Finite Group Databasebeta

← #29833821

Subgroups — #29833821

Order 15616.

1192 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#135×D61C211yesmaximal, characteristic, complemented
2C2×#16840198C211yesmaximal, characteristic, complemented
3C2×#16839995C211yesmaximal, characteristic, complemented
4#21447741C212yesmaximal, complemented
5#21447739C212yesmaximal, complemented
6#21447734C212yesmaximal, complemented
7#21447729C212yesmaximal, complemented
8#21447532C211yesmaximal, characteristic, complemented
9#21447531C211yesmaximal, characteristic, complemented
10#21447429C211yesmaximal, characteristic, complemented
11C2×C61×#509C211yesmaximal, characteristic, complemented
12C2×#41×D61—21no—
13C2×#14519800C2211yescharacteristic, complemented
14C2×#14519799C2211yescharacteristic
15C2×#14519798—21no—
16C2×#14519797—21no—
17C2×#14519796C2211yescharacteristic, complemented
18#135×D61—24no—
19#135×D61—24no—
20C2×Q16×D61C2211yescharacteristic, complemented
21C2×SD16×D61—21no—
22C2×D8×D61C2211yescharacteristic, complemented
23C22×C8×D61C2211yescharacteristic
24C2×#14519748—21no—
25#16840198—24no—
26C2×#14519721C2212yescomplemented
27C2×#14519718C2212yescomplemented
28C2×#14519717C2211yescharacteristic, complemented
29#16839995—21no—
30#16839995—21no—
31#16839995C2212yes—
32#16839995C2212yescomplemented
33C2×#14519711C2211yescharacteristic, complemented
34C2×#14519698—21no—
35#16839567C2212yescomplemented
36#16839566C2212yescomplemented
37#16839564C2212yescomplemented
38#16839563C2212yescomplemented
39#16839541C2212yescomplemented
40#16839541C2212yescomplemented
41C2×#135×C61C2211yescharacteristic
42C22×D8×C61C2211yescharacteristic, complemented
43C61×#509—24no—
44C2×SD32×C61C2212yescomplemented
45C2×D16×C61C2212yescomplemented
46C2×M32×C61C2211yescharacteristic, complemented
47C2×#11758226—21no—
48C2×#11758222—21no—
49C2×#11758222—41no—
50C2×#11758224—21no—
51C2×#11758224—41no—
52#41×D61—28no—
53#41×D61—44no—
54C2×Q8×D61—21no—
55C22×C22⋊D61—41no—
56C2×D4×D61—21no—
57C2×D4×D61—41no—
58C22×C4×D61—41no—
59C22×C4×D61D411yescharacteristic
60#14519800—22no—
61#14519800—22no—
62#14519799—22no—
63#14519799D414yescomplemented
64#14519798—28no—
65#14519797—28no—
66#14519796—22no—
67#14519796—22no—
68C2×Dic244C2311yescharacteristic
69C2×#11758202—21no—
70Q16×D61—22no—
71Q16×D61—22no—
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75SD16×D61—28no—
76C2×D488C2311yescharacteristic
77C2×#11758199—21no—
78D8×D61—22no—
79D8×D61—22no—
80C22×#92418D411yescharacteristic
81C2×C8×D61—22no—
82C2×C8×D61—22no—
83C2×C8×D61D411yescharacteristic
84C2×C8×D61D411yescharacteristic
85C2×C8×D61C2311yescharacteristic
86C2×C8×D61C2311yescharacteristic
87#14519748—22no—
88#14519748—22no—
89#14519748—41no—
90#14519748—41no—
91C2×C4⋊Dic61—21no—
92C2×C22⋊Dic61—41no—
93#14519721—24no—
94#14519718—24no—
95#14519717—22no—
96C2×#11758185C2312yes—
97#14519711—21no—
98#14519711C2312yes—
99#14519698—22no—
100#14519698—22no—
101C2×#41×C61—21no—
102C22×D4×C61—21no—
103#135×C61—22no—
104#135×C61D414yescomplemented
105C2×Q16×C61C2311yescharacteristic
106C2×SD16×C61—21no—
107C2×D8×C61—22no—
108C2×D8×C61—22no—
109C2×D8×C61—21no—
110C2×D8×C61C2312yes—
111C2×D8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113SD32×C61—24no—
114D16×C61—24no—
115M32×C61—22no—
116C2×C16×C61C2312yes—
117C24×C61—41no—
118C23×D61—41no—
119C23×D61D811yescharacteristic
120#41×C61—24no—
121#41×C61—42no—
122C2×Q8×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—22no—
125C2×D4×C61—41no—
126C2×D4×C61—42no—
127C2×D4×C61—41no—
128C2×D4×C61—42no—
129C22×C4×C61—41no—
130C22×C4×C61C2×D411yescharacteristic
131C2×C22⋊D61—41no—
132C2×C22⋊D61—41no—
133C2×C22⋊D61—42no—
134C2×C22⋊D61—42no—
135C2×C22⋊D61—41no—
136C2×C22⋊D61—84no—
137C22×Dic61—41no—
138C22×Dic61—41no—
139C22×Dic61D811yescharacteristic
140#11758226—24no—
141#11758226—42no—
142Q8×D61—24no—
143Q8×D61—42no—
144#11758224—24no—
145#11758224—42no—
146#11758224—48no—
147D4×D61—24no—
148D4×D61—42no—
149D4×D61—48no—
150#11758222—24no—
151#11758222—48no—
152#11758222—42no—
153C2×D244—21no—
154C2×D244—41no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—48no—
158C2×C4×D61—41no—
159C2×C4×D61—41no—
160C2×C4×D61—41no—
161C2×C4×D61D811yescharacteristic
162C2×C4×D61D811yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×Dic122—21no—
166C2×Dic122—41no—
167Q16×C61—21no—
168Q16×C61C2×D412yescomplemented
169SD16×C61—24no—
170D8×C61—21no—
171D8×C61—22no—
172D8×C61—22no—
173D8×C61—42no—
174D8×C61C2×D412yescomplemented
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D412yes—
177C2×C8×C61C2×D412yes—
178C2×C8×C61C2411yescharacteristic
179C22⋊Dic61—42no—
180C22⋊Dic61—42no—
181#11758202—24no—
182#11758201—24no—
183#11758200—24no—
184#11758199—24no—
185C4⋊Dic61—22no—
186C4⋊Dic61—41no—
187C2×#92418—22no—
188C2×#92418C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190Dic244—21no—
191Dic244C2×D412yescomplemented
192D488—21no—
193D488C2×D412yescomplemented
194#11758190—24no—
195C8×D61—22no—
196C8×D61—22no—
197C8×D61—21no—
198C8×D61—21no—
199C8×D61—21no—
200C8×D61—21no—
201C16×C61—22no—
202#11758185—22no—
203C23×C61—41no—
204C23×C61—42no—
205C23×C61—82no—
206C23×C61—82no—
207C23×C61—82no—
208C23×C61C2×D811yescharacteristic
209C22×D61—22no—
210C22×D61—22no—
211C22×D61—41no—
212C22×D61—44no—
213C22×D61—41no—
214C22×D61—82no—
215C22×D61—82no—
216C22×D61C2×D811yescharacteristic
217C22×D61C2×D811yescharacteristic
218Q8×C61—22no—
219Q8×C61—41no—
220D4×C61—22no—
221D4×C61—44no—
222D4×C61—42no—
223D4×C61—41no—
224D4×C61—42no—
225D4×C61—82no—
226C2×C4×C61—41no—
227C2×C4×C61—44no—
228C2×C4×C61#12012yes—
229C2×C4×C61C2×D811yescharacteristic
230C2×C4×C61AGL(1,ℤ/8)12yes—
231C2×C4×C61C22×D411yescharacteristic
232C22⋊D61—44no—
233C22⋊D61—44no—
234C22⋊D61—44no—
235C22⋊D61—84no—
236C22⋊D61—84no—
237C2×Dic61—22no—
238C2×Dic61—44no—
239C2×Dic61—41no—
240C2×Dic61—42no—
241C2×Dic61—82no—
242C2×Dic61C2×D811yescharacteristic
243C2×Dic61C2×D811yescharacteristic
244D244—22no—
245D244—44no—
246D244—41no—
247C4×D61—21no—
248C4×D61—21no—
249C4×D61—21no—
250C4×D61—21no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—44no—
254C4×D61—44no—
255C4×D61—44no—
256Dic122—22no—
257Dic122—44no—
258Dic122—41no—
259C8×C61#12012yes—
260C8×C61C22×D411yescharacteristic
261C8×C61C22×D411yescharacteristic
262#92418—21no—
263#92418—21no—
264#33520—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—42no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—82no—
271C22×C61—82no—
272C22×C61—82no—
273C22×C61—164no—
274C22×C61#44712yes—
275C22×C61C22×D811yescharacteristic
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—21no—
280C2×D61—21no—
281C2×D61—44no—
282C2×D61—42no—
283C2×D61—88no—
284C2×D61—81no—
285C2×D61—81no—
286C2×D61#50912yescomplemented
287C4×C61—42no—
288C4×C61#44712yes—
289C4×C61C2×#12011yescharacteristic
290C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
291Dic61—21no—
292Dic61—21no—
293Dic61—42no—
294Dic61—82no—
295C22×#135—611no—
296C2×#509—611no—
297C2×#509D6111yescharacteristic, complemented
298C2×#466—611no—
299#2186—612no—
300#2186—612no—
301#2185—612no—
302#2184—612no—
303#2139—611no—
304#2139—611no—
305#2112—611no—
306C2×C61—22no—
307C2×C61—82no—
308C2×C61—162no—
309C2×C61C2×#44711yescharacteristic
310C2×C61C2×#50911yescharacteristic
311C2×C61#341411yescharacteristic
312D61—22no—
313D61—41no—
314D61—82no—
315C22×#41—1221no—
316C2×#135—611no—
317C2×#135—611no—
318C2×#135—611no—
319C2×#135—1221no—
320C2×#135—1221no—
321C2×#135—1224no—
322C2×#135—1224no—
323C2×#135C2×D6111yescharacteristic
324C22×Q16—611no—
325C22×SD16—1221no—
326C22×D8—611no—
327C22×D8C2×D6111yescharacteristic, complemented
328C23×C8—611no—
329C2×#121—1221no—
330#509—24no—
331#509—1224no—
332C2×SD32—612no—
333C2×SD32C2×D6112yescomplemented
334C2×D16—612no—
335C2×D16C2×D6112yescomplemented
336C2×M32—611no—
337C2×M32C2×D6111yescharacteristic, complemented
338#466—612no—
339#466—612no—
340#466—1221no—
341#466—1221no—
342C2×#107—611no—
343C2×#102—1221no—
344#358—612no—
345#358—612no—
346#357—612no—
347#357—612no—
348C22⋊C16—612no—
349C22⋊C16—612no—
350C61#3352011yescharacteristic, complemented, Sylow
351C2×#41—21no—
352C2×#41—1221no—
353C2×#41—1221no—
354C2×#41—1228no—
355C2×#41—1221no—
356C2×#41—2441no—
357C2×#41—2441no—
358C2×#41—2444no—
359C22×Q8—1221no—
360C22×D4—21no—
361C22×D4—1221no—
362C22×D4—2441no—
363C22×D4—2441no—
364C23×C4—611no—
365C23×C4—2441no—
366#135—22no—
367#135—614no—
368#135—1222no—
369#135—1222no—
370#135—1228no—
371#135—1228no—
372#135—1222no—
373#135—1222no—
374#135—1222no—
375#135C22⋊D6114yescomplemented
376C2×Q16—611no—
377C2×Q16—1221no—
378C2×Q16—1222no—
379C2×Q16—1222no—
380C2×Q16C22×D6111yescharacteristic
381C2×SD16—21no—
382C2×SD16—1221no—
383C2×SD16—1221no—
384C2×SD16—1228no—
385C2×SD16—1221no—
386C2×D8—22no—
387C2×D8—21no—
388C2×D8—22no—
389C2×D8—611no—
390C2×D8—1221no—
391C2×D8—1222no—
392C2×D8—1222no—
393C2×D8C22×D6112yes—
394C2×D8C22×D6111yescharacteristic
395C22×C8—611no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—611no—
400C22×C8—1222no—
401C22×C8—1222no—
402C22×C8C22×D6111yescharacteristic
403#121—1222no—
404#121—1222no—
405#121—2441no—
406#121—2441no—
407C2×C4⋊C4—1221no—
408C2×C22⋊C4—2441no—
409SD32—24no—
410SD32—1224no—
411D16—24no—
412D16—1224no—
413M32—22no—
414M32—1222no—
415C2×C16—612no—
416C2×C16C22×D6112yes—
417#107—612no—
418#107—1221no—
419#102—1222no—
420#102—1222no—
421C24—41no—
422C24—611no—
423C24—2441no—
424#41—24no—
425#41—42no—
426#41—1224no—
427#41—1224no—
428#41—1224no—
429#41—2448no—
430#41—2448no—
431#41—2442no—
432#41—2442no—
433#41—2442no—
434C2×Q8—21no—
435C2×Q8—1224no—
436C2×Q8—1221no—
437C2×Q8—2441no—
438C2×Q8—2442no—
439C2×D4—21no—
440C2×D4—22no—
441C2×D4—41no—
442C2×D4—42no—
443C2×D4—41no—
444C2×D4—42no—
445C2×D4—1224no—
446C2×D4—1221no—
447C2×D4—4884no—
448C2×D4—2441no—
449C2×D4—2441no—
450C2×D4—2442no—
451C2×D4—2448no—
452C2×D4—2442no—
453C2×D4—2442no—
454C2×D4—2441no—
455C2×D4—2441no—
456C22×C4—41no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—2441no—
465C22×C4—2448no—
466C22×C4—2441no—
467C22×C4—2441no—
468C22×C4—2441no—
469C22×C4—2441no—
470C22×C4D4×D6111yescharacteristic
471Q16—21no—
472Q16—612no—
473Q16—1224no—
474Q16—1221no—
475Q16C2×C22⋊D6112yescomplemented
476SD16—24no—
477SD16—1224no—
478SD16—1224no—
479SD16—1224no—
480D8—21no—
481D8—22no—
482D8—22no—
483D8—42no—
484D8—612no—
485D8—1224no—
486D8—1221no—
487D8C2×C22⋊D6112yescomplemented
488C2×C8—611no—
489C2×C8—611no—
490C2×C8—1222no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1222no—
496C2×C8—1222no—
497C2×C8D4×D6112yes—
498C2×C8D4×D6111yescharacteristic
499C2×C8C2×C22⋊D6112yes—
500C2×C8C23×D6111yescharacteristic
501C4⋊C4—1222no—
502C4⋊C4—2441no—
503C22⋊C4—2442no—
504C22⋊C4—2442no—
505C16—22no—
506C16—1222no—
507C23—41no—
508C23—42no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—611no—
513C23—611no—
514C23—1222no—
515C23—1222no—
516C23—4882no—
517C23—4882no—
518C23—2441no—
519C23—2444no—
520C23—2441no—
521C23D8×D6111yescharacteristic
522Q8—22no—
523Q8—41no—
524Q8—1222no—
525Q8—2444no—
526Q8—2441no—
527D4—22no—
528D4—44no—
529D4—42no—
530D4—41no—
531D4—42no—
532D4—82no—
533D4—1222no—
534D4—4884no—
535D4—4884no—
536D4—2444no—
537D4—2444no—
538D4—2444no—
539D4—2444no—
540D4—2441no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—611no—
544C2×C4—611no—
545C2×C4—1222no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—4882no—
553C2×C4—2444no—
554C2×C4—2444no—
555C2×C4—2444no—
556C2×C4—2442no—
557C2×C4—2441no—
558C2×C4—2444no—
559C2×C4#1451974512yes—
560C2×C4D8×D6111yescharacteristic
561C2×C4#1451980112yes—
562C2×C4C2×D4×D6111yescharacteristic
563C8—1221no—
564C8—1221no—
565C8#1451974512yes—
566C8C2×D4×D6111yescharacteristic
567C8C22×C22⋊D6111yescharacteristic
568C22—21no—
569C22—21no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—164no—
577C22—612no—
578C22—1221no—
579C22—1221no—
580C22—1221no—
581C22—1221no—
582C22—1221no—
583C22—4888no—
584C22—4881no—
585C22—4881no—
586C22—2444no—
587C22—2442no—
588C22#1683989412yes—
589C22C2×D8×D6111yescharacteristic
590C4—42no—
591C4—1221no—
592C4—1221no—
593C4—4882no—
594C4—2442no—
595C4#1683989412yes—
596C4C2×#1451974511yescharacteristic
597C4C2×#1451980111yescharacteristic
598C2—22no—
599C2—82no—
600C2—162no—
601C2—1222no—
602C2—4882no—
603C2—2441no—
604C2C2×#1683989411yescharacteristic
605C2#509×D6111yescharacteristic
606C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#509C2
socleC22×C61C22×D8
hypercenterC2×#509D61
hypocenterC61#33520
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. C2×#135×D61C2semidirect
2. C2×#16840198C2semidirect
3. C2×#16839995C2semidirect
4. #21447741C2semidirect
5. #21447739C2semidirect
6. #21447734C2semidirect
7. #21447729C2semidirect
8. #21447532C2semidirect
9. #21447531C2semidirect
10. #21447429C2semidirect
11. C2×C61×#509C2semidirect
13. C2×#14519800C22semidirect
17. C2×#14519796C22semidirect
20. C2×Q16×D61C22semidirect
22. C2×D8×D61C22semidirect
26. C2×#14519721C22semidirect
27. C2×#14519718C22semidirect
28. C2×#14519717C22semidirect
32. #16839995C22semidirect
33. C2×#14519711C22semidirect
35. #16839567C22semidirect
36. #16839566C22semidirect
37. #16839564C22semidirect
38. #16839563C22semidirect
39. #16839541C22semidirect
40. #16839541C22semidirect
42. C22×D8×C61C22semidirect
44. C2×SD32×C61C22semidirect
45. C2×D16×C61C22semidirect
46. C2×M32×C61C22semidirect
63. #14519799D4semidirect
104. #135×C61D4semidirect
168. Q16×C61C2×D4semidirect
174. D8×C61C2×D4semidirect
191. Dic244C2×D4semidirect
193. D488C2×D4semidirect
286. C2×D61#509semidirect
297. C2×#509D61semidirect
327. C22×D8C2×D61semidirect
333. C2×SD32C2×D61semidirect
335. C2×D16C2×D61semidirect
337. C2×M32C2×D61semidirect
350. C61#33520semidirect
375. #135C22⋊D61semidirect
475. Q16C2×C22⋊D61semidirect
487. D8C2×C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#135×D612
C2×#168401982
C2×#168399952
#214477412
#214477392
#214477342
#214477292
#214475322
#214475312
#214474292
C2×C61×#5092
#3352061