Finite Group Databasebeta

← #29662788

Subgroups — #29662788

Order 15616.

760 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#21456276C211yesmaximal, characteristic, complemented
2#21455819C211yesmaximal, characteristic
3D4×#11758194C211yesmaximal, characteristic, complemented
4#21445306C212yesmaximal, complemented
5#21445286C212yesmaximal, complemented
6#21445182C212yesmaximal, complemented
7#21445162C212yesmaximal, complemented
8#21444569C211yesmaximal, characteristic, complemented
9#21444494C211yesmaximal, characteristic, complemented
10#21444491C211yesmaximal, characteristic, complemented
11C2×C61×#474C211yesmaximal, characteristic, complemented
12C2×#14519808—21no—
13C2×#14519808—21no—
14C22×#11758194—21no—
15#16840312C2211yescharacteristic
16#16840154—22no—
17#16840141C2212yescomplemented
18#16840136C2212yescomplemented
19#16840075—21no—
20#16840042C2211yescharacteristic
21#16840042C2211yescharacteristic
22#16839957C2211yescharacteristic
23#16839957C2211yescharacteristic
24#16839896—21no—
25C4×#11758199—21no—
26C4×#11758199—21no—
27#16839840C2212yes—
28#16839834—22no—
29D4×#92418C2212yes—
30#16839815C2211yescharacteristic
31#16839786C2211yescharacteristic
32#16839726—21no—
33C4×#11758194C2211yescharacteristic
34#16839514C2212yescomplemented
35#16839512C2212yescomplemented
36#16839508C2212yescomplemented
37#16839508C2212yescomplemented
38#16839507C2212yescomplemented
39#16839505C2212yescomplemented
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×C4×D4×C61C2211yescharacteristic
42C61×#474—21no—
43C61×#474—21no—
44C61×#474C2212yes—
45C61×#474C2212yes—
46C2×#107×C61—21no—
47C2×C4⋊C8×C61C2211yescharacteristic
48C2×#102×C61—21no—
49C2×#11758222—21no—
50#14519808—44no—
51#14519808—44no—
52C2×#11758202—22no—
53C2×#11758202—22no—
54C2×#11758201—22no—
55C2×#11758201—22no—
56C2×#11758199—21no—
57C2×#11758199—21no—
58C2×#11758194—21no—
59C2×#11758194—22no—
60C2×#11758194—44no—
61C2×#11758194D411yescharacteristic
62C2×#11758194D411yescharacteristic
63C22×#92418—22no—
64#14519777C2311yescharacteristic
65#14519774C2311yescharacteristic
66#14519765—21no—
67#14519751—21no—
68#14519751—21no—
69C4×D244—21no—
70C4×D244—21no—
71#14519710—21no—
72#14519710—21no—
73C4⋊#92418—21no—
74C4⋊#92418C2311yescharacteristic
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2312yes—
77#14519698—21no—
78#14519698—21no—
79#14519697—21no—
80#14519697—21no—
81C22⋊#92418—21no—
82C22⋊#92418—22no—
83C22⋊#92418—22no—
84#14519688D412yes—
85C4×#92418—21no—
86C4×#92418C2312yes—
87C22×Q8×C61—21no—
88C22×D4×C61D411yescharacteristic
89C23×C4×C61—21no—
90C22×C8×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61—21no—
93C4⋊Q8×C61—21no—
94C4⋊Q8×C61C2312yes—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61D411yescharacteristic
98C4×D4×C61D411yescharacteristic
99C4×D4×C61C2311yescharacteristic
100C4×D4×C61C2311yescharacteristic
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C42×C61C2311yescharacteristic
106#107×C61—22no—
107#107×C61—22no—
108C4⋊C8×C61—21no—
109C4⋊C8×C61C2312yes—
110#102×C61—22no—
111#102×C61—22no—
112C24×C61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—41no—
115C2×Q8×C61—42no—
116C2×Q8×C61—42no—
117C2×D4×C61—22no—
118C2×D4×C61—21no—
119C2×D4×C61—41no—
120C2×D4×C61C2×D411yescharacteristic
121C2×D4×C61C2×D411yescharacteristic
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—42no—
128C22×C4×C61—42no—
129C22×C4×C61—41no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61#4111yescharacteristic
133C2×C22⋊D61—41no—
134#11758222—42no—
135#11758222—42no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×C4×D61—41no—
140C2×Dic122—21no—
141C2×C8×C61—22no—
142C2×C8×C61—42no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—41no—
147C4⋊C4×C61—41no—
148C4⋊C4×C61C2×D411yescharacteristic
149C4⋊C4×C61C2×D411yescharacteristic
150C22⋊C4×C61—21no—
151C22⋊C4×C61—21no—
152C22⋊C4×C61—21no—
153C22⋊C4×C61—21no—
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C42×C61#4111yescharacteristic
157C42×C61C2411yescharacteristic
158#11758202—44no—
159#11758202—44no—
160#11758201—44no—
161#11758201—44no—
162#11758199—42no—
163#11758199—42no—
164#11758198—41no—
165#11758198—41no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168#11758194—22no—
169#11758194—42no—
170#11758194—44no—
171#11758194D814yescomplemented
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—22no—
175C2×#92418—44no—
176C2×#92418C2×D412yes—
177C2×#92418C2×D412yes—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—81no—
183C23×C61#12111yescharacteristic
184C22×D61—41no—
185Q8×C61—42no—
186Q8×C61—42no—
187Q8×C61—82no—
188D4×C61—21no—
189D4×C61—21no—
190D4×C61—21no—
191D4×C61—42no—
192D4×C61#13712yescomplemented
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—41no—
198C2×C4×C61—42no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—82no—
207C2×C4×C61#12111yescharacteristic
208C2×C4×C61#12111yescharacteristic
209C2×C4×C61#12111yescharacteristic
210C2×C4×C61C2×D812yes—
211C2×C4×C61AGL(1,ℤ/8)12yes—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C22⋊D61—82no—
216C2×Dic61—41no—
217D244—41no—
218D244—41no—
219D244—41no—
220D244—41no—
221D244—81no—
222C4×D61—82no—
223Dic122—41no—
224Dic122—41no—
225C8×C61—42no—
226#92418—22no—
227#92418—42no—
228#92418—41no—
229#92418—41no—
230#92418C2×D814yes—
231#19163—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—22no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61—41no—
238C22×C61—41no—
239C22×C61—41no—
240C22×C61—84no—
241C22×C61#45912yes—
242C22×C61C2×#12111yescharacteristic
243C2×D61—81no—
244C2×D61—81no—
245C2×D61—81no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—41no—
250C4×C61—41no—
251C4×C61—82no—
252C4×C61#45912yes—
253C4×C61C22×D811yescharacteristic
254C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
255C4×C61C2×#13711yescharacteristic
256C4×C61#57511yescharacteristic
257Dic61—81no—
258#3152—611no—
259#3100—611no—
260C2×#474D6111yescharacteristic, complemented
261D4×M16—611no—
262#1689—612no—
263#1685—612no—
264#1669—612no—
265#1665—612no—
266#1565—611no—
267#1547—611no—
268#1546—611no—
269C2×C61—22no—
270C2×C61—42no—
271C2×C61—81no—
272C2×C61C2×#45911yescharacteristic
273C2×C61#305611yescharacteristic
274C2×C61#306911yescharacteristic
275D61—161no—
276C2×#137—1221no—
277C2×#137—1221no—
278C22×M16—1221no—
279#532—611no—
280C2×C4⋊Q8C2×D6111yescharacteristic
281C2×C4×D4C2×D6111yescharacteristic
282#497—1222no—
283#494—612no—
284#492—612no—
285#480—1221no—
286#474—21no—
287#474—21no—
288#474—611no—
289#474—611no—
290#474C2×D6112yes—
291#474C2×D6112yes—
292#459—611no—
293#459—611no—
294#447—1221no—
295C4×D8—1221no—
296C4×D8—1221no—
297#436—612no—
298#435—1222no—
299C8×D4—612no—
300#431—611no—
301C2×#107—21no—
302#423—611no—
303C2×C4⋊C8C2×D6111yescharacteristic
304C2×#102—21no—
305#407—1221no—
306C4×M16—611no—
307#335—612no—
308#334—612no—
309#332—612no—
310#332—612no—
311#332—612no—
312#331—612no—
313C61#1916311yescharacteristic, complemented, Sylow
314C2×#41—1221no—
315C22×Q8—21no—
316C22×D4C22⋊D6111yescharacteristic
317C23×C4—21no—
318#137—2444no—
319#137—2444no—
320C2×Q16—1222no—
321C2×Q16—1222no—
322C2×SD16—1222no—
323C2×SD16—1222no—
324C2×D8—1221no—
325C2×D8—1221no—
326C2×M16—611no—
327C2×M16—611no—
328C2×M16—1222no—
329C2×M16—1221no—
330C2×M16—2444no—
331C22×C8—21no—
332C22×C8—1222no—
333C4⋊Q8—21no—
334C4⋊Q8—21no—
335C4⋊Q8—21no—
336C4⋊Q8—611no—
337C4⋊Q8C22×D6112yes—
338#127—611no—
339#124—1221no—
340#121—1221no—
341#121—1221no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—1221no—
345C4×D4—1221no—
346C4×D4C22⋊D6111yescharacteristic
347C4×D4C22⋊D6111yescharacteristic
348C4×D4C22×D6111yescharacteristic
349C4×D4C22×D6111yescharacteristic
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4C22⋊D6111yescharacteristic
353C2×C22⋊C4—21no—
354C2×C42C22×D6111yescharacteristic
355#107—22no—
356#107—22no—
357#107—1221no—
358#107—1221no—
359C4⋊C8—21no—
360C4⋊C8—612no—
361C4⋊C8—611no—
362C4⋊C8—611no—
363C4⋊C8—1221no—
364C4⋊C8C22×D6112yes—
365#102—22no—
366#102—22no—
367#102—1221no—
368#102—1221no—
369#102—1221no—
370#102—1221no—
371C22⋊C8—1222no—
372C22⋊C8—1222no—
373C22⋊C8—1221no—
374#97—612no—
375C4×C8—612no—
376C4×C8—1221no—
377C24—21no—
378#41—2442no—
379#41—2442no—
380C2×Q8—22no—
381C2×Q8—41no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—1221no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—41no—
388C2×D4—1221no—
389C2×D4—1221no—
390C2×D4—1221no—
391C2×D4—2441no—
392C2×D4C2×C22⋊D6111yescharacteristic
393C2×D4C2×C22⋊D6111yescharacteristic
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—2441no—
403C22×C4D4×D6111yescharacteristic
404C22×C4#1175822411yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406Q16—2444no—
407Q16—2444no—
408SD16—2444no—
409SD16—2444no—
410D8—2442no—
411D8—2442no—
412M16—614no—
413M16—1222no—
414M16—2444no—
415M16—2442no—
416C2×C8—22no—
417C2×C8—42no—
418C2×C8—612no—
419C2×C8—612no—
420C2×C8—1222no—
421C2×C8—1221no—
422C2×C8—1221no—
423C2×C8—2444no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—1221no—
430C4⋊C4—1221no—
431C4⋊C4C2×C22⋊D6111yescharacteristic
432C4⋊C4C2×C22⋊D6111yescharacteristic
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—2441no—
438C22⋊C4—2441no—
439C42D4×D6111yescharacteristic
440C42#1175822411yescharacteristic
441C42C2×C22⋊D6111yescharacteristic
442C42C23×D6111yescharacteristic
443C23—21no—
444C23—21no—
445C23—42no—
446C23—42no—
447C23—81no—
448C23—2441no—
449C23#1451974911yescharacteristic
450Q8—42no—
451Q8—42no—
452Q8—82no—
453Q8—2441no—
454Q8—2441no—
455D4—21no—
456D4—21no—
457D4—21no—
458D4—42no—
459D4—4881no—
460D4—4882no—
461D4—2441no—
462D4—2441no—
463D4—2441no—
464D4—2441no—
465D4#1451980812yescomplemented
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—82no—
480C2×C4—4882no—
481C2×C4—2441no—
482C2×C4#1451974911yescharacteristic
483C2×C4#1451974911yescharacteristic
484C2×C4#1451974911yescharacteristic
485C2×C4D8×D6112yes—
486C2×C4#1451980112yes—
487C2×C4C2×D4×D6111yescharacteristic
488C2×C4C22×C22⋊D6111yescharacteristic
489C2×C4C2×#1175822411yescharacteristic
490C8—42no—
491C8—614no—
492C8—1222no—
493C8—2442no—
494C8—2441no—
495C8—2441no—
496C22—21no—
497C22—21no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—84no—
505C22—4881no—
506C22—4881no—
507C22—4881no—
508C22#1683995512yes—
509C22C2×#1451974911yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—41no—
514C4—41no—
515C4—82no—
516C4—4881no—
517C4#1683995512yes—
518C4C2×D8×D6111yescharacteristic
519C4C2×#1451980111yescharacteristic
520C4C2×#1451980811yescharacteristic
521C4#1684056011yescharacteristic
522C2—22no—
523C2—42no—
524C2—81no—
525C2—9761no—
526C2C2×#1683995511yescharacteristic
527C2#2145544511yescharacteristic
528C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#474C2
socleC22×C61C2×#121
prime radical#21455819C2
hypercenterC2×#474D61
hypocenterC61#19163
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. #21456276C2semidirect
3. D4×#11758194C2semidirect
4. #21445306C2semidirect
5. #21445286C2semidirect
6. #21445182C2semidirect
7. #21445162C2semidirect
8. #21444569C2semidirect
9. #21444494C2semidirect
10. #21444491C2semidirect
11. C2×C61×#474C2semidirect
17. #16840141C22semidirect
18. #16840136C22semidirect
34. #16839514C22semidirect
35. #16839512C22semidirect
36. #16839508C22semidirect
37. #16839508C22semidirect
38. #16839507C22semidirect
39. #16839505C22semidirect
171. #11758194D8semidirect
192. D4×C61#137semidirect
260. C2×#474D61semidirect
313. C61#19163semidirect
465. D4#14519808semidirect

Maximal subgroups

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

subgroupindex
#214562762
#214558192
D4×#117581942
#214453062
#214452862
#214451822
#214451622
#214445692
#214444942
#214444912
C2×C61×#4742
#1916361