Finite Group Databasebeta

← #29645568

Subgroups — #29645568

Order 15616.

1122 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#21444089C212yesmaximal
2#21444089C211yesmaximal, characteristic
3#21444088C211yesmaximal, characteristic
4#21444083C212yesmaximal
5#21444072C212yesmaximal
6#21444071C212yesmaximal
7#21444071C212yesmaximal
8#21444070C212yesmaximal
9C61×#1445C211yesmaximal, characteristic
10C23×Dic122C2211yescharacteristic
11C22×C4⋊Dic61C2212yes—
12C22×C4⋊Dic61C2211yescharacteristic
13C22×#11758196C2212yes—
14C22×#11758196C2212yes—
15C2×#14519684C2212yes—
16C2×#14519684C2212yes—
17C2×#14519684C2212yes—
18C2×#14519684C2212yes—
19C2×#14519684C2212yes—
20C2×#14519684C2211yescharacteristic
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2211yescharacteristic
25C2×#14519684C2212yes—
26C22×C4⋊C4×C61C2211yescharacteristic
27C2×#95×C61C2211yescharacteristic
28C2×#95×C61C2212yes—
29C2×#95×C61C2212yes—
30C2×#95×C61C2211yescharacteristic
31C22×Dic122—42no—
32C22×Dic122—41no—
33C22×Dic122—42no—
34C22×Dic122—41no—
35C22×Dic122Q814yes—
36C23×Dic61C2312yes—
37C23×Dic61C2312yes—
38C23×Dic61C2312yes—
39C23×Dic61C2312yes—
40C2×C4⋊Dic61—24no—
41C2×C4⋊Dic61—41no—
42C2×C4⋊Dic61—41no—
43C2×C4⋊Dic61—42no—
44C2×C4⋊Dic61—42no—
45C2×C4⋊Dic61Q814yescomplemented
46C2×#11758196—24no—
47C2×#11758196—42no—
48C2×#11758196—42no—
49C2×#11758196—42no—
50C2×#11758196—42no—
51C2×#11758196—42no—
52#14519684—24no—
53#14519684—42no—
54#14519684—42no—
55#14519684—42no—
56#14519684—42no—
57#14519684—42no—
58#14519684—42no—
59#14519684—42no—
60#14519684—41no—
61#14519684—42no—
62#14519684—41no—
63C23×C4×C61C2312yes—
64C23×C4×C61C2311yescharacteristic
65C23×C4×C61C2312yes—
66C23×C4×C61C2311yescharacteristic
67C23×C4×C61C2311yescharacteristic
68C2×C4⋊C4×C61—41no—
69C2×C4⋊C4×C61—41no—
70C2×C4⋊C4×C61Q814yescomplemented
71#95×C61—41no—
72#95×C61—42no—
73#95×C61—41no—
74#95×C61—42no—
75C24×C61C2411yescharacteristic
76C22×C4×C61—22no—
77C22×C4×C61—22no—
78C22×C4×C61—22no—
79C22×C4×C61—21no—
80C22×C4×C61—22no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—22no—
85C22×C4×C61—22no—
86C22×C4×C61—21no—
87C22×C4×C61—22no—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—22no—
91C22×C4×C61—21no—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61—22no—
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61C2×Q812yes—
102C22×C4×C61C2×Q814yes—
103C22×C4×C61C2×Q812yes—
104C22×C4×C61C2×Q812yes—
105C22×Dic61—22no—
106C22×Dic61—22no—
107C22×Dic61—22no—
108C22×Dic61—22no—
109C22×Dic61—22no—
110C22×Dic61—22no—
111C22×Dic61—22no—
112C22×Dic61—22no—
113C22×Dic61—22no—
114C22×Dic61—22no—
115C22×Dic61—22no—
116C22×Dic61—22no—
117C22×Dic61—22no—
118C22×Dic61—22no—
119C22×Dic61—22no—
120C22×Dic61—22no—
121C22×Dic61—22no—
122C22×Dic61—22no—
123C22×Dic61—22no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61C2×Q814yes—
131C22×Dic61C2×Q814yes—
132C22×Dic61#4114yes—
133C2×Dic122—48no—
134C2×Dic122—44no—
135C2×Dic122—44no—
136C2×Dic122—48no—
137C2×Dic122—44no—
138C4⋊C4×C61—44no—
139C4⋊Dic61—44no—
140C4⋊Dic61—48no—
141#11758196—48no—
142#11758196—48no—
143C23×C61C22×Q811yescharacteristic
144C23×C61C22×Q811yescharacteristic
145C23×C61C22×Q811yescharacteristic
146C23×C61C2×#4112yes—
147C23×C61H(5,𝔽2)11yescharacteristic
148C23×C61H(5,𝔽2)11yescharacteristic
149C23×C61H(5,𝔽2)12yes—
150C23×C61H(5,𝔽2)12yes—
151C23×C61H(5,𝔽2)11yescharacteristic
152C23×C61H(5,𝔽2)11yescharacteristic
153C23×C612−1+412yes—
154C2×C4×C61—22no—
155C2×C4×C61—22no—
156C2×C4×C61—24no—
157C2×C4×C61—24no—
158C2×C4×C61—22no—
159C2×C4×C61—24no—
160C2×C4×C61—24no—
161C2×C4×C61—24no—
162C2×C4×C61—24no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61—41no—
167C2×C4×C61—42no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—42no—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×Dic61—24no—
184C2×Dic61—24no—
185C2×Dic61—24no—
186C2×Dic61—24no—
187C2×Dic61—24no—
188C2×Dic61—24no—
189C2×Dic61—24no—
190C2×Dic61—24no—
191C2×Dic61—24no—
192C2×Dic61—42no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211Dic122—416no—
212#17306—611nomaximal, Sylow
213C22×C61#53812yes—
214C22×C61#53812yes—
215C22×C61#53912yes—
216C22×C61#54311yescharacteristic
217C22×C61#54312yes—
218C22×C61#54311yescharacteristic
219C22×C61#54311yescharacteristic
220C22×C61#54412yes—
221C22×C61#54412yes—
222C22×C61#55412yes—
223C22×C61#55412yes—
224C22×C61#55511yescharacteristic
225C22×C61#55712yes—
226C22×C61Q8211yescharacteristic
227C22×C61Q8211yescharacteristic
228C22×C61Q8211yescharacteristic
229C22×C61#56011yescharacteristic
230C22×C61#56012yes—
231C22×C61#56011yescharacteristic
232C22×C61H(3,𝔽4)11yescharacteristic
233C22×C61#56212yes—
234C22×C61#56212yes—
235C22×C61#56311yescharacteristic
236C4×C61—42no—
237C4×C61—42no—
238C4×C61—42no—
239C4×C61—44no—
240C4×C61—82no—
241Dic61—44no—
242Dic61—44no—
243Dic61—44no—
244Dic61—82no—
245#1446—611no—
246#1446—612no—
247#1445—611no—
248#1445D6111yescharacteristic
249#1444—612no—
250#1441—612no—
251#1441—612no—
252#1441—612no—
253#1441—612no—
254C2×C61#278411yescharacteristic
255C2×C61#278411yescharacteristic
256C2×C61#279212yes—
257C2×C61#280612yes—
258C2×C61#280612yes—
259C2×C61#280911yescharacteristic
260C2×C61#281111yescharacteristic
261C2×C61#282412yes—
262C2×C61#284111yescharacteristic
263C2×C61#284111yescharacteristic
264C2×C61#284211yescharacteristic
265C23×Q8—611no—
266C22×C4⋊C4—611no—
267C22×C4⋊C4—612no—
268C22×C4⋊C4—612no—
269C22×C4⋊C4—612no—
270C22×C4⋊C4C2×D6111yescharacteristic
271C2×#95—612no—
272C2×#95—612no—
273C2×#95—611no—
274C2×#95—612no—
275C2×#95—612no—
276C2×#95—612no—
277C2×#95—612no—
278C2×#95—612no—
279C2×#95—612no—
280C2×#95—611no—
281C2×#95—612no—
282C2×#95C2×D6111yescharacteristic
283C2×#95C2×D6112yes—
284C2×#95C2×D6112yes—
285C2×#95C2×D6111yescharacteristic
286C61#1730611yescharacteristic, complemented, Sylow
287C22×Q8—614no—
288C22×Q8—2442no—
289C22×Q8—2441no—
290C22×Q8—2442no—
291C22×Q8—2441no—
292C23×C4—612no—
293C23×C4—612no—
294C23×C4—612no—
295C23×C4—612no—
296C23×C4C22×D6112yes—
297C23×C4C22×D6112yes—
298C23×C4C22×D6111yescharacteristic
299C23×C4C22×D6111yescharacteristic
300C23×C4C22×D6111yescharacteristic
301C2×C4⋊C4—41no—
302C2×C4⋊C4—41no—
303C2×C4⋊C4—614no—
304C2×C4⋊C4—1224no—
305C2×C4⋊C4—1224no—
306C2×C4⋊C4—2442no—
307C2×C4⋊C4—2442no—
308C2×C4⋊C4—2442no—
309C2×C4⋊C4—2442no—
310C2×C4⋊C4—2442no—
311C2×C4⋊C4—2441no—
312C2×C4⋊C4—2441no—
313C2×C4⋊C4—2442no—
314C2×C4⋊C4—2442no—
315C2×C4⋊C4Dic12214yescomplemented
316#95—42no—
317#95—41no—
318#95—41no—
319#95—42no—
320#95—1224no—
321#95—2442no—
322#95—2442no—
323#95—2442no—
324#95—2442no—
325#95—2442no—
326#95—2441no—
327#95—2441no—
328#95—2442no—
329#95—2442no—
330#95—2442no—
331C24C23×D6111yescharacteristic
332C2×Q8—2448no—
333C2×Q8—2444no—
334C2×Q8—2444no—
335C2×Q8—2448no—
336C2×Q8—2444no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—614no—
363C22×C4—614no—
364C22×C4—614no—
365C22×C4—1222no—
366C22×C4—1222no—
367C22×C4—1222no—
368C22×C4—1222no—
369C22×C4—1222no—
370C22×C4—1222no—
371C22×C4—1222no—
372C22×C4—1222no—
373C22×C4—1222no—
374C22×C4—1222no—
375C22×C4—1222no—
376C22×C4—1222no—
377C22×C4—1222no—
378C22×C4—1222no—
379C22×C4—1222no—
380C22×C4—1222no—
381C22×C4—1222no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4—1222no—
386C22×C4—1222no—
387C22×C4—1222no—
388C22×C4—1222no—
389C22×C4—1222no—
390C22×C4C2×Dic12214yes—
391C22×C4C2×Dic12212yes—
392C22×C4Q8×D6112yes—
393C22×C4Q8×D6112yes—
394C4⋊C4—44no—
395C4⋊C4—2444no—
396C4⋊C4—2448no—
397C4⋊C4—2448no—
398C4⋊C4—2448no—
399C23C2×Q8×D6111yescharacteristic
400C23C2×Q8×D6111yescharacteristic
401C23C22×Dic12211yescharacteristic
402C23#41×D6112yes—
403C23#1451982212yes—
404C23#1451982211yescharacteristic
405C23#1451982211yescharacteristic
406C23#1451982312yes—
407C23#1451982311yescharacteristic
408C23#1451982311yescharacteristic
409C23#1451982412yes—
410Q8—24416no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—1224no—
441C2×C4—1224no—
442C2×C4—1224no—
443C2×C4—1224no—
444C2×C4—1224no—
445C2×C4—1224no—
446C2×C4—1224no—
447C2×C4—1224no—
448C2×C4—1224no—
449C2×C4—2442no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4—2442no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4—2442no—
457C2×C4—2442no—
458C2×C4—2442no—
459C2×C4—2442no—
460C2×C4—2442no—
461C2×C4—2442no—
462C2×C4—2442no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2442no—
468C22#1684034412yes—
469C22#1684035112yes—
470C22#1684035312yes—
471C22#1684038711yescharacteristic
472C22#1684038712yes—
473C22#1684038711yescharacteristic
474C22#1684038811yescharacteristic
475C22#1684039012yes—
476C22#1684039112yes—
477C22#1684046812yes—
478C22#1684047112yes—
479C22#1684047411yescharacteristic
480C22#1684048912yes—
481C22#1684049211yescharacteristic
482C22Q8×Dic12211yescharacteristic
483C22Q8×Dic12211yescharacteristic
484C22#1684049911yescharacteristic
485C22#1684050212yes—
486C22#1684050211yescharacteristic
487C22#1684050411yescharacteristic
488C22#1684050912yes—
489C22#1684050912yes—
490C22#1684051511yescharacteristic
491C4—42no—
492C4—42no—
493C4—44no—
494C4—42no—
495C4—82no—
496C4—4882no—
497C4—2444no—
498C4—2444no—
499C4—2444no—
500C2#2145351911yescharacteristic
501C2#2145352011yescharacteristic
502C2#2145361612yes—
503C2#2145377712yes—
504C2#2145378012yes—
505C2#2145380211yescharacteristic
506C2#2145381211yescharacteristic
507C2#2145389712yes—
508C2#2145407611yescharacteristic
509C2#2145408111yescharacteristic
510C2#2145408411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutator, socle, prime radical, cosocleC24×C61C24
FittingC61×#1445C2
hypercenter#1445D61
hypocenterC61#17306
LCSG ▷ C24×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
45. C2×C4⋊Dic61Q8semidirect
70. C2×C4⋊C4×C61Q8semidirect
286. C61#17306semidirect
315. C2×C4⋊C4Dic122semidirect

Maximal subgroups

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

subgroupindex
#214440892
#214440882
#214440832
#214440722
#214440712
#214440702
C61×#14452
#1730661