Finite Group Databasebeta

← #29690032

Subgroups — #29690032

Order 15616.

1058 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459211C211yesmaximal, characteristic
2#21454455C211yesmaximal, characteristic, complemented
3#21448299C211yesmaximal, characteristic, complemented
4#21446764C211yesmaximal, characteristic, complemented
5#21446738C211yesmaximal, characteristic, complemented
6#21446737C211yesmaximal, characteristic, complemented
7#21446266C212yesmaximal, complemented
8#21446261C212yesmaximal, complemented
9#21445838C212yesmaximal, complemented
10#21445838C212yesmaximal, complemented
11C61×#1986C211yesmaximal, characteristic, complemented
12#16840386C412yescomplemented
13#16840386C412yescomplemented
14#16840375C412yescomplemented
15#16840368C412yescomplemented
16#16840361—21no—
17#16840361—21no—
18#16840352—21no—
19#16840352—21no—
20#16840312C2211yescharacteristic
21#16840299—22no—
22#16840299—22no—
23#16840274C2212yes—
24#16840230C2212yes—
25C2×#14519734C2211yescharacteristic
26C2×#14519734C2211yescharacteristic
27C2×C42×D61C2211yescharacteristic
28#16839824—22no—
29#16839821—22no—
30#16839785C2212yes—
31C2×#14519705C2212yes—
32C2×#14519705C2212yes—
33C2×#14519703C2212yes—
34C2×#14519703C2212yes—
35#16839725C2212yes—
36C4×#11758194C2211yescharacteristic
37#16839659—21no—
38#16839657—21no—
39#16839636—21no—
40#16839633—21no—
41C4×#11758196C2211yescharacteristic
42C4×#11758196C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44#16839521C2212yes—
45#16839521C2212yes—
46C61×#533C2211yescharacteristic
47C4×M16×C61C2211yescharacteristic
48C4×C4⋊C4×C61C2211yescharacteristic
49C61×#339C2212yes—
50C61×#339C2212yes—
51C2×#11758222C2312yes—
52C22×C4×D61—21no—
53C22×C4×D61C2311yescharacteristic
54C2×#11758194C2312yes—
55#14519777C2×C411yescharacteristic, complemented
56#14519774C2×C411yescharacteristic, complemented
57#14519772—22no—
58#14519772—22no—
59#14519772—22no—
60#14519772—22no—
61#14519768—22no—
62#14519768—22no—
63#14519766—22no—
64#14519766—22no—
65#14519765C2×C412yescomplemented
66#14519762—22no—
67#14519762—22no—
68#14519762—22no—
69#14519762—22no—
70#14519761—22no—
71#14519757C2×C412yes—
72#14519757C2×C412yes—
73#14519753—22no—
74#14519753—22no—
75#14519752—22no—
76#14519752—22no—
77#14519752C2×C412yes—
78#14519752C2×C412yes—
79#14519751—21no—
80#14519751—21no—
81#14519747—22no—
82#14519742C2×C412yescomplemented
83#14519742C2×C412yescomplemented
84C4×D244—22no—
85C4×D244—22no—
86C4×D244—21no—
87C4×D244—21no—
88#14519739C2×C412yescomplemented
89#14519739C2×C412yescomplemented
90#14519737—22no—
91#14519737—22no—
92#14519734—22no—
93#14519734—42no—
94#14519734C2×C412yes—
95#14519734C2×C412yes—
96#14519734D412yes—
97#14519734D412yes—
98C2×C4⋊Dic61—21no—
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196D411yescharacteristic
103C2×#11758196D411yescharacteristic
104C2×#11758198—21no—
105C2×#11758198D411yescharacteristic
106C2×#11758198D411yescharacteristic
107C2×C4×Dic61—21no—
108C2×C4×Dic61C2311yescharacteristic
109C2×C4×Dic61C2311yescharacteristic
110C2×C4×Dic61C2311yescharacteristic
111C42×D61—22no—
112C42×D61—42no—
113#14519706—24no—
114#14519705—42no—
115#14519705—42no—
116#14519703—42no—
117#14519703—42no—
118#14519689—24no—
119#14519688—21no—
120C4×#92418—21no—
121#14519684—21no—
122#14519684—21no—
123#14519684—21no—
124C2×M16×C61C2312yes—
125C4⋊Q8×C61C2×C411yescharacteristic, complemented
126C4⋊Q8×C61C2×C411yescharacteristic, complemented
127#125×C61C2×C412yescomplemented
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—21no—
130C4×Q8×C61—21no—
131C4×Q8×C61—21no—
132#117×C61C2312yes—
133C2×C4⋊C4×C61—21no—
134C2×C42×C61C2311yescharacteristic
135C2×C42×C61C2311yescharacteristic
136C2×C42×C61C2311yescharacteristic
137#97×C61—21no—
138C4×C8×C61—21no—
139#95×C61—21no—
140C23×D61C2×D411yescharacteristic
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C2×C22⋊D61—22no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154#11758222—22no—
155#11758222—22no—
156#11758222—22no—
157#11758222—22no—
158C2×D244—22no—
159C2×D244—22no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244C22×C412yes—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61C22×C412yes—
172C2×C4×D61C2×D412yes—
173C2×Dic122C22×C412yes—
174M16×C61—22no—
175M16×C61—22no—
176C2×C8×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—42no—
182C4⋊C4×C61C22×C412yes—
183C4⋊C4×C61C22×C412yes—
184C22⋊C4×C61—22no—
185C42×C61—22no—
186C42×C61—22no—
187C42×C61—22no—
188C42×C61C22×C411yescharacteristic
189C42×C61C22×C411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192#11758198—22no—
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—42no—
201#11758198C22⋊C412yes—
202#11758198C22⋊C412yes—
203#11758198C22⋊C412yes—
204#11758198C22⋊C412yes—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—22no—
208C4⋊Dic61—22no—
209C4⋊Dic61—42no—
210#11758196—22no—
211#11758196—22no—
212#11758196—42no—
213#11758196—42no—
214#11758196—42no—
215#11758196C22⋊C412yes—
216#11758196C22⋊C412yes—
217#11758196C22⋊C412yes—
218#11758196C22⋊C412yes—
219C4×Dic61—22no—
220C4×Dic61—22no—
221C4×Dic61—22no—
222C4×Dic61—22no—
223C4×Dic61—42no—
224C4×Dic61C22×C412yes—
225C4×Dic61C2×D412yes—
226#11758194—22no—
227#11758194—22no—
228C2×#92418—22no—
229C23×C61C22×D411yescharacteristic
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—22no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61C2×C22⋊C412yes—
236Q8×C61—81no—
237Q8×C61—81no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61H(5,𝔽2)12yes—
255C2×C4×C612−1+412yes—
256C22⋊D61—42no—
257C22⋊D61—42no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61C2×C22⋊C412yes—
270C2×Dic61C2×C22⋊C412yes—
271C2×Dic61C2×C22⋊C412yes—
272D244—22no—
273D244—22no—
274D244—81no—
275D244—81no—
276D244—82no—
277D244—82no—
278C4×D61—24no—
279C4×D61—24no—
280C4×D61—48no—
281C4×D61—42no—
282C4×D61—42no—
283C4×D61—82no—
284C4×D61—82no—
285Dic122—22no—
286Dic122—22no—
287C8×C61—42no—
288#92418—42no—
289#21901—611nomaximal, Sylow
290C22×C61—21no—
291C22×C61—21no—
292C22×C61C22×C22⋊C411yescharacteristic
293C22×C61#53511yescharacteristic
294C22×C61#53611yescharacteristic
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—42no—
308C4×C61—42no—
309C4×C61—42no—
310C4×C61—81no—
311C4×C61#52012yes—
312C4×C61#53712yes—
313Dic61—22no—
314Dic61—22no—
315Dic61—22no—
316Dic61—42no—
317Dic61—42no—
318Dic61—81no—
319#3455—611no—
320#2921—611no—
321#2296—611no—
322#1986—611no—
323#1986D6111yescharacteristic, complemented
324#1980—611no—
325#1980—611no—
326#1897—612no—
327#1896—612no—
328#1806—612no—
329#1806—612no—
330C2×C61—21no—
331C2×C61—21no—
332C2×C61#231111yescharacteristic
333C2×C61#290111yescharacteristic
334C2×C61#290111yescharacteristic
335D61—42no—
336D61—82no—
337#542—612no—
338#542—612no—
339#541—612no—
340#540—612no—
341#539—1221no—
342#539—1221no—
343#538—1221no—
344#538—1221no—
345#533C2×D6111yescharacteristic
346#532—611no—
347#529—1222no—
348#529—1222no—
349#525—612no—
350C4×#41—612no—
351C2×#117—611no—
352C2×#117—611no—
353C22×C42—611no—
354#433—1222no—
355#433—1222no—
356#423—612no—
357C2×#104—612no—
358C2×#104—612no—
359C2×#104—612no—
360C2×#104—612no—
361#407—612no—
362C4×M16—611no—
363C4×M16C2×D6111yescharacteristic
364#389—1221no—
365#389—1221no—
366#385—1221no—
367#385—1221no—
368C4×C4⋊C4—611no—
369C4×C4⋊C4—611no—
370C4×C4⋊C4—611no—
371C4×C4⋊C4C2×D6111yescharacteristic
372#339—612no—
373#339—612no—
374#339C2×D6112yes—
375#339C2×D6112yes—
376C61#2190111yescharacteristic, complemented, Sylow
377C2×#41—612no—
378C23×C4—611no—
379C23×C4—1221no—
380C2×M16—612no—
381C2×M16C22×D6112yes—
382C4⋊Q8—611no—
383C4⋊Q8C4×D6111yescharacteristic, complemented
384C4⋊Q8C4×D6111yescharacteristic, complemented
385#127—611no—
386#126—1222no—
387#126—1222no—
388#126—1222no—
389#126—1222no—
390#125—1222no—
391#125—1222no—
392#125C4×D6112yescomplemented
393#124—612no—
394#124—1222no—
395#124—1222no—
396#123—1222no—
397#123—1222no—
398#123—1222no—
399#123—1222no—
400#123—1222no—
401C22⋊Q8—21no—
402C22⋊Q8—21no—
403C22⋊Q8—612no—
404C22⋊Q8—612no—
405C22⋊Q8—1222no—
406C22⋊Q8—1222no—
407#121—612no—
408#121—612no—
409#121—1222no—
410#121—1222no—
411#121—1221no—
412#121—1221no—
413#120—1222no—
414C4×Q8—21no—
415C4×Q8—21no—
416C4×Q8—612no—
417C4×Q8—612no—
418C4×D4—612no—
419C4×D4—612no—
420C4×D4—1222no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1221no—
425C4×D4—1221no—
426#117—612no—
427#117—612no—
428#117—612no—
429#117—612no—
430#117—1222no—
431#117—2442no—
432#117C22×D6112yes—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C22⋊C4—611no—
441C2×C22⋊C4—611no—
442C2×C22⋊C4—1221no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—1222no—
447C2×C42—1221no—
448C2×C42—2442no—
449C2×C42C22×D6111yescharacteristic
450C2×C42C22×D6111yescharacteristic
451C2×C42C22×D6111yescharacteristic
452C4⋊C8—1224no—
453#104—2442no—
454#104—2442no—
455#104—2442no—
456#104—2442no—
457C22⋊C8—1224no—
458#97—21no—
459#97—1221no—
460C4×C8—21no—
461C4×C8—1221no—
462#95—21no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466C24—611no—
467#41—1222no—
468#41—1222no—
469#41—1222no—
470#41—1222no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—612no—
474C2×D4—612no—
475C2×D4—1222no—
476C2×D4—1222no—
477C2×D4—1222no—
478C2×D4—1221no—
479C2×D4—1221no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—612no—
486C22×C4—612no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1221no—
492C22×C4—1222no—
493C22×C4—1221no—
494C22×C4—2442no—
495C22×C4—2442no—
496C22×C4—2442no—
497C22×C4D4×D6111yescharacteristic
498C22×C4D4×D6111yescharacteristic
499C22×C4C23×D6111yescharacteristic
500M16—22no—
501M16—22no—
502M16—1222no—
503M16—1222no—
504C2×C8—22no—
505C2×C8—1222no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—42no—
511C4⋊C4—612no—
512C4⋊C4—612no—
513C4⋊C4—612no—
514C4⋊C4—612no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—2442no—
522C4⋊C4—2442no—
523C4⋊C4—2442no—
524C4⋊C4—2442no—
525C4⋊C4C2×C4×D6112yes—
526C4⋊C4C2×C4×D6112yes—
527C22⋊C4—22no—
528C22⋊C4—612no—
529C22⋊C4—612no—
530C22⋊C4—612no—
531C22⋊C4—612no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C22⋊C4—2442no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—612no—
545C42—612no—
546C42—1222no—
547C42—1222no—
548C42—1222no—
549C42—1222no—
550C42—2442no—
551C42C2×C4×D6111yescharacteristic
552C42C2×C4×D6111yescharacteristic
553C42D4×D6111yescharacteristic
554C42D4×D6111yescharacteristic
555C23—612no—
556C23—1221no—
557C23—1221no—
558C23—1222no—
559C23—2441no—
560C23—2441no—
561C23C2×D4×D6111yescharacteristic
562Q8—81no—
563Q8—81no—
564Q8—1222no—
565Q8—1222no—
566D4—1222no—
567D4—1222no—
568D4—4881no—
569D4—4881no—
570D4—4882no—
571D4—4882no—
572D4—2442no—
573D4—2442no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—612no—
585C2×C4—612no—
586C2×C4—612no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1224no—
590C2×C4—1224no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—4882no—
594C2×C4—4882no—
595C2×C4—2448no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4C22⋊C4×D6111yescharacteristic
606C2×C4C22⋊C4×D6111yescharacteristic
607C2×C4C22⋊C4×D6111yescharacteristic
608C2×C4C22⋊C4×D6111yescharacteristic
609C2×C4C22×C4×D6111yescharacteristic
610C2×C4C2×D4×D6111yescharacteristic
611C2×C4#1451982312yes—
612C2×C4#1451982512yes—
613C8—42no—
614C8—2442no—
615C22—21no—
616C22—21no—
617C22—1222no—
618C22—1222no—
619C22—1222no—
620C22—2442no—
621C22—2442no—
622C22—2442no—
623C22—2442no—
624C22—2442no—
625C22C2×C22⋊C4×D6111yescharacteristic
626C22#1684033011yescharacteristic
627C22#1684033711yescharacteristic
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—81no—
636C4—1222no—
637C4—1222no—
638C4—1222no—
639C4—4881no—
640C4—2442no—
641C4—2442no—
642C4#1684024412yes—
643C4#1684034212yes—
644C2—21no—
645C2—21no—
646C2—4882no—
647C2—2442no—
648C2#2144841011yescharacteristic
649C2#2145436311yescharacteristic
650C2#2145436511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1986C2
socleC22×C61C22×C22⋊C4
prime radical#21459211C2
cosocleC22×C4×C61C24
hypercenter#1986D61
hypocenterC61#21901
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
2. #21454455C2semidirect
3. #21448299C2semidirect
4. #21446764C2semidirect
5. #21446738C2semidirect
6. #21446737C2semidirect
7. #21446266C2semidirect
8. #21446261C2semidirect
9. #21445838C2semidirect
10. #21445838C2semidirect
11. C61×#1986C2semidirect
12. #16840386C4semidirect
13. #16840386C4semidirect
14. #16840375C4semidirect
15. #16840368C4semidirect
55. #14519777C2×C4semidirect
56. #14519774C2×C4semidirect
65. #14519765C2×C4semidirect
82. #14519742C2×C4semidirect
83. #14519742C2×C4semidirect
88. #14519739C2×C4semidirect
89. #14519739C2×C4semidirect
125. C4⋊Q8×C61C2×C4semidirect
126. C4⋊Q8×C61C2×C4semidirect
127. #125×C61C2×C4semidirect
323. #1986D61semidirect
376. C61#21901semidirect
383. C4⋊Q8C4×D61semidirect
384. C4⋊Q8C4×D61semidirect
392. #125C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214592112
#214544552
#214482992
#214467642
#214467382
#214467372
#214462662
#214462612
#214458382
C61×#19862
#2190161