Finite Group Databasebeta

← #29662819

Subgroups — #29662819

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#21455823C211yesmaximal, characteristic, complemented
2C2×#16839954C211yesmaximal, characteristic
3M16×D244C211yesmaximal, characteristic, complemented
4#21445377C212yesmaximal, complemented
5#21445349C212yesmaximal, complemented
6#21445253C212yesmaximal, complemented
7#21445225C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444509C211yesmaximal, characteristic, complemented
10#21444487C211yesmaximal, characteristic, complemented
11C61×#3138C211yesmaximal, characteristic, complemented
12C22×#11758199—21no—
13C2×M16×D61—21no—
14C2×C4×D244C2211yescharacteristic
15#16840090—21no—
16#16840049C2211yescharacteristic, complemented
17#16840049C2211yescharacteristic, complemented
18#16839964C2211yescharacteristic, complemented
19#16839964C2211yescharacteristic, complemented
20#16839954—21no—
21#16839954—21no—
22#16839954C2212yes—
23#16839954C2212yes—
24#16839912—21no—
25C4×#11758201—21no—
26C4×#11758201—21no—
27#16839841C2212yescomplemented
28#16839835—22no—
29C8×D244C2212yescomplemented
30#16839815C2211yescharacteristic
31#16839786C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33C2×#14519698—21no—
34#16839725—21no—
35#16839515C2212yescomplemented
36#16839513C2212yescomplemented
37#16839511C2212yescomplemented
38#16839508C2212yescomplemented
39#16839506C2212yescomplemented
40#16839506C2212yescomplemented
41C61×#533C2211yescharacteristic
42C2×#127×C61C2211yescharacteristic
43C61×#490—22no—
44C61×#486C2212yescomplemented
45C61×#486C2212yescomplemented
46C61×#418—21no—
47C61×#418—21no—
48C4×M16×C61C2211yescharacteristic
49C22×D244D411yescharacteristic
50C22×C4×D61—21no—
51C2×#11758201—21no—
52C2×#11758201—21no—
53C2×#11758199—22no—
54C2×#11758199—22no—
55C2×#11758199—44no—
56C2×#11758194—21no—
57C2×#11758189—22no—
58M16×D61—44no—
59C22×#92418—21no—
60C2×C8×D61—22no—
61C4×D244—21no—
62C4×D244—21no—
63C4×D244D411yescharacteristic
64C4×D244D411yescharacteristic
65C4×D244C2311yescharacteristic
66C4×D244C2311yescharacteristic
67C2×C4⋊Dic61D411yescharacteristic
68C2×#11758198—21no—
69#14519708—21no—
70#14519708—21no—
71C4⋊#92418—21no—
72C4⋊#92418—21no—
73C4⋊#92418C2312yes—
74C4⋊#92418C2311yescharacteristic
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2312yes—
77#14519701—21no—
78#14519701—21no—
79#14519698—22no—
80#14519698—22no—
81#14519697—21no—
82#14519697—21no—
83C22⋊#92418—21no—
84#14519689—22no—
85#14519689—22no—
86C4×#92418—21no—
87C22×D4×C61—21no—
88C22×D4×C61—21no—
89C22×D4×C61—21no—
90C2×M16×C61D411yescharacteristic
91C2×M16×C61D411yescharacteristic
92C4⋊Q8×C61C2311yescharacteristic
93C4⋊Q8×C61C2311yescharacteristic
94#127×C61—21no—
95#127×C61—21no—
96#127×C61—21no—
97#127×C61C2312yes—
98#125×C61—21no—
99C22⋊Q8×C61—21no—
100C22⋊Q8×C61—21no—
101C4×Q8×C61—21no—
102C4×Q8×C61—21no—
103#117×C61—21no—
104C2×C42×C61C2311yescharacteristic
105#102×C61—22no—
106#102×C61—22no—
107#102×C61—22no—
108#102×C61—22no—
109#97×C61D412yes—
110C4×C8×C61C2312yes—
111C24×C61—41no—
112C23×D61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—42no—
119C2×D4×C61—42no—
120C2×D4×C61—41no—
121C2×D4×C61—41no—
122C2×D4×C61—41no—
123C2×D4×C61—42no—
124C2×D4×C61—42no—
125C2×D4×C61—82no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61#4111yescharacteristic
129C22×Dic61—21no—
130C2×D244—22no—
131C2×D244—21no—
132C2×D244—41no—
133C2×D244C2×D411yescharacteristic
134C2×D244C2×D411yescharacteristic
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—42no—
140C2×C4×D61—42no—
141M16×C61—22no—
142M16×C61—22no—
143C2×C8×C61C2×D412yes—
144C2×C8×C61C2×D412yes—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—41no—
150C4⋊C4×C61—41no—
151C22⋊C4×C61—41no—
152C42×C61—41no—
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61#4111yescharacteristic
156C42×C61C2411yescharacteristic
157#11758201—42no—
158#11758201—42no—
159#11758199—44no—
160#11758199—44no—
161#11758198—21no—
162#11758198—21no—
163#11758198—21no—
164#11758198—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61C2×D411yescharacteristic
167C4⋊Dic61C2×D411yescharacteristic
168#11758194—42no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—42no—
174#11758189—44no—
175C8×D61—44no—
176C23×C61—42no—
177C23×C61—82no—
178C23×C61—82no—
179C23×C61—82no—
180C23×C61#12111yescharacteristic
181C22×D61—21no—
182C22×D61—21no—
183C22×D61—42no—
184C22×D61—42no—
185C22×D61—81no—
186Q8×C61—41no—
187Q8×C61—41no—
188Q8×C61—81no—
189D4×C61—42no—
190D4×C61—42no—
191D4×C61—42no—
192D4×C61—42no—
193D4×C61—82no—
194D4×C61—82no—
195D4×C61—82no—
196D4×C61—82no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61#12111yescharacteristic
203C2×C4×C61#12111yescharacteristic
204C2×C4×C61#12111yescharacteristic
205C2×C4×C61#13512yes—
206C2×C4×C61AGL(1,ℤ/8)12yes—
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—42no—
213D244—21no—
214D244—21no—
215D244—21no—
216D244—42no—
217D244AGL(1,ℤ/8)12yes—
218C4×D61—41no—
219C4×D61—41no—
220C4×D61—41no—
221C4×D61—41no—
222C4×D61—42no—
223C4×D61—42no—
224C8×C61—22no—
225C8×C61—22no—
226#92418—42no—
227#92418—42no—
228#92418—41no—
229#92418—41no—
230#19166—611nomaximal, Sylow
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—82no—
234C22×C61—82no—
235C22×C61—82no—
236C22×C61—164no—
237C22×C61#46312yes—
238C22×C61C2×#12111yescharacteristic
239C2×D61—22no—
240C2×D61—41no—
241C2×D61—41no—
242C2×D61—41no—
243C2×D61—41no—
244C2×D61—41no—
245C2×D61—84no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—81no—
250C4×C61—81no—
251C4×C61#46312yes—
252C4×C61C2×#13511yescharacteristic
253C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
255C4×C61#57511yescharacteristic
256Dic61—41no—
257Dic61—41no—
258#3138D6111yescharacteristic, complemented
259#3101—611no—
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—162no—
271C2×C61C2×#46311yescharacteristic
272C2×C61#305511yescharacteristic
273C2×C61#306811yescharacteristic
274D61—42no—
275D61—81no—
276C22×D8—1221no—
277C22×M16—1221no—
278#533C2×D6111yescharacteristic
279C2×#127C2×D6111yescharacteristic
280C2×C4×D4—611no—
281#490—22no—
282#486C2×D6112yescomplemented
283#486C2×D6112yescomplemented
284#483—1221no—
285#476—611no—
286#476—611no—
287#461—611no—
288#461—611no—
289#459—612no—
290#459—612no—
291#459—1221no—
292#459—1221no—
293#450—1221no—
294C4×SD16—1221no—
295C4×SD16—1221no—
296#436—612no—
297#435—1222no—
298C8×D4—612no—
299#431—611no—
300#423—611no—
301C2×C4⋊C8—611no—
302#418—21no—
303#418—21no—
304C2×#102—1221no—
305#407—1221no—
306C4×M16C2×D6111yescharacteristic
307#335—612no—
308#334—612no—
309#333—612no—
310#332—612no—
311#331—612no—
312#331—612no—
313C61#1916611yescharacteristic, complemented, Sylow
314C22×D4—21no—
315C22×D4—21no—
316C22×D4—21no—
317C22×D4—611no—
318C23×C4—1221no—
319C2×SD16—1221no—
320C2×SD16—1221no—
321C2×D8—1222no—
322C2×D8—1222no—
323C2×D8—2444no—
324C2×M16—1221no—
325C2×M16—1222no—
326C2×M16—2444no—
327C2×M16C22⋊D6111yescharacteristic
328C2×M16C22⋊D6111yescharacteristic
329C22×C8—1221no—
330C22×C8—1222no—
331C4⋊Q8C22×D6111yescharacteristic
332C4⋊Q8C22×D6111yescharacteristic
333#127—21no—
334#127—21no—
335#127—21no—
336#127C22×D6112yes—
337#125—21no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×D4—611no—
343C4×D4—611no—
344C4×D4—611no—
345C4×D4—611no—
346C4×D4—1221no—
347C4×D4—1221no—
348#117—21no—
349C2×C4⋊C4—611no—
350C2×C22⋊C4—1221no—
351C2×C42C22×D6111yescharacteristic
352#106—1221no—
353#106—1221no—
354C4⋊C8—612no—
355C4⋊C8—611no—
356C4⋊C8—611no—
357C4⋊C8—612no—
358C4⋊C8—1221no—
359C4⋊C8—1221no—
360#103—1221no—
361#103—1221no—
362#102—22no—
363#102—22no—
364#102—22no—
365#102—22no—
366#102—1221no—
367#102—1221no—
368#102—1222no—
369#102—1222no—
370C22⋊C8—1221no—
371C22⋊C8—1222no—
372C22⋊C8—1222no—
373#97C22⋊D6112yes—
374C4×C8—1221no—
375C4×C8C22×D6112yes—
376C24—41no—
377C24—1221no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—82no—
391C2×D4—611no—
392C2×D4—611no—
393C2×D4—1222no—
394C2×D4—1221no—
395C2×D4—2441no—
396C22×C4—1221no—
397C22×C4—1221no—
398C22×C4—1221no—
399C22×C4—1221no—
400C22×C4—1221no—
401C22×C4—2442no—
402C22×C4—2442no—
403C22×C4D4×D6111yescharacteristic
404C22×C4#1175822411yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406SD16—2442no—
407SD16—2442no—
408D8—2444no—
409D8—2444no—
410M16—22no—
411M16—22no—
412M16—2444no—
413M16—2442no—
414C2×C8—1221no—
415C2×C8—1221no—
416C2×C8—1222no—
417C2×C8—1222no—
418C2×C8—2444no—
419C2×C8—2442no—
420C2×C8C2×C22⋊D6112yes—
421C2×C8C2×C22⋊D6112yes—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—611no—
429C4⋊C4—611no—
430C4⋊C4—1221no—
431C22⋊C4—41no—
432C22⋊C4—1221no—
433C22⋊C4—1221no—
434C22⋊C4—1221no—
435C22⋊C4—1221no—
436C42—41no—
437C42D4×D6111yescharacteristic
438C42#1175822411yescharacteristic
439C42C2×C22⋊D6111yescharacteristic
440C42C23×D6111yescharacteristic
441C23—42no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—1221no—
446C23—1221no—
447C23—4881no—
448C23—2442no—
449C23—2442no—
450C23#1451974811yescharacteristic
451Q8—41no—
452Q8—41no—
453Q8—81no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—82no—
459D4—82no—
460D4—82no—
461D4—82no—
462D4—612no—
463D4—1221no—
464D4—1221no—
465D4—1221no—
466D4—2442no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—1221no—
473C2×C4—1221no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4#1451974811yescharacteristic
482C2×C4#1451974811yescharacteristic
483C2×C4#1451974811yescharacteristic
484C2×C4#1451979712yes—
485C2×C4#1451980212yes—
486C2×C4C2×D4×D6111yescharacteristic
487C2×C4C22×C22⋊D6111yescharacteristic
488C2×C4C2×#1175822411yescharacteristic
489C8—22no—
490C8—22no—
491C8—2442no—
492C8—2441no—
493C8—2442no—
494C8—2441no—
495C22—21no—
496C22—21no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—164no—
501C22—1222no—
502C22—4884no—
503C22—2441no—
504C22—2441no—
505C22—2441no—
506C22—2441no—
507C22—2441no—
508C22#1683997412yes—
509C22C2×#1451974811yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—81no—
514C4—81no—
515C4—2441no—
516C4—2441no—
517C4#1683997412yes—
518C4C2×#1451979711yescharacteristic
519C4AGL(1,ℤ/8)×D6111yescharacteristic
520C4C2×#1451980211yescharacteristic
521C4#1684055811yescharacteristic
522C2—22no—
523C2—162no—
524C2—4881no—
525C2—2442no—
526C2C2×#1683997411yescharacteristic
527C2#2145543211yescharacteristic
528C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3138C2
socleC22×C61C2×#121
prime radicalC2×#16839954C2
hypercenter#3138D61
hypocenterC61#19166
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
1. #21455823C2semidirect
3. M16×D244C2semidirect
4. #21445377C2semidirect
5. #21445349C2semidirect
6. #21445253C2semidirect
7. #21445225C2semidirect
8. #21444571C2semidirect
9. #21444509C2semidirect
10. #21444487C2semidirect
11. C61×#3138C2semidirect
16. #16840049C22semidirect
17. #16840049C22semidirect
18. #16839964C22semidirect
19. #16839964C22semidirect
27. #16839841C22semidirect
29. C8×D244C22semidirect
35. #16839515C22semidirect
36. #16839513C22semidirect
37. #16839511C22semidirect
38. #16839508C22semidirect
39. #16839506C22semidirect
40. #16839506C22semidirect
44. C61×#486C22semidirect
45. C61×#486C22semidirect
258. #3138D61semidirect
282. #486C2×D61semidirect
283. #486C2×D61semidirect
313. C61#19166semidirect

Maximal subgroups

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

subgroupindex
#214558232
C2×#168399542
M16×D2442
#214453772
#214453492
#214452532
#214452252
#214445712
#214445092
#214444872
C61×#31382
#1916661