Finite Group Databasebeta

← #29689223

Subgroups — #29689223

Order 15616.

1262 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459606C211yesmaximal, characteristic
2#21454301C211yesmaximal, characteristic, complemented
3#21454281C211yesmaximal, characteristic, complemented
4#21446353C212yesmaximal, complemented
5#21446346C211yesmaximal, characteristic, complemented
6#21446344C211yesmaximal, characteristic, complemented
7#21446221C212yesmaximal, complemented
8#21446203C211yesmaximal, characteristic, complemented
9#21445922C212yesmaximal, complemented
10#21445788C212yesmaximal, complemented
11C61×#1884C211yesmaximal, characteristic, complemented
12C2×#14519824C2211yescharacteristic
13C2×#14519823C2211yescharacteristic
14#16840529C2211yescharacteristic, complemented
15#16840421—22no—
16#16840408—22no—
17#16840401—24no—
18#16840337C412yescomplemented
19#16840333C2211yescharacteristic
20#16840326C2211yescharacteristic
21#16840325C412yescomplemented
22#16840311C412yescomplemented
23C2×#14519765C2211yescharacteristic
24#124×D61C412yescomplemented
25#16840239C2212yescomplemented
26C4×#11758222C2211yescharacteristic
27#16839779C2212yes—
28#16839768C2212yes—
29#16839766C2212yes—
30#16839744—22no—
31#16839744C2212yes—
32C2×#14519695—22no—
33C2×#14519695C2212yescomplemented
34#16839736C2211yescharacteristic, complemented
35#16839735C2212yes—
36#16839734—24no—
37#16839522C2212yescomplemented
38#16839519C2212yescomplemented
39#16839518C2212yescomplemented
40C61×#568C2211yescharacteristic, complemented
41C61×#532C2211yescharacteristic
42C61×#421C2212yes—
43C61×#410C2211yescharacteristic, complemented
44C61×#339C2212yescomplemented
45#14519824—42no—
46#14519824—42no—
47#14519823—42no—
48#14519823—42no—
49C2×#11758226C2×C412yes—
50C2×#11758222—21no—
51C2×#11758222D412yes—
52C2×#11758222C2311yescharacteristic
53C2×#11758224C2×C412yes—
54C2×#11758224D414yes—
55C22×Dic122D412yes—
56C22×Dic122C2311yescharacteristic
57C2×Q8×D61C2×C412yes—
58C22×D244—21no—
59C22×D244C2311yescharacteristic
60C2×D4×D61—22no—
61C2×D4×D61C2×C412yes—
62#14519784—42no—
63C2×#11758194—22no—
64C2×#11758194C2312yes—
65#14519775C2×C412yes—
66#14519773C2×C412yes—
67#14519766C2×C412yes—
68#14519765—22no—
69#14519765C2×C412yes—
70#14519765C2×C412yes—
71#14519764C2×C412yes—
72#14519763—24no—
73#14519762—24no—
74#14519758—22no—
75#14519756—22no—
76#14519755—24no—
77#14519751—22no—
78#14519750—24no—
79#14519749—24no—
80#14519745—22no—
81#14519745—24no—
82C4×Dic122—21no—
83C4×Dic122—42no—
84#14519742—42no—
85C4×D244—21no—
86C4×C22⋊D61—22no—
87#14519737—24no—
88#14519736C2312yes—
89#14519736C2311yescharacteristic
90#14519734C2×C412yes—
91#14519733—24no—
92C2×#11758198—22no—
93C22⋊C4×D61—24no—
94C42×D61C2×C412yes—
95#14519704—42no—
96#14519699—42no—
97#14519697—42no—
98#14519695—22no—
99#14519695—24no—
100#14519693—22no—
101#14519692—22no—
102#14519691—24no—
103C2×#41×C61C2311yescharacteristic
104C2×#41×C61C2311yescharacteristic
105#131×C61—42no—
106C2×M16×C61—22no—
107C2×M16×C61C2312yes—
108C4⋊Q8×C61C2×C411yescharacteristic
109#127×C61C2×C411yescharacteristic
110#124×C61C2×C411yescharacteristic
111#124×C61C2×C411yescharacteristic
112#121×C61—22no—
113C4×D4×C61—22no—
114#117×C61C2312yes—
115C2×C42×C61C2311yescharacteristic
116#104×C61—42no—
117#99×C61—22no—
118C23×D61—21no—
119C23×D61—22no—
120#41×C61—22no—
121#41×C61—22no—
122#41×C61—22no—
123#41×C61—22no—
124C2×Q8×C61C22×C411yescharacteristic
125C2×Q8×C61C22×C411yescharacteristic
126C2×D4×C61—21no—
127C2×D4×C61—22no—
128C2×D4×C61C22×C411yescharacteristic
129C2×D4×C61C22×C411yescharacteristic
130C2×D4×C61C2×D412yes—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2411yescharacteristic
135C2×C22⋊D61—24no—
136C2×C22⋊D61—22no—
137C2×C22⋊D61C22⋊C414yes—
138C2×C22⋊D61C2×D412yes—
139C22×Dic61C22⋊C414yes—
140C22×Dic61C2×D412yes—
141#11758226—44no—
142Q8×D61—44no—
143#11758224—44no—
144#11758224—48no—
145D4×D61—44no—
146D4×D61—48no—
147#11758222—22no—
148#11758222—22no—
149#11758222—42no—
150#11758222—42no—
151#11758222—42no—
152#11758222—42no—
153C2×D244—22no—
154C2×D244—22no—
155C2×D244—22no—
156C2×D244—21no—
157C2×D244—42no—
158C2×D244—42no—
159C2×D244C22×C412yes—
160C2×D244C2×D411yescharacteristic
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61C22⋊C414yes—
164C2×C4×D61C22×C412yes—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—42no—
168C2×Dic122—42no—
169C2×Dic122C22⋊C414yes—
170C2×Dic122C22×C412yes—
171C2×Dic122C2×D411yescharacteristic
172C2×Dic122C2×D412yes—
173M16×C61—22no—
174M16×C61—22no—
175M16×C61—42no—
176C2×C8×C61—22no—
177C2×C8×C61—42no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C42×C61—21no—
184C42×C61—22no—
185C42×C61C22×C411yescharacteristic
186C42×C61C22×C411yescharacteristic
187C22⋊Dic61—21no—
188C22⋊Dic61—24no—
189C22⋊Dic61—22no—
190#11758198—22no—
191#11758198—24no—
192#11758198—24no—
193#11758198—24no—
194C4⋊Dic61—21no—
195C4⋊Dic61—22no—
196#11758196—22no—
197#11758196—24no—
198#11758196—44no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61—44no—
202C4×Dic61C22×C412yes—
203#11758194—22no—
204#11758194—22no—
205#11758194—42no—
206C2×#92418—22no—
207C2×#92418—42no—
208C23×C61—21no—
209C23×C61C2×C22⋊C412yes—
210C23×C61C22×D411yescharacteristic
211C22×D61—22no—
212C22×D61—44no—
213C22×D61—42no—
214C22×D61—84no—
215C22×D61—82no—
216C22×D61C2×C22⋊C412yes—
217Q8×C61—22no—
218Q8×C61—22no—
219D4×C61—24no—
220D4×C61—22no—
221D4×C61—22no—
222D4×C61—42no—
223D4×C61—82no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—42no—
232C2×C4×C61—81no—
233C2×C4×C61C2×C22⋊C412yes—
234C2×C4×C61C23×C411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61H(5,𝔽2)12yes—
237C2×C4×C61H(5,𝔽2)12yes—
238C22⋊D61—24no—
239C22⋊D61—44no—
240C22⋊D61—44no—
241C22⋊D61—84no—
242C2×Dic61—24no—
243C2×Dic61—22no—
244C2×Dic61—44no—
245C2×Dic61—42no—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C2×C22⋊C414yes—
248D244—24no—
249D244—21no—
250D244—21no—
251D244—44no—
252D244—41no—
253D244—41no—
254C4×D61—24no—
255C4×D61—44no—
256C4×D61—44no—
257C4×D61—82no—
258Dic122—24no—
259Dic122—22no—
260Dic122—22no—
261Dic122—21no—
262Dic122—21no—
263Dic122—44no—
264C8×C61—42no—
265#92418—42no—
266#21785—611nomaximal, Sylow
267C22×C61—22no—
268C22×C61—41no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—41no—
272C22×C61C22×C22⋊C411yescharacteristic
273C22×C61#53412yes—
274C2×D61—22no—
275C2×D61—42no—
276C2×D61—44no—
277C2×D61—42no—
278C2×D61—164no—
279C2×D61—162no—
280C4×C61—22no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—82no—
285C4×C61#51812yes—
286C4×C61#53411yescharacteristic
287C4×C61#53411yescharacteristic
288Dic61—24no—
289Dic61—22no—
290Dic61—42no—
291Dic61—82no—
292#3500—611no—
293#2889—611no—
294#2884—611no—
295#1912—612no—
296#1910—611no—
297#1910—611no—
298#1888—612no—
299#1884—611no—
300#1884D6111yescharacteristic, complemented
301#1827—612no—
302#1795—612no—
303C2×C61—21no—
304C2×C61—22no—
305C2×C61—42no—
306C2×C61—81no—
307C2×C61#230811yescharacteristic
308D61—42no—
309D61—82no—
310C2×2−1+4—611no—
311C2×H(5,𝔽2)—611no—
312#568—611no—
313#568C2×D6111yescharacteristic, complemented
314#548—1222no—
315#546—1222no—
316#546—1224no—
317#536—611no—
318#536—612no—
319#535—611no—
320#535—612no—
321#532—612no—
322#532C2×D6111yescharacteristic
323C2×#124—612no—
324C2×#124—611no—
325#519—612no—
326C4×#41—611no—
327#421—612no—
328#421C2×D6112yes—
329#418—612no—
330#418—612no—
331#413—612no—
332#413—1222no—
333C2×#101—612no—
334C2×#101—1222no—
335#410—611no—
336#410—612no—
337#410—1224no—
338#410C2×D6111yescharacteristic, complemented
339#339—612no—
340#339C2×D6112yescomplemented
341H(3,ℤ/4)—612no—
342H(3,ℤ/4)—612no—
343C61#2178511yescharacteristic, complemented, Sylow
3442−1+4—2442no—
3452−1+4—2442no—
346H(5,𝔽2)—2442no—
347H(5,𝔽2)—2442no—
348C2×#41—612no—
349C2×#41—614no—
350C2×#41—612no—
351C2×#41—611no—
352C2×#41—612no—
353C2×#41—1221no—
354C2×#41C22×D6111yescharacteristic
355C2×#41C22×D6111yescharacteristic
356C22×Q8—612no—
357C22×Q8—612no—
358C22×Q8—611no—
359C22×D4—612no—
360C22×D4—611no—
361C22×D4—1221no—
362C22×D4—1222no—
363#131—42no—
364#131—2442no—
365C2×M16—22no—
366C2×M16—612no—
367C2×M16—1222no—
368C2×M16C22×D6112yes—
369C4⋊Q8—612no—
370C4⋊Q8C4×D6111yescharacteristic
371#127—612no—
372#127C4×D6111yescharacteristic
373#124—612no—
374#124—612no—
375#124—612no—
376#124—612no—
377#124—1222no—
378#124—1224no—
379#124C4×D6111yescharacteristic
380#124C4×D6111yescharacteristic
381#123—1224no—
382#123—1222no—
383C22⋊Q8—1224no—
384C22⋊Q8—1222no—
385#121—22no—
386#121—1224no—
387#121—1224no—
388#121—1222no—
389#120—1224no—
390#120—1222no—
391C4×Q8—1221no—
392C4×Q8—2442no—
393C4×Q8—2442no—
394C4×D4—22no—
395C4×D4—1221no—
396C4×D4—1224no—
397C4×D4—1222no—
398#117—611no—
399#117—612no—
400#117—612no—
401#117—1224no—
402#117C22×D6112yes—
403C2×C22⋊C4—1224no—
404C2×C22⋊C4—1222no—
405C2×C42—612no—
406C2×C42C22×D6111yescharacteristic
407#104—42no—
408#104—2442no—
409#102—2442no—
410#102—2442no—
411#101—1224no—
412#101—1222no—
413#100—1222no—
414#99—22no—
415#99—1224no—
416#99—1222no—
417C24—1222no—
418C24—1221no—
419#41—22no—
420#41—22no—
421#41—22no—
422#41—22no—
423#41—1222no—
424#41—1222no—
425#41—2442no—
426#41—2442no—
427#41—2442no—
428#41—2442no—
429#41—2448no—
430#41—2444no—
431#41—2444no—
432C2×Q8—612no—
433C2×Q8—611no—
434C2×Q8—612no—
435C2×Q8—614no—
436C2×Q8—1222no—
437C2×Q8—1222no—
438C2×Q8—2442no—
439C2×Q8—2442no—
440C2×Q8—2444no—
441C2×Q8C2×C4×D6111yescharacteristic
442C2×Q8C2×C4×D6111yescharacteristic
443C2×D4—21no—
444C2×D4—22no—
445C2×D4—612no—
446C2×D4—614no—
447C2×D4—612no—
448C2×D4—611no—
449C2×D4—1222no—
450C2×D4—1222no—
451C2×D4—1224no—
452C2×D4—1221no—
453C2×D4—1222no—
454C2×D4—1222no—
455C2×D4—2442no—
456C2×D4—2442no—
457C2×D4—2444no—
458C2×D4—2448no—
459C2×D4C2×C4×D6111yescharacteristic
460C2×D4C2×C4×D6111yescharacteristic
461C2×D4D4×D6112yes—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—612no—
465C22×C4—614no—
466C22×C4—612no—
467C22×C4—614no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4D4×D6112yes—
471C22×C4C23×D6111yescharacteristic
472M16—22no—
473M16—22no—
474M16—42no—
475M16—1222no—
476M16—1222no—
477M16—2442no—
478C2×C8—22no—
479C2×C8—42no—
480C2×C8—1222no—
481C2×C8—2442no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—1222no—
485C4⋊C4—1224no—
486C4⋊C4—1221no—
487C4⋊C4—1222no—
488C4⋊C4—2444no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—1224no—
493C22⋊C4—1224no—
494C22⋊C4—1222no—
495C22⋊C4—1224no—
496C22⋊C4—1221no—
497C22⋊C4—1224no—
498C22⋊C4—1222no—
499C42—21no—
500C42—22no—
501C42—612no—
502C42—1222no—
503C42—1222no—
504C42—2444no—
505C42C2×C4×D6111yescharacteristic
506C42C2×C4×D6111yescharacteristic
507C23—21no—
508C23—612no—
509C23—1222no—
510C23—4882no—
511C23—4884no—
512C23—2444no—
513C23—2442no—
514C23C22⋊C4×D6112yes—
515C23C2×D4×D6111yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—1224no—
519Q8—1222no—
520Q8—1221no—
521Q8—1222no—
522Q8—1221no—
523Q8—2444no—
524D4—24no—
525D4—22no—
526D4—22no—
527D4—42no—
528D4—82no—
529D4—1224no—
530D4—1224no—
531D4—1221no—
532D4—1221no—
533D4—4884no—
534D4—2444no—
535D4—2444no—
536D4—2444no—
537D4—2441no—
538D4—2441no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—81no—
548C2×C4—612no—
549C2×C4—614no—
550C2×C4—1224no—
551C2×C4—1224no—
552C2×C4—1222no—
553C2×C4—4882no—
554C2×C4—2444no—
555C2×C4—2444no—
556C2×C4—2444no—
557C2×C4—2442no—
558C2×C4C22⋊C4×D6112yes—
559C2×C4C22×C4×D6111yescharacteristic
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4#1451982212yes—
562C2×C4#1451982312yes—
563C8—42no—
564C8—2442no—
565C22—22no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—41no—
570C22—1222no—
571C22—9762no—
572C22—9764no—
573C22—2444no—
574C22—2442no—
575C22—2442no—
576C22C2×C22⋊C4×D6111yescharacteristic
577C22#1684032012yes—
578C4—22no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—82no—
583C4—1222no—
584C4—1224no—
585C4—4882no—
586C4—2442no—
587C4#1684023412yes—
588C4#1684032211yescharacteristic
589C4#1684032311yescharacteristic
590C2—21no—
591C2—22no—
592C2—42no—
593C2—81no—
594C2—4882no—
595C2—2442no—
596C2#2144839311yescharacteristic
subgroupG/H
centerC2#21448393
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1884C2
socleC2×C61#2308
prime radical#21459606C2
cosocleC22×C4×C61C24
hypercenter#1884D61
hypocenterC61#21785
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1884

Complements

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

NH ≅ G/Ntype
2. #21454301C2semidirect
3. #21454281C2semidirect
4. #21446353C2semidirect
5. #21446346C2semidirect
6. #21446344C2semidirect
7. #21446221C2semidirect
8. #21446203C2semidirect
9. #21445922C2semidirect
10. #21445788C2semidirect
11. C61×#1884C2semidirect
14. #16840529C22semidirect
18. #16840337C4semidirect
21. #16840325C4semidirect
22. #16840311C4semidirect
24. #124×D61C4semidirect
25. #16840239C22semidirect
33. C2×#14519695C22semidirect
34. #16839736C22semidirect
37. #16839522C22semidirect
38. #16839519C22semidirect
39. #16839518C22semidirect
40. C61×#568C22semidirect
43. C61×#410C22semidirect
44. C61×#339C22semidirect
300. #1884D61semidirect
313. #568C2×D61semidirect
338. #410C2×D61semidirect
340. #339C2×D61semidirect
343. C61#21785semidirect

Maximal subgroups

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

subgroupindex
#214596062
#214543012
#214542812
#214463532
#214463462
#214463442
#214462212
#214462032
#214459222
#214457882
C61×#18842
#2178561