Finite Group Databasebeta

← #29648577

Subgroups — #29648577

Order 15616.

1018 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459116C211yesmaximal, characteristic
2#21454451C211yesmaximal, characteristic, complemented
3#423×D61C211yesmaximal, characteristic, complemented
4#21444557C211yesmaximal, characteristic, complemented
5#21444552C211yesmaximal, characteristic, complemented
6#21444552C211yesmaximal, characteristic, complemented
7#21444548C212yesmaximal, complemented
8#21444540C212yesmaximal, complemented
9#21444475C212yesmaximal, complemented
10#21444467C212yesmaximal, complemented
11C61×#1561C211yesmaximal, characteristic, complemented
12C2×M16×D61—21no—
13#16840317C2211yescharacteristic
14#16840317C2211yescharacteristic
15#16840313C2211yescharacteristic
16#16840313C2211yescharacteristic
17#16840312C2211yescharacteristic
18#16840299—21no—
19#16840299—21no—
20#16840298—21no—
21#16840298—21no—
22#16840297C412yes—
23#16840297C412yes—
24#16840275—21no—
25#16840275—21no—
26#16840274C412yes—
27C4×#11758226—21no—
28C4×#11758226—21no—
29#16840230C412yes—
30C2×C42×D61C2211yescharacteristic
31#16839818—21no—
32#16839818C2212yes—
33#16839814—21no—
34#16839814C2212yes—
35#16839786C2211yescharacteristic
36#16839786C2211yescharacteristic
37#16839785—21no—
38#16839784—21no—
39#16839784C2212yes—
40C4⋊C8×D61—21no—
41C4⋊C8×D61C2212yes—
42#16839725—21no—
43M16×Dic61—21no—
44#16839510C2212yescomplemented
45#16839510C2212yescomplemented
46#16839508C2212yescomplemented
47#16839508C2212yescomplemented
48#16839505C2212yescomplemented
49#16839505C2212yescomplemented
50C61×#533C2211yescharacteristic
51C61×#423C2211yescharacteristic
52C61×#423C2211yescharacteristic
53C61×#332C2212yescomplemented
54C61×#332C2212yescomplemented
55C2×#11758226—21no—
56C2×#11758226—21no—
57C2×#11758222C2×C412yes—
58C22×C4×D61C2×C411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C2×#11758194—21no—
62C2×#11758194—21no—
63C2×#11758189—22no—
64M16×D61—44no—
65C2×C8×D61—22no—
66#14519777C2311yescharacteristic
67#14519774C2311yescharacteristic
68#14519772—22no—
69#14519772—22no—
70#14519768—21no—
71#14519768—21no—
72#14519766—21no—
73#14519766—21no—
74#14519765C2×C412yes—
75#14519762—22no—
76#14519762—22no—
77#14519761—22no—
78#14519757C2×C412yes—
79#14519757C2×C412yes—
80#14519753—21no—
81#14519753—21no—
82#14519752—21no—
83#14519752—21no—
84#14519752C2×C412yes—
85#14519752C2×C412yes—
86#14519751—21no—
87#14519751—21no—
88#14519747—22no—
89Q8×Dic61—21no—
90Q8×Dic61—21no—
91#14519742C2×C412yes—
92#14519742C2×C412yes—
93C4×D244—22no—
94C4×D244—22no—
95C4×D244—21no—
96C4×D244—21no—
97#14519739—22no—
98#14519739—22no—
99#14519739—21no—
100#14519739—21no—
101#14519739C2×C412yes—
102#14519739C2×C412yes—
103#14519737—22no—
104#14519737—22no—
105#14519735—21no—
106#14519735—21no—
107#14519735—22no—
108#14519735—22no—
109#14519734—22no—
110C2×C4×Dic61C2×C411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C42×D61—22no—
114C42×D61C2×C411yescharacteristic
115C42×D61C2×C411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61C2311yescharacteristic
121C42×D61C2311yescharacteristic
122C4⋊#92418—21no—
123C4⋊#92418—21no—
124C4⋊#92418C2312yes—
125C4⋊#92418C2312yes—
126#14519706—22no—
127#14519706—22no—
128#14519689—22no—
129#14519689—22no—
130#14519687—22no—
131C8×Dic61—22no—
132C2×M16×C61—21no—
133C2×M16×C61—21no—
134C4⋊Q8×C61C2311yescharacteristic
135C4⋊Q8×C61C2311yescharacteristic
136#125×C61C2×C412yes—
137C22⋊Q8×C61—21no—
138C22⋊Q8×C61—21no—
139C4×Q8×C61—21no—
140C4×Q8×C61—21no—
141#117×C61C2×C412yes—
142C2×C42×C61C2311yescharacteristic
143C4⋊C8×C61—21no—
144C4⋊C8×C61—21no—
145C4⋊C8×C61C2312yes—
146C4⋊C8×C61C2312yes—
147C23×D61C22⋊C411yescharacteristic
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C22×C4×C61C22×C411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C2×C22⋊D61—22no—
154C22×Dic61C22⋊C411yescharacteristic
155C22×Dic61C22⋊C411yescharacteristic
156C22×Dic61C22⋊C411yescharacteristic
157#11758226—42no—
158#11758226—42no—
159#11758226—42no—
160#11758226—42no—
161#11758222—44no—
162C2×D244—22no—
163C2×D244—22no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244C22×C412yes—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—42no—
173C2×C4×D61—41no—
174C2×C4×D61—41no—
175C2×C4×D61C22⋊C411yescharacteristic
176C2×C4×D61C22⋊C411yescharacteristic
177C2×C4×D61C22⋊C411yescharacteristic
178C2×C4×D61C22⋊C411yescharacteristic
179C2×C4×D61C22⋊C411yescharacteristic
180C2×C4×D61C22⋊C411yescharacteristic
181C2×C4×D61C22×C411yescharacteristic
182C2×C4×D61C22×C411yescharacteristic
183C2×C4×D61C2×D411yescharacteristic
184C2×C4×D61C2×D411yescharacteristic
185C2×C4×D61C2×D411yescharacteristic
186C2×C4×D61C2×D411yescharacteristic
187C2×Dic122C22×C412yes—
188M16×C61—42no—
189M16×C61—42no—
190C2×C8×C61—22no—
191C2×C8×C61—22no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—22no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61C22×C412yes—
197C4⋊C4×C61C22×C412yes—
198C22⋊C4×C61—22no—
199C42×C61—22no—
200C42×C61C22×C411yescharacteristic
201C42×C61C2×D411yescharacteristic
202C42×C61C2×D411yescharacteristic
203C42×C61C2411yescharacteristic
204#11758198—22no—
205#11758198—22no—
206#11758198—22no—
207#11758198—22no—
208#11758198—22no—
209#11758198—22no—
210#11758198—22no—
211#11758198—22no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—22no—
215C4⋊Dic61—22no—
216#11758196—22no—
217#11758196—22no—
218C4×Dic61—22no—
219C4×Dic61—22no—
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22⋊C411yescharacteristic
224C4×Dic61C22⋊C411yescharacteristic
225C4×Dic61C22⋊C411yescharacteristic
226C4×Dic61C22×C411yescharacteristic
227C4×Dic61C22×C411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229C4×Dic61C2×D411yescharacteristic
230C4×Dic61C2×D411yescharacteristic
231C4×Dic61C2×D411yescharacteristic
232#11758194—42no—
233#11758194—42no—
234C2×#92418—22no—
235C2×#92418—22no—
236#11758189—44no—
237C8×D61—44no—
238C23×C61C2×C22⋊C411yescharacteristic
239C22×D61—22no—
240C22×D61—22no—
241C22×D61—41no—
242C22×D61—41no—
243C22×D61C2×C22⋊C411yescharacteristic
244C22×D61C2×C22⋊C411yescharacteristic
245Q8×C61—41no—
246Q8×C61—41no—
247Q8×C61—41no—
248Q8×C61—41no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C2×C22⋊C411yescharacteristic
259C2×C4×C61C23×C411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C22⋊D61—44no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61C2×C22⋊C411yescharacteristic
268C2×Dic61C2×C22⋊C411yescharacteristic
269C2×Dic61C2×C22⋊C411yescharacteristic
270C2×Dic61C2×C22⋊C411yescharacteristic
271C2×Dic61C2×C22⋊C411yescharacteristic
272C2×Dic61C2×C22⋊C411yescharacteristic
273D244—42no—
274D244—44no—
275D244—44no—
276D244—41no—
277D244—41no—
278D244—41no—
279D244—41no—
280C4×D61—22no—
281C4×D61—21no—
282C4×D61—21no—
283C4×D61—21no—
284C4×D61—21no—
285C4×D61—21no—
286C4×D61—21no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—41no—
291C4×D61—41no—
292C4×D61—41no—
293C4×D61—41no—
294C4×D61—44no—
295C4×D61—44no—
296C4×D61—44no—
297C4×D61AGL(1,ℤ/8)12yes—
298C4×D61#13712yes—
299Dic122—42no—
300C8×C61—42no—
301C8×C61—42no—
302#92418—42no—
303#92418—42no—
304#17737—611nomaximal, Sylow
305C22×C61—21no—
306C22×C61—21no—
307C22×C61—21no—
308C22×C61C22×C22⋊C411yescharacteristic
309C2×D61—21no—
310C2×D61—21no—
311C2×D61—21no—
312C2×D61—21no—
313C2×D61—21no—
314C2×D61—44no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61#41712yes—
319C4×C61—21no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—42no—
323C4×C61—42no—
324C4×C61#41312yes—
325C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
326C4×C61C2×#13711yescharacteristic
327C4×C61#57511yescharacteristic
328C4×C61#57511yescharacteristic
329Dic61—21no—
330Dic61—21no—
331Dic61—21no—
332Dic61—21no—
333Dic61—21no—
334Dic61—42no—
335Dic61#41712yes—
336#3444—611no—
337#2920—611no—
338C2×#423—611no—
339#1561—611no—
340#1561D6111yescharacteristic, complemented
341#1560—611no—
342#1560—611no—
343#1559—612no—
344#1557—612no—
345#1542—612no—
346#1540—612no—
347C2×C61—41no—
348C2×C61C2×#41311yescharacteristic
349C2×C61C2×#41711yescharacteristic
350C2×C61#289611yescharacteristic
351D61—22no—
352D61—41no—
353D61—82no—
354C22×M16—1221no—
355#533—611no—
356#533—611no—
357#533C2×D6111yescharacteristic
358#532—611no—
359#532—611no—
360#532—611no—
361#529—612no—
362#529—612no—
363#529—1221no—
364#529—1221no—
365#529—1221no—
366#529—1221no—
367#525—612no—
368#525—1221no—
369#525—1221no—
370C4×#41—612no—
371C4×#41—1221no—
372C4×#41—1221no—
373C22×C42—611no—
374#432—612no—
375#432—1221no—
376#431—612no—
377#431—1221no—
378#423—611no—
379#423—611no—
380#423—612no—
381#423—1221no—
382#423—1221no—
383#423C2×D6111yescharacteristic
384#423C2×D6111yescharacteristic
385C2×C4⋊C8—612no—
386C2×C4⋊C8—1221no—
387#407—1221no—
388C4×M16—1221no—
389#333—612no—
390#333—612no—
391#332—612no—
392#332—612no—
393#332C2×D6112yescomplemented
394#332C2×D6112yescomplemented
395#331—612no—
396#331—612no—
397C61#1773711yescharacteristic, complemented, Sylow
398C2×#41—612no—
399C2×#41—1221no—
400C2×#41—1221no—
401C23×C4—611no—
402C23×C4—611no—
403C23×C4—611no—
404C2×M16—21no—
405C2×M16—21no—
406C2×M16—1222no—
407C2×M16—1221no—
408C2×M16—1221no—
409C2×M16—2444no—
410C22×C8—1222no—
411C4⋊Q8—611no—
412C4⋊Q8C22×D6111yescharacteristic
413C4⋊Q8C22×D6111yescharacteristic
414#127—611no—
415#126—1222no—
416#126—1222no—
417#125—1221no—
418#125—1221no—
419#125C4×D6112yes—
420#124—612no—
421#124—1221no—
422#124—1221no—
423#123—1222no—
424#123—1222no—
425#123—1222no—
426C22⋊Q8—21no—
427C22⋊Q8—21no—
428C22⋊Q8—612no—
429C22⋊Q8—612no—
430C22⋊Q8—1221no—
431C22⋊Q8—1221no—
432#121—612no—
433#121—612no—
434#121—1221no—
435#121—1221no—
436#121—1221no—
437#121—1221no—
438#120—1222no—
439C4×Q8—21no—
440C4×Q8—21no—
441C4×Q8—612no—
442C4×Q8—612no—
443C4×Q8—1221no—
444C4×Q8—1221no—
445C4×D4—612no—
446C4×D4—612no—
447C4×D4—1222no—
448C4×D4—1222no—
449C4×D4—1221no—
450C4×D4—1221no—
451C4×D4—1222no—
452C4×D4—1222no—
453C4×D4—1222no—
454C4×D4—1222no—
455C4×D4—1221no—
456C4×D4—1221no—
457#117—1222no—
458#117—1222no—
459#117—1222no—
460#117—1221no—
461#117—1221no—
462#117C4×D6112yes—
463C2×C42—611no—
464C2×C42—611no—
465C2×C42—611no—
466C2×C42—611no—
467C2×C42—611no—
468C2×C42—611no—
469C2×C42—611no—
470C2×C42—611no—
471C2×C42—611no—
472C2×C42—611no—
473C2×C42—611no—
474C2×C42—1222no—
475C2×C42C22×D6111yescharacteristic
476C4⋊C8—21no—
477C4⋊C8—21no—
478C4⋊C8—612no—
479C4⋊C8—612no—
480C4⋊C8—1222no—
481C4⋊C8—1222no—
482C4⋊C8—1221no—
483C4⋊C8—1221no—
484C4⋊C8C22×D6112yes—
485C4⋊C8C22×D6112yes—
486C22⋊C8—1222no—
487C22⋊C8—1222no—
488#97—1222no—
489C4×C8—1222no—
490C24—611no—
491#41—2444no—
492#41—2442no—
493#41—2442no—
494#41—2442no—
495#41—2442no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—612no—
499C2×D4—612no—
500C2×D4—1222no—
501C2×D4—1222no—
502C2×D4—1222no—
503C2×D4—1221no—
504C2×D4—1221no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—611no—
519C22×C4—611no—
520C22×C4—1222no—
521C22×C4—1222no—
522C22×C4—1222no—
523C22×C4—1222no—
524C22×C4—1222no—
525C22×C4—2442no—
526C22×C4—2441no—
527C22×C4—2441no—
528C22×C4C2×C4×D6111yescharacteristic
529C22×C4D4×D6111yescharacteristic
530C22×C4D4×D6111yescharacteristic
531M16—42no—
532M16—42no—
533M16—2444no—
534M16—2442no—
535M16—2442no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—1222no—
539C2×C8—1222no—
540C2×C8—2444no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—1222no—
546C4⋊C4—1222no—
547C4⋊C4—1222no—
548C4⋊C4—1222no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4C2×C4×D6112yes—
552C4⋊C4C2×C4×D6112yes—
553C22⋊C4—22no—
554C22⋊C4—1222no—
555C22⋊C4—1222no—
556C22⋊C4—1222no—
557C22⋊C4—1222no—
558C22⋊C4—1222no—
559C22⋊C4—1222no—
560C22⋊C4—1222no—
561C22⋊C4—1222no—
562C42—22no—
563C42—611no—
564C42—611no—
565C42—611no—
566C42—611no—
567C42—611no—
568C42—611no—
569C42—611no—
570C42—611no—
571C42—611no—
572C42—611no—
573C42—611no—
574C42—611no—
575C42—1222no—
576C42—1222no—
577C42C2×C4×D6111yescharacteristic
578C42D4×D6111yescharacteristic
579C42D4×D6111yescharacteristic
580C42C23×D6111yescharacteristic
581C23—611no—
582C23—611no—
583C23—1222no—
584C23—1222no—
585C23—2441no—
586C23—2441no—
587C23C22⋊C4×D6111yescharacteristic
588Q8—41no—
589Q8—41no—
590Q8—41no—
591Q8—41no—
592Q8—2442no—
593D4—2442no—
594D4—2444no—
595D4—2444no—
596D4—2444no—
597D4—2441no—
598D4—2441no—
599D4—2441no—
600D4—2441no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—611no—
609C2×C4—612no—
610C2×C4—612no—
611C2×C4—611no—
612C2×C4—611no—
613C2×C4—611no—
614C2×C4—611no—
615C2×C4—611no—
616C2×C4—1222no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—1221no—
625C2×C4—1221no—
626C2×C4—1221no—
627C2×C4—1222no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2444no—
633C2×C4—2444no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2444no—
637C2×C4C22⋊C4×D6111yescharacteristic
638C2×C4C22⋊C4×D6111yescharacteristic
639C2×C4C22⋊C4×D6111yescharacteristic
640C2×C4C22×C4×D6111yescharacteristic
641C2×C4C2×D4×D6111yescharacteristic
642C2×C4C2×D4×D6111yescharacteristic
643C8—42no—
644C8—42no—
645C8—2442no—
646C8—2442no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—612no—
651C22—1221no—
652C22—1221no—
653C22—1221no—
654C22—1221no—
655C22—1221no—
656C22—2444no—
657C22—2442no—
658C22—2442no—
659C22—2442no—
660C22C2×C22⋊C4×D6111yescharacteristic
661C4—21no—
662C4—21no—
663C4—21no—
664C4—42no—
665C4—42no—
666C4—612no—
667C4—1221no—
668C4—1221no—
669C4—1221no—
670C4—1221no—
671C4—1221no—
672C4—2442no—
673C4#1683974512yes—
674C4AGL(1,ℤ/8)×D6111yescharacteristic
675C4#137×D6111yescharacteristic
676C4#1684055811yescharacteristic
677C4#1684056111yescharacteristic
678C2—41no—
679C2—1222no—
680C2—4882no—
681C2—2441no—
682C2C2×#1683974511yescharacteristic
683C2#417×D6111yescharacteristic
684C2#2145433511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1561C2
socleC22×C61C22×C22⋊C4
prime radical#21459116C2
cosocleC42×C61C24
hypercenter#1561D61
hypocenterC61#17737
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

Complements

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

NH ≅ G/Ntype
2. #21454451C2semidirect
3. #423×D61C2semidirect
4. #21444557C2semidirect
5. #21444552C2semidirect
6. #21444552C2semidirect
7. #21444548C2semidirect
8. #21444540C2semidirect
9. #21444475C2semidirect
10. #21444467C2semidirect
11. C61×#1561C2semidirect
44. #16839510C22semidirect
45. #16839510C22semidirect
46. #16839508C22semidirect
47. #16839508C22semidirect
48. #16839505C22semidirect
49. #16839505C22semidirect
53. C61×#332C22semidirect
54. C61×#332C22semidirect
340. #1561D61semidirect
393. #332C2×D61semidirect
394. #332C2×D61semidirect
397. C61#17737semidirect

Maximal subgroups

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

subgroupindex
#214591162
#214544512
#423×D612
#214445572
#214445522
#214445482
#214445402
#214444752
#214444672
C61×#15612
#1773761