Finite Group Databasebeta

← C4×#15689116 #26214026

Subgroups — C4×#15689116 #26214026

Order 12032.

1670 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625352C211yesmaximal, characteristic
2C2×#15689116C212yesmaximal
3C2×C4×C8×D47C211yesmaximal, characteristic, complemented
4C4×#13093982C211yesmaximal, characteristic, complemented
5C4×#13093947C211yesmaximal, characteristic, complemented
6C4×#13093949C211yesmaximal, characteristic, complemented
7#20243207C212yesmaximal, complemented
8#102×Dic47C212yesmaximal, complemented
9C4×#13093945C211yesmaximal, characteristic, complemented
10#20243103C212yesmaximal, complemented
11C4×#102×C47C211yesmaximal, characteristic, complemented
12C22×#10625352C2211yescharacteristic
13C22×C8×D47C2212yes—
14C4×#10625352—24no—
15C4×#10625352C418yescomplemented
16C2×C4×Dic94C2211yescharacteristic
17C2×#13093989C2211yescharacteristic
18C2×D4×Dic47C2211yescharacteristic
19C2×C4×C22⋊D47—21no—
20C2×(C22⋊D47)⋊C4—21no—
21C2×#13093982C2211yescharacteristic
22C2×#13093982C2212yes—
23C2×#13093980—21no—
24C22×C4×Dic47—21no—
25C2×C42×D47C2211yescharacteristic
26#15689116C4116yesdirect product factor
27C2×#13093949C2212yes—
28C2×#13093947C2212yes—
29C2×#13093945C2212yes—
30C2×C4×C47⋊C8C2211yescharacteristic
31C2×C8×Dic47C2212yes—
32C4×C8×D47—24no—
33C4⋊C4×Dic47C2212yes—
34C4×C4⋊Dic47C2211yescharacteristic
35C4×(C2×D47)⋊C4—21no—
36#15688970—21no—
37#15688968—21no—
38C42×Dic47—21no—
39#15688886C2212yes—
40#15688885C2212yes—
41#15688885C2212yes—
42C2×C4×D4×C47C2211yescharacteristic
43C2×#102×C47C2212yes—
44C2×C4×C8×C47C2211yescharacteristic
45C4×C4⋊C4×C47C2211yescharacteristic
46C47×#340C2212yes—
47C2×#10625352—22no—
48C2×#10625352—24no—
49C2×#10625352C2×C414yes—
50C2×#10625352C2×C418yes—
51C22×Dic94C2311yescharacteristic
52C22×C22⋊D47—21no—
53C23×Dic47—21no—
54C23×Dic47—21no—
55C22×C4×D47D411yescharacteristic
56C22×C4×D47D411yescharacteristic
57C22×C4×D47C2311yescharacteristic
58C22×C47⋊C8C2312yes—
59C2×C8×D47—24no—
60C2×C8×D47—24no—
61C2×C8×D47C2×C418yescomplemented
62C4×Dic94—22no—
63C4×Dic94C2×C414yes—
64#13093989—22no—
65#13093989C2×C414yes—
66D4×Dic47—22no—
67D4×Dic47C2×C414yes—
68C4×C22⋊D47—28no—
69(C22⋊D47)⋊C4—28no—
70#13093982—22no—
71#13093982—24no—
72#13093982C2×C414yes—
73#13093982C2×C418yescomplemented
74#13093980—28no—
75C2×C4⋊Dic47C2312yes—
76C2×C4⋊Dic47C2311yescharacteristic
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×C22⋊Dic47—21no—
80C2×(C2×D47)⋊C4—22no—
81C2×(C2×D47)⋊C4—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—22no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47—28no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47—22no—
89C2×C4×Dic47D411yescharacteristic
90C2×C4×Dic47D411yescharacteristic
91C2×C4×Dic47C2311yescharacteristic
92C42×D47C2×C414yes—
93C42×D47D414yes—
94#13093949C2×C418yescomplemented
95#13093947C2×C418yescomplemented
96#13093945C2×C418yescomplemented
97C4×C47⋊C8—22no—
98C8×Dic47—24no—
99C47⋊#95—21no—
100C47⋊#95—22no—
101C22×D4×C47C2311yescharacteristic
102C23×C4×C47—21no—
103C22×C8×C47C2312yes—
104C4×D4×C47—22no—
105C4×D4×C47C2×C414yes—
106C2×C4⋊C4×C47C2312yes—
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C22⋊C4×C47—21no—
109C2×C42×C47—21no—
110C2×C42×C47C2311yescharacteristic
111#102×C47C2×C418yescomplemented
112C4×C8×C47—22no—
113#95×C47—21no—
114C24×C47—21no—
115C23×D47C2×D411yescharacteristic
116C2×D4×C47—21no—
117C2×D4×C47—22no—
118C2×D4×C47C22×C412yes—
119C2×D4×C47C22×C414yes—
120C22×C4×C47—22no—
121C22×C4×C47—21no—
122C22×C4×C47—24no—
123C22×C4×C47—42no—
124C22×C4×C47—42no—
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2411yescharacteristic
128C2×C22⋊D47—24no—
129C2×C22⋊D47—28no—
130C22×Dic47—21no—
131C22×Dic47—24no—
132C22×Dic47—28no—
133C22×Dic47—22no—
134C22×Dic47—21no—
135C22×Dic47—24no—
136C22×Dic47—44no—
137C22×Dic47C2×D411yescharacteristic
138C22×Dic47C2×D411yescharacteristic
139C22×Dic47C2×D411yescharacteristic
140#10625352—28no—
141#10625352—216no—
142#10625352C42116yescomplemented
143C2×C4×D47—22no—
144C2×C4×D47—22no—
145C2×C4×D47—24no—
146C2×C4×D47C22⋊C414yes—
147C2×C4×D47C22⋊C414yes—
148C2×C4×D47C22×C412yes—
149C2×C4×D47C22×C414yes—
150C2×C4×D47C2×D412yes—
151C2×C4×D47#4114yes—
152C2×Dic94—21no—
153C2×Dic94—22no—
154C2×Dic94C22×C414yes—
155C2×Dic94C22×C412yes—
156C2×C8×C47—22no—
157C2×C8×C47—22no—
158C2×C8×C47C22×C414yes—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—22no—
161C4⋊C4×C47C22×C412yes—
162C4⋊C4×C47C22×C414yes—
163C22⋊C4×C47—24no—
164C42×C47—42no—
165C42×C47C22×C412yes—
166C42×C47C2×D412yes—
167C22⋊Dic47—24no—
168(C2×D47)⋊C4—24no—
169(C2×D47)⋊C4—28no—
170C4⋊Dic47—21no—
171C4⋊Dic47—22no—
172C4⋊Dic47C22×C414yes—
173C4⋊Dic47C22×C412yes—
174Dic47⋊C4—24no—
175Dic47⋊C4—24no—
176C4×Dic47—24no—
177C4×Dic47—24no—
178C4×Dic47—24no—
179C4×Dic47—28no—
180C4×Dic47—42no—
181C4×Dic47—44no—
182C4×Dic47C22⋊C414yes—
183C4×Dic47C22⋊C414yes—
184C4×Dic47C22×C412yes—
185C4×Dic47C2×D412yes—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8C22×C414yes—
189C8×D47—216no—
190C23×C47—22no—
191C23×C47—24no—
192C23×C47—42no—
193C23×C47—41no—
194C23×C47—41no—
195C23×C47C22×D411yescharacteristic
196C22×D47—22no—
197C22×D47—21no—
198C22×D47—21no—
199C22×D47C2×C22⋊C412yes—
200C22×D47C4×D414yes—
201D4×C47—22no—
202D4×C47—24no—
203D4×C47C2×C4214yes—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—24no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
211C2×C4×C47—48no—
212C2×C4×C47C2×C22⋊C412yes—
213C2×C4×C47C2×C22⋊C412yes—
214C2×C4×C47#13512yes—
215C2×C4×C47#13512yes—
216C2×C4×C47C23×C411yescharacteristic
217C2×C4×C47C23×C412yes—
218C2×C4×C47C22×D411yescharacteristic
219C2×C4×C47C2×#4112yes—
220C22⋊D47—216no—
221C2×Dic47—22no—
222C2×Dic47—24no—
223C2×Dic47—21no—
224C2×Dic47—21no—
225C2×Dic47—216no—
226C2×Dic47—22no—
227C2×Dic47—24no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—48no—
232C2×Dic47C2×C22⋊C412yes—
233C2×Dic47C2×C22⋊C412yes—
234C2×Dic47C2×C22⋊C412yes—
235C2×Dic47C4×D414yes—
236C2×Dic47#13514yes—
237C4×D47—28no—
238C4×D47—28no—
239C4×D47C2×C4214yes—
240C4×D47#11714yes—
241C4×D47C4×D418yes—
242Dic94—22no—
243Dic94—24no—
244Dic94C2×C4214yes—
245C8×C47—24no—
246C47⋊C8—24no—
247C4×#416—471nomaximal, Sylow
248C22×C47—24no—
249C22×C47—42no—
250C22×C47—42no—
251C22×C47—44no—
252C22×C47—44no—
253C22×C47C22×C22⋊C411yescharacteristic
254C22×C47C2×C4×D412yes—
255C22×C47C2×#13512yes—
256C22×C47C2×#13511yescharacteristic
257C22×C47C2×#13511yescharacteristic
258C2×D47—22no—
259C2×D47—22no—
260C2×D47—24no—
261C2×D47—24no—
262C2×D47C4×C22⋊C414yes—
263C4×C47—22no—
264C4×C47—42no—
265C4×C47—44no—
266C4×C47#41614yescomplemented
267C4×C47C22×C4211yescharacteristic
268C4×C47C2×#11711yescharacteristic
269C4×C47C2×C4×D412yes—
270Dic47—22no—
271Dic47—24no—
272Dic47—42no—
273Dic47—44no—
274Dic47C4×C22⋊C414yes—
275Dic47#41614yescomplemented
276C2×C4×#41—471no—
277C2×#416—472no—
278C22×C4×C8—471no—
279C4×#117—471no—
280C4×#103—471no—
281C4×#103—471no—
282C4×#103—472no—
283C4×#102—472no—
284C4×#102—471no—
285C4×#102D4711yescharacteristic, complemented
286#1735—472no—
287C2×C47—44no—
288C2×C47C2×C4×C22⋊C411yescharacteristic
289C2×C47C2×#41611yescharacteristic
290C2×C47C2×#41611yescharacteristic
291C2×C47C4×#13512yes—
292C2×C47C4×#13512yes—
293D47—24no—
294C22×#41—471no—
295C23×C8—472no—
296C4×#41—478no—
297C4×#41—944no—
298C2×C4×Q8—471no—
299C2×C4×Q8—471no—
300C2×C4×D4—471no—
301C2×C4×D4—941no—
302C2×C4×D4—941no—
303C2×C4×D4C2×D4711yescharacteristic
304C2×#117—472no—
305C2×#117—471no—
306C2×#117—941no—
307C22×C42—471no—
308C22×C42—941no—
309#416—4716no—
310C2×#103—472no—
311C2×#103—472no—
312C2×#102—472no—
313C2×#102C2×D4712yes—
314C2×C4×C8—472no—
315C2×C4×C8—471no—
316C2×C4×C8—944no—
317C2×C4×C8C2×D4711yescharacteristic
318C4×C4⋊C4—472no—
319C4×C4⋊C4—471no—
320C4×C4⋊C4C2×D4711yescharacteristic
321C4×C22⋊C4—941no—
322C4×C22⋊C4—941no—
323#376—941no—
324C43—941no—
325#340—472no—
326#340—472no—
327#340—472no—
328#340C2×D4712yes—
329C47C4×#41611yescharacteristic, complemented, Sylow
330C2×#41—474no—
331C2×#41—478no—
332C2×#41—942no—
333C2×#41—944no—
334C22×Q8—471no—
335C22×D4—941no—
336C22×D4C22×D4711yescharacteristic
337C23×C4—21no—
338C23×C4—471no—
339C23×C4—471no—
340C23×C4—471no—
341C23×C4—941no—
342C23×C4—941no—
343C22×C8—472no—
344C22×C8—478no—
345C22×C8—944no—
346C22×C8—944no—
347C22×C8C22×D4712yes—
348C4×Q8—474no—
349C4×Q8—474no—
350C4×Q8—942no—
351C4×Q8—942no—
352C4×D4—22no—
353C4×D4—474no—
354C4×D4—942no—
355C4×D4—948no—
356C4×D4—948no—
357C4×D4C4×D4714yes—
358#117—474no—
359#117—478no—
360#117—948no—
361#117—942no—
362#117—944no—
363C2×C4⋊C4—471no—
364C2×C4⋊C4—472no—
365C2×C4⋊C4—941no—
366C2×C4⋊C4—941no—
367C2×C4⋊C4C22×D4711yescharacteristic
368C2×C4⋊C4C22×D4712yes—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—941no—
371C2×C22⋊C4—942no—
372C2×C22⋊C4—941no—
373C2×C42—21no—
374C2×C42—471no—
375C2×C42—471no—
376C2×C42—471no—
377C2×C42—474no—
378C2×C42—474no—
379C2×C42—948no—
380C2×C42—941no—
381C2×C42—942no—
382C2×C42—941no—
383C2×C42—941no—
384C2×C42—942no—
385C2×C42—941no—
386C2×C42C22×D4711yescharacteristic
387#103—478no—
388#103—478no—
389#102—478no—
390#102C4×D4718yescomplemented
391C4×C8—22no—
392C4×C8—944no—
393C4×C8—942no—
394#95—21no—
395#95—942no—
396#95—941no—
397C24—21no—
398C24—471no—
399#41—4716no—
400#41—948no—
401#41—9416no—
402C2×Q8—474no—
403C2×Q8—472no—
404C2×Q8—941no—
405C2×Q8—942no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—944no—
409C2×D4—948no—
410C2×D4C2×C4×D4714yes—
411C2×D4C2×C4×D4712yes—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—471no—
418C22×C4—471no—
419C22×C4—472no—
420C22×C4—474no—
421C22×C4—472no—
422C22×C4—474no—
423C22×C4—471no—
424C22×C4—474no—
425C22×C4—474no—
426C22×C4—944no—
427C22×C4—944no—
428C22×C4—948no—
429C22×C4—944no—
430C22×C4—941no—
431C22×C4—942no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—941no—
435C22×C4—1884no—
436C22×C4D4×D4711yescharacteristic
437C22×C4D4×D4711yescharacteristic
438C22×C4C23×D4711yescharacteristic
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—474no—
442C2×C8—9416no—
443C2×C8—942no—
444C2×C8—942no—
445C2×C8C2×C4×D4714yes—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—474no—
449C4⋊C4—472no—
450C4⋊C4—944no—
451C4⋊C4—944no—
452C4⋊C4—941no—
453C4⋊C4—942no—
454C4⋊C4C2×C4×D4714yes—
455C4⋊C4C2×C4×D4712yes—
456C22⋊C4—24no—
457C22⋊C4—948no—
458C22⋊C4—944no—
459C22⋊C4—944no—
460C42—42no—
461C42—474no—
462C42—474no—
463C42—472no—
464C42—472no—
465C42—944no—
466C42—944no—
467C42—948no—
468C42—944no—
469C42—1884no—
470C42—1882no—
471C42C2×C4×D4712yes—
472C42D4×D4712yes—
473C23—22no—
474C23—24no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—472no—
479C23—474no—
480C23—942no—
481C23—941no—
482C23—941no—
483C23C2×D4×D4711yescharacteristic
484Q8—474no—
485Q8—942no—
486Q8—944no—
487D4—22no—
488D4—24no—
489D4—9416no—
490D4C42×D4714yes—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—48no—
499C2×C4—472no—
500C2×C4—472no—
501C2×C4—474no—
502C2×C4—474no—
503C2×C4—474no—
504C2×C4—474no—
505C2×C4—478no—
506C2×C4—472no—
507C2×C4—948no—
508C2×C4—9416no—
509C2×C4—942no—
510C2×C4—944no—
511C2×C4—941no—
512C2×C4—941no—
513C2×C4—948no—
514C2×C4—942no—
515C2×C4—944no—
516C2×C4—1888no—
517C2×C4—1882no—
518C2×C4—1882no—
519C2×C4—1882no—
520C2×C4C22⋊C4×D4712yes—
521C2×C4C22⋊C4×D4712yes—
522C2×C4#1309404312yes—
523C2×C4#1309404512yes—
524C2×C4C22×C4×D4711yescharacteristic
525C2×C4C22×C4×D4712yes—
526C2×C4C2×D4×D4711yescharacteristic
527C2×C4#41×D4712yes—
528C8—24no—
529C8—944no—
530C22—24no—
531C22—42no—
532C22—42no—
533C22—44no—
534C22—44no—
535C22—474no—
536C22—942no—
537C22—942no—
538C22—944no—
539C22—944no—
540C22C2×C22⋊C4×D4711yescharacteristic
541C22C4×D4×D4712yes—
542C22#135×D4712yes—
543C22C2×#1309404311yescharacteristic
544C22C2×#1309404511yescharacteristic
545C4—22no—
546C4—42no—
547C4—44no—
548C4—474no—
549C4—474no—
550C4—944no—
551C4—942no—
552C4—1882no—
553C4—1884no—
554C4#1568911614yesdirect product factor
555C4C2×C42×D4711yescharacteristic
556C4#117×D4711yescharacteristic
557C4C4×D4×D4712yes—
558C2—44no—
559C2—944no—
560C2C4×C22⋊C4×D4711yescharacteristic
561C2#416×D4711yescharacteristic
562C2C2×#1568911611yescharacteristic
563C2C4×#1309404312yes—
564C2C4×#1309404512yes—
subgroupG/H
centerC22×C4D4×D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC4×#102×C47C2
socleC23×C47C22×D4
prime radicalC22×#10625352C22
cosocleC22×C4×C47C24
hypercenterC4×#102D47
hypocenterC47C4×#416
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
3. C2×C4×C8×D47C2semidirect
4. C4×#13093982C2semidirect
5. C4×#13093947C2semidirect
6. C4×#13093949C2semidirect
7. #20243207C2semidirect
8. #102×Dic47C2semidirect
9. C4×#13093945C2semidirect
10. #20243103C2semidirect
11. C4×#102×C47C2semidirect
15. C4×#10625352C4semidirect
26. #15689116C4direct (N × H)
61. C2×C8×D47C2×C4semidirect
73. #13093982C2×C4semidirect
94. #13093949C2×C4semidirect
95. #13093947C2×C4semidirect
96. #13093945C2×C4semidirect
111. #102×C47C2×C4semidirect
142. #10625352C42semidirect
266. C4×C47#416semidirect
275. Dic47#416semidirect
285. C4×#102D47semidirect
329. C47C4×#416semidirect
390. #102C4×D47semidirect
554. C4#15689116direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#106253522
C2×#156891162
C2×C4×C8×D472
C4×#130939822
C4×#130939472
C4×#130939492
#202432072
#102×Dic472
C4×#130939452
#202431032
C4×#102×C472
C4×#41647