Finite Group Databasebeta

← #29832945

Subgroups — #29832945

Order 15616.

1114 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21460588C211yesmaximal, characteristic
2C2×#16840180C211yesmaximal, characteristic, complemented
3#21454427C211yesmaximal, characteristic, complemented
4#21447548C211yesmaximal, characteristic, complemented
5#21447545C214yesmaximal, complemented
6C2×#16839569C211yesmaximal, characteristic, complemented
7C2×#16839568C211yesmaximal, characteristic, complemented
8#359×D61C214yesmaximal, complemented
9C61×#2143C211yesmaximal, characteristic, complemented
10#16840598—21no—
11#16840576C412yes—
12#16840575C412yes—
13#16840569—22no—
14#16840568—22no—
15#16840564C412yes—
16#16840563C412yes—
17#16840560—21no—
18#16840559—22no—
19C2×#14519799C2211yescharacteristic
20#135×D61C2214yes—
21#16840528—21no—
22C2×#14519783C2211yescharacteristic
23#16840180—22no—
24#16840180—22no—
25C2×#14519716C2212yescomplemented
26C22×#11758185C2211yescharacteristic, complemented
27C2×C16×D61C2212yescomplemented
28#16839810—21no—
29#16839809—22no—
30C2×#14519713C2211yescharacteristic, complemented
31#108×D61C2214yescomplemented
32#16839570C2214yescomplemented
33#16839569C2212yescomplemented
34#16839569C2212yescomplemented
35#16839568C2212yescomplemented
36#16839568C2212yescomplemented
37C2×#135×C61C2211yescharacteristic
38C2×M32×C61C2211yescharacteristic, complemented
39C2×#108×C61C2211yescharacteristic, complemented
40C61×#359C2214yescomplemented
41#14519825—22no—
42#14519824—22no—
43#14519824—41no—
44#14519823—22no—
45#14519823—41no—
46#14519822—22no—
47#14519822—42no—
48C2×#11758222—21no—
49C2×#11758222—42no—
50C2×#11758222D411yescharacteristic
51#41×D61—24no—
52#41×D61—42no—
53#14519809—24no—
54#14519808—22no—
55#14519807—22no—
56#14519806—24no—
57#14519804—24no—
58#14519803—22no—
59#14519802—22no—
60#14519801—24no—
61#14519800C2×C414yes—
62#14519799—21no—
63#14519799—21no—
64#14519799C2×C412yes—
65#14519799C2312yes—
66#14519798—24no—
67#14519797—24no—
68#14519796C2×C414yes—
69C2×Dic244C2×C411yescharacteristic
70Q16×D61C2×C414yes—
71C2×#11758190—21no—
72SD16×D61—24no—
73C2×D488C2×C411yescharacteristic
74D8×D61C2×C414yes—
75#14519785—42no—
76#14519783—42no—
77#14519783C2×C412yes—
78#14519783C2×C412yes—
79#14519783D412yes—
80#14519783D412yes—
81C2×#11758194—21no—
82C2×#11758194D411yescharacteristic
83C2×#11758189D412yes—
84M16×D61—24no—
85C2×C8×D61C2312yes—
86#14519716—24no—
87C2×#11758185—21no—
88C2×#11758185—21no—
89C2×#11758185C2311yescharacteristic
90C2×#11758185C2311yescharacteristic
91C16×D61—24no—
92#14519713—21no—
93#14519713C2312yes—
94#14519712—24no—
95C2×#41×C61—21no—
96#135×C61—21no—
97#135×C61—21no—
98#135×C61C2×C412yes—
99#135×C61C2312yes—
100C2×Q16×C61C2×C411yescharacteristic
101C2×SD16×C61—21no—
102C2×D8×C61C2×C411yescharacteristic
103C2×M16×C61—21no—
104C22×C8×C61C2311yescharacteristic
105M32×C61—21no—
106M32×C61—21no—
107C2×C16×C61C2312yes—
108#108×C61—21no—
109#108×C61C2312yes—
110#41×C61—22no—
111#41×C61—22no—
112#41×C61—41no—
113#41×C61—41no—
114C2×Q8×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—41no—
117C22×C4×C61—41no—
118C22×C4×C61C2×D411yescharacteristic
119C2×C22⋊D61—41no—
120C2×C22⋊D61—44no—
121C2×C22⋊D61SD1612yes—
122#11758226—24no—
123#11758226—42no—
124Q8×D61—24no—
125Q8×D61—42no—
126#11758224—24no—
127#11758224—44no—
128#11758224—44no—
129#11758224—42no—
130D4×D61—24no—
131D4×D61—44no—
132D4×D61—44no—
133D4×D61—42no—
134#11758222—22no—
135#11758222—22no—
136#11758222—41no—
137#11758222—41no—
138#11758222—44no—
139#11758222—42no—
140#11758222—44no—
141#11758222—42no—
142#11758222C22⋊C412yes—
143#11758222C22⋊C412yes—
144#11758222D812yes—
145#11758222D812yes—
146C2×D244—21no—
147C2×D244—42no—
148C2×D244SD1611yescharacteristic
149C2×C4×D61—41no—
150C2×C4×D61—44no—
151C2×C4×D61C2×D412yes—
152C2×Dic122—21no—
153C2×Dic122—42no—
154C2×Dic122SD1611yescharacteristic
155Q16×C61—21no—
156Q16×C61C22×C412yes—
157SD16×C61—22no—
158SD16×C61—22no—
159D8×C61—21no—
160D8×C61C22×C412yes—
161M16×C61—22no—
162M16×C61—41no—
163C2×C8×C61—41no—
164C2×C8×C61C22×C411yescharacteristic
165C2×C8×C61C22×C411yescharacteristic
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2×D411yescharacteristic
169C2×C8×C61C2411yescharacteristic
170#11758202—24no—
171#11758201—24no—
172#11758200—24no—
173#11758199—24no—
174#11758194—22no—
175#11758194—41no—
176#11758194C22⋊C412yes—
177#11758194C22⋊C412yes—
178C2×#92418—41no—
179C2×#92418C2×D412yes—
180Dic244—21no—
181Dic244C22×C412yes—
182D488—21no—
183D488C22×C412yes—
184#11758190—22no—
185#11758190—22no—
186#11758189—44no—
187#11758189C22⋊C414yes—
188#11758189C22⋊C414yes—
189C8×D61—44no—
190C8×D61C22×C414yes—
191C8×D61C2×D414yes—
192C16×C61—22no—
193#11758185—21no—
194#11758185—21no—
195C23×C61—41no—
196C23×C61C2×SD1611yescharacteristic
197C22×D61—41no—
198C22×D61—44no—
199C22×D61C2×SD1612yes—
200Q8×C61—22no—
201Q8×C61—41no—
202D4×C61—22no—
203D4×C61—42no—
204D4×C61—42no—
205D4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61C2×C22⋊C411yescharacteristic
210C2×C4×C61C2×C22⋊C411yescharacteristic
211C2×C4×C61C2×D811yescharacteristic
212C2×C4×C61C2×D811yescharacteristic
213C2×C4×C61C2×SD1611yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C22⋊D61—42no—
216C22⋊D61—42no—
217C22⋊D61—44no—
218C22⋊D61—44no—
219C22⋊D61#10214yes—
220C22⋊D61#10214yes—
221C2×Dic61—41no—
222C2×Dic61—44no—
223C2×Dic61C2×SD1612yes—
224D244—22no—
225D244—44no—
226D244—41no—
227D244#10212yes—
228D244#10212yes—
229C4×D61—44no—
230C4×D61—42no—
231C4×D61—42no—
232C4×D61—44no—
233C4×D61C2×C22⋊C414yes—
234C4×D61C2×D814yes—
235Dic122—22no—
236Dic122—41no—
237Dic122—44no—
238Dic122#10212yes—
239Dic122#10212yes—
240C8×C61—42no—
241C8×C61C2×C22⋊C411yescharacteristic
242C8×C61C2×C22⋊C411yescharacteristic
243C8×C61C23×C411yescharacteristic
244C8×C61C22×D411yescharacteristic
245#92418—42no—
246#92418C2×C22⋊C414yes—
247#33385—611nomaximal, Sylow
248C22×C61—41no—
249C22×C61—42no—
250C22×C61—161no—
251C22×C61C2×#10211yescharacteristic
252C22×C61C2×#10211yescharacteristic
253C22×C61C22×SD1611yescharacteristic
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—164no—
257C2×D61—161no—
258C2×D61C2×#10214yes—
259C4×C61—42no—
260C4×C61C2×#10211yescharacteristic
261C4×C61C2×#10211yescharacteristic
262C4×C61C22×C22⋊C411yescharacteristic
263C4×C61C22×D811yescharacteristic
264Dic61—42no—
265Dic61C2×#10214yes—
266#3584—611no—
267C2×#504—611no—
268#2916—611no—
269#2143—611no—
270#2143—614no—
271#2143D6111yescharacteristic, complemented
272C2×#359—611no—
273C2×#359—614no—
274C2×#359—611no—
275C2×C61—21no—
276C2×C61—21no—
277C2×C61—21no—
278C2×C61—82no—
279C2×C61C22×#10211yescharacteristic
280D61—24no—
281D61—82no—
282#585—1221no—
283#578—612no—
284#578—612no—
285#577—1222no—
286#577—1222no—
287#576—612no—
288#576—612no—
289#575—1222no—
290#575—1221no—
291C2×#135—614no—
292C2×#135—611no—
293C2×#135C2×D6111yescharacteristic
294#568—1221no—
295C2×#131—611no—
296#504—1222no—
297#504—1222no—
298C2×M32—612no—
299C2×M32C2×D6111yescharacteristic, complemented
300C22×C16—611no—
301C22×C16—612no—
302#430—1222no—
303#430—1221no—
304C2×#108—611no—
305C2×#108—614no—
306C2×#108C2×D6111yescharacteristic, complemented
307#359—612no—
308#359—612no—
309#359—614no—
310#359—612no—
311#359—612no—
312#359C2×D6114yescomplemented
313C61#3338511yescharacteristic, complemented, Sylow
3142−1+4—1222no—
3152−1+4—1222no—
3162−1+4—2441no—
317H(5,𝔽2)—1222no—
318H(5,𝔽2)—1222no—
319H(5,𝔽2)—2442no—
320H(5,𝔽2)—2441no—
321C2×#41—21no—
322C2×#41—611no—
323C2×#41—1224no—
324C2×#41—1221no—
325C2×#41—2442no—
326C2×#41—2442no—
327#137—1224no—
328#137—1224no—
329#137—1222no—
330#137—1222no—
331AGL(1,ℤ/8)—1224no—
332AGL(1,ℤ/8)—1224no—
333AGL(1,ℤ/8)—1222no—
334AGL(1,ℤ/8)—1222no—
335#135—21no—
336#135—21no—
337#135—614no—
338#135—614no—
339#135—612no—
340#135—612no—
341#135—1224no—
342#135—1224no—
343#135—1221no—
344#135—1221no—
345#135C4×D6112yes—
346#135C22×D6112yes—
347C2×Q16—614no—
348C2×Q16—611no—
349C2×Q16C4×D6111yescharacteristic
350C2×SD16—21no—
351C2×SD16—1224no—
352C2×SD16—1221no—
353C2×D8—614no—
354C2×D8—611no—
355C2×D8C4×D6111yescharacteristic
356#131—612no—
357#131—612no—
358#131—612no—
359#131—612no—
360#131—2442no—
361#131—2442no—
362C2×M16—21no—
363C2×M16—611no—
364C2×M16—612no—
365C2×M16—1224no—
366C2×M16—1221no—
367C22×C8—612no—
368C22×C8C22×D6111yescharacteristic
369M32—21no—
370M32—21no—
371M32—1224no—
372C2×C16—611no—
373C2×C16—611no—
374C2×C16—1224no—
375C2×C16—1221no—
376C2×C16—1221no—
377C2×C16C22×D6112yes—
378#108—21no—
379#108—612no—
380#108—1224no—
381#108—1221no—
382#108C22×D6112yes—
383#41—22no—
384#41—22no—
385#41—41no—
386#41—41no—
387#41—612no—
388#41—612no—
389#41—612no—
390#41—612no—
391#41—1224no—
392#41—1224no—
393#41—1222no—
394#41—1222no—
395#41—2444no—
396#41—2444no—
397#41—2442no—
398#41—2442no—
399#41—2444no—
400#41—2444no—
401#41—2442no—
402#41—2442no—
403#41—2441no—
404#41—2441no—
405C2×Q8—21no—
406C2×Q8—611no—
407C2×Q8—1224no—
408C2×Q8—1221no—
409C2×Q8—2442no—
410C2×Q8—2442no—
411C2×D4—21no—
412C2×D4—41no—
413C2×D4—612no—
414C2×D4—611no—
415C2×D4—1224no—
416C2×D4—1221no—
417C2×D4—2442no—
418C2×D4—2444no—
419C2×D4—2442no—
420C2×D4—2444no—
421C2×D4—2444no—
422C2×D4—2441no—
423C22×C4—41no—
424C22×C4—612no—
425C22×C4—2444no—
426C22×C4—2441no—
427C22×C4D4×D6111yescharacteristic
428Q16—21no—
429Q16—612no—
430Q16—1224no—
431Q16—1221no—
432Q16C2×C4×D6112yes—
433SD16—22no—
434SD16—22no—
435SD16—1224no—
436SD16—1224no—
437SD16—1222no—
438SD16—1222no—
439D8—21no—
440D8—612no—
441D8—1224no—
442D8—1221no—
443D8C2×C4×D6112yes—
444M16—22no—
445M16—41no—
446M16—612no—
447M16—612no—
448M16—614no—
449M16—614no—
450M16—1222no—
451M16—2444no—
452M16—2441no—
453C2×C8—41no—
454C2×C8—612no—
455C2×C8—614no—
456C2×C8—614no—
457C2×C8—2444no—
458C2×C8—2441no—
459C2×C8C2×C4×D6111yescharacteristic
460C2×C8C2×C4×D6111yescharacteristic
461C2×C8D4×D6111yescharacteristic
462C2×C8D4×D6111yescharacteristic
463C2×C8D4×D6111yescharacteristic
464C2×C8C23×D6111yescharacteristic
465C16—22no—
466C16—1221no—
467C16—1221no—
468C23—41no—
469C23—612no—
470C23—2444no—
471C23—2441no—
472C23SD16×D6111yescharacteristic
473Q8—22no—
474Q8—41no—
475Q8—612no—
476Q8—612no—
477Q8—1222no—
478Q8—2444no—
479Q8—2441no—
480D4—22no—
481D4—42no—
482D4—42no—
483D4—41no—
484D4—614no—
485D4—614no—
486D4—612no—
487D4—612no—
488D4—1222no—
489D4—2444no—
490D4—2444no—
491D4—2444no—
492D4—2442no—
493D4—2442no—
494D4—2441no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—612no—
499C2×C4—614no—
500C2×C4—614no—
501C2×C4—2444no—
502C2×C4—2444no—
503C2×C4—2444no—
504C2×C4—2441no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4C22⋊C4×D6111yescharacteristic
508C2×C4C22⋊C4×D6111yescharacteristic
509C2×C4D8×D6111yescharacteristic
510C2×C4D8×D6111yescharacteristic
511C2×C4SD16×D6111yescharacteristic
512C2×C4C2×D4×D6111yescharacteristic
513C8—42no—
514C8—614no—
515C8—2442no—
516C8C22⋊C4×D6111yescharacteristic
517C8C22⋊C4×D6111yescharacteristic
518C8C22×C4×D6111yescharacteristic
519C8C2×D4×D6111yescharacteristic
520C22—41no—
521C22—42no—
522C22—161no—
523C22—614no—
524C22—9761no—
525C22—9764no—
526C22—2442no—
527C22—2442no—
528C22#102×D6111yescharacteristic
529C22#102×D6111yescharacteristic
530C22C2×SD16×D6111yescharacteristic
531C4—42no—
532C4—614no—
533C4—2442no—
534C4#102×D6111yescharacteristic
535C4#102×D6111yescharacteristic
536C4C2×C22⋊C4×D6111yescharacteristic
537C4C2×D8×D6111yescharacteristic
538C2—21no—
539C2—21no—
540C2—21no—
541C2—82no—
542C2—1224no—
543C2—4882no—
544C2C2×#102×D6111yescharacteristic
subgroupG/H
centerC4C2×D8×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2143C2
socleC2×C61C22×#102
prime radical#21460588C2
cosocleC2×C8×C61C24
hypercenter#2143D61
hypocenterC61#33385
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2143

Complements

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

NH ≅ G/Ntype
2. C2×#16840180C2semidirect
3. #21454427C2semidirect
4. #21447548C2semidirect
5. #21447545C2semidirect
6. C2×#16839569C2semidirect
7. C2×#16839568C2semidirect
8. #359×D61C2semidirect
9. C61×#2143C2semidirect
25. C2×#14519716C22semidirect
26. C22×#11758185C22semidirect
27. C2×C16×D61C22semidirect
30. C2×#14519713C22semidirect
31. #108×D61C22semidirect
32. #16839570C22semidirect
33. #16839569C22semidirect
34. #16839569C22semidirect
35. #16839568C22semidirect
36. #16839568C22semidirect
38. C2×M32×C61C22semidirect
39. C2×#108×C61C22semidirect
40. C61×#359C22semidirect
271. #2143D61semidirect
299. C2×M32C2×D61semidirect
306. C2×#108C2×D61semidirect
312. #359C2×D61semidirect
313. C61#33385semidirect

Maximal subgroups

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

subgroupindex
#214605882
C2×#168401802
#214544272
#214475482
#214475452
C2×#168395692
C2×#168395682
#359×D612
C61×#21432
#3338561