Finite Group Databasebeta

← #29657063

Subgroups — #29657063

Order 15616.

964 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456826C211yesmaximal, characteristic, complemented
2#21456553C211yesmaximal, characteristic, complemented
3#21455819C211yesmaximal, characteristic, complemented
4#21445089C212yesmaximal, complemented
5#21445012C212yesmaximal, complemented
6#21444929C212yesmaximal, complemented
7#21444884C212yesmaximal, complemented
8#21444524C211yesmaximal, characteristic, complemented
9#21444262C211yesmaximal, characteristic, complemented
10#21444240C211yesmaximal, characteristic, complemented
11C61×#3506C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13#16840158C2211yescharacteristic, complemented
14#16840154—21no—
15#16840138C2211yescharacteristic, complemented
16#16840124—21no—
17#16840110C2211yescharacteristic, complemented
18#16840108C2211yescharacteristic, complemented
19#16840075—21no—
20#16840042C2212yes—
21#16840013—22no—
22#16839972C2212yescomplemented
23#16839962C2212yescomplemented
24#16839957C2212yes—
25#16839922—22no—
26#16839896—21no—
27#16839872—22no—
28C4×#11758199—22no—
29#16839819C2211yescharacteristic
30#16839815C2211yescharacteristic
31C4×#11758194C2211yescharacteristic
32#16839508C2212yescomplemented
33#16839506C2212yescomplemented
34Q8⋊#92418C2212yescomplemented
35Q8⋊#92418C2212yescomplemented
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C61×#559—22no—
39C61×#556C2212yes—
40C61×#555—21no—
41C61×#553—22no—
42C61×#550C2212yes—
43C61×#547—21no—
44C61×#538—22no—
45C61×#535—21no—
46C61×#533C2211yescharacteristic
47C61×#532C2211yescharacteristic
48C61×#525—22no—
49C61×#520—21no—
50C2×C4×D4×C61C2211yescharacteristic
51C2×#11758222—21no—
52C2×#11758202—22no—
53C2×#11758201—22no—
54C2×#11758199—22no—
55C2×#11758199—22no—
56C2×#11758194—21no—
57#14519777C2311yescharacteristic
58#14519774C2311yescharacteristic
59#14519765—21no—
60#14519751—21no—
61#14519751—21no—
62C4×D244—21no—
63C4×D244—21no—
64#14519710—22no—
65#14519708—22no—
66C4⋊#92418C2312yes—
67C4⋊#92418C2312yes—
68#14519700—22no—
69#14519698—22no—
70#14519698—22no—
71#14519697—22no—
72#14519697—22no—
73#14519697—22no—
74C22⋊#92418—21no—
75C22⋊#92418—21no—
76#14519688—21no—
77#14519688—21no—
78C4×#92418—21no—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C2×#41×C61—22no—
82C2×#41×C61—21no—
83C22×D4×C61D411yescharacteristic
84C23×C4×C61—21no—
85C4⋊Q8×C61D411yescharacteristic
86C4⋊Q8×C61D411yescharacteristic
87C4⋊Q8×C61C2311yescharacteristic
88#127×C61—22no—
89#127×C61C2311yescharacteristic
90#126×C61—24no—
91#125×C61—21no—
92#124×C61—24no—
93#124×C61—22no—
94#124×C61—21no—
95#123×C61—24no—
96#123×C61—24no—
97C22⋊Q8×C61—22no—
98C22⋊Q8×C61—21no—
99C22⋊Q8×C61D412yes—
100#121×C61—22no—
101#121×C61—22no—
102#121×C61—24no—
103#121×C61—21no—
104#121×C61—22no—
105#121×C61D412yes—
106#120×C61—24no—
107C4×Q8×C61—22no—
108C4×Q8×C61—21no—
109C4×Q8×C61C2312yes—
110C4×D4×C61—24no—
111C4×D4×C61—24no—
112C4×D4×C61—22no—
113C4×D4×C61—21no—
114C4×D4×C61—21no—
115C4×D4×C61—21no—
116C4×D4×C61D412yes—
117C4×D4×C61D412yes—
118C4×D4×C61C2312yes—
119#117×C61—22no—
120#117×C61—21no—
121#117×C61—22no—
122C2×C4⋊C4×C61—22no—
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C22⋊C4×C61—21no—
125C2×C42×C61C2311yescharacteristic
126C24×C61—21no—
127#41×C61—44no—
128#41×C61—82no—
129#41×C61—82no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61C2×D412yes—
133C2×D4×C61—22no—
134C2×D4×C61—24no—
135C2×D4×C61—24no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—42no—
140C2×D4×C61C2×D412yes—
141C2×D4×C61C2×D412yes—
142C22×C4×C61—24no—
143C22×C4×C61—21no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61—82no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61—41no—
152#11758222—82no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×C4×D61—41no—
157C2×Dic122—21no—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61C2×D412yes—
167C4⋊C4×C61C2×D412yes—
168C4⋊C4×C61C2×D412yes—
169C22⋊C4×C61—24no—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C42×C61—24no—
178C42×C61—22no—
179C42×C61C2×D411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2411yescharacteristic
183#11758202—82no—
184#11758201—82no—
185#11758199—44no—
186#11758199—44no—
187#11758198—41no—
188#11758198—41no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191#11758194—42no—
192C2×#92418—21no—
193C2×#92418—22no—
194C2×#92418—21no—
195C2×#92418—22no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—42no—
199C23×C61—81no—
200C23×C61—81no—
201C23×C61#12011yescharacteristic
202C22×D61—41no—
203Q8×C61—22no—
204Q8×C61—22no—
205Q8×C61—42no—
206Q8×C61—81no—
207D4×C61—22no—
208D4×C61—22no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61—44no—
212D4×C61—42no—
213D4×C61—42no—
214D4×C61—84no—
215D4×C61—84no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—41no—
223C2×C4×C61—44no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—82no—
227C2×C4×C61—81no—
228C2×C4×C61—84no—
229C2×C4×C61#12011yescharacteristic
230C2×C4×C61#12011yescharacteristic
231C2×C4×C61#12011yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C22⋊D61—82no—
236C2×Dic61—41no—
237D244—42no—
238D244—42no—
239D244—42no—
240C4×D61—82no—
241Dic122—42no—
242#92418—22no—
243#92418—41no—
244#92418—42no—
245#92418—42no—
246#18614—611nomaximal, Sylow
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—84no—
257C22×C61C2×#12011yescharacteristic
258C2×D61—81no—
259C2×D61—81no—
260C2×D61—81no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—42no—
266C4×C61—42no—
267C4×C61—42no—
268C4×C61—42no—
269C4×C61#57512yes—
270C4×C61#57612yes—
271Dic61—81no—
272#3506D6111yescharacteristic, complemented
273#3216—611no—
274#3184—611no—
275#3100—611no—
276#1652—612no—
277#1641—612no—
278#1628—612no—
279#1621—612no—
280#1553—611no—
281#1490—611no—
282#1485—611no—
283C2×C61—41no—
284C2×C61—82no—
285C2×C61—82no—
286C2×C61#300511yescharacteristic
287C2×C61#300811yescharacteristic
288C2×C61#301811yescharacteristic
289D61—161no—
290#559—22no—
291#556C2×D6112yes—
292#555—21no—
293#553—22no—
294#550C2×D6112yes—
295#547—21no—
296#538—22no—
297#535—21no—
298#533C2×D6111yescharacteristic
299#532—611no—
300#532C2×D6111yescharacteristic
301#525—22no—
302#520—21no—
303C2×C4×D4C2×D6111yescharacteristic
304#498—611no—
305#497—1221no—
306#493—611no—
307#489—1221no—
308#486—611no—
309#486—611no—
310#480—1221no—
311#474—612no—
312#469—1222no—
313#462—612no—
314#460—612no—
315#459—612no—
316#452—1222no—
317#447—1221no—
318#442—1222no—
319C4×D8—1222no—
320#432—611no—
321#431—611no—
322C4×M16—611no—
323#332—612no—
324#331—612no—
325Q8⋊C8—612no—
326Q8⋊C8—612no—
327D4⋊C8—612no—
328D4⋊C8—612no—
329C61#1861411yescharacteristic, complemented, Sylow
330C2×#41—22no—
331C2×#41—21no—
332C2×#41—1221no—
333C22×D4C22⋊D6111yescharacteristic
334C23×C4—21no—
335C2×Q16—1222no—
336C2×SD16—1222no—
337C2×D8—1222no—
338C2×D8—1222no—
339C2×M16—1221no—
340C4⋊Q8—611no—
341C4⋊Q8C22⋊D6111yescharacteristic
342C4⋊Q8C22⋊D6111yescharacteristic
343C4⋊Q8C22×D6111yescharacteristic
344#127—22no—
345#127—611no—
346#127C22×D6111yescharacteristic
347#126—24no—
348#125—21no—
349#124—24no—
350#124—22no—
351#124—21no—
352#124—1221no—
353#123—24no—
354#123—24no—
355C22⋊Q8—22no—
356C22⋊Q8—21no—
357C22⋊Q8C22⋊D6112yes—
358#121—22no—
359#121—22no—
360#121—24no—
361#121—21no—
362#121—22no—
363#121—1221no—
364#121—1221no—
365#121C22⋊D6112yes—
366#120—24no—
367C4×Q8—22no—
368C4×Q8—21no—
369C4×Q8C22×D6112yes—
370C4×D4—24no—
371C4×D4—24no—
372C4×D4—22no—
373C4×D4—21no—
374C4×D4—21no—
375C4×D4—21no—
376C4×D4—1221no—
377C4×D4—1221no—
378C4×D4C22⋊D6112yes—
379C4×D4C22⋊D6112yes—
380C4×D4C22×D6112yes—
381#117—22no—
382#117—21no—
383#117—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4C22⋊D6111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C42C22×D6111yescharacteristic
388#107—1222no—
389#106—1222no—
390C4⋊C8—612no—
391C4⋊C8—612no—
392#103—1222no—
393#102—1222no—
394#102—1222no—
395#102—1222no—
396#102—1222no—
397#102—1222no—
398C22⋊C8—1221no—
399C22⋊C8—1221no—
400#97—1221no—
401#97—1221no—
402C4×C8—611no—
403C4×C8—611no—
404C4×C8—1221no—
405C24—21no—
406#41—44no—
407#41—82no—
408#41—82no—
409#41—4882no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—1221no—
413C2×Q8C2×C22⋊D6112yes—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—1221no—
422C2×D4—1221no—
423C2×D4—1221no—
424C2×D4—2441no—
425C2×D4C2×C22⋊D6112yes—
426C2×D4C2×C22⋊D6112yes—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—82no—
433C22×C4—2441no—
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C2×C22⋊D6111yescharacteristic
437Q16—4882no—
438SD16—4882no—
439D8—2444no—
440D8—2444no—
441M16—2442no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1222no—
445C2×C8—1222no—
446C4⋊C4—24no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4C2×C22⋊D6112yes—
457C4⋊C4C2×C22⋊D6112yes—
458C4⋊C4C2×C22⋊D6112yes—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—2441no—
468C22⋊C4—2441no—
469C42—24no—
470C42—22no—
471C42C2×C22⋊D6111yescharacteristic
472C42C2×C22⋊D6111yescharacteristic
473C42C2×C22⋊D6111yescharacteristic
474C42C23×D6111yescharacteristic
475C23—22no—
476C23—22no—
477C23—42no—
478C23—81no—
479C23—81no—
480C23—2441no—
481C23#1451974611yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—42no—
485Q8—81no—
486Q8—2442no—
487D4—22no—
488D4—22no—
489D4—42no—
490D4—42no—
491D4—44no—
492D4—42no—
493D4—42no—
494D4—84no—
495D4—84no—
496D4—4882no—
497D4—2442no—
498D4—2442no—
499D4—2442no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—84no—
513C2×C4—4882no—
514C2×C4—2441no—
515C2×C4#1451974611yescharacteristic
516C2×C4#1451974611yescharacteristic
517C2×C4#1451974611yescharacteristic
518C2×C4C22×C22⋊D6111yescharacteristic
519C2×C4C22×C22⋊D6111yescharacteristic
520C2×C4C22×C22⋊D6111yescharacteristic
521C8—1222no—
522C8—2441no—
523C8—2442no—
524C8—2442no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—84no—
535C22—4881no—
536C22—4881no—
537C22—4881no—
538C22C2×#1451974611yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—4881no—
548C4#1684056012yes—
549C4#1684056512yes—
550C2—41no—
551C2—82no—
552C2—82no—
553C2—9761no—
554C2#2145506111yescharacteristic
555C2#2145508711yescharacteristic
556C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3506C2
socleC22×C61C2×#120
hypercenter#3506D61
hypocenterC61#18614
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #21456826C2semidirect
2. #21456553C2semidirect
3. #21455819C2semidirect
4. #21445089C2semidirect
5. #21445012C2semidirect
6. #21444929C2semidirect
7. #21444884C2semidirect
8. #21444524C2semidirect
9. #21444262C2semidirect
10. #21444240C2semidirect
11. C61×#3506C2semidirect
13. #16840158C22semidirect
15. #16840138C22semidirect
17. #16840110C22semidirect
18. #16840108C22semidirect
22. #16839972C22semidirect
23. #16839962C22semidirect
32. #16839508C22semidirect
33. #16839506C22semidirect
34. Q8⋊#92418C22semidirect
35. Q8⋊#92418C22semidirect
272. #3506D61semidirect
329. C61#18614semidirect

Maximal subgroups

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

subgroupindex
#214568262
#214565532
#214558192
#214450892
#214450122
#214449292
#214448842
#214445242
#214442622
#214442402
C61×#35062
#1861461