Finite Group Databasebeta

← #26214068

Subgroups — #26214068

Order 12032.

1706 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625354C211yesmaximal, characteristic
2C2×#15689123C212yesmaximal
3C2×#15689075C211yesmaximal, characteristic, complemented
4C4×C4⋊C4×D47C211yesmaximal, characteristic, complemented
5#20243218C211yesmaximal, characteristic, complemented
6#20243216C211yesmaximal, characteristic, complemented
7#20243214C211yesmaximal, characteristic, complemented
8#103×Dic47C212yesmaximal, complemented
9#20243202C212yesmaximal, complemented
10#20243106C212yesmaximal, complemented
11C47×#1763C211yesmaximal, characteristic, complemented
12C22×#10625354C2211yescharacteristic
13C22×#10625324C2212yes—
14C4×#10625354—24no—
15C4×#10625354C418yescomplemented
16C2×Q8×Dic47C2211yescharacteristic
17C2×C4×D188—21no—
18C2×C4×D188C2211yescharacteristic
19C2×#13093986—21no—
20C2×#13093986C2211yescharacteristic
21C2×#13093982—21no—
22C2×#13093982C2211yescharacteristic
23C2×C4⋊C4×D47C2212yes—
24C2×C42×D47—21no—
25C2×C42×D47—21no—
26C2×C42×D47C2211yescharacteristic
27#15689123C4116yescomplemented
28C2×#13093948C2212yes—
29C2×#13093946C2212yes—
30C2×#13093944C2212yes—
31#15689075—24no—
32C2×#13093935C2211yescharacteristic
33C2×C8×Dic47C2212yes—
34C4⋊C4×Dic47C2212yes—
35C4×Dic47⋊C4—21no—
36C4×C4⋊Dic47C2211yescharacteristic
37#15688974—21no—
38#95×D47—21no—
39#15688886C2212yes—
40#15688885C2212yes—
41#15688885C2212yes—
42C2×C4×Q8×C47C2211yescharacteristic
43C2×#103×C47C2212yes—
44C2×#97×C47C2211yescharacteristic
45C4×C4⋊C4×C47C2211yescharacteristic
46C47×#340C2212yes—
47C2×#10625354—22no—
48C2×#10625354—24no—
49C2×#10625354C2×C414yes—
50C2×#10625354C2×C418yes—
51C22×D188—21no—
52C22×D188C2311yescharacteristic
53C22×C4×D47—21no—
54C22×C4×D47—22no—
55C22×C4×D47—21no—
56C22×C4×D47—21no—
57C22×C4×D47D411yescharacteristic
58C22×C4×D47D411yescharacteristic
59C22×C4×D47C2311yescharacteristic
60C2×#10625324—24no—
61C2×#10625324—24no—
62C2×#10625324C2×C418yescomplemented
63C22×C47⋊C8C2312yes—
64Q8×Dic47—22no—
65Q8×Dic47C2×C414yescomplemented
66C4×D188—28no—
67C4×D188—22no—
68C4×D188C2×C414yescomplemented
69#13093986—28no—
70#13093986—22no—
71#13093986C2×C414yescomplemented
72#13093982—22no—
73#13093982—28no—
74#13093982C2×C414yes—
75C2×C4⋊Dic47—21no—
76C2×C4⋊Dic47C2312yes—
77C2×C4⋊Dic47C2311yescharacteristic
78C2×Dic47⋊C4—22no—
79C4⋊C4×D47—24no—
80C4⋊C4×D47C2×C418yescomplemented
81C2×(C2×D47)⋊C4—21no—
82C2×(C2×D47)⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47—22no—
88C2×C4×Dic47D411yescharacteristic
89C2×C4×Dic47D411yescharacteristic
90C2×C4×Dic47C2311yescharacteristic
91C42×D47—28no—
92C42×D47—44no—
93C42×D47C2×C414yes—
94C42×D47D414yes—
95#13093948C2×C418yescomplemented
96#13093946C2×C418yescomplemented
97#13093944C2×C418yescomplemented
98#13093935—22no—
99C8×Dic47—24no—
100C47⋊#95—21no—
101C47⋊#95—22no—
102C22×Q8×C47C2311yescharacteristic
103C22×C8×C47C2312yes—
104C4×Q8×C47—22no—
105C4×Q8×C47C2×C414yescomplemented
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47C2312yes—
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C42×C47—21no—
110C2×C42×C47—21no—
111C2×C42×C47C2311yescharacteristic
112#103×C47C2×C418yescomplemented
113#97×C47—22no—
114#95×C47—21no—
115C23×D47—21no—
116C23×D47C2×D411yescharacteristic
117C2×Q8×C47—21no—
118C2×Q8×C47—22no—
119C2×Q8×C47C22×C414yes—
120C2×Q8×C47C22×C412yes—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—22no—
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2411yescharacteristic
127C22×Dic47—22no—
128C22×Dic47—21no—
129C22×Dic47C2×D411yescharacteristic
130C22×Dic47C2×D411yescharacteristic
131C22×Dic47C2×D411yescharacteristic
132#10625354—28no—
133#10625354—216no—
134#10625354C42116yescomplemented
135C2×D188—24no—
136C2×D188—28no—
137C2×D188—21no—
138C2×D188—22no—
139C2×D188C22×C414yes—
140C2×D188C22×C412yes—
141C2×C4×D47—24no—
142C2×C4×D47—28no—
143C2×C4×D47—22no—
144C2×C4×D47—22no—
145C2×C4×D47—28no—
146C2×C4×D47—24no—
147C2×C4×D47—24no—
148C2×C4×D47—42no—
149C2×C4×D47—44no—
150C2×C4×D47—44no—
151C2×C4×D47—44no—
152C2×C4×D47—42no—
153C2×C4×D47—42no—
154C2×C4×D47C22⋊C414yes—
155C2×C4×D47C22⋊C414yes—
156C2×C4×D47C22×C412yes—
157C2×C4×D47C22×C414yes—
158C2×C4×D47C2×D412yes—
159C2×C4×D47#4114yes—
160C2×C8×C47—22no—
161C2×C8×C47—22no—
162C2×C8×C47C22×C414yes—
163C4⋊C4×C47—24no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—22no—
166C4⋊C4×C47C22×C414yes—
167C4⋊C4×C47C22×C412yes—
168C42×C47—24no—
169C42×C47—42no—
170C42×C47C22×C412yes—
171C42×C47C2×D412yes—
172(C2×D47)⋊C4—24no—
173(C2×D47)⋊C4—24no—
174(C2×D47)⋊C4—24no—
175C4⋊Dic47—21no—
176C4⋊Dic47—22no—
177C4⋊Dic47—24no—
178C4⋊Dic47C22×C414yes—
179C4⋊Dic47C22×C412yes—
180Dic47⋊C4—28no—
181C4×Dic47—24no—
182C4×Dic47—24no—
183C4×Dic47—42no—
184C4×Dic47—44no—
185C4×Dic47C22⋊C414yes—
186C4×Dic47C22⋊C414yes—
187C4×Dic47C22×C412yes—
188C4×Dic47C2×D412yes—
189C2×C47⋊C8—22no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8C22×C414yes—
192#10625324—216no—
193C23×C47C22×D411yescharacteristic
194C22×D47—21no—
195C22×D47—21no—
196C22×D47—22no—
197C22×D47—22no—
198C22×D47—24no—
199C22×D47—41no—
200C22×D47—41no—
201C22×D47—42no—
202C22×D47C2×C22⋊C412yes—
203C22×D47C4×D414yes—
204Q8×C47—22no—
205Q8×C47—24no—
206Q8×C47C2×C4214yescomplemented
207C2×C4×C47—22no—
208C2×C4×C47—24no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—24no—
214C2×C4×C47—42no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—42no—
218C2×C4×C47C2×C22⋊C412yes—
219C2×C4×C47C2×C22⋊C412yes—
220C2×C4×C47C23×C411yescharacteristic
221C2×C4×C47C23×C412yes—
222C2×C4×C47C22×D411yescharacteristic
223C2×C4×C47C2×#4112yes—
224C2×Dic47—22no—
225C2×Dic47—22no—
226C2×Dic47—24no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47C2×C22⋊C412yes—
231C2×Dic47C2×C22⋊C412yes—
232C2×Dic47C2×C22⋊C412yes—
233C2×Dic47C4×D414yes—
234C2×Dic47AGL(1,ℤ/8)12yes—
235C2×Dic47#13712yes—
236D188—216no—
237D188—22no—
238D188—24no—
239D188C2×C4214yescomplemented
240C4×D47—216no—
241C4×D47—28no—
242C4×D47—44no—
243C4×D47—48no—
244C4×D47—416no—
245C4×D47—48no—
246C4×D47C2×C4214yes—
247C4×D47#11714yes—
248C4×D47C4×D418yes—
249C8×C47—24no—
250C47⋊C8—24no—
251C4×#417—471nomaximal, Sylow
252C22×C47C22×C22⋊C411yescharacteristic
253C22×C47C2×C4×D412yes—
254C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C47C2×#13711yescharacteristic
256C22×C47#57512yes—
257C2×D47—22no—
258C2×D47—22no—
259C2×D47—24no—
260C2×D47—24no—
261C2×D47—24no—
262C2×D47—42no—
263C2×D47—42no—
264C2×D47—44no—
265C2×D47—44no—
266C2×D47C4×C22⋊C414yes—
267C4×C47—22no—
268C4×C47—22no—
269C4×C47—24no—
270C4×C47—42no—
271C4×C47—44no—
272C4×C47C22×C4211yescharacteristic
273C4×C47C2×#11711yescharacteristic
274C4×C47C2×C4×D412yes—
275Dic47—22no—
276Dic47—42no—
277Dic47—44no—
278Dic47C4×C22⋊C414yes—
279Dic47#41714yescomplemented
280C2×C4×#41—471no—
281C2×#417—472no—
282C2×C4×M16—471no—
283C2×C4×C4⋊C4—471no—
284#1763—471no—
285#1763D4711yescharacteristic, complemented
286#1762—471no—
287#1762—471no—
288C4×#103—472no—
289C4×#102—472no—
290#1736—472no—
291C2×C47C2×C4×C22⋊C411yescharacteristic
292C2×C47C2×#41711yescharacteristic
293C2×C47#289611yescharacteristic
294C2×C47C4×AGL(1,ℤ/8)12yes—
295C2×C47C4×#13712yes—
296D47—24no—
297D47—44no—
298C22×#41—471no—
299C22×M16—472no—
300C4×#41—478no—
301C4×#41—944no—
302C2×C4×Q8—471no—
303C2×C4×Q8C2×D4711yescharacteristic
304C2×C4×D4—471no—
305C2×C4×D4—471no—
306C2×C4×D4—941no—
307C2×C4×D4—941no—
308C2×#117—471no—
309C2×#117—941no—
310C22×C4⋊C4—472no—
311C22×C42—471no—
312C22×C42—941no—
313C22×C42—941no—
314#417—4716no—
315C2×#103—472no—
316C2×#103C2×D4712yes—
317C2×#102—472no—
318C2×#102—472no—
319C4×M16—944no—
320C2×#97—471no—
321C2×#97C2×D4711yescharacteristic
322C2×C4×C8—472no—
323C4×C4⋊C4—472no—
324C4×C4⋊C4—471no—
325C4×C4⋊C4—941no—
326C4×C4⋊C4—941no—
327C4×C4⋊C4C2×D4711yescharacteristic
328C2×#95—941no—
329#340—472no—
330#340—472no—
331#340—472no—
332#340C2×D4712yes—
333C47C4×#41711yescharacteristic, complemented, Sylow
334C2×#41—474no—
335C2×#41—478no—
336C2×#41—942no—
337C2×#41—944no—
338C22×Q8C22×D4711yescharacteristic
339C22×D4—471no—
340C22×D4—941no—
341C23×C4—471no—
342C23×C4—471no—
343C23×C4—471no—
344C23×C4—941no—
345C23×C4—941no—
346C23×C4—942no—
347C23×C4—941no—
348C2×M16—478no—
349C2×M16—944no—
350C2×M16—944no—
351C22×C8—472no—
352C22×C8C22×D4712yes—
353C4×Q8—22no—
354C4×Q8—474no—
355C4×Q8—942no—
356C4×Q8C4×D4714yescomplemented
357C4×D4—474no—
358C4×D4—474no—
359C4×D4—948no—
360C4×D4—942no—
361C4×D4—948no—
362C4×D4—942no—
363#117—474no—
364#117—948no—
365#117—942no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—472no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—478no—
370C2×C4⋊C4—944no—
371C2×C4⋊C4—942no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4C22×D4712yes—
374C2×C4⋊C4C22×D4711yescharacteristic
375C2×C22⋊C4—941no—
376C2×C22⋊C4—941no—
377C2×C22⋊C4—941no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—471no—
381C2×C42—471no—
382C2×C42—471no—
383C2×C42—474no—
384C2×C42—474no—
385C2×C42—948no—
386C2×C42—941no—
387C2×C42—941no—
388C2×C42—942no—
389C2×C42—941no—
390C2×C42—1884no—
391C2×C42C22×D4711yescharacteristic
392#103—478no—
393#103C4×D4718yescomplemented
394#102—478no—
395#102—478no—
396#97—22no—
397#97—942no—
398C4×C8—944no—
399#95—21no—
400#95—942no—
401#95—941no—
402C24—471no—
403C24—941no—
404#41—4716no—
405#41—948no—
406#41—9416no—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8C2×C4×D4714yes—
410C2×Q8C2×C4×D4712yes—
411C2×D4—474no—
412C2×D4—472no—
413C2×D4—944no—
414C2×D4—948no—
415C2×D4—941no—
416C2×D4—942no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—471no—
421C22×C4—471no—
422C22×C4—472no—
423C22×C4—474no—
424C22×C4—472no—
425C22×C4—474no—
426C22×C4—471no—
427C22×C4—474no—
428C22×C4—474no—
429C22×C4—944no—
430C22×C4—942no—
431C22×C4—942no—
432C22×C4—948no—
433C22×C4—944no—
434C22×C4—948no—
435C22×C4—942no—
436C22×C4—941no—
437C22×C4—944no—
438C22×C4—1882no—
439C22×C4—1884no—
440C22×C4—1882no—
441C22×C4—1882no—
442C22×C4—1884no—
443C22×C4—1884no—
444C22×C4D4×D4711yescharacteristic
445C22×C4D4×D4711yescharacteristic
446C22×C4C23×D4711yescharacteristic
447M16—9416no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—474no—
451C2×C8—942no—
452C2×C8—942no—
453C2×C8C2×C4×D4714yes—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—474no—
458C4⋊C4—472no—
459C4⋊C4—948no—
460C4⋊C4—944no—
461C4⋊C4—941no—
462C4⋊C4—942no—
463C4⋊C4C2×C4×D4714yes—
464C4⋊C4C2×C4×D4712yes—
465C22⋊C4—944no—
466C22⋊C4—944no—
467C22⋊C4—944no—
468C42—24no—
469C42—42no—
470C42—474no—
471C42—474no—
472C42—472no—
473C42—472no—
474C42—944no—
475C42—944no—
476C42—1882no—
477C42—1884no—
478C42C2×C4×D4712yes—
479C42D4×D4712yes—
480C23—472no—
481C23—474no—
482C23—941no—
483C23—941no—
484C23—942no—
485C23—942no—
486C23—944no—
487C23—1881no—
488C23—1881no—
489C23—1882no—
490C23C2×D4×D4711yescharacteristic
491Q8—22no—
492Q8—24no—
493Q8C42×D4714yescomplemented
494D4—474no—
495D4—9416no—
496D4—942no—
497D4—944no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—472no—
510C2×C4—472no—
511C2×C4—474no—
512C2×C4—472no—
513C2×C4—472no—
514C2×C4—474no—
515C2×C4—474no—
516C2×C4—478no—
517C2×C4—472no—
518C2×C4—948no—
519C2×C4—9416no—
520C2×C4—942no—
521C2×C4—942no—
522C2×C4—944no—
523C2×C4—1884no—
524C2×C4—1888no—
525C2×C4—1888no—
526C2×C4—18816no—
527C2×C4—1882no—
528C2×C4—1882no—
529C2×C4—1882no—
530C2×C4C22⋊C4×D4712yes—
531C2×C4C22⋊C4×D4712yes—
532C2×C4C22×C4×D4711yescharacteristic
533C2×C4C22×C4×D4712yes—
534C2×C4C2×D4×D4711yescharacteristic
535C2×C4#41×D4712yes—
536C8—24no—
537C8—944no—
538C22—474no—
539C22—942no—
540C22—944no—
541C22—944no—
542C22—942no—
543C22—944no—
544C22—1882no—
545C22—1884no—
546C22—1884no—
547C22—1882no—
548C22C2×C22⋊C4×D4711yescharacteristic
549C22C4×D4×D4712yes—
550C22AGL(1,ℤ/8)×D4711yescharacteristic
551C22#137×D4711yescharacteristic
552C22#1568992212yes—
553C4—22no—
554C4—24no—
555C4—22no—
556C4—44no—
557C4—42no—
558C4—474no—
559C4—474no—
560C4—942no—
561C4—1882no—
562C4—1884no—
563C4C2×C42×D4711yescharacteristic
564C4#117×D4711yescharacteristic
565C4C4×D4×D4712yes—
566C2—944no—
567C2—1884no—
568C2C4×C22⋊C4×D4711yescharacteristic
569C2#417×D4711yescharacteristic
570C2#2025190611yescharacteristic
571C2#2025219312yes—
572C2#2025220412yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC47×#1763C2
socleC23×C47C22×D4
prime radicalC22×#10625354C22
cosocleC22×C4×C47C24
hypercenter#1763D47
hypocenterC47C4×#417
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
3. C2×#15689075C2semidirect
4. C4×C4⋊C4×D47C2semidirect
5. #20243218C2semidirect
6. #20243216C2semidirect
7. #20243214C2semidirect
8. #103×Dic47C2semidirect
9. #20243202C2semidirect
10. #20243106C2semidirect
11. C47×#1763C2semidirect
15. C4×#10625354C4semidirect
27. #15689123C4semidirect
62. C2×#10625324C2×C4semidirect
65. Q8×Dic47C2×C4semidirect
68. C4×D188C2×C4semidirect
71. #13093986C2×C4semidirect
80. C4⋊C4×D47C2×C4semidirect
95. #13093948C2×C4semidirect
96. #13093946C2×C4semidirect
97. #13093944C2×C4semidirect
105. C4×Q8×C47C2×C4semidirect
112. #103×C47C2×C4semidirect
134. #10625354C42semidirect
206. Q8×C47C2×C42semidirect
239. D188C2×C42semidirect
279. Dic47#417semidirect
285. #1763D47semidirect
333. C47C4×#417semidirect
356. C4×Q8C4×D47semidirect
393. #103C4×D47semidirect
493. Q8C42×D47semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#106253542
C2×#156891232
C2×#156890752
C4×C4⋊C4×D472
#202432182
#202432162
#202432142
#103×Dic472
#202432022
#202431062
C47×#17632
C4×#41747