Finite Group Databasebeta

← #29681785

Subgroups — #29681785

Order 15616.

1830 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459152C211yesmaximal, characteristic
2C2×#16840230C211yesmaximal, characteristic, complemented
3C2×#16839780C211yesmaximal, characteristic, complemented
4C22×#14519703C211yesmaximal, characteristic, complemented
5#21454263C211yesmaximal, characteristic, complemented
6#21446266C214yesmaximal, complemented
7#21446261C214yesmaximal, complemented
8C2×#16839521C211yesmaximal, characteristic, complemented
9C61×#1733C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758222C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13#16840343C414yescomplemented
14#16840338—21no—
15#16840338C412yescomplemented
16#16840331—21no—
17#16840331C412yescomplemented
18#16840324—22no—
19#16840312—22no—
20#16840274—22no—
21#16840274C2214yes—
22#16840230—24no—
23#16840230—24no—
24C2×#14519742—21no—
25C2×#14519739—21no—
26C2×#14519737—22no—
27C2×#14519734C2212yescomplemented
28C22×C4×Dic61C2211yescharacteristic
29C2×C42×D61C2212yescomplemented
30#16839780—24no—
31C2×#14519705C2214yes—
32C2×#14519703—24no—
33C2×#14519703C2212yes—
34C2×#14519703C2212yes—
35#16839730—22no—
36#16839728—22no—
37#16839725C2214yescomplemented
38#16839521C2212yescomplemented
39#16839521C2212yescomplemented
40C22×M16×C61C2211yescharacteristic
41C2×#117×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C61×#339C2214yescomplemented
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—21no—
49C2×#11758222—21no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222—22no—
53C2×#11758222—22no—
54C2×#11758222C2×C412yes—
55C2×#11758222C2×C412yes—
56C2×#11758222C2×C412yescomplemented
57C2×#11758222D412yes—
58C2×#11758222D412yes—
59C2×#11758222C2312yes—
60C22×Dic122C2×C411yescharacteristic, complemented
61C22×Dic122D411yescharacteristic
62C22×D244C2×C411yescharacteristic, complemented
63C22×D244D411yescharacteristic
64C22×C22⋊D61—21no—
65C22×C22⋊D61D412yes—
66C23×Dic61—21no—
67C22×C4×D61—22no—
68C22×C4×D61—21no—
69C22×C4×D61C2312yes—
70C2×#11758194—21no—
71C2×#11758194—22no—
72C2×#11758194—22no—
73C2×#11758194C2312yes—
74C22×#92418—21no—
75#14519777—22no—
76#14519774—22no—
77#14519765—22no—
78#14519765—22no—
79#14519761—24no—
80#14519761—24no—
81#14519757—24no—
82#14519757C2×C418yescomplemented
83#14519752—24no—
84#14519752C2×C418yescomplemented
85#14519751—28no—
86#14519747—24no—
87#14519747—24no—
88#14519742—28no—
89C4×D244—28no—
90#14519739—28no—
91#14519737—216no—
92#14519734—24no—
93#14519734—24no—
94C2×#11758196—22no—
95C2×#11758198—22no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—22no—
98C2×C4×Dic61—22no—
99C2×C4×Dic61D414yescomplemented
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102C42×D61—24no—
103C42×D61—24no—
104#14519705—28no—
105#14519703—24no—
106#14519703—24no—
107#14519689—28no—
108C23×C4×C61C2311yescharacteristic
109C2×M16×C61—21no—
110C2×M16×C61—22no—
111C2×M16×C61—22no—
112C2×M16×C61C2312yes—
113C22×C8×C61—21no—
114#117×C61—22no—
115#117×C61—21no—
116#117×C61—21no—
117#117×C61C2×C412yescomplemented
118#117×C61D414yescomplemented
119#117×C61C2312yes—
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61C2×C412yescomplemented
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C42×C61—21no—
125C2×C42×C61C2312yes—
126C24×C61C2×D411yescharacteristic
127C23×D61—21no—
128C23×D61C2×D412yes—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61C22×C411yescharacteristic
132C22×C4×C61C22×C411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61#4112yes—
139C22×C4×C61#4112yes—
140C22×C4×C61C2411yescharacteristic
141C2×C22⋊D61—24no—
142C2×C22⋊D61—24no—
143C2×C22⋊D61—22no—
144C2×C22⋊D61—22no—
145C2×C22⋊D61—42no—
146C2×C22⋊D61—42no—
147C2×C22⋊D61C22⋊C414yes—
148C2×C22⋊D61C22⋊C414yes—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—24no—
152C22×Dic61—21no—
153C22×Dic61—44no—
154C22×Dic61C2×D412yes—
155#11758222—24no—
156#11758222—24no—
157#11758222—24no—
158#11758222—24no—
159#11758222—24no—
160#11758222—24no—
161#11758222—24no—
162#11758222—24no—
163#11758222—24no—
164#11758222—24no—
165#11758222—22no—
166#11758222—22no—
167#11758222—22no—
168#11758222—22no—
169#11758222—42no—
170#11758222—42no—
171#11758222—42no—
172#11758222—42no—
173C2×D244—22no—
174C2×D244—22no—
175C2×D244—22no—
176C2×D244—22no—
177C2×D244—21no—
178C2×D244—28no—
179C2×D244C22⋊C412yes—
180C2×D244C22⋊C412yes—
181C2×D244C22×C412yes—
182C2×C4×D61—22no—
183C2×C4×D61—22no—
184C2×C4×D61—22no—
185C2×C4×D61—22no—
186C2×C4×D61—28no—
187C2×C4×D61—22no—
188C2×C4×D61—22no—
189C2×C4×D61—48no—
190C2×C4×D61—42no—
191C2×C4×D61—42no—
192C2×C4×D61C22×C414yes—
193C2×C4×D61C2×D414yes—
194C2×Dic122—22no—
195C2×Dic122—22no—
196C2×Dic122—22no—
197C2×Dic122—22no—
198C2×Dic122—21no—
199C2×Dic122C22⋊C412yes—
200C2×Dic122C22⋊C412yes—
201C2×Dic122C22×C412yes—
202M16×C61—22no—
203M16×C61—22no—
204M16×C61—42no—
205C2×C8×C61—22no—
206C2×C8×C61—42no—
207C4⋊C4×C61—22no—
208C4⋊C4×C61—24no—
209C4⋊C4×C61C22×C414yes—
210C22⋊C4×C61—22no—
211C22⋊C4×C61—24no—
212C22⋊C4×C61—22no—
213C42×C61—22no—
214C42×C61—22no—
215C42×C61—22no—
216C42×C61C2×D414yes—
217#11758198—28no—
218#11758198—28no—
219#11758198—28no—
220C4⋊Dic61—28no—
221#11758196—28no—
222C4×Dic61—28no—
223C4×Dic61—24no—
224C4×Dic61—21no—
225C4×Dic61—21no—
226C4×Dic61C2×D412yes—
227C4×Dic61C2×D412yes—
228#11758194—22no—
229#11758194—22no—
230#11758194—42no—
231C2×#92418—22no—
232C2×#92418—42no—
233C23×C61—42no—
234C23×C61C2×C22⋊C411yescharacteristic
235C23×C61C2×C22⋊C411yescharacteristic
236C23×C61#12012yes—
237C23×C61#12312yes—
238C23×C61C22×D411yescharacteristic
239C22×D61—22no—
240C22×D61—22no—
241C22×D61—22no—
242C22×D61—44no—
243C22×D61—42no—
244C22×D61—82no—
245C22×D61C2×C22⋊C414yes—
246C2×C4×C61—24no—
247C2×C4×C61—24no—
248C2×C4×C61—44no—
249C2×C4×C61—82no—
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C4×D412yes—
253C2×C4×C61C4×D412yes—
254C2×C4×C61C4×D412yes—
255C2×C4×C61C4×D412yes—
256C2×C4×C61#12012yes—
257C2×C4×C61#12112yes—
258C2×C4×C61#12112yes—
259C2×C4×C61#12112yes—
260C2×C4×C61#12112yes—
261C2×C4×C61#12312yes—
262C2×C4×C61C23×C411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C22⋊D61—28no—
269C22⋊D61—28no—
270C22⋊D61—42no—
271C22⋊D61—42no—
272C22⋊D61—44no—
273C2×Dic61—24no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—22no—
279C2×Dic61—48no—
280C2×Dic61—42no—
281C2×Dic61—42no—
282C2×Dic61C2×C22⋊C414yes—
283D244—22no—
284D244—22no—
285D244—24no—
286D244—24no—
287D244—42no—
288D244—88no—
289C4×D61—24no—
290C4×D61—24no—
291C4×D61—24no—
292C4×D61—24no—
293C4×D61—416no—
294C4×D61—42no—
295C4×D61—42no—
296C4×D61—44no—
297Dic122—24no—
298Dic122—24no—
299Dic122—22no—
300Dic122—22no—
301Dic122—42no—
302C8×C61—42no—
303#92418—42no—
304#20980—611nomaximal, Sylow
305C22×C61—22no—
306C22×C61—22no—
307C22×C61—22no—
308C22×C61—42no—
309C22×C61—42no—
310C22×C61#38612yes—
311C22×C61#38612yes—
312C22×C61C22×C22⋊C411yescharacteristic
313C22×C61C2×#12011yescharacteristic
314C22×C61C2×#12311yescharacteristic
315C2×D61—28no—
316C2×D61—24no—
317C2×D61—44no—
318C2×D61—44no—
319C2×D61—44no—
320C2×D61—42no—
321C2×D61—84no—
322C4×C61—44no—
323C4×C61—44no—
324C4×C61#38612yes—
325C4×C61#38612yes—
326C4×C61C2×C4×D411yescharacteristic
327C4×C61C2×C4×D411yescharacteristic
328C4×C61C2×#12111yescharacteristic
329C4×C61C2×#12111yescharacteristic
330Dic61—24no—
331Dic61—24no—
332Dic61—42no—
333Dic61—42no—
334#3449—611no—
335C2×C4×#41—611no—
336C2×#421—611no—
337C22×#104—611no—
338#2879—611no—
339#1897—614no—
340#1896—614no—
341#1733D6111yescharacteristic, complemented
342C2×#339—611no—
343C2×C61—21no—
344C2×C61—21no—
345C2×C61—22no—
346C2×C61—22no—
347C2×C61C2×#38611yescharacteristic
348C2×C61#290112yes—
349D61—44no—
350D61—82no—
351C22×#41—611no—
352C22×#41—611no—
353C22×M16—611no—
354C22×M16C2×D6111yescharacteristic
355#537—614no—
356#536—612no—
357#536—1221no—
358#535—612no—
359#535—1221no—
360#534—1222no—
361#532—1222no—
362#525—614no—
363#525—1222no—
364C4×#41—1224no—
365C4×#41—1224no—
366C2×C4×Q8—1221no—
367C2×C4×D4—1222no—
368C2×C4×D4—1221no—
369C2×#117—612no—
370C2×#117C2×D6111yescharacteristic
371C2×#117C2×D6111yescharacteristic
372C22×C42—611no—
373C22×C42—612no—
374#421—1224no—
375C2×#104—612no—
376C2×#104—612no—
377C2×#104—614no—
378C2×#104—1224no—
379#408—1222no—
380#408—1222no—
381#407—614no—
382#339—612no—
383#339—612no—
384#339C2×D6114yescomplemented
385C61#2098011yescharacteristic, complemented, Sylow
386C2×#41—612no—
387C2×#41—612no—
388C2×#41—612no—
389C2×#41—612no—
390C2×#41—612no—
391C2×#41—612no—
392C2×#41—1222no—
393C2×#41—1222no—
394C2×#41—1222no—
395C2×#41—1221no—
396C2×#41—1221no—
397C2×#41—1222no—
398C2×#41—1222no—
399C2×#41—1222no—
400C2×#41—1222no—
401C2×#41—1222no—
402C22×Q8—611no—
403C22×Q8—611no—
404C22×D4—612no—
405C22×D4—611no—
406C22×D4—611no—
407C22×D4—1221no—
408C23×C4—612no—
409C23×C4—1222no—
410C23×C4—1221no—
411C23×C4—1221no—
412C23×C4C22×D6111yescharacteristic
413C2×M16—21no—
414C2×M16—22no—
415C2×M16—22no—
416C2×M16—612no—
417C2×M16—1221no—
418C2×M16—1222no—
419C2×M16—1222no—
420C2×M16C22×D6112yes—
421C22×C8—21no—
422C22×C8—1221no—
423C4⋊Q8—1222no—
424#127—1222no—
425#124—1222no—
426#124—1222no—
427#123—1224no—
428#123—1224no—
429C22⋊Q8—618no—
430C22⋊Q8—1224no—
431#121—618no—
432#121—1224no—
433#121—1228no—
434#120—1224no—
435#120—1224no—
436C4×Q8—1228no—
437C4×D4—1228no—
438C4×D4—12216no—
439C4×D4—1228no—
440#117—22no—
441#117—21no—
442#117—21no—
443#117—1224no—
444#117—1224no—
445#117C4×D6112yescomplemented
446#117D24414yescomplemented
447#117C22×D6112yes—
448C2×C4⋊C4—21no—
449C2×C4⋊C4—1222no—
450C2×C4⋊C4C4×D6112yescomplemented
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—1222no—
454C2×C42—21no—
455C2×C42—614no—
456C2×C42—611no—
457C2×C42—611no—
458C2×C42—1224no—
459C2×C42—1222no—
460C2×C42—1222no—
461C2×C42—1221no—
462C2×C42—1224no—
463C2×C42C22×D6112yes—
464#104—1224no—
465#104—1224no—
466#104—1228no—
467C22⋊C8—1228no—
468C24—612no—
469C24—1221no—
470C24D4×D6111yescharacteristic
471#41—1224no—
472#41—1224no—
473#41—1224no—
474#41—1224no—
475#41—1224no—
476#41—1224no—
477#41—1224no—
478#41—1224no—
479#41—1222no—
480#41—1222no—
481#41—1222no—
482#41—1222no—
483#41—1224no—
484#41—1224no—
485#41—2442no—
486#41—2442no—
487#41—2442no—
488#41—2442no—
489C2×Q8—612no—
490C2×Q8—612no—
491C2×Q8—612no—
492C2×Q8—1221no—
493C2×Q8—1222no—
494C2×Q8—1222no—
495C2×Q8—1222no—
496C2×Q8—1222no—
497C2×D4—614no—
498C2×D4—614no—
499C2×D4—612no—
500C2×D4—612no—
501C2×D4—612no—
502C2×D4—1228no—
503C2×D4—1224no—
504C2×D4—1224no—
505C2×D4—1222no—
506C2×D4—1222no—
507C2×D4—1221no—
508C2×D4—1222no—
509C2×D4—1222no—
510C2×D4—1222no—
511C2×D4—1222no—
512C2×D4—2442no—
513C2×D4—2442no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—612no—
517C22×C4—614no—
518C22×C4—614no—
519C22×C4—1228no—
520C22×C4—1221no—
521C22×C4—1222no—
522C22×C4—1222no—
523C22×C4—1222no—
524C22×C4—1222no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—1224no—
528C22×C4—1222no—
529C22×C4—1222no—
530C22×C4—2448no—
531C22×C4—2444no—
532C22×C4—2442no—
533C22×C4—2442no—
534C22×C4C2×C4×D6111yescharacteristic
535C22×C4C2×C4×D6111yescharacteristic
536C22×C4C2×D24412yes—
537C22×C4#1175822212yes—
538C22×C4D4×D6112yes—
539C22×C4D4×D6111yescharacteristic
540C22×C4D4×D6111yescharacteristic
541C22×C4D4×D6111yescharacteristic
542C22×C4#1175822612yes—
543C22×C4C23×D6111yescharacteristic
544M16—22no—
545M16—22no—
546M16—42no—
547M16—1222no—
548M16—1222no—
549M16—2442no—
550C2×C8—22no—
551C2×C8—42no—
552C2×C8—1222no—
553C2×C8—2442no—
554C4⋊C4—22no—
555C4⋊C4—24no—
556C4⋊C4—1228no—
557C4⋊C4—1228no—
558C4⋊C4C2×C4×D6114yes—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—24no—
562C22⋊C4—1228no—
563C22⋊C4—1228no—
564C22⋊C4—1228no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—612no—
569C42—612no—
570C42—1228no—
571C42—1221no—
572C42—1221no—
573C42—1224no—
574C42C2×D24414yes—
575C23—42no—
576C23—614no—
577C23—1222no—
578C23—1222no—
579C23—1222no—
580C23—4882no—
581C23—2444no—
582C23—2442no—
583C23C22⋊C4×D6111yescharacteristic
584C23C22⋊C4×D6111yescharacteristic
585C23#1451974712yes—
586C23#1451976212yes—
587C23C2×D4×D6111yescharacteristic
588Q8—1224no—
589Q8—1224no—
590Q8—1222no—
591Q8—1222no—
592Q8—2442no—
593D4—1228no—
594D4—1228no—
595D4—1222no—
596D4—1222no—
597D4—1224no—
598D4—1224no—
599D4—4888no—
600D4—2442no—
601D4—2442no—
602D4—2444no—
603D4—2442no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—44no—
607C2×C4—82no—
608C2×C4—614no—
609C2×C4—1224no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—1224no—
614C2×C4—1224no—
615C2×C4—1224no—
616C2×C4—1224no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—24416no—
620C2×C4—2448no—
621C2×C4—2442no—
622C2×C4—2442no—
623C2×C4—2442no—
624C2×C4—2442no—
625C2×C4—2444no—
626C2×C4C22⋊C4×D6111yescharacteristic
627C2×C4C22⋊C4×D6111yescharacteristic
628C2×C4#1451973912yes—
629C2×C4#1451973912yes—
630C2×C4C4×D24412yes—
631C2×C4C4×D24412yes—
632C2×C4#1451974712yes—
633C2×C4#1451975012yes—
634C2×C4#1451975012yes—
635C2×C4#1451975212yes—
636C2×C4#1451975212yes—
637C2×C4#1451976212yes—
638C2×C4C22×C4×D6111yescharacteristic
639C2×C4C2×D4×D6111yescharacteristic
640C2×C4C2×D4×D6111yescharacteristic
641C2×C4C22×D24411yescharacteristic
642C2×C4C2×#1175822211yescharacteristic
643C2×C4C2×#1175822611yescharacteristic
644C8—42no—
645C8—2442no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—42no—
650C22—42no—
651C22—1228no—
652C22—1224no—
653C22—4884no—
654C22—2444no—
655C22—2444no—
656C22—2444no—
657C22—2442no—
658C22#1683964012yes—
659C22#1683964012yes—
660C22C2×C22⋊C4×D6111yescharacteristic
661C22C2×#1451974711yescharacteristic
662C22C2×#1451976211yescharacteristic
663C4—44no—
664C4—44no—
665C4—1224no—
666C4—1224no—
667C4—2442no—
668C4—2442no—
669C4#1683964012yes—
670C4#1683964012yes—
671C4C2×#1451973911yescharacteristic
672C4C2×C4×D24411yescharacteristic
673C4C2×#1451975011yescharacteristic
674C4C2×#1451975211yescharacteristic
675C2—21no—
676C2—22no—
677C2—21no—
678C2—22no—
679C2—4882no—
680C2—2444no—
681C2C2×#1683964011yescharacteristic
682C2#2145436512yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1733C2
socleC22×C61C22×C22⋊C4
prime radical#21459152C2
cosocleC22×C4×C61C24
hypercenter#1733D61
hypocenterC61#20980
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
2. C2×#16840230C2semidirect
3. C2×#16839780C2semidirect
4. C22×#14519703C2semidirect
5. #21454263C2semidirect
6. #21446266C2semidirect
7. #21446261C2semidirect
8. C2×#16839521C2semidirect
9. C61×#1733C2semidirect
13. #16840343C4semidirect
15. #16840338C4semidirect
17. #16840331C4semidirect
27. C2×#14519734C22semidirect
29. C2×C42×D61C22semidirect
37. #16839725C22semidirect
38. #16839521C22semidirect
39. #16839521C22semidirect
43. C61×#339C22semidirect
56. C2×#11758222C2×C4semidirect
60. C22×Dic122C2×C4semidirect
62. C22×D244C2×C4semidirect
82. #14519757C2×C4semidirect
84. #14519752C2×C4semidirect
99. C2×C4×Dic61D4semidirect
117. #117×C61C2×C4semidirect
118. #117×C61D4semidirect
121. C2×C4⋊C4×C61C2×C4semidirect
341. #1733D61semidirect
384. #339C2×D61semidirect
385. C61#20980semidirect
445. #117C4×D61semidirect
446. #117D244semidirect
450. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214591522
C2×#168402302
C2×#168397802
C22×#145197032
#214542632
#214462662
#214462612
C2×#168395212
C61×#17332
#2098061