Finite Group Databasebeta

← #29646690

Subgroups — #29646690

Order 15616.

984 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459235C211yesmaximal, characteristic
2#21454937C211yesmaximal, characteristic, complemented
3#21454775C211yesmaximal, characteristic, complemented
4#21444286C212yesmaximal, complemented
5#21444283C212yesmaximal, complemented
6#21444278C212yesmaximal, complemented
7#21444268C212yesmaximal, complemented
8#21444211C211yesmaximal, characteristic, complemented
9#21444201C211yesmaximal, characteristic, complemented
10C2×#16839489C211yesmaximal, characteristic, complemented
11C61×#1477C211yesmaximal, characteristic, complemented
12#16840390C411yescharacteristic
13#16840384C412yes—
14#16840383—21no—
15#16840367—21no—
16#16840363C412yes—
17#16840345—22no—
18#16840340C411yescharacteristic
19#16840331C411yescharacteristic
20#16840326C411yescharacteristic
21#16840312—21no—
22#16840309C2212yes—
23#16840299C2212yes—
24#16840296—21no—
25#16840271—22no—
26C4×#11758222C2211yescharacteristic
27C2×C4×D244C2211yescharacteristic
28#16839889—21no—
29#16839886—21no—
30#16839839—21no—
31#16839833—22no—
32#16839827—21no—
33#16839825—22no—
34#16839818C2212yescomplemented
35#16839815C2211yescharacteristic
36#16839813C2212yescomplemented
37#16839784C2212yescomplemented
38C2×C4×#92418C2211yescharacteristic
39C4×C8×D61C2212yescomplemented
40Q8⋊#92418C2211yescharacteristic, complemented
41Q8⋊#92418C2211yescharacteristic, complemented
42#16839491C2212yes—
43#16839491C2212yes—
44D4⋊#92418C2211yescharacteristic, complemented
45D4⋊#92418C2211yescharacteristic, complemented
46#16839489C2212yes—
47#16839489C2212yes—
48C4×#41×C61C2211yescharacteristic
49C61×#431C2211yescharacteristic
50C2×C4⋊C8×C61C2211yescharacteristic
51C61×Q8⋊C8C2212yescomplemented
52C61×D4⋊C8C2212yescomplemented
53C2×#11758222—22no—
54C2×#11758222C2×C411yescharacteristic
55C22×D244C2×C411yescharacteristic
56C22×C4×D61—21no—
57C2×#11758189—22no—
58C22×#92418—21no—
59C2×C8×D61—22no—
60#14519777—21no—
61#14519776C2×C412yes—
62#14519774—21no—
63#14519773C2×C412yes—
64#14519772—22no—
65#14519771—22no—
66#14519769—21no—
67#14519768C2×C412yes—
68#14519766C2×C412yes—
69#14519765—21no—
70#14519765—22no—
71#14519763—22no—
72#14519762—22no—
73#14519762—22no—
74#14519760—22no—
75#14519759—22no—
76#14519756—21no—
77#14519755—22no—
78#14519754—22no—
79#14519753C2×C412yes—
80#14519752C2×C412yes—
81#14519752C2×C412yes—
82#14519751—21no—
83#14519751—22no—
84#14519751—21no—
85#14519751—21no—
86#14519750—22no—
87#14519750—22no—
88#14519749—22no—
89#14519748C2×C412yes—
90#14519747—22no—
91#14519745—22no—
92C4×Dic122D411yescharacteristic
93C4×Dic122D411yescharacteristic
94C4×D244—22no—
95C4×D244—21no—
96C4×D244—21no—
97C4×D244C2×C412yes—
98C4×D244D411yescharacteristic
99C4×D244D411yescharacteristic
100C4×D244C2312yes—
101C4×C22⋊D61—22no—
102C4×C22⋊D61D412yes—
103C4×C22⋊D61D412yes—
104#14519736C2×C411yescharacteristic
105#14519734—22no—
106#14519734C2×C412yes—
107C2×C4⋊Dic61C2×C411yescharacteristic
108C2×#11758198—21no—
109C42×D61C2312yes—
110C4⋊#92418C2311yescharacteristic
111C4⋊#92418C2311yescharacteristic
112#14519706—22no—
113#14519706—22no—
114C22⋊#92418—21no—
115#14519689—22no—
116#14519689—22no—
117#14519687—22no—
118C4×#92418—21no—
119C4×#92418—21no—
120C4×#92418C2311yescharacteristic
121C4×#92418C2311yescharacteristic
122C8×Dic61—22no—
123C2×#41×C61C2×C411yescharacteristic
124C22×C8×C61—21no—
125C4×Q8×C61C2×C411yescharacteristic
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61C2×C411yescharacteristic
130C4×D4×C61C2311yescharacteristic
131#117×C61—21no—
132#117×C61C2×C411yescharacteristic
133C2×C42×C61—21no—
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2312yes—
138C22⋊C8×C61—21no—
139C4×C8×C61C2312yes—
140C23×D61—21no—
141#41×C61—22no—
142#41×C61—22no—
143C2×Q8×C61C22×C411yescharacteristic
144C2×D4×C61—21no—
145C2×D4×C61C22×C411yescharacteristic
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61C22⋊C412yes—
156C22×Dic61—21no—
157#11758222—22no—
158#11758222—22no—
159#11758222—82no—
160#11758222—82no—
161C2×D244—22no—
162C2×D244—22no—
163C2×D244—21no—
164C2×D244—22no—
165C2×D244—22no—
166C2×D244C22⋊C411yescharacteristic
167C2×D244C22×C412yes—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—81no—
171C2×C4×D61—81no—
172C2×C4×D61C22×C412yes—
173C2×C4×D61C2×D412yes—
174C2×C4×D61C2×D412yes—
175C2×Dic122—22no—
176C2×Dic122C22⋊C411yescharacteristic
177C2×C8×C61—22no—
178C2×C8×C61—22no—
179C2×C8×C61—81no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61C22×C411yescharacteristic
183C4⋊C4×C61C22×C411yescharacteristic
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61C22×C411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2411yescharacteristic
194C22⋊Dic61—22no—
195C22⋊Dic61C22⋊C412yes—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198C22⋊C412yes—
204#11758198C22⋊C412yes—
205C4⋊Dic61—21no—
206C4⋊Dic61—22no—
207C4⋊Dic61C22⋊C411yescharacteristic
208C4⋊Dic61C22⋊C411yescharacteristic
209C4⋊Dic61C22×C412yes—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196C22⋊C412yes—
214#11758196C22⋊C412yes—
215C4×Dic61C22×C412yes—
216C4×Dic61C2×D412yes—
217C4×Dic61C2×D412yes—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—42no—
223#11758189—82no—
224C8×D61—44no—
225C23×C61—21no—
226C23×C61C2×C22⋊C411yescharacteristic
227C22×D61—22no—
228C22×D61—42no—
229C22×D61—81no—
230C22×D61—81no—
231C22×D61C2×C22⋊C412yes—
232Q8×C61—21no—
233Q8×C61—21no—
234D4×C61—21no—
235D4×C61—21no—
236D4×C61—42no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—42no—
246C2×C4×C61—81no—
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61#13112yes—
251C2×C4×C61#13112yes—
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C22⋊D61—24no—
256C22⋊D61—82no—
257C22⋊D61—82no—
258C22⋊D61—82no—
259C2×Dic61—22no—
260C2×Dic61—81no—
261C2×Dic61C2×C22⋊C412yes—
262C2×Dic61C2×C22⋊C412yes—
263C2×Dic61C2×C22⋊C412yes—
264D244—22no—
265D244—22no—
266D244—22no—
267D244—42no—
268D244—82no—
269D244—82no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—22no—
274C4×D61—42no—
275C4×D61—42no—
276C4×D61—42no—
277C4×D61—82no—
278Dic122—22no—
279Dic122—82no—
280C8×C61—42no—
281C8×C61—42no—
282#92418—22no—
283#92418—41no—
284#92418—41no—
285#92418—41no—
286#17483—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61#40812yes—
293C22×C61C22×C22⋊C411yescharacteristic
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—84no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61#40812yes—
308C4×C61C2×#13111yescharacteristic
309C4×C61C2×#13111yescharacteristic
310C4×C61#57611yescharacteristic
311C4×C61#57711yescharacteristic
312Dic61—22no—
313Dic61—22no—
314Dic61—22no—
315Dic61—42no—
316#3460—611no—
317#2989—611no—
318C8×#41—611no—
319#1496—612no—
320#1495—612no—
321#1494—612no—
322#1492—612no—
323#1478—611no—
324#1477D6111yescharacteristic, complemented
325#1476—611no—
326C2×D4⋊C8—611no—
327C2×C61—22no—
328C2×C61—81no—
329C2×C61C2×#40811yescharacteristic
330C2×C61#289711yescharacteristic
331C2×C61#290111yescharacteristic
332D61—42no—
333D61—82no—
334#544—611no—
335#542—612no—
336#542—1221no—
337#540—612no—
338#540—1221no—
339#538—1222no—
340#537—611no—
341#535—611no—
342#535—611no—
343#532—612no—
344#532—1221no—
345#529—612no—
346#529—1221no—
347#525—1222no—
348C4×#41—611no—
349C4×#41C2×D6111yescharacteristic
350C2×C4×D4—611no—
351#446—1221no—
352C8×Q8—1221no—
353#436—1221no—
354#435—1222no—
355C8×D4—1222no—
356C8×D4—1221no—
357#432—612no—
358#431—612no—
359#431—611no—
360#431C2×D6111yescharacteristic
361#423—612no—
362C2×C4⋊C8C2×D6111yescharacteristic
363C2×C4×C8—611no—
364C2×C4×C8—612no—
365Q8⋊C8—611no—
366Q8⋊C8—611no—
367Q8⋊C8C2×D6112yescomplemented
368D4⋊C8—611no—
369D4⋊C8—611no—
370D4⋊C8—612no—
371D4⋊C8—612no—
372D4⋊C8—612no—
373D4⋊C8—612no—
374D4⋊C8C2×D6112yescomplemented
375C61#1748311yescharacteristic, complemented, Sylow
376C2×#41—611no—
377C2×#41—1222no—
378C2×#41C4×D6111yescharacteristic
379C22×D4—611no—
380C23×C4—1221no—
381C2×M16—1222no—
382C22×C8—21no—
383C22×C8—1222no—
384C22×C8—1221no—
385C4⋊Q8—612no—
386C4⋊Q8—1221no—
387#127—612no—
388#127—1221no—
389#126—1222no—
390#126—1222no—
391#125—612no—
392#125—1221no—
393#124—612no—
394#124—1222no—
395#124—1221no—
396#124—1222no—
397#123—1222no—
398#123—1222no—
399#123—1222no—
400#123—1222no—
401C22⋊Q8—612no—
402C22⋊Q8—1222no—
403C22⋊Q8—1222no—
404C22⋊Q8—1221no—
405#121—612no—
406#121—612no—
407#121—612no—
408#121—1222no—
409#121—1222no—
410#121—1222no—
411#121—1221no—
412#121—1221no—
413#121—1221no—
414#121—1222no—
415#120—1222no—
416#120—1222no—
417C4×Q8—611no—
418C4×Q8—611no—
419C4×Q8C4×D6111yescharacteristic
420C4×Q8C22×D6111yescharacteristic
421C4×D4—21no—
422C4×D4—21no—
423C4×D4—612no—
424C4×D4—612no—
425C4×D4—612no—
426C4×D4—611no—
427C4×D4—611no—
428C4×D4—612no—
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—1221no—
432C4×D4—1221no—
433C4×D4C4×D6111yescharacteristic
434C4×D4C22×D6111yescharacteristic
435#117—21no—
436#117—611no—
437#117—612no—
438#117—1222no—
439#117C4×D6111yescharacteristic
440C2×C4⋊C4—611no—
441C2×C22⋊C4—1221no—
442C2×C42—21no—
443C2×C42—612no—
444C2×C42C22×D6111yescharacteristic
445C4⋊C8—21no—
446C4⋊C8—21no—
447C4⋊C8—611no—
448C4⋊C8—611no—
449C4⋊C8—1222no—
450C4⋊C8—1222no—
451C4⋊C8C22×D6112yes—
452C22⋊C8—21no—
453C22⋊C8—1222no—
454C22⋊C8—1222no—
455C22⋊C8—1221no—
456#97—1222no—
457C4×C8—611no—
458C4×C8—611no—
459C4×C8—1222no—
460C4×C8—1221no—
461C4×C8—1221no—
462C4×C8C22×D6112yes—
463C24—1221no—
464#41—22no—
465#41—22no—
466#41—1222no—
467#41—1222no—
468#41—4882no—
469#41—4882no—
470C2×Q8—611no—
471C2×Q8—1222no—
472C2×Q8C2×C4×D6111yescharacteristic
473C2×D4—21no—
474C2×D4—612no—
475C2×D4—612no—
476C2×D4—611no—
477C2×D4—1222no—
478C2×D4—1222no—
479C2×D4—1222no—
480C2×D4—1222no—
481C2×D4—1222no—
482C2×D4—1221no—
483C2×D4—1222no—
484C2×D4—1222no—
485C2×D4C2×C4×D6111yescharacteristic
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—612no—
490C22×C4—612no—
491C22×C4—612no—
492C22×C4—1221no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—4881no—
496C22×C4—4881no—
497C22×C4C2×C4×D6111yescharacteristic
498C22×C4D4×D6111yescharacteristic
499C22×C4D4×D6111yescharacteristic
500M16—4882no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—81no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—2444no—
509C2×C8—2442no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—612no—
513C4⋊C4—612no—
514C4⋊C4—611no—
515C4⋊C4—611no—
516C4⋊C4—612no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1221no—
522C4⋊C4C2×C4×D6111yescharacteristic
523C4⋊C4C2×C4×D6111yescharacteristic
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C22⋊C4—612no—
528C22⋊C4—612no—
529C22⋊C4—612no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—612no—
542C42—612no—
543C42—612no—
544C42C2×C4×D6111yescharacteristic
545C42D4×D6111yescharacteristic
546C42D4×D6111yescharacteristic
547C42C23×D6111yescharacteristic
548C23—21no—
549C23—612no—
550C23—1222no—
551C23—4881no—
552C23—4881no—
553C23—2442no—
554C23C22⋊C4×D6111yescharacteristic
555Q8—21no—
556Q8—21no—
557Q8—1222no—
558Q8—4882no—
559D4—21no—
560D4—21no—
561D4—42no—
562D4—1224no—
563D4—1222no—
564D4—1222no—
565D4—1222no—
566D4—4882no—
567D4—4882no—
568D4—4882no—
569D4—4882no—
570D4—4882no—
571D4—2442no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—81no—
582C2×C4—612no—
583C2×C4—612no—
584C2×C4—612no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—4882no—
591C2×C4—4881no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4C22⋊C4×D6111yescharacteristic
596C2×C4C22⋊C4×D6111yescharacteristic
597C2×C4C22⋊C4×D6111yescharacteristic
598C2×C4#1451978312yes—
599C2×C4#1451978512yes—
600C2×C4C22×C4×D6111yescharacteristic
601C2×C4C2×D4×D6111yescharacteristic
602C2×C4C2×D4×D6111yescharacteristic
603C8—42no—
604C8—42no—
605C8—1222no—
606C8—2441no—
607C8—2441no—
608C8—2441no—
609C22—21no—
610C22—21no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—1222no—
615C22—1222no—
616C22—1222no—
617C22—4884no—
618C22—2442no—
619C22—2442no—
620C22—2442no—
621C22#1683972712yes—
622C22C2×C22⋊C4×D6111yescharacteristic
623C4—21no—
624C4—21no—
625C4—21no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—1222no—
630C4—1222no—
631C4—1222no—
632C4—2442no—
633C4#1683972712yes—
634C4C2×#1451978311yescharacteristic
635C4C2×#1451978511yescharacteristic
636C4#1684056611yescharacteristic
637C4#1684057111yescharacteristic
638C2—22no—
639C2—81no—
640C2—4882no—
641C2—2442no—
642C2C2×#1683972711yescharacteristic
643C2#2145434311yescharacteristic
644C2#2145436511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1477C2
socleC22×C61C22×C22⋊C4
prime radical#21459235C2
cosocleC42×C61C24
hypercenter#1477D61
hypocenterC61#17483
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1477

Complements

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

NH ≅ G/Ntype
2. #21454937C2semidirect
3. #21454775C2semidirect
4. #21444286C2semidirect
5. #21444283C2semidirect
6. #21444278C2semidirect
7. #21444268C2semidirect
8. #21444211C2semidirect
9. #21444201C2semidirect
10. C2×#16839489C2semidirect
11. C61×#1477C2semidirect
34. #16839818C22semidirect
36. #16839813C22semidirect
37. #16839784C22semidirect
39. C4×C8×D61C22semidirect
40. Q8⋊#92418C22semidirect
41. Q8⋊#92418C22semidirect
44. D4⋊#92418C22semidirect
45. D4⋊#92418C22semidirect
51. C61×Q8⋊C8C22semidirect
52. C61×D4⋊C8C22semidirect
324. #1477D61semidirect
367. Q8⋊C8C2×D61semidirect
374. D4⋊C8C2×D61semidirect
375. C61#17483semidirect

Maximal subgroups

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

subgroupindex
#214592352
#214549372
#214547752
#214442862
#214442832
#214442782
#214442682
#214442112
#214442012
C2×#168394892
C61×#14772
#1748361