Finite Group Databasebeta

← #29689217

Subgroups — #29689217

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#21459613C211yesmaximal, characteristic
2#21454296C211yesmaximal, characteristic, complemented
3#21454278C211yesmaximal, characteristic, complemented
4#21446348C212yesmaximal, complemented
5#21446338C212yesmaximal, complemented
6#21446204C212yesmaximal, complemented
7#21446199C212yesmaximal, complemented
8#21445924C211yesmaximal, characteristic, complemented
9#21445920C211yesmaximal, characteristic, complemented
10#21445794C211yesmaximal, characteristic, complemented
11C61×#1796C211yesmaximal, characteristic, complemented
12C2×#14519823C2211yescharacteristic
13#16840528C2212yes—
14#16840423—22no—
15#16840400—24no—
16#16840400—22no—
17#16840336C412yescomplemented
18#16840336C2211yescharacteristic
19#16840326C412yescomplemented
20#16840326C2211yescharacteristic
21#16840308C412yescomplemented
22#16840308C2211yescharacteristic
23C2×#14519766C412yescomplemented
24C2×#14519766C2211yescharacteristic
25#16840236C2212yes—
26#41×Dic61C2211yescharacteristic
27#16839779C2212yescomplemented
28#16839771C2212yescomplemented
29#16839746C2211yescharacteristic
30#16839745—22no—
31C2×#14519693C2212yes—
32#100×D61—22no—
33#16839736C2211yescharacteristic
34#16839735C2212yes—
35#99×D61—24no—
36#16839523C2212yescomplemented
37#16839521C2212yescomplemented
38#16839519C2212yescomplemented
39#16839518C2212yescomplemented
40C2×2−1+4×C61C2211yescharacteristic
41C61×#421C2212yescomplemented
42C61×#419C2212yescomplemented
43C61×#413C2211yescharacteristic
44C61×#410C2211yescharacteristic
45#14519823—42no—
46#14519823—42no—
47C2×#11758226D412yes—
48C2×#11758222C2×C412yes—
49C2×#11758222C2311yescharacteristic
50C22×D244C2×C412yes—
51C22×D244C2311yescharacteristic
52C2×D4×D61D414yes—
53C2×D4×D61D412yes—
54#14519785—42no—
55#14519783—42no—
56C2×#11758194C2312yes—
57M16×D61—24no—
58#14519775C2×C412yescomplemented
59#14519775C2×C411yescharacteristic, complemented
60#14519773C2×C412yes—
61#14519773C2×C411yescharacteristic, complemented
62#14519766—24no—
63#14519766—22no—
64#14519766C2×C412yes—
65#14519766C2×C412yes—
66#14519766C2×C412yes—
67#14519766C2×C411yescharacteristic, complemented
68#14519766C2×C412yescomplemented
69#14519764C2×C411yescharacteristic, complemented
70#14519760—24no—
71#14519760—22no—
72#14519753—24no—
73#14519753—22no—
74#14519751—24no—
75#14519751—22no—
76#14519749—24no—
77#14519749—22no—
78#14519745—24no—
79#14519745—22no—
80Q8×Dic61—21no—
81C4×D244—42no—
82D4×Dic61—22no—
83D4×Dic61—21no—
84#14519739—42no—
85C4×C22⋊D61—24no—
86C4×C22⋊D61—22no—
87#14519736C2×C412yescomplemented
88#14519736C2312yes—
89#14519736C2311yescharacteristic
90C2×#11758198—24no—
91C2×#11758198—22no—
92C22⋊C4×D61—24no—
93C2×C4×Dic61C2×C412yescomplemented
94C2×C4×Dic61C2311yescharacteristic
95#14519704—42no—
96#14519701—42no—
97#14519696—21no—
98#14519694—21no—
99#14519693—24no—
100#14519692—22no—
101#14519691—24no—
1022−1+4×C61—42no—
1032−1+4×C61—42no—
104C2×#41×C61—21no—
105C2×#41×C61—22no—
106C2×#41×C61C2×C412yescomplemented
107C2×#41×C61C2311yescharacteristic
108C2×#41×C61C2311yescharacteristic
109C22×Q8×C61—21no—
110C22×Q8×C61C2×C412yescomplemented
111C22×Q8×C61C2311yescharacteristic
112C2×M16×C61C2312yes—
113#117×C61C2312yes—
114#104×C61—42no—
115#103×C61—42no—
116#101×C61—21no—
117#100×C61—21no—
118#99×C61—22no—
119C23×D61C22⋊C414yes—
120C23×D61C2×D412yes—
121#41×C61—22no—
122#41×C61—22no—
123#41×C61—22no—
124#41×C61—22no—
125#41×C61—42no—
126#41×C61—42no—
127#41×C61—48no—
128#41×C61—44no—
129C2×Q8×C61—22no—
130C2×Q8×C61—21no—
131C2×Q8×C61—22no—
132C2×Q8×C61—22no—
133C2×Q8×C61—42no—
134C2×Q8×C61—42no—
135C2×Q8×C61—44no—
136C2×Q8×C61C22×C411yescharacteristic
137C2×Q8×C61C22×C412yes—
138C2×Q8×C61C2×D411yescharacteristic
139C2×D4×C61—22no—
140C2×D4×C61—21no—
141C2×D4×C61C22×C411yescharacteristic
142C2×D4×C61C2×D412yes—
143C2×D4×C61C2×D411yescharacteristic
144C22×C4×C61—22no—
145C22×C4×C61—21no—
146C22×C4×C61—22no—
147C22×C4×C61—21no—
148C22×C4×C61C22×C412yes—
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—24no—
151C2×C22⋊D61—22no—
152C2×C22⋊D61C22⋊C414yes—
153C2×C22⋊D61C2×D412yes—
154C22×Dic61—22no—
155C22×Dic61—21no—
156#11758226—44no—
157D4×D61—44no—
158D4×D61—48no—
159#11758222—22no—
160#11758222—22no—
161#11758222—42no—
162#11758222—42no—
163C2×D244—22no—
164C2×D244—22no—
165C2×D244—42no—
166C2×D244—42no—
167C2×D244C22⋊C414yes—
168C2×D244C22×C411yescharacteristic
169C2×D244C22×C412yes—
170C2×D244C2×D412yes—
171C2×C4×D61—44no—
172C2×C4×D61C22⋊C414yes—
173C2×C4×D61C2×D412yes—
174C2×Dic122C22×C411yescharacteristic
175M16×C61—22no—
176M16×C61—22no—
177C2×C8×C61—22no—
178C4⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C42×C61—22no—
181C22⋊Dic61—22no—
182C22⋊Dic61—21no—
183C22⋊Dic61—22no—
184C22⋊Dic61—22no—
185#11758198—24no—
186#11758198—22no—
187#11758198—24no—
188#11758198—24no—
189#11758198—24no—
190#11758198—44no—
191C4⋊Dic61—22no—
192C4⋊Dic61—21no—
193C4⋊Dic61—22no—
194#11758196—24no—
195#11758196—22no—
196C4×Dic61—22no—
197C4×Dic61—21no—
198C4×Dic61—22no—
199C4×Dic61C22×C411yescharacteristic
200C4×Dic61C22×C412yes—
201C4×Dic61C22×C411yescharacteristic
202#11758194—22no—
203#11758194—22no—
204C2×#92418—22no—
205#11758189—44no—
206C8×D61—44no—
207C23×C61—21no—
208C23×C61C2×C22⋊C412yes—
209C23×C61C22×D411yescharacteristic
210C22×D61—24no—
211C22×D61—44no—
212C22×D61—82no—
213C22×D61—84no—
214C22×D61C2×C22⋊C414yes—
215C22×D61C2×C22⋊C412yes—
216Q8×C61—22no—
217Q8×C61—22no—
218Q8×C61—22no—
219Q8×C61—21no—
220Q8×C61—21no—
221Q8×C61—44no—
222Q8×C61—42no—
223D4×C61—24no—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61—22no—
227D4×C61—44no—
228D4×C61—42no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—44no—
234C2×C4×C61—44no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61C2×C22⋊C412yes—
239C2×C4×C61C23×C411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61H(5,𝔽2)12yes—
242C2×C4×C61H(5,𝔽2)12yes—
243C22⋊D61—24no—
244C22⋊D61—44no—
245C22⋊D61—84no—
246C22⋊D61—82no—
247C2×Dic61—21no—
248C2×Dic61—22no—
249C2×Dic61—21no—
250C2×Dic61—42no—
251C2×Dic61—81no—
252C2×Dic61—82no—
253C2×Dic61C2×C22⋊C412yes—
254D244—22no—
255D244—22no—
256D244—21no—
257D244—21no—
258D244—24no—
259D244—44no—
260C4×D61—24no—
261C4×D61—44no—
262C4×D61—164no—
263Dic122—21no—
264Dic122—21no—
265C8×C61—42no—
266#92418—42no—
267#21784—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—41no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—41no—
273C22×C61C22×C22⋊C411yescharacteristic
274C22×C61#53412yes—
275C2×D61—22no—
276C2×D61—24no—
277C2×D61—44no—
278C2×D61—42no—
279C2×D61—84no—
280C2×D61—164no—
281C4×C61—22no—
282C4×C61—42no—
283C4×C61—41no—
284C4×C61—82no—
285C4×C61#51811yescharacteristic
286C4×C61#51811yescharacteristic
287C4×C61#53412yes—
288Dic61—22no—
289Dic61—41no—
290Dic61—42no—
291Dic61—41no—
292Dic61—82no—
293#3500—611no—
294#2888—611no—
295#2883—611no—
296#1911—612no—
297#1909—612no—
298#1885—612no—
299#1884—612no—
300#1827—611no—
301#1827—611no—
302#1796—611no—
303#1796D6111yescharacteristic, complemented
304C2×C61—21no—
305C2×C61—22no—
306C2×C61—42no—
307C2×C61—81no—
308C2×C61#230811yescharacteristic
309D61—44no—
310D61—42no—
311C2×2−1+4C2×D6111yescharacteristic
312C2×H(5,𝔽2)—611no—
313#568—612no—
314#548—1222no—
315#546—1224no—
316#546—1222no—
317#536—612no—
318#536—611no—
319#535—612no—
320#535—611no—
321#532—612no—
322#532—611no—
323C2×#124—612no—
324C2×#124—611no—
325#518—612no—
326C4×#41—611no—
327#421—612no—
328#421C2×D6112yescomplemented
329#419—612no—
330#419C2×D6112yescomplemented
331#413—611no—
332#413—1222no—
333#413C2×D6111yescharacteristic
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)—612no—
344C61#2178411yescharacteristic, complemented, Sylow
3452−1+4—42no—
3462−1+4—42no—
347H(5,𝔽2)—2442no—
348H(5,𝔽2)—2442no—
349C2×#41—22no—
350C2×#41—21no—
351C2×#41—612no—
352C2×#41—612no—
353C2×#41—611no—
354C2×#41C4×D6112yescomplemented
355C2×#41C22×D6111yescharacteristic
356C2×#41C22×D6111yescharacteristic
357C22×Q8—21no—
358C22×Q8C4×D6112yescomplemented
359C22×Q8C22×D6111yescharacteristic
360C22×D4—614no—
361C22×D4—612no—
362C22×D4—612no—
363C22×D4—611no—
364#131—2442no—
365#131—2442no—
366C2×M16—612no—
367C2×M16—1224no—
368C2×M16C22×D6112yes—
369C4⋊Q8—611no—
370C4⋊Q8—612no—
371#127—612no—
372#127—611no—
373#124—612no—
374#124—611no—
375#124—612no—
376#124—612no—
377#124—612no—
378#124—611no—
379#124—1222no—
380#124—1224no—
381#123—1222no—
382#123—1224no—
383C22⋊Q8—1222no—
384C22⋊Q8—1224no—
385#121—1222no—
386#121—1224no—
387#121—1222no—
388#121—1224no—
389#120—1222no—
390#120—1224no—
391C4×Q8—1221no—
392C4×D4—1222no—
393C4×D4—1224no—
394C4×D4—1221no—
395C4×D4—1222no—
396C4×D4—2442no—
397C4×D4—2442no—
398#117—612no—
399#117—611no—
400#117—612no—
401#117C22×D6112yes—
402C2×C22⋊C4—1224no—
403C2×C22⋊C4—1222no—
404C2×C22⋊C4—1224no—
405C2×C42—612no—
406C2×C42—611no—
407#104—42no—
408#104—2442no—
409#103—42no—
410#103—2442no—
411#101—21no—
412#101—1221no—
413#100—21no—
414#100—1224no—
415#100—1221no—
416#99—22no—
417#99—1224no—
418#99—1222no—
419C24—612no—
420C24—614no—
421#41—22no—
422#41—22no—
423#41—22no—
424#41—22no—
425#41—42no—
426#41—42no—
427#41—48no—
428#41—44no—
429#41—1222no—
430#41—1222no—
431#41—2442no—
432#41—2442no—
433#41—2444no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—44no—
441C2×Q8—611no—
442C2×Q8C2×C4×D6111yescharacteristic
443C2×Q8C2×C4×D6112yes—
444C2×Q8D4×D6111yescharacteristic
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—612no—
448C2×D4—614no—
449C2×D4—612no—
450C2×D4—611no—
451C2×D4—612no—
452C2×D4—614no—
453C2×D4—1224no—
454C2×D4—1222no—
455C2×D4—1222no—
456C2×D4—1222no—
457C2×D4—2442no—
458C2×D4—2442no—
459C2×D4—2444no—
460C2×D4—2448no—
461C2×D4C2×C4×D6111yescharacteristic
462C2×D4D4×D6111yescharacteristic
463C2×D4D4×D6112yes—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—612no—
469C22×C4—614no—
470C22×C4—1221no—
471C22×C4—1222no—
472C22×C4—2444no—
473C22×C4C2×C4×D6112yes—
474C22×C4C23×D6111yescharacteristic
475M16—22no—
476M16—22no—
477M16—1222no—
478M16—1222no—
479M16—2444no—
480C2×C8—22no—
481C2×C8—1222no—
482C2×C8—2444no—
483C4⋊C4—22no—
484C4⋊C4—1224no—
485C4⋊C4—1222no—
486C4⋊C4—1221no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C22⋊C4—22no—
490C22⋊C4—1224no—
491C22⋊C4—1222no—
492C22⋊C4—1224no—
493C22⋊C4—1224no—
494C22⋊C4—1224no—
495C22⋊C4—1222no—
496C22⋊C4—1221no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C22⋊C4—2444no—
500C42—22no—
501C42—611no—
502C42—612no—
503C42—611no—
504C42—1221no—
505C42—1222no—
506C42—1222no—
507C23—21no—
508C23—612no—
509C23—614no—
510C23—1224no—
511C23—4884no—
512C23—4882no—
513C23—2444no—
514C23C22⋊C4×D6112yes—
515C23C2×D4×D6111yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—22no—
519Q8—21no—
520Q8—21no—
521Q8—44no—
522Q8—42no—
523Q8—1221no—
524Q8—1221no—
525D4—24no—
526D4—21no—
527D4—21no—
528D4—22no—
529D4—44no—
530D4—42no—
531D4—1224no—
532D4—1222no—
533D4—1222no—
534D4—1221no—
535D4—1221no—
536D4—1224no—
537D4—4882no—
538D4—4884no—
539D4—2444no—
540D4—2444no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—612no—
551C2×C4—1224no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1222no—
555C2×C4—9764no—
556C2×C4—4881no—
557C2×C4—4882no—
558C2×C4—2444no—
559C2×C4—2442no—
560C2×C4C22⋊C4×D6112yes—
561C2×C4C22×C4×D6111yescharacteristic
562C2×C4C2×D4×D6111yescharacteristic
563C2×C4#1451982212yes—
564C2×C4#1451982212yes—
565C8—42no—
566C8—2442no—
567C22—22no—
568C22—41no—
569C22—42no—
570C22—42no—
571C22—41no—
572C22—1222no—
573C22—1224no—
574C22—9764no—
575C22—4884no—
576C22—2442no—
577C22—2444no—
578C22C2×C22⋊C4×D6111yescharacteristic
579C22#1684032212yes—
580C4—22no—
581C4—42no—
582C4—41no—
583C4—82no—
584C4—1222no—
585C4—4882no—
586C4—2441no—
587C4—2441no—
588C4—2442no—
589C4#1684023511yescharacteristic
590C4#1684023511yescharacteristic
591C4#1684032212yes—
592C2—22no—
593C2—21no—
594C2—42no—
595C2—81no—
596C2—2442no—
597C2—2444no—
598C2#2144839511yescharacteristic
subgroupG/H
centerC2#21448395
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1796C2
socleC2×C61#2308
prime radical#21459613C2
cosocleC22×C4×C61C24
hypercenter#1796D61
hypocenterC61#21784
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1796

Complements

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

NH ≅ G/Ntype
2. #21454296C2semidirect
3. #21454278C2semidirect
4. #21446348C2semidirect
5. #21446338C2semidirect
6. #21446204C2semidirect
7. #21446199C2semidirect
8. #21445924C2semidirect
9. #21445920C2semidirect
10. #21445794C2semidirect
11. C61×#1796C2semidirect
17. #16840336C4semidirect
19. #16840326C4semidirect
21. #16840308C4semidirect
23. C2×#14519766C4semidirect
27. #16839779C22semidirect
28. #16839771C22semidirect
36. #16839523C22semidirect
37. #16839521C22semidirect
38. #16839519C22semidirect
39. #16839518C22semidirect
41. C61×#421C22semidirect
42. C61×#419C22semidirect
58. #14519775C2×C4semidirect
59. #14519775C2×C4semidirect
61. #14519773C2×C4semidirect
67. #14519766C2×C4semidirect
68. #14519766C2×C4semidirect
69. #14519764C2×C4semidirect
87. #14519736C2×C4semidirect
93. C2×C4×Dic61C2×C4semidirect
106. C2×#41×C61C2×C4semidirect
110. C22×Q8×C61C2×C4semidirect
303. #1796D61semidirect
328. #421C2×D61semidirect
330. #419C2×D61semidirect
344. C61#21784semidirect
354. C2×#41C4×D61semidirect
358. C22×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214596132
#214542962
#214542782
#214463482
#214463382
#214462042
#214461992
#214459242
#214459202
#214457942
C61×#17962
#2178461