Finite Group Databasebeta

← #29684082

Subgroups — #29684082

Order 15616.

1706 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840311C211yesmaximal, characteristic
2C2×#16839784C211yesmaximal, characteristic, complemented
3C2×#16839762C212yesmaximal, complemented
4C2×#16839760C212yesmaximal, complemented
5C4×C4⋊C4×D61C211yesmaximal, characteristic, complemented
6#21446752C211yesmaximal, characteristic, complemented
7#21446751C211yesmaximal, characteristic, complemented
8#21446749C211yesmaximal, characteristic, complemented
9#21446744C212yesmaximal, complemented
10#21446740C212yesmaximal, complemented
11C61×#1983C211yesmaximal, characteristic, complemented
12C22×#11758226C2211yescharacteristic
13C22×#11758224C2211yescharacteristic
14C22×#11758189C2212yes—
15#16840311—24no—
16#16840311C418yescomplemented
17C2×#14519775C2211yescharacteristic
18C2×#14519773C2211yescharacteristic
19C2×#14519765C2211yescharacteristic
20C2×#14519750—21no—
21C2×#14519750—21no—
22C2×#14519737—21no—
23C2×#14519737—21no—
24C2×C4⋊C4×D61C2212yes—
25C2×C42×D61—21no—
26C2×C42×D61C2211yescharacteristic
27#16839784—24no—
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519706C2212yes—
30#16839762—28no—
31#16839760—28no—
32C2×#14519702C2212yes—
33C2×#14519701C2212yes—
34C2×#14519700C2212yes—
35C2×#14519699C2212yes—
36C2×#14519698C2212yes—
37C2×#14519697C2212yes—
38C4⋊C4×Dic61C2212yes—
39C4×#11758196—21no—
40C4×C4⋊Dic61C2211yescharacteristic
41#16839613—21no—
42#95×D61—21no—
43C2×#124×C61C2211yescharacteristic
44C2×C4⋊C8×C61C2211yescharacteristic
45C2×#103×C61C2212yes—
46C2×#102×C61C2212yes—
47C4×C4⋊C4×C61C2211yescharacteristic
48C2×#11758226—22no—
49C2×#11758226—24no—
50C2×#11758226—24no—
51C2×#11758226C2×C414yes—
52C2×#11758224—24no—
53C2×#11758224—22no—
54C2×#11758224—24no—
55C2×#11758224C2×C414yes—
56C22×Dic122C2311yescharacteristic
57C22×D244—21no—
58C22×D244C2311yescharacteristic
59C22×C22⋊D61—21no—
60C22×C22⋊D61—21no—
61C22×C22⋊D61—21no—
62C23×Dic61—21no—
63C22×C4×D61—22no—
64C22×C4×D61—21no—
65C22×C4×D61D411yescharacteristic
66C22×C4×D61D411yescharacteristic
67C22×C4×D61C2311yescharacteristic
68C2×#11758189—24no—
69C2×#11758189—24no—
70C2×#11758189—24no—
71C22×#92418C2312yes—
72#14519775—22no—
73#14519775C2×C414yescomplemented
74#14519773—22no—
75#14519773C2×C414yescomplemented
76#14519765—22no—
77#14519765C2×C414yescomplemented
78#14519750—28no—
79#14519750—28no—
80#14519737—28no—
81#14519737—28no—
82C2×C4⋊Dic61C2312yes—
83C2×#11758196—22no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C4⋊C4×D61—24no—
87C4⋊C4×D61D418yescomplemented
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—22no—
92C2×C4×Dic61D411yescharacteristic
93C2×C4×Dic61D411yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95C42×D61—44no—
96C42×D61C2×C414yes—
97C42×D61D414yes—
98C4⋊#92418—22no—
99#14519706—24no—
100#14519702—24no—
101#14519701—24no—
102#14519700—24no—
103#14519699—24no—
104#14519698—24no—
105#14519697—24no—
106#14519684—21no—
107#14519684—22no—
108C22×Q8×C61C2311yescharacteristic
109C22×D4×C61C2311yescharacteristic
110C22×C8×C61C2312yes—
111#124×C61—22no—
112#124×C61C2×C414yescomplemented
113C2×C4⋊C4×C61C2312yes—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118C4⋊C8×C61—22no—
119#103×C61—24no—
120#102×C61—24no—
121#95×C61—21no—
122C24×C61—21no—
123C23×D61—21no—
124C23×D61C2×D411yescharacteristic
125C2×Q8×C61—21no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61C22×C412yes—
129C2×D4×C61—22no—
130C2×D4×C61—21no—
131C2×D4×C61—22no—
132C2×D4×C61C22×C412yes—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2411yescharacteristic
138C2×C22⋊D61—24no—
139C2×C22⋊D61—24no—
140C2×C22⋊D61—24no—
141C2×C22⋊D61—44no—
142C2×C22⋊D61—44no—
143C2×C22⋊D61—42no—
144C2×C22⋊D61—42no—
145C22×Dic61—22no—
146C22×Dic61—21no—
147C22×Dic61—24no—
148C22×Dic61—44no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152#11758226—216no—
153#11758226—44no—
154#11758226—44no—
155#11758224—216no—
156#11758224—44no—
157#11758224—44no—
158C2×D244—22no—
159C2×D244—22no—
160C2×D244—24no—
161C2×D244—21no—
162C2×D244—44no—
163C2×D244C22×C412yes—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—28no—
167C2×C4×D61—24no—
168C2×C4×D61—42no—
169C2×C4×D61—44no—
170C2×C4×D61—44no—
171C2×C4×D61—44no—
172C2×C4×D61C22⋊C414yes—
173C2×C4×D61C22⋊C414yes—
174C2×C4×D61C22×C412yes—
175C2×C4×D61C2×D414yes—
176C2×C4×D61C2×D412yes—
177C2×C4×D61#4114yes—
178C2×Dic122—22no—
179C2×Dic122—22no—
180C2×Dic122—21no—
181C2×Dic122C22×C412yes—
182C2×C8×C61—22no—
183C2×C8×C61—22no—
184C2×C8×C61—22no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61C2×D414yes—
187C22⋊C4×C61—24no—
188C22⋊C4×C61—24no—
189C42×C61—42no—
190C42×C61C22×C412yes—
191C42×C61C2×D412yes—
192#11758198—24no—
193#11758198—24no—
194C4⋊Dic61—22no—
195C4⋊Dic61C2×D414yes—
196#11758196—24no—
197#11758196—24no—
198#11758196—28no—
199C4×Dic61—42no—
200C4×Dic61—44no—
201C4×Dic61C22⋊C414yes—
202C4×Dic61C22⋊C414yes—
203C4×Dic61C22×C412yes—
204C4×Dic61C2×D412yes—
205C2×#92418—22no—
206C2×#92418—22no—
207C2×#92418—22no—
208#11758189—48no—
209C23×C61—22no—
210C23×C61—42no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—42no—
214C23×C61C22×D411yescharacteristic
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—22no—
218C22×D61—22no—
219C22×D61—42no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—42no—
223C22×D61C2×C22⋊C412yes—
224C22×D61#12114yes—
225Q8×C61—24no—
226Q8×C61—41no—
227Q8×C61—41no—
228D4×C61—24no—
229D4×C61—41no—
230D4×C61—41no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—24no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61C2×C22⋊C412yes—
240C2×C4×C61C2×C22⋊C412yes—
241C2×C4×C61C23×C411yescharacteristic
242C2×C4×C61C22×D412yes—
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C2×#4112yes—
245C22⋊D61—48no—
246C22⋊D61—48no—
247C22⋊D61—48no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—24no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—48no—
255C2×Dic61C2×C22⋊C412yes—
256C2×Dic61C2×C22⋊C412yes—
257C2×Dic61C2×C22⋊C412yes—
258C2×Dic61#12114yes—
259C2×Dic61AGL(1,ℤ/8)12yes—
260C2×Dic61#13712yes—
261D244—24no—
262D244—48no—
263D244—41no—
264D244—41no—
265C4×D61—28no—
266C4×D61—44no—
267C4×D61—416no—
268C4×D61—48no—
269C4×D61C4×D418yes—
270C4×D61#12414yes—
271C4×D61#12714yes—
272Dic122—24no—
273Dic122—41no—
274Dic122—41no—
275C8×C61—42no—
276#92418—42no—
277#21258—611nomaximal, Sylow
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—82no—
282C22×C61—82no—
283C22×C61C22×C22⋊C411yescharacteristic
284C22×C61C2×#12112yes—
285C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C61C2×#13711yescharacteristic
287C22×C61#57512yes—
288C2×D61—22no—
289C2×D61—22no—
290C2×D61—24no—
291C2×D61—24no—
292C2×D61—42no—
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—82no—
296C2×D61—82no—
297C2×D61#39014yes—
298C4×C61—22no—
299C4×C61—22no—
300C4×C61—44no—
301C4×C61—42no—
302C4×C61C2×C4×D412yes—
303C4×C61C2×#12411yescharacteristic
304C4×C61C2×#12711yescharacteristic
305Dic61—22no—
306Dic61—44no—
307Dic61—42no—
308Dic61#39014yes—
309Dic61#41714yescomplemented
310C2×#532—611no—
311C2×#423—611no—
312C2×#417—612no—
313C2×#417—612no—
314C2×C4×C4⋊C4—611no—
315#1983—611no—
316#1983—611no—
317#1983—611no—
318#1983D6111yescharacteristic, complemented
319#1982—612no—
320#1981—612no—
321C2×C61—82no—
322C2×C61C2×#39011yescharacteristic
323C2×C61C2×#41711yescharacteristic
324C2×C61#289611yescharacteristic
325C2×C61#303612yes—
326C2×C61#303712yes—
327D61—24no—
328D61—82no—
329C22×#41—611no—
330C22×#41—611no—
331C22×M16—612no—
332#532—618no—
333#532—1224no—
334C2×C4⋊Q8—611no—
335C2×#127—611no—
336C2×#124—611no—
337C2×#124C2×D6111yescharacteristic
338C2×#121—1221no—
339C2×#121—1221no—
340C2×C4×D4—1221no—
341C2×C4×D4—1221no—
342C22×C4⋊C4—612no—
343C22×C42—611no—
344C22×C42—1221no—
345#423—1224no—
346C2×C4⋊C8—612no—
347C2×C4⋊C8—611no—
348C2×C4⋊C8C2×D6111yescharacteristic
349#417—1228no—
350#417—1228no—
351C2×#103—612no—
352C2×#103—612no—
353C2×#103—612no—
354C2×#103C2×D6112yes—
355C2×#102—612no—
356C2×#102—612no—
357C2×#102—612no—
358C2×#102C2×D6112yes—
359C4×C4⋊C4—612no—
360C4×C4⋊C4—611no—
361C4×C4⋊C4—1221no—
362C4×C4⋊C4—1221no—
363C4×C4⋊C4C2×D6111yescharacteristic
364C2×#95—1221no—
365C61#2125811yescharacteristic, complemented, Sylow
366C2×#41—614no—
367C2×#41—614no—
368C2×#41—1222no—
369C2×#41—1222no—
370C2×#41—1224no—
371C2×#41—1224no—
372C2×#41—1224no—
373C2×#41—1224no—
374C22×Q8—611no—
375C22×Q8C22×D6111yescharacteristic
376C22×D4—611no—
377C22×D4—1221no—
378C22×D4—1221no—
379C22×D4—1221no—
380C22×D4—1221no—
381C22×D4C22×D6111yescharacteristic
382C23×C4—611no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—1221no—
386C23×C4—1222no—
387C23×C4—1221no—
388C2×M16—1224no—
389C2×M16—1224no—
390C2×M16—1224no—
391C22×C8—612no—
392C22×C8C22×D6112yes—
393C4⋊Q8—614no—
394C4⋊Q8—1222no—
395#127—614no—
396#127—1222no—
397#124—22no—
398#124—614no—
399#124—1222no—
400#124C4×D6114yescomplemented
401#121—1228no—
402#121—1228no—
403C4×D4—1228no—
404C4×D4—1228no—
405C2×C4⋊C4—612no—
406C2×C4⋊C4—618no—
407C2×C4⋊C4—1224no—
408C2×C4⋊C4—1222no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4C22×D6112yes—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C42—21no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—614no—
421C2×C42—614no—
422C2×C42—1222no—
423C2×C42—1221no—
424C2×C42—2444no—
425C2×C42C22×D6111yescharacteristic
426C4⋊C8—22no—
427C4⋊C8—1224no—
428C4⋊C8—1222no—
429#103—24no—
430#103—1224no—
431#103—1224no—
432#103—1224no—
433#102—24no—
434#102—1224no—
435#102—1224no—
436#102—1224no—
437#95—21no—
438#95—1222no—
439#95—1221no—
440C24—21no—
441C24—611no—
442C24—1221no—
443#41—12216no—
444#41—12216no—
445#41—2444no—
446#41—2444no—
447#41—2444no—
448#41—2444no—
449C2×Q8—21no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—612no—
453C2×Q8—1221no—
454C2×Q8—1222no—
455C2×Q8—1222no—
456C2×Q8C2×C4×D6112yes—
457C2×D4—22no—
458C2×D4—21no—
459C2×D4—22no—
460C2×D4—612no—
461C2×D4—1224no—
462C2×D4—1224no—
463C2×D4—1224no—
464C2×D4—1224no—
465C2×D4—1222no—
466C2×D4—1221no—
467C2×D4—1222no—
468C2×D4—2444no—
469C2×D4—2442no—
470C2×D4—2442no—
471C2×D4—2444no—
472C2×D4—2444no—
473C2×D4C2×C4×D6112yes—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—614no—
481C22×C4—614no—
482C22×C4—611no—
483C22×C4—614no—
484C22×C4—614no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1224no—
488C22×C4—1228no—
489C22×C4—1224no—
490C22×C4—1222no—
491C22×C4—1221no—
492C22×C4—2442no—
493C22×C4—2444no—
494C22×C4—2444no—
495C22×C4—2444no—
496C22×C4—2444no—
497C22×C4D4×D6111yescharacteristic
498C22×C4D4×D6111yescharacteristic
499C22×C4C23×D6111yescharacteristic
500M16—2448no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1222no—
507C4⋊C4—22no—
508C4⋊C4—614no—
509C4⋊C4—1228no—
510C4⋊C4—1224no—
511C4⋊C4—1224no—
512C4⋊C4—1222no—
513C4⋊C4D4×D6114yes—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—1224no—
517C22⋊C4—1224no—
518C42—42no—
519C42—612no—
520C42—612no—
521C42—614no—
522C42—614no—
523C42—2442no—
524C42—2444no—
525C42C2×C4×D6112yes—
526C42D4×D6112yes—
527C23—22no—
528C23—42no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—612no—
533C23—614no—
534C23—1221no—
535C23—1221no—
536C23—1222no—
537C23—1222no—
538C23—2442no—
539C23—2441no—
540C23—2441no—
541C23—2442no—
542C23C2×D4×D6111yescharacteristic
543Q8—24no—
544Q8—41no—
545Q8—41no—
546Q8—1224no—
547Q8—2441no—
548Q8—2441no—
549D4—24no—
550D4—41no—
551D4—41no—
552D4—1224no—
553D4—2448no—
554D4—2448no—
555D4—2448no—
556D4—2448no—
557D4—2441no—
558D4—2441no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—612no—
568C2×C4—612no—
569C2×C4—614no—
570C2×C4—612no—
571C2×C4—612no—
572C2×C4—618no—
573C2×C4—614no—
574C2×C4—614no—
575C2×C4—612no—
576C2×C4—1228no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1224no—
580C2×C4—2444no—
581C2×C4—2448no—
582C2×C4—24416no—
583C2×C4—2448no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4C22⋊C4×D6112yes—
588C2×C4C22⋊C4×D6112yes—
589C2×C4C22×C4×D6111yescharacteristic
590C2×C4C2×D4×D6112yes—
591C2×C4C2×D4×D6111yescharacteristic
592C2×C4#41×D6112yes—
593C8—42no—
594C8—2442no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—82no—
599C22—82no—
600C22—614no—
601C22—1224no—
602C22—1224no—
603C22—1222no—
604C22—1222no—
605C22—4882no—
606C22—4882no—
607C22—2442no—
608C22—2442no—
609C22—2442no—
610C22C2×C22⋊C4×D6111yescharacteristic
611C22#121×D6112yes—
612C22AGL(1,ℤ/8)×D6111yescharacteristic
613C22#137×D6111yescharacteristic
614C22#1684055912yes—
615C4—22no—
616C4—22no—
617C4—42no—
618C4—44no—
619C4—614no—
620C4—614no—
621C4—1222no—
622C4—2442no—
623C4—2444no—
624C4C4×D4×D6112yes—
625C4#124×D6111yescharacteristic
626C4#127×D6111yescharacteristic
627C2—82no—
628C2—1224no—
629C2—4882no—
630C2#390×D6111yescharacteristic
631C2#417×D6111yescharacteristic
632C2#2145433411yescharacteristic
633C2#2145528112yes—
634C2#2145528912yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1983C2
socleC23×C61C22×D4
prime radicalC2×#16840311C2
cosocleC22×C4×C61C24
hypercenter#1983D61
hypocenterC61#21258
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1983

Complements

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

NH ≅ G/Ntype
2. C2×#16839784C2semidirect
3. C2×#16839762C2semidirect
4. C2×#16839760C2semidirect
5. C4×C4⋊C4×D61C2semidirect
6. #21446752C2semidirect
7. #21446751C2semidirect
8. #21446749C2semidirect
9. #21446744C2semidirect
10. #21446740C2semidirect
11. C61×#1983C2semidirect
16. #16840311C4semidirect
73. #14519775C2×C4semidirect
75. #14519773C2×C4semidirect
77. #14519765C2×C4semidirect
87. C4⋊C4×D61D4semidirect
112. #124×C61C2×C4semidirect
309. Dic61#417semidirect
318. #1983D61semidirect
365. C61#21258semidirect
400. #124C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403112
C2×#168397842
C2×#168397622
C2×#168397602
C4×C4⋊C4×D612
#214467522
#214467512
#214467492
#214467442
#214467402
C61×#19832
#2125861