Finite Group Databasebeta

← #29648575

Subgroups — #29648575

Order 15616.

1018 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459114C211yesmaximal, characteristic
2#21454449C211yesmaximal, characteristic, complemented
3#423×D61C211yesmaximal, characteristic, complemented
4#21444555C212yesmaximal, complemented
5#21444551C212yesmaximal, complemented
6#21444550C211yesmaximal, characteristic, complemented
7#21444547C211yesmaximal, characteristic, complemented
8#21444541C211yesmaximal, characteristic, complemented
9#21444475C212yesmaximal, complemented
10#21444465C212yesmaximal, complemented
11C61×#1557C211yesmaximal, characteristic, complemented
12C2×M16×D61—21no—
13#16840319C2211yescharacteristic
14#16840317C2211yescharacteristic
15#16840317C2211yescharacteristic
16#16840310C2211yescharacteristic
17#16840310C2211yescharacteristic
18#16840294—21no—
19#16840294C412yes—
20#16840293—21no—
21#16840293C412yes—
22#16840291—21no—
23#16840291—21no—
24#16840272—21no—
25#16840272C412yes—
26#16840269—21no—
27#41×Dic61—21no—
28C4×#11758224—21no—
29C4×#11758224C412yes—
30C2×C42×D61C2211yescharacteristic
31#16839819C2211yescharacteristic
32#16839817—21no—
33#16839814C2212yes—
34#16839812—21no—
35#16839786C2211yescharacteristic
36#16839785—21no—
37#16839784—21no—
38#16839784C2212yes—
39#16839784C2212yes—
40C4⋊C8×D61—21no—
41C4⋊C8×D61C2212yes—
42#16839724—21no—
43#16839714—21no—
44#16839510C2212yescomplemented
45#16839510C2212yescomplemented
46#16839509C2212yescomplemented
47#16839508C2212yescomplemented
48#16839507C2212yescomplemented
49#16839506C2212yescomplemented
50C61×#532C2211yescharacteristic
51C61×#432C2211yescharacteristic
52C61×#423C2211yescharacteristic
53C61×#332C2212yescomplemented
54C61×#331C2212yescomplemented
55C2×#11758224—21no—
56C2×#11758224C2×C412yes—
57C22×C4×D61C2×C411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C2×#11758194—21no—
61C2×#11758189—22no—
62C2×#11758189—22no—
63M16×D61—44no—
64C2×C8×D61—22no—
65#14519777C2311yescharacteristic
66#14519777C2311yescharacteristic
67#14519770—22no—
68#14519770—22no—
69#14519769—21no—
70#14519768—21no—
71#14519768C2×C412yes—
72#14519764—21no—
73#14519764C2×C412yes—
74#14519759—22no—
75#14519759—22no—
76#14519758—22no—
77#14519757—21no—
78#14519757C2×C412yes—
79#14519756—21no—
80#14519756C2×C412yes—
81#14519753—21no—
82#14519753—21no—
83#14519748—21no—
84#14519748—21no—
85#14519748—21no—
86#14519748C2×C412yes—
87#14519745—22no—
88Q8×Dic61—21no—
89Q8×Dic61—21no—
90C4×Dic122—21no—
91C4×Dic122C2×C412yes—
92#14519742—21no—
93#14519742C2×C412yes—
94D4×Dic61—22no—
95D4×Dic61—22no—
96D4×Dic61—21no—
97D4×Dic61—21no—
98D4×Dic61—21no—
99D4×Dic61C2×C412yes—
100C4×C22⋊D61—22no—
101C4×C22⋊D61—22no—
102#14519737—22no—
103#14519737—22no—
104#14519736—21no—
105#14519736—22no—
106#14519735—21no—
107#14519735C2×C412yes—
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⋊#92418C2312yes—
124C4⋊#92418C2312yes—
125#14519706—22no—
126#14519706—22no—
127#14519706—22no—
128C22⋊#92418—21no—
129#14519689—22no—
130#14519688—21no—
131C8×Dic61—22no—
132C2×#41×C61—21no—
133C2×M16×C61—21no—
134C4⋊Q8×C61C2311yescharacteristic
135#127×C61C2311yescharacteristic
136#124×C61—21no—
137#121×C61—21no—
138#121×C61C2×C412yes—
139C4×D4×C61—21no—
140C4×D4×C61C2×C412yes—
141C2×C42×C61C2311yescharacteristic
142C4⋊C8×C61—21no—
143C4⋊C8×C61C2312yes—
144C4⋊C8×C61C2312yes—
145C22⋊C8×C61—21no—
146#97×C61—21no—
147C23×D61C22⋊C411yescharacteristic
148#41×C61—44no—
149C2×Q8×C61—21no—
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—44no—
166#11758224—42no—
167#11758224—42no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×C4×D61C22⋊C411yescharacteristic
173C2×C4×D61C22⋊C411yescharacteristic
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×D61C2×D411yescharacteristic
181C2×C4×D61C2×D411yescharacteristic
182C2×C4×D61C2×D411yescharacteristic
183C2×C4×D61C2×D411yescharacteristic
184C2×Dic122—21no—
185C2×Dic122C22×C412yes—
186M16×C61—42no—
187C2×C8×C61—22no—
188C2×C8×C61—22no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61C22×C412yes—
191C22⋊C4×C61—22no—
192C22⋊C4×C61—22no—
193C42×C61C22×C411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C42×C61C2411yescharacteristic
197C22⋊Dic61—22no—
198C22⋊Dic61—22no—
199C22⋊Dic61—22no—
200C22⋊Dic61—22no—
201C22⋊Dic61—22no—
202#11758198—22no—
203#11758198—22no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—22no—
208C4⋊Dic61—22no—
209C4⋊Dic61C22×C412yes—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196—22no—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61—22no—
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22×C411yescharacteristic
224C4×Dic61C22×C411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229#11758194—42no—
230C2×#92418—22no—
231C2×#92418—22no—
232#11758189—44no—
233#11758189—44no—
234C8×D61—44no—
235C23×C61—22no—
236C23×C61C2×C22⋊C411yescharacteristic
237C22×D61—22no—
238C22×D61—41no—
239C22×D61—41no—
240C22×D61C2×C22⋊C411yescharacteristic
241C22×D61C2×C22⋊C411yescharacteristic
242Q8×C61—42no—
243D4×C61—42no—
244D4×C61—44no—
245D4×C61—41no—
246D4×C61—41no—
247D4×C61—42no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—41no—
253C2×C4×C61—44no—
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C23×C411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C22⋊D61—44no—
261C22⋊D61—44no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—41no—
267C2×Dic61—44no—
268C2×Dic61—44no—
269C2×Dic61C2×C22⋊C411yescharacteristic
270C2×Dic61C2×C22⋊C411yescharacteristic
271C2×Dic61C2×C22⋊C411yescharacteristic
272C2×Dic61C2×C22⋊C411yescharacteristic
273C2×Dic61C2×C22⋊C411yescharacteristic
274C2×Dic61C2×C22⋊C411yescharacteristic
275C4×D61—22no—
276C4×D61—21no—
277C4×D61—21no—
278C4×D61—22no—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—41no—
284C4×D61—41no—
285C4×D61—41no—
286C4×D61—41no—
287C4×D61AGL(1,ℤ/8)12yes—
288C4×D61#13712yes—
289Dic122—41no—
290Dic122—41no—
291Dic122—42no—
292C8×C61—42no—
293C8×C61—42no—
294#92418—42no—
295#92418—42no—
296#17737—611nomaximal, Sylow
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—21no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61C22×C22⋊C411yescharacteristic
304C2×D61—21no—
305C2×D61—21no—
306C2×D61—21no—
307C2×D61—21no—
308C2×D61—21no—
309C2×D61—44no—
310C2×D61#41712yes—
311C4×C61—21no—
312C4×C61—21no—
313C4×C61—21no—
314C4×C61—21no—
315C4×C61—42no—
316C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
317C4×C61C2×#13711yescharacteristic
318C4×C61#57512yes—
319Dic61—21no—
320Dic61—21no—
321Dic61—21no—
322Dic61—21no—
323Dic61—42no—
324Dic61—42no—
325Dic61#41312yes—
326Dic61#41712yes—
327#3444—611no—
328#2920—611no—
329C2×#423—611no—
330#1561—612no—
331#1560—612no—
332#1559—611no—
333#1559—611no—
334#1557—611no—
335#1557D6111yescharacteristic, complemented
336#1542—612no—
337#1540—612no—
338C2×C61—41no—
339C2×C61—82no—
340C2×C61C2×#41311yescharacteristic
341C2×C61C2×#41711yescharacteristic
342C2×C61#289611yescharacteristic
343D61—22no—
344D61—41no—
345C22×M16—1221no—
346#533—611no—
347#533—611no—
348#533—611no—
349#532—611no—
350#532—611no—
351#532C2×D6111yescharacteristic
352#529—612no—
353#529—612no—
354#529—1221no—
355#529—1221no—
356#529—1221no—
357#529—1221no—
358#525—612no—
359#525—1221no—
360#525—1221no—
361C4×#41—612no—
362C4×#41—1221no—
363C4×#41—1221no—
364C22×C42—611no—
365#432—611no—
366#432—1221no—
367#432C2×D6111yescharacteristic
368#431—612no—
369#431—1221no—
370#423—611no—
371#423—612no—
372#423—612no—
373#423—1221no—
374#423—1221no—
375#423C2×D6111yescharacteristic
376C2×C4⋊C8—612no—
377C2×C4⋊C8—1221no—
378#407—1221no—
379C4×M16—1221no—
380#333—612no—
381#333—612no—
382#332—612no—
383#332—612no—
384#332—612no—
385#332C2×D6112yescomplemented
386#331—612no—
387#331C2×D6112yescomplemented
388C61#1773711yescharacteristic, complemented, Sylow
389C2×#41—21no—
390C2×#41—612no—
391C2×#41—1221no—
392C23×C4—611no—
393C23×C4—611no—
394C23×C4—611no—
395C2×M16—21no—
396C2×M16—1222no—
397C2×M16—1222no—
398C2×M16—1221no—
399C2×M16—2444no—
400C22×C8—1222no—
401C4⋊Q8—611no—
402C4⋊Q8—611no—
403C4⋊Q8C22×D6111yescharacteristic
404#127C22×D6111yescharacteristic
405#126—1222no—
406#126—1222no—
407#125—612no—
408#125—1221no—
409#125—1221no—
410#124—21no—
411#124—612no—
412#124—1221no—
413#123—1222no—
414#123—1222no—
415#123—1222no—
416C22⋊Q8—612no—
417C22⋊Q8—612no—
418C22⋊Q8—1221no—
419C22⋊Q8—1221no—
420C22⋊Q8—1221no—
421C22⋊Q8—1221no—
422#121—21no—
423#121—612no—
424#121—1221no—
425#121—1221no—
426#121—1221no—
427#121C4×D6112yes—
428#120—1222no—
429C4×Q8—612no—
430C4×Q8—612no—
431C4×Q8—1221no—
432C4×Q8—1221no—
433C4×Q8—1221no—
434C4×Q8—1221no—
435C4×D4—21no—
436C4×D4—612no—
437C4×D4—1221no—
438C4×D4—1221no—
439C4×D4—1221no—
440C4×D4—1222no—
441C4×D4—1222no—
442C4×D4—1222no—
443C4×D4—1222no—
444C4×D4—1222no—
445C4×D4—1222no—
446C4×D4C4×D6112yes—
447#117—612no—
448#117—1222no—
449#117—1222no—
450#117—1222no—
451#117—1221no—
452#117—1221no—
453C2×C42—611no—
454C2×C42—611no—
455C2×C42—611no—
456C2×C42—611no—
457C2×C42—611no—
458C2×C42—611no—
459C2×C42—611no—
460C2×C42—611no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42—1222no—
465C2×C42C22×D6111yescharacteristic
466C4⋊C8—21no—
467C4⋊C8—612no—
468C4⋊C8—612no—
469C4⋊C8—1222no—
470C4⋊C8—1222no—
471C4⋊C8—1222no—
472C4⋊C8—1221no—
473C4⋊C8C22×D6112yes—
474C4⋊C8C22×D6112yes—
475C22⋊C8—21no—
476C22⋊C8—1222no—
477C22⋊C8—1221no—
478#97—21no—
479#97—1221no—
480C4×C8—1222no—
481C24—611no—
482#41—44no—
483#41—2442no—
484#41—2442no—
485#41—2444no—
486C2×Q8—21no—
487C2×Q8—612no—
488C2×Q8—1221no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—1222no—
493C2×D4—1222no—
494C2×D4C2×C4×D6112yes—
495C22×C4—22no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—1222no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—2442no—
516C22×C4—2442no—
517C22×C4C2×C4×D6111yescharacteristic
518C22×C4D4×D6111yescharacteristic
519C22×C4D4×D6111yescharacteristic
520M16—42no—
521M16—2444no—
522M16—2444no—
523M16—2442no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—1222no—
527C2×C8—1222no—
528C2×C8—2444no—
529C4⋊C4—21no—
530C4⋊C4—612no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1222no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1222no—
539C4⋊C4—1222no—
540C4⋊C4C2×C4×D6112yes—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—1222no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4—1222no—
547C22⋊C4—1222no—
548C22⋊C4—1222no—
549C22⋊C4—1222no—
550C42—611no—
551C42—611no—
552C42—611no—
553C42—611no—
554C42—611no—
555C42—611no—
556C42—611no—
557C42—611no—
558C42—611no—
559C42—611no—
560C42—611no—
561C42—611no—
562C42—1222no—
563C42—1222no—
564C42—1222no—
565C42C2×C4×D6111yescharacteristic
566C42D4×D6111yescharacteristic
567C42D4×D6111yescharacteristic
568C42C23×D6111yescharacteristic
569C23—22no—
570C23—611no—
571C23—611no—
572C23—1222no—
573C23—2441no—
574C23—2441no—
575C23C22⋊C4×D6111yescharacteristic
576Q8—42no—
577Q8—2442no—
578Q8—2441no—
579Q8—2441no—
580D4—42no—
581D4—44no—
582D4—41no—
583D4—41no—
584D4—42no—
585D4—2444no—
586D4—2444no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—611no—
594C2×C4—611no—
595C2×C4—612no—
596C2×C4—612no—
597C2×C4—611no—
598C2×C4—611no—
599C2×C4—611no—
600C2×C4—611no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2444no—
618C2×C4—2444no—
619C2×C4—2441no—
620C2×C4C22⋊C4×D6111yescharacteristic
621C2×C4C22⋊C4×D6111yescharacteristic
622C2×C4C22⋊C4×D6111yescharacteristic
623C2×C4C22×C4×D6111yescharacteristic
624C2×C4C2×D4×D6111yescharacteristic
625C2×C4C2×D4×D6111yescharacteristic
626C8—42no—
627C8—42no—
628C8—2442no—
629C8—2442no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—612no—
637C22—1221no—
638C22—1221no—
639C22—1221no—
640C22—1221no—
641C22—1221no—
642C22—2444no—
643C22C2×C22⋊C4×D6111yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—42no—
649C4—612no—
650C4—612no—
651C4—1221no—
652C4—1221no—
653C4—1221no—
654C4—1221no—
655C4—2442no—
656C4—2442no—
657C4AGL(1,ℤ/8)×D6111yescharacteristic
658C4#137×D6111yescharacteristic
659C4#1684055812yes—
660C2—41no—
661C2—82no—
662C2—1222no—
663C2—2441no—
664C2#413×D6111yescharacteristic
665C2#417×D6111yescharacteristic
666C2#2145433311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1557C2
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC42×C61C24
hypercenter#1557D61
hypocenterC61#17737
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1557

Complements

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

NH ≅ G/Ntype
2. #21454449C2semidirect
3. #423×D61C2semidirect
4. #21444555C2semidirect
5. #21444551C2semidirect
6. #21444550C2semidirect
7. #21444547C2semidirect
8. #21444541C2semidirect
9. #21444475C2semidirect
10. #21444465C2semidirect
11. C61×#1557C2semidirect
44. #16839510C22semidirect
45. #16839510C22semidirect
46. #16839509C22semidirect
47. #16839508C22semidirect
48. #16839507C22semidirect
49. #16839506C22semidirect
53. C61×#332C22semidirect
54. C61×#331C22semidirect
335. #1557D61semidirect
385. #332C2×D61semidirect
387. #331C2×D61semidirect
388. C61#17737semidirect

Maximal subgroups

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

subgroupindex
#214591142
#214544492
#423×D612
#214445552
#214445512
#214445502
#214445472
#214445412
#214444752
#214444652
C61×#15572
#1773761