Finite Group Databasebeta

← #29680150

Subgroups — #29680150

Order 15616.

1258 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459018C211yesmaximal, characteristic
2#21454455C211yesmaximal, characteristic, complemented
3#21448314C211yesmaximal, characteristic, complemented
4#21445917C212yesmaximal, complemented
5#21445838C212yesmaximal, complemented
6#21445837C212yesmaximal, complemented
7#21445761C212yesmaximal
8#21445630C211yesmaximal, characteristic, complemented
9C4×#14519703C211yesmaximal, characteristic, complemented
10C4×#14519704C211yesmaximal, characteristic, complemented
11C61×#1759C211yesmaximal, characteristic, complemented
12C2×#41×D61C2211yescharacteristic
13#16840244C412yescomplemented
14#16840241—22no—
15#16840241C412yescomplemented
16#16840238C412yescomplemented
17#16840234—22no—
18#16840234C412yescomplemented
19#41×Dic61C2211yescharacteristic
20C4×#11758222—21no—
21C4×#11758222C2211yescharacteristic
22#16840230—21no—
23#16840230C2211yescharacteristic
24C2×#14519734—21no—
25C2×#14519734C2211yescharacteristic
26#117×D61—21no—
27#117×D61C2211yescharacteristic
28#16839824—21no—
29#16839824—21no—
30#16839821—21no—
31#16839821—21no—
32#16839785C2212yescomplemented
33C2×#14519705C2212yes—
34C2×#14519704C2212yes—
35C2×#14519703C2212yes—
36#16839725C2212yes—
37C4×#11758194C2211yescharacteristic, complemented
38#16839651—22no—
39#16839644—22no—
40C4×#11758196C2212yescomplemented
41C4×#11758198C2212yes—
42C42×Dic61C2211yescharacteristic, complemented
43#16839523C2212yescomplemented
44#16839522C2212yescomplemented
45#16839521C2212yescomplemented
46C4×#41×C61C2211yescharacteristic
47C2×#104×C61C2212yes—
48C4×M16×C61C2211yescharacteristic, complemented
49C61×#376C2211yescharacteristic, complemented
50C61×#339C2212yescomplemented
51C2×#11758226C2×C411yescharacteristic
52C2×#11758222—21no—
53C2×#11758222C2311yescharacteristic
54C2×#11758224—21no—
55C2×#11758224C2×C411yescharacteristic
56#41×D61—24no—
57#41×D61—24no—
58C2×Q8×D61C2×C411yescharacteristic
59C2×D4×D61—21no—
60C2×D4×D61C2×C411yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61C2311yescharacteristic
63C2×#11758194C2312yes—
64Q8×Dic61C2×C412yes—
65C4×Dic122—22no—
66C4×Dic122C2×C412yes—
67#14519742—22no—
68#14519742C2×C412yes—
69C4×D244—22no—
70C4×D244C2×C412yes—
71D4×Dic61—22no—
72D4×Dic61C2×C412yes—
73#14519739—22no—
74#14519739C2×C412yes—
75C4×C22⋊D61—22no—
76C4×C22⋊D61—22no—
77C4×C22⋊D61—22no—
78#14519737—22no—
79#14519737—22no—
80#14519737—22no—
81#14519736—21no—
82#14519736C2311yescharacteristic
83#14519735—22no—
84#14519735C2×C412yes—
85#14519734—21no—
86#14519734—21no—
87#14519734—22no—
88#14519734—22no—
89#14519734—21no—
90#14519734C2×C412yes—
91#14519734C2×C412yes—
92#14519734D411yescharacteristic
93#14519734D411yescharacteristic
94#14519734D411yescharacteristic
95#14519734D411yescharacteristic
96#14519733—21no—
97#14519733—22no—
98C2×#11758196—22no—
99C2×#11758196—22no—
100C2×#11758196D412yes—
101C4⋊C4×D61—22no—
102C4⋊C4×D61C2×C412yes—
103C2×#11758198—22no—
104C2×#11758198—22no—
105C2×#11758198D412yes—
106C22⋊C4×D61—21no—
107C22⋊C4×D61—22no—
108C2×C4×Dic61—21no—
109C2×C4×Dic61—22no—
110C2×C4×Dic61C2312yes—
111C2×C4×Dic61C2311yescharacteristic
112C42×D61—21no—
113C42×D61—21no—
114C42×D61—21no—
115#14519706—22no—
116#14519706—22no—
117#14519705—24no—
118#14519704—24no—
119#14519703—24no—
120#14519689—22no—
121#14519689—22no—
122#14519688—21no—
123C4×#92418—21no—
124#14519684—22no—
125#14519684—22no—
126C2×#41×C61C2311yescharacteristic
127C2×M16×C61C2312yes—
128C4×Q8×C61C2×C412yes—
129C4×D4×C61—22no—
130C4×D4×C61C2×C412yes—
131#117×C61—21no—
132#117×C61C2311yescharacteristic
133C2×C42×C61—21no—
134C2×C42×C61C2312yes—
135C2×C42×C61C2311yescharacteristic
136#104×C61—24no—
137#97×C61—21no—
138C4×C8×C61—21no—
139#95×C61—22no—
140C23×D61—21no—
141C23×D61C2×D411yescharacteristic
142#41×C61—22no—
143#41×C61#4114yes—
144C2×Q8×C61C22×C411yescharacteristic
145C2×D4×C61—21no—
146C2×D4×C61C22×C411yescharacteristic
147C22×C4×C61—21no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2411yescharacteristic
152C2×C22⋊D61—21no—
153C2×C22⋊D61—21no—
154C2×C22⋊D61—21no—
155C22×Dic61—21no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D412yes—
160#11758226—22no—
161#11758226—22no—
162Q8×D61—22no—
163Q8×D61—22no—
164#11758224—28no—
165#11758224—22no—
166#11758224—22no—
167D4×D61—28no—
168D4×D61—22no—
169D4×D61—22no—
170#11758222—28no—
171#11758222—22no—
172#11758222#4114yes—
173C2×D244—21no—
174C2×D244C22×C411yescharacteristic
175C2×C4×D61—28no—
176C2×C4×D61—21no—
177C2×C4×D61—21no—
178C2×C4×D61—22no—
179C2×C4×D61—22no—
180C2×C4×D61—21no—
181C2×C4×D61—22no—
182C2×C4×D61—22no—
183C2×C4×D61—22no—
184C2×C4×D61C22×C411yescharacteristic
185C2×C4×D61C22×C411yescharacteristic
186C2×C4×D61C2×D411yescharacteristic
187C2×C4×D61C2×D411yescharacteristic
188C2×Dic122—21no—
189C2×Dic122C22×C411yescharacteristic
190M16×C61—22no—
191M16×C61—22no—
192C2×C8×C61—22no—
193C4⋊C4×C61—22no—
194C4⋊C4×C61C22×C412yes—
195C22⋊C4×C61—21no—
196C22⋊C4×C61—22no—
197C42×C61—22no—
198C42×C61—21no—
199C42×C61—22no—
200C42×C61—22no—
201C42×C61C22×C412yes—
202C42×C61C2×D411yescharacteristic
203C42×C61C2×D411yescharacteristic
204C22⋊Dic61—21no—
205C22⋊Dic61—22no—
206#11758198—22no—
207#11758198—22no—
208#11758198—22no—
209#11758198—44no—
210#11758198C22⋊C412yes—
211#11758198C22⋊C412yes—
212#11758198C22⋊C412yes—
213#11758198C22⋊C412yes—
214C4⋊Dic61—22no—
215C4⋊Dic61C22×C412yes—
216#11758196—22no—
217#11758196—22no—
218#11758196—22no—
219#11758196—44no—
220#11758196C22⋊C412yes—
221#11758196C22⋊C412yes—
222#11758196C22⋊C412yes—
223#11758196C22⋊C412yes—
224C4×Dic61—22no—
225C4×Dic61—21no—
226C4×Dic61—22no—
227C4×Dic61—22no—
228C4×Dic61—22no—
229C4×Dic61—22no—
230C4×Dic61—22no—
231C4×Dic61—44no—
232C4×Dic61C22×C412yes—
233C4×Dic61C2×D411yescharacteristic
234C4×Dic61C2×D411yescharacteristic
235#11758194—22no—
236#11758194—22no—
237C2×#92418—22no—
238C23×C61—21no—
239C23×C61C22×D411yescharacteristic
240C22×D61—21no—
241C22×D61—24no—
242C22×D61—22no—
243C22×D61—42no—
244C22×D61—82no—
245C22×D61C2×C22⋊C411yescharacteristic
246C22×D61C2×C22⋊C411yescharacteristic
247Q8×C61—21no—
248Q8×C61#11712yes—
249D4×C61—24no—
250D4×C61—21no—
251D4×C61#11712yes—
252C2×C4×C61—21no—
253C2×C4×C61—24no—
254C2×C4×C61—22no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61—42no—
258C2×C4×C61—42no—
259C2×C4×C61—82no—
260C2×C4×C61C2×C22⋊C412yes—
261C2×C4×C61C2×C22⋊C412yes—
262C2×C4×C61C23×C411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C2×#4112yes—
265C2×C4×C61C2×#4112yes—
266C22⋊D61—24no—
267C22⋊D61—24no—
268C22⋊D61—24no—
269C2×Dic61—24no—
270C2×Dic61—21no—
271C2×Dic61—22no—
272C2×Dic61—24no—
273C2×Dic61—22no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—82no—
278C2×Dic61C2×C22⋊C411yescharacteristic
279C2×Dic61C2×C22⋊C412yes—
280C2×Dic61C2×C22⋊C412yes—
281C2×Dic61C2×C22⋊C411yescharacteristic
282D244—24no—
283D244—21no—
284D244#11712yes—
285C4×D61—24no—
286C4×D61—24no—
287C4×D61—21no—
288C4×D61—21no—
289C4×D61—21no—
290C4×D61—21no—
291C4×D61—22no—
292C4×D61—24no—
293C4×D61—22no—
294C4×D61—82no—
295C4×D61—82no—
296C4×D61—82no—
297Dic122—24no—
298Dic122—21no—
299Dic122#11712yes—
300C8×C61—42no—
301#92418—42no—
302#20749—611nomaximal, Sylow
303C22×C61—22no—
304C22×C61—22no—
305C22×C61—42no—
306C22×C61C22×C22⋊C411yescharacteristic
307C22×C61#52512yes—
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—21no—
311C2×D61—21no—
312C2×D61—22no—
313C2×D61—42no—
314C2×D61—82no—
315C2×D61—84no—
316C4×C61—22no—
317C4×C61—22no—
318C4×C61—22no—
319C4×C61—42no—
320C4×C61—82no—
321C4×C61C2×#11711yescharacteristic
322C4×C61C2×#11711yescharacteristic
323C4×C61#52512yes—
324Dic61—22no—
325Dic61—22no—
326Dic61—21no—
327Dic61—22no—
328Dic61—21no—
329Dic61—44no—
330Dic61—42no—
331#3428—611no—
332#2921—611no—
333#2298—611no—
334#1826—612no—
335#1806—612no—
336#1806—612no—
337#1789—612no—
338#1759—611no—
339#1759D6111yescharacteristic, complemented
340C4×#104—611no—
341C4×#104—611no—
342C2×C61—22no—
343C2×C61—42no—
344C2×C61#228011yescharacteristic
345C2×C61#290112yes—
346D61—41no—
347D61—41no—
348D61—42no—
349C22×#41—611no—
350#520—612no—
351#520—612no—
352#520—1222no—
353#519—612no—
354#518—612no—
355#518—1222no—
356C4×#41—611no—
357C4×#41—611no—
358C4×#41—611no—
359C4×#41—1221no—
360C4×#41—1221no—
361C4×#41C2×D6111yescharacteristic
362C2×#117—611no—
363C2×#117—611no—
364C2×#117—1221no—
365C2×#117—1221no—
366#433—1221no—
367#433—1221no—
368#433—1221no—
369#433—1221no—
370#423—612no—
371C2×#104—612no—
372C2×#104—612no—
373C2×#104—612no—
374C2×#104C2×D6112yes—
375#407—612no—
376C4×M16—611no—
377C4×M16C2×D6111yescharacteristic, complemented
378#388—1222no—
379#387—1222no—
380C4×C4⋊C4—612no—
381C4×C22⋊C4—612no—
382#376C2×D6111yescharacteristic, complemented
383C43—611no—
384#339—612no—
385#339—612no—
386#339—612no—
387#339C2×D6112yescomplemented
388C61#2074911yescharacteristic, complemented, Sylow
389C2×#41—611no—
390C2×#41—611no—
391C2×#41—611no—
392C2×#41—1221no—
393C2×#41—1221no—
394C2×#41—1224no—
395C2×#41—1224no—
396C2×#41C22×D6111yescharacteristic
397C22×Q8—611no—
398C22×D4—611no—
399C22×D4—1221no—
400C23×C4—611no—
401C23×C4—1221no—
402C2×M16—612no—
403C2×M16C22×D6112yes—
404C4×Q8—612no—
405C4×Q8—612no—
406C4×Q8—612no—
407C4×Q8—1222no—
408C4×Q8—1222no—
409C4×Q8C4×D6112yes—
410C4×D4—22no—
411C4×D4—612no—
412C4×D4—612no—
413C4×D4—612no—
414C4×D4—1222no—
415C4×D4—1222no—
416C4×D4—1222no—
417C4×D4—1222no—
418C4×D4—1222no—
419C4×D4—1222no—
420C4×D4—1222no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4C4×D6112yes—
424#117—21no—
425#117—612no—
426#117—612no—
427#117—611no—
428#117—611no—
429#117—611no—
430#117—611no—
431#117—611no—
432#117—612no—
433#117—1221no—
434#117—1222no—
435#117—1222no—
436#117—1221no—
437#117—1221no—
438#117—1222no—
439#117—1221no—
440#117—1222no—
441#117—1221no—
442#117C22×D6111yescharacteristic
443C2×C4⋊C4—612no—
444C2×C4⋊C4—612no—
445C2×C4⋊C4—1222no—
446C2×C4⋊C4—1222no—
447C2×C4⋊C4—1222no—
448C2×C22⋊C4—612no—
449C2×C22⋊C4—1222no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1222no—
452C2×C22⋊C4—1222no—
453C2×C42—21no—
454C2×C42—611no—
455C2×C42—612no—
456C2×C42—1221no—
457C2×C42—1221no—
458C2×C42—1221no—
459C2×C42—1221no—
460C2×C42—1222no—
461C2×C42C22×D6112yes—
462C2×C42C22×D6111yescharacteristic
463C4⋊C8—1222no—
464C4⋊C8—1222no—
465#104—24no—
466#104—1224no—
467#104—1224no—
468#104—1224no—
469C22⋊C8—1222no—
470C22⋊C8—1222no—
471#97—21no—
472#97—1221no—
473C4×C8—21no—
474C4×C8—1221no—
475#95—22no—
476#95—1222no—
477#95—1222no—
478C24—611no—
479C24—1221no—
480#41—22no—
481#41—614no—
482#41—1228no—
483#41—1222no—
484#41—1228no—
485#41—1222no—
486#41—1222no—
487#41—1222no—
488#41—1222no—
489#41#1175822214yes—
490C2×Q8—611no—
491C2×Q8—1221no—
492C2×Q8—1222no—
493C2×Q8—1222no—
494C2×Q8C2×C4×D6111yescharacteristic
495C2×D4—21no—
496C2×D4—611no—
497C2×D4—1221no—
498C2×D4—1221no—
499C2×D4—1222no—
500C2×D4—1228no—
501C2×D4—1221no—
502C2×D4—1221no—
503C2×D4—1222no—
504C2×D4C2×C4×D6111yescharacteristic
505C22×C4—21no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—611no—
509C22×C4—612no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—1221no—
515C22×C4—1222no—
516C22×C4—1221no—
517C22×C4—1228no—
518C22×C4—1222no—
519C22×C4—1221no—
520C22×C4—1222no—
521C22×C4—1221no—
522C22×C4—1222no—
523C22×C4—1222no—
524C22×C4—1222no—
525C22×C4—1222no—
526C22×C4D4×D6112yes—
527C22×C4C23×D6111yescharacteristic
528M16—22no—
529M16—22no—
530M16—1222no—
531M16—1222no—
532C2×C8—22no—
533C2×C8—1222no—
534C4⋊C4—22no—
535C4⋊C4—612no—
536C4⋊C4—612no—
537C4⋊C4—612no—
538C4⋊C4—612no—
539C4⋊C4—612no—
540C4⋊C4—1222no—
541C4⋊C4—1222no—
542C4⋊C4—1222no—
543C4⋊C4—1222no—
544C4⋊C4—2444no—
545C4⋊C4C2×C4×D6112yes—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—612no—
549C22⋊C4—612no—
550C22⋊C4—612no—
551C22⋊C4—612no—
552C22⋊C4—1222no—
553C22⋊C4—1222no—
554C22⋊C4—1222no—
555C22⋊C4—1221no—
556C22⋊C4—1222no—
557C22⋊C4—2444no—
558C42—21no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—612no—
563C42—611no—
564C42—611no—
565C42—1222no—
566C42—1222no—
567C42—1222no—
568C42—1221no—
569C42—1222no—
570C42—1222no—
571C42—1222no—
572C42—2444no—
573C42C2×C4×D6112yes—
574C42D4×D6111yescharacteristic
575C42D4×D6111yescharacteristic
576C23—21no—
577C23—611no—
578C23—611no—
579C23—1224no—
580C23—1222no—
581C23—1221no—
582C23—4882no—
583C23—2442no—
584C23C2×D4×D6111yescharacteristic
585Q8—21no—
586Q8—612no—
587Q8—1224no—
588Q8—1221no—
589Q8#1451973412yes—
590D4—24no—
591D4—21no—
592D4—612no—
593D4—1224no—
594D4—1224no—
595D4—1224no—
596D4—1224no—
597D4—1221no—
598D4#1451973412yes—
599C2×C4—21no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—82no—
607C2×C4—612no—
608C2×C4—612no—
609C2×C4—611no—
610C2×C4—611no—
611C2×C4—1224no—
612C2×C4—1224no—
613C2×C4—1224no—
614C2×C4—1221no—
615C2×C4—1222no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1224no—
623C2×C4—1224no—
624C2×C4—1222no—
625C2×C4—4882no—
626C2×C4—4882no—
627C2×C4—4882no—
628C2×C4—4882no—
629C2×C4—2442no—
630C2×C4—2442no—
631C2×C4—2442no—
632C2×C4C22⋊C4×D6112yes—
633C2×C4C22⋊C4×D6112yes—
634C2×C4C22×C4×D6111yescharacteristic
635C2×C4C2×D4×D6111yescharacteristic
636C2×C4#41×D6112yes—
637C2×C4C2×#1175822212yes—
638C8—42no—
639C8—2442no—
640C22—22no—
641C22—22no—
642C22—42no—
643C22—1221no—
644C22—1222no—
645C22—1221no—
646C22—1222no—
647C22—1222no—
648C22—4882no—
649C22—4884no—
650C22—2442no—
651C22C2×C22⋊C4×D6111yescharacteristic
652C22#1684027112yes—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—42no—
657C4—82no—
658C4—1221no—
659C4—1221no—
660C4—1222no—
661C4—1222no—
662C4—1222no—
663C4—2444no—
664C4—2442no—
665C4#117×D6111yescharacteristic
666C4C2×#1451973411yescharacteristic
667C4#1684027112yes—
668C2—22no—
669C2—42no—
670C2—2441no—
671C2—2441no—
672C2—2442no—
673C2#2144820011yescharacteristic
674C2#2145436512yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1759C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#41×D61C22
cosocleC22×C4×C61C24
hypercenter#1759D61
hypocenterC61#20749
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1759

Complements

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

NH ≅ G/Ntype
2. #21454455C2semidirect
3. #21448314C2semidirect
4. #21445917C2semidirect
5. #21445838C2semidirect
6. #21445837C2semidirect
8. #21445630C2semidirect
9. C4×#14519703C2semidirect
10. C4×#14519704C2semidirect
11. C61×#1759C2semidirect
13. #16840244C4semidirect
15. #16840241C4semidirect
16. #16840238C4semidirect
18. #16840234C4semidirect
32. #16839785C22semidirect
37. C4×#11758194C22semidirect
40. C4×#11758196C22semidirect
42. C42×Dic61C22semidirect
43. #16839523C22semidirect
44. #16839522C22semidirect
45. #16839521C22semidirect
48. C4×M16×C61C22semidirect
49. C61×#376C22semidirect
50. C61×#339C22semidirect
339. #1759D61semidirect
377. C4×M16C2×D61semidirect
382. #376C2×D61semidirect
387. #339C2×D61semidirect
388. C61#20749semidirect

Maximal subgroups

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

subgroupindex
#214590182
#214544552
#214483142
#214459172
#214458382
#214458372
#214457612
#214456302
C4×#145197032
C4×#145197042
C61×#17592
#2074961