Finite Group Databasebeta

← C4×#15689279 #26221880

Subgroups — C4×#15689279 #26221880

Order 12032.

1448 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689279C211yesmaximal, characteristic
2C2×C4×#10625337C212yesmaximal, complemented
3C4×#13094003C211yesmaximal, characteristic
4#20243729C212yesmaximal, complemented
5#20243695C211yesmaximal, characteristic
6#20243642C212yesmaximal, complemented
7#20243576C211yesmaximal, characteristic
8#20243323C211yesmaximal, characteristic
9C4×#13093948C212yesmaximal, complemented
10C4×C22⋊(C47⋊C8)C211yesmaximal, characteristic
11C22×C4×Q8×C47C211yesmaximal, characteristic
12C22×#10625337C2212yes—
13C2×#13094003C2211yescharacteristic
14C2×C4×Dic94C2211yescharacteristic
15#15689279C418yesdirect product factor
16C4×#10625337—28no—
17C2×#13093957C2212yes—
18C2×#13093948C2212yes—
19C2×#13093948C2212yes—
20C2×#13093948C2212yes—
21C2×#13093947C2212yes—
22C2×C22⋊(C47⋊C8)C2211yescharacteristic
23C2×C22⋊(C47⋊C8)C2211yescharacteristic
24C2×C4×C47⋊C8C2212yes—
25#15689029C2211yescharacteristic
26#15689021C2211yescharacteristic
27#15689007—21no—
28#15688996—21no—
29#15688983C2211yescharacteristic
30C4×Dic47⋊C4—21no—
31C4×C4⋊Dic47C2211yescharacteristic
32C4×C22⋊Dic47—21no—
33#15688885C2212yes—
34#15688885C2212yes—
35#15688878C2212yes—
36C23×Q8×C47C2211yescharacteristic
37C2×C4×Q8×C47—28no—
38C2×C4×Q8×C47—21no—
39C2×C4×Q8×C47—21no—
40C2×C4×Q8×C47C2212yes—
41C2×C4×Q8×C47C2212yes—
42C22×C4⋊C4×C47—21no—
43C22×C4⋊C4×C47C2211yescharacteristic
44C22×C42×C47—21no—
45C22×C42×C47C2211yescharacteristic
46C22×Dic94C2311yescharacteristic
47C2×#10625337—24no—
48C2×#10625337—24no—
49C2×#10625337C2×C418yescomplemented
50C22×C47⋊C8C2312yes—
51C22×C47⋊C8C2312yes—
52#13094003—22no—
53#13094003C2×C414yes—
54C4×Dic94—24no—
55C2×C4⋊Dic47C2311yescharacteristic
56C2×C4⋊Dic47C2311yescharacteristic
57C2×C4⋊Dic47C2311yescharacteristic
58C2×Dic47⋊C4—21no—
59C2×Dic47⋊C4—21no—
60C2×Dic47⋊C4—21no—
61C2×C22⋊Dic47—21no—
62C2×C22⋊Dic47—21no—
63C2×C4×Dic47—21no—
64C2×C4×Dic47—21no—
65#13093957—24no—
66#13093948—24no—
67#13093948—24no—
68#13093948C2×C418yescomplemented
69#13093947—24no—
70C22⋊(C47⋊C8)—22no—
71C22⋊(C47⋊C8)C2×C414yes—
72C4×C47⋊C8—24no—
73C47⋊#95—21no—
74C47⋊#95—21no—
75C47⋊#95—21no—
76C47⋊#95—21no—
77C47⋊#95—21no—
78C22×Q8×C47—24no—
79C22×Q8×C47—24no—
80C22×Q8×C47—21no—
81C22×Q8×C47—22no—
82C22×Q8×C47C2×C414yes—
83C22×Q8×C47C2312yes—
84C23×C4×C47—21no—
85C23×C4×C47—21no—
86C23×C4×C47D411yescharacteristic
87C23×C4×C47D411yescharacteristic
88C23×C4×C47C2311yescharacteristic
89C4×Q8×C47—28no—
90C4×Q8×C47—28no—
91C4×Q8×C47—24no—
92C4×Q8×C47—24no—
93C4×Q8×C47—24no—
94C4×Q8×C47D418yes—
95C2×C4⋊C4×C47—28no—
96C2×C4⋊C4×C47—22no—
97C2×C4⋊C4×C47—22no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47—24no—
100C2×C4⋊C4×C47C2312yes—
101C2×C42×C47—28no—
102C2×C42×C47—22no—
103C2×C42×C47—22no—
104C2×C42×C47—24no—
105C2×C42×C47D411yescharacteristic
106C2×C42×C47D411yescharacteristic
107C2×C42×C47C2311yescharacteristic
108C2×C42×C47C2311yescharacteristic
109C24×C47C2×D411yescharacteristic
110C2×Q8×C47—24no—
111C2×Q8×C47—216no—
112C2×Q8×C47—24no—
113C2×Q8×C47—216no—
114C2×Q8×C47—24no—
115C2×Q8×C47—22no—
116C2×Q8×C47—22no—
117C2×Q8×C47—22no—
118C2×Q8×C47C22×C414yes—
119C2×Q8×C47C2×D414yes—
120C2×Q8×C47#4114yes—
121C22×C4×C47—24no—
122C22×C4×C47—24no—
123C22×C4×C47—24no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—24no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—41no—
134C22×C4×C47—42no—
135C22×C4×C47—42no—
136C22×C4×C47Q1614yescomplemented
137C22×C4×C47C22×C412yes—
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47#4112yes—
144C22×C4×C47C2411yescharacteristic
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—21no—
148C22×Dic47—21no—
149C2×Dic94—22no—
150C2×Dic94—22no—
151C2×Dic94—21no—
152C2×Dic94C22×C412yes—
153C4⋊C4×C47—28no—
154C4⋊C4×C47—24no—
155C4⋊C4×C47—28no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47C2×D414yes—
158C42×C47—28no—
159C42×C47—28no—
160C42×C47—22no—
161C42×C47—22no—
162C42×C47—22no—
163C42×C47C2×D414yes—
164C22⋊Dic47—24no—
165C22⋊Dic47—42no—
166#10625337—216no—
167C4⋊Dic47—21no—
168C4⋊Dic47—22no—
169C4⋊Dic47—22no—
170C4⋊Dic47C22×C412yes—
171Dic47⋊C4—24no—
172Dic47⋊C4—42no—
173Dic47⋊C4—42no—
174C4×Dic47—42no—
175C4×Dic47—42no—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8C22×C414yes—
182C23×C47—21no—
183C23×C47—21no—
184C23×C47—21no—
185C23×C47—21no—
186C23×C47C4×D412yes—
187C23×C47C2×Q1612yes—
188C23×C47#13512yes—
189C23×C47C22×D411yescharacteristic
190Q8×C47—216no—
191Q8×C47—24no—
192Q8×C47—24no—
193Q8×C47—24no—
194Q8×C47C4×D418yes—
195C2×C4×C47—216no—
196C2×C4×C47—24no—
197C2×C4×C47—21no—
198C2×C4×C47—24no—
199C2×C4×C47—21no—
200C2×C4×C47—24no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—24no—
206C2×C4×C47—24no—
207C2×C4×C47—28no—
208C2×C4×C47—21no—
209C2×C4×C47—24no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—44no—
213C2×C4×C47—48no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47C4×D412yes—
217C2×C4×C47#12012yes—
218C2×C4×C47#12012yes—
219C2×C4×C47C2×Q1612yes—
220C2×C4×C47C2×Q1612yes—
221C2×C4×C47#13712yes—
222C2×C4×C47C23×C411yescharacteristic
223C2×C4×C47C22×D412yes—
224C2×C4×C47C2×#4112yes—
225C2×C4×C47C2×#4111yescharacteristic
226C2×Dic47—22no—
227C2×Dic47—22no—
228C2×Dic47—41no—
229C2×Dic47—41no—
230C2×Dic47—41no—
231C2×Dic47—41no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238Dic94—24no—
239Dic94—42no—
240C47⋊C8—24no—
241C47⋊C8—42no—
242C4×#451—471nomaximal, Sylow
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—22no—
246C22×C47—22no—
247C22×C47—22no—
248C22×C47—22no—
249C22×C47C4×Q1614yescomplemented
250C22×C47C2×C4×D411yescharacteristic
251C22×C47C2×#12011yescharacteristic
252C22×C47#52511yescharacteristic
253C22×C47C22×Q1611yescharacteristic
254C22×C47C2×#13511yescharacteristic
255C22×C47C2×#13711yescharacteristic
256C22×C47#57511yescharacteristic
257C4×C47—22no—
258C4×C47—22no—
259C4×C47—28no—
260C4×C47—22no—
261C4×C47—41no—
262C4×C47—44no—
263C4×C47#45114yescomplemented
264C4×C47C2×C4×D412yes—
265C4×C47#52512yes—
266Dic47—42no—
267Dic47—42no—
268Dic47—81no—
269Dic47—81no—
270C22×C4×Q8D4711yescharacteristic
271C2×#451—471no—
272C2×C4×Q16—472no—
273C4×C22⋊Q8—471no—
274#1878—472no—
275#1872—471no—
276#1863—472no—
277#1852—471no—
278#1787—471no—
279C4×#103—472no—
280C4×C22⋊C8—471no—
281C2×C47—24no—
282C2×C47C4×#12011yescharacteristic
283C2×C47C2×C4×Q1611yescharacteristic
284C2×C47C4×#13711yescharacteristic
285C2×C47C2×#45111yescharacteristic
286C2×C47#300211yescharacteristic
287C2×C47#310411yescharacteristic
288C2×C47#310811yescharacteristic
289C23×Q8C2×D4711yescharacteristic
290C22×Q16—472no—
291C2×C22⋊Q8—471no—
292C2×C4×Q8—28no—
293C2×C4×Q8—21no—
294C2×C4×Q8—21no—
295C2×C4×Q8—471no—
296C2×C4×Q8C2×D4712yes—
297C2×C4×Q8C2×D4712yes—
298C22×C4⋊C4—21no—
299C22×C4⋊C4C2×D4711yescharacteristic
300C22×C42—21no—
301C22×C42C2×D4711yescharacteristic
302#451—478no—
303C4×Q16—948no—
304C2×#107—472no—
305C2×#103—472no—
306C2×#103—472no—
307C2×#103—472no—
308C2×#103—472no—
309C2×C22⋊C8—471no—
310C2×C22⋊C8—471no—
311C2×C4×C8—472no—
312#391—471no—
313#389—471no—
314#387—941no—
315#385—941no—
316#380—471no—
317C4×C4⋊C4—471no—
318C4×C4⋊C4—941no—
319C4×C22⋊C4—941no—
320#340—472no—
321#340—472no—
322#336—472no—
323C47C4×#45111yescharacteristic, complemented, Sylow
324C22×Q8—24no—
325C22×Q8—24no—
326C22×Q8—22no—
327C22×Q8—21no—
328C22×Q8—471no—
329C22×Q8C4×D4714yes—
330C22×Q8C22×D4712yes—
331C23×C4—21no—
332C23×C4—21no—
333C23×C4C22⋊D4711yescharacteristic
334C23×C4C22⋊D4711yescharacteristic
335C23×C4C22×D4711yescharacteristic
336C2×Q16—478no—
337C2×Q16—944no—
338C2×Q16—944no—
339C22×C8—472no—
340C22×C8—472no—
341C22⋊Q8—474no—
342C22⋊Q8—942no—
343C4×Q8—28no—
344C4×Q8—28no—
345C4×Q8—24no—
346C4×Q8—24no—
347C4×Q8—24no—
348C4×Q8—944no—
349C4×Q8C22⋊D4718yes—
350C2×C4⋊C4—28no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—471no—
356C2×C4⋊C4—471no—
357C2×C4⋊C4—471no—
358C2×C4⋊C4—941no—
359C2×C4⋊C4—941no—
360C2×C4⋊C4—941no—
361C2×C4⋊C4C22×D4712yes—
362C2×C22⋊C4—941no—
363C2×C22⋊C4—941no—
364C2×C42—28no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—24no—
368C2×C42—941no—
369C2×C42—941no—
370C2×C42C22⋊D4711yescharacteristic
371C2×C42C22⋊D4711yescharacteristic
372C2×C42C22×D4711yescharacteristic
373C2×C42C22×D4711yescharacteristic
374#107—944no—
375#103—478no—
376#103—944no—
377#103—944no—
378#103—944no—
379C22⋊C8—474no—
380C22⋊C8—942no—
381C4×C8—944no—
382#95—941no—
383#95—941no—
384#95—941no—
385#95—941no—
386#95—941no—
387C24C2×C22⋊D4711yescharacteristic
388C2×Q8—24no—
389C2×Q8—216no—
390C2×Q8—24no—
391C2×Q8—216no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—472no—
397C2×Q8—941no—
398C2×Q8—942no—
399C2×Q8—942no—
400C2×Q8C2×C4×D4714yes—
401C2×Q8#1062535014yes—
402C2×Q8C2×C22⋊D4714yes—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—941no—
419C22×C4—941no—
420C22×C4—941no—
421C22×C4—941no—
422C22×C4#1062533714yescomplemented
423C22×C4C2×C4×D4712yes—
424C22×C4#1062535012yes—
425C22×C4C2×C22⋊D4711yescharacteristic
426C22×C4C2×C22⋊D4711yescharacteristic
427C22×C4C2×C22⋊D4711yescharacteristic
428C22×C4C2×C22⋊D4711yescharacteristic
429C22×C4C2×C22⋊D4711yescharacteristic
430C22×C4C23×D4711yescharacteristic
431Q16—9416no—
432C2×C8—474no—
433C2×C8—942no—
434C2×C8—942no—
435C2×C8—942no—
436C2×C8—942no—
437C2×C8—942no—
438C4⋊C4—28no—
439C4⋊C4—24no—
440C4⋊C4—28no—
441C4⋊C4—22no—
442C4⋊C4—472no—
443C4⋊C4—944no—
444C4⋊C4—941no—
445C4⋊C4—942no—
446C4⋊C4—942no—
447C4⋊C4—1882no—
448C4⋊C4—1882no—
449C4⋊C4C2×C22⋊D4714yes—
450C22⋊C4—944no—
451C22⋊C4—1882no—
452C42—28no—
453C42—28no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—1882no—
458C42—1882no—
459C42C2×C22⋊D4714yes—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23C4×C22⋊D4712yes—
465C23C2×#1062533712yes—
466C23#1309404412yes—
467C23C22×C22⋊D4711yescharacteristic
468Q8—216no—
469Q8—24no—
470Q8—24no—
471Q8—24no—
472Q8—944no—
473Q8—1882no—
474Q8C4×C22⋊D4718yes—
475C2×C4—216no—
476C2×C4—24no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—44no—
493C2×C4—48no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—942no—
497C2×C4—942no—
498C2×C4—1881no—
499C2×C4—1881no—
500C2×C4—1881no—
501C2×C4—1881no—
502C2×C4—1881no—
503C2×C4—1881no—
504C2×C4—1881no—
505C2×C4—1881no—
506C2×C4—1881no—
507C2×C4—1881no—
508C2×C4C4×C22⋊D4712yes—
509C2×C4#1309399312yes—
510C2×C4#1309399312yes—
511C2×C4C2×#1062533712yes—
512C2×C4C2×#1062533712yes—
513C2×C4#1309405512yes—
514C2×C4C22×C4×D4711yescharacteristic
515C2×C4C22×C22⋊D4712yes—
516C2×C4C2×#1062535012yes—
517C2×C4C2×#1062535011yescharacteristic
518C8—944no—
519C8—1882no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22C4×#1062533714yescomplemented
527C22C2×C4×C22⋊D4711yescharacteristic
528C22C2×#1309399311yescharacteristic
529C22#1568963411yescharacteristic
530C22C22×#1062533711yescharacteristic
531C22C2×#1309404411yescharacteristic
532C22C2×#1309405511yescharacteristic
533C22#1568992111yescharacteristic
534C4—28no—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—41no—
539C4—44no—
540C4—3761no—
541C4—3761no—
542C4—1882no—
543C4—1882no—
544C4#1568927914yesdirect product factor
545C4C2×C4×C22⋊D4712yes—
546C4#1568963412yes—
547C2—24no—
548C2C4×#1309399311yescharacteristic
549C2C2×C4×#1062533711yescharacteristic
550C2C4×#1309405511yescharacteristic
551C2C2×#1568927911yescharacteristic
552C2#2025260611yescharacteristic
553C2#2025342711yescharacteristic
554C2#2025346811yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×C4×Q8×C47C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenterC22×C4×Q8D47
hypocenterC47C4×#451
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C4×#10625337C2semidirect
4. #20243729C2semidirect
6. #20243642C2semidirect
9. C4×#13093948C2semidirect
15. #15689279C4direct (N × H)
49. C2×#10625337C2×C4semidirect
68. #13093948C2×C4semidirect
136. C22×C4×C47Q16semidirect
249. C22×C47C4×Q16semidirect
263. C4×C47#451semidirect
323. C47C4×#451semidirect
422. C22×C4#10625337semidirect
526. C22C4×#10625337semidirect
544. C4#15689279direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156892792
C2×C4×#106253372
C4×#130940032
#202437292
#202436952
#202436422
#202435762
#202433232
C4×#130939482
C4×C22⋊(C47⋊C8)2
C22×C4×Q8×C472
C4×#45147