Finite Group Databasebeta

← #29679814

Subgroups — #29679814

Order 15616.

1470 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D61C211yesmaximal, characteristic
2C2×#16839724C211yesmaximal, characteristic
3#21448201C211yesmaximal, characteristic
4#21445915C211yesmaximal, characteristic, complemented
5#21445914C211yesmaximal, characteristic, complemented
6#21445913C212yesmaximal, complemented
7#21445912C212yesmaximal, complemented
8#21445911C211yesmaximal, characteristic, complemented
9#21445537C212yesmaximal, complemented
10#21445537C212yesmaximal, complemented
11C61×#1825C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C22×#11758189C2212yes—
14C2×#14519736C2211yescharacteristic
15C2×#14519735C2212yes—
16C2×#14519735C2211yescharacteristic
17C2×#14519735C2211yescharacteristic
18C2×#14519734—21no—
19C2×#14519733—21no—
20#117×D61—24no—
21#117×D61C418yescomplemented
22C2×C4⋊C4×D61C2211yescharacteristic
23C2×C4⋊C4×D61C2211yescharacteristic
24C2×C22⋊C4×D61—21no—
25C2×C42×D61—21no—
26#16839724—24no—
27C2×C22⋊#92418C2211yescharacteristic
28C2×#14519689C2212yes—
29#16839624—21no—
30#16839622C2211yescharacteristic
31#16839619C2212yes—
32#16839617—21no—
33C22⋊C4×Dic61—21no—
34#16839609—21no—
35#16839525C2212yescomplemented
36#16839525C2212yescomplemented
37#16839524C2212yescomplemented
38#16839524C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C2×#117×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2211yescharacteristic
43C61×#380C2211yescharacteristic
44C61×#340C2212yescomplemented
45C61×#340C2212yescomplemented
46C24×D61D411yescharacteristic
47C23×Dic61D411yescharacteristic
48C22×C4×D61—28no—
49C22×C4×D61—21no—
50C22×C4×D61—21no—
51C22×C4×D61C2×C414yes—
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61D414yes—
55C22×C4×D61C2311yescharacteristic
56C22×C4×D61C2311yescharacteristic
57C2×#11758189—24no—
58C2×#11758189—28no—
59C22×#92418C2312yes—
60#14519736—22no—
61#14519736C2×C414yes—
62#14519735—22no—
63#14519735—22no—
64#14519735—28no—
65#14519735C2×C414yes—
66#14519735C2×C414yes—
67#14519734—28no—
68#14519733—28no—
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C4⋊C4×D61—22no—
75C4⋊C4×D61—22no—
76C4⋊C4×D61C2×C414yes—
77C4⋊C4×D61C2×C414yes—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—22no—
83C2×#11758198—22no—
84C22⋊C4×D61—28no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—22no—
89C42×D61—28no—
90C22⋊#92418—22no—
91#14519689—24no—
92#14519684—21no—
93#14519684—22no—
94C23×C4×C61C2311yescharacteristic
95C22×C8×C61C2312yes—
96#117×C61—22no—
97#117×C61C2×C414yes—
98C2×C4⋊C4×C61C2312yes—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C42×C61—21no—
104C22⋊C8×C61—22no—
105#95×C61—21no—
106C24×C61C2×D411yescharacteristic
107C23×D61—24no—
108C23×D61—42no—
109C23×D61—41no—
110C23×D61—41no—
111C23×D61C22⋊C414yes—
112C23×D61C2×D411yescharacteristic
113C23×D61C2×D411yescharacteristic
114C22×C4×C61—24no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61C22×C412yes—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—24no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—22no—
126C22×Dic61C22⋊C414yes—
127C22×Dic61C2×D411yescharacteristic
128C22×Dic61C2×D411yescharacteristic
129C2×C4×D61—24no—
130C2×C4×D61—24no—
131C2×C4×D61—24no—
132C2×C4×D61—216no—
133C2×C4×D61—216no—
134C2×C4×D61—24no—
135C2×C4×D61—24no—
136C2×C4×D61—42no—
137C2×C4×D61—44no—
138C2×C4×D61—42no—
139C2×C4×D61C22⋊C414yes—
140C2×C4×D61C22⋊C414yes—
141C2×C4×D61C22×C412yes—
142C2×C4×D61C22×C412yes—
143C2×C4×D61C2×D412yes—
144C2×C4×D61C2×D412yes—
145C2×C4×D61#4118yes—
146C2×C8×C61—22no—
147C2×C8×C61—24no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61C22×C412yes—
152C4⋊C4×C61C22×C412yes—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—42no—
155C42×C61—24no—
156C22⋊Dic61—24no—
157C22⋊Dic61—42no—
158#11758198—24no—
159#11758198—24no—
160#11758198—44no—
161#11758198—44no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—24no—
165C4⋊Dic61C22×C412yes—
166C4⋊Dic61C22×C412yes—
167#11758196—24no—
168#11758196—24no—
169C4×Dic61—24no—
170C4×Dic61—24no—
171C4×Dic61—24no—
172C4×Dic61—44no—
173C2×#92418—22no—
174C2×#92418—24no—
175#11758189—48no—
176C23×C61—22no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—22no—
180C23×C61C2×C22⋊C412yes—
181C23×C61C22×D411yescharacteristic
182C22×D61—21no—
183C22×D61—22no—
184C22×D61—21no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—21no—
188C22×D61—21no—
189C22×D61—28no—
190C22×D61—44no—
191C22×D61—44no—
192C22×D61—48no—
193C22×D61—48no—
194C22×D61C2×C22⋊C412yes—
195C22×D61C2×C22⋊C412yes—
196C22×D61#12314yes—
197C2×C4×C61—22no—
198C2×C4×C61—24no—
199C2×C4×C61—24no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—41no—
203C2×C4×C61—42no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61C2×C22⋊C412yes—
209C2×C4×C61C23×C411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C2×#4114yes—
212C2×Dic61—22no—
213C2×Dic61—28no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—41no—
217C2×Dic61—42no—
218C2×Dic61—41no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61C2×C22⋊C412yes—
223C2×Dic61C2×C22⋊C412yes—
224C2×Dic61#12314yes—
225C4×D61—28no—
226C4×D61—24no—
227C4×D61—24no—
228C4×D61—28no—
229C4×D61—48no—
230C4×D61—48no—
231C4×D61#11718yes—
232C4×D61#12318yes—
233C8×C61—42no—
234#92418—42no—
235#20701—611nomaximal, Sylow
236C22×C61—22no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61C22×C22⋊C411yescharacteristic
242C22×C61C2×#12312yes—
243C22×C61#57511yescharacteristic
244C22×C61#57512yes—
245C22×C61#57511yescharacteristic
246C2×D61—24no—
247C2×D61—24no—
248C2×D61—22no—
249C2×D61—22no—
250C2×D61—22no—
251C2×D61—22no—
252C2×D61—22no—
253C2×D61—416no—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61#38114yes—
257C4×C61—22no—
258C4×C61—42no—
259C4×C61—42no—
260C4×C61—82no—
261C4×C61C2×#11712yes—
262C4×C61C2×#12312yes—
263Dic61—22no—
264Dic61—42no—
265Dic61—42no—
266Dic61—82no—
267Dic61#38114yes—
268C22×#117—611no—
269C2×#407—611no—
270#2280—611no—
271#1825—611no—
272#1825—611no—
273#1825—612no—
274#1825—612no—
275#1825—611no—
276#1825D6111yescharacteristic, complemented
277#1736—612no—
278#1736—612no—
279C2×C61—42no—
280C2×C61C2×#38111yescharacteristic
281C2×C61#289611yescharacteristic
282C2×C61#289611yescharacteristic
283C2×C61#309312yes—
284C2×C61#309312yes—
285D61—24no—
286D61—42no—
287C24×C4—611no—
288C22×M16—612no—
289C2×#117—611no—
290C2×#117—618no—
291C2×#117—611no—
292C2×#117—611no—
293C2×#117—612no—
294C2×#117—1221no—
295C2×#117—1221no—
296C2×#117—1224no—
297C2×#117C2×D6111yescharacteristic
298C22×C4⋊C4—611no—
299C22×C4⋊C4—611no—
300C22×C22⋊C4—1221no—
301C22×C42—1221no—
302#407—1224no—
303C2×C22⋊C8—611no—
304C2×C22⋊C8—612no—
305C2×C22⋊C8C2×D6111yescharacteristic
306#381—1221no—
307#380—611no—
308#380—612no—
309#380C2×D6111yescharacteristic
310#379—1221no—
311C4×C22⋊C4—1221no—
312C4×C22⋊C4—1221no—
313#340—612no—
314#340—612no—
315#340—612no—
316#340—612no—
317#340—612no—
318#340—612no—
319#340C2×D6112yescomplemented
320#340C2×D6112yescomplemented
321C61#2070111yescharacteristic, complemented, Sylow
322C25—611no—
323C23×C4—611no—
324C23×C4—611no—
325C23×C4—614no—
326C23×C4—611no—
327C23×C4—614no—
328C23×C4—611no—
329C23×C4—611no—
330C23×C4—1228no—
331C23×C4—1221no—
332C23×C4—1221no—
333C23×C4C22×D6111yescharacteristic
334C2×M16—1224no—
335C2×M16—1228no—
336C22×C8—612no—
337C22×C8C22×D6112yes—
338#117—22no—
339#117—614no—
340#117—614no—
341#117—614no—
342#117—1228no—
343#117—1228no—
344#117—1222no—
345#117—1222no—
346#117—1222no—
347#117—1228no—
348#117C4×D6114yes—
349C2×C4⋊C4—611no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—614no—
352C2×C4⋊C4—614no—
353C2×C4⋊C4—612no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1222no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4C22×D6112yes—
359C2×C4⋊C4C22×D6111yescharacteristic
360C2×C4⋊C4C22×D6111yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—1228no—
364C2×C22⋊C4—1221no—
365C2×C22⋊C4—1221no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1221no—
370C2×C42—21no—
371C2×C42—1228no—
372C2×C42—1221no—
373C2×C42—1221no—
374C2×C42—1221no—
375C2×C42—1222no—
376C22⋊C8—22no—
377C22⋊C8—1224no—
378C22⋊C8—1222no—
379#95—21no—
380#95—1222no—
381#95—1221no—
382C24—611no—
383C24—611no—
384C24—614no—
385C24—1224no—
386C24—2441no—
387C24—2442no—
388C24—2441no—
389C24D4×D6111yescharacteristic
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—611no—
395C22×C4—612no—
396C22×C4—612no—
397C22×C4—618no—
398C22×C4—611no—
399C22×C4—614no—
400C22×C4—612no—
401C22×C4—612no—
402C22×C4—614no—
403C22×C4—614no—
404C22×C4—1224no—
405C22×C4—12216no—
406C22×C4—12216no—
407C22×C4—1224no—
408C22×C4—1224no—
409C22×C4—1224no—
410C22×C4—1224no—
411C22×C4—1224no—
412C22×C4—1221no—
413C22×C4—1221no—
414C22×C4—1222no—
415C22×C4—2442no—
416C22×C4—2444no—
417C22×C4—2442no—
418C22×C4C2×C4×D6112yes—
419C22×C4D4×D6112yes—
420C22×C4D4×D6111yescharacteristic
421C22×C4C23×D6111yescharacteristic
422M16—2448no—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—1222no—
426C2×C8—1224no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—24no—
430C4⋊C4—612no—
431C4⋊C4—612no—
432C4⋊C4—1224no—
433C4⋊C4—1224no—
434C4⋊C4—1221no—
435C4⋊C4—1221no—
436C4⋊C4—1224no—
437C4⋊C4C2×C4×D6112yes—
438C4⋊C4C2×C4×D6112yes—
439C22⋊C4—24no—
440C22⋊C4—42no—
441C22⋊C4—1224no—
442C22⋊C4—1224no—
443C22⋊C4—1224no—
444C22⋊C4—2444no—
445C22⋊C4—2444no—
446C22⋊C4—2442no—
447C42—24no—
448C42—1224no—
449C42—1224no—
450C42—1224no—
451C42—2444no—
452C23—22no—
453C23—21no—
454C23—22no—
455C23—21no—
456C23—612no—
457C23—614no—
458C23—612no—
459C23—1228no—
460C23—1221no—
461C23—1222no—
462C23—1221no—
463C23—1222no—
464C23—1221no—
465C23—1222no—
466C23—1221no—
467C23—2444no—
468C23—2444no—
469C23—2448no—
470C23—2448no—
471C23C22⋊C4×D6112yes—
472C23C2×D4×D6111yescharacteristic
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—612no—
485C2×C4—614no—
486C2×C4—618no—
487C2×C4—618no—
488C2×C4—612no—
489C2×C4—1228no—
490C2×C4—1228no—
491C2×C4—1228no—
492C2×C4—1222no—
493C2×C4—1224no—
494C2×C4—1224no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—2448no—
498C2×C4—2448no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4—2441no—
503C2×C4—2442no—
504C2×C4—2441no—
505C2×C4C22⋊C4×D6112yes—
506C2×C4C22×C4×D6111yescharacteristic
507C2×C4C2×D4×D6111yescharacteristic
508C2×C4#41×D6114yes—
509C8—42no—
510C8—2442no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—42no—
515C22—42no—
516C22—614no—
517C22—1224no—
518C22—1224no—
519C22—1222no—
520C22—1222no—
521C22—1222no—
522C22—1222no—
523C22—1222no—
524C22—24416no—
525C22—2442no—
526C22—2442no—
527C22C2×C22⋊C4×D6111yescharacteristic
528C22#123×D6112yes—
529C22#1684055811yescharacteristic
530C22#1684055912yes—
531C22#1684056111yescharacteristic
532C4—22no—
533C4—42no—
534C4—42no—
535C4—82no—
536C4—614no—
537C4—1222no—
538C4—4882no—
539C4—2442no—
540C4—2442no—
541C4#117×D6112yes—
542C4#123×D6112yes—
543C2—42no—
544C2—1224no—
545C2—2442no—
546C2#381×D6111yescharacteristic
547C2#2145433411yescharacteristic
548C2#2145433511yescharacteristic
549C2#2145574712yes—
550C2#2145575112yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1825C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1825D61
hypocenterC61#20701
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
4. #21445915C2semidirect
5. #21445914C2semidirect
6. #21445913C2semidirect
7. #21445912C2semidirect
8. #21445911C2semidirect
9. #21445537C2semidirect
10. #21445537C2semidirect
11. C61×#1825C2semidirect
21. #117×D61C4semidirect
35. #16839525C22semidirect
36. #16839525C22semidirect
37. #16839524C22semidirect
38. #16839524C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
44. C61×#340C22semidirect
45. C61×#340C22semidirect
276. #1825D61semidirect
319. #340C2×D61semidirect
320. #340C2×D61semidirect
321. C61#20701semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D612
C2×#168397242
#214482012
#214459152
#214459142
#214459132
#214459122
#214459112
#214455372
C61×#18252
#2070161