Finite Group Databasebeta

← C4×#15689118 #26214028

Subgroups — C4×#15689118 #26214028

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×#10625354C211yesmaximal, characteristic
2C2×#15689118C212yesmaximal
3C2×C4×C8×D47C211yesmaximal, characteristic, complemented
4C4×#13093982C211yesmaximal, characteristic, complemented
5#103×Dic47C212yesmaximal, complemented
6C4×#13093948C211yesmaximal, characteristic, complemented
7C4×#13093944C211yesmaximal, characteristic, complemented
8C4×#13093946C211yesmaximal, characteristic, complemented
9#20243202C212yesmaximal, complemented
10#20243103C212yesmaximal, complemented
11C4×#103×C47C211yesmaximal, characteristic, complemented
12C22×#10625354C2211yescharacteristic
13C22×C8×D47C2212yes—
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×#13093982C2212yes—
24C2×C42×D47—21no—
25C2×C42×D47C2211yescharacteristic
26#15689118C4116yesdirect product factor
27C2×#13093948C2212yes—
28C2×#13093946C2212yes—
29C2×#13093944C2212yes—
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×Q8×C47C2211yescharacteristic
43C2×#103×C47C2212yes—
44C2×C4×C8×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—21no—
55C22×C4×D47—21no—
56C22×C4×D47D411yescharacteristic
57C22×C4×D47D411yescharacteristic
58C22×C4×D47C2311yescharacteristic
59C22×C47⋊C8C2312yes—
60C2×C8×D47—24no—
61C2×C8×D47—24no—
62C2×C8×D47C2×C418yescomplemented
63Q8×Dic47—22no—
64Q8×Dic47C2×C414yes—
65C4×D188—28no—
66C4×D188—22no—
67C4×D188C2×C414yes—
68#13093986—28no—
69#13093986—22no—
70#13093986C2×C414yes—
71#13093982—22no—
72#13093982—24no—
73#13093982—28no—
74#13093982C2×C414yes—
75#13093982C2×C418yescomplemented
76C2×C4⋊Dic47—21no—
77C2×C4⋊Dic47C2312yes—
78C2×C4⋊Dic47C2311yescharacteristic
79C2×(C2×D47)⋊C4—22no—
80C2×(C2×D47)⋊C4—21no—
81C2×(C2×D47)⋊C4—21no—
82C2×(C2×D47)⋊C4—21no—
83C2×C4×Dic47—22no—
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×D47C2×C414yes—
93C42×D47D414yes—
94#13093948C2×C418yescomplemented
95#13093946C2×C418yescomplemented
96#13093944C2×C418yescomplemented
97C4×C47⋊C8—22no—
98C8×Dic47—24no—
99C47⋊#95—21no—
100C47⋊#95—22no—
101C22×Q8×C47C2311yescharacteristic
102C22×C8×C47C2312yes—
103C4×Q8×C47—22no—
104C4×Q8×C47C2×C414yes—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47C2312yes—
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C42×C47—21no—
109C2×C42×C47—21no—
110C2×C42×C47C2311yescharacteristic
111#103×C47C2×C418yescomplemented
112C4×C8×C47—22no—
113#95×C47—21no—
114C23×D47—21no—
115C23×D47C2×D411yescharacteristic
116C2×Q8×C47—21no—
117C2×Q8×C47—22no—
118C2×Q8×C47C22×C414yes—
119C2×Q8×C47C22×C412yes—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—22no—
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2411yescharacteristic
126C22×Dic47—22no—
127C22×Dic47—21no—
128C22×Dic47C2×D411yescharacteristic
129C22×Dic47C2×D411yescharacteristic
130C22×Dic47C2×D411yescharacteristic
131#10625354—28no—
132#10625354—216no—
133#10625354C42116yescomplemented
134C2×D188—24no—
135C2×D188—28no—
136C2×D188—21no—
137C2×D188—22no—
138C2×D188C22×C414yes—
139C2×D188C22×C412yes—
140C2×C4×D47—22no—
141C2×C4×D47—22no—
142C2×C4×D47—24no—
143C2×C4×D47—28no—
144C2×C4×D47—24no—
145C2×C4×D47—24no—
146C2×C4×D47—24no—
147C2×C4×D47—42no—
148C2×C4×D47—42no—
149C2×C4×D47—44no—
150C2×C4×D47C22⋊C414yes—
151C2×C4×D47C22⋊C414yes—
152C2×C4×D47C22×C412yes—
153C2×C4×D47C22×C414yes—
154C2×C4×D47C2×D412yes—
155C2×C4×D47#4114yes—
156C2×C8×C47—22no—
157C2×C8×C47—22no—
158C2×C8×C47C22×C414yes—
159C4⋊C4×C47—24no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—22no—
162C4⋊C4×C47C22×C414yes—
163C4⋊C4×C47C22×C412yes—
164C42×C47—24no—
165C42×C47—42no—
166C42×C47C22×C412yes—
167C42×C47C2×D412yes—
168(C2×D47)⋊C4—24no—
169(C2×D47)⋊C4—28no—
170(C2×D47)⋊C4—24no—
171(C2×D47)⋊C4—24no—
172C4⋊Dic47—21no—
173C4⋊Dic47—22no—
174C4⋊Dic47—24no—
175C4⋊Dic47C22×C414yes—
176C4⋊Dic47C22×C412yes—
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×C47C22×D411yescharacteristic
191C22×D47—22no—
192C22×D47—21no—
193C22×D47—21no—
194C22×D47—22no—
195C22×D47—24no—
196C22×D47—42no—
197C22×D47—41no—
198C22×D47—41no—
199C22×D47C2×C22⋊C412yes—
200C22×D47C4×D414yes—
201Q8×C47—22no—
202Q8×C47—24no—
203Q8×C47C2×C4214yes—
204C2×C4×C47—22no—
205C2×C4×C47—24no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—24no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
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—
220C2×Dic47—21no—
221C2×Dic47—21no—
222C2×Dic47—22no—
223C2×Dic47—24no—
224C2×Dic47—42no—
225C2×Dic47—42no—
226C2×Dic47—41no—
227C2×Dic47—41no—
228C2×Dic47C2×C22⋊C412yes—
229C2×Dic47C2×C22⋊C412yes—
230C2×Dic47C2×C22⋊C412yes—
231C2×Dic47C4×D414yes—
232C2×Dic47#13514yes—
233D188—216no—
234D188—22no—
235D188—24no—
236D188C2×C4214yes—
237C4×D47—28no—
238C4×D47—216no—
239C4×D47—28no—
240C4×D47—48no—
241C4×D47—48no—
242C4×D47C2×C4214yes—
243C4×D47#11714yes—
244C4×D47C4×D418yes—
245C8×C47—24no—
246C47⋊C8—24no—
247C4×#416—471nomaximal, Sylow
248C22×C47C22×C22⋊C411yescharacteristic
249C22×C47C2×C4×D412yes—
250C22×C47C2×#13512yes—
251C22×C47C2×#13511yescharacteristic
252C22×C47C2×#13511yescharacteristic
253C2×D47—22no—
254C2×D47—22no—
255C2×D47—24no—
256C2×D47—24no—
257C2×D47—24no—
258C2×D47—42no—
259C2×D47—42no—
260C2×D47—44no—
261C2×D47—44no—
262C2×D47C4×C22⋊C414yes—
263C4×C47—22no—
264C4×C47—24no—
265C4×C47—42no—
266C4×C47—44no—
267C4×C47#41614yescomplemented
268C4×C47C22×C4211yescharacteristic
269C4×C47C2×#11711yescharacteristic
270C4×C47C2×C4×D412yes—
271Dic47—22no—
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—472no—
281C4×#103—471no—
282C4×#103D4711yescharacteristic, complemented
283C4×#102—471no—
284C4×#102—471no—
285C4×#102—472no—
286#1735—472no—
287C2×C47C2×C4×C22⋊C411yescharacteristic
288C2×C47C2×#41611yescharacteristic
289C2×C47C2×#41611yescharacteristic
290C2×C47C4×#13512yes—
291C2×C47C4×#13512yes—
292D47—24no—
293D47—44no—
294C22×#41—471no—
295C23×C8—472no—
296C4×#41—478no—
297C4×#41—944no—
298C2×C4×Q8—471no—
299C2×C4×Q8C2×D4711yescharacteristic
300C2×C4×D4—471no—
301C2×C4×D4—471no—
302C2×C4×D4—941no—
303C2×C4×D4—941no—
304C2×#117—472no—
305C2×#117—471no—
306C2×#117—941no—
307C22×C42—471no—
308C22×C42—941no—
309#416—4716no—
310C2×#103—472no—
311C2×#103C2×D4712yes—
312C2×#102—472no—
313C2×#102—472no—
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×Q8C22×D4711yescharacteristic
335C22×D4—471no—
336C22×D4—941no—
337C23×C4—471no—
338C23×C4—471no—
339C23×C4—471no—
340C23×C4—941no—
341C23×C4—941no—
342C23×C4—941no—
343C22×C8—472no—
344C22×C8—478no—
345C22×C8—944no—
346C22×C8—944no—
347C22×C8C22×D4712yes—
348C4×Q8—22no—
349C4×Q8—474no—
350C4×Q8—942no—
351C4×Q8C4×D4714yes—
352C4×D4—474no—
353C4×D4—474no—
354C4×D4—948no—
355C4×D4—942no—
356C4×D4—948no—
357C4×D4—942no—
358#117—474no—
359#117—478no—
360#117—948no—
361#117—942no—
362#117—944no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—471no—
365C2×C4⋊C4—472no—
366C2×C4⋊C4—941no—
367C2×C4⋊C4C22×D4711yescharacteristic
368C2×C4⋊C4C22×D4712yes—
369C2×C22⋊C4—941no—
370C2×C22⋊C4—941no—
371C2×C22⋊C4—942no—
372C2×C22⋊C4—941no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—471no—
376C2×C42—471no—
377C2×C42—471no—
378C2×C42—474no—
379C2×C42—474no—
380C2×C42—948no—
381C2×C42—941no—
382C2×C42—942no—
383C2×C42—941no—
384C2×C42—942no—
385C2×C42—941no—
386C2×C42C22×D4711yescharacteristic
387#103—478no—
388#103C4×D4718yescomplemented
389#102—478no—
390#102—478no—
391C4×C8—22no—
392C4×C8—944no—
393C4×C8—942no—
394#95—21no—
395#95—942no—
396#95—941no—
397C24—471no—
398C24—941no—
399#41—4716no—
400#41—948no—
401#41—9416no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8C2×C4×D4714yes—
405C2×Q8C2×C4×D4712yes—
406C2×D4—474no—
407C2×D4—472no—
408C2×D4—944no—
409C2×D4—948no—
410C2×D4—941no—
411C2×D4—942no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—471no—
416C22×C4—471no—
417C22×C4—472no—
418C22×C4—474no—
419C22×C4—472no—
420C22×C4—474no—
421C22×C4—471no—
422C22×C4—474no—
423C22×C4—474no—
424C22×C4—944no—
425C22×C4—944no—
426C22×C4—948no—
427C22×C4—944no—
428C22×C4—942no—
429C22×C4—942no—
430C22×C4—942no—
431C22×C4—941no—
432C22×C4—944no—
433C22×C4—1884no—
434C22×C4—1882no—
435C22×C4—1882no—
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—24no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—474no—
450C4⋊C4—472no—
451C4⋊C4—944no—
452C4⋊C4—941no—
453C4⋊C4—942no—
454C4⋊C4C2×C4×D4714yes—
455C4⋊C4C2×C4×D4712yes—
456C22⋊C4—944no—
457C22⋊C4—948no—
458C22⋊C4—944no—
459C22⋊C4—944no—
460C42—24no—
461C42—42no—
462C42—474no—
463C42—474no—
464C42—472no—
465C42—472no—
466C42—944no—
467C42—944no—
468C42—948no—
469C42—1884no—
470C42—1882no—
471C42C2×C4×D4712yes—
472C42D4×D4712yes—
473C23—472no—
474C23—474no—
475C23—942no—
476C23—941no—
477C23—941no—
478C23—942no—
479C23—944no—
480C23—1882no—
481C23—1881no—
482C23—1881no—
483C23C2×D4×D4711yescharacteristic
484Q8—22no—
485Q8—24no—
486Q8C42×D4714yes—
487D4—474no—
488D4—9416no—
489D4—942no—
490D4—944no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
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—941no—
510C2×C4—941no—
511C2×C4—948no—
512C2×C4—942no—
513C2×C4—944no—
514C2×C4—1888no—
515C2×C4—1882no—
516C2×C4—1881no—
517C2×C4—1882no—
518C2×C4—1881no—
519C2×C4—1888no—
520C2×C4C22⋊C4×D4712yes—
521C2×C4C22⋊C4×D4712yes—
522C2×C4#1309404512yes—
523C2×C4#1309404712yes—
524C2×C4C22×C4×D4711yescharacteristic
525C2×C4C22×C4×D4712yes—
526C2×C4C2×D4×D4711yescharacteristic
527C2×C4#41×D4712yes—
528C8—24no—
529C8—944no—
530C22—474no—
531C22—944no—
532C22—942no—
533C22—942no—
534C22—944no—
535C22—944no—
536C22—1884no—
537C22—1882no—
538C22—1882no—
539C22—1884no—
540C22C2×C22⋊C4×D4711yescharacteristic
541C22C4×D4×D4712yes—
542C22#135×D4712yes—
543C22C2×#1309404511yescharacteristic
544C22C2×#1309404711yescharacteristic
545C4—24no—
546C4—22no—
547C4—42no—
548C4—44no—
549C4—474no—
550C4—474no—
551C4—942no—
552C4—1884no—
553C4—1882no—
554C4#1568911814yesdirect product factor
555C4C2×C42×D4711yescharacteristic
556C4#117×D4711yescharacteristic
557C4C4×D4×D4712yes—
558C2—944no—
559C2—1884no—
560C2C4×C22⋊C4×D4711yescharacteristic
561C2#416×D4711yescharacteristic
562C2C2×#1568911811yescharacteristic
563C2C4×#1309404512yes—
564C2C4×#1309404712yes—
subgroupG/H
centerC22×C4D4×D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC4×#103×C47C2
socleC23×C47C22×D4
prime radicalC22×#10625354C22
cosocleC22×C4×C47C24
hypercenterC4×#103D47
hypocenterC47C4×#416
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
3. C2×C4×C8×D47C2semidirect
4. C4×#13093982C2semidirect
5. #103×Dic47C2semidirect
6. C4×#13093948C2semidirect
7. C4×#13093944C2semidirect
8. C4×#13093946C2semidirect
9. #20243202C2semidirect
10. #20243103C2semidirect
11. C4×#103×C47C2semidirect
15. C4×#10625354C4semidirect
26. #15689118C4direct (N × H)
62. C2×C8×D47C2×C4semidirect
75. #13093982C2×C4semidirect
94. #13093948C2×C4semidirect
95. #13093946C2×C4semidirect
96. #13093944C2×C4semidirect
111. #103×C47C2×C4semidirect
133. #10625354C42semidirect
267. C4×C47#416semidirect
275. Dic47#416semidirect
282. C4×#103D47semidirect
329. C47C4×#416semidirect
388. #103C4×D47semidirect
554. C4#15689118direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#106253542
C2×#156891182
C2×C4×C8×D472
C4×#130939822
#103×Dic472
C4×#130939482
C4×#130939442
C4×#130939462
#202432022
#202431032
C4×#103×C472
C4×#41647