Finite Group Databasebeta

← #29662863

Subgroups — #29662863

Order 15616.

766 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455846C211yesmaximal, characteristic, complemented
2C2×#16839959C211yesmaximal, characteristic, complemented
3M16×Dic122C211yesmaximal, characteristic, complemented
4#21445300C212yesmaximal, complemented
5#21445288C212yesmaximal, complemented
6#21445176C212yesmaximal, complemented
7#21445164C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444496C211yesmaximal, characteristic, complemented
10#21444487C211yesmaximal, characteristic, complemented
11C61×#3151C211yesmaximal, characteristic
12C22×#11758200—21no—
13C2×C4×Dic122C2211yescharacteristic
14#16840078—21no—
15#16840044C2211yescharacteristic, complemented
16#16840044C2211yescharacteristic, complemented
17#16839959—21no—
18#16839959—21no—
19#16839959C2212yes—
20#16839959C2212yes—
21#16839959C2211yescharacteristic, complemented
22#16839959C2211yescharacteristic, complemented
23#16839900—21no—
24#16839892C2212yescomplemented
25#16839890—22no—
26C8×Dic122C2212yescomplemented
27C4×#11758200—21no—
28C4×#11758200—21no—
29#16839815C2211yescharacteristic
30#16839786C2211yescharacteristic
31#16839785—21no—
32C2×C4⋊#92418C2211yescharacteristic
33C2×#14519698—21no—
34M16×Dic61—21no—
35#16839515C2212yescomplemented
36#16839513C2212yescomplemented
37#16839508C2212yescomplemented
38#16839506C2212yes—
39#16839506C2212yes—
40#16839506C2212yescomplemented
41C2×AGL(1,ℤ/8)×C61—21no—
42C2×AGL(1,ℤ/8)×C61—21no—
43C61×#532C2211yescharacteristic
44C2×#127×C61C2211yescharacteristic
45C61×#496—22no—
46C61×#493C2212yes—
47C61×#492C2212yes—
48C4×M16×C61C2211yescharacteristic
49C22×Dic122D411yescharacteristic
50C2×#11758200—22no—
51C2×#11758200—22no—
52C2×#11758200—21no—
53C2×#11758200—21no—
54C2×#11758200—44no—
55C2×#11758194—21no—
56C22×#92418—21no—
57C4×Dic122—21no—
58C4×Dic122—21no—
59C4×Dic122D411yescharacteristic
60C4×Dic122D411yescharacteristic
61C4×Dic122C2311yescharacteristic
62C4×Dic122C2311yescharacteristic
63C2×C4⋊Dic61D411yescharacteristic
64C2×#11758196—21no—
65C2×C4×Dic61—21no—
66#14519708—21no—
67#14519708—21no—
68C4⋊#92418—21no—
69C4⋊#92418—21no—
70C4⋊#92418C2312yes—
71C4⋊#92418C2311yescharacteristic
72C4⋊#92418C2311yescharacteristic
73C4⋊#92418C2312yes—
74#14519706—22no—
75#14519706—22no—
76#14519700—21no—
77#14519700—21no—
78#14519698—22no—
79#14519698—22no—
80#14519698—21no—
81#14519698—21no—
82C22⋊#92418—21no—
83#14519687—22no—
84C4×#92418—21no—
85C8×Dic61—22no—
86C2×#41×C61—21no—
87C22×D4×C61—21no—
88C22×D4×C61—21no—
89C22×D4×C61—21no—
90AGL(1,ℤ/8)×C61—44no—
91AGL(1,ℤ/8)×C61—44no—
92C2×SD16×C61—22no—
93C2×SD16×C61—22no—
94C2×D8×C61—22no—
95C2×D8×C61—22no—
96C2×M16×C61D411yescharacteristic
97C2×M16×C61D411yescharacteristic
98C4⋊Q8×C61C2311yescharacteristic
99#127×C61—21no—
100#127×C61—21no—
101#127×C61—21no—
102#127×C61C2312yes—
103#127×C61C2311yescharacteristic
104#124×C61—21no—
105#121×C61—21no—
106#121×C61—21no—
107C4×D4×C61—21no—
108C4×D4×C61—21no—
109C2×C42×C61C2311yescharacteristic
110#97×C61D412yes—
111C4×C8×C61C2312yes—
112C24×C61—41no—
113#41×C61—42no—
114#41×C61—42no—
115C2×Q8×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—21no—
121C2×D4×C61—22no—
122C2×D4×C61—42no—
123C2×D4×C61—42no—
124C2×D4×C61—41no—
125C2×D4×C61—41no—
126C2×D4×C61—41no—
127C2×D4×C61—41no—
128C2×D4×C61—42no—
129C2×D4×C61—42no—
130C2×D4×C61—82no—
131C22×C4×C61—41no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61#4111yescharacteristic
135C22×Dic61—21no—
136C22×Dic61—21no—
137C2×Dic122—22no—
138C2×Dic122—21no—
139C2×Dic122—41no—
140C2×Dic122C2×D411yescharacteristic
141C2×Dic122C2×D411yescharacteristic
142SD16×C61—44no—
143SD16×C61—44no—
144D8×C61—44no—
145D8×C61—44no—
146M16×C61—22no—
147M16×C61SD1614yescomplemented
148C2×C8×C61C2×D412yes—
149C2×C8×C61C2×D412yes—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C22⋊C4×C61—41no—
153C22⋊C4×C61—41no—
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C42×C61#4111yescharacteristic
157C42×C61C2411yescharacteristic
158#11758200—42no—
159#11758200—44no—
160#11758200—42no—
161#11758200—44no—
162C4⋊Dic61—21no—
163C4⋊Dic61C2×D411yescharacteristic
164C4⋊Dic61C2×D411yescharacteristic
165#11758196—21no—
166#11758196—21no—
167#11758196—21no—
168#11758196—21no—
169C4×Dic61—21no—
170C4×Dic61—21no—
171C4×Dic61—21no—
172C4×Dic61—21no—
173#11758194—42no—
174C2×#92418—22no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—22no—
178C2×#92418—42no—
179C23×C61—41no—
180C23×C61—42no—
181C23×C61—82no—
182C23×C61—82no—
183C23×C61—82no—
184C23×C61#12111yescharacteristic
185Q8×C61—41no—
186Q8×C61—41no—
187D4×C61—41no—
188D4×C61—42no—
189D4×C61—42no—
190D4×C61—41no—
191D4×C61—41no—
192D4×C61—42no—
193D4×C61—41no—
194D4×C61—42no—
195D4×C61—81no—
196D4×C61—82no—
197D4×C61—82no—
198D4×C61—82no—
199D4×C61—82no—
200D4×C61—82no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—82no—
206C2×C4×C61#12111yescharacteristic
207C2×C4×C61#12111yescharacteristic
208C2×C4×C61#12111yescharacteristic
209C2×C4×C61C2×SD1612yes—
210C2×C4×C61AGL(1,ℤ/8)12yes—
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—42no—
219C2×Dic61—81no—
220Dic122—21no—
221Dic122—21no—
222Dic122—21no—
223Dic122—42no—
224Dic122AGL(1,ℤ/8)12yescomplemented
225C8×C61—22no—
226C8×C61C2×SD1614yescomplemented
227#92418—42no—
228#92418—42no—
229#92418—41no—
230#92418—41no—
231#19170—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—81no—
235C22×C61—82no—
236C22×C61—82no—
237C22×C61—81no—
238C22×C61—82no—
239C22×C61—81no—
240C22×C61—164no—
241C22×C61#46012yes—
242C22×C61C2×#12111yescharacteristic
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—81no—
247C4×C61#46012yes—
248C4×C61C22×SD1611yescharacteristic
249C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
250C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
251C4×C61#57511yescharacteristic
252Dic61—22no—
253Dic61—41no—
254Dic61—41no—
255Dic61—41no—
256#3151D6111yescharacteristic
257#3103—611no—
258C2×#460—611no—
259Q8×M16—611no—
260#1688—612no—
261#1686—612no—
262#1668—612no—
263#1666—612no—
264#1565—611no—
265#1547—611no—
266#1545—611no—
267C2×C61—22no—
268C2×C61—161no—
269C2×C61—162no—
270C2×C61C2×#46011yescharacteristic
271C2×C61#305511yescharacteristic
272C2×C61#306811yescharacteristic
273C2×AGL(1,ℤ/8)—21no—
274C2×AGL(1,ℤ/8)—21no—
275C22×SD16—1221no—
276#532C2×D6111yescharacteristic
277C2×#127C2×D6111yescharacteristic
278C2×C4×Q8—611no—
279#496—22no—
280#493C2×D6112yes—
281#492C2×D6112yes—
282#481—1221no—
283#475—611no—
284#475—611no—
285#460—612no—
286#460—612no—
287#460—611no—
288#460—611no—
289#460—1221no—
290#460—1221no—
291#448—1221no—
292#446—612no—
293#446—1222no—
294C8×Q8—612no—
295C4×SD16—1221no—
296C4×SD16—1221no—
297#431—611no—
298#423—611no—
299#423—1221no—
300C2×C4⋊C8—611no—
301C2×#102—1221no—
302C4×M16—1221no—
303C4×M16C2×D6111yescharacteristic
304#335—612no—
305#334—612no—
306#332—612no—
307#331—612no—
308#331—612no—
309#331—612no—
310C61#1917011yescharacteristic, complemented, Sylow
311C2×#41—21no—
312C22×Q8—611no—
313C22×D4—21no—
314C22×D4—21no—
315C22×D4—21no—
316AGL(1,ℤ/8)—44no—
317AGL(1,ℤ/8)—44no—
318C2×SD16—22no—
319C2×SD16—22no—
320C2×SD16—1222no—
321C2×SD16—1222no—
322C2×SD16—1221no—
323C2×SD16—1221no—
324C2×SD16—2444no—
325C2×D8—22no—
326C2×D8—22no—
327C2×M16—1221no—
328C2×M16C22⋊D6111yescharacteristic
329C2×M16C22⋊D6111yescharacteristic
330C22×C8—1221no—
331C4⋊Q8C22×D6111yescharacteristic
332#127—21no—
333#127—21no—
334#127—21no—
335#127C22×D6112yes—
336#127C22×D6111yescharacteristic
337#124—21no—
338#121—21no—
339#121—21no—
340C4×Q8—611no—
341C4×Q8—611no—
342C4×Q8—611no—
343C4×Q8—611no—
344C4×Q8—1221no—
345C4×Q8—1221no—
346C4×D4—21no—
347C4×D4—21no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—1221no—
350C2×C42—1221no—
351C2×C42C22×D6111yescharacteristic
352#106—1221no—
353#106—1221no—
354C4⋊C8—612no—
355C4⋊C8—611no—
356C4⋊C8—611no—
357C4⋊C8—612no—
358C4⋊C8—1222no—
359C4⋊C8—1222no—
360C4⋊C8—1221no—
361C4⋊C8—1221no—
362#103—1221no—
363#103—1221no—
364#102—1222no—
365#102—1222no—
366#102—1221no—
367#102—1221no—
368C22⋊C8—1221no—
369#97—1222no—
370#97C22⋊D6112yes—
371C4×C8—1222no—
372C4×C8—1221no—
373C4×C8C22×D6112yes—
374C24—41no—
375#41—42no—
376#41—42no—
377C2×Q8—21no—
378C2×Q8—611no—
379C2×Q8—611no—
380C2×Q8—1222no—
381C2×Q8—1221no—
382C2×Q8—2441no—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—82no—
398C22×C4—41no—
399C22×C4—1221no—
400C22×C4—1221no—
401C22×C4D4×D6111yescharacteristic
402C22×C4#1175822411yescharacteristic
403C22×C4C2×C22⋊D6111yescharacteristic
404SD16—44no—
405SD16—44no—
406SD16—2442no—
407SD16—2444no—
408SD16—2442no—
409SD16—2444no—
410D8—44no—
411D8—44no—
412M16—22no—
413M16—2442no—
414M16#1175820014yescomplemented
415C2×C8—1221no—
416C2×C8—1221no—
417C2×C8—1222no—
418C2×C8—1222no—
419C2×C8—2442no—
420C2×C8C2×C22⋊D6112yes—
421C2×C8C2×C22⋊D6112yes—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—611no—
425C4⋊C4—611no—
426C4⋊C4—1221no—
427C4⋊C4—1221no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—1221no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C42—1221no—
434C42—1221no—
435C42—1221no—
436C42—1221no—
437C42D4×D6111yescharacteristic
438C42#1175822411yescharacteristic
439C42C2×C22⋊D6111yescharacteristic
440C42C23×D6111yescharacteristic
441C23—41no—
442C23—42no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23#1451974811yescharacteristic
447Q8—41no—
448Q8—41no—
449Q8—612no—
450Q8—1221no—
451Q8—1221no—
452Q8—1221no—
453Q8—2442no—
454D4—41no—
455D4—42no—
456D4—42no—
457D4—41no—
458D4—41no—
459D4—42no—
460D4—41no—
461D4—42no—
462D4—81no—
463D4—82no—
464D4—82no—
465D4—82no—
466D4—82no—
467D4—82no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—82no—
473C2×C4—1221no—
474C2×C4—1221no—
475C2×C4—1221no—
476C2×C4—1221no—
477C2×C4—4881no—
478C2×C4—2442no—
479C2×C4#1451974811yescharacteristic
480C2×C4#1451974811yescharacteristic
481C2×C4#1451974811yescharacteristic
482C2×C4C2×#1175820012yes—
483C2×C4#1451980312yes—
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4C22×C22⋊D6111yescharacteristic
486C2×C4C2×#1175822411yescharacteristic
487C8—22no—
488C8—2442no—
489C8—2441no—
490C8—2442no—
491C8—2441no—
492C8C2×#1175820014yescomplemented
493C22—21no—
494C22—21no—
495C22—82no—
496C22—81no—
497C22—81no—
498C22—82no—
499C22—82no—
500C22—81no—
501C22—164no—
502C22#1683995912yes—
503C22C2×#1451974811yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—81no—
508C4—1222no—
509C4—2441no—
510C4—2441no—
511C4—2441no—
512C4#1683995912yes—
513C4C22×#1175820011yescharacteristic
514C4AGL(1,ℤ/8)×D6111yescharacteristic
515C4C2×#1451980311yescharacteristic
516C4#1684055811yescharacteristic
517C2—22no—
518C2—162no—
519C2—161no—
520C2C2×#1683995911yescharacteristic
521C2#2145543211yescharacteristic
522C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3151C2
socleC22×C61C2×#121
hypercenter#3151D61
hypocenterC61#19170
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3151

Complements

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

NH ≅ G/Ntype
1. #21455846C2semidirect
2. C2×#16839959C2semidirect
3. M16×Dic122C2semidirect
4. #21445300C2semidirect
5. #21445288C2semidirect
6. #21445176C2semidirect
7. #21445164C2semidirect
8. #21444571C2semidirect
9. #21444496C2semidirect
10. #21444487C2semidirect
15. #16840044C22semidirect
16. #16840044C22semidirect
21. #16839959C22semidirect
22. #16839959C22semidirect
24. #16839892C22semidirect
26. C8×Dic122C22semidirect
35. #16839515C22semidirect
36. #16839513C22semidirect
37. #16839508C22semidirect
40. #16839506C22semidirect
147. M16×C61SD16semidirect
224. Dic122AGL(1,ℤ/8)semidirect
226. C8×C61C2×SD16semidirect
310. C61#19170semidirect
414. M16#11758200semidirect
492. C8C2×#11758200semidirect

Maximal subgroups

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

subgroupindex
#214558462
C2×#168399592
M16×Dic1222
#214453002
#214452882
#214451762
#214451642
#214445712
#214444962
#214444872
C61×#31512
#1917061