Finite Group Databasebeta

← #29662844

Subgroups — #29662844

Order 15616.

696 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456212C211yesmaximal, characteristic, complemented
2#21455815C211yesmaximal, characteristic
3C2×#16840053C211yesmaximal, characteristic, complemented
4#21454554C211yesmaximal, characteristic, complemented
5#21445393C212yesmaximal, complemented
6#21445351C212yesmaximal, complemented
7#21445269C212yesmaximal, complemented
8#21445227C212yesmaximal, complemented
9#21444569C211yesmaximal, characteristic, complemented
10#21444501C211yesmaximal, characteristic, complemented
11C61×#1547C211yesmaximal, characteristic, complemented
12C2×M16×D61—21no—
13C2×#14519777C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16840121—22no—
16#16840112C2212yescomplemented
17#16840108C2212yescomplemented
18#16840073—21no—
19#16840053—21no—
20#16840053—21no—
21#16840053C2212yes—
22#16840053C2212yes—
23#16840039C2211yescharacteristic
24#16840039C2211yescharacteristic
25#16839954C2211yescharacteristic
26#16839954C2211yescharacteristic
27#16839895—21no—
28C4×#11758199—21no—
29C4×#11758199—21no—
30#16839839C2212yes—
31#16839832—22no—
32#16839827C2212yes—
33#16839815C2211yescharacteristic
34C2×#14519709—21no—
35#16839763—21no—
36#16839763—21no—
37C2×#14519699—21no—
38#16839725—21no—
39C4×#11758194C2211yescharacteristic
40#16839514C2212yescomplemented
41#16839512C2212yescomplemented
42#16839510C2212yescomplemented
43#16839510C2212yescomplemented
44C61×#532C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#332C2212yescomplemented
48C61×#331C2212yescomplemented
49C22×Dic122—21no—
50C22×D244D411yescharacteristic
51C22×C4×D61—21no—
52C2×#11758199—21no—
53C2×#11758199—21no—
54C2×#11758194D411yescharacteristic
55C2×#11758194D411yescharacteristic
56C2×#11758189—22no—
57M16×D61—44no—
58C2×C8×D61—22no—
59#14519777—21no—
60#14519777—21no—
61#14519777—21no—
62#14519777C2312yes—
63C4×D244—21no—
64C4×D244—21no—
65C4×D244D411yescharacteristic
66C4×D244D411yescharacteristic
67C4×D244C2311yescharacteristic
68C4×D244C2311yescharacteristic
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×#11758198—21no—
73#14519710—21no—
74#14519710—21no—
75#14519709—22no—
76#14519709—22no—
77C4⋊#92418C2311yescharacteristic
78C4⋊#92418C2311yescharacteristic
79#14519701—22no—
80#14519701—22no—
81#14519699—22no—
82#14519699—22no—
83#14519698—21no—
84#14519698—21no—
85#14519698—22no—
86#14519698—22no—
87#14519697—21no—
88#14519697—21no—
89C22⋊#92418—21no—
90#14519689—22no—
91#14519689—22no—
92#14519688D412yes—
93C4×#92418—21no—
94C4×#92418C2312yes—
95C2×#41×C61—21no—
96C2×M16×C61—21no—
97C22×C8×C61—21no—
98C4⋊Q8×C61C2311yescharacteristic
99#127×C61C2311yescharacteristic
100#124×C61—21no—
101#121×C61—21no—
102#121×C61—21no—
103C4×D4×C61—21no—
104C4×D4×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—21no—
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109C4⋊C8×C61C2312yes—
110C23×D61—21no—
111#41×C61—42no—
112#41×C61—42no—
113C2×Q8×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—41no—
118C22×C4×C61—41no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—41no—
124C2×D244—22no—
125C2×D244—21no—
126C2×D244—41no—
127C2×D244C2×D411yescharacteristic
128C2×D244C2×D411yescharacteristic
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—42no—
134C2×C4×D61—42no—
135C2×Dic122—22no—
136C2×Dic122—41no—
137C2×Dic122—82no—
138M16×C61—42no—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C2×C8×C61—42no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C22⋊C4×C61—41no—
145C22⋊C4×C61—41no—
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C42×C61#4111yescharacteristic
149C42×C61C2411yescharacteristic
150#11758199—42no—
151#11758199—42no—
152#11758198—21no—
153#11758198—21no—
154#11758198—21no—
155#11758198—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—41no—
160C4⋊Dic61—41no—
161C4⋊Dic61C2×D411yescharacteristic
162C4⋊Dic61C2×D411yescharacteristic
163#11758194—22no—
164#11758194—22no—
165C2×#92418—21no—
166C2×#92418—21no—
167C2×#92418C2×D412yes—
168C2×#92418C2×D412yes—
169#11758189—44no—
170C8×D61—44no—
171C23×C61—41no—
172C23×C61#12111yescharacteristic
173C22×D61—21no—
174C22×D61—21no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—81no—
178Q8×C61—41no—
179Q8×C61—41no—
180D4×C61—41no—
181D4×C61—41no—
182D4×C61—41no—
183D4×C61—41no—
184D4×C61—81no—
185D4×C61—82no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—82no—
191C2×C4×C61#12111yescharacteristic
192C2×C4×C61#12111yescharacteristic
193C2×C4×C61#12111yescharacteristic
194C2×C4×C61#13512yes—
195C2×C4×C61AGL(1,ℤ/8)12yes—
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61—82no—
204D244—21no—
205D244—21no—
206D244—21no—
207D244—42no—
208D244#13712yes—
209C4×D61—41no—
210C4×D61—41no—
211C4×D61—41no—
212C4×D61—41no—
213C4×D61—42no—
214C4×D61—42no—
215Dic122—82no—
216Dic122—82no—
217C8×C61—42no—
218C8×C61—42no—
219#92418—22no—
220#92418—22no—
221#92418—41no—
222#92418—41no—
223#19168—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61—81no—
229C22×C61#46312yes—
230C22×C61C2×#12111yescharacteristic
231C2×D61—22no—
232C2×D61—41no—
233C2×D61—41no—
234C2×D61—41no—
235C2×D61—41no—
236C2×D61—41no—
237C2×D61—84no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—81no—
242C4×C61#46312yes—
243C4×C61C2×#13511yescharacteristic
244C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
245C4×C61C2×#13711yescharacteristic
246C4×C61#57511yescharacteristic
247Dic61—41no—
248Dic61—41no—
249Dic61—82no—
250#3141—611no—
251#3100—611no—
252C2×#476—611no—
253D4×M16—611no—
254#1703—612no—
255#1697—612no—
256#1683—612no—
257#1677—612no—
258#1565—611no—
259#1548—611no—
260#1547D6111yescharacteristic, complemented
261C2×C61—22no—
262C2×C61—161no—
263C2×C61C2×#46311yescharacteristic
264C2×C61#305611yescharacteristic
265C2×C61#306811yescharacteristic
266D61—42no—
267D61—81no—
268C22×M16—1221no—
269#532C2×D6111yescharacteristic
270C2×C4⋊Q8—611no—
271C2×C4×D4—611no—
272#489—1222no—
273#487—612no—
274#486—612no—
275#480—1221no—
276#476—612no—
277#476—612no—
278#476—1221no—
279#476—1221no—
280#474—611no—
281#474—611no—
282#459—611no—
283#459—611no—
284#447—1221no—
285C4×D8—1221no—
286C4×D8—1221no—
287#436—612no—
288#435—1222no—
289C8×D4—612no—
290#431—611no—
291C2×#106—1221no—
292#423C2×D6111yescharacteristic
293C2×C4⋊C8C2×D6111yescharacteristic
294#417—1221no—
295#417—1221no—
296C2×#102—1221no—
297#407—1221no—
298C4×M16—611no—
299#335—612no—
300#334—612no—
301#333—612no—
302#333—612no—
303#332C2×D6112yescomplemented
304#331C2×D6112yescomplemented
305C61#1916811yescharacteristic, complemented, Sylow
306C2×#41—21no—
307C22×Q8—1221no—
308C22×D4—611no—
309C23×C4—1221no—
310C2×D8—1221no—
311C2×D8—1221no—
312C2×M16—21no—
313C2×M16—611no—
314C2×M16—611no—
315C2×M16—1222no—
316C2×M16—2444no—
317C22×C8—21no—
318C22×C8—1222no—
319C4⋊Q8—612no—
320C4⋊Q8—1221no—
321C4⋊Q8—1221no—
322C4⋊Q8—1221no—
323C4⋊Q8C22×D6111yescharacteristic
324#127C22×D6111yescharacteristic
325#124—21no—
326#121—21no—
327#121—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—611no—
331C4×D4—611no—
332C4×D4—611no—
333C4×D4—611no—
334C4×D4—1221no—
335C4×D4—1221no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—1221no—
339C2×C22⋊C4—1221no—
340C2×C42C22×D6111yescharacteristic
341#107—1221no—
342#107—1221no—
343#106—1222no—
344#106—1222no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8—611no—
348C4⋊C8—611no—
349C4⋊C8C22×D6112yes—
350C4⋊C8C22×D6112yes—
351#103—1222no—
352#103—1222no—
353#102—1221no—
354#102—1221no—
355#102—1221no—
356#102—1221no—
357#102—1222no—
358#102—1222no—
359#102—1222no—
360#102—1222no—
361C22⋊C8—1222no—
362C22⋊C8—1222no—
363C22⋊C8—1221no—
364#97—612no—
365C4×C8—612no—
366C4×C8—1221no—
367C24—1221no—
368#41—42no—
369#41—42no—
370C2×Q8—21no—
371C2×Q8—1222no—
372C2×Q8—4882no—
373C2×Q8—2441no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—41no—
378C2×D4—611no—
379C2×D4—611no—
380C2×D4—1222no—
381C2×D4—1221no—
382C2×D4—2441no—
383C22×C4—41no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4—1221no—
387C22×C4—1221no—
388C22×C4—1221no—
389C22×C4—2441no—
390C22×C4—2442no—
391C22×C4—2442no—
392C22×C4#1175822211yescharacteristic
393C22×C4D4×D6111yescharacteristic
394C22×C4D4×D6111yescharacteristic
395D8—2442no—
396D8—2442no—
397M16—42no—
398M16—1222no—
399M16—1222no—
400M16—2444no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—42no—
404C2×C8—612no—
405C2×C8—612no—
406C2×C8—1221no—
407C2×C8—1221no—
408C2×C8—2444no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—611no—
412C4⋊C4—611no—
413C4⋊C4—1221no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C4⋊C4—2441no—
417C4⋊C4—2441no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—1221no—
421C22⋊C4—1221no—
422C22⋊C4—1221no—
423C22⋊C4—1221no—
424C42#1175822211yescharacteristic
425C42D4×D6111yescharacteristic
426C42D4×D6111yescharacteristic
427C42C23×D6111yescharacteristic
428C23—41no—
429C23—1221no—
430C23—1221no—
431C23—4881no—
432C23—2442no—
433C23—2442no—
434C23#1451975011yescharacteristic
435Q8—41no—
436Q8—41no—
437Q8—4882no—
438Q8—4882no—
439D4—41no—
440D4—41no—
441D4—41no—
442D4—41no—
443D4—81no—
444D4—82no—
445D4—612no—
446D4—1221no—
447D4—1221no—
448D4—1221no—
449D4—2442no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—82no—
455C2×C4—1221no—
456C2×C4—1221no—
457C2×C4—4882no—
458C2×C4—2442no—
459C2×C4—2442no—
460C2×C4—2442no—
461C2×C4—2442no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4#1451975011yescharacteristic
467C2×C4#1451975011yescharacteristic
468C2×C4#1451975011yescharacteristic
469C2×C4#1451979812yes—
470C2×C4#1451980112yes—
471C2×C4C2×D4×D6111yescharacteristic
472C2×C4C2×D4×D6111yescharacteristic
473C2×C4C2×#1175822211yescharacteristic
474C8—42no—
475C8—42no—
476C8—1222no—
477C8—1222no—
478C8—2441no—
479C8—2441no—
480C22—21no—
481C22—21no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—1222no—
486C22—4884no—
487C22—2441no—
488C22—2441no—
489C22—2441no—
490C22—2441no—
491C22—2441no—
492C22#1683997612yes—
493C22C2×#1451975011yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—81no—
498C4—4882no—
499C4—2441no—
500C4—2441no—
501C4#1683997612yes—
502C4C2×#1451979811yescharacteristic
503C4C2×#1451980111yescharacteristic
504C4#137×D6111yescharacteristic
505C4#1684055811yescharacteristic
506C2—22no—
507C2—161no—
508C2—4881no—
509C2—2442no—
510C2C2×#1683997611yescharacteristic
511C2#2145545211yescharacteristic
512C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1547C2
socleC22×C61C2×#121
prime radical#21455815C2
hypercenter#1547D61
hypocenterC61#19168
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1547

Complements

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

NH ≅ G/Ntype
1. #21456212C2semidirect
3. C2×#16840053C2semidirect
4. #21454554C2semidirect
5. #21445393C2semidirect
6. #21445351C2semidirect
7. #21445269C2semidirect
8. #21445227C2semidirect
9. #21444569C2semidirect
10. #21444501C2semidirect
11. C61×#1547C2semidirect
16. #16840112C22semidirect
17. #16840108C22semidirect
40. #16839514C22semidirect
41. #16839512C22semidirect
42. #16839510C22semidirect
43. #16839510C22semidirect
47. C61×#332C22semidirect
48. C61×#331C22semidirect
260. #1547D61semidirect
303. #332C2×D61semidirect
304. #331C2×D61semidirect
305. C61#19168semidirect

Maximal subgroups

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

subgroupindex
#214562122
#214558152
C2×#168400532
#214545542
#214453932
#214453512
#214452692
#214452272
#214445692
#214445012
C61×#15472
#1916861