Finite Group Databasebeta

← #29689218

Subgroups — #29689218

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#21459611C211yesmaximal, characteristic
2#21454295C211yesmaximal, characteristic, complemented
3#21454279C211yesmaximal, characteristic, complemented
4#21446352C211yesmaximal, characteristic, complemented
5#21446340C212yesmaximal, complemented
6#21446207C211yesmaximal, characteristic, complemented
7#21446205C211yesmaximal, characteristic, complemented
8#21446199C212yesmaximal, complemented
9#21445921C212yesmaximal, complemented
10#21445793C212yesmaximal, complemented
11C61×#1911C211yesmaximal, characteristic, complemented
12C2×#14519824C2211yescharacteristic
13C2×#14519822C2211yescharacteristic
14#16840528C2212yes—
15#16840418—22no—
16#16840409—22no—
17#16840399—24no—
18#16840338C2211yescharacteristic
19#16840332C412yescomplemented
20#16840325C412yescomplemented
21#16840310C412yescomplemented
22#16840308C2211yescharacteristic
23C2×#14519764C2211yescharacteristic
24#124×D61C412yescomplemented
25#16840237C2211yescharacteristic
26#16840230C2211yescharacteristic
27#16839779C2212yescomplemented
28#16839772C2212yescomplemented
29#16839770C2212yescomplemented
30#16839744C2212yes—
31#16839743—22no—
32C2×#14519694C2211yescharacteristic
33#100×D61—22no—
34#16839735C2212yes—
35#16839735C2212yes—
36C2×#14519691—24no—
37#16839523C2212yescomplemented
38#16839521C2212yescomplemented
39#16839518C2212yescomplemented
40C61×#535C2211yescharacteristic
41C61×#518C2211yescharacteristic
42C61×#421C2212yescomplemented
43C2×#100×C61C2211yescharacteristic
44C61×H(3,ℤ/4)C2212yescomplemented
45#14519824—42no—
46#14519824—42no—
47#14519822—42no—
48#14519822—42no—
49C2×#11758222—21no—
50C2×#11758222C2311yescharacteristic
51C2×#11758222C2311yescharacteristic
52C2×#11758224—22no—
53C2×#11758224C2×C412yes—
54C2×#11758224C2×C412yescomplemented
55C2×#11758224D412yes—
56C22×Dic122—21no—
57C22×Dic122C2311yescharacteristic
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×C412yescomplemented
67#14519775C2×C411yescharacteristic, complemented
68#14519773C2×C411yescharacteristic, complemented
69#14519766C2×C411yescharacteristic, complemented
70#14519765C2×C412yescomplemented
71#14519764—22no—
72#14519764C2×C412yes—
73#14519764C2×C412yes—
74#14519764C2×C411yescharacteristic, complemented
75#14519763—24no—
76#14519761—22no—
77#14519759—24no—
78#14519757—22no—
79#14519754—24no—
80#14519749—24no—
81#14519749—22no—
82#14519748—24no—
83#14519745—24no—
84#14519742—21no—
85D4×Dic61—24no—
86D4×Dic61—42no—
87#14519739—21no—
88C4×C22⋊D61—22no—
89#14519737—22no—
90#14519736C2312yes—
91#14519734C2×C412yescomplemented
92C2×C22⋊Dic61—22no—
93C2×C22⋊Dic61—22no—
94C22⋊C4×D61—24no—
95C2×C4×Dic61C2311yescharacteristic
96C42×D61C2×C412yescomplemented
97#14519704—42no—
98#14519702—42no—
99#14519700—42no—
100#14519695—22no—
101#14519694—22no—
102#14519693—22no—
103#14519691—24no—
104#14519691—24no—
105C2×#41×C61C2311yescharacteristic
106C22×D4×C61C2311yescharacteristic
107C2×M16×C61C2312yes—
108#127×C61C2×C412yes—
109#124×C61C2×C412yes—
110#121×C61—22no—
111#120×C61—22no—
112C4×D4×C61—42no—
113#117×C61C2312yes—
114#117×C61C2311yescharacteristic
115C2×C22⋊C4×C61—22no—
116#104×C61—42no—
117#100×C61—22no—
118C24×C61C2×D412yes—
119C23×D61C22⋊C414yes—
120#41×C61—22no—
121#41×C61—22no—
122C2×Q8×C61C22×C411yescharacteristic
123C2×D4×C61—21no—
124C2×D4×C61—24no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61C22×C411yescharacteristic
128C2×D4×C61C22×C412yes—
129C2×D4×C61C2×D412yes—
130C22×C4×C61—21no—
131C22×C4×C61—42no—
132C22×C4×C61C2411yescharacteristic
133C2×C22⋊D61—22no—
134C2×C22⋊D61—22no—
135C2×C22⋊D61—44no—
136C2×C22⋊D61—44no—
137C2×C22⋊D61C22⋊C414yes—
138C2×C22⋊D61C22⋊C414yes—
139C2×C22⋊D61C2×D412yes—
140C2×C22⋊D61C2×D412yes—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—22no—
144C22×Dic61—42no—
145C22×Dic61C22⋊C414yes—
146Q8×D61—44no—
147#11758224—44no—
148#11758224—44no—
149#11758224—44no—
150#11758224—44no—
151#11758224—44no—
152D4×D61—44no—
153D4×D61—44no—
154#11758222—22no—
155#11758222—22no—
156#11758222—22no—
157#11758222—22no—
158#11758222—44no—
159C2×D244C22×C411yescharacteristic
160C2×D244C2×D411yescharacteristic
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61C22×C412yes—
164C2×C4×D61C2×D412yes—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—21no—
168C2×Dic122—22no—
169C2×Dic122—44no—
170C2×Dic122C22×C412yes—
171C2×Dic122C22×C411yescharacteristic
172C2×Dic122C2×D411yescharacteristic
173M16×C61—22no—
174M16×C61—22no—
175C2×C8×C61—22no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—22no—
178C22⋊C4×C61—24no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—42no—
182C42×C61—22no—
183C42×C61C22×C412yes—
184C22⋊Dic61—24no—
185C22⋊Dic61—24no—
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—42no—
189#11758198—22no—
190#11758198—24no—
191#11758198—22no—
192C4⋊Dic61—24no—
193C4⋊Dic61—22no—
194#11758196—22no—
195#11758196—22no—
196C4×Dic61—21no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61C22×C411yescharacteristic
200C4×Dic61C22×C411yescharacteristic
201#11758194—22no—
202#11758194—22no—
203C2×#92418—22no—
204#11758189—44no—
205C8×D61—44no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—21no—
209C23×C61—82no—
210C23×C61C2×C22⋊C414yes—
211C23×C61C22×D411yescharacteristic
212C22×D61—44no—
213C22×D61—84no—
214C22×D61C2×C22⋊C412yes—
215C22×D61C2×C22⋊C412yes—
216Q8×C61—22no—
217D4×C61—24no—
218D4×C61—22no—
219D4×C61—24no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—84no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
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—44no—
239C22⋊D61—82no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—22no—
243C2×Dic61—44no—
244C2×Dic61—44no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—42no—
248C2×Dic61—81no—
249C2×Dic61—162no—
250C2×Dic61C2×C22⋊C412yes—
251C2×Dic61C2×C22⋊C412yes—
252D244—22no—
253D244—22no—
254C4×D61—24no—
255C4×D61—24no—
256C4×D61—44no—
257C4×D61—82no—
258Dic122—22no—
259Dic122—22no—
260Dic122—24no—
261Dic122—44no—
262Dic122—42no—
263C8×C61—42no—
264#92418—42no—
265#21784—611nomaximal, Sylow
266C22×C61—24no—
267C22×C61—41no—
268C22×C61—44no—
269C22×C61—42no—
270C22×C61—41no—
271C22×C61—82no—
272C22×C61—82no—
273C22×C61C22×C22⋊C411yescharacteristic
274C22×C61#53411yescharacteristic
275C22×C61#53411yescharacteristic
276C2×D61—22no—
277C2×D61—22no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—164no—
281C4×C61—42no—
282C4×C61—41no—
283C4×C61—82no—
284C4×C61#51812yes—
285C4×C61#53412yes—
286Dic61—22no—
287Dic61—22no—
288Dic61—41no—
289Dic61—41no—
290Dic61—42no—
291Dic61—82no—
292#3500—611no—
293#2888—611no—
294#2883—611no—
295#1911—611no—
296#1911D6111yescharacteristic, complemented
297#1909—612no—
298#1885—611no—
299#1885—611no—
300#1884—612no—
301#1827—612no—
302#1796—612no—
303C2×C61—21no—
304C2×C61—21no—
305C2×C61—21no—
306C2×C61—44no—
307C2×C61—81no—
308C2×C61#230811yescharacteristic
309D61—42no—
310D61—42no—
311C2×2−1+4—611no—
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#535C2×D6111yescharacteristic
321#532—611no—
322#532—612no—
323C2×#124—611no—
324C2×#124—612no—
325#518—611no—
326#518C2×D6111yescharacteristic
327C4×#41—611no—
328#421—612no—
329#421C2×D6112yescomplemented
330#419—612no—
331#419—612no—
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#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—611no—
353C2×#41—612no—
354C2×#41—1221no—
355C2×#41—1222no—
356C2×#41C22×D6111yescharacteristic
357C22×Q8—612no—
358C22×Q8—611no—
359C22×Q8—1221no—
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—611no—
370C4⋊Q8—612no—
371#127—611no—
372#127C4×D6112yes—
373#124—611no—
374#124—612no—
375#124—612no—
376#124—611no—
377#124—612no—
378#124—1224no—
379#124—1222no—
380#124C4×D6112yes—
381#123—1224no—
382#123—1222no—
383C22⋊Q8—1224no—
384C22⋊Q8—1222no—
385#121—22no—
386#121—1224no—
387#121—1222no—
388#121—1224no—
389#120—22no—
390#120—1224no—
391C4×Q8—1221no—
392C4×D4—42no—
393C4×D4—1222no—
394C4×D4—1221no—
395C4×D4—1222no—
396C4×D4—1224no—
397C4×D4—2442no—
398#117—612no—
399#117—612no—
400#117C22×D6112yes—
401#117C22×D6111yescharacteristic
402C2×C22⋊C4—22no—
403C2×C22⋊C4—1224no—
404C2×C22⋊C4—1222no—
405C2×C22⋊C4—1222no—
406C2×C42—611no—
407C2×C42—612no—
408#104—42no—
409#104—2442no—
410#103—2442no—
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—22no—
421#41—22no—
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—611no—
433C2×Q8—611no—
434C2×Q8—612no—
435C2×Q8—1222no—
436C2×Q8—1221no—
437C2×Q8—1222no—
438C2×Q8—1222no—
439C2×Q8—2444no—
440C2×Q8—2444no—
441C2×Q8C2×C4×D6111yescharacteristic
442C2×D4—21no—
443C2×D4—24no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—612no—
447C2×D4—612no—
448C2×D4—614no—
449C2×D4—614no—
450C2×D4—611no—
451C2×D4—611no—
452C2×D4—1222no—
453C2×D4—1222no—
454C2×D4—2444no—
455C2×D4—2444no—
456C2×D4—2444no—
457C2×D4—2444no—
458C2×D4C2×C4×D6111yescharacteristic
459C2×D4C2×C4×D6112yes—
460C2×D4D4×D6112yes—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—614no—
464C22×C4—612no—
465C22×C4—612no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1221no—
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—21no—
482C4⋊C4—22no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—1224no—
486C4⋊C4—1222no—
487C22⋊C4—21no—
488C22⋊C4—24no—
489C22⋊C4—22no—
490C22⋊C4—42no—
491C22⋊C4—1222no—
492C22⋊C4—1224no—
493C22⋊C4—1222no—
494C22⋊C4—1222no—
495C22⋊C4—1224no—
496C22⋊C4—1224no—
497C22⋊C4—1222no—
498C22⋊C4—2442no—
499C42—22no—
500C42—611no—
501C42—611no—
502C42—1222no—
503C42—1221no—
504C42—1222no—
505C42C2×C4×D6112yes—
506C23—22no—
507C23—22no—
508C23—21no—
509C23—82no—
510C23—612no—
511C23—612no—
512C23—4884no—
513C23—2444no—
514C23C22⋊C4×D6114yes—
515C23C2×D4×D6111yescharacteristic
516Q8—22no—
517Q8—1222no—
518Q8—1222no—
519Q8—1224no—
520Q8—2444no—
521Q8—2442no—
522D4—24no—
523D4—22no—
524D4—24no—
525D4—41no—
526D4—41no—
527D4—84no—
528D4—1222no—
529D4—1222no—
530D4—1222no—
531D4—1222no—
532D4—1222no—
533D4—1222no—
534D4—4882no—
535D4—2444no—
536D4—2444no—
537D4—2444no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—162no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1222no—
550C2×C4—9762no—
551C2×C4—4882no—
552C2×C4—4881no—
553C2×C4—2444no—
554C2×C4—2444no—
555C2×C4—2444no—
556C2×C4—2442no—
557C2×C4—2441no—
558C2×C4—2441no—
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—42no—
569C22—41no—
570C22—82no—
571C22—82no—
572C22—1222no—
573C22—1222no—
574C22—9764no—
575C22—2442no—
576C22—2442no—
577C22C2×C22⋊C4×D6111yescharacteristic
578C22#1684032211yescharacteristic
579C22#1684032311yescharacteristic
580C4—42no—
581C4—41no—
582C4—82no—
583C4—1222no—
584C4—1222no—
585C4—4882no—
586C4—2441no—
587C4—2441no—
588C4—2442no—
589C4#1684023412yes—
590C4#1684032012yes—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—44no—
595C2—81no—
596C2—2442no—
597C2—2442no—
598C2#2144839311yescharacteristic
subgroupG/H
centerC2#21448393
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1911C2
socleC2×C61#2308
prime radical#21459611C2
cosocleC22×C4×C61C24
hypercenter#1911D61
hypocenterC61#21784
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1911

Complements

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

NH ≅ G/Ntype
2. #21454295C2semidirect
3. #21454279C2semidirect
4. #21446352C2semidirect
5. #21446340C2semidirect
6. #21446207C2semidirect
7. #21446205C2semidirect
8. #21446199C2semidirect
9. #21445921C2semidirect
10. #21445793C2semidirect
11. C61×#1911C2semidirect
19. #16840332C4semidirect
20. #16840325C4semidirect
21. #16840310C4semidirect
24. #124×D61C4semidirect
27. #16839779C22semidirect
28. #16839772C22semidirect
29. #16839770C22semidirect
37. #16839523C22semidirect
38. #16839521C22semidirect
39. #16839518C22semidirect
42. C61×#421C22semidirect
44. C61×H(3,ℤ/4)C22semidirect
54. C2×#11758224C2×C4semidirect
58. C2×Q8×D61C2×C4semidirect
66. #14519777C2×C4semidirect
67. #14519775C2×C4semidirect
68. #14519773C2×C4semidirect
69. #14519766C2×C4semidirect
70. #14519765C2×C4semidirect
74. #14519764C2×C4semidirect
91. #14519734C2×C4semidirect
96. C42×D61C2×C4semidirect
296. #1911D61semidirect
329. #421C2×D61semidirect
343. H(3,ℤ/4)C2×D61semidirect
344. C61#21784semidirect

Maximal subgroups

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

subgroupindex
#214596112
#214542952
#214542792
#214463522
#214463402
#214462072
#214462052
#214461992
#214459212
#214457932
C61×#19112
#2178461