Finite Group Databasebeta

← #29734366

Subgroups — #29734366

Order 15616.

1114 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840291C211yesmaximal, characteristic
2C2×#16840168C211yesmaximal, characteristic, complemented
3#21454166C211yesmaximal, characteristic, complemented
4#21447013C212yesmaximal, complemented
5#21446999C212yesmaximal
6#21446788C211yesmaximal, characteristic, complemented
7#21446636C211yesmaximal, characteristic, complemented
8#21446078C212yesmaximal, complemented
9#21446071C212yesmaximal, complemented
10#21445582C211yesmaximal, characteristic, complemented
11C2×C61×#489C211yesmaximal, characteristic, complemented
12#16840291—24no—
13#16840291—24no—
14C2×#14519770—21no—
15C2×#14519769C2211yescharacteristic
16C2×#14519759—21no—
17C2×#14519753—21no—
18C2×#14519753C2211yescharacteristic
19C2×#14519748C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22C2×#14519737—21no—
23C2×#14519733—21no—
24C2×C42×D61C2211yescharacteristic
25#16840168—24no—
26C2×#14519710C2212yes—
27C2×#14519708C2212yes—
28C2×#14519701C2212yes—
29C2×#14519698C2212yes—
30C2×#14519689C2212yes—
31C2×#14519688C2211yescharacteristic
32#16839669—21no—
33#16839658—21no—
34#16839658C2212yes—
35#16839632C2211yescharacteristic
36#16839619—21no—
37#16839619C2212yes—
38#16839525C2212yes—
39#16839525C2212yes—
40#16839516C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×#124×C61C2211yescharacteristic
43C61×#489—24no—
44C2×#103×C61C2212yes—
45C2×#102×C61C2212yes—
46C2×#97×C61C2211yescharacteristic
47C22×C22⋊D61—21no—
48C23×Dic61—21no—
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C22×#92418C2312yes—
53#14519770—28no—
54#14519769—22no—
55#14519769D414yes—
56#14519759—28no—
57#14519753—22no—
58#14519753—22no—
59#14519753—44no—
60#14519748—22no—
61#14519748—22no—
62Q8×Dic61—22no—
63Q8×Dic61—22no—
64D4×Dic61—22no—
65D4×Dic61—22no—
66#14519737—28no—
67#14519733—28no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—22no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—21no—
81C2×#11758198—22no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C42×D61—22no—
87C42×D61—22no—
88#14519710—24no—
89#14519708—24no—
90#14519701—24no—
91#14519698—24no—
92#14519689—24no—
93#14519688—22no—
94#14519684—22no—
95#14519684—22no—
96C22×Q8×C61—21no—
97C22×Q8×C61C2311yescharacteristic
98C22×D4×C61C2311yescharacteristic
99C22×C8×C61C2312yes—
100C4⋊Q8×C61—22no—
101C4⋊Q8×C61—22no—
102#124×C61—22no—
103#124×C61D414yes—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61C2312yes—
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61—21no—
108C2×C42×C61C2311yescharacteristic
109#103×C61—24no—
110#102×C61—24no—
111#97×C61—22no—
112C24×C61—21no—
113C23×D61C2×D411yescharacteristic
114C2×Q8×C61—21no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—42no—
118C2×Q8×C61—42no—
119C2×Q8×C61—42no—
120C2×Q8×C61C2×D412yes—
121C2×D4×C61—21no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61C2×D412yes—
125C22×C4×C61—21no—
126C22×C4×C61—22no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—24no—
131C2×C22⋊D61—42no—
132C2×C22⋊D61—42no—
133C22×Dic61—24no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C22×Dic61—42no—
138C22×Dic61—42no—
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—41no—
151C2×C4×D61—41no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—42no—
158C22⋊C4×C61—24no—
159C22⋊C4×C61—24no—
160C42×C61C2×D412yes—
161C42×C61C2×D412yes—
162C22⋊Dic61—24no—
163C22⋊Dic61—24no—
164#11758198—24no—
165#11758198—44no—
166C4⋊Dic61—24no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—42no—
172C4⋊Dic61—42no—
173C4⋊Dic61C2×D412yes—
174C4⋊Dic61C2×D412yes—
175#11758196—24no—
176#11758196—24no—
177#11758196—24no—
178#11758196—44no—
179C4×Dic61—24no—
180C4×Dic61—22no—
181C4×Dic61—22no—
182C4×Dic61—21no—
183C4×Dic61—21no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418#4114yes—
187C23×C61—22no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61C22×D411yescharacteristic
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61AGL(1,ℤ/8)12yes—
202Q8×C61—24no—
203Q8×C61—42no—
204Q8×C61—84no—
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×C61#12012yes—
218C2×C4×C61#12012yes—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4112yes—
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—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—48no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61#13712yes—
243C4×D61—44no—
244C4×D61—44no—
245C4×D61—42no—
246C4×D61—42no—
247C8×C61—42no—
248#92418—24no—
249#25679—611nomaximal, Sylow
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—81no—
254C22×C61—81no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61C2×#12011yescharacteristic
258C22×C61#53211yescharacteristic
259C22×C61#53611yescharacteristic
260C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C61C2×#13711yescharacteristic
262C22×C61#57511yescharacteristic
263C22×C61#57511yescharacteristic
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—41no—
268C2×D61—41no—
269C2×D61—41no—
270C2×D61—41no—
271C4×C61—22no—
272C4×C61—22no—
273C4×C61—42no—
274C4×C61—82no—
275C4×C61#54712yes—
276C4×C61#54812yes—
277Dic61—22no—
278Dic61—41no—
279Dic61—41no—
280Dic61—42no—
281Dic61—42no—
282Dic61—82no—
283C2×#529—611no—
284C2×#501—611no—
285C2×#489D6111yescharacteristic, complemented
286#2857—611no—
287#2035—612no—
288#2033—612no—
289#1992—611no—
290#1959—611no—
291#1864—612no—
292#1863—612no—
293#1749—611no—
294C2×C61—82no—
295C2×C61#243211yescharacteristic
296C2×C61#300611yescharacteristic
297C2×C61#300711yescharacteristic
298C2×C61#317211yescharacteristic
299C2×C61#318011yescharacteristic
300C2×C61#323811yescharacteristic
301C2×C61#324011yescharacteristic
302D61—42no—
303C2×C4⋊Q8C2×D6111yescharacteristic
304#529—1224no—
305#529—1224no—
306C2×#126—1221no—
307C2×#125—611no—
308C2×#124C2×D6111yescharacteristic
309C2×#123—1221no—
310C2×C22⋊Q8—611no—
311C2×C22⋊Q8—1221no—
312C2×#121—611no—
313C2×C4×Q8—611no—
314C2×C4×D4—611no—
315C2×C4×D4—1221no—
316C2×#117—1221no—
317C22×C42—611no—
318#501—1224no—
319#489—24no—
320C2×#107—612no—
321C2×#106—612no—
322C2×#103—612no—
323C2×#103C2×D6112yes—
324C2×#102—612no—
325C2×#102C2×D6112yes—
326C2×C22⋊C8—612no—
327C2×#97—611no—
328C2×#97C2×D6111yescharacteristic
329#391—1221no—
330#389—612no—
331#389—1221no—
332#384—611no—
333#380—612no—
334#380—1221no—
335#340—612no—
336#340—612no—
337#336—612no—
338C61#2567911yescharacteristic, complemented, Sylow
339C22×Q8—21no—
340C22×Q8C22×D6111yescharacteristic
341C22×D4—1221no—
342C22×D4C22×D6111yescharacteristic
343C23×C4—611no—
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—1221no—
347C22×C8—612no—
348C22×C8C22×D6112yes—
349C4⋊Q8—22no—
350C4⋊Q8—22no—
351#126—1228no—
352#125—614no—
353#125—1222no—
354#124—22no—
355#124C22⋊D6114yes—
356#123—1228no—
357C22⋊Q8—1222no—
358C22⋊Q8—1222no—
359C22⋊Q8—2444no—
360#121—1222no—
361#121—1222no—
362C4×Q8—1222no—
363C4×Q8—1222no—
364C4×D4—1228no—
365C4×D4—1222no—
366C4×D4—1222no—
367#117—1228no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—612no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4C22×D6112yes—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1222no—
386C2×C42—611no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—1221no—
390C2×C42—1222no—
391C2×C42—1222no—
392C2×C42C22×D6111yescharacteristic
393#107—1224no—
394#106—1224no—
395#103—24no—
396#103—1224no—
397#102—24no—
398#102—1224no—
399C22⋊C8—1224no—
400#97—22no—
401#97—1222no—
402#95—1222no—
403#95—1222no—
404C24—21no—
405C24—611no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8C2×C22⋊D6112yes—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—1224no—
417C2×D4—2442no—
418C2×D4—2442no—
419C2×D4C2×C22⋊D6112yes—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1224no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—2441no—
438C22×C4—2441no—
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—614no—
448C2×C8—1222no—
449C2×C8—1222no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—612no—
454C4⋊C4—612no—
455C4⋊C4—1224no—
456C4⋊C4—1224no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1224no—
460C4⋊C4—1224no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—2442no—
464C4⋊C4—2442no—
465C4⋊C4—2444no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—1224no—
469C22⋊C4—1224no—
470C22⋊C4—1224no—
471C22⋊C4—2444no—
472C42—1224no—
473C42—1221no—
474C42—1221no—
475C42—1222no—
476C42—1222no—
477C42D4×D6112yes—
478C42C2×C22⋊D6112yes—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—612no—
487C23—1221no—
488C23—1221no—
489C23—1221no—
490C23—1221no—
491C23—1221no—
492C23—1221no—
493C23C2×D4×D6111yescharacteristic
494Q8—24no—
495Q8—42no—
496Q8—84no—
497D4—24no—
498D4—42no—
499D4—2448no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—612no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—2444no—
522C2×C4—2444no—
523C2×C4—2448no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4#1451974512yes—
534C2×C4#1451974512yes—
535C2×C4C2×D4×D6111yescharacteristic
536C2×C4C22×C22⋊D6111yescharacteristic
537C2×C4#41×D6112yes—
538C2×C4#1451982412yes—
539C8—42no—
540C8—1224no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—1222no—
549C22—1222no—
550C22—1222no—
551C22—2441no—
552C22—2441no—
553C22—2441no—
554C22—2441no—
555C22C2×#1451974511yescharacteristic
556C22#1684031111yescharacteristic
557C22#1684033211yescharacteristic
558C22AGL(1,ℤ/8)×D6111yescharacteristic
559C22#137×D6111yescharacteristic
560C22#1684055811yescharacteristic
561C22#1684056111yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—82no—
566C4—1222no—
567C4—4882no—
568C4—2441no—
569C4—2442no—
570C4—2442no—
571C4—2441no—
572C4#1684041112yes—
573C4#1684042012yes—
574C2—82no—
575C2—2442no—
576C2#2144945011yescharacteristic
577C2#2145506711yescharacteristic
578C2#2145507511yescharacteristic
579C2#2145642811yescharacteristic
580C2#2145652011yescharacteristic
581C2#2145703611yescharacteristic
582C2#2145705611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#489C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
hypercenterC2×#489D61
hypocenterC61#25679
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
2. C2×#16840168C2semidirect
3. #21454166C2semidirect
4. #21447013C2semidirect
6. #21446788C2semidirect
7. #21446636C2semidirect
8. #21446078C2semidirect
9. #21446071C2semidirect
10. #21445582C2semidirect
11. C2×C61×#489C2semidirect
285. C2×#489D61semidirect
338. C61#25679semidirect

Maximal subgroups

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

subgroupindex
C2×#168402912
C2×#168401682
#214541662
#214470132
#214469992
#214467882
#214466362
#214460782
#214460712
#214455822
C2×C61×#4892
#2567961