Finite Group Databasebeta

← #29648573

Subgroups — #29648573

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#21459114C211yesmaximal, characteristic
2#21454451C211yesmaximal, characteristic, complemented
3#423×D61C211yesmaximal, characteristic, complemented
4#21444556C211yesmaximal, characteristic, complemented
5#21444556C211yesmaximal, characteristic, complemented
6#21444554C211yesmaximal, characteristic, complemented
7#21444546C212yesmaximal, complemented
8#21444538C212yesmaximal, complemented
9#21444473C212yesmaximal, complemented
10#21444465C212yesmaximal, complemented
11C61×#1560C211yesmaximal, characteristic, complemented
12C2×M16×D61—21no—
13#16840319C2211yescharacteristic
14#16840317C2211yescharacteristic
15#16840317C2211yescharacteristic
16#16840310C2211yescharacteristic
17#16840310C2211yescharacteristic
18#16840294—21no—
19#16840294—21no—
20#16840293—21no—
21#16840293—21no—
22#16840291C412yes—
23#16840291C412yes—
24#16840272—21no—
25#16840272—21no—
26#16840269C412yes—
27#41×Dic61C412yes—
28C4×#11758224—21no—
29C4×#11758224—21no—
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#16839511C2212yescomplemented
45#16839509C2212yescomplemented
46#16839509C2212yescomplemented
47#16839507C2212yescomplemented
48#16839507C2212yescomplemented
49#16839506C2212yescomplemented
50C61×#532C2211yescharacteristic
51C61×#423C2211yescharacteristic
52C61×#423C2211yescharacteristic
53C61×#333C2212yescomplemented
54C61×#331C2212yescomplemented
55C2×#11758224—21no—
56C2×#11758224—21no—
57C22×C4×D61C2×C411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C2×#11758194—21no—
61C2×#11758194—21no—
62C2×#11758189—22no—
63M16×D61—44no—
64C2×C8×D61—22no—
65#14519777C2311yescharacteristic
66#14519777C2311yescharacteristic
67#14519770—22no—
68#14519770—22no—
69#14519769C2×C412yes—
70#14519768—21no—
71#14519768—21no—
72#14519764—21no—
73#14519764—21no—
74#14519759—22no—
75#14519759—22no—
76#14519758—22no—
77#14519757—21no—
78#14519757—21no—
79#14519756—21no—
80#14519756—21no—
81#14519753C2×C412yes—
82#14519753C2×C412yes—
83#14519748—21no—
84#14519748—21no—
85#14519748C2×C412yes—
86#14519748C2×C412yes—
87#14519745—22no—
88Q8×Dic61C2×C412yes—
89Q8×Dic61C2×C412yes—
90C4×Dic122—21no—
91C4×Dic122—21no—
92#14519742—21no—
93#14519742—21no—
94D4×Dic61—22no—
95D4×Dic61—22no—
96D4×Dic61—21no—
97D4×Dic61—21no—
98D4×Dic61C2×C412yes—
99D4×Dic61C2×C412yes—
100C4×C22⋊D61—22no—
101C4×C22⋊D61—22no—
102#14519737—22no—
103#14519737—22no—
104#14519736—22no—
105#14519736C2×C412yes—
106#14519735—21no—
107#14519735—21no—
108#14519733—22no—
109#14519733—22no—
110C2×C4×Dic61—22no—
111C2×C4×Dic61C2×C411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61D411yescharacteristic
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×#41×C61C2×C412yes—
133C2×M16×C61—21no—
134C2×M16×C61—21no—
135C4⋊Q8×C61C2311yescharacteristic
136#127×C61C2311yescharacteristic
137#124×C61C2×C412yes—
138#121×C61—21no—
139#121×C61—21no—
140C4×D4×C61—21no—
141C4×D4×C61—21no—
142C2×C42×C61C2311yescharacteristic
143C4⋊C8×C61—21no—
144C4⋊C8×C61—21no—
145C4⋊C8×C61C2312yes—
146C4⋊C8×C61C2312yes—
147C23×D61C22⋊C411yescharacteristic
148#41×C61—44no—
149C2×Q8×C61C22×C412yes—
150C2×D4×C61—22no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61C22×C412yes—
154C22×C4×C61—22no—
155C22×C4×C61C22×C411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C2×C22⋊D61—22no—
159C2×C22⋊D61—22no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61C22⋊C411yescharacteristic
163C22×Dic61C22⋊C411yescharacteristic
164C22×Dic61C22⋊C411yescharacteristic
165#11758224—42no—
166#11758224—42no—
167#11758224—42no—
168#11758224—42no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—42no—
172C2×C4×D61—41no—
173C2×C4×D61—41no—
174C2×C4×D61C22⋊C411yescharacteristic
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×D61C2×D411yescharacteristic
183C2×C4×D61C2×D411yescharacteristic
184C2×C4×D61C2×D411yescharacteristic
185C2×C4×D61C2×D411yescharacteristic
186C2×Dic122—21no—
187C2×Dic122—21no—
188M16×C61—42no—
189M16×C61—42no—
190C2×C8×C61—22no—
191C2×C8×C61—22no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C22⋊C4×C61—22no—
195C22⋊C4×C61—22no—
196C42×C61C22×C411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2×D411yescharacteristic
199C42×C61C2411yescharacteristic
200C22⋊Dic61—22no—
201C22⋊Dic61—22no—
202C22⋊Dic61—22no—
203C22⋊Dic61—22no—
204C22⋊Dic61—22no—
205#11758198—22no—
206#11758198—22no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—22no—
210C4⋊Dic61—22no—
211C4⋊Dic61C22×C412yes—
212C4⋊Dic61C22×C412yes—
213#11758196—22no—
214#11758196—22no—
215#11758196—22no—
216#11758196—22no—
217C4×Dic61—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×C61—22no—
239C23×C61C2×C22⋊C411yescharacteristic
240C22×D61—22no—
241C22×D61—41no—
242C22×D61—41no—
243C22×D61C2×C22⋊C411yescharacteristic
244C22×D61C2×C22⋊C411yescharacteristic
245Q8×C61—42no—
246D4×C61—42no—
247D4×C61—44no—
248D4×C61—41no—
249D4×C61—41no—
250D4×C61—41no—
251D4×C61—41no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—44no—
259C2×C4×C61C2×C22⋊C411yescharacteristic
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C2×C22⋊C411yescharacteristic
262C2×C4×C61C23×C411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C22⋊D61—44no—
266C22⋊D61—44no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—44no—
273C2×Dic61—44no—
274C2×Dic61C2×C22⋊C411yescharacteristic
275C2×Dic61C2×C22⋊C411yescharacteristic
276C2×Dic61C2×C22⋊C411yescharacteristic
277C2×Dic61C2×C22⋊C411yescharacteristic
278C2×Dic61C2×C22⋊C411yescharacteristic
279C2×Dic61C2×C22⋊C411yescharacteristic
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×D61AGL(1,ℤ/8)12yes—
295C4×D61#13712yes—
296Dic122—41no—
297Dic122—41no—
298Dic122—41no—
299Dic122—41no—
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×C61—42no—
309C22×C61—42no—
310C22×C61—42no—
311C22×C61C22×C22⋊C411yescharacteristic
312C2×D61—21no—
313C2×D61—21no—
314C2×D61—21no—
315C2×D61—21no—
316C2×D61—21no—
317C2×D61—44no—
318C2×D61#41712yes—
319C4×C61—21no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—42no—
323C4×C61#41312yes—
324C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
325C4×C61C2×#13711yescharacteristic
326C4×C61#57511yescharacteristic
327C4×C61#57511yescharacteristic
328Dic61—21no—
329Dic61—21no—
330Dic61—21no—
331Dic61—21no—
332Dic61—21no—
333Dic61—42no—
334Dic61—42no—
335Dic61#41712yes—
336#3444—611no—
337#2920—611no—
338C2×#423—611no—
339#1561—611no—
340#1561—611no—
341#1560—611no—
342#1560D6111yescharacteristic, complemented
343#1559—612no—
344#1557—612no—
345#1542—612no—
346#1540—612no—
347C2×C61—41no—
348C2×C61—82no—
349C2×C61C2×#41311yescharacteristic
350C2×C61C2×#41711yescharacteristic
351C2×C61#289611yescharacteristic
352D61—22no—
353D61—41no—
354C22×M16—1221no—
355#533—611no—
356#533—611no—
357#533—611no—
358#532—611no—
359#532—611no—
360#532C2×D6111yescharacteristic
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#333C2×D6112yescomplemented
391#332—612no—
392#332—612no—
393#332—612no—
394#332—612no—
395#331—612no—
396#331C2×D6112yescomplemented
397C61#1773711yescharacteristic, complemented, Sylow
398C2×#41—1221no—
399C2×#41—1221no—
400C2×#41C4×D6112yes—
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⋊Q8—611no—
413C4⋊Q8C22×D6111yescharacteristic
414#127C22×D6111yescharacteristic
415#126—1222no—
416#126—1222no—
417#125—612no—
418#125—1221no—
419#125—1221no—
420#124—1221no—
421#124—1221no—
422#124C4×D6112yes—
423#123—1222no—
424#123—1222no—
425#123—1222no—
426C22⋊Q8—612no—
427C22⋊Q8—612no—
428C22⋊Q8—1221no—
429C22⋊Q8—1221no—
430C22⋊Q8—1221no—
431C22⋊Q8—1221no—
432#121—21no—
433#121—21no—
434#121—612no—
435#121—612no—
436#121—1221no—
437#121—1221no—
438#120—1222no—
439C4×Q8—612no—
440C4×Q8—612no—
441C4×Q8—1221no—
442C4×Q8—1221no—
443C4×Q8—1221no—
444C4×Q8—1221no—
445C4×D4—21no—
446C4×D4—21no—
447C4×D4—612no—
448C4×D4—612no—
449C4×D4—1221no—
450C4×D4—1221no—
451C4×D4—1222no—
452C4×D4—1222no—
453C4×D4—1222no—
454C4×D4—1222no—
455C4×D4—1222no—
456C4×D4—1222no—
457#117—612no—
458#117—1222no—
459#117—1222no—
460#117—1222no—
461#117—1221no—
462#117—1221no—
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—44no—
492#41—2442no—
493#41—2442no—
494#41—2442no—
495#41—2442no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8C2×C4×D6112yes—
499C2×D4—22no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—1222no—
503C2×D4—1222no—
504C2×D4C2×C4×D6112yes—
505C22×C4—22no—
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—611no—
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—612no—
544C4⋊C4—612no—
545C4⋊C4—1222no—
546C4⋊C4—1222no—
547C4⋊C4—1222no—
548C4⋊C4—1222no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1222no—
552C4⋊C4—1222no—
553C22⋊C4—22no—
554C22⋊C4—22no—
555C22⋊C4—1222no—
556C22⋊C4—1222no—
557C22⋊C4—1222no—
558C22⋊C4—1222no—
559C22⋊C4—1222no—
560C22⋊C4—1222no—
561C22⋊C4—1222no—
562C42—611no—
563C42—611no—
564C42—611no—
565C42—611no—
566C42—611no—
567C42—611no—
568C42—611no—
569C42—611no—
570C42—611no—
571C42—611no—
572C42—611no—
573C42—611no—
574C42—1222no—
575C42—1222no—
576C42—1222no—
577C42C2×C4×D6111yescharacteristic
578C42D4×D6111yescharacteristic
579C42D4×D6111yescharacteristic
580C42C23×D6111yescharacteristic
581C23—22no—
582C23—611no—
583C23—611no—
584C23—1222no—
585C23—2441no—
586C23—2441no—
587C23C22⋊C4×D6111yescharacteristic
588Q8—42no—
589Q8—2441no—
590Q8—2441no—
591Q8—2441no—
592Q8—2441no—
593D4—42no—
594D4—44no—
595D4—41no—
596D4—41no—
597D4—41no—
598D4—41no—
599D4—2444no—
600D4—2444no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—44no—
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—1222no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
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—42no—
651C22—42no—
652C22—42no—
653C22—612no—
654C22—1221no—
655C22—1221no—
656C22—1221no—
657C22—1221no—
658C22—1221no—
659C22—2444no—
660C22C2×C22⋊C4×D6111yescharacteristic
661C4—21no—
662C4—21no—
663C4—21no—
664C4—42no—
665C4—612no—
666C4—1221no—
667C4—1221no—
668C4—1221no—
669C4—1221no—
670C4—1221no—
671C4—2442no—
672C4—2442no—
673C4#1683974312yes—
674C4AGL(1,ℤ/8)×D6111yescharacteristic
675C4#137×D6111yescharacteristic
676C4#1684055811yescharacteristic
677C4#1684056111yescharacteristic
678C2—41no—
679C2—82no—
680C2—1222no—
681C2—2441no—
682C2C2×#1683974311yescharacteristic
683C2#417×D6111yescharacteristic
684C2#2145433511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1560C2
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC42×C61C24
hypercenter#1560D61
hypocenterC61#17737
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1560

Complements

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

NH ≅ G/Ntype
2. #21454451C2semidirect
3. #423×D61C2semidirect
4. #21444556C2semidirect
5. #21444556C2semidirect
6. #21444554C2semidirect
7. #21444546C2semidirect
8. #21444538C2semidirect
9. #21444473C2semidirect
10. #21444465C2semidirect
11. C61×#1560C2semidirect
44. #16839511C22semidirect
45. #16839509C22semidirect
46. #16839509C22semidirect
47. #16839507C22semidirect
48. #16839507C22semidirect
49. #16839506C22semidirect
53. C61×#333C22semidirect
54. C61×#331C22semidirect
342. #1560D61semidirect
390. #333C2×D61semidirect
396. #331C2×D61semidirect
397. C61#17737semidirect

Maximal subgroups

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

subgroupindex
#214591142
#214544512
#423×D612
#214445562
#214445542
#214445462
#214445382
#214444732
#214444652
C61×#15602
#1773761