Finite Group Databasebeta

← #29656893

Subgroups — #29656893

Order 15616.

944 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456826C211yesmaximal, characteristic, complemented
2#21456538C211yesmaximal, characteristic, complemented
3#21455819C211yesmaximal, characteristic
4#21445068C212yesmaximal, complemented
5#21445033C212yesmaximal, complemented
6#21444922C212yesmaximal, complemented
7#21444891C212yesmaximal, complemented
8#21444529C211yesmaximal, characteristic, complemented
9#21444262C211yesmaximal, characteristic, complemented
10#21444240C211yesmaximal, characteristic, complemented
11C61×#3501C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13#16840165C2211yescharacteristic, complemented
14#16840154—21no—
15#16840136C2211yescharacteristic, complemented
16#16840124—21no—
17#16840112C2211yescharacteristic, complemented
18#16840110C2211yescharacteristic, complemented
19#16840075—21no—
20#16840042C2212yes—
21#16840010—22no—
22#16839972C2212yescomplemented
23#16839962C2212yescomplemented
24#16839957C2212yes—
25#16839922—22no—
26#16839896—21no—
27#16839860—22no—
28C4×#11758199—22no—
29#16839819C2211yescharacteristic
30#16839815C2211yescharacteristic
31C4×#11758194C2211yescharacteristic
32#16839511C2212yescomplemented
33#16839508C2212yescomplemented
34Q8⋊#92418C2212yescomplemented
35Q8⋊#92418C2212yescomplemented
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C61×#554—21no—
39C61×#554—22no—
40C61×#554—22no—
41C61×#554C2212yes—
42D4×Q8×C61—21no—
43D4×Q8×C61C2212yes—
44C61×#544—21no—
45C61×#543—22no—
46C61×#533C2211yescharacteristic
47C61×#533C2211yescharacteristic
48C2×C22⋊Q8×C61—22no—
49C61×#518—21no—
50C2×C4×D4×C61C2211yescharacteristic
51C2×#11758222—21no—
52C2×#11758202—22no—
53C2×#11758201—22no—
54C2×#11758201—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#14519710—22no—
66C4⋊#92418C2312yes—
67C4⋊#92418C2312yes—
68#14519701—22no—
69#14519700—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
81C22×Q8×C61—22no—
82C22×D4×C61D411yescharacteristic
83C23×C4×C61—21no—
84C4⋊Q8×C61—22no—
85C4⋊Q8×C61—22no—
86C4⋊Q8×C61D411yescharacteristic
87C4⋊Q8×C61D411yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89C4⋊Q8×C61C2311yescharacteristic
90#125×C61—22no—
91#125×C61—22no—
92#125×C61—21no—
93#125×C61—21no—
94C22⋊Q8×C61—24no—
95C22⋊Q8×C61—24no—
96C22⋊Q8×C61—24no—
97C22⋊Q8×C61—24no—
98C22⋊Q8×C61—24no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61—24no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61D412yes—
105C22⋊Q8×C61D412yes—
106C4×Q8×C61—21no—
107C4×Q8×C61—21no—
108C4×Q8×C61D412yes—
109C4×Q8×C61C2312yes—
110C4×D4×C61—22no—
111C4×D4×C61—22no—
112C4×D4×C61—22no—
113C4×D4×C61—22no—
114C4×D4×C61—21no—
115C4×D4×C61—21no—
116C4×D4×C61D412yes—
117C4×D4×C61C2312yes—
118#117×C61—21no—
119#117×C61—21no—
120C2×C4⋊C4×C61—24no—
121C2×C4⋊C4×C61—22no—
122C2×C4⋊C4×C61D411yescharacteristic
123C2×C22⋊C4×C61—24no—
124C2×C22⋊C4×C61—21no—
125C2×C42×C61C2311yescharacteristic
126C24×C61—21no—
127C2×Q8×C61—24no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—82no—
131C2×Q8×C61—82no—
132C2×Q8×C61C2×D412yes—
133C2×Q8×C61C2×D412yes—
134C2×D4×C61—24no—
135C2×D4×C61—21no—
136C2×D4×C61C2×D412yes—
137C22×C4×C61—24no—
138C22×C4×C61—24no—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—82no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C2×C22⋊D61—41no—
147#11758222—82no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×C4×D61—41no—
152C2×Dic122—21no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—21no—
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—21no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61C2×D412yes—
168C4⋊C4×C61C2×D412yes—
169C4⋊C4×C61C2×D412yes—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C42×C61—22no—
177C42×C61—22no—
178C42×C61C2×D411yescharacteristic
179C42×C61C2×D411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C42×C61C2411yescharacteristic
182#11758202—82no—
183#11758201—44no—
184#11758201—82no—
185#11758199—44no—
186#11758198—41no—
187#11758198—41no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190#11758194—42no—
191C2×#92418—21no—
192C2×#92418—22no—
193C2×#92418—21no—
194C2×#92418—22no—
195C23×C61—24no—
196C23×C61—22no—
197C23×C61—81no—
198C23×C61—81no—
199C23×C61#12011yescharacteristic
200C22×D61—41no—
201Q8×C61—22no—
202Q8×C61—22no—
203Q8×C61—42no—
204Q8×C61—42no—
205Q8×C61—81no—
206Q8×C61—84no—
207D4×C61—24no—
208D4×C61—22no—
209D4×C61#13714yes—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—41no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—82no—
221C2×C4×C61—81no—
222C2×C4×C61—84no—
223C2×C4×C61—84no—
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61#12011yescharacteristic
226C2×C4×C61#12011yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C22⋊D61—82no—
231C2×Dic61—41no—
232D244—42no—
233D244—42no—
234D244—42no—
235C4×D61—82no—
236Dic122—42no—
237#92418—22no—
238#92418—41no—
239#92418—42no—
240#92418—42no—
241#18600—611nomaximal, Sylow
242C22×C61—21no—
243C22×C61—24no—
244C22×C61—21no—
245C22×C61—21no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—84no—
249C22×C61C2×#12011yescharacteristic
250C2×D61—81no—
251C2×D61—81no—
252C2×D61—81no—
253C4×C61—21no—
254C4×C61—21no—
255C4×C61—21no—
256C4×C61—21no—
257C4×C61—42no—
258C4×C61—42no—
259C4×C61—42no—
260C4×C61—42no—
261C4×C61—42no—
262C4×C61C2×#13712yes—
263C4×C61#57612yes—
264Dic61—81no—
265#3501D6111yescharacteristic, complemented
266#3216—611no—
267#3182—611no—
268#3100—611no—
269#1649—612no—
270#1644—612no—
271#1627—612no—
272#1622—612no—
273#1554—611no—
274#1490—611no—
275#1485—611no—
276C2×C61—41no—
277C2×C61—44no—
278C2×C61#300411yescharacteristic
279C2×C61#300811yescharacteristic
280C2×C61#301811yescharacteristic
281D61—161no—
282#554—21no—
283#554—22no—
284#554—22no—
285#554C2×D6112yes—
286D4×Q8—21no—
287D4×Q8C2×D6112yes—
288#544—21no—
289#543—22no—
290#533C2×D6111yescharacteristic
291#533C2×D6111yescharacteristic
292#532—611no—
293C2×C22⋊Q8—22no—
294#518—21no—
295C2×C4×D4C2×D6111yescharacteristic
296#500—611no—
297#497—1221no—
298#492—611no—
299#489—1221no—
300#487—611no—
301#486—611no—
302#480—1221no—
303#474—612no—
304#468—1222no—
305#462—612no—
306#460—612no—
307#459—612no—
308#452—1222no—
309#447—1221no—
310#440—1222no—
311C4×D8—1222no—
312#432—611no—
313#431—611no—
314C4×M16—611no—
315#333—612no—
316#332—612no—
317Q8⋊C8—612no—
318Q8⋊C8—612no—
319D4⋊C8—612no—
320D4⋊C8—612no—
321C61#1860011yescharacteristic, complemented, Sylow
322C2×#41—1221no—
323C22×Q8—22no—
324C22×D4C22⋊D6111yescharacteristic
325C23×C4—21no—
326C2×Q16—1222no—
327C2×SD16—1222no—
328C2×SD16—1222no—
329C2×D8—1222no—
330C2×M16—1221no—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333C4⋊Q8—611no—
334C4⋊Q8C22⋊D6111yescharacteristic
335C4⋊Q8C22⋊D6111yescharacteristic
336C4⋊Q8C22×D6111yescharacteristic
337C4⋊Q8C22×D6111yescharacteristic
338#127—611no—
339#125—22no—
340#125—22no—
341#125—21no—
342#125—21no—
343#124—1221no—
344C22⋊Q8—24no—
345C22⋊Q8—24no—
346C22⋊Q8—24no—
347C22⋊Q8—24no—
348C22⋊Q8—22no—
349C22⋊Q8—24no—
350C22⋊Q8—21no—
351C22⋊Q8—21no—
352C22⋊Q8—22no—
353C22⋊Q8—24no—
354C22⋊Q8C22⋊D6112yes—
355C22⋊Q8C22⋊D6112yes—
356#121—1221no—
357#121—1221no—
358C4×Q8—21no—
359C4×Q8—21no—
360C4×Q8C22⋊D6112yes—
361C4×Q8C22×D6112yes—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4—1221no—
369C4×D4—1221no—
370C4×D4C22⋊D6112yes—
371C4×D4C22×D6112yes—
372#117—21no—
373#117—21no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4C22⋊D6111yescharacteristic
377C2×C22⋊C4—24no—
378C2×C22⋊C4—21no—
379C2×C42C22×D6111yescharacteristic
380#107—1222no—
381#107—1222no—
382C4⋊C8—612no—
383C4⋊C8—612no—
384#103—1222no—
385#103—1222no—
386#102—1222no—
387#102—1222no—
388#102—1222no—
389#102—1222no—
390C22⋊C8—1221no—
391C22⋊C8—1221no—
392#97—1221no—
393#97—1221no—
394C4×C8—611no—
395C4×C8—611no—
396C4×C8—1221no—
397C24—21no—
398#41—4882no—
399C2×Q8—24no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—82no—
403C2×Q8—82no—
404C2×Q8—1221no—
405C2×Q8C2×C22⋊D6112yes—
406C2×Q8C2×C22⋊D6112yes—
407C2×D4—24no—
408C2×D4—21no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—2441no—
413C2×D4C2×C22⋊D6112yes—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—82no—
420C22×C4—2441no—
421C22×C4C2×C22⋊D6111yescharacteristic
422C22×C4C2×C22⋊D6111yescharacteristic
423C22×C4C2×C22⋊D6111yescharacteristic
424Q16—4882no—
425SD16—4882no—
426SD16—2444no—
427D8—2444no—
428M16—2442no—
429C2×C8—1221no—
430C2×C8—1221no—
431C2×C8—1222no—
432C2×C8—1222no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—1221no—
448C4⋊C4—1221no—
449C4⋊C4C2×C22⋊D6112yes—
450C4⋊C4C2×C22⋊D6112yes—
451C4⋊C4C2×C22⋊D6112yes—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—2441no—
459C22⋊C4—2441no—
460C42—22no—
461C42—22no—
462C42C2×C22⋊D6111yescharacteristic
463C42C2×C22⋊D6111yescharacteristic
464C42C2×C22⋊D6111yescharacteristic
465C42C23×D6111yescharacteristic
466C23—24no—
467C23—22no—
468C23—81no—
469C23—81no—
470C23—2441no—
471C23#1451974611yescharacteristic
472Q8—22no—
473Q8—22no—
474Q8—42no—
475Q8—42no—
476Q8—81no—
477Q8—84no—
478Q8—2442no—
479D4—24no—
480D4—22no—
481D4—4882no—
482D4—2442no—
483D4—2442no—
484D4—2442no—
485D4#1451980814yes—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—84no—
499C2×C4—84no—
500C2×C4—4882no—
501C2×C4—2441no—
502C2×C4#1451974611yescharacteristic
503C2×C4#1451974611yescharacteristic
504C2×C4#1451974611yescharacteristic
505C2×C4C22×C22⋊D6111yescharacteristic
506C2×C4C22×C22⋊D6111yescharacteristic
507C2×C4C22×C22⋊D6111yescharacteristic
508C8—1222no—
509C8—2441no—
510C8—2442no—
511C8—2442no—
512C22—21no—
513C22—21no—
514C22—24no—
515C22—21no—
516C22—42no—
517C22—42no—
518C22—84no—
519C22—4881no—
520C22—4881no—
521C22—4881no—
522C22C2×#1451974611yescharacteristic
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—4881no—
533C4C2×#1451980812yes—
534C4#1684056512yes—
535C2—41no—
536C2—44no—
537C2—9761no—
538C2#2145505311yescharacteristic
539C2#2145508711yescharacteristic
540C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radical#21455819C2
hypercenter#3501D61
hypocenterC61#18600
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21456826C2semidirect
2. #21456538C2semidirect
4. #21445068C2semidirect
5. #21445033C2semidirect
6. #21444922C2semidirect
7. #21444891C2semidirect
8. #21444529C2semidirect
9. #21444262C2semidirect
10. #21444240C2semidirect
11. C61×#3501C2semidirect
13. #16840165C22semidirect
15. #16840136C22semidirect
17. #16840112C22semidirect
18. #16840110C22semidirect
22. #16839972C22semidirect
23. #16839962C22semidirect
32. #16839511C22semidirect
33. #16839508C22semidirect
34. Q8⋊#92418C22semidirect
35. Q8⋊#92418C22semidirect
265. #3501D61semidirect
321. C61#18600semidirect

Maximal subgroups

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

subgroupindex
#214568262
#214565382
#214558192
#214450682
#214450332
#214449222
#214448912
#214445292
#214442622
#214442402
C61×#35012
#1860061