Finite Group Databasebeta

← #30615720

Subgroups — #30615720

Order 15616.

2102 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840829C211yesmaximal, characteristic, complemented
2C22×#14519836C212yesmaximal, complemented
3#21462291C211yesmaximal, characteristic, complemented
4#21461843C214yesmaximal, complemented
5#21461841C214yesmaximal, complemented
6#21461516C212yesmaximal, complemented
7C2×#16840323C211yesmaximal, characteristic
8#16840829—28no—
9C2×D4×#6923—24no—
10C2×#14519856C2212yescomplemented
11C22×C22⋊#6923C2212yes—
12C22×C22⋊#6923C2212yes—
13C22×C22⋊#6923C2212yescomplemented
14C2×#14519836—28no—
15C2×#14519836C2218yes—
16#16840710—24no—
17#16840705C2212yescomplemented
18#16840703C2212yescomplemented
19#16840640C2218yescomplemented
20C23×D244C2211yescharacteristic
21C22×D4×D61C2211yescharacteristic
22#16840323—24no—
23#16840323—24no—
24C2×#14519762C2212yes—
25C2×#14519760C411yescharacteristic, complemented
26C2×#14519752C412yescomplemented
27C2×#14519750C412yescomplemented
28C2×#14519747C411yescharacteristic, complemented
29C2×#14519747C2212yes—
30C2×#14519745C411yescharacteristic, complemented
31C2×C22⋊C4×D61C2211yescharacteristic
32C2×#123×C61C411yescharacteristic, complemented
33C23×#6923—22no—
34C23×#6923—22no—
35C23×#6923—22no—
36D4×#6923—232no—
37#14519856—28no—
38C2×C4⋊#6923—22no—
39C2×C22⋊#6923—22no—
40C2×C22⋊#6923—24no—
41C2×C22⋊#6923—22no—
42C2×C22⋊#6923—28no—
43C2×C22⋊#6923—24no—
44C2×C22⋊#6923—24no—
45C2×C22⋊#6923—22no—
46C2×C22⋊#6923—24no—
47C2×C22⋊#6923D418yescomplemented
48C2×C22⋊#6923C2314yes—
49C2×C22⋊#6923C2314yes—
50C2×C4×#6923—22no—
51#14519836—216no—
52(C2×#6923)⋊C4—24no—
53C24×D61C2×C411yescharacteristic, complemented
54C24×D61C2311yescharacteristic
55C24×D61C2311yescharacteristic
56C22×D244—24no—
57C22×D244—22no—
58C22×D244—22no—
59C22×D244—24no—
60C22×D244C2×C411yescharacteristic
61C22×D244C2×C412yescomplemented
62C22×D244C2×C412yescomplemented
63C22×D244C2312yes—
64C22×C22⋊D61C2×C411yescharacteristic
65C22×C22⋊D61C2×C411yescharacteristic
66C22×C22⋊D61C2×C412yescomplemented
67C2×D4×D61—24no—
68C2×D4×D61—24no—
69C2×D4×D61D414yes—
70C2×D4×D61D414yes—
71C22×C4×D61C2×C412yescomplemented
72C22×C4×D61C2311yescharacteristic
73#14519762—24no—
74#14519762—24no—
75#14519760—24no—
76#14519752—28no—
77#14519750—28no—
78#14519747—24no—
79#14519747—24no—
80#14519747—24no—
81#14519745—24no—
82C2×#11758196C2×C412yescomplemented
83C2×C22⋊Dic61C2×C411yescharacteristic, complemented
84C2×#11758198C2×C412yescomplemented
85C2×#11758198C2×C412yescomplemented
86C2×#11758198C2×C412yescomplemented
87C2×#11758198C2312yes—
88C22⋊C4×D61—22no—
89C22⋊C4×D61—22no—
90C22×D4×C61C2×C411yescharacteristic
91C23×C4×C61C2×C411yescharacteristic, complemented
92#123×C61—24no—
93C2×C4⋊C4×C61C2×C412yescomplemented
94C2×C22⋊C4×C61C2×C411yescharacteristic, complemented
95C2×C22⋊C4×C61C2×C412yescomplemented
96C24×C61C22×C411yescharacteristic
97C24×C61C22×C411yescharacteristic
98C23×D61—22no—
99C23×D61—22no—
100C23×D61—22no—
101C23×D61—24no—
102C23×D61—82no—
103C23×D61—82no—
104C23×D61—82no—
105C23×D61C22×C412yes—
106C23×D61C22×C411yescharacteristic
107C23×D61C22×C411yescharacteristic
108C23×D61C2×D412yes—
109C23×D61C2×D412yes—
110C23×D61C2×D412yes—
111C23×D61C2×D412yes—
112C23×D61C2×D414yes—
113C23×D61#4114yes—
114C23×D61C2411yescharacteristic
115C22×#6923—24no—
116C22×#6923—28no—
117C22×#6923—28no—
118C22×#6923—24no—
119C22×#6923—44no—
120C22×#6923—84no—
121C22×#6923—84no—
122C22×#6923—84no—
123C2×D4×C61—24no—
124C2×D4×C61C22⋊C414yes—
125C22×C4×C61—22no—
126C22×C4×C61—24no—
127C22×C4×C61C22×C412yes—
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C22×C412yes—
130C2×C22⋊D61—24no—
131C2×C22⋊D61—24no—
132C2×C22⋊D61—24no—
133C2×C22⋊D61—82no—
134C2×C22⋊D61C22⋊C414yes—
135C2×C22⋊D61C22⋊C414yes—
136C22×Dic61C22×C411yescharacteristic
137C22×Dic61C22×C412yes—
138D4×D61—28no—
139D4×D61—28no—
140D4×D61—28no—
141D4×D61—28no—
142C2×D244—24no—
143C2×D244—24no—
144C2×D244—28no—
145C2×D244—24no—
146C2×D244—22no—
147C2×D244—22no—
148C2×D244—24no—
149C2×D244—24no—
150C2×D244—44no—
151C2×D244—44no—
152C2×D244—44no—
153C2×D244—44no—
154C2×D244—82no—
155C2×D244C22⋊C414yes—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—24no—
159C2×C4×D61—82no—
160C2×C4×D61C2×D412yes—
161C2×C4×D61C2×D412yes—
162C22⋊#6923—28no—
163C22⋊#6923—28no—
164C22⋊#6923—24no—
165C22⋊#6923—24no—
166C22⋊#6923—24no—
167C22⋊#6923—44no—
168C22⋊#6923—84no—
169C22⋊#6923C2×D418yes—
170C4⋊#6923—28no—
171C4×#6923—28no—
172C4⋊C4×C61—24no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—24no—
175C22⋊Dic61—22no—
176#11758198—24no—
177#11758198—22no—
178#11758198—22no—
179#11758198—24no—
180#11758198—24no—
181#11758196—24no—
182C23×C61—22no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61C2×C22⋊C412yes—
186C23×C61C2×C22⋊C412yes—
187C23×C61C4×D414yes—
188C23×C61C23×C411yescharacteristic
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—24no—
193C22×D61—22no—
194C22×D61—24no—
195C22×D61—42no—
196C22×D61—44no—
197C22×D61—44no—
198C22×D61—44no—
199C22×D61—42no—
200C22×D61—44no—
201C22×D61—42no—
202C22×D61—44no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—84no—
206C22×D61—84no—
207C22×D61—84no—
208C22×D61—84no—
209C22×D61—84no—
210C22×D61—88no—
211C22×D61C2×C22⋊C412yes—
212C22×D61C2×C22⋊C412yes—
213C22×D61C4×D414yes—
214C22×D61#12014yes—
215C22×D61#12114yes—
216C22×D61#12114yes—
217C22×D61#12314yes—
218C22×D61C22×D411yescharacteristic
219C22×D61C22×D411yescharacteristic
220C22×D61C22×D412yes—
221C22×D61C2×#4112yes—
222C2×#6923—28no—
223C2×#6923—48no—
224C2×#6923—48no—
225C2×#6923—44no—
226C2×#6923—88no—
227C2×#6923—88no—
228C2×#6923—88no—
229D4×C61—28no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—24no—
234C2×C4×C61—44no—
235C2×C4×C61—42no—
236C2×C4×C61C2×C22⋊C412yes—
237C22⋊D61—28no—
238C22⋊D61—28no—
239C22⋊D61—84no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—42no—
243C2×Dic61C2×C22⋊C412yes—
244D244—28no—
245D244—42no—
246D244—44no—
247D244—42no—
248D244—48no—
249D244—84no—
250C4×D61—22no—
251C4×D61—24no—
252C4×D61—22no—
253C4×D61—84no—
254#20984—611nomaximal, Sylow
255C22×C61—22no—
256C22×C61—24no—
257C22×C61—24no—
258C22×C61—22no—
259C22×C61—44no—
260C22×C61—82no—
261C22×C61#38614yes—
262C22×C61C22×C22⋊C411yescharacteristic
263C22×C61C2×C4×D412yes—
264C2×D61—24no—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—24no—
268C2×D61—22no—
269C2×D61—44no—
270C2×D61—44no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—44no—
274C2×D61—44no—
275C2×D61—44no—
276C2×D61—44no—
277C2×D61—44no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—84no—
282C2×D61—82no—
283C2×D61—88no—
284C2×D61—84no—
285C2×D61#38614yes—
286C2×D61C2×#12011yescharacteristic
287C2×D61C2×#12112yes—
288C2×D61C2×#12311yescharacteristic
289#6923—48no—
290#6923—84no—
291C4×C61—22no—
292C4×C61—42no—
293C4×C61—42no—
294Dic61—22no—
295Dic61—42no—
296C2×#534—611no—
297C2×#518—611no—
298C22×#99—612no—
299#2308—611no—
300#1899—614no—
301#1898—614no—
302C2×#342—612no—
303C2×C61—22no—
304C2×C61—24no—
305C2×C61—42no—
306C2×C61C2×#38611yescharacteristic
307C2×C61#288312yes—
308D61—21no—
309D61—22no—
310D61—21no—
311D61—24no—
312D61—42no—
313D61—82no—
314C23×D4—611no—
315C23×D4—611no—
316#534—1224no—
317#534—1224no—
318C2×#123—611no—
319C2×#123—612no—
320C2×#123#692311yescharacteristic, complemented
321C2×#121—612no—
322C2×#121—612no—
323C2×#120—611no—
324C2×#120—611no—
325C2×#120—612no—
326#518—1228no—
327C2×C4×D4—1224no—
328C2×#117—612no—
329C22×C22⋊C4—612no—
330C22×C22⋊C4—611no—
331C22×C22⋊C4—612no—
332C22×C22⋊C4—612no—
333C2×#99—618no—
334C2×#99—1228no—
335#386—1224no—
336#381—612no—
337#379—612no—
338#342—618no—
339C61#2098411yescharacteristic, complemented, Sylow
340C25—611no—
341C25—611no—
342C25—611no—
343C22×D4—611no—
344C22×D4—611no—
345C22×D4—614no—
346C22×D4—614no—
347C22×D4—612no—
348C22×D4—611no—
349C22×D4—612no—
350C22×D4—612no—
351C22×D4—612no—
352C22×D4—1222no—
353C22×D4—1222no—
354C22×D4—1224no—
355C22×D4—1224no—
356C22×D4—1224no—
357C22×D4—1224no—
358C22×D4C2×#692311yescharacteristic
359C23×C4—611no—
360C23×C4—612no—
361C23×C4—1222no—
362C23×C4—1222no—
363C23×C4—1222no—
364C23×C4C2×#692311yescharacteristic, complemented
365#123—24no—
366#123—1224no—
367#123—1224no—
368#123—1224no—
369#121—1228no—
370#121—1228no—
371#120—1224no—
372#120—1224no—
373#120—1224no—
374#120—1224no—
375C4×D4—12232no—
376#117—1228no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4C2×#692312yescomplemented
380C2×C22⋊C4—614no—
381C2×C22⋊C4—618no—
382C2×C22⋊C4—611no—
383C2×C22⋊C4—612no—
384C2×C22⋊C4—612no—
385C2×C22⋊C4—612no—
386C2×C22⋊C4—612no—
387C2×C22⋊C4—614no—
388C2×C22⋊C4—1222no—
389C2×C22⋊C4—1222no—
390C2×C22⋊C4—1228no—
391C2×C22⋊C4—1222no—
392C2×C22⋊C4—1222no—
393C2×C22⋊C4—1224no—
394C2×C22⋊C4—1224no—
395C2×C22⋊C4—1224no—
396C2×C22⋊C4—1224no—
397C2×C22⋊C4—1222no—
398C2×C22⋊C4C2×#692311yescharacteristic, complemented
399C2×C22⋊C4C2×#692312yescomplemented
400C2×C42—1222no—
401#99—12216no—
402#95—1224no—
403C24—614no—
404C24—614no—
405C24—612no—
406C24—612no—
407C24—611no—
408C24—611no—
409C24—612no—
410C24—612no—
411C24—611no—
412C24—612no—
413C24—1222no—
414C24—1222no—
415C24—1222no—
416C24—1224no—
417C24—4882no—
418C24—4882no—
419C24—4882no—
420C24C22×#692311yescharacteristic
421C24C22×#692311yescharacteristic
422C2×D4—24no—
423C2×D4—614no—
424C2×D4—614no—
425C2×D4—614no—
426C2×D4—1224no—
427C2×D4—1224no—
428C2×D4—1224no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1224no—
432C2×D4—1228no—
433C2×D4—1224no—
434C2×D4—1228no—
435C2×D4—1228no—
436C2×D4—1228no—
437C2×D4—1228no—
438C2×D4—1224no—
439C2×D4—1224no—
440C2×D4—1224no—
441C2×D4—4882no—
442C2×D4—4882no—
443C2×D4—2444no—
444C2×D4—2444no—
445C2×D4—2444no—
446C2×D4—2444no—
447C2×D4C22⋊#692314yes—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—611no—
451C22×C4—612no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1224no—
457C22×C4—1228no—
458C22×C4—1228no—
459C22×C4—1224no—
460C22×C4—1224no—
461C22×C4—4882no—
462C22×C4—4884no—
463C22×C4—4884no—
464C22×C4—4884no—
465C22×C4—2444no—
466C22×C4C22×#692312yes—
467C22×C4C22×#692311yescharacteristic
468C22×C4C22×#692312yes—
469C4⋊C4—24no—
470C4⋊C4—1224no—
471C4⋊C4—1228no—
472C22⋊C4—22no—
473C22⋊C4—24no—
474C22⋊C4—618no—
475C22⋊C4—1224no—
476C22⋊C4—1222no—
477C22⋊C4—1222no—
478C22⋊C4—1224no—
479C22⋊C4—1224no—
480C22⋊C4—1228no—
481C22⋊C4—1228no—
482C22⋊C4—1222no—
483C22⋊C4—1224no—
484C22⋊C4—1224no—
485C22⋊C4—1224no—
486C22⋊C4—4884no—
487C22⋊C4—2444no—
488C42—1228no—
489C23—22no—
490C23—42no—
491C23—42no—
492C23—612no—
493C23—611no—
494C23—611no—
495C23—614no—
496C23—614no—
497C23—614no—
498C23—614no—
499C23—614no—
500C23—612no—
501C23—612no—
502C23—612no—
503C23—1224no—
504C23—1222no—
505C23—1222no—
506C23—1222no—
507C23—1224no—
508C23—1222no—
509C23—4884no—
510C23—4888no—
511C23—4884no—
512C23—4884no—
513C23—4884no—
514C23—4884no—
515C23—2444no—
516C23—2442no—
517C23—2442no—
518C23—2444no—
519C23—2444no—
520C23—2444no—
521C23—2442no—
522C23—2442no—
523C23—2442no—
524C23—2444no—
525C23C2×C22⋊#692312yes—
526C23C2×C22⋊#692312yes—
527C23D4×#692314yes—
528C23C23×#692311yescharacteristic
529D4—28no—
530D4—1228no—
531D4—1228no—
532D4—1228no—
533D4—4884no—
534D4—4884no—
535D4—2442no—
536D4—2444no—
537D4—2442no—
538D4—2448no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—612no—
546C2×C4—1222no—
547C2×C4—1224no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1228no—
551C2×C4—1222no—
552C2×C4—4884no—
553C2×C4—4888no—
554C2×C4—4888no—
555C2×C4—4888no—
556C2×C4—2444no—
557C2×C4—2448no—
558C2×C4—2448no—
559C2×C4—2442no—
560C2×C4C2×C22⋊#692312yes—
561C22—22no—
562C22—24no—
563C22—24no—
564C22—22no—
565C22—44no—
566C22—82no—
567C22—611no—
568C22—612no—
569C22—611no—
570C22—614no—
571C22—1224no—
572C22—1222no—
573C22—1222no—
574C22—1224no—
575C22—1222no—
576C22—4884no—
577C22—4888no—
578C22—4882no—
579C22—4884no—
580C22—2444no—
581C22—2444no—
582C22—2442no—
583C22—2442no—
584C22—2444no—
585C22—2444no—
586C22—2444no—
587C22—2444no—
588C22—2444no—
589C22—2442no—
590C22—2442no—
591C22—2442no—
592C22#1684071014yes—
593C22C22×C22⋊#692311yescharacteristic
594C22C2×D4×#692312yes—
595C4—22no—
596C4—42no—
597C4—42no—
598C4—1222no—
599C4—4884no—
600C4—2448no—
601C4—2442no—
602C2—22no—
603C2—24no—
604C2—42no—
605C2—1222no—
606C2—1221no—
607C2—1221no—
608C2—1224no—
609C2—4882no—
610C2—2442no—
611C2C2×#1684071011yescharacteristic
612C2#2146241912yes—
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#123×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840323C2
cosocleC23×D61C24
hypercenterC2×#123#6923
hypocenterC61#20984
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#16840829C2semidirect
2. C22×#14519836C2semidirect
3. #21462291C2semidirect
4. #21461843C2semidirect
5. #21461841C2semidirect
6. #21461516C2semidirect
10. C2×#14519856C22semidirect
13. C22×C22⋊#6923C22semidirect
17. #16840705C22semidirect
18. #16840703C22semidirect
19. #16840640C22semidirect
25. C2×#14519760C4semidirect
26. C2×#14519752C4semidirect
27. C2×#14519750C4semidirect
28. C2×#14519747C4semidirect
30. C2×#14519745C4semidirect
32. C2×#123×C61C4semidirect
47. C2×C22⋊#6923D4semidirect
53. C24×D61C2×C4semidirect
61. C22×D244C2×C4semidirect
62. C22×D244C2×C4semidirect
66. C22×C22⋊D61C2×C4semidirect
71. C22×C4×D61C2×C4semidirect
82. C2×#11758196C2×C4semidirect
83. C2×C22⋊Dic61C2×C4semidirect
84. C2×#11758198C2×C4semidirect
85. C2×#11758198C2×C4semidirect
86. C2×#11758198C2×C4semidirect
91. C23×C4×C61C2×C4semidirect
93. C2×C4⋊C4×C61C2×C4semidirect
94. C2×C22⋊C4×C61C2×C4semidirect
95. C2×C22⋊C4×C61C2×C4semidirect
320. C2×#123#6923semidirect
339. C61#20984semidirect
364. C23×C4C2×#6923semidirect
379. C2×C4⋊C4C2×#6923semidirect
398. C2×C22⋊C4C2×#6923semidirect
399. C2×C22⋊C4C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168408292
C22×#145198362
#214622912
#214618432
#214618412
#214615162
C2×#168403232
#2098461