158C4⋊C4×C47C22×C412yes—
159C4⋊C4×C47C2×Q814yes—
160C22⋊C4×C47—24no—
161C42×C47—24no—
162C22⋊Dic47—24no—
163C4⋊Dic47—24no—
164C4⋊Dic47—21no—
165C4⋊Dic47—21no—
166C4⋊Dic47—22no—
167C4⋊Dic47—24no—
168C4⋊Dic47—24no—
169C4⋊Dic47C22×C412yes—
170C4⋊Dic47C22×C412yes—
171C4⋊Dic47C2×Q814yes—
172Dic47⋊C4—24no—
173Dic47⋊C4—24no—
174Dic47⋊C4—24no—
175Dic47⋊C4—24no—
176C4×Dic47—24no—
177C4×Dic47—44no—
178C4×Dic47—44no—
179C4×Dic47C22⋊C414yes—
180C4×Dic47C22⋊C414yes—
181C4×Dic47C22×C412yes—
182C4×Dic47C22×C412yes—
183C4×Dic47C2×D412yes—
184C4×Dic47C2×D412yes—
185#10625329—44no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—22no—
189C23×C47—21no—
190C23×C47—21no—
191C23×C47—21no—
192C23×C47—21no—
193C23×C47C2×C22⋊C412yes—
194C23×C47C22⋊Q812yes—
195C23×C47C22⋊Q812yes—
196C23×C47C22×D411yescharacteristic
197C2×C4×C47—22no—
198C2×C4×C47—22no—
199C2×C4×C47—24no—
200C2×C4×C47—48no—
201C2×C4×C47—42no—
202C2×C4×C47—41no—
203C2×C4×C47—41no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47—42no—
207C2×C4×C47C2×C22⋊C412yes—
208C2×C4×C47C4×Q814yes—
209C2×C4×C47C4×Q814yes—
210C2×C4×C47C22⋊Q812yes—
211C2×C4×C47C22⋊Q812yes—
212C2×C4×C47#12414yes—
213C2×C4×C47C4⋊Q814yes—
214C2×C4×C47C23×C411yescharacteristic
215C2×C4×C47C22×D411yescharacteristic
216C2×C4×C47C22×Q811yescharacteristic
217C2×C4×C47C22×Q811yescharacteristic
218C2×C4×C47C2×#4111yescharacteristic
219C2×C4×C47C2×#4111yescharacteristic
220C2×Dic47—21no—
221C2×Dic47—21no—
222C2×Dic47—21no—
223C2×Dic47—21no—
224C2×Dic47—21no—
225C2×Dic47—21no—
226C2×Dic47—28no—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—24no—
230C2×Dic47—44no—
231C2×Dic47—48no—
232C2×Dic47—42no—
233C2×Dic47—41no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237C2×Dic47—42no—
238C2×Dic47C2×C22⋊C412yes—
239C2×Dic47C2×C22⋊C412yes—
240C2×Dic47C2×C22⋊C412yes—
241C2×Dic47C2×C22⋊C412yes—
242C2×Dic47AGL(1,ℤ/8)12yes—
243C2×Dic47#13712yes—
244Dic94—48no—
245Dic94—48no—
246C8×C47—42no—
247C47⋊C8—42no—
248#21312—471nomaximal, Sylow
249C22×C47—22no—
250C22×C47—22no—
251C22×C47—22no—
252C22×C47—22no—
253C22×C47—22no—
254C22×C47—22no—
255C22×C47#39114yes—
256C22×C47C22×C22⋊C411yescharacteristic
257C22×C47C2×C22⋊Q811yescharacteristic
258C22×C47C2×C22⋊Q811yescharacteristic
259C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C47C2×#13711yescharacteristic
261C22×C47#57511yescharacteristic
262C22×C47#57511yescharacteristic
263C4×C47—44no—
264C4×C47—42no—
265C4×C47—42no—
266C4×C47#39114yes—
267C4×C47C2×C4×Q811yescharacteristic
268C4×C47C2×C4×Q811yescharacteristic
269C4×C47C2×#12411yescharacteristic
270C4×C47C2×C4⋊Q811yescharacteristic
271Dic47—22no—
272Dic47—22no—
273Dic47—22no—
274Dic47—44no—
275Dic47—42no—
276Dic47—42no—
277Dic47#41714yes—
278C2×#533—471no—
279C2×#423—471no—
280C2×C4×C4⋊C4—471no—
281#1989—472no—
282#1989—472no—
283#1988—472no—
284#1987—472no—
285#1736—471no—
286#1736—471no—
287#1736—471no—
288#1736D4711yescharacteristic
289C2×C47—24no—
290C2×C47C2×#39111yescharacteristic
291C2×C47C2×#41711yescharacteristic
292C2×C47#289611yescharacteristic
293C2×C47#307711yescharacteristic
294C2×C47#307711yescharacteristic
295C2×C47#307811yescharacteristic
296C2×C47#307811yescharacteristic
297C22×M16—471no—
298C22×M16C2×D4711yescharacteristic
299#533—478no—
300#533—944no—
301C2×C4⋊Q8—471no—
302C2×C4⋊Q8—471no—
303C2×#125—471no—
304C2×#125—471no—
305C2×C22⋊Q8—941no—
306C2×C22⋊Q8—941no—
307C2×C4×Q8—941no—
308C2×C4×Q8—941no—
309C2×#117—471no—
310C2×#117C2×D4711yescharacteristic
311C22×C4⋊C4—471no—
312C22×C4⋊C4C2×D4711yescharacteristic
313C22×C42—471no—
314C22×C42—941no—
315#423—944no—
316C2×C4⋊C8—472no—
317C2×C4⋊C8—472no—
318C4×C4⋊C4—472no—
319C4×C4⋊C4—472no—
320C4×C4⋊C4—941no—
321C4×C4⋊C4—941no—
322C2×#95—941no—
323#340—472no—
324#340—472no—
325#340—472no—
326#340—472no—
327#340—472no—
328#340—472no—
329#340C2×D4712yes—
330#340C2×D4712yes—
331C47#2131211yescharacteristic, complemented, Sylow
332C22×Q8—941no—
333C22×Q8—941no—
334C23×C4—21no—
335C23×C4—471no—
336C23×C4—471no—
337C23×C4—941no—
338C23×C4C22×D4711yescharacteristic
339C2×M16—22no—
340C2×M16—22no—
341C2×M16—22no—
342C2×M16—942no—
343C2×M16—942no—
344C2×M16—942no—
345C22×C8—472no—
346C22×C8C22×D4712yes—
347C4⋊Q8—474no—
348C4⋊Q8—474no—
349C4⋊Q8—942no—
350C4⋊Q8—942no—
351#125—474no—
352#125—474no—
353#125—942no—
354#125—942no—
355C22⋊Q8—948no—
356C22⋊Q8—948no—
357C4×Q8—948no—
358C4×Q8—948no—
359#117—22no—
360#117—474no—
361#117—942no—
362#117C4×D4714yes—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—474no—
366C2×C4⋊C4—472no—
367C2×C4⋊C4—471no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—941no—
370C2×C4⋊C4—941no—
371C2×C4⋊C4—941no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—942no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4Dic9414yescomplemented
378C2×C4⋊C4C22×D4712yes—
379C2×C4⋊C4C22×D4711yescharacteristic
380C2×C4⋊C4C22×D4711yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—941no—
383C2×C42—21no—
384C2×C42—474no—
385C2×C42—471no—
386C2×C42—471no—
387C2×C42—474no—
388C2×C42—471no—
389C2×C42—471no—
390C2×C42—941no—
391C2×C42—942no—
392C2×C42—942no—
393C2×C42—1884no—
394C4⋊C8—944no—
395C4⋊C8—944no—
396#95—942no—
397#95—942no—
398C24D4×D4711yescharacteristic
399C2×Q8—944no—
400C2×Q8—944no—
401C2×Q8—1882no—
402C2×Q8—1882no—
403C2×Q8—1882no—
404C2×Q8—1882no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—471no—
412C22×C4—474no—
413C22×C4—471no—
414C22×C4—471no—
415C22×C4—471no—
416C22×C4—474no—
417C22×C4—942no—
418C22×C4—942no—
419C22×C4—944no—
420C22×C4—942no—
421C22×C4—941no—
422C22×C4—941no—
423C22×C4—1881no—
424C22×C4—1881no—
425C22×C4—1882no—
426C22×C4—1882no—
427C22×C4C2×Dic9412yes—
428C22×C4C2×C4×D4712yes—
429C22×C4#1062535012yes—
430C22×C4D4×D4711yescharacteristic
431C22×C4D4×D4712yes—
432C22×C4#1062535212yes—
433C22×C4Q8×D4712yes—
434C22×C4C23×D4711yescharacteristic
435M16—44no—
436M16—1884no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—942no—
441C2×C8—942no—
442C2×C8—942no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—472no—
448C4⋊C4—472no—
449C4⋊C4—474no—
450C4⋊C4—944no—
451C4⋊C4—944no—
452C4⋊C4—944no—
453C4⋊C4—944no—
454C4⋊C4—944no—
455C4⋊C4—944no—
456C4⋊C4—944no—
457C4⋊C4—941no—
458C4⋊C4—941no—
459C4⋊C4—942no—
460C4⋊C4C2×Dic9414yes—
461C4⋊C4C2×C4×D4712yes—
462C4⋊C4C2×C4×D4712yes—
463C22⋊C4—24no—
464C22⋊C4—944no—
465C42—24no—
466C42—474no—
467C42—474no—
468C42—472no—
469C42—472no—
470C42—472no—
471C42—472no—
472C42—944no—
473C42—1884no—
474C42—1884no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23C22⋊C4×D4712yes—
480C23#1309400112yes—
481C23#1309400212yes—
482C23C2×D4×D4711yescharacteristic
483Q8—1888no—
484Q8—1888no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—48no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—472no—
496C2×C4—472no—
497C2×C4—472no—
498C2×C4—472no—
499C2×C4—472no—
500C2×C4—472no—
501C2×C4—948no—
502C2×C4—941no—
503C2×C4—941no—
504C2×C4—941no—
505C2×C4—941no—
506C2×C4—941no—
507C2×C4—941no—
508C2×C4—942no—
509C2×C4—942no—
510C2×C4—944no—
511C2×C4—1884no—
512C2×C4—1888no—
513C2×C4—1882no—
514C2×C4—1881no—
515C2×C4—1881no—
516C2×C4—1881no—
517C2×C4—1882no—
518C2×C4—1881no—
519C2×C4C22⋊C4×D4712yes—
520C2×C4#1309398914yes—
521C2×C4C4×Dic9414yes—
522C2×C4#1309400112yes—
523C2×C4#1309400212yes—
524C2×C4#1309401014yes—
525C2×C4#1309402314yes—
526C2×C4C22×C4×D4711yescharacteristic
527C2×C4C2×D4×D4711yescharacteristic
528C2×C4C2×Q8×D4711yescharacteristic
529C2×C4C22×Dic9411yescharacteristic
530C2×C4C2×#1062535211yescharacteristic
531C2×C4C2×#1062535011yescharacteristic
532C8—42no—
533C8—1882no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22#1568902714yes—
541C22C2×C22⋊C4×D4711yescharacteristic
542C22C2×#1309400111yescharacteristic
543C22C2×#1309400211yescharacteristic
544C22AGL(1,ℤ/8)×D4711yescharacteristic
545C22#137×D4711yescharacteristic
546C22#1568991911yescharacteristic
547C22#1568992211yescharacteristic
548C4—42no—
549C4—42no—
550C4—44no—
551C4—474no—
552C4—942no—
553C4—942no—
554C4—942no—
555C4—1882no—
556C4—1884no—
557C4—1882no—
558C4#1568902714yes—
559C4C2×#1309398911yescharacteristic
560C4C2×C4×Dic9411yescharacteristic
561C4C2×#1309401011yescharacteristic
562C4C2×#1309402311yescharacteristic
563C2—24no—
564C2C2×#1568902711yescharacteristic
565C2#417×D4711yescharacteristic
566C2#2025190411yescharacteristic
567C2#2025318211yescharacteristic
568C2#2025318411yescharacteristic
569C2#2025319011yescharacteristic
570C2#2025319211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1736C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#1736D47
hypocenterC47#21312
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #20244344C2semidirect
5. #20244342C2semidirect
6. #20244338C2semidirect
7. #20244334C2semidirect
14. #15689676C4semidirect
57. #13094023C2×C4semidirect
58. #13094023C2×C4semidirect
61. #13094015C2×C4semidirect
62. #13094015C2×C4semidirect
73. C2×C4⋊Dic47Q8semidirect
106. C2×C4⋊C4×C47Q8semidirect
331. C47#21312semidirect
377. C2×C4⋊C4Dic94semidirect

Maximal subgroups

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

subgroupindex
C2×#156896762
C2×#156891462
C2×C4⋊C4×Dic472
#202443442
#202443422
#202443382
#202443342
#202431092
#202431082
#202431072
C47×#17362
#2131247