Finite Group Databasebeta

← #29646756

Subgroups — #29646756

Order 15616.

1036 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840299C211yesmaximal, characteristic
2#21454447C211yesmaximal, characteristic, complemented
3#423×D61C211yesmaximal, characteristic, complemented
4#21444283C214yesmaximal, complemented
5#21444248C211yesmaximal, characteristic, complemented
6#21444245C211yesmaximal, characteristic, complemented
7#21444243C211yesmaximal, characteristic, complemented
8#21444231C214yesmaximal, complemented
9C61×#1487C211yesmaximal, characteristic, complemented
10C2×M16×D61—21no—
11#16840299—21no—
12#16840299—21no—
13#16840299—21no—
14#16840299—21no—
15#16840299C412yes—
16#16840299C412yes—
17#16840299C2212yes—
18#16840299C2212yes—
19C2×#14519772—21no—
20C2×#14519768C411yescharacteristic
21C2×#14519766C411yescharacteristic
22C2×#14519762—21no—
23C2×#14519753C411yescharacteristic
24C2×#14519752C411yescharacteristic
25C2×C4×D244—21no—
26C2×C4×D244C2211yescharacteristic
27C2×#14519734—21no—
28C2×C42×D61C2211yescharacteristic
29#16839818—21no—
30#16839816—21no—
31#16839815C2211yescharacteristic
32#16839813C2212yescomplemented
33#16839786C2211yescharacteristic
34#16839785—21no—
35#16839784—21no—
36#16839784—21no—
37#16839784C2212yescomplemented
38C4⋊C8×D61—21no—
39C4⋊C8×D61C2212yescomplemented
40#16839724—21no—
41C4×#11758189C2212yescomplemented
42Q8⋊#92418C2214yescomplemented
43#16839491C2214yescomplemented
44#16839489C2214yescomplemented
45C2×C4×Q8×C61C2211yescharacteristic
46C61×#431C2211yescharacteristic
47C61×#423C2211yescharacteristic
48C61×Q8⋊C8C2214yescomplemented
49C22×D244—21no—
50C22×D244C2×C411yescharacteristic
51C22×C4×D61—21no—
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61D411yescharacteristic
55C2×#11758194—21no—
56C2×#11758189—22no—
57C2×#11758189—22no—
58M16×D61—44no—
59C2×C8×D61—22no—
60#14519772—22no—
61#14519772—22no—
62#14519772—22no—
63#14519772—22no—
64#14519768—21no—
65#14519768—21no—
66#14519768C2×C412yes—
67#14519768C2×C412yes—
68#14519766—21no—
69#14519766—21no—
70#14519766C2×C412yes—
71#14519766C2×C412yes—
72#14519762—22no—
73#14519762—22no—
74#14519762—22no—
75#14519762—22no—
76#14519753—21no—
77#14519753—21no—
78#14519753C2×C412yes—
79#14519753C2×C412yes—
80#14519752—21no—
81#14519752—21no—
82#14519752C2×C412yes—
83#14519752C2×C412yes—
84C4×D244—22no—
85C4×D244—22no—
86C4×D244—22no—
87C4×D244—22no—
88C4×D244—21no—
89C4×D244—21no—
90C4×D244C2×C412yes—
91C4×D244C2312yes—
92#14519734—22no—
93#14519734—22no—
94#14519734—22no—
95#14519734—22no—
96C2×C4⋊Dic61—21no—
97C2×C4⋊Dic61C2×C411yescharacteristic
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758198—21no—
101C2×#11758198—21no—
102C2×#11758198—21no—
103C2×#11758198—21no—
104C2×#11758198—21no—
105C2×C4×Dic61C2×C411yescharacteristic
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C42×D61C2×C411yescharacteristic
109C42×D61C2×C411yescharacteristic
110C42×D61D411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61D411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61C2311yescharacteristic
115C42×D61C2311yescharacteristic
116C4⋊#92418—21no—
117C4⋊#92418—21no—
118C4⋊#92418C2312yes—
119#14519706—22no—
120#14519706—22no—
121#14519706—22no—
122C22⋊#92418—21no—
123#14519689—22no—
124#14519687—22no—
125C4×#92418C2312yes—
126C22×Q8×C61C2×C411yescharacteristic
127C2×M16×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61C2×C412yes—
131C4×Q8×C61C2312yes—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61C2×C411yescharacteristic
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2312yes—
139C22⋊C8×C61—21no—
140C4×C8×C61C2312yes—
141C23×D61—21no—
142C23×D61C22⋊C411yescharacteristic
143C2×Q8×C61—21no—
144C2×Q8×C61—42no—
145C2×Q8×C61C22×C412yes—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61C22⋊C411yescharacteristic
153C22×Dic61C22⋊C411yescharacteristic
154C22×Dic61C22⋊C411yescharacteristic
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—21no—
158C2×D244—42no—
159C2×D244—44no—
160C2×D244C22×C412yes—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
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×D61C22×C411yescharacteristic
176C2×C4×D61C22×C411yescharacteristic
177C2×C4×D61C2×D411yescharacteristic
178C2×C4×D61C2×D411yescharacteristic
179C2×C4×D61C2×D411yescharacteristic
180C2×C4×D61C2×D411yescharacteristic
181M16×C61—42no—
182C2×C8×C61—22no—
183C2×C8×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61C22×C412yes—
188C42×C61—22no—
189C42×C61—22no—
190C42×C61C22×C411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2411yescharacteristic
194#11758198—22no—
195#11758198—22no—
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—
210#11758196—22no—
211#11758196—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—
227#11758189—44no—
228#11758189—44no—
229C8×D61—44no—
230C23×C61C2×C22⋊C411yescharacteristic
231C22×D61—22no—
232C22×D61—22no—
233C22×D61—22no—
234C22×D61—41no—
235C22×D61—42no—
236C22×D61—42no—
237C22×D61—42no—
238C22×D61C2×C22⋊C411yescharacteristic
239C22×D61C2×C22⋊C411yescharacteristic
240Q8×C61—24no—
241Q8×C61—42no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—41no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61AGL(1,ℤ/8)12yes—
252C2×C4×C61#13712yes—
253C2×C4×C61C23×C411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—42no—
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C2×C22⋊C411yescharacteristic
265C2×Dic61C2×C22⋊C411yescharacteristic
266C2×Dic61C2×C22⋊C411yescharacteristic
267D244—24no—
268D244—42no—
269D244—44no—
270D244—44no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—21no—
276C4×D61—21no—
277C4×D61—21no—
278C4×D61—21no—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—41no—
282C4×D61—41no—
283C4×D61—41no—
284C4×D61—41no—
285C4×D61—42no—
286C4×D61—42no—
287C4×D61—44no—
288C8×C61—42no—
289C8×C61—42no—
290#92418—42no—
291#92418—42no—
292#17492—611nomaximal, Sylow
293C22×C61—21no—
294C22×C61—21no—
295C22×C61#41712yes—
296C22×C61C22×C22⋊C411yescharacteristic
297C2×D61—21no—
298C2×D61—21no—
299C2×D61—21no—
300C2×D61—21no—
301C2×D61—21no—
302C2×D61—44no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—84no—
307C2×D61#42112yes—
308C4×C61—21no—
309C4×C61—21no—
310C4×C61—21no—
311C4×C61—42no—
312C4×C61—42no—
313C4×C61#41712yes—
314C4×C61#56812yes—
315C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
316C4×C61C2×#13711yescharacteristic
317Dic61—21no—
318Dic61—21no—
319Dic61—21no—
320Dic61—21no—
321Dic61—21no—
322Dic61—21no—
323Dic61—42no—
324C2×#529—611no—
325#2920—611no—
326C2×#423—611no—
327#1495—614no—
328#1487—611no—
329#1487D6111yescharacteristic, complemented
330#1486—611no—
331#1486—611no—
332#1483—614no—
333C2×C61—22no—
334C2×C61#287911yescharacteristic
335C2×C61C2×#41711yescharacteristic
336C2×C61C2×#42111yescharacteristic
337D61—22no—
338D61—41no—
339D61—82no—
340C22×M16—1221no—
341#529—612no—
342#529—612no—
343#529—612no—
344#529—612no—
345#529—1221no—
346#529—1221no—
347#529—1221no—
348#529—1221no—
349C2×#126—1221no—
350C2×#125—611no—
351C2×#124—611no—
352C2×#123—1221no—
353C2×C22⋊Q8—611no—
354C2×#121—611no—
355C2×C4×Q8C2×D6111yescharacteristic
356C2×C4×D4—611no—
357C2×C4×D4—1221no—
358C2×#117—1221no—
359C22×C42—611no—
360#432—1221no—
361#432—1221no—
362#431—611no—
363#431—612no—
364#431C2×D6111yescharacteristic
365#423—611no—
366#423—612no—
367#423—1221no—
368#423—1221no—
369#423—1221no—
370#423C2×D6111yescharacteristic
371C2×C4⋊C8—612no—
372C2×C4⋊C8—1221no—
373#407—1221no—
374C4×M16—612no—
375Q8⋊C8—614no—
376Q8⋊C8C2×D6114yescomplemented
377D4⋊C8—614no—
378D4⋊C8—614no—
379C61#1749211yescharacteristic, complemented, Sylow
380C22×Q8C4×D6111yescharacteristic
381C22×D4—611no—
382C22×D4—1221no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—1221no—
387C2×M16—21no—
388C2×M16—1222no—
389C2×M16—1222no—
390C2×M16—1221no—
391C2×M16—2444no—
392C22×C8—1222no—
393#126—1222no—
394#126—1222no—
395#126—1222no—
396#126—1222no—
397#125—612no—
398#125—612no—
399#125—1221no—
400#125—1221no—
401#124—612no—
402#124—612no—
403#124—1221no—
404#124—1221no—
405#123—1222no—
406#123—1222no—
407#123—1222no—
408#123—1222no—
409C22⋊Q8—612no—
410C22⋊Q8—612no—
411C22⋊Q8—1221no—
412C22⋊Q8—1221no—
413#121—612no—
414#121—612no—
415#121—1221no—
416#121—1221no—
417C4×Q8—21no—
418C4×Q8—21no—
419C4×Q8C4×D6112yes—
420C4×Q8C22×D6112yes—
421C4×D4—612no—
422C4×D4—612no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1222no—
427C4×D4—1221no—
428C4×D4—1221no—
429#117—1222no—
430#117—1222no—
431#117—1222no—
432#117—1222no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4C4×D6111yescharacteristic
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C42—21no—
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—611no—
454C2×C42—611no—
455C2×C42—611no—
456C2×C42C22×D6111yescharacteristic
457C4⋊C8—21no—
458C4⋊C8—21no—
459C4⋊C8—612no—
460C4⋊C8—1222no—
461C4⋊C8—1222no—
462C4⋊C8—1222no—
463C4⋊C8—1221no—
464C4⋊C8—1221no—
465C4⋊C8C22×D6112yes—
466C22⋊C8—21no—
467C22⋊C8—1222no—
468C22⋊C8—1221no—
469#97—1222no—
470C4×C8—612no—
471C4×C8C22×D6112yes—
472C24—611no—
473C24—1221no—
474C2×Q8—21no—
475C2×Q8—42no—
476C2×Q8C2×C4×D6112yes—
477C2×D4—612no—
478C2×D4—1222no—
479C2×D4—1222no—
480C2×D4—1221no—
481C2×D4—2442no—
482C2×D4—2444no—
483C22×C4—21no—
484C22×C4—21no—
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—611no—
499C22×C4—611no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4—1221no—
503C22×C4—1222no—
504C22×C4—1222no—
505C22×C4—2442no—
506C22×C4—2442no—
507C22×C4—2442no—
508C22×C4—2442no—
509C22×C4C2×C4×D6111yescharacteristic
510C22×C4D4×D6111yescharacteristic
511C22×C4D4×D6111yescharacteristic
512M16—42no—
513M16—2444no—
514M16—2444no—
515M16—2442no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—1222no—
519C2×C8—1222no—
520C2×C8—2444no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—612no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1221no—
532C4⋊C4C2×C4×D6112yes—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4—1222no—
542C22⋊C4—1222no—
543C42—22no—
544C42—22no—
545C42—611no—
546C42—611no—
547C42—611no—
548C42—611no—
549C42—611no—
550C42—611no—
551C42—611no—
552C42—611no—
553C42—611no—
554C42—611no—
555C42—611no—
556C42—611no—
557C42C2×C4×D6111yescharacteristic
558C42D4×D6111yescharacteristic
559C42D4×D6111yescharacteristic
560C42C23×D6111yescharacteristic
561C23—611no—
562C23—611no—
563C23—1222no—
564C23—1222no—
565C23—1222no—
566C23—2441no—
567C23—2442no—
568C23—2442no—
569C23—2442no—
570C23C22⋊C4×D6111yescharacteristic
571Q8—24no—
572Q8—42no—
573D4—1224no—
574D4—2442no—
575D4—2444no—
576D4—2444no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—611no—
584C2×C4—611no—
585C2×C4—611no—
586C2×C4—611no—
587C2×C4—611no—
588C2×C4—611no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1222no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2442no—
608C2×C4—2444no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4C22⋊C4×D6111yescharacteristic
612C2×C4C22⋊C4×D6111yescharacteristic
613C2×C4C22⋊C4×D6111yescharacteristic
614C2×C4#1451980412yes—
615C2×C4#1451980912yes—
616C2×C4C22×C4×D6111yescharacteristic
617C2×C4C2×D4×D6111yescharacteristic
618C2×C4C2×D4×D6111yescharacteristic
619C8—42no—
620C8—42no—
621C8—2442no—
622C8—2442no—
623C22—21no—
624C22—21no—
625C22—612no—
626C22—1221no—
627C22—1221no—
628C22—1221no—
629C22—1221no—
630C22—1221no—
631C22—4884no—
632C22—2444no—
633C22—2442no—
634C22—2442no—
635C22—2442no—
636C22#1683976212yes—
637C22C2×C22⋊C4×D6111yescharacteristic
638C4—21no—
639C4—21no—
640C4—21no—
641C4—42no—
642C4—42no—
643C4—1221no—
644C4—1221no—
645C4—1221no—
646C4—1221no—
647C4—1221no—
648C4—1221no—
649C4—2442no—
650C4#1683976212yes—
651C4#1684052812yes—
652C4C2×#1451980411yescharacteristic
653C4C2×#1451980911yescharacteristic
654C2—22no—
655C2—1222no—
656C2—4882no—
657C2—2441no—
658C2#2145426211yescharacteristic
659C2C2×#1683976211yescharacteristic
660C2#421×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1487C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519752C4
cosocleC42×C61C24
hypercenter#1487D61
hypocenterC61#17492
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
2. #21454447C2semidirect
3. #423×D61C2semidirect
4. #21444283C2semidirect
5. #21444248C2semidirect
6. #21444245C2semidirect
7. #21444243C2semidirect
8. #21444231C2semidirect
9. C61×#1487C2semidirect
32. #16839813C22semidirect
37. #16839784C22semidirect
39. C4⋊C8×D61C22semidirect
41. C4×#11758189C22semidirect
42. Q8⋊#92418C22semidirect
43. #16839491C22semidirect
44. #16839489C22semidirect
48. C61×Q8⋊C8C22semidirect
329. #1487D61semidirect
376. Q8⋊C8C2×D61semidirect
379. C61#17492semidirect

Maximal subgroups

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

subgroupindex
C2×#168402992
#214544472
#423×D612
#214442832
#214442482
#214442452
#214442432
#214442312
C61×#14872
#1749261