Finite Group Databasebeta

← #29690031

Subgroups — #29690031

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#21459208C211yesmaximal, characteristic
2#21454455C211yesmaximal, characteristic, complemented
3#21448299C211yesmaximal, characteristic, complemented
4#21446763C211yesmaximal, characteristic, complemented
5#21446762C211yesmaximal, characteristic, complemented
6#21446739C211yesmaximal, characteristic, complemented
7#21446264C212yesmaximal, complemented
8#21446259C212yesmaximal, complemented
9#21445837C212yesmaximal, complemented
10#21445837C212yesmaximal, complemented
11C61×#1980C211yesmaximal, characteristic, complemented
12#16840381C412yescomplemented
13#16840381C412yescomplemented
14#16840373C412yescomplemented
15#16840366C412yescomplemented
16#16840357—21no—
17#16840357—21no—
18#16840349—21no—
19#16840349—21no—
20#16840319C2211yescharacteristic
21#16840292—22no—
22#16840292—22no—
23#16840269C2212yes—
24#41×Dic61C2212yes—
25C2×#14519734C2211yescharacteristic
26C2×#14519734C2211yescharacteristic
27C2×C42×D61C2211yescharacteristic
28#16839824—22no—
29#16839821—22no—
30#16839785C2212yes—
31C2×#14519704C2212yes—
32C2×#14519704C2212yes—
33#16839725C2212yes—
34C4×#11758194C2211yescharacteristic
35#16839659—21no—
36#16839657—21no—
37#16839636—21no—
38#16839633—21no—
39C4×#11758196C2211yescharacteristic
40C4×#11758196C2211yescharacteristic
41C4×C4⋊Dic61C2211yescharacteristic
42#16839523C2212yes—
43#16839523C2212yes—
44#16839522C2212yes—
45#16839522C2212yes—
46C61×#532C2211yescharacteristic
47C2×#104×C61C2212yes—
48C2×#104×C61C2212yes—
49C4×M16×C61C2211yescharacteristic
50C4×C4⋊C4×C61C2211yescharacteristic
51C22×C4×D61—21no—
52C22×C4×D61C2311yescharacteristic
53C2×#11758194C2312yes—
54#14519777C2×C411yescharacteristic, complemented
55#14519777C2×C411yescharacteristic, complemented
56#14519772—22no—
57#14519772—22no—
58#14519770—22no—
59#14519770—22no—
60#14519769C2×C412yescomplemented
61#14519767—22no—
62#14519767—22no—
63#14519763—22no—
64#14519763—22no—
65#14519761—22no—
66#14519761—22no—
67#14519758—22no—
68#14519758—22no—
69#14519758—22no—
70#14519756—21no—
71#14519756—21no—
72#14519754—22no—
73#14519754—22no—
74#14519753C2×C412yes—
75#14519753C2×C412yes—
76#14519749—22no—
77#14519749—22no—
78#14519748C2×C412yes—
79#14519748C2×C412yes—
80#14519745—22no—
81Q8×Dic61C2×C412yescomplemented
82Q8×Dic61C2×C412yescomplemented
83C4×Dic122—21no—
84C4×Dic122—21no—
85D4×Dic61—22no—
86D4×Dic61—22no—
87D4×Dic61C2×C412yescomplemented
88D4×Dic61C2×C412yescomplemented
89C4×C22⋊D61—22no—
90C4×C22⋊D61—22no—
91#14519736—22no—
92#14519736C2312yes—
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—22no—
108C2×C4×Dic61—21no—
109C2×C4×Dic61C2311yescharacteristic
110C2×C4×Dic61C2311yescharacteristic
111C2×C4×Dic61C2311yescharacteristic
112C42×D61—42no—
113#14519706—24no—
114#14519704—42no—
115#14519704—42no—
116#14519689—24no—
117#14519688—21no—
118C4×#92418—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122C2×#41×C61C2312yes—
123C2×M16×C61C2312yes—
124C4⋊Q8×C61C2×C411yescharacteristic, complemented
125#127×C61C2×C411yescharacteristic, complemented
126#124×C61C2×C412yescomplemented
127#121×C61—21no—
128#121×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C2×C4⋊C4×C61—21no—
132C2×C42×C61C2311yescharacteristic
133C2×C42×C61C2311yescharacteristic
134C2×C42×C61C2311yescharacteristic
135#104×C61—42no—
136#104×C61—42no—
137#97×C61—21no—
138C4×C8×C61—21no—
139#95×C61—21no—
140C23×D61C2×D411yescharacteristic
141#41×C61—22no—
142#41×C61—22no—
143#41×C61—22no—
144#41×C61—22no—
145C2×Q8×C61C22×C412yes—
146C2×D4×C61—22no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61C22×C412yes—
150C22×C4×C61—21no—
151C22×C4×C61—22no—
152C22×C4×C61—21no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C2×C22⋊D61—22no—
157C2×C22⋊D61—22no—
158C22×Dic61—21no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—21no—
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×C4×D61C22×C412yes—
171C2×C4×D61C2×D412yes—
172C2×Dic122—21no—
173C2×Dic122—21no—
174M16×C61—22no—
175M16×C61—22no—
176C2×C8×C61—22no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—42no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C42×C61—22no—
183C42×C61—22no—
184C42×C61C22×C411yescharacteristic
185C42×C61C22×C411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193#11758198—22no—
194#11758198—22no—
195#11758198—42no—
196#11758198C22⋊C412yes—
197#11758198C22⋊C412yes—
198#11758198C22⋊C412yes—
199#11758198C22⋊C412yes—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—22no—
203C4⋊Dic61—22no—
204C4⋊Dic61—42no—
205C4⋊Dic61C22×C412yes—
206C4⋊Dic61C22×C412yes—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—42no—
212#11758196—42no—
213#11758196—42no—
214#11758196C22⋊C412yes—
215#11758196C22⋊C412yes—
216#11758196C22⋊C412yes—
217#11758196C22⋊C412yes—
218C4×Dic61—22no—
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×C61—22no—
230C23×C61C22×D411yescharacteristic
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61C2×C22⋊C412yes—
236Q8×C61—22no—
237Q8×C61—22no—
238D4×C61—22no—
239D4×C61—22no—
240D4×C61—42no—
241D4×C61—42no—
242D4×C61—81no—
243D4×C61—81no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C2×C22⋊C411yescharacteristic
259C2×C4×C61C23×C411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61H(5,𝔽2)12yes—
262C2×C4×C612−1+412yes—
263C22⋊D61—82no—
264C22⋊D61—82no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—82no—
278C2×Dic61—82no—
279C2×Dic61C2×C22⋊C412yes—
280C2×Dic61C2×C22⋊C412yes—
281C2×Dic61C2×C22⋊C412yes—
282C4×D61—24no—
283C4×D61—24no—
284C4×D61—48no—
285Dic122—81no—
286Dic122—81no—
287C8×C61—42no—
288#92418—42no—
289#21901—611nomaximal, Sylow
290C22×C61—21no—
291C22×C61—21no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61C22×C22⋊C411yescharacteristic
296C22×C61#53511yescharacteristic
297C22×C61#53611yescharacteristic
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—22no—
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—81no—
310C4×C61#52012yes—
311C4×C61#53712yes—
312Dic61—22no—
313Dic61—22no—
314Dic61—22no—
315Dic61—42no—
316Dic61—42no—
317Dic61—42no—
318Dic61—81no—
319#3455—611no—
320#2921—611no—
321#2296—611no—
322#1986—611no—
323#1986—611no—
324#1980—611no—
325#1980D6111yescharacteristic, complemented
326#1897—612no—
327#1896—612no—
328#1806—612no—
329#1806—612no—
330C2×C61—21no—
331C2×C61—21no—
332C2×C61—82no—
333C2×C61#231111yescharacteristic
334C2×C61#290111yescharacteristic
335C2×C61#290111yescharacteristic
336D61—42no—
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#533—611no—
346#532C2×D6111yescharacteristic
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×#104C2×D6112yes—
360C2×#104C2×D6112yes—
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#339—612no—
375#339—612no—
376C61#2190111yescharacteristic, complemented, Sylow
377C2×#41C22×D6112yes—
378C23×C4—611no—
379C23×C4—1221no—
380C2×M16—612no—
381C2×M16C22×D6112yes—
382C4⋊Q8—611no—
383C4⋊Q8—611no—
384C4⋊Q8C4×D6111yescharacteristic, complemented
385#127C4×D6111yescharacteristic, complemented
386#126—1222no—
387#126—1222no—
388#126—1222no—
389#126—1222no—
390#125—612no—
391#125—1222no—
392#125—1222no—
393#124—1222no—
394#124—1222no—
395#124C4×D6112yescomplemented
396#123—1222no—
397#123—1222no—
398#123—1222no—
399#123—1222no—
400#123—1222no—
401C22⋊Q8—612no—
402C22⋊Q8—612no—
403C22⋊Q8—1222no—
404C22⋊Q8—1222no—
405C22⋊Q8—1221no—
406C22⋊Q8—1221no—
407#121—21no—
408#121—21no—
409#121—612no—
410#121—612no—
411#121—1222no—
412#121—1222no—
413#120—1222no—
414C4×Q8—612no—
415C4×Q8—612no—
416C4×Q8—1221no—
417C4×Q8—1221no—
418C4×D4—21no—
419C4×D4—21no—
420C4×D4—612no—
421C4×D4—612no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426#117—612no—
427#117—612no—
428#117—612no—
429#117—612no—
430#117—612no—
431#117—1222no—
432#117—2442no—
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—42no—
454#104—42no—
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—22no—
468#41—22no—
469#41—22no—
470#41—22no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8C2×C4×D6112yes—
474C2×D4—22no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—1222no—
478C2×D4—1222no—
479C2×D4C2×C4×D6112yes—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—612no—
487C22×C4—612no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1221no—
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—21no—
507C4⋊C4—21no—
508C4⋊C4—42no—
509C4⋊C4—612no—
510C4⋊C4—612no—
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—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—2442no—
524C4⋊C4—2442no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—612no—
530C22⋊C4—612no—
531C22⋊C4—612no—
532C22⋊C4—612no—
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—612no—
544C42—612no—
545C42—1222no—
546C42—1222no—
547C42—1222no—
548C42—1222no—
549C42—1222no—
550C42—2442no—
551C42C2×C4×D6111yescharacteristic
552C42C2×C4×D6111yescharacteristic
553C42D4×D6111yescharacteristic
554C42D4×D6111yescharacteristic
555C23—22no—
556C23—612no—
557C23—1221no—
558C23—1221no—
559C23—2441no—
560C23—2441no—
561C23C2×D4×D6111yescharacteristic
562Q8—22no—
563Q8—22no—
564Q8—4881no—
565Q8—4881no—
566D4—22no—
567D4—22no—
568D4—42no—
569D4—42no—
570D4—81no—
571D4—81no—
572D4—4882no—
573D4—4882no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—612no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1224no—
591C2×C4—1224no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—4882no—
596C2×C4—4882no—
597C2×C4—2448no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
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#1451982212yes—
612C2×C4#1451982412yes—
613C8—42no—
614C8—2442no—
615C22—21no—
616C22—21no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—1222no—
621C22—1222no—
622C22—1222no—
623C22—2442no—
624C22—2442no—
625C22C2×C22⋊C4×D6111yescharacteristic
626C22#1684032511yescharacteristic
627C22#1684033211yescharacteristic
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—42no—
633C4—42no—
634C4—81no—
635C4—1222no—
636C4—1222no—
637C4—1222no—
638C4—4881no—
639C4—2442no—
640C4—2442no—
641C4—2442no—
642C4#1684024112yes—
643C4#1684034112yes—
644C2—21no—
645C2—21no—
646C2—82no—
647C2—2442no—
648C2#2144840911yescharacteristic
649C2#2145436311yescharacteristic
650C2#2145436511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1980C2
socleC22×C61C22×C22⋊C4
prime radical#21459208C2
cosocleC22×C4×C61C24
hypercenter#1980D61
hypocenterC61#21901
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
2. #21454455C2semidirect
3. #21448299C2semidirect
4. #21446763C2semidirect
5. #21446762C2semidirect
6. #21446739C2semidirect
7. #21446264C2semidirect
8. #21446259C2semidirect
9. #21445837C2semidirect
10. #21445837C2semidirect
11. C61×#1980C2semidirect
12. #16840381C4semidirect
13. #16840381C4semidirect
14. #16840373C4semidirect
15. #16840366C4semidirect
54. #14519777C2×C4semidirect
55. #14519777C2×C4semidirect
60. #14519769C2×C4semidirect
81. Q8×Dic61C2×C4semidirect
82. Q8×Dic61C2×C4semidirect
87. D4×Dic61C2×C4semidirect
88. D4×Dic61C2×C4semidirect
124. C4⋊Q8×C61C2×C4semidirect
125. #127×C61C2×C4semidirect
126. #124×C61C2×C4semidirect
325. #1980D61semidirect
376. C61#21901semidirect
384. C4⋊Q8C4×D61semidirect
385. #127C4×D61semidirect
395. #124C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214592082
#214544552
#214482992
#214467632
#214467622
#214467392
#214462642
#214462592
#214458372
C61×#19802
#2190161