Finite Group Databasebeta

← #29689738

Subgroups — #29689738

Order 15616.

1114 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×#16840314C211yesmaximal, characteristic
2C2×#16839816C211yesmaximal, characteristic
3C2×#16839627C211yesmaximal, characteristic
4#21446790C212yesmaximal, complemented
5#21446637C212yesmaximal, complemented
6#21446127C211yesmaximal, characteristic
7#21446126C211yesmaximal, characteristic
8#21446124C211yesmaximal, characteristic
9#21445833C212yesmaximal, complemented
10#21445832C212yesmaximal, complemented
11C61×#1872C211yesmaximal, characteristic
12#16840314—24no—
13#16840314C418yescomplemented
14C2×#14519776C2211yescharacteristic
15C2×#14519775C2211yescharacteristic
16C2×#14519767—21no—
17C2×#14519756C2211yescharacteristic
18C2×#14519754—21no—
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519742—21no—
21C2×#14519742C2211yescharacteristic
22C2×#14519733—21no—
23C22×C4⋊Dic61C2211yescharacteristic
24C22×C4×Dic61C2211yescharacteristic
25#16839816—24no—
26C2×#14519706C2212yes—
27C2×#14519702C2212yes—
28C2×#14519701C2212yes—
29C2×#14519700C2212yes—
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519687C2212yes—
32#16839627—21no—
33#16839627—21no—
34#16839627C2212yes—
35#16839627C2212yes—
36C2×#14519684—21no—
37C2×#14519684—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C22⋊Q8×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C23×Dic61—21no—
49C23×Dic61D411yescharacteristic
50C23×Dic61D411yescharacteristic
51C22×#92418C2312yes—
52#14519776—22no—
53#14519776C2×C414yescomplemented
54#14519775—22no—
55#14519775C2×C414yescomplemented
56#14519767—28no—
57#14519756—22no—
58#14519756C2×C414yes—
59#14519754—28no—
60Q8×Dic61—22no—
61Q8×Dic61C2×C414yescomplemented
62#14519742—28no—
63#14519742—22no—
64#14519742C2×C414yescomplemented
65#14519733—28no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—42no—
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61C2×C414yes—
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61D414yes—
83C2×C4×Dic61C2311yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85#14519706—24no—
86#14519702—42no—
87#14519701—42no—
88#14519700—42no—
89C22⋊#92418—22no—
90#14519687—24no—
91#14519684—22no—
92#14519684—22no—
93#14519684—22no—
94#14519684—22no—
95C22×Q8×C61C2311yescharacteristic
96C23×C4×C61—21no—
97C23×C4×C61C2311yescharacteristic
98C22×C8×C61C2312yes—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61C2×C414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—42no—
104C2×C4⋊C4×C61C2312yes—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C22⋊C4×C61—21no—
107#103×C61—42no—
108C22⋊C8×C61—22no—
109C24×C61C2×D411yescharacteristic
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61C22×C412yes—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61C22×C412yes—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2411yescharacteristic
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—22no—
131C22×Dic61—41no—
132C22×Dic61—41no—
133C22×Dic61—42no—
134C22×Dic61—42no—
135C22×Dic61—42no—
136C22×Dic61C22⋊C414yes—
137C22×Dic61C22⋊C414yes—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C2×Dic122—22no—
143C2×Dic122—22no—
144C2×Dic122—21no—
145C2×Dic122—24no—
146C2×Dic122—42no—
147C2×Dic122—42no—
148C2×Dic122C22×C412yes—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—42no—
157C4⋊C4×C61C22×C412yes—
158C22⋊C4×C61—24no—
159C22⋊Dic61—24no—
160C4⋊Dic61—21no—
161C4⋊Dic61—24no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61—42no—
165C4⋊Dic61C22×C412yes—
166#11758196—24no—
167#11758196—24no—
168#11758196—24no—
169#11758196—24no—
170#11758196—24no—
171C4×Dic61—24no—
172C4×Dic61—24no—
173C4×Dic61C22⋊C414yes—
174C4×Dic61C22⋊C414yes—
175C4×Dic61C22×C412yes—
176C4×Dic61C22×C412yes—
177C4×Dic61C2×D412yes—
178C4×Dic61C2×D412yes—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C23×C61—21no—
183C23×C61—21no—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61C2×C22⋊C412yes—
187C23×C61#13712yes—
188C23×C61#13712yes—
189C23×C61C22×D411yescharacteristic
190Q8×C61—24no—
191Q8×C61—42no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—24no—
195C2×C4×C61—24no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—84no—
206C2×C4×C61C2×C22⋊C412yes—
207C2×C4×C61C23×C411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61H(5,𝔽2)12yes—
210C2×C4×C612−1+412yes—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—28no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—44no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—84no—
231C2×Dic61C2×C22⋊C412yes—
232C2×Dic61C2×C22⋊C412yes—
233C2×Dic61C2×C22⋊C412yes—
234C2×Dic61C2×C22⋊C412yes—
235Dic122—24no—
236Dic122—48no—
237Dic122—42no—
238C8×C61—42no—
239#92418—42no—
240#21866—611nomaximal, Sylow
241C22×C61—22no—
242C22×C61—22no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61#41914yes—
248C22×C61C22×C22⋊C411yescharacteristic
249C22×C61#53511yescharacteristic
250C22×C61#53611yescharacteristic
251C22×C61C2×#13711yescharacteristic
252C22×C61C2×#13711yescharacteristic
253C22×C61#57511yescharacteristic
254C22×C61#57511yescharacteristic
255C4×C61—22no—
256C4×C61—42no—
257C4×C61—42no—
258C4×C61—82no—
259C4×C61#52012yes—
260C4×C61#53712yes—
261Dic61—22no—
262Dic61—22no—
263Dic61—22no—
264Dic61—22no—
265Dic61—42no—
266Dic61—42no—
267Dic61—82no—
268C2×#533—611no—
269C2×#432—611no—
270C2×#382—611no—
271#1993—612no—
272#1960—612no—
273#1872—611no—
274#1872—611no—
275#1872—611no—
276#1872D6111yescharacteristic
277#1805—612no—
278#1805—612no—
279C2×C61—24no—
280C2×C61#231111yescharacteristic
281C2×C61C2×#41911yescharacteristic
282C2×C61#289611yescharacteristic
283C2×C61#321511yescharacteristic
284C2×C61#321511yescharacteristic
285C2×C61#323611yescharacteristic
286C2×C61#323611yescharacteristic
287#533—618no—
288#533—1224no—
289C2×C4⋊Q8—611no—
290C2×C4⋊Q8—611no—
291C2×#125—1221no—
292C2×C22⋊Q8—611no—
293C2×C22⋊Q8—1221no—
294C2×C22⋊Q8C2×D6111yescharacteristic
295C2×C4×Q8—611no—
296C2×C4×Q8—611no—
297C2×C4×Q8—1221no—
298C2×#117—1221no—
299C22×C4⋊C4—611no—
300C22×C4⋊C4C2×D6111yescharacteristic
301C22×C42—611no—
302#432—1224no—
303C2×C4⋊C8—612no—
304C2×#103—612no—
305C2×#103—612no—
306C2×#103—612no—
307C2×#103C2×D6112yes—
308C2×C22⋊C8—611no—
309C2×C22⋊C8C2×D6111yescharacteristic
310C2×#97—612no—
311#382—612no—
312#382—612no—
313#382—1221no—
314#382—1221no—
315C2×#95—1221no—
316C2×#95—1221no—
317#340—612no—
318#340—612no—
319#340—612no—
320#340C2×D6112yes—
321C61#2186611yescharacteristic, complemented, Sylow
322C22×Q8—611no—
323C22×Q8—1221no—
324C22×Q8C22×D6111yescharacteristic
325C23×C4—21no—
326C23×C4—611no—
327C23×C4—611no—
328C23×C4—1221no—
329C23×C4C22×D6111yescharacteristic
330C22×C8—612no—
331C22×C8C22×D6112yes—
332C4⋊Q8—614no—
333C4⋊Q8—614no—
334C4⋊Q8—1222no—
335C4⋊Q8—1222no—
336#125—1228no—
337C22⋊Q8—22no—
338C22⋊Q8—614no—
339C22⋊Q8—1228no—
340C22⋊Q8—1222no—
341C22⋊Q8C4×D6114yes—
342C4×Q8—614no—
343C4×Q8—614no—
344C4×Q8—1222no—
345C4×Q8—1228no—
346C4×Q8—1222no—
347#117—1228no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—612no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—2442no—
361C2×C4⋊C4C22×D6112yes—
362C2×C4⋊C4C22×D6111yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—1221no—
365C2×C42—614no—
366C2×C42—611no—
367C2×C42—611no—
368C2×C42—614no—
369C2×C42—611no—
370C2×C42—611no—
371C2×C42—1221no—
372C2×C42—1221no—
373C4⋊C8—1224no—
374#103—42no—
375#103—2442no—
376#103—2442no—
377#103—2442no—
378C22⋊C8—22no—
379C22⋊C8—1222no—
380#97—1224no—
381#95—1222no—
382#95—1222no—
383#95—1222no—
384#95—1222no—
385C24D4×D6111yescharacteristic
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—612no—
390C2×Q8—1224no—
391C2×Q8—1221no—
392C2×Q8—1222no—
393C2×Q8—1222no—
394C2×Q8—2442no—
395C2×Q8—2442no—
396C2×Q8C2×C4×D6112yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—611no—
408C22×C4—611no—
409C22×C4—614no—
410C22×C4—614no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—1221no—
414C22×C4—1221no—
415C22×C4—1222no—
416C22×C4—2441no—
417C22×C4—2441no—
418C22×C4—2442no—
419C22×C4—2442no—
420C22×C4—2442no—
421C22×C4C2×C4×D6112yes—
422C22×C4D4×D6112yes—
423C22×C4D4×D6111yescharacteristic
424C22×C4C23×D6111yescharacteristic
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—1222no—
429C2×C8—1222no—
430C2×C8—1222no—
431C4⋊C4—21no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—42no—
436C4⋊C4—612no—
437C4⋊C4—1224no—
438C4⋊C4—1224no—
439C4⋊C4—1224no—
440C4⋊C4—1224no—
441C4⋊C4—1224no—
442C4⋊C4—1221no—
443C4⋊C4—1224no—
444C4⋊C4—1222no—
445C4⋊C4—1222no—
446C4⋊C4—2442no—
447C4⋊C4C2×C4×D6112yes—
448C22⋊C4—24no—
449C22⋊C4—1224no—
450C42—612no—
451C42—612no—
452C42—612no—
453C42—612no—
454C42—614no—
455C42—614no—
456C42—1224no—
457C42—1224no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23C22⋊C4×D6112yes—
463C23#1451980612yes—
464C23#1451980912yes—
465C23C2×D4×D6111yescharacteristic
466Q8—24no—
467Q8—42no—
468Q8—1224no—
469Q8—2448no—
470Q8—2442no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—84no—
485C2×C4—612no—
486C2×C4—612no—
487C2×C4—612no—
488C2×C4—612no—
489C2×C4—1228no—
490C2×C4—1221no—
491C2×C4—1221no—
492C2×C4—1221no—
493C2×C4—1221no—
494C2×C4—1221no—
495C2×C4—1221no—
496C2×C4—1221no—
497C2×C4—1221no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—4884no—
501C2×C4—2444no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4C22⋊C4×D6112yes—
510C2×C4C22×C4×D6111yescharacteristic
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4#1451982212yes—
513C2×C4#1451982412yes—
514C8—42no—
515C8—2442no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22#1683976914yes—
523C22C2×C22⋊C4×D6111yescharacteristic
524C22#1684032711yescharacteristic
525C22#1684033411yescharacteristic
526C22C2×#1451980611yescharacteristic
527C22C2×#1451980911yescharacteristic
528C22#1684055811yescharacteristic
529C22#1684056111yescharacteristic
530C4—22no—
531C4—42no—
532C4—42no—
533C4—82no—
534C4—1222no—
535C4—1222no—
536C4—1222no—
537C4—1222no—
538C4—4882no—
539C4—2442no—
540C4—2442no—
541C4#1684024112yes—
542C4#1684033912yes—
543C2—24no—
544C2#2144840611yescharacteristic
545C2C2×#1683976911yescharacteristic
546C2#2145433511yescharacteristic
547C2#2145681911yescharacteristic
548C2#2145682111yescharacteristic
549C2#2145702211yescharacteristic
550C2#2145702411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1872C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1872D61
hypocenterC61#21866
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
4. #21446790C2semidirect
5. #21446637C2semidirect
9. #21445833C2semidirect
10. #21445832C2semidirect
13. #16840314C4semidirect
53. #14519776C2×C4semidirect
55. #14519775C2×C4semidirect
61. Q8×Dic61C2×C4semidirect
64. #14519742C2×C4semidirect
321. C61#21866semidirect

Maximal subgroups

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

subgroupindex
C2×#168403142
C2×#168398162
C2×#168396272
#214467902
#214466372
#214461272
#214461262
#214461242
#214458332
#214458322
C61×#18722
#2186661