Finite Group Databasebeta

← #29597595

Subgroups — #29597595

Order 15616.

666 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454298C211yesmaximal, characteristic, complemented
2#21447511C211yesmaximal, characteristic, complemented
3#21447469C211yesmaximal, characteristic, complemented
4#21445789C211yesmaximal, characteristic, complemented
5#21443948C211yesmaximal, characteristic, complemented
6#21443942C211yesmaximal, characteristic, complemented
7C61×#3008C211yesmaximal, characteristic
8#16840529—21no—
9#16839779—21no—
10#16839767—21no—
11#16839746—22no—
12#16839746C2211yescharacteristic
13#16839746C2211yescharacteristic
14#16839746C2211yescharacteristic
15C2×#14519694—22no—
16C2×#14519694—21no—
17#16839736C2211yescharacteristic
18#16839559—22no—
19#16839549—22no—
20C2×H(5,𝔽2)×C61C2211yescharacteristic
21C2×AGL(1,ℤ/8)×C61C411yescharacteristic, complemented
22C22×D8×C61C411yescharacteristic, complemented
23C61×#535C2211yescharacteristic
24C61×#449—21no—
25C61×#449—21no—
26C61×#447—21no—
27C61×#447—21no—
28C61×#418C411yescharacteristic, complemented
29C61×#416C411yescharacteristic, complemented
30C61×#408C2211yescharacteristic
31#14519784—42no—
32#14519784—42no—
33C2×#11758194—24no—
34C2×#11758194—21no—
35C2×#11758194—21no—
36C2×#11758194—21no—
37#14519736—21no—
38#14519704—42no—
39#14519698—42no—
40#14519696—21no—
41#14519696—21no—
42#14519696D412yes—
43#14519694—24no—
44#14519694—21no—
45#14519694—21no—
46#14519694D412yes—
47#14519692—21no—
48#14519692—21no—
49H(5,𝔽2)×C61—42no—
50H(5,𝔽2)×C61—42no—
51C2×#41×C61—21no—
52C2×#41×C61C2×C411yescharacteristic
53C2×#41×C61D411yescharacteristic
54C2×#41×C61D411yescharacteristic
55C2×#41×C61C2311yescharacteristic
56C22×D4×C61—21no—
57C22×D4×C61—21no—
58C22×D4×C61—21no—
59C22×D4×C61C2×C411yescharacteristic
60C22×D4×C61C2×C411yescharacteristic
61C22×D4×C61D411yescharacteristic
62C22×D4×C61D411yescharacteristic
63AGL(1,ℤ/8)×C61—42no—
64C2×SD16×C61—21no—
65C2×D8×C61—21no—
66C2×D8×C61—21no—
67C2×D8×C61—21no—
68C2×D8×C61—42no—
69C2×M16×C61C2×C411yescharacteristic
70C22×C8×C61C2×C411yescharacteristic
71#127×C61—21no—
72#124×C61—21no—
73#121×C61—21no—
74#121×C61—21no—
75#120×C61—21no—
76#120×C61—21no—
77#117×C61C2×C411yescharacteristic
78#103×C61—21no—
79#102×C61—21no—
80#102×C61—21no—
81#102×C61—21no—
82C22⋊C8×C61—21no—
83C22⋊C8×C61—21no—
84C24×C61—21no—
85C24×C61—21no—
86C24×C61—21no—
87C24×C61—21no—
88#41×C61—22no—
89#41×C61—22no—
90#41×C61—22no—
91#41×C61—22no—
92#41×C61—41no—
93#41×C61—41no—
94#41×C61—41no—
95#41×C61—41no—
96#41×C61—44no—
97C2×Q8×C61C22⋊C411yescharacteristic
98C2×Q8×C61C2×D411yescharacteristic
99C2×D4×C61—21no—
100C2×D4×C61—22no—
101C2×D4×C61—22no—
102C2×D4×C61—21no—
103C2×D4×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—41no—
110C2×D4×C61—41no—
111C2×D4×C61—41no—
112C2×D4×C61—41no—
113C2×D4×C61—44no—
114C2×D4×C61—41no—
115C2×D4×C61—41no—
116C2×D4×C61—44no—
117C2×D4×C61—44no—
118C2×D4×C61—41no—
119C2×D4×C61—41no—
120C2×D4×C61C22⋊C411yescharacteristic
121C2×D4×C61C22⋊C411yescharacteristic
122C2×D4×C61C22⋊C411yescharacteristic
123C2×D4×C61C2×D411yescharacteristic
124C2×D4×C61#4111yescharacteristic
125C2×D4×C61#4111yescharacteristic
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132SD16×C61—42no—
133D8×C61—42no—
134D8×C61—42no—
135D8×C61—42no—
136M16×C61—41no—
137C2×C8×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61—41no—
140C4⋊C4×C61—21no—
141C22⋊C4×C61—21no—
142C22⋊C4×C61—41no—
143C22⋊C4×C61—41no—
144C42×C61—21no—
145C22⋊Dic61—41no—
146C4⋊Dic61—41no—
147C4×Dic61—41no—
148#11758194—22no—
149#11758194—22no—
150#11758194—44no—
151#11758194—41no—
152#11758194—41no—
153#11758194—41no—
154#11758194—41no—
155C2×#92418—44no—
156C2×#92418—41no—
157C2×#92418—41no—
158C2×#92418—41no—
159C23×C61—21no—
160C23×C61—21no—
161C23×C61—21no—
162C23×C61—42no—
163C23×C61—42no—
164C23×C61—42no—
165C23×C61—41no—
166C23×C61—81no—
167C23×C61—161no—
168C23×C61—161no—
169C23×C61—161no—
170C23×C61—161no—
171C23×C61C4×D411yescharacteristic
172C23×C61#12111yescharacteristic
173C23×C61#12311yescharacteristic
174Q8×C61—21no—
175Q8×C61—41no—
176Q8×C61#9912yes—
177D4×C61—24no—
178D4×C61—21no—
179D4×C61—21no—
180D4×C61—21no—
181D4×C61—21no—
182D4×C61—21no—
183D4×C61—41no—
184D4×C61—41no—
185D4×C61—41no—
186D4×C61—42no—
187D4×C61—42no—
188D4×C61—42no—
189D4×C61—42no—
190D4×C61—42no—
191D4×C61—41no—
192D4×C61—41no—
193D4×C61—81no—
194D4×C61—161no—
195D4×C61—161no—
196D4×C61#9912yes—
197C2×C4×C61—24no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—41no—
208C2×C4×C61C2×C22⋊C411yescharacteristic
209C2×C4×C61C4×D411yescharacteristic
210C2×C4×C61#12011yescharacteristic
211C2×C4×C61#12111yescharacteristic
212C2×Dic61—81no—
213C8×C61—41no—
214C8×C61—41no—
215#92418—42no—
216#92418—42no—
217#92418—81no—
218#13099—611nomaximal, Sylow
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—22no—
222C22×C61—21no—
223C22×C61—41no—
224C22×C61—41no—
225C22×C61—41no—
226C22×C61—41no—
227C22×C61—41no—
228C22×C61—81no—
229C22×C61—81no—
230C22×C61—81no—
231C22×C61—81no—
232C22×C61—81no—
233C22×C61—81no—
234C22×C61—162no—
235C22×C61—162no—
236C22×C61—162no—
237C22×C61—162no—
238C22×C61#38611yescharacteristic
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—22no—
242C4×C61—21no—
243C4×C61—81no—
244C4×C61C2×#9911yescharacteristic
245C4×C61C2×#9911yescharacteristic
246Dic61—161no—
247#3008D6111yescharacteristic
248#2888—611no—
249#2134—611no—
250#2122—611no—
251#1795—611no—
252#1395—611no—
253#1392—611no—
254C2×C61—21no—
255C2×C61—41no—
256C2×C61—41no—
257C2×C61—41no—
258C2×C61—42no—
259C2×C61—81no—
260C2×C61—81no—
261C2×C61—161no—
262C2×C61#189811yescharacteristic
263C2×H(5,𝔽2)C2×D6111yescharacteristic
264C2×AGL(1,ℤ/8)Dic6111yescharacteristic, complemented
265C22×D8Dic6111yescharacteristic, complemented
266#568—1221no—
267#535C2×D6111yescharacteristic
268#449—21no—
269#449—21no—
270#447—21no—
271#447—21no—
272#421—1221no—
273#418—1221no—
274#418Dic6111yescharacteristic, complemented
275#416Dic6111yescharacteristic, complemented
276#413—611no—
277#413—611no—
278#413—611no—
279#413—1222no—
280C2×#100—1222no—
281C2×#100—1221no—
282#410—611no—
283#408C2×D6111yescharacteristic
284#355—1222no—
285AΣL(1,𝔽16)—1222no—
286C61#1309911yescharacteristic, complemented, Sylow
287H(5,𝔽2)—42no—
288H(5,𝔽2)—42no—
289C2×#41—21no—
290C2×#41C2×Dic6111yescharacteristic
291C2×#41C22⋊D6111yescharacteristic
292C2×#41C22⋊D6111yescharacteristic
293C2×#41C22×D6111yescharacteristic
294C22×D4—21no—
295C22×D4—21no—
296C22×D4—21no—
297C22×D4C2×Dic6111yescharacteristic
298C22×D4C2×Dic6111yescharacteristic
299C22×D4C22⋊D6111yescharacteristic
300C22×D4C22⋊D6111yescharacteristic
301AGL(1,ℤ/8)—42no—
302C2×SD16—21no—
303C2×D8—21no—
304C2×D8—21no—
305C2×D8—21no—
306C2×D8—42no—
307#131—2442no—
308#131—2442no—
309C2×M16—1224no—
310C2×M16—1221no—
311C2×M16—1221no—
312C2×M16—1221no—
313C2×M16C2×Dic6111yescharacteristic
314C22×C8C2×Dic6111yescharacteristic
315#127—21no—
316#124—21no—
317#121—21no—
318#121—21no—
319#120—21no—
320#120—21no—
321#117—1221no—
322#117C2×Dic6111yescharacteristic
323#104—2442no—
324#103—21no—
325#102—21no—
326#102—21no—
327#102—21no—
328#102—2442no—
329#101—612no—
330#101—1221no—
331#101—1221no—
332#100—612no—
333#100—1224no—
334#100—1221no—
335#100—1221no—
336#99—1221no—
337#99—1221no—
338C22⋊C8—21no—
339C22⋊C8—21no—
340C24—21no—
341C24—21no—
342C24—21no—
343C24—21no—
344#41—22no—
345#41—22no—
346#41—22no—
347#41—22no—
348#41—41no—
349#41—41no—
350#41—41no—
351#41—41no—
352#41—44no—
353C2×Q8C22⋊Dic6111yescharacteristic
354C2×Q8D4×D6111yescharacteristic
355C2×D4—21no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—44no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4C22⋊Dic6111yescharacteristic
377C2×D4C22⋊Dic6111yescharacteristic
378C2×D4C22⋊Dic6111yescharacteristic
379C2×D4D4×D6111yescharacteristic
380C2×D4#1175822411yescharacteristic
381C2×D4#1175822411yescharacteristic
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4C22×Dic6111yescharacteristic
386C22×C4C2×C22⋊D6111yescharacteristic
387C22×C4C2×C22⋊D6111yescharacteristic
388SD16—42no—
389D8—42no—
390D8—42no—
391D8—42no—
392M16—41no—
393M16—1222no—
394M16—1222no—
395M16—2441no—
396M16—2441no—
397M16—2444no—
398M16—2441no—
399M16—2441no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—41no—
403C2×C8—2441no—
404C2×C8—2444no—
405C2×C8—2441no—
406C2×C8—2441no—
407C4⋊C4—21no—
408C4⋊C4—2441no—
409C22⋊C4—21no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—2441no—
413C42—21no—
414C42—2441no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—41no—
422C23—81no—
423C23—161no—
424C23—161no—
425C23—161no—
426C23—161no—
427C23D4×Dic6111yescharacteristic
428C23#1451974911yescharacteristic
429C23#1451975811yescharacteristic
430Q8—21no—
431Q8—41no—
432Q8#1451969212yes—
433D4—24no—
434D4—21no—
435D4—21no—
436D4—21no—
437D4—21no—
438D4—21no—
439D4—41no—
440D4—41no—
441D4—41no—
442D4—42no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—42no—
447D4—41no—
448D4—41no—
449D4—81no—
450D4—161no—
451D4—161no—
452D4#1451969212yes—
453C2×C4—24no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—4881no—
465C2×C4C2×C22⋊Dic6111yescharacteristic
466C2×C4D4×Dic6111yescharacteristic
467C2×C4#1451974511yescharacteristic
468C2×C4#1451974911yescharacteristic
469C8—41no—
470C8—41no—
471C8—4881no—
472C8—2442no—
473C8—2442no—
474C22—21no—
475C22—21no—
476C22—22no—
477C22—21no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—162no—
490C22—162no—
491C22—162no—
492C22—162no—
493C22#1683964211yescharacteristic
494C4—21no—
495C4—21no—
496C4—22no—
497C4—21no—
498C4—81no—
499C4—9761no—
500C4C2×#1451969211yescharacteristic
501C4C2×#1451969211yescharacteristic
502C2—21no—
503C2—41no—
504C2—41no—
505C2—41no—
506C2—42no—
507C2—81no—
508C2—81no—
509C2—161no—
510C2#2144627311yescharacteristic
subgroupG/H
centerC2#21446273
commutatorC22×C4×C61C22×C4
FrattiniC2×#41C22×D61
Fitting, prime radicalC61×#3008C2
socleC2×C61#1898
cosocleC2×#41×C61C23
hypercenter#3008D61
hypocenterC61#13099
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ #3008

Complements

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

NH ≅ G/Ntype
1. #21454298C2semidirect
2. #21447511C2semidirect
3. #21447469C2semidirect
4. #21445789C2semidirect
5. #21443948C2semidirect
6. #21443942C2semidirect
21. C2×AGL(1,ℤ/8)×C61C4semidirect
22. C22×D8×C61C4semidirect
28. C61×#418C4semidirect
29. C61×#416C4semidirect
264. C2×AGL(1,ℤ/8)Dic61semidirect
265. C22×D8Dic61semidirect
274. #418Dic61semidirect
275. #416Dic61semidirect
286. C61#13099semidirect

Maximal subgroups

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

subgroupindex
#214542982
#214475112
#214474692
#214457892
#214439482
#214439422
C61×#30082
#1309961