Finite Group Databasebeta

← #29685147

Subgroups — #29685147

Order 15616.

1182 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840297C211yesmaximal, characteristic
2C2×#16839818C211yesmaximal, characteristic, complemented
3C4×#14519735C211yesmaximal, characteristic, complemented
4#21446777C211yesmaximal, characteristic, complemented
5#21446751C211yesmaximal, characteristic, complemented
6#21446749C211yesmaximal, characteristic, complemented
7#21446140C212yesmaximal
8#21446130C212yesmaximal, complemented
9#21445648C212yesmaximal, complemented
10#21445643C212yesmaximal, complemented
11C61×#1989C211yesmaximal, characteristic, complemented
12#16840297—24no—
13#16840297C418yescomplemented
14C2×#14519772—21no—
15C2×#14519765C2211yescharacteristic
16C2×#14519762—21no—
17C2×#14519757C2211yescharacteristic
18C2×#14519752C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×#14519739—21no—
21C2×#14519739C2211yescharacteristic
22C2×#14519735—21no—
23C2×#14519735C2212yes—
24C2×C42×D61C2211yescharacteristic
25#16839818—24no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519702C2212yes—
28C2×#14519700C2212yes—
29C2×#14519699C2212yes—
30C2×#14519697C2212yes—
31C2×#14519689C2212yes—
32C2×#14519687C2212yes—
33C4⋊C4×Dic61C2212yes—
34C4×C4⋊Dic61C2211yescharacteristic
35C4×#11758198—21no—
36#16839609—21no—
37#16839607—21no—
38C42×Dic61—21no—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#125×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C4×C4⋊C4×C61C2211yescharacteristic
44C61×#340C2212yes—
45C61×#340C2212yes—
46C22×Dic122C2311yescharacteristic
47C22×D244—21no—
48C22×D244C2311yescharacteristic
49C22×C4×D61—21no—
50C22×C4×D61D411yescharacteristic
51C22×C4×D61D411yescharacteristic
52C22×C4×D61C2311yescharacteristic
53C22×#92418C2312yes—
54#14519772—28no—
55#14519765—22no—
56#14519765C2×C414yes—
57#14519762—28no—
58#14519757—22no—
59#14519757C2×C414yes—
60#14519752—22no—
61#14519752C2×C414yes—
62#14519742—22no—
63#14519742C2×C414yes—
64#14519739—28no—
65#14519739—22no—
66#14519739C2×C414yes—
67#14519735—24no—
68#14519735—24no—
69#14519735—28no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C2×#11758196—21no—
73C2×#11758198—22no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—22no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C42×D61C2×C414yes—
87C42×D61D414yes—
88C4⋊#92418—22no—
89#14519702—24no—
90#14519700—24no—
91#14519699—24no—
92#14519697—24no—
93#14519689—24no—
94#14519687—24no—
95#14519684—21no—
96#14519684—22no—
97C22×C8×C61C2312yes—
98#125×C61—22no—
99#125×C61C2×C414yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2312yes—
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—22no—
108#95×C61—21no—
109C23×D61—21no—
110C23×D61C2×D411yescharacteristic
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61C2×D411yescharacteristic
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—24no—
125C2×D244—21no—
126C2×D244—44no—
127C2×D244C22×C412yes—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—24no—
131C2×C4×D61—42no—
132C2×C4×D61—41no—
133C2×C4×D61—41no—
134C2×C4×D61—42no—
135C2×C4×D61—42no—
136C2×C4×D61C22⋊C414yes—
137C2×C4×D61C22⋊C414yes—
138C2×C4×D61C22×C412yes—
139C2×C4×D61C2×D412yes—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—21no—
143C2×Dic122C22×C412yes—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61C22×C412yes—
153C4⋊C4×C61C22×C412yes—
154C4⋊C4×C61#4114yes—
155C42×C61—42no—
156C42×C61C22×C412yes—
157C42×C61C2×D412yes—
158#11758198—28no—
159#11758198—24no—
160#11758198—24no—
161#11758198—24no—
162#11758198—24no—
163#11758198—24no—
164C4⋊Dic61—22no—
165C4⋊Dic61—24no—
166C4⋊Dic61#4114yes—
167#11758196—24no—
168C4×Dic61—28no—
169C4×Dic61—24no—
170C4×Dic61—44no—
171C4×Dic61—42no—
172C4×Dic61C22⋊C414yes—
173C4×Dic61C22⋊C414yes—
174C4×Dic61C22×C412yes—
175C4×Dic61C2×D412yes—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C23×C61C22×D411yescharacteristic
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—21no—
183C22×D61—21no—
184C22×D61—22no—
185C22×D61—42no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—42no—
189C22×D61C2×C22⋊C412yes—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—24no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61C2×C22⋊C412yes—
202C2×C4×C61C2×C22⋊C412yes—
203C2×C4×C61C23×C411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C2×#4112yes—
206C2×C4×C61C2×#4112yes—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—22no—
212C2×Dic61—24no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61C2×C22⋊C412yes—
217C2×Dic61C2×C22⋊C412yes—
218C2×Dic61C2×C22⋊C412yes—
219D244—24no—
220D244—48no—
221D244—41no—
222D244—41no—
223C4×D61—24no—
224C4×D61—22no—
225C4×D61—22no—
226C4×D61—44no—
227C4×D61—44no—
228C4×D61—48no—
229Dic122—24no—
230Dic122—41no—
231Dic122—41no—
232C8×C61—42no—
233#92418—42no—
234#21396—611nomaximal, Sylow
235C22×C61C22×C22⋊C411yescharacteristic
236C22×C61#52512yes—
237C22×C61#57511yescharacteristic
238C22×C61#57512yes—
239C22×C61#57511yescharacteristic
240C2×D61—22no—
241C2×D61—22no—
242C2×D61—22no—
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—42no—
247C2×D61—42no—
248C2×D61—82no—
249C2×D61—82no—
250C4×C61—22no—
251C4×C61—22no—
252C4×C61—44no—
253C4×C61—42no—
254C4×C61—42no—
255C4×C61C4×#4112yes—
256C4×C61#53211yescharacteristic
257C4×C61#53211yescharacteristic
258Dic61—22no—
259Dic61—22no—
260Dic61—22no—
261Dic61—44no—
262Dic61—42no—
263C2×#529—611no—
264C2×#432—611no—
265C4×#117—611no—
266#1989—611no—
267#1989D6111yescharacteristic, complemented
268#1983—611no—
269#1983—611no—
270#1875—612no—
271#1873—612no—
272#1763—612no—
273#1762—612no—
274C2×C61#229711yescharacteristic
275C2×C61#289611yescharacteristic
276C2×C61#289611yescharacteristic
277C2×C61#311112yes—
278C2×C61#311112yes—
279D61—42no—
280D61—82no—
281#529—618no—
282#529—1224no—
283C2×#126—1221no—
284C2×#125C2×D6111yescharacteristic
285C2×#124—611no—
286C2×#123—1221no—
287C2×C22⋊Q8—611no—
288C2×#121—611no—
289C2×C4×Q8—611no—
290C2×C4×D4—611no—
291C2×C4×D4—1221no—
292C2×#117—612no—
293C2×#117—1221no—
294C22×C42—611no—
295#432—1224no—
296C2×C4⋊C8—611no—
297C2×C4⋊C8C2×D6111yescharacteristic
298C2×#103—612no—
299C2×#103—612no—
300C2×#102—612no—
301C2×#102—612no—
302C2×C22⋊C8—612no—
303C2×#97—612no—
304C4×C4⋊C4—612no—
305C4×C4⋊C4—611no—
306C4×C4⋊C4C2×D6111yescharacteristic
307C4×C22⋊C4—1221no—
308C4×C22⋊C4—1221no—
309#376—1221no—
310C43—1221no—
311#340—612no—
312#340—612no—
313#340C2×D6112yes—
314#340C2×D6112yes—
315C61#2139611yescharacteristic, complemented, Sylow
316C22×Q8—611no—
317C22×D4—611no—
318C22×D4—1221no—
319C23×C4—611no—
320C23×C4—611no—
321C23×C4—611no—
322C23×C4—1221no—
323C22×C8—612no—
324C22×C8C22×D6112yes—
325#126—1228no—
326#125—22no—
327#125C4×D6114yes—
328#124—614no—
329#124—1222no—
330#123—1228no—
331C22⋊Q8—614no—
332C22⋊Q8—1222no—
333#121—614no—
334#121—1222no—
335C4×Q8—614no—
336C4×Q8—1222no—
337C4×D4—614no—
338C4×D4—1222no—
339C4×D4—1228no—
340#117—1228no—
341#117—1224no—
342#117—1224no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—612no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4C22×D6111yescharacteristic
349C2×C4⋊C4C22×D6111yescharacteristic
350C2×C4⋊C4C22×D6112yes—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1222no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1221no—
357C2×C42—21no—
358C2×C42—611no—
359C2×C42—611no—
360C2×C42—611no—
361C2×C42—614no—
362C2×C42—614no—
363C2×C42—1222no—
364C2×C42—1221no—
365C2×C42—1222no—
366C2×C42—1221no—
367C2×C42C22×D6111yescharacteristic
368C4⋊C8—22no—
369C4⋊C8—1222no—
370#103—1224no—
371#103—1224no—
372#102—1224no—
373#102—1224no—
374C22⋊C8—1224no—
375#97—1224no—
376#95—21no—
377#95—1222no—
378#95—1221no—
379C24—611no—
380C24—1221no—
381C2×Q8—612no—
382C2×Q8—1221no—
383C2×Q8—1222no—
384C2×Q8—1222no—
385C2×D4—612no—
386C2×D4—1224no—
387C2×D4—1221no—
388C2×D4—1222no—
389C2×D4—1222no—
390C2×D4—2444no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—611no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—612no—
398C22×C4—612no—
399C22×C4—614no—
400C22×C4—614no—
401C22×C4—1224no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1221no—
406C22×C4—2441no—
407C22×C4—2442no—
408C22×C4—2441no—
409C22×C4—2442no—
410C22×C4—2442no—
411C22×C4D4×D6111yescharacteristic
412C22×C4D4×D6111yescharacteristic
413C22×C4C23×D6111yescharacteristic
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—1222no—
418C2×C8—1222no—
419C2×C8—1222no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—614no—
426C4⋊C4—1224no—
427C4⋊C4—1224no—
428C4⋊C4—1222no—
429C4⋊C4C2×C4×D6112yes—
430C4⋊C4C2×C4×D6112yes—
431C4⋊C4#1175822614yes—
432C22⋊C4—1224no—
433C22⋊C4—1224no—
434C22⋊C4—1224no—
435C22⋊C4—1228no—
436C22⋊C4—1224no—
437C22⋊C4—1224no—
438C42—42no—
439C42—612no—
440C42—612no—
441C42—614no—
442C42—614no—
443C42—1224no—
444C42—1228no—
445C42—2442no—
446C42—2444no—
447C42C2×C4×D6112yes—
448C42D4×D6112yes—
449C23—612no—
450C23—1222no—
451C23—1221no—
452C23—1222no—
453C23—1221no—
454C23—1222no—
455C23—2442no—
456C23—2441no—
457C23—2442no—
458C23—2441no—
459C23C2×D4×D6111yescharacteristic
460Q8—1224no—
461Q8—2441no—
462Q8—2441no—
463D4—1224no—
464D4—2448no—
465D4—2441no—
466D4—2441no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—612no—
479C2×C4—612no—
480C2×C4—612no—
481C2×C4—1222no—
482C2×C4—1221no—
483C2×C4—1222no—
484C2×C4—1221no—
485C2×C4—1224no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—1224no—
490C2×C4—2444no—
491C2×C4—2444no—
492C2×C4—2448no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4C22⋊C4×D6112yes—
497C2×C4C22⋊C4×D6112yes—
498C2×C4C22×C4×D6111yescharacteristic
499C2×C4C2×D4×D6111yescharacteristic
500C2×C4#41×D6112yes—
501C2×C4C2×#1175822612yes—
502C8—42no—
503C8—2442no—
504C22—1222no—
505C22—1222no—
506C22—1222no—
507C22—4882no—
508C22—4882no—
509C22—2442no—
510C22—2442no—
511C22—2442no—
512C22—2442no—
513C22—2442no—
514C22C2×C22⋊C4×D6111yescharacteristic
515C22#1684027512yes—
516C22#1684055811yescharacteristic
517C22#1684055912yes—
518C22#1684056111yescharacteristic
519C4—22no—
520C4—22no—
521C4—44no—
522C4—42no—
523C4—42no—
524C4—1222no—
525C4—1222no—
526C4—1222no—
527C4—2444no—
528C4—2442no—
529C4C4×#1175822612yes—
530C4#1684031111yescharacteristic
531C4#1684031311yescharacteristic
532C2—4882no—
533C2—2442no—
534C2#2144830811yescharacteristic
535C2#2145433411yescharacteristic
536C2#2145433511yescharacteristic
537C2#2145593312yes—
538C2#2145593512yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1989C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
cosocleC22×C4×C61C24
hypercenter#1989D61
hypocenterC61#21396
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
2. C2×#16839818C2semidirect
3. C4×#14519735C2semidirect
4. #21446777C2semidirect
5. #21446751C2semidirect
6. #21446749C2semidirect
8. #21446130C2semidirect
9. #21445648C2semidirect
10. #21445643C2semidirect
11. C61×#1989C2semidirect
13. #16840297C4semidirect
267. #1989D61semidirect
315. C61#21396semidirect

Maximal subgroups

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

subgroupindex
C2×#168402972
C2×#168398182
C4×#145197352
#214467772
#214467512
#214467492
#214461402
#214461302
#214456482
#214456432
C61×#19892
#2139661