Finite Group Databasebeta

← #29689749

Subgroups — #29689749

Order 15616.

1234 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840291C211yesmaximal, characteristic
2C2×#16839817C211yesmaximal, characteristic, complemented
3#384×D61C211yesmaximal, characteristic, complemented
4#21446788C211yesmaximal, characteristic, complemented
5#21446787C211yesmaximal, characteristic, complemented
6#21446636C211yesmaximal, characteristic, complemented
7#21446133C212yesmaximal
8#21446123C212yesmaximal, complemented
9#21445822C212yesmaximal, complemented
10#21445817C212yesmaximal, complemented
11C61×#1959C211yesmaximal, characteristic, complemented
12#16840291—24no—
13#16840291C418yescomplemented
14C2×#14519770—21no—
15C2×#14519769C2211yescharacteristic
16C2×#14519759—21no—
17C2×#14519753C2211yescharacteristic
18C2×#14519748C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×#14519737—21no—
22C2×#14519733—21no—
23C2×C4⋊C4×D61—21no—
24C2×C4⋊C4×D61—21no—
25C2×C4⋊C4×D61C2212yes—
26C2×C42×D61C2211yescharacteristic
27#16839817—24no—
28C2×#14519706C2212yes—
29C2×#14519701C2212yes—
30C2×#14519698C2212yes—
31C2×#14519689C2212yes—
32C2×#14519688C2211yescharacteristic
33#16839632C2211yescharacteristic
34#16839631—21no—
35#16839631—21no—
36#16839631C2212yes—
37#16839525C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#124×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C2×#102×C61C2212yes—
44C2×#97×C61C2211yescharacteristic
45C61×#384C2211yescharacteristic
46C22×C22⋊D61—21no—
47C23×Dic61—21no—
48C22×C4×D61—22no—
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C22×#92418C2312yes—
53#14519770—28no—
54#14519769—22no—
55#14519769C2×C414yescomplemented
56#14519759—28no—
57#14519753—22no—
58#14519753C2×C414yes—
59#14519748—22no—
60#14519748C2×C414yes—
61Q8×Dic61—22no—
62Q8×Dic61C2×C414yescomplemented
63D4×Dic61—22no—
64D4×Dic61C2×C414yescomplemented
65#14519737—28no—
66#14519733—28no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—22no—
77C2×#11758196—22no—
78C2×#11758196—22no—
79C4⋊C4×D61—42no—
80C4⋊C4×D61—42no—
81C4⋊C4×D61—44no—
82C4⋊C4×D61—44no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×#11758198—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90C42×D61C2×C414yes—
91C42×D61D414yes—
92#14519706—24no—
93#14519701—42no—
94#14519698—42no—
95#14519689—24no—
96#14519688—22no—
97C22×Q8×C61C2311yescharacteristic
98C22×D4×C61C2311yescharacteristic
99C22×C8×C61C2312yes—
100#124×C61—22no—
101#124×C61C2×C414yescomplemented
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2312yes—
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61C2311yescharacteristic
108#103×C61—42no—
109#102×C61—42no—
110#97×C61—22no—
111C24×C61—21no—
112C23×D61C2×D411yescharacteristic
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61C22×C412yes—
117C2×D4×C61—21no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61C22×C412yes—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—24no—
127C2×C22⋊D61—42no—
128C2×C22⋊D61—42no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—24no—
132C22×Dic61—22no—
133C22×Dic61—42no—
134C22×Dic61—42no—
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—44no—
145C2×C4×D61—44no—
146C2×C4×D61—44no—
147C2×C4×D61C22⋊C414yes—
148C2×C4×D61C22⋊C414yes—
149C2×C4×D61C22×C412yes—
150C2×C4×D61C2×D412yes—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—42no—
157C4⋊C4×C61—42no—
158C22⋊C4×C61—24no—
159C22⋊C4×C61—24no—
160C42×C61C22×C412yes—
161C42×C61C2×D412yes—
162C22⋊Dic61—24no—
163C22⋊Dic61—24no—
164#11758198—24no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—24no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61—42no—
171C4⋊Dic61—42no—
172C4⋊Dic61C22×C412yes—
173C4⋊Dic61C22×C412yes—
174#11758196—24no—
175#11758196—24no—
176#11758196—24no—
177#11758196—44no—
178#11758196—44no—
179#11758196—44no—
180C4×Dic61—24no—
181C4×Dic61C22⋊C414yes—
182C4×Dic61C22⋊C414yes—
183C4×Dic61C22×C412yes—
184C4×Dic61C2×D412yes—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C23×C61—22no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61C22×D411yescharacteristic
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61C2×C22⋊C412yes—
201C22×D61AGL(1,ℤ/8)12yes—
202C22×D61#13712yes—
203Q8×C61—24no—
204Q8×C61—42no—
205D4×C61—24no—
206D4×C61—42no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61C2×C22⋊C412yes—
218C2×C4×C61C2×C22⋊C412yes—
219C2×C4×C61C23×C411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61H(5,𝔽2)12yes—
222C2×C4×C612−1+412yes—
223C22⋊D61—48no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—48no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61C2×C22⋊C412yes—
239C2×Dic61C2×C22⋊C412yes—
240C2×Dic61C2×C22⋊C412yes—
241C2×Dic61AGL(1,ℤ/8)12yes—
242C2×Dic61#13712yes—
243C4×D61—28no—
244C4×D61—24no—
245C4×D61—24no—
246C4×D61—28no—
247C4×D61—88no—
248C8×C61—42no—
249#92418—42no—
250#21867—611nomaximal, Sylow
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—81no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61—81no—
258C22×C61C22×C22⋊C411yescharacteristic
259C22×C61#53511yescharacteristic
260C22×C61#53611yescharacteristic
261C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C61C2×#13711yescharacteristic
264C22×C61C2×#13711yescharacteristic
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—22no—
270C2×D61—22no—
271C2×D61#41714yes—
272C4×C61—22no—
273C4×C61—22no—
274C4×C61—42no—
275C4×C61—82no—
276C4×C61#52012yes—
277C4×C61#53712yes—
278Dic61—22no—
279Dic61—22no—
280Dic61—42no—
281Dic61—42no—
282Dic61—82no—
283Dic61#41714yes—
284C2×#529—611no—
285C2×#432—611no—
286C2×#384—611no—
287#1992—611no—
288#1992—611no—
289#1959—611no—
290#1959D6111yescharacteristic, complemented
291#1874—612no—
292#1872—612no—
293#1803—612no—
294#1802—612no—
295C2×C61—82no—
296C2×C61#231111yescharacteristic
297C2×C61C2×#41711yescharacteristic
298C2×C61C2×#41711yescharacteristic
299C2×C61#321711yescharacteristic
300C2×C61#321811yescharacteristic
301C2×C61#323811yescharacteristic
302C2×C61#323911yescharacteristic
303D61—24no—
304#529—618no—
305#529—1224no—
306C2×#126—1221no—
307C2×#125—611no—
308C2×#124C2×D6111yescharacteristic
309C2×#123—1221no—
310C2×C22⋊Q8—611no—
311C2×#121—611no—
312C2×C4×Q8—611no—
313C2×C4×D4—611no—
314C2×C4×D4—1221no—
315C2×#117—1221no—
316C22×C4⋊C4—612no—
317C22×C4⋊C4—1221no—
318C22×C4⋊C4—1221no—
319C22×C42—611no—
320#432—1224no—
321C2×C4⋊C8—612no—
322C2×#103—612no—
323C2×#103C2×D6112yes—
324C2×#102—612no—
325C2×#102C2×D6112yes—
326C2×C22⋊C8—612no—
327C2×#97—611no—
328C2×#97C2×D6111yescharacteristic
329#384—612no—
330#384—611no—
331#384—1221no—
332#384—1221no—
333#384C2×D6111yescharacteristic
334#340—612no—
335#340—612no—
336#340—612no—
337#340—612no—
338C61#2186711yescharacteristic, complemented, Sylow
339C22×Q8C22×D6111yescharacteristic
340C22×D4—1221no—
341C22×D4C22×D6111yescharacteristic
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—611no—
345C23×C4—1222no—
346C23×C4—1221no—
347C22×C8—612no—
348C22×C8C22×D6112yes—
349#126—1228no—
350#125—614no—
351#125—1222no—
352#124—22no—
353#124C4×D6114yescomplemented
354#123—1228no—
355C22⋊Q8—614no—
356C22⋊Q8—1222no—
357#121—614no—
358#121—1222no—
359C4×Q8—614no—
360C4×Q8—1222no—
361C4×D4—614no—
362C4×D4—1222no—
363C4×D4—1228no—
364#117—1228no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—2442no—
380C2×C4⋊C4—2442no—
381C2×C4⋊C4—2444no—
382C2×C4⋊C4—2444no—
383C2×C4⋊C4C22×D6112yes—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—614no—
393C2×C42—614no—
394C2×C42—1221no—
395C2×C42C22×D6111yescharacteristic
396C4⋊C8—1224no—
397#103—42no—
398#103—2442no—
399#102—42no—
400#102—2442no—
401C22⋊C8—1224no—
402#97—22no—
403#97—1222no—
404C24—21no—
405C24—611no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8C2×C4×D6112yes—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—1224no—
414C2×D4—2442no—
415C2×D4—2442no—
416C2×D4C2×C4×D6112yes—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—612no—
423C22×C4—612no—
424C22×C4—614no—
425C22×C4—614no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1224no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1222no—
436C22×C4—2444no—
437C22×C4—2444no—
438C22×C4—2444no—
439C22×C4—2442no—
440C22×C4—2442no—
441C22×C4D4×D6111yescharacteristic
442C22×C4D4×D6111yescharacteristic
443C22×C4C23×D6111yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—1222no—
448C2×C8—1222no—
449C2×C8—1222no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—612no—
455C4⋊C4—612no—
456C4⋊C4—1224no—
457C4⋊C4—1224no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1224no—
461C4⋊C4—1224no—
462C4⋊C4—1222no—
463C4⋊C4—1222no—
464C4⋊C4—2442no—
465C4⋊C4—2442no—
466C4⋊C4—2444no—
467C4⋊C4—2444no—
468C4⋊C4—2444no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C22⋊C4—1224no—
474C42—612no—
475C42—612no—
476C42—614no—
477C42—614no—
478C42—1224no—
479C42C2×C4×D6112yes—
480C42D4×D6112yes—
481C23—22no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—612no—
489C23—612no—
490C23—612no—
491C23—1221no—
492C23—1221no—
493C23—1221no—
494C23—1221no—
495C23C2×D4×D6111yescharacteristic
496Q8—24no—
497Q8—42no—
498D4—24no—
499D4—42no—
500D4—2448no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—612no—
512C2×C4—612no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—1228no—
517C2×C4—1228no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—4888no—
527C2×C4—2448no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4C22⋊C4×D6112yes—
536C2×C4C22⋊C4×D6112yes—
537C2×C4C22×C4×D6111yescharacteristic
538C2×C4C2×D4×D6111yescharacteristic
539C2×C4#1451982212yes—
540C2×C4#1451982412yes—
541C8—42no—
542C8—2442no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—614no—
551C22—1222no—
552C22—1222no—
553C22—1222no—
554C22—1222no—
555C22—1222no—
556C22—1222no—
557C22C2×C22⋊C4×D6111yescharacteristic
558C22#1684032511yescharacteristic
559C22#1684033211yescharacteristic
560C22AGL(1,ℤ/8)×D6111yescharacteristic
561C22AGL(1,ℤ/8)×D6111yescharacteristic
562C22#137×D6111yescharacteristic
563C22#137×D6111yescharacteristic
564C4—22no—
565C4—22no—
566C4—42no—
567C4—82no—
568C4—614no—
569C4—1222no—
570C4—1222no—
571C4—4882no—
572C4—2442no—
573C4—2442no—
574C4#1684024112yes—
575C4#1684034112yes—
576C2—82no—
577C2—1224no—
578C2#2144840911yescharacteristic
579C2#417×D6111yescharacteristic
580C2#417×D6111yescharacteristic
581C2#2145683311yescharacteristic
582C2#2145684211yescharacteristic
583C2#2145703611yescharacteristic
584C2#2145704511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1959C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
cosocleC22×C4×C61C24
hypercenter#1959D61
hypocenterC61#21867
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1959

Complements

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

NH ≅ G/Ntype
2. C2×#16839817C2semidirect
3. #384×D61C2semidirect
4. #21446788C2semidirect
5. #21446787C2semidirect
6. #21446636C2semidirect
8. #21446123C2semidirect
9. #21445822C2semidirect
10. #21445817C2semidirect
11. C61×#1959C2semidirect
13. #16840291C4semidirect
55. #14519769C2×C4semidirect
62. Q8×Dic61C2×C4semidirect
64. D4×Dic61C2×C4semidirect
101. #124×C61C2×C4semidirect
290. #1959D61semidirect
338. C61#21867semidirect
353. #124C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402912
C2×#168398172
#384×D612
#214467882
#214467872
#214466362
#214461332
#214461232
#214458222
#214458172
C61×#19592
#2186761