Finite Group Databasebeta

← #30552386

Subgroups — #30552386

Order 15616.

1690 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459224C211yesmaximal, characteristic, complemented
2#21457111C218yesmaximal, complemented
3#21457096C218yesmaximal, complemented
4#21456361C212yesmaximal, complemented
5#21456205C212yesmaximal, complemented
6#21455763C212yesmaximal, complemented
7#21455755C212yesmaximal
8#21455624C212yesmaximal, complemented
9#21455616C212yesmaximal, complemented
10#21454457C211yesmaximal, characteristic, complemented
11C2×C61×#542C211yesmaximal, characteristic
12#16840393—21no—
13#16840393—21no—
14#16840390C2214yescomplemented
15#16840390C2214yescomplemented
16#16840388—21no—
17#16840388—21no—
18#16840319C2212yes—
19#16840319C2212yes—
20#16840315—24no—
21C2×#14519736C2212yes—
22#16840168C2218yescomplemented
23#16840165C2214yescomplemented
24#16840158C2214yescomplemented
25#16840121C2214yescomplemented
26#16840121C2214yescomplemented
27#16840116C2218yescomplemented
28#16840100C2218yescomplemented
29#16840095C2214yescomplemented
30#16840090C2214yescomplemented
31#16840083C2218yes—
32#16840078C2214yes—
33#16840073C2214yes—
34#16840069C2218yescomplemented
35#16840062C2218yescomplemented
36#16840057C2214yescomplemented
37#16840052C2214yescomplemented
38#16840047C2214yescomplemented
39#16840042C2214yescomplemented
40#16839823C2218yescomplemented
41#16839802C2214yes—
42#16839802C2214yes—
43#16839786C2212yes—
44#16839771C2214yes—
45#16839767C2214yes—
46#16839726C2212yes—
47C4×#11758194C2212yes—
48C61×#542—24no—
49C61×#542C2218yes—
50C2×C4⋊Q8×C61C2211yescharacteristic
51C2×#126×C61—21no—
52C2×#124×C61—21no—
53C2×#124×C61C2211yescharacteristic
54C2×#123×C61—21no—
55C2×C22⋊Q8×C61—21no—
56C2×C22⋊Q8×C61C2211yescharacteristic
57C2×#121×C61C2211yescharacteristic
58C2×C4×Q8×C61C2211yescharacteristic
59C2×C4×D4×C61C2211yescharacteristic
60C2×#117×C61C2211yescharacteristic
61C2×#11758194C2314yes—
62#14519777C2312yes—
63#14519777C2312yes—
64#14519777C2312yes—
65#14519777C2312yes—
66#14519776—24no—
67#14519776—24no—
68#14519769C2314yes—
69#14519769C2314yes—
70#14519768—24no—
71#14519768—24no—
72#14519756—22no—
73#14519756—22no—
74#14519756—22no—
75#14519756—22no—
76#14519754—24no—
77#14519754—24no—
78#14519754—24no—
79#14519754—24no—
80C4×Dic122—22no—
81C4×Dic122—22no—
82C4×Dic122—22no—
83C4×Dic122—22no—
84#14519736—22no—
85#14519736—22no—
86#14519736C2314yes—
87#14519736C2314yes—
88C2×C4⋊Dic61C2312yes—
89C2×C4⋊Dic61C2312yes—
90C2×C22⋊Dic61—22no—
91C2×C4×Dic61—22no—
92#14519710C2318yes—
93#14519710C2318yes—
94#14519708C2318yes—
95#14519708C2318yes—
96C4⋊#92418C2314yes—
97C4⋊#92418C2314yes—
98#14519701C2318yes—
99#14519701C2318yes—
100#14519698C2318yes—
101#14519698C2318yes—
102C22⋊#92418C2314yes—
103C22⋊#92418C2314yes—
104#14519688C2314yes—
105C4×#92418C2314yes—
106C22×Q8×C61—21no—
107C22×Q8×C61C2311yescharacteristic
108C22×D4×C61—21no—
109C22×D4×C61C2311yescharacteristic
110C23×C4×C61—21no—
111C23×C4×C61C2311yescharacteristic
112C4⋊Q8×C61—22no—
113C4⋊Q8×C61C2314yes—
114#126×C61—28no—
115#124×C61—28no—
116#124×C61—22no—
117#124×C61C2314yes—
118#123×C61—28no—
119C22⋊Q8×C61—22no—
120C22⋊Q8×C61—28no—
121C22⋊Q8×C61C2314yes—
122#121×C61—22no—
123#121×C61C2314yes—
124C4×Q8×C61—22no—
125C4×Q8×C61C2314yes—
126C4×D4×C61—22no—
127C4×D4×C61C2314yes—
128#117×C61D414yes—
129#117×C61C2314yes—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61D411yescharacteristic
134C2×C4⋊C4×C61D411yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C4⋊C4×C61C2311yescharacteristic
137C2×C22⋊C4×C61—21no—
138C2×C22⋊C4×C61—21no—
139C2×C22⋊C4×C61—21no—
140C2×C22⋊C4×C61—21no—
141C2×C22⋊C4×C61—21no—
142C2×C22⋊C4×C61D411yescharacteristic
143C2×C22⋊C4×C61D411yescharacteristic
144C2×C42×C61—21no—
145C2×C42×C61C2311yescharacteristic
146C2×C42×C61C2311yescharacteristic
147C24×C61—21no—
148C24×C61C2×D411yescharacteristic
149C2×Q8×C61—21no—
150C2×Q8×C61—24no—
151C2×Q8×C61—24no—
152C2×Q8×C61—82no—
153C2×Q8×C61C2412yes—
154C2×D4×C61—24no—
155C2×D4×C61—21no—
156C2×D4×C61—24no—
157C2×D4×C61—82no—
158C2×D4×C61C2412yes—
159C22×C4×C61—24no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61—24no—
164C22×C4×C61—82no—
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D412yes—
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2×D411yescharacteristic
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61C2412yes—
172C22×C4×C61C2411yescharacteristic
173C22×Dic61—22no—
174C22×Dic61—22no—
175C2×Dic122—22no—
176C2×Dic122—22no—
177C2×Dic122—22no—
178C2×Dic122—22no—
179C4⋊C4×C61—24no—
180C4⋊C4×C61—24no—
181C4⋊C4×C61—24no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61C2×D414yes—
185C4⋊C4×C61C2×D414yes—
186C4⋊C4×C61C2412yes—
187C4⋊C4×C61C2412yes—
188C22⋊C4×C61—24no—
189C22⋊C4×C61—24no—
190C22⋊C4×C61—24no—
191C22⋊C4×C61—24no—
192C22⋊C4×C61—24no—
193C22⋊C4×C61C2×D414yes—
194C22⋊C4×C61C2×D414yes—
195C42×C61—24no—
196C42×C61C2×D412yes—
197C42×C61C2×D412yes—
198C42×C61C2412yes—
199C42×C61C2412yes—
200C22⋊Dic61—24no—
201C22⋊Dic61—24no—
202C4⋊Dic61—22no—
203C4⋊Dic61—22no—
204C4⋊Dic61—22no—
205C4⋊Dic61—22no—
206C4⋊Dic61—22no—
207C4⋊Dic61—22no—
208C4⋊Dic61C2414yes—
209C4⋊Dic61C2414yes—
210#11758196—24no—
211#11758196—24no—
212#11758196—24no—
213#11758196—24no—
214C4×Dic61—24no—
215C4×Dic61—24no—
216#11758194—28no—
217C2×#92418C2418yes—
218C23×C61—22no—
219C23×C61—22no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—42no—
223C23×C61—42no—
224C23×C61—42no—
225C23×C61C22×D412yes—
226C23×C61C22×D411yescharacteristic
227Q8×C61—24no—
228Q8×C61—42no—
229Q8×C61—84no—
230D4×C61—24no—
231D4×C61—42no—
232D4×C61—84no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—24no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—84no—
247C2×C4×C61C22×D412yes—
248C2×C4×C61C22×D412yes—
249C2×C4×C61C22×D412yes—
250C2×C4×C61C22×D412yes—
251C2×C4×C61C22×D412yes—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C612−1+414yes—
254C2×C4×C61C2511yescharacteristic
255C2×Dic61—24no—
256C2×Dic61—24no—
257C2×Dic61—82no—
258C2×Dic61—82no—
259Dic122—82no—
260Dic122—82no—
261Dic122—82no—
262Dic122—82no—
263#92418—28no—
264#60870—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—82no—
274C22×C61—84no—
275C22×C61#53612yes—
276C22×C61C23×D411yescharacteristic
277C4×C61—22no—
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—22no—
281C4×C61—22no—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61—42no—
285C4×C61#53612yes—
286C4×C61C2×2−1+412yes—
287Dic61—44no—
288Dic61—44no—
289#3458—611no—
290C2×#542D6111yescharacteristic
291#3244—618no—
292#3243—618no—
293#3165—612no—
294#3140—612no—
295#3095—612no—
296#3094—612no—
297#3078—612no—
298#3077—612no—
299#2921—611no—
300C2×C61—22no—
301C2×C61—42no—
302C2×C61—82no—
303C2×C61C2×#53611yescharacteristic
304C2×C61#358511yescharacteristic
305C2×C61#358611yescharacteristic
306#544—614no—
307#544—614no—
308#544—1221no—
309#544—1221no—
310#543—1221no—
311#543—1221no—
312#542—24no—
313#542C2×D6118yes—
314#533—612no—
315#533—612no—
316#533—1224no—
317C2×C4⋊Q8C2×D6111yescharacteristic
318C2×#126—21no—
319C2×#124—21no—
320C2×#124C2×D6111yescharacteristic
321C2×#123—21no—
322C2×C22⋊Q8—21no—
323C2×C22⋊Q8C2×D6111yescharacteristic
324C2×#121C2×D6111yescharacteristic
325C2×C4×Q8C2×D6111yescharacteristic
326C2×C4×D4C2×D6111yescharacteristic
327C2×#117—612no—
328C2×#117C2×D6111yescharacteristic
329#501—618no—
330#500—614no—
331#498—614no—
332#489—614no—
333#489—614no—
334#488—618no—
335#485—618no—
336#484—614no—
337#483—614no—
338#482—618no—
339#481—614no—
340#480—614no—
341#479—618no—
342#478—618no—
343#477—614no—
344#476—614no—
345#475—614no—
346#474—614no—
347#433—618no—
348#428—614no—
349#428—614no—
350#423—612no—
351#419—614no—
352#418—614no—
353#407—612no—
354C4×M16—612no—
355C61#6087011yescharacteristic, complemented, Sylow
356C22×Q8—21no—
357C22×Q8C22×D6111yescharacteristic
358C22×D4—21no—
359C22×D4C22×D6111yescharacteristic
360C23×C4—21no—
361C23×C4C22×D6111yescharacteristic
362C2×M16—614no—
363C4⋊Q8—22no—
364C4⋊Q8—612no—
365C4⋊Q8—612no—
366C4⋊Q8—612no—
367C4⋊Q8—612no—
368C4⋊Q8—1224no—
369C4⋊Q8—1224no—
370C4⋊Q8C22×D6114yes—
371#126—28no—
372#125—614no—
373#125—614no—
374#125—1224no—
375#125—1224no—
376#124—28no—
377#124—22no—
378#124C22×D6114yes—
379#123—28no—
380C22⋊Q8—22no—
381C22⋊Q8—28no—
382C22⋊Q8—1222no—
383C22⋊Q8—1222no—
384C22⋊Q8—1222no—
385C22⋊Q8—1222no—
386C22⋊Q8—1224no—
387C22⋊Q8—1224no—
388C22⋊Q8—1224no—
389C22⋊Q8—1224no—
390C22⋊Q8C22×D6114yes—
391#121—22no—
392#121C22×D6114yes—
393C4×Q8—22no—
394C4×Q8—1222no—
395C4×Q8—1222no—
396C4×Q8—1222no—
397C4×Q8—1222no—
398C4×Q8C22×D6114yes—
399C4×D4—22no—
400C4×D4C22×D6114yes—
401#117—614no—
402#117—614no—
403#117—1222no—
404#117—1222no—
405#117C22⋊D6114yes—
406#117C22×D6114yes—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—612no—
412C2×C4⋊C4C22⋊D6111yescharacteristic
413C2×C4⋊C4C22⋊D6111yescharacteristic
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1222no—
422C2×C22⋊C4C22⋊D6111yescharacteristic
423C2×C22⋊C4C22⋊D6111yescharacteristic
424C2×C42—21no—
425C2×C42—1222no—
426C2×C42C22×D6111yescharacteristic
427C2×C42C22×D6111yescharacteristic
428#107—618no—
429#107—618no—
430#106—618no—
431#106—618no—
432C4⋊C8—614no—
433C4⋊C8—614no—
434#103—618no—
435#103—618no—
436#102—618no—
437#102—618no—
438C22⋊C8—614no—
439C22⋊C8—614no—
440#97—614no—
441C4×C8—614no—
442C24—21no—
443C24C2×C22⋊D6111yescharacteristic
444C2×Q8—21no—
445C2×Q8—24no—
446C2×Q8—24no—
447C2×Q8—82no—
448C2×Q8—1222no—
449C2×Q8—1222no—
450C2×Q8—1222no—
451C2×Q8—1222no—
452C2×Q8C23×D6112yes—
453C2×D4—24no—
454C2×D4—21no—
455C2×D4—24no—
456C2×D4—82no—
457C2×D4C23×D6112yes—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—82no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4C2×C22⋊D6112yes—
467C22×C4C2×C22⋊D6111yescharacteristic
468C22×C4C2×C22⋊D6111yescharacteristic
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C23×D6112yes—
473C22×C4C23×D6111yescharacteristic
474M16—1228no—
475C2×C8—618no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—614no—
482C4⋊C4—614no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—1224no—
486C4⋊C4—1224no—
487C4⋊C4—1224no—
488C4⋊C4—1224no—
489C4⋊C4—1222no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4C2×C22⋊D6114yes—
494C4⋊C4C2×C22⋊D6114yes—
495C4⋊C4C23×D6112yes—
496C4⋊C4C23×D6112yes—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—1224no—
503C22⋊C4—1224no—
504C22⋊C4C2×C22⋊D6114yes—
505C22⋊C4C2×C22⋊D6114yes—
506C42—24no—
507C42—1224no—
508C42—1224no—
509C42C2×C22⋊D6112yes—
510C42C2×C22⋊D6112yes—
511C42C23×D6112yes—
512C42C23×D6112yes—
513C23—22no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23C22×C22⋊D6112yes—
521C23C22×C22⋊D6111yescharacteristic
522Q8—24no—
523Q8—42no—
524Q8—84no—
525Q8—4882no—
526Q8—4882no—
527Q8—4882no—
528Q8—4882no—
529D4—24no—
530D4—42no—
531D4—84no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—84no—
546C2×C4—1224no—
547C2×C4—1224no—
548C2×C4—4882no—
549C2×C4—4882no—
550C2×C4C22×C22⋊D6112yes—
551C2×C4C22×C22⋊D6112yes—
552C2×C4C22×C22⋊D6112yes—
553C2×C4C22×C22⋊D6112yes—
554C2×C4C22×C22⋊D6112yes—
555C2×C4C22×C22⋊D6111yescharacteristic
556C2×C4#1451982414yes—
557C2×C4C24×D6111yescharacteristic
558C8—1228no—
559C22—21no—
560C22—21no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—82no—
568C22—84no—
569C22#1684033312yes—
570C22C23×C22⋊D6111yescharacteristic
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—2444no—
580C4—2444no—
581C4#1684033312yes—
582C4C2×#1451982412yes—
583C2—22no—
584C2—42no—
585C2—82no—
586C2C2×#1684033311yescharacteristic
587C2#2146059911yescharacteristic
588C2#2146060611yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#542C2
socleC22×C61C23×D4
prime radicalC2×#121×C61C22
hypercenterC2×#542D61
hypocenterC61#60870
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
1. #21459224C2semidirect
2. #21457111C2semidirect
3. #21457096C2semidirect
4. #21456361C2semidirect
5. #21456205C2semidirect
6. #21455763C2semidirect
8. #21455624C2semidirect
9. #21455616C2semidirect
10. #21454457C2semidirect
14. #16840390C22semidirect
15. #16840390C22semidirect
22. #16840168C22semidirect
23. #16840165C22semidirect
24. #16840158C22semidirect
25. #16840121C22semidirect
26. #16840121C22semidirect
27. #16840116C22semidirect
28. #16840100C22semidirect
29. #16840095C22semidirect
30. #16840090C22semidirect
34. #16840069C22semidirect
35. #16840062C22semidirect
36. #16840057C22semidirect
37. #16840052C22semidirect
38. #16840047C22semidirect
39. #16840042C22semidirect
40. #16839823C22semidirect
355. C61#60870semidirect

Maximal subgroups

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

subgroupindex
#214592242
#214571112
#214570962
#214563612
#214562052
#214557632
#214557552
#214556242
#214556162
#214544572
C2×C61×#5422
#6087061