Finite Group Databasebeta

← #29689220

Subgroups — #29689220

Order 15616.

1278 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459609C211yesmaximal, characteristic
2#21454295C211yesmaximal, characteristic, complemented
3#21454279C211yesmaximal, characteristic, complemented
4#21446351C211yesmaximal, characteristic, complemented
5#21446349C211yesmaximal, characteristic, complemented
6#21446338C212yesmaximal, complemented
7#21446208C211yesmaximal, characteristic, complemented
8#21446201C212yesmaximal, complemented
9#21445922C212yesmaximal, complemented
10#21445791C212yesmaximal, complemented
11C61×#1885C211yesmaximal, characteristic, complemented
12C2×#14519825C2211yescharacteristic
13C2×#14519822C2211yescharacteristic
14#16840528C2212yes—
15#16840420—22no—
16#16840405—22no—
17#16840402—24no—
18#16840336C2211yescharacteristic
19#16840332C412yescomplemented
20#16840330C412yescomplemented
21#16840329C2211yescharacteristic
22#16840312C2211yescharacteristic
23#16840311C412yescomplemented
24#124×D61C412yescomplemented
25#16840237C2211yescharacteristic
26C4×#11758222C2211yescharacteristic
27#16839780C2212yescomplemented
28#16839778C2212yescomplemented
29#16839771C2212yescomplemented
30#16839744C2212yes—
31#16839743—22no—
32C2×#14519694C2211yescharacteristic
33#100×D61—22no—
34#16839735C2212yes—
35#16839735C2212yes—
36C2×#14519691—24no—
37#16839522C2212yescomplemented
38#16839519C2212yescomplemented
39#16839518C2212yescomplemented
40C2×#124×C61C2211yescharacteristic
41C61×#518C2211yescharacteristic
42C61×#419C2212yescomplemented
43C2×#100×C61C2211yescharacteristic
44C61×#339C2212yescomplemented
45#14519825—42no—
46#14519825—42no—
47#14519822—42no—
48#14519822—42no—
49C2×#11758226—21no—
50C2×#11758226C2×C412yescomplemented
51C2×#11758222—22no—
52C2×#11758222C2311yescharacteristic
53C2×#11758222C2311yescharacteristic
54C2×#11758222C2311yescharacteristic
55C2×#11758224C2×C412yes—
56C2×#11758224D412yes—
57C2×Q8×D61—21no—
58C2×Q8×D61C2×C412yescomplemented
59C22×C22⋊D61D414yes—
60C2×D4×D61C2×C412yes—
61C2×D4×D61D412yes—
62#14519785—42no—
63#14519783—42no—
64C2×#11758194C2312yes—
65M16×D61—24no—
66#14519777C2×C411yescharacteristic, complemented
67#14519775C2×C412yescomplemented
68#14519774C2×C411yescharacteristic, complemented
69#14519773C2×C412yes—
70#14519766C2×C412yescomplemented
71#14519765C2×C411yescharacteristic, complemented
72#14519765C2×C411yescharacteristic, complemented
73#14519764C2×C412yes—
74#14519763—24no—
75#14519760—22no—
76#14519759—24no—
77#14519754—24no—
78#14519753—22no—
79#14519751—22no—
80#14519750—24no—
81#14519750—24no—
82#14519748—22no—
83#14519747—24no—
84#14519746—22no—
85C4×Dic122—21no—
86C4×D244—21no—
87C4×D244—22no—
88D4×Dic61—42no—
89C4×C22⋊D61—22no—
90#14519737—24no—
91#14519736C2312yes—
92#14519736C2311yescharacteristic
93#14519734C2×C412yescomplemented
94C2×C22⋊Dic61—22no—
95C22⋊C4×D61—24no—
96C42×D61C2×C412yescomplemented
97#14519705—42no—
98#14519703—42no—
99#14519701—42no—
100#14519695—22no—
101#14519694—22no—
102#14519693—22no—
103#14519691—24no—
104#14519691—24no—
105C22×Q8×C61C2311yescharacteristic
106C22×D4×C61C2311yescharacteristic
107C2×M16×C61C2312yes—
108#124×C61—22no—
109#124×C61C2×C412yes—
110#124×C61C2×C412yes—
111C4×D4×C61—42no—
112#117×C61C2312yes—
113C2×C22⋊C4×C61—22no—
114C2×C22⋊C4×C61—22no—
115C2×C42×C61C2311yescharacteristic
116#103×C61—42no—
117#100×C61—22no—
118C24×C61C2×D412yes—
119C23×D61C22⋊C414yes—
120C2×Q8×C61—21no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61C22×C412yes—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61C22×C412yes—
127C2×D4×C61C2×D412yes—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—42no—
131C22×C4×C61C2411yescharacteristic
132C2×C22⋊D61—21no—
133C2×C22⋊D61—24no—
134C2×C22⋊D61—44no—
135C2×C22⋊D61—44no—
136C2×C22⋊D61C22⋊C414yes—
137C2×C22⋊D61C22⋊C414yes—
138C2×C22⋊D61C2×D412yes—
139C2×C22⋊D61C2×D412yes—
140C22×Dic61—42no—
141C22×Dic61C22⋊C414yes—
142#11758226—44no—
143#11758226—44no—
144Q8×D61—44no—
145Q8×D61—44no—
146#11758224—44no—
147#11758224—44no—
148D4×D61—44no—
149D4×D61—44no—
150#11758222—22no—
151#11758222—22no—
152#11758222—22no—
153#11758222—22no—
154#11758222—22no—
155#11758222—22no—
156#11758222—44no—
157#11758222—44no—
158C2×D244—22no—
159C2×D244—22no—
160C2×D244C22×C411yescharacteristic
161C2×D244C22×C411yescharacteristic
162C2×D244C2×D411yescharacteristic
163C2×C4×D61—22no—
164C2×C4×D61—21no—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61C22×C412yes—
168C2×C4×D61C2×D412yes—
169C2×Dic122—22no—
170C2×Dic122C22×C411yescharacteristic
171C2×Dic122C22×C411yescharacteristic
172C2×Dic122C2×D411yescharacteristic
173M16×C61—22no—
174M16×C61—22no—
175C2×C8×C61—22no—
176C4⋊C4×C61—22no—
177C22⋊C4×C61—24no—
178C22⋊C4×C61—24no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—42no—
181C42×C61—21no—
182C42×C61—22no—
183C42×C61C22×C411yescharacteristic
184C42×C61C22×C411yescharacteristic
185C22⋊Dic61—24no—
186C22⋊Dic61—21no—
187C22⋊Dic61—22no—
188C22⋊Dic61—42no—
189#11758198—22no—
190#11758198—24no—
191#11758198—22no—
192#11758198—22no—
193C4⋊Dic61—21no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196#11758196—22no—
197#11758196—24no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200C4×Dic61C22×C412yes—
201#11758194—22no—
202#11758194—22no—
203C2×#92418—22no—
204#11758189—44no—
205C8×D61—44no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—82no—
209C23×C61C2×C22⋊C414yes—
210C23×C61C22×D411yescharacteristic
211C22×D61—21no—
212C22×D61—44no—
213C22×D61—84no—
214C22×D61C2×C22⋊C412yes—
215C22×D61C2×C22⋊C412yes—
216Q8×C61—24no—
217Q8×C61—42no—
218D4×C61—24no—
219D4×C61—24no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61—81no—
227C2×C4×C61—162no—
228C2×C4×C61C23×C411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61H(5,𝔽2)12yes—
231C2×C4×C61H(5,𝔽2)12yes—
232C22⋊D61—22no—
233C22⋊D61—22no—
234C22⋊D61—22no—
235C22⋊D61—22no—
236C22⋊D61—44no—
237C22⋊D61—44no—
238C22⋊D61—41no—
239C22⋊D61—41no—
240C22⋊D61—84no—
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—44no—
244C2×Dic61—42no—
245C2×Dic61—162no—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C2×C22⋊C412yes—
248D244—22no—
249D244—22no—
250D244—22no—
251D244—44no—
252D244—82no—
253C4×D61—24no—
254C4×D61—24no—
255C4×D61—44no—
256C4×D61—44no—
257C4×D61—42no—
258C4×D61—82no—
259Dic122—22no—
260Dic122—22no—
261Dic122—22no—
262Dic122—44no—
263C8×C61—42no—
264#92418—42no—
265#21784—611nomaximal, Sylow
266C22×C61—24no—
267C22×C61—41no—
268C22×C61—44no—
269C22×C61—82no—
270C22×C61—82no—
271C22×C61C22×C22⋊C411yescharacteristic
272C22×C61#53411yescharacteristic
273C22×C61#53411yescharacteristic
274C2×D61—22no—
275C2×D61—22no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—41no—
280C2×D61—164no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—42no—
284C4×C61—82no—
285C4×C61#51812yes—
286C4×C61#53412yes—
287Dic61—22no—
288Dic61—22no—
289Dic61—42no—
290Dic61—41no—
291Dic61—82no—
292#3500—611no—
293#2888—611no—
294#2883—611no—
295#1911—611no—
296#1911—611no—
297#1909—612no—
298#1885—611no—
299#1885D6111yescharacteristic, complemented
300#1884—612no—
301#1827—612no—
302#1796—612no—
303C2×C61—21no—
304C2×C61—21no—
305C2×C61—21no—
306C2×C61—44no—
307C2×C61#230811yescharacteristic
308D61—42no—
309D61—42no—
310D61—81no—
311C2×2−1+4—611no—
312C2×H(5,𝔽2)—611no—
313#568—612no—
314#548—1222no—
315#546—1224no—
316#546—1222no—
317#536—611no—
318#536—612no—
319#535—611no—
320#535—612no—
321#532—612no—
322#532—611no—
323C2×#124—612no—
324C2×#124C2×D6111yescharacteristic
325#518—611no—
326#518C2×D6111yescharacteristic
327C4×#41—611no—
328#421—612no—
329#421—612no—
330#419—612no—
331#419C2×D6112yescomplemented
332#413—612no—
333#413—1222no—
334C2×#100—611no—
335C2×#100—1222no—
336C2×#100C2×D6111yescharacteristic
337#410—612no—
338#410—612no—
339C2×#99—1224no—
340#339—612no—
341#339C2×D6112yescomplemented
342H(3,ℤ/4)—612no—
343H(3,ℤ/4)—612no—
344C61#2178411yescharacteristic, complemented, Sylow
3452−1+4—2442no—
3462−1+4—2442no—
347H(5,𝔽2)—2442no—
348H(5,𝔽2)—2442no—
349C2×#41—612no—
350C2×#41—612no—
351C2×#41—611no—
352C2×#41—611no—
353C2×#41—612no—
354C2×#41—611no—
355C2×#41—1222no—
356C2×#41—1221no—
357C22×Q8—612no—
358C22×Q8—1221no—
359C22×Q8C22×D6111yescharacteristic
360C22×D4—614no—
361C22×D4—612no—
362C22×D4—612no—
363C22×D4C22×D6111yescharacteristic
364#131—2442no—
365#131—2442no—
366C2×M16—612no—
367C2×M16—1224no—
368C2×M16C22×D6112yes—
369C4⋊Q8—612no—
370C4⋊Q8—611no—
371#127—612no—
372#127—611no—
373#124—22no—
374#124—612no—
375#124—612no—
376#124—611no—
377#124—611no—
378#124—1224no—
379#124C4×D6112yes—
380#124C4×D6112yes—
381#123—1224no—
382#123—1222no—
383C22⋊Q8—1224no—
384C22⋊Q8—1222no—
385#121—1224no—
386#121—1224no—
387#121—1222no—
388#121—1222no—
389#120—1224no—
390#120—1222no—
391C4×Q8—1221no—
392C4×D4—42no—
393C4×D4—1222no—
394C4×D4—1221no—
395C4×D4—1224no—
396C4×D4—1222no—
397C4×D4—2442no—
398#117—611no—
399#117—612no—
400#117—612no—
401#117C22×D6112yes—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—1224no—
405C2×C22⋊C4—1222no—
406C2×C42—612no—
407C2×C42C22×D6111yescharacteristic
408#104—2442no—
409#104—2442no—
410#103—42no—
411#103—2442no—
412#101—1222no—
413#100—22no—
414#100—1222no—
415#100—1222no—
416#99—1224no—
417#99—1224no—
418C24—614no—
419C24D4×D6112yes—
420#41—1222no—
421#41—1222no—
422#41—1222no—
423#41—1222no—
424#41—1222no—
425#41—1222no—
426#41—2444no—
427#41—2444no—
428#41—2444no—
429#41—2444no—
430#41—2444no—
431#41—2444no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—611no—
436C2×Q8—611no—
437C2×Q8—611no—
438C2×Q8—1222no—
439C2×Q8—2444no—
440C2×Q8—2444no—
441C2×Q8C2×C4×D6112yes—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—612no—
445C2×D4—612no—
446C2×D4—614no—
447C2×D4—614no—
448C2×D4—611no—
449C2×D4—611no—
450C2×D4—611no—
451C2×D4—1222no—
452C2×D4—1222no—
453C2×D4—1224no—
454C2×D4—1221no—
455C2×D4—2444no—
456C2×D4—2444no—
457C2×D4—2444no—
458C2×D4—2444no—
459C2×D4C2×C4×D6112yes—
460C2×D4D4×D6112yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—42no—
464C22×C4—614no—
465C22×C4—612no—
466C22×C4—612no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1221no—
471C22×C4—2442no—
472C22×C4C23×D6111yescharacteristic
473M16—22no—
474M16—22no—
475M16—1222no—
476M16—1222no—
477M16—2444no—
478C2×C8—22no—
479C2×C8—1222no—
480C2×C8—2444no—
481C4⋊C4—22no—
482C4⋊C4—1222no—
483C4⋊C4—1224no—
484C4⋊C4—1222no—
485C4⋊C4—1221no—
486C4⋊C4—1222no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—22no—
490C22⋊C4—42no—
491C22⋊C4—1222no—
492C22⋊C4—1222no—
493C22⋊C4—1224no—
494C22⋊C4—1222no—
495C22⋊C4—1221no—
496C22⋊C4—1224no—
497C22⋊C4—1222no—
498C22⋊C4—2442no—
499C42—21no—
500C42—22no—
501C42—612no—
502C42—1222no—
503C42—1222no—
504C42C2×C4×D6111yescharacteristic
505C42C2×C4×D6111yescharacteristic
506C23—22no—
507C23—22no—
508C23—82no—
509C23—612no—
510C23—612no—
511C23—1221no—
512C23—4884no—
513C23—2444no—
514C23C22⋊C4×D6114yes—
515C23C2×D4×D6111yescharacteristic
516Q8—24no—
517Q8—42no—
518Q8—1222no—
519Q8—1222no—
520Q8—1222no—
521Q8—2444no—
522D4—24no—
523D4—24no—
524D4—1222no—
525D4—1222no—
526D4—1222no—
527D4—1222no—
528D4—1222no—
529D4—1222no—
530D4—1222no—
531D4—4882no—
532D4—4884no—
533D4—2444no—
534D4—2444no—
535D4—2444no—
536D4—2441no—
537D4—2441no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—81no—
545C2×C4—162no—
546C2×C4—612no—
547C2×C4—612no—
548C2×C4—1224no—
549C2×C4—1224no—
550C2×C4—1222no—
551C2×C4—1221no—
552C2×C4—9762no—
553C2×C4—4882no—
554C2×C4—2444no—
555C2×C4—2444no—
556C2×C4—2444no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4C22×C4×D6111yescharacteristic
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4#1451982212yes—
562C2×C4#1451982312yes—
563C8—42no—
564C8—2442no—
565C22—24no—
566C22—41no—
567C22—44no—
568C22—82no—
569C22—82no—
570C22—1222no—
571C22—1222no—
572C22—9764no—
573C22—2442no—
574C22—2442no—
575C22—2442no—
576C22—2441no—
577C22C2×C22⋊C4×D6111yescharacteristic
578C22#1684032211yescharacteristic
579C22#1684032311yescharacteristic
580C4—41no—
581C4—41no—
582C4—42no—
583C4—82no—
584C4—1222no—
585C4—1222no—
586C4—4882no—
587C4—2442no—
588C4—2441no—
589C4#1684023412yes—
590C4#1684032012yes—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—44no—
595C2—4881no—
596C2—2442no—
597C2—2442no—
598C2#2144839311yescharacteristic
subgroupG/H
centerC2#21448393
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1885C2
socleC2×C61#2308
prime radical#21459609C2
cosocleC22×C4×C61C24
hypercenter#1885D61
hypocenterC61#21784
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1885

Complements

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

NH ≅ G/Ntype
2. #21454295C2semidirect
3. #21454279C2semidirect
4. #21446351C2semidirect
5. #21446349C2semidirect
6. #21446338C2semidirect
7. #21446208C2semidirect
8. #21446201C2semidirect
9. #21445922C2semidirect
10. #21445791C2semidirect
11. C61×#1885C2semidirect
19. #16840332C4semidirect
20. #16840330C4semidirect
23. #16840311C4semidirect
24. #124×D61C4semidirect
27. #16839780C22semidirect
28. #16839778C22semidirect
29. #16839771C22semidirect
37. #16839522C22semidirect
38. #16839519C22semidirect
39. #16839518C22semidirect
42. C61×#419C22semidirect
44. C61×#339C22semidirect
50. C2×#11758226C2×C4semidirect
58. C2×Q8×D61C2×C4semidirect
66. #14519777C2×C4semidirect
67. #14519775C2×C4semidirect
68. #14519774C2×C4semidirect
70. #14519766C2×C4semidirect
71. #14519765C2×C4semidirect
72. #14519765C2×C4semidirect
93. #14519734C2×C4semidirect
96. C42×D61C2×C4semidirect
299. #1885D61semidirect
331. #419C2×D61semidirect
341. #339C2×D61semidirect
344. C61#21784semidirect

Maximal subgroups

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

subgroupindex
#214596092
#214542952
#214542792
#214463512
#214463492
#214463382
#214462082
#214462012
#214459222
#214457912
C61×#18852
#2178461