Finite Group Databasebeta

← #29597623

Subgroups — #29597623

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454283C211yesmaximal, characteristic, complemented
2#21447491C211yesmaximal, characteristic, complemented
3#21447469C211yesmaximal, characteristic, complemented
4#21445794C211yesmaximal, characteristic, complemented
5#21443948C211yesmaximal, characteristic, complemented
6#21443940C211yesmaximal, characteristic, complemented
7C61×#3010C211yesmaximal, characteristic
8#16840240—21no—
9#16839779—21no—
10#16839771—21no—
11#16839746C2211yescharacteristic
12#16839736—22no—
13#16839736—22no—
14#16839736—21no—
15#16839736C2211yescharacteristic
16#16839736C2211yescharacteristic
17#16839736C2211yescharacteristic
18#16839555—22no—
19#16839549—22no—
20C2×2−1+4×C61C2211yescharacteristic
21C2×AGL(1,ℤ/8)×C61C411yescharacteristic, complemented
22C22×SD16×C61C411yescharacteristic, complemented
23C61×#535C2211yescharacteristic
24C61×#449—21no—
25C61×#449—21no—
26C61×#448—21no—
27C61×#448—21no—
28C61×#419C411yescharacteristic, complemented
29C61×#416C411yescharacteristic, complemented
30C61×#408C2211yescharacteristic
31C2×#11758194—21no—
32Q8×Dic61—42no—
33Q8×Dic61—42no—
34#14519736—24no—
35#14519736—21no—
36#14519736—21no—
37#14519736—21no—
38#14519704—42no—
39#14519701—42no—
40#14519696—21no—
41#14519694—21no—
42#14519692—24no—
43#14519692—21no—
44#14519692—21no—
45#14519692—21no—
46#14519692—21no—
47#14519692D412yes—
48#14519692D412yes—
492−1+4×C61—42no—
502−1+4×C61—42no—
51C2×#41×C61—21no—
52C2×#41×C61—21no—
53C2×#41×C61—21no—
54C2×#41×C61C2×C411yescharacteristic
55C2×#41×C61D411yescharacteristic
56C2×#41×C61D411yescharacteristic
57C2×#41×C61C2311yescharacteristic
58C22×Q8×C61—21no—
59C22×Q8×C61C2×C411yescharacteristic
60C22×Q8×C61D411yescharacteristic
61C22×Q8×C61D411yescharacteristic
62C22×D4×C61C2×C411yescharacteristic
63AGL(1,ℤ/8)×C61—42no—
64C2×SD16×C61—21no—
65C2×SD16×C61—21no—
66C2×SD16×C61—21no—
67C2×SD16×C61—42no—
68C2×D8×C61—21no—
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#103×C61—21no—
80#103×C61—21no—
81#102×C61—21no—
82C22⋊C8×C61—21no—
83C22⋊C8×C61—21no—
84C24×C61—21no—
85#41×C61—22no—
86#41×C61—22no—
87#41×C61—41no—
88#41×C61—41no—
89#41×C61—41no—
90#41×C61—41no—
91#41×C61—41no—
92#41×C61—41no—
93#41×C61—44no—
94#41×C61—44no—
95#41×C61—44no—
96C2×Q8×C61—22no—
97C2×Q8×C61—22no—
98C2×Q8×C61—21no—
99C2×Q8×C61—21no—
100C2×Q8×C61—21no—
101C2×Q8×C61—22no—
102C2×Q8×C61—22no—
103C2×Q8×C61—41no—
104C2×Q8×C61—41no—
105C2×Q8×C61—44no—
106C2×Q8×C61C22⋊C411yescharacteristic
107C2×Q8×C61C22⋊C411yescharacteristic
108C2×Q8×C61C22⋊C411yescharacteristic
109C2×Q8×C61#4111yescharacteristic
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—41no—
116C2×D4×C61—41no—
117C2×D4×C61—41no—
118C2×D4×C61—41no—
119C2×D4×C61C22⋊C411yescharacteristic
120C2×D4×C61C2×D411yescharacteristic
121C2×D4×C61C2×D411yescharacteristic
122C2×D4×C61#4111yescharacteristic
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
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—
133SD16×C61—42no—
134SD16×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—22no—
146C22⋊Dic61—22no—
147C22⋊Dic61—41no—
148C4⋊Dic61—44no—
149C4⋊Dic61—41no—
150C4⋊Dic61—41no—
151C4⋊Dic61—41no—
152C4×Dic61—44no—
153C4×Dic61—41no—
154C4×Dic61—41no—
155C4×Dic61—41no—
156#11758194—41no—
157#11758194—41no—
158C2×#92418—41no—
159C23×C61—21no—
160C23×C61—41no—
161C23×C61—81no—
162C23×C61—161no—
163C23×C61C4×D411yescharacteristic
164C23×C61#12011yescharacteristic
165C23×C61#12111yescharacteristic
166Q8×C61—24no—
167Q8×C61—21no—
168Q8×C61—21no—
169Q8×C61—21no—
170Q8×C61—21no—
171Q8×C61—41no—
172Q8×C61—42no—
173Q8×C61—42no—
174Q8×C61#9912yes—
175Q8×C61#9912yes—
176D4×C61—21no—
177D4×C61—21no—
178D4×C61—41no—
179D4×C61—41no—
180D4×C61—41no—
181D4×C61—42no—
182D4×C61—42no—
183D4×C61—42no—
184D4×C61—41no—
185D4×C61—41no—
186D4×C61—81no—
187D4×C61—161no—
188D4×C61—161no—
189C2×C4×C61—24no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—41no—
205C2×C4×C61C2×C22⋊C411yescharacteristic
206C2×C4×C61C4×D411yescharacteristic
207C2×C4×C61#12111yescharacteristic
208C2×C4×C61#12311yescharacteristic
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—81no—
212C8×C61—41no—
213C8×C61—41no—
214#92418—81no—
215#13103—611nomaximal, Sylow
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—41no—
220C22×C61—41no—
221C22×C61—41no—
222C22×C61—41no—
223C22×C61—81no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61—81no—
227C22×C61—162no—
228C22×C61#38611yescharacteristic
229C4×C61—21no—
230C4×C61—22no—
231C4×C61—22no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—41no—
235C4×C61—81no—
236C4×C61C2×#9911yescharacteristic
237C4×C61C2×#9911yescharacteristic
238Dic61—82no—
239Dic61—82no—
240Dic61—161no—
241#3010D6111yescharacteristic
242#2884—611no—
243#2128—611no—
244#2122—611no—
245#1796—611no—
246#1395—611no—
247#1391—611no—
248C2×C61—21no—
249C2×C61—41no—
250C2×C61—41no—
251C2×C61—41no—
252C2×C61—81no—
253C2×C61—161no—
254C2×C61#189811yescharacteristic
255C2×2−1+4C2×D6111yescharacteristic
256C2×AGL(1,ℤ/8)Dic6111yescharacteristic, complemented
257C22×SD16Dic6111yescharacteristic, complemented
258#535C2×D6111yescharacteristic
259#519—1221no—
260#449—21no—
261#449—21no—
262#448—21no—
263#448—21no—
264#421—1221no—
265#419—1221no—
266#419Dic6111yescharacteristic, complemented
267#416Dic6111yescharacteristic, complemented
268#413—611no—
269#410—611no—
270#410—611no—
271#410—611no—
272#410—1222no—
273#410—1222no—
274#410—1221no—
275#408C2×D6111yescharacteristic
276#353—1222no—
277AΣL(1,𝔽16)—1222no—
278C61#1310311yescharacteristic, complemented, Sylow
2792−1+4—42no—
2802−1+4—42no—
281C2×#41—21no—
282C2×#41—21no—
283C2×#41—21no—
284C2×#41C2×Dic6111yescharacteristic
285C2×#41C22⋊D6111yescharacteristic
286C2×#41C22⋊D6111yescharacteristic
287C2×#41C22×D6111yescharacteristic
288C22×Q8—21no—
289C22×Q8C2×Dic6111yescharacteristic
290C22×Q8C22⋊D6111yescharacteristic
291C22×Q8C22⋊D6111yescharacteristic
292C22×D4C2×Dic6111yescharacteristic
293AGL(1,ℤ/8)—42no—
294C2×SD16—21no—
295C2×SD16—21no—
296C2×SD16—21no—
297C2×SD16—42no—
298C2×D8—21no—
299C2×M16—1221no—
300C2×M16C2×Dic6111yescharacteristic
301C22×C8C2×Dic6111yescharacteristic
302#127—21no—
303#124—21no—
304#121—21no—
305#121—21no—
306#120—21no—
307#120—21no—
308C4×Q8—2442no—
309C4×Q8—2442no—
310#117—1224no—
311#117—1221no—
312#117—1221no—
313#117—1221no—
314#117C2×Dic6111yescharacteristic
315#104—2442no—
316#103—21no—
317#103—21no—
318#103—21no—
319#103—2442no—
320#102—21no—
321#101—1221no—
322#100—1221no—
323#99—612no—
324#99—612no—
325#99—1224no—
326#99—1221no—
327#99—1221no—
328#99—1221no—
329#99—1221no—
330C22⋊C8—21no—
331C22⋊C8—21no—
332C24—21no—
333#41—22no—
334#41—22no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—41no—
340#41—41no—
341#41—44no—
342#41—44no—
343#41—44no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—44no—
354C2×Q8C22⋊Dic6111yescharacteristic
355C2×Q8C22⋊Dic6111yescharacteristic
356C2×Q8C22⋊Dic6111yescharacteristic
357C2×Q8#1175822411yescharacteristic
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4C22⋊Dic6111yescharacteristic
368C2×D4D4×D6111yescharacteristic
369C2×D4D4×D6111yescharacteristic
370C2×D4#1175822411yescharacteristic
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4C22×Dic6111yescharacteristic
378C22×C4C2×C22⋊D6111yescharacteristic
379C22×C4C2×C22⋊D6111yescharacteristic
380SD16—42no—
381SD16—42no—
382SD16—42no—
383D8—42no—
384M16—41no—
385M16—2441no—
386M16—2441no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—41no—
390C2×C8—2441no—
391C4⋊C4—21no—
392C4⋊C4—2441no—
393C4⋊C4—2444no—
394C4⋊C4—2441no—
395C4⋊C4—2441no—
396C22⋊C4—21no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—1222no—
400C22⋊C4—1222no—
401C22⋊C4—2441no—
402C42—21no—
403C42—2441no—
404C42—2444no—
405C42—2441no—
406C42—2441no—
407C23—21no—
408C23—41no—
409C23—81no—
410C23—161no—
411C23D4×Dic6111yescharacteristic
412C23#1451974511yescharacteristic
413C23#1451974911yescharacteristic
414Q8—24no—
415Q8—21no—
416Q8—21no—
417Q8—21no—
418Q8—21no—
419Q8—41no—
420Q8—42no—
421Q8—42no—
422Q8#1451969212yes—
423Q8#1451969212yes—
424D4—21no—
425D4—21no—
426D4—41no—
427D4—41no—
428D4—41no—
429D4—42no—
430D4—42no—
431D4—42no—
432D4—41no—
433D4—41no—
434D4—81no—
435D4—161no—
436D4—161no—
437C2×C4—24no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—4881no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4C2×C22⋊Dic6111yescharacteristic
457C2×C4D4×Dic6111yescharacteristic
458C2×C4#1451974911yescharacteristic
459C2×C4#1451975811yescharacteristic
460C8—41no—
461C8—41no—
462C8—4881no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—162no—
475C22#1683964211yescharacteristic
476C4—21no—
477C4—22no—
478C4—22no—
479C4—21no—
480C4—21no—
481C4—41no—
482C4—81no—
483C4—9761no—
484C4—4882no—
485C4—4882no—
486C4C2×#1451969211yescharacteristic
487C4C2×#1451969211yescharacteristic
488C2—21no—
489C2—41no—
490C2—41no—
491C2—41no—
492C2—81no—
493C2—161no—
494C2#2144627311yescharacteristic
subgroupG/H
centerC2#21446273
commutatorC22×C4×C61C22×C4
FrattiniC2×#41C22×D61
Fitting, prime radicalC61×#3010C2
socleC2×C61#1898
cosocleC2×#41×C61C23
hypercenter#3010D61
hypocenterC61#13103
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ #3010

Complements

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

NH ≅ G/Ntype
1. #21454283C2semidirect
2. #21447491C2semidirect
3. #21447469C2semidirect
4. #21445794C2semidirect
5. #21443948C2semidirect
6. #21443940C2semidirect
21. C2×AGL(1,ℤ/8)×C61C4semidirect
22. C22×SD16×C61C4semidirect
28. C61×#419C4semidirect
29. C61×#416C4semidirect
256. C2×AGL(1,ℤ/8)Dic61semidirect
257. C22×SD16Dic61semidirect
266. #419Dic61semidirect
267. #416Dic61semidirect
278. C61#13103semidirect

Maximal subgroups

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

subgroupindex
#214542832
#214474912
#214474692
#214457942
#214439482
#214439402
C61×#30102
#1310361