Finite Group Databasebeta

← #29609203

Subgroups — #29609203

Order 15616.

1406 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839623C211yesmaximal, characteristic
2#21444080C212yesmaximal, complemented
3#21444074C212yesmaximal, complemented
4#21444062C212yesmaximal
5#21444056C211yesmaximal, characteristic
6#21444050C212yesmaximal, complemented
7#21444046C212yesmaximal, complemented
8#21444041C212yesmaximal
9C61×#1437C211yesmaximal, characteristic
10C24×Dic61C2211yescharacteristic
11C22×#11758196C2212yes—
12C22×C22⋊Dic61C2212yes—
13C22×C22⋊Dic61C2211yescharacteristic
14C22×C4×Dic61C2211yescharacteristic
15#16839623C418yescomplemented
16C2×#14519684C2212yes—
17C2×#14519684C2212yes—
18C2×#14519684C2212yes—
19C2×#14519684C2212yes—
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2211yescharacteristic
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C22×C22⋊C4×C61C2212yes—
28C22×C22⋊C4×C61C2211yescharacteristic
29C22×C42×C61C2211yescharacteristic
30C2×#95×C61C2212yes—
31C2×#95×C61C2211yescharacteristic
32C23×Dic61—24no—
33C23×Dic61—41no—
34C23×Dic61—41no—
35C23×Dic61—42no—
36C23×Dic61C2×C414yes—
37C23×Dic61C2312yes—
38C23×Dic61C2312yes—
39C23×Dic61C2312yes—
40C23×Dic61C2312yes—
41C2×#11758196—24no—
42C2×#11758196—42no—
43C2×#11758196—42no—
44C2×C22⋊Dic61—24no—
45C2×C22⋊Dic61—24no—
46C2×C22⋊Dic61—42no—
47C2×C22⋊Dic61—42no—
48C2×C22⋊Dic61C2×C414yes—
49C2×C4×Dic61—41no—
50C2×C4×Dic61—42no—
51#14519684—24no—
52#14519684—24no—
53#14519684—42no—
54#14519684—42no—
55#14519684—42no—
56#14519684—42no—
57#14519684—42no—
58#14519684—42no—
59#14519684—41no—
60#14519684C2×C418yes—
61#14519684C2×C418yes—
62C25×C61C2311yescharacteristic
63C23×C4×C61C2312yes—
64C23×C4×C61C2311yescharacteristic
65C23×C4×C61C2311yescharacteristic
66C23×C4×C61C2312yes—
67C2×C22⋊C4×C61—24no—
68C2×C22⋊C4×C61—24no—
69C2×C22⋊C4×C61—41no—
70C2×C22⋊C4×C61—41no—
71C2×C22⋊C4×C61—42no—
72C2×C22⋊C4×C61C2×C414yes—
73C2×C42×C61—24no—
74C2×C42×C61—41no—
75#95×C61—42no—
76#95×C61—41no—
77C24×C61—22no—
78C24×C61—22no—
79C24×C61—21no—
80C24×C61—21no—
81C24×C61—22no—
82C24×C61—22no—
83C24×C61—22no—
84C24×C61C22×C412yes—
85C24×C61#4114yes—
86C24×C61C2411yescharacteristic
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—21no—
95C22×C4×C61—22no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61C22×C412yes—
109C22×C4×C61C22×C412yes—
110C22×C4×C61#4114yes—
111C22×Dic61—28no—
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—44no—
138C22×Dic61—44no—
139C22×Dic61—48no—
140C22×Dic61—48no—
141C22×Dic61C22×C414yes—
142C22×Dic61C22×C414yes—
143C22×Dic61#4114yes—
144C22⋊C4×C61—44no—
145C22⋊C4×C61—48no—
146C42×C61—44no—
147C22⋊Dic61—44no—
148C22⋊Dic61—48no—
149#11758196—48no—
150C4×Dic61—44no—
151C23×C61—24no—
152C23×C61—22no—
153C23×C61—22no—
154C23×C61—24no—
155C23×C61—24no—
156C23×C61—24no—
157C23×C61—24no—
158C23×C61—24no—
159C23×C61—24no—
160C23×C61—24no—
161C23×C61—42no—
162C23×C61—42no—
163C23×C61—42no—
164C23×C61—42no—
165C23×C61—41no—
166C23×C61—42no—
167C23×C61—41no—
168C23×C61—41no—
169C23×C61—42no—
170C23×C61—41no—
171C23×C61#11714yes—
172C23×C61C23×C411yescharacteristic
173C23×C61C2×#4112yes—
174C23×C61C2×#4112yes—
175C23×C61C2×#4112yes—
176C23×C61H(5,𝔽2)12yes—
177C23×C61H(5,𝔽2)11yescharacteristic
178C23×C61H(5,𝔽2)11yescharacteristic
179C23×C61H(5,𝔽2)12yes—
180C23×C612−1+412yes—
181C2×C4×C61—24no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—24no—
185C2×C4×C61—24no—
186C2×C4×C61—24no—
187C2×C4×C61—24no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—41no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—41no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
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—416no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234#14507—611nomaximal, Sylow
235C22×C61—24no—
236C22×C61—28no—
237C22×C61—44no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—44no—
241C22×C61—44no—
242C22×C61—44no—
243C22×C61—44no—
244C22×C61C2×#11711yescharacteristic
245C22×C61C4×#4112yes—
246C22×C61#51811yescharacteristic
247C22×C61#51811yescharacteristic
248C22×C61#52012yes—
249C22×C61#53812yes—
250C22×C61#53812yes—
251C22×C61#53812yes—
252C22×C61#53912yes—
253C22×C61#54012yes—
254C22×C61#54212yes—
255C22×C61#55112yes—
256C22×C61#55112yes—
257C22×C61#55111yescharacteristic
258C22×C61#55212yes—
259C22×C61#55212yes—
260C22×C61#55312yes—
261C22×C61#55911yescharacteristic
262C22×C61#56011yescharacteristic
263C22×C61#56212yes—
264C22×C61#56311yescharacteristic
265C4×C61—42no—
266C4×C61—42no—
267C4×C61—44no—
268C4×C61—82no—
269Dic61—44no—
270Dic61—44no—
271Dic61—44no—
272Dic61—82no—
273C2×#381—611no—
274#1443—612no—
275#1442—612no—
276#1439—612no—
277#1437—611no—
278#1437D6111yescharacteristic
279#1436—612no—
280#1434—612no—
281C22⋊#95—612no—
282C2×C61—44no—
283C2×C61#231211yescharacteristic
284C2×C61#233512yes—
285C2×C61#234312yes—
286C2×C61#237411yescharacteristic
287C2×C61#237611yescharacteristic
288C2×C61#274312yes—
289C2×C61#274912yes—
290C2×C61#275112yes—
291C2×C61#276512yes—
292C24×C4—611no—
293C22×C4⋊C4—612no—
294C22×C22⋊C4—611no—
295C22×C22⋊C4—612no—
296C22×C22⋊C4C2×D6112yes—
297C22×C22⋊C4C2×D6111yescharacteristic
298C22×C42—611no—
299C22×C42C2×D6111yescharacteristic
300#381—618no—
301C2×#95—612no—
302C2×#95—612no—
303C2×#95—612no—
304C2×#95—612no—
305C2×#95—612no—
306C2×#95—612no—
307C2×#95—612no—
308C2×#95—612no—
309C2×#95—612no—
310C2×#95—611no—
311C2×#95—612no—
312C2×#95C2×D6112yes—
313C2×#95C2×D6111yescharacteristic
314C61#1450711yescharacteristic, complemented, Sylow
315C25C22×D6111yescharacteristic
316C23×C4—612no—
317C23×C4—612no—
318C23×C4—612no—
319C23×C4—614no—
320C23×C4—612no—
321C23×C4—1224no—
322C23×C4—2441no—
323C23×C4—2441no—
324C23×C4—2442no—
325C23×C4C22×D6112yes—
326C23×C4C22×D6111yescharacteristic
327C23×C4C22×D6111yescharacteristic
328C23×C4C22×D6112yes—
329C2×C4⋊C4—1224no—
330C2×C4⋊C4—2442no—
331C2×C4⋊C4—2442no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—41no—
335C2×C22⋊C4—41no—
336C2×C22⋊C4—42no—
337C2×C22⋊C4—614no—
338C2×C22⋊C4—1224no—
339C2×C22⋊C4—1224no—
340C2×C22⋊C4—2442no—
341C2×C22⋊C4—2442no—
342C2×C22⋊C4C4×D6114yes—
343C2×C42—24no—
344C2×C42—41no—
345C2×C42—2442no—
346C2×C42—2441no—
347#95—42no—
348#95—41no—
349#95—618no—
350#95—618no—
351#95—1224no—
352#95—1224no—
353#95—2442no—
354#95—2441no—
355#95—2442no—
356#95—2442no—
357#95—2442no—
358#95—2442no—
359#95—2442no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—21no—
364C24—21no—
365C24—22no—
366C24—22no—
367C24C2×C4×D6112yes—
368C24#1175822214yes—
369C24C23×D6111yescharacteristic
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—614no—
392C22×C4—614no—
393C22×C4—614no—
394C22×C4—1222no—
395C22×C4—1222no—
396C22×C4—1222no—
397C22×C4—1222no—
398C22×C4—1222no—
399C22×C4—1222no—
400C22×C4—1222no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1228no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—2444no—
421C22×C4—2444no—
422C22×C4—2448no—
423C22×C4—2448no—
424C22×C4C2×C4×D6112yes—
425C22×C4C2×C4×D6112yes—
426C22×C4#1175822414yes—
427C4⋊C4—2448no—
428C22⋊C4—44no—
429C22⋊C4—48no—
430C22⋊C4—2444no—
431C22⋊C4—2448no—
432C42—44no—
433C42—2444no—
434C23—24no—
435C23—24no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—42no—
445C23—41no—
446C23—41no—
447C23—42no—
448C23—41no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—41no—
454C23#1451973414yes—
455C23C22×C4×D6111yescharacteristic
456C23#41×D6112yes—
457C23C2×#1175822412yes—
458C23C2×#1175822212yes—
459C23#1451982211yescharacteristic
460C23#1451982211yescharacteristic
461C23#1451982212yes—
462C23#1451982312yes—
463C23#1451982412yes—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—1224no—
489C2×C4—1224no—
490C2×C4—1224no—
491C2×C4—1224no—
492C2×C4—1224no—
493C2×C4—1224no—
494C2×C4—1224no—
495C2×C4—1224no—
496C2×C4—1224no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4—24416no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C22—24no—
518C22—28no—
519C22—42no—
520C22—44no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—42no—
526C22C2×#1451973411yescharacteristic
527C22C4×#1175822412yes—
528C22#1684023511yescharacteristic
529C22#1684023611yescharacteristic
530C22#1684024112yes—
531C22#1684034512yes—
532C22#1684034512yes—
533C22#1684034612yes—
534C22#1684035612yes—
535C22#1684036512yes—
536C22#1684037812yes—
537C22#1684043812yes—
538C22#1684044112yes—
539C22#1684044311yescharacteristic
540C22#1684044512yes—
541C22#1684045212yes—
542C22#1684045912yes—
543C22#1684049611yescharacteristic
544C22#1684049911yescharacteristic
545C22#1684050712yes—
546C22#1684051211yescharacteristic
547C4—42no—
548C4—44no—
549C4—42no—
550C4—82no—
551C4—4882no—
552C4—2444no—
553C4—2444no—
554C4—2444no—
555C2—44no—
556C2#2144841411yescharacteristic
557C2#2144861412yes—
558C2#2144867312yes—
559C2#2144895211yescharacteristic
560C2#2144896911yescharacteristic
561C2#2145296212yes—
562C2#2145304012yes—
563C2#2145307012yes—
564C2#2145325212yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1437C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenter#1437D61
hypocenterC61#14507
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. #21444080C2semidirect
3. #21444074C2semidirect
6. #21444050C2semidirect
7. #21444046C2semidirect
15. #16839623C4semidirect
314. C61#14507semidirect

Maximal subgroups

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

subgroupindex
C2×#168396232
#214440802
#214440742
#214440622
#214440562
#214440502
#214440462
#214440412
C61×#14372
#1450761