Finite Group Databasebeta

← #29887238

Subgroups — #29887238

Order 15616.

1770 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454106C212yesmaximal
2#21452973C214yesmaximal, complemented
3#21452904C214yesmaximal, complemented
4#21452393C214yesmaximal, complemented
5#21452270C214yesmaximal, complemented
6#21451933C211yesmaximal, characteristic, complemented
7#21451789C211yesmaximal, characteristic, complemented
8#21450320C214yesmaximal, complemented
9#21450059C214yesmaximal
10#21448552C212yesmaximal, complemented
11C2×C61×#381C211yesmaximal, characteristic
12C2×#14519770C2214yes—
13C2×#14519756C2212yes—
14C2×#14519756C2212yes—
15C2×#14519756C2212yes—
16#16839704C2214yescomplemented
17#16839703C2214yescomplemented
18#16839703C2214yescomplemented
19#16839702C2214yescomplemented
20#16839696C2214yescomplemented
21#16839696C2214yescomplemented
22#16839694C2214yescomplemented
23#16839690C2214yes—
24#16839690C2214yes—
25#16839687C2214yes—
26#16839685C2214yes—
27#16839683C2214yescomplemented
28#16839666C2214yescomplemented
29#16839659C2214yescomplemented
30#16839659C2214yescomplemented
31#16839656C2214yes—
32#16839655C2214yes—
33#16839650C2214yes—
34#16839649C2214yescomplemented
35#16839647C2214yescomplemented
36#16839645C2214yescomplemented
37#16839644C2214yescomplemented
38#16839635C2214yes—
39#16839635C2214yes—
40#16839635C2214yes—
41#16839635C2214yes—
42#16839635C2214yes—
43#16839626C2214yescomplemented
44#16839622C2212yes—
45#16839622C2212yes—
46#16839622C2212yes—
47#16839622C2214yes—
48#16839618C2214yes—
49C4×C22⋊Dic61C2214yes—
50C24×C4×C61C2211yescharacteristic
51C22×C22⋊C4×C61C2212yes—
52C61×#381C2214yes—
53C61×#381C2214yes—
54C2×#95×C61C2211yescharacteristic
55C2×#95×C61C2212yes—
56C2×#95×C61C2211yescharacteristic
57C22×Dic122C2312yes—
58#14519770—48no—
59#14519756—28no—
60#14519756—44no—
61#14519756—44no—
62C2×C4⋊Dic61C2312yes—
63C2×C4⋊Dic61C2314yes—
64C2×C4⋊Dic61C2314yes—
65C2×C4⋊Dic61C2312yes—
66C2×C4⋊Dic61C2312yes—
67C2×#11758196C2314yes—
68C2×#11758196C2314yes—
69C2×#11758196C2314yes—
70C2×#11758196C2314yes—
71C2×#11758196C2314yes—
72C2×#11758196C2314yes—
73C2×#11758196C2314yes—
74C2×C22⋊Dic61C2314yes—
75C2×C22⋊Dic61C2314yes—
76C2×C22⋊Dic61C2314yes—
77C2×C22⋊Dic61C2314yes—
78C2×C22⋊Dic61C2314yes—
79C2×C22⋊Dic61C2314yes—
80C2×C4×Dic61C2314yes—
81C2×C4×Dic61C2314yes—
82#14519684C2314yes—
83#14519684C2314yes—
84#14519684C2314yes—
85#14519684C2314yes—
86#14519684C2314yes—
87#14519684C2314yes—
88#14519684C2314yes—
89#14519684C2314yes—
90#14519684C2314yes—
91#14519684C2314yes—
92#14519684C2314yes—
93C25×C61C2311yescharacteristic
94C23×C4×C61—24no—
95C23×C4×C61—44no—
96C23×C4×C61C2312yes—
97C23×C4×C61C2312yes—
98C23×C4×C61C2311yescharacteristic
99C23×C4×C61C2312yes—
100C23×C4×C61C2312yes—
101C23×C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—44no—
103C2×C22⋊C4×C61C2314yes—
104C2×C22⋊C4×C61C2314yes—
105#95×C61C2314yes—
106#95×C61C2314yes—
107#95×C61C2314yes—
108#95×C61C2314yes—
109C24×C61—22no—
110C24×C61—24no—
111C24×C61—24no—
112C24×C61—41no—
113C24×C61—41no—
114C24×C61C2412yes—
115C24×C61C2411yescharacteristic
116C22×C4×C61—22no—
117C22×C4×C61—24no—
118C22×C4×C61—24no—
119C22×C4×C61—24no—
120C22×C4×C61—24no—
121C22×C4×C61—24no—
122C22×C4×C61—24no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61—42no—
129C22×C4×C61—42no—
130C22×C4×C61—42no—
131C22×C4×C61—416no—
132C22×C4×C61#4114yes—
133C22×C4×C61C2414yes—
134C22×C4×C61C2414yes—
135C22×C4×C61C2411yescharacteristic
136C22×C4×C61C2411yescharacteristic
137C22×C4×C61C2412yes—
138C22×Dic61C2414yes—
139C22×Dic61C2414yes—
140C22×Dic61C2414yes—
141C22×Dic61C2414yes—
142C2×Dic122—24no—
143C2×Dic122—42no—
144C2×Dic122—42no—
145C22⋊C4×C61—28no—
146C22⋊C4×C61—44no—
147C22⋊Dic61—28no—
148C22⋊Dic61—28no—
149C22⋊Dic61—28no—
150C22⋊Dic61—28no—
151C22⋊Dic61—44no—
152C22⋊Dic61—44no—
153C4⋊Dic61—24no—
154C4⋊Dic61—44no—
155C4⋊Dic61—42no—
156C4⋊Dic61—42no—
157C4⋊Dic61—44no—
158#11758196—28no—
159#11758196—44no—
160#11758196—44no—
161#11758196—44no—
162#11758196—44no—
163#11758196—44no—
164#11758196—44no—
165C4×Dic61—44no—
166C4×Dic61—44no—
167C23×C61—22no—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—28no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—48no—
178C23×C61—48no—
179C23×C61—44no—
180C23×C61—44no—
181C23×C61C2×#4112yes—
182C23×C61H(5,𝔽2)14yes—
183C23×C612−1+414yes—
184C23×C61C2511yescharacteristic
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—24no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—24no—
193C2×C4×C61—24no—
194C2×C4×C61—24no—
195C2×C4×C61—24no—
196C2×C4×C61—24no—
197C2×C4×C61—44no—
198C2×C4×C61—48no—
199C2×C4×C61—48no—
200C2×C4×C61—44no—
201C2×C4×C61—84no—
202C2×C4×C61—84no—
203C2×C4×C61C2×#4114yes—
204C2×Dic61—24no—
205C2×Dic61—24no—
206C2×Dic61—24no—
207C2×Dic61—24no—
208C2×Dic61—24no—
209C2×Dic61—24no—
210C2×Dic61—24no—
211C2×Dic61—24no—
212C2×Dic61—24no—
213C2×Dic61—24no—
214C2×Dic61—24no—
215C2×Dic61—24no—
216Dic122—84no—
217#36732—611nomaximal, Sylow
218C22×C61—22no—
219C22×C61—22no—
220C22×C61—24no—
221C22×C61—24no—
222C22×C61—24no—
223C22×C61—24no—
224C22×C61—24no—
225C22×C61—416no—
226C22×C61—42no—
227C22×C61—42no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61#54114yes—
231C22×C61C22×#4111yescharacteristic
232C22×C61C2×H(5,𝔽2)12yes—
233C22×C61C2×2−1+412yes—
234C22×C61#58511yescharacteristic
235C22×C61#58511yescharacteristic
236C4×C61—24no—
237C4×C61—41no—
238C4×C61—41no—
239C4×C61—44no—
240C4×C61—44no—
241Dic61—44no—
242Dic61—44no—
243Dic61—44no—
244Dic61—44no—
245#2848—612no—
246#2744—614no—
247#2739—614no—
248#2698—614no—
249#2689—614no—
250#2659—611no—
251#2646—611no—
252#2505—614no—
253#2480—614no—
254#2329—612no—
255C2×#381D6111yescharacteristic
256C2×C61—24no—
257C2×C61—44no—
258C2×C61C2×#54111yescharacteristic
259C2×C61#349011yescharacteristic
260C2×C61#349311yescharacteristic
261C2×C61#353512yes—
262C2×C61#353512yes—
263C24×C4C2×D6111yescharacteristic
264C2×#126—614no—
265C2×C22⋊Q8—612no—
266C2×C22⋊Q8—612no—
267C2×C22⋊Q8—612no—
268C22×C22⋊C4C2×D6112yes—
269#400—614no—
270#400—614no—
271#400—614no—
272#400—614no—
273#398—614no—
274#398—614no—
275#398—614no—
276#397—614no—
277#397—614no—
278#397—614no—
279#396—614no—
280#395—614no—
281#391—614no—
282#389—614no—
283#389—614no—
284#388—614no—
285#388—614no—
286#388—614no—
287#387—614no—
288#387—614no—
289#387—614no—
290#387—614no—
291#385—614no—
292#385—614no—
293#385—614no—
294#385—614no—
295#385—614no—
296#382—614no—
297#381C2×D6114yes—
298#381C2×D6114yes—
299#380—612no—
300#380—612no—
301#380—612no—
302#380—614no—
303#379—614no—
304C4×C22⋊C4—614no—
305C2×#95C2×D6111yescharacteristic
306C2×#95C2×D6111yescharacteristic
307C2×#95C2×D6112yes—
308C61#3673211yescharacteristic, complemented, Sylow
309C25C22×D6111yescharacteristic
310C22×Q8—612no—
311C23×C4—24no—
312C23×C4—44no—
313C23×C4C22×D6112yes—
314C23×C4C22×D6112yes—
315C23×C4C22×D6112yes—
316C23×C4C22×D6112yes—
317C23×C4C22×D6111yescharacteristic
318C23×C4C22×D6111yescharacteristic
319#126—2448no—
320C22⋊Q8—1228no—
321C22⋊Q8—2444no—
322C22⋊Q8—2444no—
323C2×C4⋊C4—612no—
324C2×C4⋊C4—614no—
325C2×C4⋊C4—612no—
326C2×C4⋊C4—612no—
327C2×C4⋊C4—614no—
328C2×C4⋊C4—614no—
329C2×C4⋊C4—614no—
330C2×C4⋊C4—614no—
331C2×C4⋊C4—614no—
332C2×C4⋊C4—614no—
333C2×C4⋊C4—614no—
334C2×C4⋊C4—614no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—614no—
337C2×C22⋊C4—614no—
338C2×C22⋊C4—614no—
339C2×C22⋊C4—614no—
340C2×C22⋊C4—614no—
341C2×C22⋊C4—614no—
342C2×C22⋊C4C22×D6114yes—
343C2×C22⋊C4C22×D6114yes—
344C2×C42—614no—
345C2×C42—614no—
346#95—614no—
347#95—614no—
348#95—614no—
349#95—614no—
350#95—614no—
351#95—614no—
352#95—614no—
353#95—614no—
354#95—614no—
355#95—614no—
356#95—614no—
357#95C22×D6114yes—
358#95C22×D6114yes—
359#95C22×D6114yes—
360#95C22×D6114yes—
361C24—22no—
362C24—24no—
363C24—24no—
364C24—41no—
365C24—41no—
366C24C23×D6112yes—
367C24C23×D6111yescharacteristic
368C2×Q8—1224no—
369C2×Q8—2442no—
370C2×Q8—2442no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—416no—
387C22×C4—614no—
388C22×C4—614no—
389C22×C4—614no—
390C22×C4—614no—
391C22×C4#1175822214yes—
392C22×C4C23×D6114yes—
393C22×C4C23×D6114yes—
394C22×C4C23×D6111yescharacteristic
395C22×C4C23×D6111yescharacteristic
396C22×C4C23×D6112yes—
397C4⋊C4—1228no—
398C4⋊C4—1224no—
399C4⋊C4—2444no—
400C4⋊C4—2444no—
401C4⋊C4—2444no—
402C4⋊C4—2444no—
403C4⋊C4—2444no—
404C4⋊C4—2444no—
405C4⋊C4—2444no—
406C4⋊C4—2442no—
407C4⋊C4—2442no—
408C4⋊C4—2444no—
409C22⋊C4—28no—
410C22⋊C4—44no—
411C22⋊C4—1228no—
412C22⋊C4—1228no—
413C22⋊C4—1228no—
414C22⋊C4—1228no—
415C22⋊C4—2444no—
416C22⋊C4—2444no—
417C42—2444no—
418C42—2444no—
419C23—22no—
420C23—21no—
421C23—21no—
422C23—28no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—48no—
430C23—48no—
431C23—44no—
432C23—44no—
433C23C2×#1175822212yes—
434C23#1451982214yes—
435C23#1451982414yes—
436C23C24×D6111yescharacteristic
437Q8—4884no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—44no—
451C2×C4—48no—
452C2×C4—48no—
453C2×C4—44no—
454C2×C4—84no—
455C2×C4—84no—
456C2×C4—1224no—
457C2×C4—1224no—
458C2×C4—1224no—
459C2×C4—1224no—
460C2×C4—1224no—
461C2×C4—1224no—
462C2×C4—1224no—
463C2×C4—1224no—
464C2×C4—1224no—
465C2×C4—1224no—
466C2×C4—1224no—
467C2×C4—1224no—
468C2×C4C2×#1175822214yes—
469C22—22no—
470C22—24no—
471C22—24no—
472C22—22no—
473C22—24no—
474C22—24no—
475C22—24no—
476C22—416no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22#1684037414yes—
482C22C22×#1175822211yescharacteristic
483C22C2×#1451982212yes—
484C22C2×#1451982412yes—
485C22#1684059811yescharacteristic
486C22#1684059811yescharacteristic
487C4—24no—
488C4—41no—
489C4—41no—
490C4—44no—
491C4—44no—
492C4—2444no—
493C4—2444no—
494C4—2444no—
495C4—2444no—
496C2—24no—
497C2—44no—
498C2C2×#1684037411yescharacteristic
499C2#2145950811yescharacteristic
500C2#2145953911yescharacteristic
501C2#2146003212yes—
502C2#2146003212yes—
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC2×C61×#381C2
prime radicalC25×C61C23
hypercenterC2×#381D61
hypocenterC61#36732
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
2. #21452973C2semidirect
3. #21452904C2semidirect
4. #21452393C2semidirect
5. #21452270C2semidirect
6. #21451933C2semidirect
7. #21451789C2semidirect
8. #21450320C2semidirect
10. #21448552C2semidirect
16. #16839704C22semidirect
17. #16839703C22semidirect
18. #16839703C22semidirect
19. #16839702C22semidirect
20. #16839696C22semidirect
21. #16839696C22semidirect
22. #16839694C22semidirect
27. #16839683C22semidirect
28. #16839666C22semidirect
29. #16839659C22semidirect
30. #16839659C22semidirect
34. #16839649C22semidirect
35. #16839647C22semidirect
36. #16839645C22semidirect
37. #16839644C22semidirect
43. #16839626C22semidirect
308. C61#36732semidirect

Maximal subgroups

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

subgroupindex
#214541062
#214529732
#214529042
#214523932
#214522702
#214519332
#214517892
#214503202
#214500592
#214485522
C2×C61×#3812
#3673261