Finite Group Databasebeta

← #29662775

Subgroups — #29662775

Order 15616.

844 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455815C211yesmaximal, characteristic, complemented
2C2×#16839954C211yesmaximal, characteristic, complemented
3M16×D244C211yesmaximal, characteristic, complemented
4#21445295C212yesmaximal, complemented
5#21445281C212yesmaximal, complemented
6#21445171C212yesmaximal, complemented
7#21445157C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444496C211yesmaximal, characteristic, complemented
10#21444487C211yesmaximal, characteristic, complemented
11C61×#3151C211yesmaximal, characteristic, complemented
12C22×#11758199—21no—
13C2×M16×D61—21no—
14C2×C4×D244C2211yescharacteristic
15#16840073—21no—
16#16840039C2211yescharacteristic, complemented
17#16840039C2211yescharacteristic, complemented
18#16839954—21no—
19#16839954—21no—
20#16839954C2212yes—
21#16839954C2212yes—
22#16839954C2211yescharacteristic, complemented
23#16839954C2211yescharacteristic, complemented
24#16839895—21no—
25C4×#11758199—21no—
26C4×#11758199—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#16839508C2212yescomplemented
38#16839506C2212yescomplemented
39#16839506C2212yescomplemented
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×#493C2212yescomplemented
47C61×#492C2212yescomplemented
48C4×M16×C61C2211yescharacteristic
49C22×D244D411yescharacteristic
50C22×C4×D61—21no—
51C2×#11758199—22no—
52C2×#11758199—22no—
53C2×#11758199—21no—
54C2×#11758199—21no—
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#14519710—21no—
70#14519710—21no—
71C4⋊#92418—21no—
72C4⋊#92418—21no—
73C4⋊#92418C2312yes—
74C4⋊#92418C2311yescharacteristic
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2312yes—
77#14519698—22no—
78#14519698—22no—
79#14519698—21no—
80#14519698—21no—
81#14519697—21no—
82#14519697—21no—
83C22⋊#92418—21no—
84#14519689—22no—
85#14519689—22no—
86C4×#92418—21no—
87C2×#41×C61—21no—
88C22×D4×C61—21no—
89C22×D4×C61—21no—
90C22×D4×C61—21no—
91AGL(1,ℤ/8)×C61—44no—
92AGL(1,ℤ/8)×C61—44no—
93C2×SD16×C61—22no—
94C2×SD16×C61—22no—
95C2×D8×C61—22no—
96C2×D8×C61—22no—
97C2×M16×C61D411yescharacteristic
98C2×M16×C61D411yescharacteristic
99C4⋊Q8×C61C2311yescharacteristic
100#127×C61—21no—
101#127×C61—21no—
102#127×C61—21no—
103#127×C61C2312yes—
104#127×C61C2311yescharacteristic
105#124×C61—21no—
106#121×C61—21no—
107#121×C61—21no—
108C4×D4×C61—21no—
109C4×D4×C61—21no—
110C2×C42×C61C2311yescharacteristic
111#97×C61D412yes—
112C4×C8×C61C2312yes—
113C24×C61—41no—
114C23×D61—21no—
115#41×C61—42no—
116#41×C61—42no—
117C2×Q8×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—21no—
123C2×D4×C61—22no—
124C2×D4×C61—42no—
125C2×D4×C61—42no—
126C2×D4×C61—41no—
127C2×D4×C61—41no—
128C2×D4×C61—41no—
129C2×D4×C61—41no—
130C2×D4×C61—42no—
131C2×D4×C61—42no—
132C2×D4×C61—82no—
133C22×C4×C61—41no—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61#4111yescharacteristic
137C22×Dic61—21no—
138C2×D244—22no—
139C2×D244—21no—
140C2×D244—41no—
141C2×D244C2×D411yescharacteristic
142C2×D244C2×D411yescharacteristic
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—42no—
148C2×C4×D61—42no—
149SD16×C61—44no—
150SD16×C61—44no—
151D8×C61—44no—
152D8×C61—44no—
153M16×C61—22no—
154M16×C61D814yescomplemented
155C2×C8×C61C2×D412yes—
156C2×C8×C61C2×D412yes—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C22⋊C4×C61—41no—
160C22⋊C4×C61—41no—
161C42×C61C2×D411yescharacteristic
162C42×C61C2×D411yescharacteristic
163C42×C61#4111yescharacteristic
164C42×C61C2411yescharacteristic
165#11758199—42no—
166#11758199—44no—
167#11758199—42no—
168#11758199—44no—
169#11758198—21no—
170#11758198—21no—
171#11758198—21no—
172#11758198—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61C2×D411yescharacteristic
175C4⋊Dic61C2×D411yescharacteristic
176#11758194—42no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—42no—
182#11758189—44no—
183C8×D61—44no—
184C23×C61—41no—
185C23×C61—42no—
186C23×C61—82no—
187C23×C61—82no—
188C23×C61—82no—
189C23×C61#12111yescharacteristic
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—81no—
195Q8×C61—41no—
196Q8×C61—41no—
197D4×C61—41no—
198D4×C61—42no—
199D4×C61—42no—
200D4×C61—41no—
201D4×C61—41no—
202D4×C61—42no—
203D4×C61—41no—
204D4×C61—42no—
205D4×C61—81no—
206D4×C61—82no—
207D4×C61—82no—
208D4×C61—82no—
209D4×C61—82no—
210D4×C61—82no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—82no—
216C2×C4×C61#12111yescharacteristic
217C2×C4×C61#12111yescharacteristic
218C2×C4×C61#12111yescharacteristic
219C2×C4×C61C2×D812yes—
220C2×C4×C61AGL(1,ℤ/8)12yes—
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—42no—
227D244—21no—
228D244—21no—
229D244—21no—
230D244—42no—
231D244AGL(1,ℤ/8)12yescomplemented
232C4×D61—41no—
233C4×D61—41no—
234C4×D61—41no—
235C4×D61—41no—
236C4×D61—42no—
237C4×D61—42no—
238C8×C61—22no—
239C8×C61C2×D814yescomplemented
240#92418—42no—
241#92418—42no—
242#92418—41no—
243#92418—41no—
244#19162—611nomaximal, Sylow
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—81no—
248C22×C61—82no—
249C22×C61—82no—
250C22×C61—81no—
251C22×C61—82no—
252C22×C61—81no—
253C22×C61—164no—
254C22×C61#45912yes—
255C22×C61C2×#12111yescharacteristic
256C2×D61—22no—
257C2×D61—41no—
258C2×D61—41no—
259C2×D61—41no—
260C2×D61—41no—
261C2×D61—41no—
262C2×D61—84no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—81no—
267C4×C61#45912yes—
268C4×C61C22×D811yescharacteristic
269C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
270C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
271C4×C61#57511yescharacteristic
272Dic61—41no—
273Dic61—41no—
274#3151D6111yescharacteristic, complemented
275#3100—611no—
276C2×#459—611no—
277D4×M16—611no—
278#1687—612no—
279#1685—612no—
280#1667—612no—
281#1665—612no—
282#1565—611no—
283#1547—611no—
284#1545—611no—
285C2×C61—22no—
286C2×C61—161no—
287C2×C61—162no—
288C2×C61C2×#45911yescharacteristic
289C2×C61#305511yescharacteristic
290C2×C61#306811yescharacteristic
291D61—42no—
292D61—81no—
293C2×AGL(1,ℤ/8)—21no—
294C2×AGL(1,ℤ/8)—21no—
295C22×D8—1221no—
296C22×M16—1221no—
297#532C2×D6111yescharacteristic
298C2×#127C2×D6111yescharacteristic
299C2×C4×D4—611no—
300#496—22no—
301#493C2×D6112yescomplemented
302#492C2×D6112yescomplemented
303#480—1221no—
304#474—611no—
305#474—611no—
306#459—612no—
307#459—612no—
308#459—611no—
309#459—611no—
310#459—1221no—
311#459—1221no—
312#447—1221no—
313C4×D8—1221no—
314C4×D8—1221no—
315#436—612no—
316#435—1222no—
317C8×D4—612no—
318#431—611no—
319#423—611no—
320C2×C4⋊C8—611no—
321C2×#102—1221no—
322#407—1221no—
323C4×M16C2×D6111yescharacteristic
324#335—612no—
325#334—612no—
326#332—612no—
327#331—612no—
328#331—612no—
329#331—612no—
330C61#1916211yescharacteristic, complemented, Sylow
331C2×#41—21no—
332C22×D4—21no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4—611no—
336C23×C4—1221no—
337AGL(1,ℤ/8)—44no—
338AGL(1,ℤ/8)—44no—
339C2×SD16—22no—
340C2×SD16—22no—
341C2×D8—22no—
342C2×D8—22no—
343C2×D8—1222no—
344C2×D8—1222no—
345C2×D8—1221no—
346C2×D8—1221no—
347C2×D8—2444no—
348C2×M16—1221no—
349C2×M16—1222no—
350C2×M16—2444no—
351C2×M16C22⋊D6111yescharacteristic
352C2×M16C22⋊D6111yescharacteristic
353C22×C8—1221no—
354C22×C8—1222no—
355C4⋊Q8C22×D6111yescharacteristic
356#127—21no—
357#127—21no—
358#127—21no—
359#127C22×D6112yes—
360#127C22×D6111yescharacteristic
361#124—21no—
362#121—21no—
363#121—21no—
364C4×D4—21no—
365C4×D4—21no—
366C4×D4—611no—
367C4×D4—611no—
368C4×D4—611no—
369C4×D4—611no—
370C4×D4—1221no—
371C4×D4—1221no—
372C2×C4⋊C4—611no—
373C2×C22⋊C4—1221no—
374C2×C42C22×D6111yescharacteristic
375#107—1221no—
376#107—1221no—
377C4⋊C8—612no—
378C4⋊C8—611no—
379C4⋊C8—611no—
380C4⋊C8—612no—
381C4⋊C8—1221no—
382C4⋊C8—1221no—
383#102—1221no—
384#102—1221no—
385#102—1222no—
386#102—1222no—
387#102—1221no—
388#102—1221no—
389C22⋊C8—1221no—
390C22⋊C8—1222no—
391C22⋊C8—1222no—
392#97C22⋊D6112yes—
393C4×C8—1221no—
394C4×C8C22×D6112yes—
395C24—41no—
396C24—1221no—
397#41—42no—
398#41—42no—
399C2×Q8—21no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—82no—
415C2×D4—611no—
416C2×D4—611no—
417C2×D4—1222no—
418C2×D4—1221no—
419C2×D4—2441no—
420C22×C4—41no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—2442no—
427C22×C4—2442no—
428C22×C4D4×D6111yescharacteristic
429C22×C4#1175822411yescharacteristic
430C22×C4C2×C22⋊D6111yescharacteristic
431SD16—44no—
432SD16—44no—
433D8—44no—
434D8—44no—
435D8—2442no—
436D8—2444no—
437D8—2442no—
438D8—2444no—
439M16—22no—
440M16—2442no—
441M16—2444no—
442M16#1175819914yescomplemented
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1222no—
446C2×C8—1222no—
447C2×C8—2442no—
448C2×C8—2444no—
449C2×C8C2×C22⋊D6112yes—
450C2×C8C2×C22⋊D6112yes—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—611no—
454C4⋊C4—611no—
455C4⋊C4—1221no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—1221no—
459C22⋊C4—1221no—
460C22⋊C4—1221no—
461C22⋊C4—1221no—
462C42D4×D6111yescharacteristic
463C42#1175822411yescharacteristic
464C42C2×C22⋊D6111yescharacteristic
465C42C23×D6111yescharacteristic
466C23—41no—
467C23—42no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—1221no—
472C23—1221no—
473C23—4881no—
474C23—2442no—
475C23—2442no—
476C23#1451974811yescharacteristic
477Q8—41no—
478Q8—41no—
479D4—41no—
480D4—42no—
481D4—42no—
482D4—41no—
483D4—41no—
484D4—42no—
485D4—41no—
486D4—42no—
487D4—81no—
488D4—82no—
489D4—82no—
490D4—82no—
491D4—82no—
492D4—82no—
493D4—612no—
494D4—1221no—
495D4—1221no—
496D4—1221no—
497D4—2442no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—82no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4#1451974811yescharacteristic
513C2×C4#1451974811yescharacteristic
514C2×C4#1451974811yescharacteristic
515C2×C4C2×#1175819912yes—
516C2×C4#1451980312yes—
517C2×C4C2×D4×D6111yescharacteristic
518C2×C4C22×C22⋊D6111yescharacteristic
519C2×C4C2×#1175822411yescharacteristic
520C8—22no—
521C8—2442no—
522C8—2442no—
523C8—2441no—
524C8—2441no—
525C8C2×#1175819914yescomplemented
526C22—21no—
527C22—21no—
528C22—82no—
529C22—81no—
530C22—81no—
531C22—82no—
532C22—82no—
533C22—81no—
534C22—164no—
535C22—1222no—
536C22—4884no—
537C22—2441no—
538C22—2441no—
539C22—2441no—
540C22—2441no—
541C22—2441no—
542C22#1683995412yes—
543C22C2×#1451974811yescharacteristic
544C4—21no—
545C4—21no—
546C4—21no—
547C4—81no—
548C4—2441no—
549C4—2441no—
550C4#1683995412yes—
551C4C22×#1175819911yescharacteristic
552C4AGL(1,ℤ/8)×D6111yescharacteristic
553C4C2×#1451980311yescharacteristic
554C4#1684055811yescharacteristic
555C2—22no—
556C2—162no—
557C2—161no—
558C2—4881no—
559C2—2442no—
560C2C2×#1683995411yescharacteristic
561C2#2145543211yescharacteristic
562C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3151C2
socleC22×C61C2×#121
hypercenter#3151D61
hypocenterC61#19162
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. #21455815C2semidirect
2. C2×#16839954C2semidirect
3. M16×D244C2semidirect
4. #21445295C2semidirect
5. #21445281C2semidirect
6. #21445171C2semidirect
7. #21445157C2semidirect
8. #21444571C2semidirect
9. #21444496C2semidirect
10. #21444487C2semidirect
11. C61×#3151C2semidirect
16. #16840039C22semidirect
17. #16840039C22semidirect
22. #16839954C22semidirect
23. #16839954C22semidirect
27. #16839841C22semidirect
29. C8×D244C22semidirect
35. #16839515C22semidirect
36. #16839513C22semidirect
37. #16839508C22semidirect
38. #16839506C22semidirect
39. #16839506C22semidirect
40. #16839506C22semidirect
46. C61×#493C22semidirect
47. C61×#492C22semidirect
154. M16×C61D8semidirect
231. D244AGL(1,ℤ/8)semidirect
239. C8×C61C2×D8semidirect
274. #3151D61semidirect
301. #493C2×D61semidirect
302. #492C2×D61semidirect
330. C61#19162semidirect
442. M16#11758199semidirect
525. C8C2×#11758199semidirect

Maximal subgroups

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

subgroupindex
#214558152
C2×#168399542
M16×D2442
#214452952
#214452812
#214451712
#214451572
#214445712
#214444962
#214444872
C61×#31512
#1916261