Finite Group Databasebeta

← #29689216

Subgroups — #29689216

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#21459608C211yesmaximal, characteristic
2#21454297C211yesmaximal, characteristic, complemented
3#21454278C211yesmaximal, characteristic, complemented
4#21446350C212yesmaximal, complemented
5#21446342C211yesmaximal, characteristic, complemented
6#21446204C212yesmaximal, complemented
7#21446202C211yesmaximal, characteristic, complemented
8#21446200C211yesmaximal, characteristic, complemented
9#21445921C212yesmaximal, complemented
10#21445791C212yesmaximal, complemented
11C61×#1909C211yesmaximal, characteristic, complemented
12C2×#14519825C2211yescharacteristic
13C2×#14519823C2211yescharacteristic
14#16840529C2211yescharacteristic
15#16840422—22no—
16#16840404—24no—
17#16840403—22no—
18#16840337C412yescomplemented
19#16840331C2211yescharacteristic
20#16840330C412yescomplemented
21#16840313C412yescomplemented
22#16840308C2211yescharacteristic
23C2×#14519766C2211yescharacteristic
24#124×D61C412yescomplemented
25#16840236C2212yes—
26#16840230C2211yescharacteristic
27#16839780C2212yescomplemented
28#16839778C2212yescomplemented
29#16839771C2212yescomplemented
30#16839744—22no—
31#16839744C2212yes—
32C2×#14519693—22no—
33C2×#14519693C2212yes—
34#16839736C2211yescharacteristic
35#16839735C2212yes—
36#99×D61—24no—
37#16839523C2212yescomplemented
38#16839521C2212yescomplemented
39#16839518C2212yescomplemented
40C61×#568C2211yescharacteristic
41C61×#536C2211yescharacteristic
42C61×#419C2212yescomplemented
43C61×#410C2211yescharacteristic
44C61×H(3,ℤ/4)C2212yescomplemented
45#14519825—42no—
46#14519825—42no—
47#14519823—42no—
48#14519823—42no—
49C2×#11758226—21no—
50C2×#11758226C2×C412yes—
51C2×#11758226C2×C412yescomplemented
52C2×#11758222—22no—
53C2×#11758222D412yes—
54C2×#11758222C2311yescharacteristic
55C2×#11758222C2311yescharacteristic
56C2×Q8×D61—21no—
57C2×Q8×D61C2×C412yescomplemented
58C22×D244D412yes—
59C22×D244C2311yescharacteristic
60C2×D4×D61C2×C412yes—
61C2×D4×D61D414yes—
62#14519784—42no—
63C2×#11758194—22no—
64C2×#11758194C2312yes—
65#14519775C2×C411yescharacteristic, complemented
66#14519774C2×C412yes—
67#14519773C2×C411yescharacteristic, complemented
68#14519766—22no—
69#14519766C2×C412yes—
70#14519766C2×C412yes—
71#14519766C2×C411yescharacteristic, complemented
72#14519765C2×C412yes—
73#14519764C2×C411yescharacteristic, complemented
74#14519763—24no—
75#14519762—24no—
76#14519755—24no—
77#14519752—24no—
78#14519752—22no—
79#14519750—24no—
80#14519749—22no—
81#14519747—24no—
82#14519747—22no—
83#14519742—21no—
84C4×D244—42no—
85#14519739—21no—
86#14519739—24no—
87#14519739—42no—
88C4×C22⋊D61—22no—
89#14519737—22no—
90#14519736C2312yes—
91#14519734C2×C412yescomplemented
92C2×#11758198—22no—
93C22⋊C4×D61—24no—
94C22⋊C4×D61—24no—
95C2×C4×Dic61C2311yescharacteristic
96C42×D61C2×C412yescomplemented
97#14519705—42no—
98#14519703—42no—
99#14519701—42no—
100#14519695—22no—
101#14519693—22no—
102#14519693—24no—
103#14519692—22no—
104#14519691—24no—
105C2×#41×C61C2311yescharacteristic
106C22×Q8×C61C2311yescharacteristic
107#131×C61—42no—
108C2×M16×C61—22no—
109C2×M16×C61C2312yes—
110C4⋊Q8×C61C2×C412yescomplemented
111#124×C61C2×C412yescomplemented
112#123×C61—22no—
113C22⋊Q8×C61—22no—
114#117×C61C2312yes—
115#117×C61C2311yescharacteristic
116#103×C61—42no—
117#99×C61—22no—
118C23×D61C22⋊C414yes—
119C23×D61C2×D412yes—
120#41×C61—22no—
121#41×C61—22no—
122C2×Q8×C61—21no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61C22×C412yes—
126C2×Q8×C61C22×C411yescharacteristic
127C2×D4×C61C22×C411yescharacteristic
128C2×D4×C61C2×D412yes—
129C22×C4×C61—21no—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2411yescharacteristic
132C2×C22⋊D61—21no—
133C2×C22⋊D61—22no—
134C2×C22⋊D61C22⋊C414yes—
135C2×C22⋊D61C2×D412yes—
136C22×Dic61—21no—
137#11758226—44no—
138#11758226—44no—
139#11758226—42no—
140#11758226—42no—
141Q8×D61—44no—
142Q8×D61—42no—
143Q8×D61—42no—
144D4×D61—44no—
145D4×D61—48no—
146#11758222—22no—
147#11758222—22no—
148#11758222—22no—
149#11758222—22no—
150#11758222—42no—
151#11758222—42no—
152#11758222—48no—
153C2×D244—22no—
154C2×D244—22no—
155C2×D244—22no—
156C2×D244—24no—
157C2×D244—42no—
158C2×D244—42no—
159C2×D244C22⋊C414yes—
160C2×D244C22×C411yescharacteristic
161C2×D244C22×C412yes—
162C2×D244C2×D411yescharacteristic
163C2×D244C2×D412yes—
164C2×C4×D61—22no—
165C2×C4×D61—21no—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168C2×C4×D61—44no—
169C2×C4×D61C22⋊C414yes—
170C2×C4×D61C22×C412yes—
171C2×Dic122—22no—
172C2×Dic122C22×C411yescharacteristic
173C2×Dic122C2×D411yescharacteristic
174M16×C61—22no—
175M16×C61—22no—
176M16×C61—42no—
177C2×C8×C61—22no—
178C2×C8×C61—42no—
179C4⋊C4×C61—24no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—22no—
182C22⋊C4×C61—24no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—22no—
185C42×C61—22no—
186C42×C61C22×C412yes—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189#11758198—24no—
190#11758198—24no—
191#11758198—24no—
192#11758198—22no—
193#11758198—22no—
194#11758198—44no—
195C4⋊Dic61—22no—
196#11758196—22no—
197#11758196—22no—
198C4×Dic61—21no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61C22×C411yescharacteristic
202C4×Dic61C22×C411yescharacteristic
203#11758194—22no—
204#11758194—22no—
205#11758194—42no—
206C2×#92418—22no—
207C2×#92418—42no—
208C23×C61C2×C22⋊C412yes—
209C23×C61C22×D411yescharacteristic
210C22×D61—24no—
211C22×D61—21no—
212C22×D61—44no—
213C22×D61—82no—
214C22×D61—84no—
215C22×D61C2×C22⋊C414yes—
216C22×D61C2×C22⋊C412yes—
217Q8×C61—22no—
218Q8×C61—24no—
219Q8×C61—41no—
220Q8×C61—41no—
221D4×C61—24no—
222D4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61C2×C22⋊C412yes—
230C2×C4×C61C23×C411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61H(5,𝔽2)12yes—
233C2×C4×C61H(5,𝔽2)12yes—
234C22⋊D61—24no—
235C22⋊D61—44no—
236C22⋊D61—42no—
237C22⋊D61—82no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—42no—
242C2×Dic61—81no—
243C2×Dic61C2×C22⋊C412yes—
244D244—24no—
245D244—22no—
246D244—22no—
247D244—22no—
248D244—21no—
249D244—21no—
250D244—44no—
251D244—44no—
252D244—84no—
253C4×D61—24no—
254C4×D61—44no—
255C4×D61—44no—
256C4×D61—44no—
257C4×D61—41no—
258C4×D61—41no—
259C4×D61—82no—
260C4×D61—164no—
261Dic122—21no—
262Dic122—21no—
263Dic122—22no—
264Dic122—44no—
265C8×C61—42no—
266#92418—42no—
267#21784—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—41no—
270C22×C61—42no—
271C22×C61C22×C22⋊C411yescharacteristic
272C22×C61#53412yes—
273C2×D61—22no—
274C2×D61—24no—
275C2×D61—42no—
276C2×D61—44no—
277C2×D61—42no—
278C2×D61—41no—
279C2×D61—84no—
280C2×D61—164no—
281C4×C61—22no—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61—82no—
285C4×C61#51812yes—
286C4×C61#53411yescharacteristic
287C4×C61#53411yescharacteristic
288Dic61—22no—
289Dic61—41no—
290Dic61—41no—
291Dic61—41no—
292Dic61—82no—
293#3500—611no—
294#2888—611no—
295#2883—611no—
296#1911—612no—
297#1909—611no—
298#1909D6111yescharacteristic, complemented
299#1885—612no—
300#1884—611no—
301#1884—611no—
302#1827—612no—
303#1796—612no—
304C2×C61—21no—
305C2×C61—22no—
306C2×C61—42no—
307C2×C61#230811yescharacteristic
308D61—44no—
309D61—42no—
310D61—81no—
311C2×2−1+4—611no—
312C2×H(5,𝔽2)—611no—
313#568—611no—
314#568C2×D6111yescharacteristic
315#548—1222no—
316#546—1222no—
317#546—1224no—
318#536—612no—
319#536C2×D6111yescharacteristic
320#535—612no—
321#535—611no—
322#532—611no—
323#532—612no—
324C2×#124—611no—
325C2×#124—612no—
326#518—612no—
327C4×#41—611no—
328#421—612no—
329#421—612no—
330#419—612no—
331#419C2×D6112yescomplemented
332#413—612no—
333#413—1222no—
334C2×#100—612no—
335C2×#100—1222no—
336#410—611no—
337#410—612no—
338#410C2×D6111yescharacteristic
339C2×#99—1224no—
340#339—612no—
341#339—612no—
342H(3,ℤ/4)—612no—
343H(3,ℤ/4)C2×D6112yescomplemented
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—612no—
353C2×#41—611no—
354C2×#41—1222no—
355C2×#41—1221no—
356C2×#41C22×D6111yescharacteristic
357C22×Q8—612no—
358C22×Q8—1221no—
359C22×Q8C22×D6111yescharacteristic
360C22×D4—612no—
361C22×D4—614no—
362C22×D4—612no—
363C22×D4—611no—
364#131—42no—
365#131—2442no—
366C2×M16—22no—
367C2×M16—612no—
368C2×M16—1222no—
369C2×M16C22×D6112yes—
370C4⋊Q8—611no—
371C4⋊Q8C4×D6112yescomplemented
372#127—611no—
373#127—612no—
374#124—612no—
375#124—612no—
376#124—611no—
377#124—611no—
378#124—612no—
379#124—1224no—
380#124—1222no—
381#124C4×D6112yescomplemented
382#123—22no—
383#123—1224no—
384C22⋊Q8—22no—
385C22⋊Q8—1224no—
386#121—1224no—
387#121—1224no—
388#121—1222no—
389#121—1222no—
390#120—1222no—
391#120—1224no—
392C4×Q8—1221no—
393C4×D4—1222no—
394C4×D4—1224no—
395C4×D4—1221no—
396C4×D4—1222no—
397C4×D4—2442no—
398C4×D4—2442no—
399#117—612no—
400#117—612no—
401#117C22×D6112yes—
402#117C22×D6111yescharacteristic
403C2×C22⋊C4—1224no—
404C2×C22⋊C4—1224no—
405C2×C22⋊C4—1222no—
406C2×C42—611no—
407C2×C42—612no—
408#104—2442no—
409#104—2442no—
410#103—42no—
411#103—2442no—
412#101—1222no—
413#100—1224no—
414#100—1222no—
415#99—22no—
416#99—1224no—
417#99—1222no—
418C24—612no—
419C24—614no—
420#41—22no—
421#41—22no—
422#41—1222no—
423#41—1222no—
424#41—1222no—
425#41—1222no—
426#41—2442no—
427#41—2442no—
428#41—2448no—
429#41—2444no—
430#41—2444no—
431#41—2442no—
432#41—2442no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—611no—
437C2×Q8—611no—
438C2×Q8—1222no—
439C2×Q8—2444no—
440C2×Q8—2442no—
441C2×Q8—2442no—
442C2×Q8C2×C4×D6112yes—
443C2×Q8C2×C4×D6111yescharacteristic
444C2×D4—612no—
445C2×D4—614no—
446C2×D4—612no—
447C2×D4—611no—
448C2×D4—611no—
449C2×D4—612no—
450C2×D4—614no—
451C2×D4—1224no—
452C2×D4—1222no—
453C2×D4—1222no—
454C2×D4—1221no—
455C2×D4—1222no—
456C2×D4—1222no—
457C2×D4—2442no—
458C2×D4—2442no—
459C2×D4—2444no—
460C2×D4—2448no—
461C2×D4C2×C4×D6111yescharacteristic
462C2×D4D4×D6112yes—
463C22×C4—21no—
464C22×C4—612no—
465C22×C4—614no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—2444no—
472C22×C4D4×D6112yes—
473C22×C4C23×D6111yescharacteristic
474M16—22no—
475M16—22no—
476M16—42no—
477M16—1222no—
478M16—1222no—
479M16—2442no—
480C2×C8—22no—
481C2×C8—42no—
482C2×C8—1222no—
483C2×C8—2442no—
484C4⋊C4—21no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C22⋊C4—21no—
491C22⋊C4—24no—
492C22⋊C4—22no—
493C22⋊C4—1224no—
494C22⋊C4—1224no—
495C22⋊C4—1224no—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C22⋊C4—1222no—
500C22⋊C4—2444no—
501C42—22no—
502C42—611no—
503C42—611no—
504C42—1222no—
505C42—1221no—
506C42—1222no—
507C42C2×C4×D6112yes—
508C23—614no—
509C23—612no—
510C23—1224no—
511C23—1221no—
512C23—4884no—
513C23—4882no—
514C23—2444no—
515C23C22⋊C4×D6112yes—
516C23C2×D4×D6111yescharacteristic
517Q8—22no—
518Q8—24no—
519Q8—41no—
520Q8—41no—
521Q8—1221no—
522Q8—1221no—
523Q8—1222no—
524Q8—2444no—
525D4—24no—
526D4—22no—
527D4—1224no—
528D4—1224no—
529D4—1222no—
530D4—1222no—
531D4—1221no—
532D4—1222no—
533D4—1221no—
534D4—4884no—
535D4—4882no—
536D4—2444no—
537D4—2444no—
538D4—2444no—
539D4—2442no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—612no—
547C2×C4—1224no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—9764no—
552C2×C4—4882no—
553C2×C4—4881no—
554C2×C4—2444no—
555C2×C4—2444no—
556C2×C4—2444no—
557C2×C4—2442no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4C22⋊C4×D6112yes—
561C2×C4C22×C4×D6111yescharacteristic
562C2×C4C2×D4×D6111yescharacteristic
563C2×C4#1451982212yes—
564C2×C4#1451982312yes—
565C8—42no—
566C8—2442no—
567C22—22no—
568C22—41no—
569C22—42no—
570C22—1222no—
571C22—1224no—
572C22—9764no—
573C22—4884no—
574C22—2442no—
575C22—2442no—
576C22—2444no—
577C22—2441no—
578C22C2×C22⋊C4×D6111yescharacteristic
579C22#1684032012yes—
580C4—22no—
581C4—42no—
582C4—42no—
583C4—82no—
584C4—1222no—
585C4—4882no—
586C4—2441no—
587C4—2441no—
588C4—2441no—
589C4#1684023412yes—
590C4#1684032211yescharacteristic
591C4#1684032311yescharacteristic
592C2—21no—
593C2—22no—
594C2—42no—
595C2—4881no—
596C2—2442no—
597C2—2444no—
598C2#2144839311yescharacteristic
subgroupG/H
centerC2#21448393
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1909C2
socleC2×C61#2308
prime radical#21459608C2
cosocleC22×C4×C61C24
hypercenter#1909D61
hypocenterC61#21784
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1909

Complements

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

NH ≅ G/Ntype
2. #21454297C2semidirect
3. #21454278C2semidirect
4. #21446350C2semidirect
5. #21446342C2semidirect
6. #21446204C2semidirect
7. #21446202C2semidirect
8. #21446200C2semidirect
9. #21445921C2semidirect
10. #21445791C2semidirect
11. C61×#1909C2semidirect
18. #16840337C4semidirect
20. #16840330C4semidirect
21. #16840313C4semidirect
24. #124×D61C4semidirect
27. #16839780C22semidirect
28. #16839778C22semidirect
29. #16839771C22semidirect
37. #16839523C22semidirect
38. #16839521C22semidirect
39. #16839518C22semidirect
42. C61×#419C22semidirect
44. C61×H(3,ℤ/4)C22semidirect
51. C2×#11758226C2×C4semidirect
57. C2×Q8×D61C2×C4semidirect
65. #14519775C2×C4semidirect
67. #14519773C2×C4semidirect
71. #14519766C2×C4semidirect
73. #14519764C2×C4semidirect
91. #14519734C2×C4semidirect
96. C42×D61C2×C4semidirect
110. C4⋊Q8×C61C2×C4semidirect
111. #124×C61C2×C4semidirect
298. #1909D61semidirect
331. #419C2×D61semidirect
343. H(3,ℤ/4)C2×D61semidirect
344. C61#21784semidirect
371. C4⋊Q8C4×D61semidirect
381. #124C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214596082
#214542972
#214542782
#214463502
#214463422
#214462042
#214462022
#214462002
#214459212
#214457912
C61×#19092
#2178461