Finite Group Databasebeta

← #29608731

Subgroups — #29608731

Order 15616.

1354 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839669C212yesmaximal
2#21444084C212yesmaximal
3#21444076C212yesmaximal
4#21444072C212yesmaximal
5#21444053C212yesmaximal
6#21444053C212yesmaximal
7#21444049C211yesmaximal, characteristic
8#21444046C211yesmaximal, characteristic
9C2×C61×#391C211yesmaximal, characteristic
10C22×C4⋊Dic61C2212yes—
11C22×C4⋊Dic61C2212yes—
12C22×C4⋊Dic61C2212yes—
13C22×C4×Dic61C2212yes—
14#16839669—28no—
15C2×#14519684C2212yes—
16C2×#14519684C2212yes—
17C2×#14519684C2212yes—
18C2×#14519684C2212yes—
19C2×#14519684C2211yescharacteristic
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2211yescharacteristic
23C2×#14519684C2211yescharacteristic
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2211yescharacteristic
27C23×Q8×C61C2211yescharacteristic
28C22×C4⋊C4×C61C2211yescharacteristic
29C22×C42×C61C2211yescharacteristic
30C61×#391C418yescomplemented
31C2×#95×C61C2212yes—
32C2×#95×C61C2212yes—
33C23×Dic61C2312yes—
34C23×Dic61C2312yes—
35C23×Dic61C2312yes—
36C23×Dic61C2312yes—
37C2×C4⋊Dic61—24no—
38C2×C4⋊Dic61—24no—
39C2×C4⋊Dic61—24no—
40C2×C4⋊Dic61—24no—
41C2×C4⋊Dic61—42no—
42C2×C4⋊Dic61—42no—
43C2×C4⋊Dic61—42no—
44C2×C4⋊Dic61—42no—
45C2×C4⋊Dic61—42no—
46C2×C4×Dic61—24no—
47C2×C4×Dic61—42no—
48C2×C4×Dic61—42no—
49#14519684—24no—
50#14519684—24no—
51#14519684—24no—
52#14519684—24no—
53#14519684—22no—
54#14519684—24no—
55#14519684—42no—
56#14519684—42no—
57#14519684—41no—
58#14519684—41no—
59#14519684—42no—
60#14519684Q814yescomplemented
61C22×Q8×C61—42no—
62C22×Q8×C61—42no—
63C22×Q8×C61C2×C414yes—
64C22×Q8×C61D414yes—
65C22×Q8×C61Q814yes—
66C23×C4×C61C2312yes—
67C23×C4×C61C2311yescharacteristic
68C23×C4×C61C2312yes—
69C23×C4×C61C2311yescharacteristic
70C23×C4×C61C2311yescharacteristic
71C2×C4⋊C4×C61—42no—
72C2×C4⋊C4×C61C2×C414yes—
73C2×C42×C61—42no—
74C2×C42×C61C2×C414yes—
75#95×C61C2×C418yes—
76#95×C61C2×C418yes—
77C24×C61C2411yescharacteristic
78C2×Q8×C61—44no—
79C2×Q8×C61—44no—
80C2×Q8×C61—48no—
81C2×Q8×C61—48no—
82C2×Q8×C61C4⋊C4116yescomplemented
83C22×C4×C61—22no—
84C22×C4×C61—22no—
85C22×C4×C61—22no—
86C22×C4×C61—22no—
87C22×C4×C61—22no—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—22no—
91C22×C4×C61—22no—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61—22no—
96C22×C4×C61—21no—
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—21no—
101C22×C4×C61C22×C412yes—
102C22×C4×C61C22×C414yes—
103C22×C4×C61C22×C414yes—
104C22×C4×C61C22×C412yes—
105C22×C4×C61C22×C412yes—
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D414yes—
108C22×C4×C61C2×Q814yes—
109C22×C4×C61C2×Q812yes—
110C22×C4×C61C2×Q812yes—
111C22×C4×C61#4112yes—
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×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61C2×Q814yes—
139C22×Dic61#4114yes—
140C4⋊C4×C61—44no—
141C42×C61—44no—
142C4⋊Dic61—48no—
143C4⋊Dic61—48no—
144C4⋊Dic61—48no—
145C4×Dic61—48no—
146C23×C61C23×C411yescharacteristic
147C23×C61C22×D411yescharacteristic
148C23×C61C22×Q811yescharacteristic
149C23×C61C22×Q811yescharacteristic
150C23×C61C2×#4111yescharacteristic
151C23×C61C2×#4112yes—
152C23×C61H(5,𝔽2)12yes—
153C23×C61H(5,𝔽2)12yes—
154C23×C61H(5,𝔽2)11yescharacteristic
155C23×C612−1+412yes—
156C23×C612−1+411yescharacteristic
157Q8×C61—416no—
158C2×C4×C61—22no—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—24no—
163C2×C4×C61—24no—
164C2×C4×C61—24no—
165C2×C4×C61—24no—
166C2×C4×C61—24no—
167C2×C4×C61—24no—
168C2×C4×C61—24no—
169C2×C4×C61—24no—
170C2×C4×C61—24no—
171C2×C4×C61—24no—
172C2×C4×C61—24no—
173C2×C4×C61—22no—
174C2×C4×C61—24no—
175C2×C4×C61—22no—
176C2×C4×C61—24no—
177C2×C4×C61—22no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61C2×C4⋊C418yes—
185C2×C4×C61C2×C4⋊C414yes—
186C2×C4×C61C4×Q814yes—
187C2×Dic61—24no—
188C2×Dic61—24no—
189C2×Dic61—24no—
190C2×Dic61—24no—
191C2×Dic61—24no—
192C2×Dic61—24no—
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—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215#14466—611nomaximal, Sylow
216C22×C61C22×C4⋊C411yescharacteristic
217C22×C61C2×C4×Q811yescharacteristic
218C22×C61C4×#4111yescharacteristic
219C22×C61#51811yescharacteristic
220C22×C61#51811yescharacteristic
221C22×C61#51911yescharacteristic
222C22×C61#52011yescharacteristic
223C22×C61#53412yes—
224C22×C61#53812yes—
225C22×C61#54212yes—
226C22×C61#54212yes—
227C22×C61#54312yes—
228C22×C61#54412yes—
229C22×C61#54812yes—
230C22×C61D4×Q811yescharacteristic
231C22×C61#55011yescharacteristic
232C22×C61#55112yes—
233C22×C61#55412yes—
234C22×C61#55712yes—
235C22×C61#55711yescharacteristic
236C22×C61Q8211yescharacteristic
237C22×C61#56212yes—
238C22×C61#56312yes—
239C4×C61—24no—
240C4×C61—28no—
241C4×C61—24no—
242C4×C61—44no—
243C4×C61—42no—
244Dic61—44no—
245Dic61—44no—
246Dic61—82no—
247Dic61—82no—
248C2×#391—612no—
249C2×#391D6111yescharacteristic
250#1444—612no—
251#1442—612no—
252#1441—612no—
253#1436—612no—
254#1436—612no—
255C4⋊#95—611no—
256#1434—611no—
257C2×C61#231511yescharacteristic
258C2×C61#233111yescharacteristic
259C2×C61#233711yescharacteristic
260C2×C61Q8×C4⋊C411yescharacteristic
261C2×C61#235111yescharacteristic
262C2×C61#236811yescharacteristic
263C2×C61#237811yescharacteristic
264C2×C61#272912yes—
265C2×C61#273012yes—
266C2×C61#279912yes—
267C2×C61#280612yes—
268C23×Q8C2×D6111yescharacteristic
269C22×C4⋊C4—612no—
270C22×C4⋊C4—612no—
271C22×C4⋊C4—612no—
272C22×C4⋊C4C2×D6111yescharacteristic
273C22×C42—612no—
274C22×C42C2×D6111yescharacteristic
275#391—1228no—
276#391Dic6118yescomplemented
277C2×#95—611no—
278C2×#95—612no—
279C2×#95—611no—
280C2×#95—612no—
281C2×#95—612no—
282C2×#95—612no—
283C2×#95—612no—
284C2×#95—612no—
285C2×#95—611no—
286C2×#95—611no—
287C2×#95—612no—
288C2×#95—612no—
289C2×#95C2×D6112yes—
290C2×#95C2×D6112yes—
291C61#1446611yescharacteristic, complemented, Sylow
292C22×Q8—42no—
293C22×Q8—42no—
294C22×Q8Dic12214yes—
295C22×Q8D24414yes—
296C22×Q8C2×Dic6114yes—
297C23×C4—612no—
298C23×C4—612no—
299C23×C4—612no—
300C23×C4—612no—
301C23×C4C22×D6112yes—
302C23×C4C22×D6111yescharacteristic
303C23×C4C22×D6112yes—
304C23×C4C22×D6111yescharacteristic
305C23×C4C22×D6111yescharacteristic
306C2×C4⋊C4—42no—
307C2×C4⋊C4—1224no—
308C2×C4⋊C4—1224no—
309C2×C4⋊C4—1224no—
310C2×C4⋊C4—1224no—
311C2×C4⋊C4—2442no—
312C2×C4⋊C4—2442no—
313C2×C4⋊C4—2442no—
314C2×C4⋊C4—2442no—
315C2×C4⋊C4—2442no—
316C2×C4⋊C4C2×Dic6114yes—
317C2×C42—42no—
318C2×C42—1224no—
319C2×C42—2442no—
320C2×C42—2442no—
321C2×C42C2×Dic6114yes—
322#95—614no—
323#95—1224no—
324#95—1224no—
325#95—1224no—
326#95—1224no—
327#95—1224no—
328#95—1222no—
329#95—2442no—
330#95—2442no—
331#95—2441no—
332#95—2441no—
333#95—2442no—
334#95C2×Dic6118yes—
335#95C2×Dic6118yes—
336C24C23×D6111yescharacteristic
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8C4⋊Dic61116yescomplemented
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—614no—
361C22×C4—614no—
362C22×C4—1222no—
363C22×C4—1222no—
364C22×C4—1222no—
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×C4C2×Dic12214yes—
389C22×C4C2×Dic12212yes—
390C22×C4C2×D24412yes—
391C22×C4C2×D24414yes—
392C22×C4Q8×D6112yes—
393C22×C4#1175822612yes—
394C22×C4C22×Dic6112yes—
395C22×C4C22×Dic6114yes—
396C22×C4C22×Dic6114yes—
397C22×C4C22×Dic6112yes—
398C22×C4C22×Dic6112yes—
399C4⋊C4—44no—
400C4⋊C4—2448no—
401C4⋊C4—2448no—
402C4⋊C4—2448no—
403C42—44no—
404C42—2448no—
405C23C23×Dic6111yescharacteristic
406C23C22×D24411yescharacteristic
407C23C2×Q8×D6111yescharacteristic
408C23C22×Dic12211yescharacteristic
409C23#41×D6112yes—
410C23C2×#1175822611yescharacteristic
411C23#1451982212yes—
412C23#1451982212yes—
413C23#1451982311yescharacteristic
414C23#1451982411yescharacteristic
415C23#1451982512yes—
416Q8—416no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
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—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4C2×C4⋊Dic6114yes—
472C2×C4C2×C4⋊Dic6118yes—
473C2×C4Q8×Dic6114yes—
474C22C22×C4⋊Dic6111yescharacteristic
475C22C2×Q8×Dic6111yescharacteristic
476C22#41×Dic6111yescharacteristic
477C22#1684023711yescharacteristic
478C22#1684023711yescharacteristic
479C22#1684024011yescharacteristic
480C22#1684024511yescharacteristic
481C22#1684032412yes—
482C22#1684034612yes—
483C22#1684037612yes—
484C22#1684038612yes—
485C22#1684038812yes—
486C22#1684038912yes—
487C22#1684042412yes—
488C22Q8×D24411yescharacteristic
489C22#1684043411yescharacteristic
490C22#1684043912yes—
491C22#1684046612yes—
492C22#1684048812yes—
493C22#1684049111yescharacteristic
494C22Q8×Dic12211yescharacteristic
495C22#1684050812yes—
496C22#1684051312yes—
497C4—24no—
498C4—24no—
499C4—28no—
500C4—42no—
501C4—44no—
502C4—4882no—
503C4—4882no—
504C4—2444no—
505C4—2444no—
506C2#2144843311yescharacteristic
507C2#2144856811yescharacteristic
508C2#2144863411yescharacteristic
509C2Q8×C4⋊Dic6111yescharacteristic
510C2#2144874011yescharacteristic
511C2#2144890411yescharacteristic
512C2#2144898511yescharacteristic
513C2#2145279912yes—
514C2#2145281412yes—
515C2#2145370512yes—
516C2#2145377912yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#391C2
socle, prime radical, cosocleC24×C61C24
hypercenterC2×#391D61
hypocenterC61#14466
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
30. C61×#391C4semidirect
60. #14519684Q8semidirect
82. C2×Q8×C61C4⋊C4semidirect
276. #391Dic61semidirect
291. C61#14466semidirect
341. C2×Q8C4⋊Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396692
#214440842
#214440762
#214440722
#214440532
#214440492
#214440462
C2×C61×#3912
#1446661