Finite Group Databasebeta

← #29646675

Subgroups — #29646675

Order 15616.

1056 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459116C211yesmaximal, characteristic
2C2×#16839813C211yesmaximal, characteristic, complemented
3#423×D61C211yesmaximal, characteristic, complemented
4#21444283C214yesmaximal, complemented
5#21444274C211yesmaximal, characteristic, complemented
6#21444271C211yesmaximal, characteristic, complemented
7#21444269C211yesmaximal, characteristic, complemented
8#21444205C214yesmaximal, complemented
9C61×#1493C211yesmaximal, characteristic, complemented
10C2×M16×D61—21no—
11#16840317C411yescharacteristic
12#16840317C411yescharacteristic
13#16840313C411yescharacteristic
14#16840313C411yescharacteristic
15#16840312C2211yescharacteristic
16#16840299—21no—
17#16840299C2212yes—
18#16840298—21no—
19#16840298C412yes—
20#16840297—21no—
21#16840297—21no—
22#16840275—21no—
23#16840275C412yes—
24#16840274—21no—
25C4×#11758226—21no—
26C4×#11758226C2212yes—
27#16840230—21no—
28C2×C42×D61C2211yescharacteristic
29#16839813—21no—
30#16839813—21no—
31#16839813C2212yes—
32#16839813C2212yes—
33#16839786C2211yescharacteristic
34#16839785—21no—
35#16839784—21no—
36#16839784C2212yes—
37C2×#14519706—21no—
38C4⋊C8×D61—21no—
39C4⋊C8×D61C2212yes—
40C2×#14519689—21no—
41C2×C4×#92418C2211yescharacteristic
42Q8⋊#92418C2214yescomplemented
43#16839491C2214yescomplemented
44#16839489C2214yescomplemented
45C61×#533C2211yescharacteristic
46C61×#423C2211yescharacteristic
47C2×C4×C8×C61C2211yescharacteristic
48C61×Q8⋊C8C2214yescomplemented
49C2×#11758226—21no—
50C2×#11758226C2×C412yes—
51C2×#11758222—21no—
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61D411yescharacteristic
55C2×#11758194—21no—
56C2×#11758189—22no—
57M16×D61—44no—
58C22×#92418—21no—
59C2×C8×D61—22no—
60#14519777C2×C411yescharacteristic
61#14519774C2×C411yescharacteristic
62#14519772—22no—
63#14519772—22no—
64#14519768—21no—
65#14519768C2×C412yes—
66#14519766—21no—
67#14519766C2×C412yes—
68#14519765—21no—
69#14519762—22no—
70#14519762—22no—
71#14519761—22no—
72#14519757—21no—
73#14519757—21no—
74#14519753—21no—
75#14519753C2×C412yes—
76#14519752—21no—
77#14519752—21no—
78#14519752—21no—
79#14519752C2×C412yes—
80#14519751—21no—
81#14519751C2×C412yes—
82#14519747—22no—
83Q8×Dic61—21no—
84Q8×Dic61C2×C412yes—
85#14519742—21no—
86#14519742—21no—
87C4×D244—22no—
88C4×D244—22no—
89C4×D244—21no—
90C4×D244C2312yes—
91#14519739—22no—
92#14519739—22no—
93#14519739—21no—
94#14519739—21no—
95#14519739—21no—
96#14519739C2×C412yes—
97#14519737—22no—
98#14519737—22no—
99#14519735—21no—
100#14519735—22no—
101#14519735—22no—
102#14519735C2×C412yes—
103#14519734—22no—
104C2×C4×Dic61C2×C411yescharacteristic
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61D411yescharacteristic
107C42×D61—22no—
108C42×D61C2×C411yescharacteristic
109C42×D61C2×C411yescharacteristic
110C42×D61D411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61D411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61C2311yescharacteristic
115C42×D61C2311yescharacteristic
116C4⋊#92418—21no—
117C4⋊#92418C2312yes—
118#14519706—22no—
119#14519706—22no—
120#14519706—22no—
121#14519706—22no—
122#14519689—22no—
123#14519689—22no—
124C4×#92418—21no—
125C4×#92418C2312yes—
126C2×M16×C61—21no—
127C22×C8×C61—21no—
128C4⋊Q8×C61C2×C411yescharacteristic
129C4⋊Q8×C61C2×C411yescharacteristic
130#125×C61—21no—
131C22⋊Q8×C61—21no—
132C22⋊Q8×C61C2×C412yes—
133C4×Q8×C61—21no—
134C4×Q8×C61C2312yes—
135#117×C61—21no—
136C2×C42×C61C2311yescharacteristic
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2312yes—
139C4×C8×C61—21no—
140C4×C8×C61C2312yes—
141C23×D61C22⋊C411yescharacteristic
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C2×C22⋊D61—22no—
148C22×Dic61C22⋊C411yescharacteristic
149C22×Dic61C22⋊C411yescharacteristic
150C22×Dic61C22⋊C411yescharacteristic
151#11758226—28no—
152#11758226—44no—
153#11758222—42no—
154#11758222—42no—
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244C22×C412yes—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—24no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—42no—
167C2×C4×D61C22⋊C411yescharacteristic
168C2×C4×D61C22⋊C411yescharacteristic
169C2×C4×D61C22⋊C411yescharacteristic
170C2×C4×D61C22⋊C411yescharacteristic
171C2×C4×D61C22⋊C411yescharacteristic
172C2×C4×D61C22⋊C411yescharacteristic
173C2×C4×D61C22×C411yescharacteristic
174C2×C4×D61C22×C411yescharacteristic
175C2×C4×D61C2×D411yescharacteristic
176C2×C4×D61C2×D411yescharacteristic
177C2×C4×D61C2×D411yescharacteristic
178C2×C4×D61C2×D411yescharacteristic
179C2×Dic122—21no—
180M16×C61—42no—
181C2×C8×C61—22no—
182C2×C8×C61—22no—
183C2×C8×C61—42no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61C22×C412yes—
190C22⋊C4×C61—22no—
191C42×C61—22no—
192C42×C61C22×C411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2411yescharacteristic
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204C4⋊Dic61—21no—
205C4⋊Dic61—22no—
206C4⋊Dic61—22no—
207C4⋊Dic61C22×C412yes—
208#11758196—22no—
209#11758196—22no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22×C411yescharacteristic
219C4×Dic61C22×C411yescharacteristic
220C4×Dic61C2×D411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222C4×Dic61C2×D411yescharacteristic
223C4×Dic61C2×D411yescharacteristic
224#11758194—42no—
225C2×#92418—22no—
226C2×#92418—22no—
227C2×#92418—42no—
228#11758189—44no—
229C8×D61—44no—
230C23×C61C2×C22⋊C411yescharacteristic
231C22×D61—22no—
232C22×D61—22no—
233C22×D61—22no—
234C22×D61—41no—
235C22×D61C2×C22⋊C411yescharacteristic
236C22×D61C2×C22⋊C411yescharacteristic
237Q8×C61—24no—
238Q8×C61—42no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61AGL(1,ℤ/8)12yes—
247C2×C4×C61#13712yes—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C22⋊D61—44no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—41no—
256C2×Dic61C2×C22⋊C411yescharacteristic
257C2×Dic61C2×C22⋊C411yescharacteristic
258C2×Dic61C2×C22⋊C411yescharacteristic
259C2×Dic61C2×C22⋊C411yescharacteristic
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262D244—24no—
263D244—44no—
264D244—44no—
265D244—42no—
266D244—42no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—21no—
270C4×D61—21no—
271C4×D61—21no—
272C4×D61—21no—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—41no—
279C4×D61—41no—
280C4×D61—44no—
281C4×D61—42no—
282C4×D61—42no—
283C4×D61—44no—
284C4×D61#13114yes—
285Dic122—42no—
286C8×C61—42no—
287C8×C61—42no—
288#92418—42no—
289#92418—42no—
290#17481—611nomaximal, Sylow
291C22×C61—21no—
292C22×C61—21no—
293C22×C61#41712yes—
294C22×C61C22×C22⋊C411yescharacteristic
295C2×D61—21no—
296C2×D61—21no—
297C2×D61—21no—
298C2×D61—21no—
299C2×D61—21no—
300C2×D61—44no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61#40812yes—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—42no—
308C4×C61—42no—
309C4×C61#41712yes—
310C4×C61#42112yes—
311C4×C61C2×#13112yes—
312C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
313C4×C61C2×#13711yescharacteristic
314Dic61—21no—
315Dic61—21no—
316Dic61—21no—
317Dic61—21no—
318Dic61—21no—
319Dic61—42no—
320Dic61#40812yes—
321#3444—611no—
322C2×#431—611no—
323C2×#423—611no—
324#1495—614no—
325#1493—611no—
326#1493D6111yescharacteristic, complemented
327#1492—611no—
328#1492—611no—
329#1477—614no—
330C2×C61—22no—
331C2×C61C2×#40811yescharacteristic
332C2×C61C2×#41711yescharacteristic
333C2×C61C2×#42111yescharacteristic
334D61—22no—
335D61—41no—
336D61—82no—
337C22×M16—1221no—
338#533—611no—
339#533—611no—
340#533C2×D6111yescharacteristic
341#532—611no—
342#532—611no—
343#532—611no—
344#529—612no—
345#529—612no—
346#529—1221no—
347#529—1221no—
348#529—1221no—
349#529—1221no—
350#525—612no—
351#525—1221no—
352#525—1221no—
353C4×#41—612no—
354C4×#41—1221no—
355C4×#41—1221no—
356C22×C42—611no—
357#431—612no—
358#431—612no—
359#431—1221no—
360#431—1221no—
361#423—611no—
362#423—612no—
363#423—1221no—
364#423—1221no—
365#423C2×D6111yescharacteristic
366C2×C4⋊C8—612no—
367C2×C4⋊C8—1221no—
368C2×C4⋊C8—1221no—
369C2×C22⋊C8—1221no—
370C2×C4×C8—611no—
371C2×C4×C8C2×D6111yescharacteristic
372Q8⋊C8—614no—
373Q8⋊C8C2×D6114yescomplemented
374D4⋊C8—614no—
375D4⋊C8—614no—
376C61#1748111yescharacteristic, complemented, Sylow
377C2×#41—612no—
378C2×#41—1221no—
379C2×#41—1221no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—611no—
383C2×M16—21no—
384C2×M16—1222no—
385C2×M16—1221no—
386C2×M16—2444no—
387C22×C8—21no—
388C22×C8—1222no—
389C22×C8—1221no—
390C4⋊Q8—611no—
391C4⋊Q8C4×D6111yescharacteristic
392C4⋊Q8C4×D6111yescharacteristic
393#127—611no—
394#126—1222no—
395#126—1222no—
396#125—21no—
397#125—612no—
398#125—1221no—
399#124—612no—
400#124—1221no—
401#124—1221no—
402#123—1222no—
403#123—1222no—
404#123—1222no—
405C22⋊Q8—21no—
406C22⋊Q8—612no—
407C22⋊Q8—1221no—
408C22⋊Q8—1221no—
409C22⋊Q8—1221no—
410C22⋊Q8C4×D6112yes—
411#121—612no—
412#121—612no—
413#121—1221no—
414#121—1221no—
415#121—1221no—
416#121—1221no—
417#120—1222no—
418C4×Q8—21no—
419C4×Q8—612no—
420C4×Q8—1221no—
421C4×Q8—1221no—
422C4×Q8—1221no—
423C4×Q8C22×D6112yes—
424C4×D4—612no—
425C4×D4—612no—
426C4×D4—1222no—
427C4×D4—1222no—
428C4×D4—1221no—
429C4×D4—1221no—
430C4×D4—1221no—
431C4×D4—1222no—
432C4×D4—1222no—
433C4×D4—1222no—
434C4×D4—1222no—
435C4×D4—1221no—
436#117—21no—
437#117—612no—
438#117—1222no—
439#117—1222no—
440#117—1222no—
441#117—1221no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—611no—
452C2×C42—611no—
453C2×C42—1222no—
454C2×C42C22×D6111yescharacteristic
455C4⋊C8—21no—
456C4⋊C8—612no—
457C4⋊C8—1222no—
458C4⋊C8—1222no—
459C4⋊C8—1222no—
460C4⋊C8—1222no—
461C4⋊C8—1221no—
462C4⋊C8C22×D6112yes—
463C22⋊C8—1222no—
464C22⋊C8—1222no—
465C4×C8—21no—
466C4×C8—612no—
467C4×C8—1221no—
468C4×C8C22×D6112yes—
469C24—611no—
470#41—1228no—
471#41—2444no—
472#41—2442no—
473#41—2442no—
474C2×Q8—21no—
475C2×Q8—1221no—
476C2×Q8C2×C4×D6112yes—
477C2×D4—612no—
478C2×D4—1222no—
479C2×D4—1222no—
480C2×D4—1222no—
481C2×D4—1221no—
482C2×D4—1221no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—1224no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4—1222no—
503C22×C4—1222no—
504C22×C4—2442no—
505C22×C4C2×C4×D6111yescharacteristic
506C22×C4D4×D6111yescharacteristic
507C22×C4D4×D6111yescharacteristic
508M16—42no—
509M16—2444no—
510M16—2442no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—42no—
514C2×C8—1222no—
515C2×C8—1222no—
516C2×C8—2444no—
517C2×C8—2442no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—612no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1221no—
529C4⋊C4C2×C4×D6112yes—
530C22⋊C4—22no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C42—22no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—611no—
548C42—611no—
549C42—611no—
550C42—611no—
551C42—611no—
552C42—1222no—
553C42—1222no—
554C42C2×C4×D6111yescharacteristic
555C42D4×D6111yescharacteristic
556C42D4×D6111yescharacteristic
557C42C23×D6111yescharacteristic
558C23—611no—
559C23—611no—
560C23—1222no—
561C23—1222no—
562C23—1222no—
563C23—2441no—
564C23C22⋊C4×D6111yescharacteristic
565Q8—24no—
566Q8—42no—
567Q8—2442no—
568D4—1224no—
569D4—2444no—
570D4—2444no—
571D4—2444no—
572D4—2442no—
573D4—2442no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—611no—
579C2×C4—614no—
580C2×C4—611no—
581C2×C4—611no—
582C2×C4—611no—
583C2×C4—611no—
584C2×C4—611no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1222no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2444no—
602C2×C4—2444no—
603C2×C4—2441no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4C22⋊C4×D6111yescharacteristic
607C2×C4C22⋊C4×D6111yescharacteristic
608C2×C4C22⋊C4×D6111yescharacteristic
609C2×C4#1451980412yes—
610C2×C4#1451980912yes—
611C2×C4C22×C4×D6111yescharacteristic
612C2×C4C2×D4×D6111yescharacteristic
613C2×C4C2×D4×D6111yescharacteristic
614C8—42no—
615C8—42no—
616C8—2442no—
617C8—2442no—
618C22—21no—
619C22—21no—
620C22—612no—
621C22—1221no—
622C22—1221no—
623C22—1221no—
624C22—1221no—
625C22—1221no—
626C22—2444no—
627C22—2442no—
628C22—2442no—
629C22—2442no—
630C22#1683976212yes—
631C22C2×C22⋊C4×D6111yescharacteristic
632C4—21no—
633C4—21no—
634C4—42no—
635C4—42no—
636C4—612no—
637C4—1221no—
638C4—1221no—
639C4—1221no—
640C4—1221no—
641C4—1221no—
642C4—2442no—
643C4#1683976212yes—
644C4#1683977712yes—
645C4#131×D6112yes—
646C4C2×#1451980411yescharacteristic
647C4C2×#1451980911yescharacteristic
648C2—22no—
649C2—1222no—
650C2—4882no—
651C2—2441no—
652C2#408×D6111yescharacteristic
653C2C2×#1683976211yescharacteristic
654C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1493C2
socleC22×C61C22×C22⋊C4
prime radical#21459116C2
cosocleC42×C61C24
hypercenter#1493D61
hypocenterC61#17481
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
2. C2×#16839813C2semidirect
3. #423×D61C2semidirect
4. #21444283C2semidirect
5. #21444274C2semidirect
6. #21444271C2semidirect
7. #21444269C2semidirect
8. #21444205C2semidirect
9. C61×#1493C2semidirect
42. Q8⋊#92418C22semidirect
43. #16839491C22semidirect
44. #16839489C22semidirect
48. C61×Q8⋊C8C22semidirect
326. #1493D61semidirect
373. Q8⋊C8C2×D61semidirect
376. C61#17481semidirect

Maximal subgroups

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

subgroupindex
#214591162
C2×#168398132
#423×D612
#214442832
#214442742
#214442712
#214442692
#214442052
C61×#14932
#1748161