Finite Group Databasebeta

← #29689167

Subgroups — #29689167

Order 15616.

1318 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459459C211yesmaximal, characteristic
2#21454280C211yesmaximal, characteristic, complemented
3#21446361C212yesmaximal, complemented
4#21446359C212yesmaximal, complemented
5#21446232C212yesmaximal, complemented
6#21446230C212yesmaximal, complemented
7#21445925C212yesmaximal, complemented
8#21445735C212yesmaximal, complemented
9C61×#2883C211yesmaximal, characteristic, complemented
10C22×#11758224C2211yescharacteristic
11#16840411—22no—
12#16840411—22no—
13#16840406—22no—
14#16840406—22no—
15#16840333C412yescomplemented
16#16840321C412yescomplemented
17#16840321C2211yescharacteristic
18#16840321C2211yescharacteristic
19#16840269C412yescomplemented
20C2×#14519758C412yescomplemented
21C2×#14519758C2211yescharacteristic
22C2×#14519758C2211yescharacteristic
23#16840237C2212yes—
24#16840237C2211yescharacteristic
25#16839736—22no—
26C2×#14519692—22no—
27C2×#14519692C2212yes—
28C2×#14519692C2212yes—
29C2×#14519691C2214yescomplemented
30#16839623C2214yescomplemented
31#16839529C2214yescomplemented
32#16839528C2214yescomplemented
33C2×H(5,𝔽2)×C61C2211yescharacteristic
34C61×#518C2212yes—
35C61×#410—22no—
36C2×#99×C61—22no—
37C2×#99×C61C2212yes—
38C2×#99×C61C2212yes—
39C2×#11758224—22no—
40C2×#11758224—22no—
41C2×#11758224—44no—
42C22×Dic122C2×C411yescharacteristic, complemented
43C22×C22⋊D61C2×C411yescharacteristic, complemented
44C22×C22⋊D61C2311yescharacteristic
45C23×Dic61C2×C411yescharacteristic, complemented
46C23×Dic61C2311yescharacteristic
47C22×C4×D61C2×C411yescharacteristic, complemented
48#14519775—22no—
49#14519764—22no—
50#14519760—22no—
51#14519760—21no—
52#14519760—21no—
53#14519758—22no—
54#14519758—22no—
55#14519758—22no—
56#14519758—22no—
57#14519758C2×C412yescomplemented
58#14519758C2×C412yescomplemented
59#14519758C2×C412yescomplemented
60#14519758C2×C412yescomplemented
61#14519758C2×C412yescomplemented
62#14519758C2×C412yescomplemented
63#14519756—22no—
64#14519756—22no—
65#14519753—22no—
66#14519749—22no—
67#14519749—22no—
68#14519749—22no—
69#14519749—22no—
70#14519748—22no—
71#14519746—22no—
72#14519746—21no—
73#14519746—21no—
74#14519745C2×C412yescomplemented
75#14519745C2×C412yescomplemented
76D4×Dic61—22no—
77D4×Dic61—22no—
78D4×Dic61—44no—
79C4×C22⋊D61—22no—
80C4×C22⋊D61—22no—
81#14519736—22no—
82#14519736C2×C412yescomplemented
83C2×#11758196—22no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×C22⋊Dic61—22no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—22no—
90C2×C22⋊Dic61C2×C412yescomplemented
91C2×C22⋊Dic61C2314yes—
92C2×C22⋊Dic61C2311yescharacteristic
93C2×C22⋊Dic61C2311yescharacteristic
94#14519692—24no—
95#14519692—24no—
96#14519691—44no—
97#14519684—44no—
98H(5,𝔽2)×C61—44no—
99C2×#41×C61C2×C412yes—
100C2×#41×C61D412yes—
101C2×#41×C61D412yes—
102C22×D4×C61C2×C412yes—
103C22×D4×C61D412yes—
104C22×D4×C61D412yes—
105C22×D4×C61C2311yescharacteristic
106C22×D4×C61C2311yescharacteristic
107C22×D4×C61C2311yescharacteristic
108C4×D4×C61—44no—
109#117×C61—22no—
110C2×C22⋊C4×C61—22no—
111C2×C22⋊C4×C61C2314yes—
112#99×C61—24no—
113#99×C61—24no—
114C24×C61C22⋊C412yes—
115C24×C61C22×C411yescharacteristic
116C24×C61C2×D411yescharacteristic
117C24×C61C2×D411yescharacteristic
118C24×C61C2411yescharacteristic
119C23×D61C22×C411yescharacteristic
120#41×C61—44no—
121#41×C61—44no—
122#41×C61—44no—
123C2×Q8×C61C22⋊C412yes—
124C2×D4×C61—24no—
125C2×D4×C61—24no—
126C2×D4×C61—24no—
127C2×D4×C61—44no—
128C2×D4×C61—44no—
129C2×D4×C61—44no—
130C2×D4×C61C22⋊C412yes—
131C2×D4×C61C22⋊C412yes—
132C2×D4×C61C22⋊C412yes—
133C2×D4×C61C22×C412yes—
134C2×D4×C61C22×C412yes—
135C2×D4×C61C2×D412yes—
136C2×D4×C61C2×D412yes—
137C2×D4×C61C2×D412yes—
138C2×D4×C61C2×D412yes—
139C22×C4×C61—24no—
140C22×C4×C61—42no—
141C22×C4×C61C22⋊C412yes—
142C22×C4×C61C22⋊C412yes—
143C22×C4×C61C22⋊C412yes—
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C2×C22⋊D61—22no—
148C2×C22⋊D61—21no—
149C2×C22⋊D61—21no—
150C2×C22⋊D61—24no—
151C2×C22⋊D61—22no—
152C2×C22⋊D61—22no—
153C2×C22⋊D61—42no—
154C22×Dic61—22no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—24no—
158C22×Dic61—22no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—24no—
162C22×Dic61—42no—
163C22×Dic61—42no—
164C22×Dic61C22×C411yescharacteristic
165C22×Dic61C22×C411yescharacteristic
166C22×Dic61C22×C411yescharacteristic
167#11758224—48no—
168#11758224—48no—
169C2×C4×D61—22no—
170C2×C4×D61—42no—
171C2×Dic122—22no—
172C2×Dic122—42no—
173C4⋊C4×C61—42no—
174C22⋊C4×C61—24no—
175C22⋊C4×C61—24no—
176C22⋊C4×C61—42no—
177C42×C61—42no—
178C22⋊Dic61—22no—
179C22⋊Dic61—22no—
180C22⋊Dic61—22no—
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183C22⋊Dic61—22no—
184C22⋊Dic61—22no—
185C22⋊Dic61—21no—
186C22⋊Dic61—21no—
187C22⋊Dic61—24no—
188C22⋊Dic61—24no—
189C22⋊Dic61—42no—
190C22⋊Dic61C22×C412yes—
191C22⋊Dic61C22×C412yes—
192#11758198—22no—
193#11758198—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61—42no—
196#11758196—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202C4×Dic61—22no—
203C4×Dic61—42no—
204C23×C61—22no—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—42no—
208C23×C61—81no—
209C23×C61—81no—
210C23×C61—81no—
211C23×C61—81no—
212C23×C61—82no—
213C23×C61C2×C22⋊C412yes—
214C23×C61C2×C22⋊C412yes—
215C23×C61C2×C22⋊C412yes—
216C23×C61C23×C411yescharacteristic
217C23×C61C22×D411yescharacteristic
218C23×C61C22×D411yescharacteristic
219C23×C61H(5,𝔽2)14yes—
220C22×D61—21no—
221C22×D61—22no—
222C22×D61—81no—
223Q8×C61—42no—
224D4×C61—24no—
225D4×C61—24no—
226D4×C61—24no—
227D4×C61—24no—
228D4×C61—24no—
229D4×C61—24no—
230D4×C61—42no—
231D4×C61—42no—
232D4×C61—42no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—44no—
240C2×C4×C61—84no—
241C2×C4×C61—162no—
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C2×C22⋊C412yes—
244C2×C4×C61C2×C22⋊C412yes—
245C22⋊D61—44no—
246C22⋊D61—42no—
247C22⋊D61—42no—
248C22⋊D61—82no—
249C22⋊D61—82no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—44no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—44no—
262C2×Dic61—81no—
263C2×Dic61—81no—
264C2×Dic61—82no—
265C2×Dic61—84no—
266C2×Dic61—162no—
267C4×D61—44no—
268Dic122—44no—
269#21777—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—44no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—82no—
281C22×C61—82no—
282C22×C61—82no—
283C22×C61—162no—
284C22×C61C22×C22⋊C411yescharacteristic
285C22×C61#51812yes—
286C22×C61#53412yes—
287C22×C61#53412yes—
288C2×D61—42no—
289C2×D61—41no—
290C2×D61—82no—
291C4×C61—22no—
292C4×C61—22no—
293C4×C61—22no—
294C4×C61—84no—
295Dic61—42no—
296Dic61—42no—
297Dic61—41no—
298Dic61—41no—
299Dic61—41no—
300Dic61—84no—
301#3485—611no—
302#2883—611no—
303#2883D6111yescharacteristic, complemented
304#1913—612no—
305#1913—612no—
306#1890—612no—
307#1890—612no—
308#1828—612no—
309#1782—612no—
310C2×C61—21no—
311C2×C61—22no—
312C2×C61—22no—
313C2×C61—22no—
314C2×C61—42no—
315C2×C61—42no—
316C2×C61—42no—
317C2×C61#230811yescharacteristic
318D61—81no—
319C2×H(5,𝔽2)C2×D6111yescharacteristic
320C22×#41—611no—
321#547—1222no—
322#547—1222no—
323#546—1222no—
324#546—1222no—
325#536—612no—
326#534—612no—
327#534—611no—
328#534—611no—
329#525—612no—
330C2×#123—612no—
331C2×#123—611no—
332C2×#123—611no—
333#518—611no—
334#518—612no—
335#518C2×D6112yes—
336#410—22no—
337#410—1222no—
338C2×#99—22no—
339C2×#99—612no—
340C2×#99—612no—
341C2×#99—614no—
342C2×#99—1222no—
343C2×#99C2×D6112yes—
344C2×#99C2×D6112yes—
345#381—614no—
346#342—614no—
347#342—614no—
348C61#2177711yescharacteristic, complemented, Sylow
349H(5,𝔽2)—44no—
350C2×#41—1222no—
351C2×#41—1222no—
352C2×#41—2444no—
353C2×#41C4×D6112yes—
354C2×#41D24412yes—
355C2×#41C22⋊D6112yes—
356C22×Q8—611no—
357C22×D4—611no—
358C22×D4—611no—
359C22×D4C4×D6112yes—
360C22×D4D24412yes—
361C22×D4C22⋊D6112yes—
362C22×D4C22×D6111yescharacteristic
363C22×D4C22×D6111yescharacteristic
364C22×D4C22×D6111yescharacteristic
365C23×C4—611no—
366C23×C4—611no—
367C23×C4—611no—
368C4⋊Q8—1222no—
369#124—1222no—
370#123—612no—
371#123—612no—
372#123—612no—
373#123—612no—
374#123—612no—
375#123—612no—
376#123—1222no—
377#123—1222no—
378#123—1222no—
379#123—1222no—
380#123—1222no—
381#123—1221no—
382#123—1221no—
383C22⋊Q8—1222no—
384C22⋊Q8—1222no—
385C22⋊Q8—1222no—
386#121—1222no—
387#121—1222no—
388#121—1222no—
389#121—1222no—
390#121—1222no—
391#120—612no—
392#120—612no—
393#120—1222no—
394#120—1221no—
395#120—1221no—
396C4×D4—44no—
397C4×D4—1222no—
398C4×D4—1222no—
399C4×D4—1222no—
400C4×D4—1222no—
401C4×D4—2444no—
402#117—22no—
403#117—612no—
404#117—1222no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—612no—
410C2×C22⋊C4—611no—
411C2×C22⋊C4—611no—
412C2×C22⋊C4—614no—
413C2×C22⋊C4—1222no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1222no—
417C2×C22⋊C4C22×D6114yes—
418#99—24no—
419#99—24no—
420#99—1224no—
421#99—1224no—
422#99—2444no—
423#95—2444no—
424C24—611no—
425C24#1175819812yes—
426C24C2×C4×D6111yescharacteristic
427C24C2×D24411yescharacteristic
428C24C2×C22⋊D6111yescharacteristic
429C24C23×D6111yescharacteristic
430#41—44no—
431#41—44no—
432#41—44no—
433#41—2448no—
434#41—2448no—
435C2×Q8—1222no—
436C2×Q8—2442no—
437C2×Q8#1175819812yes—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—1222no—
445C2×D4—1222no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1224no—
449C2×D4—1222no—
450C2×D4—2442no—
451C2×D4#1175819812yes—
452C2×D4#1175819812yes—
453C2×D4#1175819812yes—
454C2×D4C2×C4×D6112yes—
455C2×D4C2×C4×D6112yes—
456C2×D4C2×D24412yes—
457C2×D4C2×D24412yes—
458C2×D4C2×C22⋊D6112yes—
459C2×D4C2×C22⋊D6112yes—
460C22×C4—24no—
461C22×C4—42no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1222no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1224no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1224no—
474C22×C4—2442no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4#1175819812yes—
478C22×C4#1175819812yes—
479C22×C4#1175819812yes—
480C22×C4C2×C4×D6111yescharacteristic
481C22×C4C2×D24411yescharacteristic
482C22×C4C2×C22⋊D6111yescharacteristic
483C4⋊C4—42no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—1222no—
491C4⋊C4—2442no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—42no—
495C22⋊C4—612no—
496C22⋊C4—612no—
497C22⋊C4—1221no—
498C22⋊C4—1221no—
499C22⋊C4—1222no—
500C22⋊C4—1222no—
501C22⋊C4—1222no—
502C22⋊C4—1222no—
503C22⋊C4—1222no—
504C22⋊C4—1222no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—1224no—
509C22⋊C4—1224no—
510C22⋊C4—2442no—
511C42—42no—
512C42—1222no—
513C42—2442no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—42no—
518C23—81no—
519C23—81no—
520C23—81no—
521C23—81no—
522C23—82no—
523C23—1222no—
524C23—1221no—
525C23—4881no—
526C23C2×#1175819812yes—
527C23C2×#1175819812yes—
528C23C2×#1175819812yes—
529C23C22×C4×D6111yescharacteristic
530C23C22×C22⋊D6111yescharacteristic
531C23C22×D24411yescharacteristic
532C23#1451982214yes—
533Q8—42no—
534Q8—2444no—
535D4—24no—
536D4—24no—
537D4—24no—
538D4—24no—
539D4—24no—
540D4—24no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—4882no—
545D4—4882no—
546D4—2444no—
547D4—2442no—
548D4—2442no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—84no—
557C2×C4—162no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—9762no—
567C2×C4—4884no—
568C2×C4—4881no—
569C2×C4—4881no—
570C2×C4—4882no—
571C2×C4—2444no—
572C2×C4—2444no—
573C2×C4—2444no—
574C2×C4—2442no—
575C2×C4—2442no—
576C2×C4C2×#1175819812yes—
577C2×C4C2×#1175819812yes—
578C2×C4C2×#1175819812yes—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—44no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—162no—
593C22—4882no—
594C22—2442no—
595C22—2441no—
596C22C22×#1175819811yescharacteristic
597C22#1684023512yes—
598C22#1684032112yes—
599C22#1684032412yes—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—84no—
604C4—4884no—
605C4—2442no—
606C4—2442no—
607C4—2441no—
608C4—2441no—
609C4—2441no—
610C2—22no—
611C2—22no—
612C2—21no—
613C2—22no—
614C2—42no—
615C2—42no—
616C2—42no—
617C2—4881no—
618C2#2144839411yescharacteristic
subgroupG/H
centerC2#21448394
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#2883C2
socleC2×C61#2308
prime radical#21459459C2
cosocleC24×C61C24
hypercenter#2883D61
hypocenterC61#21777
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C24 ◁ #2883

Complements

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

NH ≅ G/Ntype
2. #21454280C2semidirect
3. #21446361C2semidirect
4. #21446359C2semidirect
5. #21446232C2semidirect
6. #21446230C2semidirect
7. #21445925C2semidirect
8. #21445735C2semidirect
9. C61×#2883C2semidirect
15. #16840333C4semidirect
16. #16840321C4semidirect
19. #16840269C4semidirect
20. C2×#14519758C4semidirect
29. C2×#14519691C22semidirect
30. #16839623C22semidirect
31. #16839529C22semidirect
32. #16839528C22semidirect
42. C22×Dic122C2×C4semidirect
43. C22×C22⋊D61C2×C4semidirect
45. C23×Dic61C2×C4semidirect
47. C22×C4×D61C2×C4semidirect
57. #14519758C2×C4semidirect
58. #14519758C2×C4semidirect
59. #14519758C2×C4semidirect
60. #14519758C2×C4semidirect
61. #14519758C2×C4semidirect
62. #14519758C2×C4semidirect
74. #14519745C2×C4semidirect
75. #14519745C2×C4semidirect
82. #14519736C2×C4semidirect
90. C2×C22⋊Dic61C2×C4semidirect
303. #2883D61semidirect
348. C61#21777semidirect

Maximal subgroups

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

subgroupindex
#214594592
#214542802
#214463612
#214463592
#214462322
#214462302
#214459252
#214457352
C61×#28832
#2177761