Finite Group Databasebeta

← #29662827

Subgroups — #29662827

Order 15616.

758 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456195C211yesmaximal, characteristic, complemented
2C2×#16839957C211yesmaximal, characteristic
3D4×#11758194C211yesmaximal, characteristic, complemented
4#21445382C212yesmaximal, complemented
5#21445354C212yesmaximal, complemented
6#21445258C212yesmaximal, complemented
7#21445230C212yesmaximal, complemented
8#21444572C211yesmaximal, characteristic, complemented
9#21444509C211yesmaximal, characteristic, complemented
10#21444486C211yesmaximal, characteristic, complemented
11C61×#3101C211yesmaximal, characteristic, complemented
12C22×#11758199—21no—
13C22×#11758194—21no—
14C2×#14519774C2211yescharacteristic
15#16840128—22no—
16#16840110C2212yescomplemented
17#16840110C2212yescomplemented
18#16839957—21no—
19#16839957—21no—
20#16839957C2212yes—
21#16839957C2212yes—
22#16839840C2212yescomplemented
23#16839834—22no—
24D4×#92418C2212yescomplemented
25#16839786C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27#16839766—21no—
28#16839766—21no—
29C2×#14519697—21no—
30#16839726—21no—
31C4×#11758194C2211yescharacteristic
32#16839514C2212yescomplemented
33#16839512C2212yescomplemented
34#16839511C2212yescomplemented
35#16839508C2212yescomplemented
36#16839505C2212yescomplemented
37#16839505C2212yescomplemented
38C61×#533C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C61×#483—21no—
41C61×#476C2211yescharacteristic, complemented
42C61×#476C2211yescharacteristic, complemented
43C61×#461C2211yescharacteristic, complemented
44C61×#461C2211yescharacteristic, complemented
45C61×#450—21no—
46C4×SD16×C61—21no—
47C4×SD16×C61—21no—
48C61×#431C2211yescharacteristic
49C22×D244—21no—
50C22×D244—21no—
51C22×D244—21no—
52C2×#11758199—22no—
53C2×#11758199—22no—
54C2×#11758199—44no—
55C2×#11758194—21no—
56C2×#11758194—22no—
57C2×#11758194—44no—
58C2×#11758194D411yescharacteristic
59C2×#11758194D411yescharacteristic
60C22×#92418—21no—
61C22×#92418—22no—
62#14519774—21no—
63#14519774—21no—
64#14519774—21no—
65#14519774C2312yes—
66C4⋊#92418—21no—
67C4⋊#92418—21no—
68C4⋊#92418C2312yes—
69C4⋊#92418C2312yes—
70#14519697—22no—
71#14519697—22no—
72#14519697—22no—
73#14519697—22no—
74#14519697—22no—
75#14519697—22no—
76C22⋊#92418—22no—
77C22⋊#92418—22no—
78#14519688D412yes—
79C4×#92418C2312yes—
80C22×D4×C61D411yescharacteristic
81C23×C4×C61—21no—
82C2×SD16×C61—21no—
83C2×SD16×C61—21no—
84C4⋊Q8×C61C2311yescharacteristic
85C4⋊Q8×C61C2311yescharacteristic
86#125×C61—21no—
87C22⋊Q8×C61—21no—
88C22⋊Q8×C61—21no—
89C4×Q8×C61—21no—
90C4×Q8×C61—21no—
91C4×D4×C61—21no—
92C4×D4×C61—21no—
93C4×D4×C61D411yescharacteristic
94C4×D4×C61D411yescharacteristic
95C4×D4×C61C2311yescharacteristic
96C4×D4×C61C2311yescharacteristic
97#117×C61—21no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C22⋊C4×C61—21no—
100C2×C42×C61C2311yescharacteristic
101#106×C61—21no—
102#106×C61—21no—
103C4⋊C8×C61C2311yescharacteristic
104C4⋊C8×C61C2311yescharacteristic
105#103×C61—21no—
106#103×C61—21no—
107#102×C61—21no—
108#102×C61—21no—
109C22⋊C8×C61—21no—
110C4×C8×C61—21no—
111C24×C61—21no—
112C23×D61—41no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×D4×C61—22no—
116C2×D4×C61—21no—
117C2×D4×C61—41no—
118C2×D4×C61C2×D411yescharacteristic
119C2×D4×C61C2×D411yescharacteristic
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61#4111yescharacteristic
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—42no—
134C2×D244—42no—
135C2×D244—42no—
136C2×D244—42no—
137C2×D244—41no—
138C2×D244—41no—
139C2×D244—41no—
140C2×D244—82no—
141SD16×C61—42no—
142SD16×C61—42no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—41no—
151C4⋊C4×C61—41no—
152C4⋊C4×C61C2×D411yescharacteristic
153C4⋊C4×C61C2×D411yescharacteristic
154C22⋊C4×C61—21no—
155C22⋊C4×C61—21no—
156C22⋊C4×C61—21no—
157C22⋊C4×C61—21no—
158C22⋊C4×C61—41no—
159C42×C61—41no—
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162C42×C61#4111yescharacteristic
163C42×C61C2411yescharacteristic
164#11758199—44no—
165#11758199—44no—
166#11758194—22no—
167#11758194—22no—
168#11758194—44no—
169#11758194—42no—
170C2×#92418—22no—
171C2×#92418—22no—
172C2×#92418—44no—
173C2×#92418—42no—
174C2×#92418C2×D412yes—
175C2×#92418C2×D412yes—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—81no—
181C23×C61#12111yescharacteristic
182C22×D61—42no—
183C22×D61—82no—
184C22×D61—82no—
185C22×D61—82no—
186Q8×C61—41no—
187Q8×C61—41no—
188Q8×C61—81no—
189D4×C61—21no—
190D4×C61—21no—
191D4×C61—21no—
192D4×C61—42no—
193D4×C61AGL(1,ℤ/8)12yes—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—41no—
199C2×C4×C61—42no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61#12111yescharacteristic
209C2×C4×C61#12111yescharacteristic
210C2×C4×C61#12111yescharacteristic
211C2×C4×C61#13512yes—
212C2×C4×C61AGL(1,ℤ/8)12yes—
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216D244—42no—
217D244—42no—
218D244—42no—
219D244—42no—
220D244—82no—
221D244—82no—
222D244—82no—
223D244—82no—
224C8×C61—41no—
225C8×C61—41no—
226#92418—22no—
227#92418—22no—
228#92418—42no—
229#92418—42no—
230#19166—611nomaximal, Sylow
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—22no—
234C22×C61—41no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61—41no—
238C22×C61—41no—
239C22×C61—84no—
240C22×C61#46312yes—
241C22×C61C2×#12111yescharacteristic
242C2×D61—82no—
243C2×D61—82no—
244C2×D61—82no—
245C2×D61—164no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—41no—
250C4×C61—41no—
251C4×C61—81no—
252C4×C61—81no—
253C4×C61#46312yes—
254C4×C61C2×#13511yescharacteristic
255C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
256C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
257C4×C61#57511yescharacteristic
258#3138—611no—
259#3101D6111yescharacteristic, complemented
260C2×#459—611no—
261D4×M16—611no—
262#1701—612no—
263#1697—612no—
264#1681—612no—
265#1677—612no—
266#1565—611no—
267#1550—611no—
268#1545—611no—
269C2×C61—22no—
270C2×C61—42no—
271C2×C61—81no—
272C2×C61C2×#46311yescharacteristic
273C2×C61#305511yescharacteristic
274C2×C61#306811yescharacteristic
275D61—162no—
276C22×D8—1221no—
277C22×M16—1221no—
278#533C2×D6111yescharacteristic
279C2×#127—611no—
280C2×C4×D4C2×D6111yescharacteristic
281#490—1222no—
282#486—612no—
283#486—612no—
284#483—21no—
285#476C2×D6111yescharacteristic, complemented
286#476C2×D6111yescharacteristic, complemented
287#461C2×D6111yescharacteristic, complemented
288#461C2×D6111yescharacteristic, complemented
289#459—612no—
290#459—612no—
291#459—1221no—
292#459—1221no—
293#450—21no—
294C4×SD16—21no—
295C4×SD16—21no—
296#436—612no—
297#435—1222no—
298C8×D4—612no—
299#431C2×D6111yescharacteristic
300#423—611no—
301C2×C4⋊C8—611no—
302#418—1221no—
303#418—1221no—
304C2×#102—1221no—
305#407—1221no—
306C4×M16—611no—
307#335—612no—
308#334—612no—
309#333—612no—
310#332—612no—
311#331—612no—
312#331—612no—
313C61#1916611yescharacteristic, complemented, Sylow
314C22×D4—1221no—
315C22×D4—1221no—
316C22×D4—1221no—
317C22×D4C22⋊D6111yescharacteristic
318C23×C4—21no—
319C2×SD16—21no—
320C2×SD16—21no—
321C2×D8—1222no—
322C2×D8—1222no—
323C2×D8—2444no—
324C2×M16—611no—
325C2×M16—611no—
326C2×M16—1222no—
327C2×M16—1221no—
328C2×M16—2444no—
329C22×C8—1222no—
330C22×C8—1221no—
331C4⋊Q8C22×D6111yescharacteristic
332C4⋊Q8C22×D6111yescharacteristic
333#127—612no—
334#127—1221no—
335#127—1221no—
336#127—1221no—
337#125—21no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4C22⋊D6111yescharacteristic
345C4×D4C22⋊D6111yescharacteristic
346C4×D4C22×D6111yescharacteristic
347C4×D4C22×D6111yescharacteristic
348#117—21no—
349C2×C4⋊C4C22⋊D6111yescharacteristic
350C2×C22⋊C4—21no—
351C2×C42C22×D6111yescharacteristic
352#106—21no—
353#106—21no—
354C4⋊C8—612no—
355C4⋊C8—612no—
356C4⋊C8—1221no—
357C4⋊C8—1221no—
358C4⋊C8C22×D6111yescharacteristic
359C4⋊C8C22×D6111yescharacteristic
360#103—21no—
361#103—21no—
362#102—21no—
363#102—21no—
364#102—1222no—
365#102—1222no—
366#102—1222no—
367#102—1222no—
368#102—1222no—
369#102—1222no—
370C22⋊C8—21no—
371C22⋊C8—1222no—
372C22⋊C8—1222no—
373#97—612no—
374C4×C8—21no—
375C4×C8—612no—
376C24—21no—
377C24—2441no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—41no—
383C2×D4—1222no—
384C2×D4—1222no—
385C2×D4—1222no—
386C2×D4—4882no—
387C2×D4—2442no—
388C2×D4—2442no—
389C2×D4—2441no—
390C2×D4—2441no—
391C2×D4—2441no—
392C2×D4—2442no—
393C2×D4—2442no—
394C2×D4C2×C22⋊D6111yescharacteristic
395C2×D4C2×C22⋊D6111yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4D4×D6111yescharacteristic
404C22×C4#1175822411yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406SD16—42no—
407SD16—42no—
408D8—2444no—
409D8—2444no—
410M16—1222no—
411M16—1222no—
412M16—2444no—
413M16—2442no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—612no—
417C2×C8—612no—
418C2×C8—1222no—
419C2×C8—1222no—
420C2×C8—2444no—
421C2×C8—2442no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4C2×C22⋊D6111yescharacteristic
430C4⋊C4C2×C22⋊D6111yescharacteristic
431C22⋊C4—21no—
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—41no—
436C42—41no—
437C42D4×D6111yescharacteristic
438C42#1175822411yescharacteristic
439C42C2×C22⋊D6111yescharacteristic
440C42C23×D6111yescharacteristic
441C23—21no—
442C23—21no—
443C23—42no—
444C23—42no—
445C23—81no—
446C23—4882no—
447C23—4882no—
448C23—4882no—
449C23—2442no—
450C23#1451974911yescharacteristic
451Q8—41no—
452Q8—41no—
453Q8—81no—
454D4—21no—
455D4—21no—
456D4—21no—
457D4—42no—
458D4—4882no—
459D4—4882no—
460D4—4882no—
461D4—4882no—
462D4—2442no—
463D4—2442no—
464D4—2442no—
465D4—2442no—
466D4#1451980212yes—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4#1451974911yescharacteristic
482C2×C4#1451974911yescharacteristic
483C2×C4#1451974911yescharacteristic
484C2×C4#1451979812yes—
485C2×C4#1451980412yes—
486C2×C4C2×D4×D6111yescharacteristic
487C2×C4C22×C22⋊D6111yescharacteristic
488C2×C4C2×#1175822411yescharacteristic
489C8—41no—
490C8—41no—
491C8—1222no—
492C8—1222no—
493C8—2442no—
494C8—2442no—
495C22—21no—
496C22—21no—
497C22—22no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—84no—
504C22—9764no—
505C22—4882no—
506C22—4882no—
507C22—4882no—
508C22#1683997912yes—
509C22C2×#1451974911yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—41no—
514C4—41no—
515C4—81no—
516C4—81no—
517C4#1683997912yes—
518C4C2×#1451979811yescharacteristic
519C4C2×#1451980211yescharacteristic
520C4C2×#1451980411yescharacteristic
521C4#1684056011yescharacteristic
522C2—22no—
523C2—42no—
524C2—81no—
525C2—9762no—
526C2C2×#1683997911yescharacteristic
527C2#2145544011yescharacteristic
528C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3101C2
socleC22×C61C2×#121
prime radicalC2×#16839957C2
hypercenter#3101D61
hypocenterC61#19166
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
1. #21456195C2semidirect
3. D4×#11758194C2semidirect
4. #21445382C2semidirect
5. #21445354C2semidirect
6. #21445258C2semidirect
7. #21445230C2semidirect
8. #21444572C2semidirect
9. #21444509C2semidirect
10. #21444486C2semidirect
11. C61×#3101C2semidirect
16. #16840110C22semidirect
17. #16840110C22semidirect
22. #16839840C22semidirect
24. D4×#92418C22semidirect
32. #16839514C22semidirect
33. #16839512C22semidirect
34. #16839511C22semidirect
35. #16839508C22semidirect
36. #16839505C22semidirect
37. #16839505C22semidirect
41. C61×#476C22semidirect
42. C61×#476C22semidirect
43. C61×#461C22semidirect
44. C61×#461C22semidirect
259. #3101D61semidirect
285. #476C2×D61semidirect
286. #476C2×D61semidirect
287. #461C2×D61semidirect
288. #461C2×D61semidirect
313. C61#19166semidirect

Maximal subgroups

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

subgroupindex
#214561952
C2×#168399572
D4×#117581942
#214453822
#214453542
#214452582
#214452302
#214445722
#214445092
#214444862
C61×#31012
#1916661