Finite Group Databasebeta

← #29736666

Subgroups — #29736666

Order 15616.

1320 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×C4⋊Q8×D61C211yesmaximal, characteristic, complemented
2C2×#16840121C211yesmaximal, characteristic, complemented
3#21454115C211yesmaximal, characteristic
4#21446958C212yesmaximal, complemented
5#21446953C212yesmaximal, complemented
6#21446793C211yesmaximal, characteristic, complemented
7#21446792C211yesmaximal, characteristic, complemented
8#21446133C212yesmaximal
9#21446123C212yesmaximal, complemented
10#21445582C211yesmaximal, characteristic, complemented
11C2×C61×#489C211yesmaximal, characteristic, complemented
12C22×Q8×D61—21no—
13C2×#14519777C2211yescharacteristic
14C2×#14519776—21no—
15C2×#14519776C2212yes—
16C2×#14519775—21no—
17C4⋊Q8×D61—24no—
18C4⋊Q8×D61—24no—
19C2×#14519753C2211yescharacteristic
20C2×#14519748C2211yescharacteristic
21C2×C4⋊C4×D61—21no—
22C2×C4⋊C4×D61C2212yes—
23C2×C42×D61C2211yescharacteristic
24#16840121—24no—
25C2×#14519701C2212yes—
26C2×#14519698C2212yes—
27C2×#14519689C2212yes—
28C2×#14519688C2211yescharacteristic
29#16839669C2211yescharacteristic
30#16839665C2211yescharacteristic
31#16839650—21no—
32#16839650—21no—
33#16839638—21no—
34#16839638—21no—
35#16839628C2211yescharacteristic
36#16839525C2212yes—
37#16839525C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×#124×C61C2211yescharacteristic
43C61×#489—24no—
44C2×#103×C61C2212yes—
45C2×#102×C61C2212yes—
46C2×#97×C61C2211yescharacteristic
47C22×Dic122—21no—
48C22×Dic122—22no—
49C2×Q8×D61—28no—
50C2×Q8×D61—44no—
51C2×Q8×D61—44no—
52C22×C22⋊D61—21no—
53C22×C4×D61—22no—
54C22×C4×D61D411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61C2311yescharacteristic
57C22×#92418C2312yes—
58#14519777—22no—
59#14519777D414yescomplemented
60#14519776—28no—
61#14519776—24no—
62#14519776—24no—
63#14519775—28no—
64#14519753—24no—
65#14519748—24no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×#11758196—22no—
71C2×#11758196—22no—
72C2×#11758196—21no—
73C4⋊C4×D61—28no—
74C4⋊C4×D61—24no—
75C4⋊C4×D61—24no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×#11758198—21no—
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83C42×D61—22no—
84C42×D61—22no—
85#14519701—24no—
86#14519698—24no—
87#14519689—24no—
88#14519688—22no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93C22×Q8×C61—21no—
94C22×Q8×C61C2311yescharacteristic
95C22×D4×C61C2311yescharacteristic
96C22×C8×C61C2312yes—
97C4⋊Q8×C61—22no—
98C4⋊Q8×C61D414yescomplemented
99#124×C61—22no—
100#124×C61—22no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C22⋊C4×C61—21no—
104C2×C22⋊C4×C61—21no—
105C2×C42×C61C2311yescharacteristic
106#103×C61—24no—
107#102×C61—24no—
108#97×C61—22no—
109C24×C61—21no—
110C23×D61C2×D411yescharacteristic
111C2×Q8×C61—24no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—42no—
117C2×Q8×C61—42no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—42no—
128C2×C22⋊D61—42no—
129C2×C22⋊D61—42no—
130C22×Dic61—22no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136Q8×D61—432no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—28no—
147C2×C4×D61—44no—
148C2×C4×D61—44no—
149C2×C4×D61#4114yes—
150C2×Dic122—28no—
151C2×Dic122—24no—
152C2×Dic122—44no—
153C2×Dic122—42no—
154C2×Dic122—42no—
155C2×Dic122—44no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C4⋊C4×C61—24no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61C2×D414yes—
162C22⋊C4×C61—42no—
163C22⋊C4×C61—42no—
164C42×C61C2×D412yes—
165C42×C61C2×D412yes—
166C22⋊Dic61—42no—
167C22⋊Dic61—42no—
168C22⋊Dic61—42no—
169#11758198—42no—
170C4⋊Dic61—24no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61C2×D414yes—
175#11758196—28no—
176#11758196—28no—
177#11758196—42no—
178C4×Dic61—22no—
179C4×Dic61—22no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61C22×D411yescharacteristic
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—21no—
197C22×D61#12412yes—
198C22×D61AGL(1,ℤ/8)12yes—
199C22×D61#13712yes—
200Q8×C61—48no—
201Q8×C61—42no—
202Q8×C61—42no—
203D4×C61—42no—
204D4×C61—42no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—24no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61#12012yes—
216C2×C4×C61#12012yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C2×#4112yes—
220C2×C4×C612−1+412yes—
221C22⋊D61—84no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—24no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61#12412yes—
238C2×Dic61AGL(1,ℤ/8)12yes—
239C2×Dic61#13712yes—
240C4×D61—28no—
241C4×D61—28no—
242C4×D61—24no—
243C4×D61—24no—
244C4×D61—416no—
245Dic122—416no—
246Dic122—48no—
247C8×C61—42no—
248#92418—42no—
249#25908—611nomaximal, Sylow
250C22×C61—81no—
251C22×C61—81no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61—81no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61C2×#12011yescharacteristic
258C22×C61C2×#12411yescharacteristic
259C22×C61#53611yescharacteristic
260C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C61C2×#13711yescharacteristic
263C22×C61C2×#13711yescharacteristic
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—22no—
270C2×D61#48914yescomplemented
271C4×C61—22no—
272C4×C61—22no—
273C4×C61—44no—
274C4×C61—81no—
275C4×C61#54812yes—
276C4×C61#54812yes—
277Dic61—22no—
278Dic61—22no—
279Dic61—22no—
280Dic61—44no—
281Dic61—81no—
282Dic61—81no—
283C22×C4⋊Q8—611no—
284C2×#489—611no—
285C2×#489D6111yescharacteristic, complemented
286#2850—611no—
287#2024—612no—
288#2023—612no—
289#1993—611no—
290#1993—611no—
291#1874—612no—
292#1872—612no—
293#1749—611no—
294C2×C61—161no—
295C2×C61#243611yescharacteristic
296C2×C61#300311yescharacteristic
297C2×C61#300411yescharacteristic
298C2×C61C2×#48911yescharacteristic
299C2×C61#314111yescharacteristic
300C2×C61#323811yescharacteristic
301C2×C61#323911yescharacteristic
302D61—24no—
303C23×Q8—1221no—
304C2×C4⋊Q8—612no—
305C2×C4⋊Q8—611no—
306C2×C4⋊Q8—1221no—
307C2×C4⋊Q8—1221no—
308C2×C4⋊Q8—1224no—
309C2×C4⋊Q8—1224no—
310C2×C4⋊Q8C2×D6111yescharacteristic
311C2×#124C2×D6111yescharacteristic
312C2×C22⋊Q8—611no—
313C2×#121—611no—
314C22×C4⋊C4—612no—
315C22×C4⋊C4—1221no—
316C22×C42—611no—
317#489—24no—
318#489—1224no—
319C2×#103—612no—
320C2×#103C2×D6112yes—
321C2×#102—612no—
322C2×#102C2×D6112yes—
323C2×C22⋊C8—612no—
324C2×#97—611no—
325C2×#97C2×D6111yescharacteristic
326#391—611no—
327#390—611no—
328#388—1221no—
329#388—1221no—
330#386—1221no—
331#386—1221no—
332#382—611no—
333#340—612no—
334#340—612no—
335#340—612no—
336#340—612no—
337#336—612no—
338C61#2590811yescharacteristic, complemented, Sylow
339C22×Q8—21no—
340C22×Q8—1222no—
341C22×Q8—1228no—
342C22×Q8—1221no—
343C22×Q8—2444no—
344C22×Q8—2444no—
345C22×Q8C22×D6111yescharacteristic
346C22×D4—1221no—
347C22×D4C22×D6111yescharacteristic
348C23×C4—611no—
349C23×C4—611no—
350C23×C4—611no—
351C23×C4—1222no—
352C22×C8—612no—
353C22×C8C22×D6112yes—
354C4⋊Q8—22no—
355C4⋊Q8—614no—
356C4⋊Q8—1224no—
357C4⋊Q8—1228no—
358C4⋊Q8—1228no—
359C4⋊Q8—1224no—
360C4⋊Q8—1222no—
361C4⋊Q8C22⋊D6114yescomplemented
362#124—22no—
363#124—22no—
364C22⋊Q8—1224no—
365#121—1224no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—1224no—
371C2×C4⋊C4—1228no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1224no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4C22×D6112yes—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C42—611no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—1222no—
388C2×C42—1222no—
389C2×C42C22×D6111yescharacteristic
390#103—24no—
391#103—1224no—
392#102—24no—
393#102—1224no—
394C22⋊C8—1224no—
395#97—22no—
396#97—1222no—
397#95—1221no—
398#95—1221no—
399#95—1221no—
400#95—1221no—
401C24—21no—
402C24—611no—
403C2×Q8—24no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—1228no—
411C2×Q8—1224no—
412C2×Q8—2442no—
413C2×Q8—2442no—
414C2×Q8—2444no—
415C2×Q8—2444no—
416C2×Q8—24432no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—2442no—
422C2×D4—2442no—
423C2×D4—2442no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—614no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1228no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—2444no—
444C22×C4—2444no—
445C22×C4D4×D6111yescharacteristic
446C22×C4D4×D6111yescharacteristic
447C22×C4C23×D6111yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—1222no—
452C2×C8—1222no—
453C2×C8—1222no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—614no—
457C4⋊C4—1228no—
458C4⋊C4—1228no—
459C4⋊C4—1222no—
460C4⋊C4—1224no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—2442no—
464C4⋊C4C2×C22⋊D6114yes—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—2442no—
468C22⋊C4—2442no—
469C22⋊C4—2442no—
470C22⋊C4—2442no—
471C42—1221no—
472C42—1221no—
473C42—1222no—
474C42—1222no—
475C42D4×D6112yes—
476C42C2×C22⋊D6112yes—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—612no—
485C23—612no—
486C23—612no—
487C23—1221no—
488C23—1221no—
489C23—1221no—
490C23—1221no—
491C23C2×D4×D6111yescharacteristic
492Q8—48no—
493Q8—42no—
494Q8—42no—
495Q8—2448no—
496Q8—24416no—
497D4—42no—
498D4—42no—
499D4—4884no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—612no—
511C2×C4—612no—
512C2×C4—612no—
513C2×C4—1228no—
514C2×C4—1228no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1224no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—24416no—
525C2×C4—2441no—
526C2×C4—2441no—
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—2442no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—614no—
549C22—1222no—
550C22—1222no—
551C22—1222no—
552C22—1222no—
553C22—1222no—
554C22—1222no—
555C22C2×#1451974511yescharacteristic
556C22#124×D6111yescharacteristic
557C22#1684033211yescharacteristic
558C22AGL(1,ℤ/8)×D6111yescharacteristic
559C22AGL(1,ℤ/8)×D6111yescharacteristic
560C22#137×D6111yescharacteristic
561C22#137×D6111yescharacteristic
562C4—22no—
563C4—22no—
564C4—44no—
565C4—81no—
566C4—1222no—
567C4—1222no—
568C4—1222no—
569C4—4881no—
570C4—4881no—
571C4—2444no—
572C4#1684041912yes—
573C4#1684042012yes—
574C2—161no—
575C2—1224no—
576C2#2144947411yescharacteristic
577C2#2145504411yescharacteristic
578C2#2145505111yescharacteristic
579C2#489×D6111yescharacteristic
580C2#2145621111yescharacteristic
581C2#2145703611yescharacteristic
582C2#2145704511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#489C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
hypercenterC2×#489D61
hypocenterC61#25908
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
1. C2×C4⋊Q8×D61C2semidirect
2. C2×#16840121C2semidirect
4. #21446958C2semidirect
5. #21446953C2semidirect
6. #21446793C2semidirect
7. #21446792C2semidirect
9. #21446123C2semidirect
10. #21445582C2semidirect
11. C2×C61×#489C2semidirect
59. #14519777D4semidirect
98. C4⋊Q8×C61D4semidirect
270. C2×D61#489semidirect
285. C2×#489D61semidirect
338. C61#25908semidirect
361. C4⋊Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D612
C2×#168401212
#214541152
#214469582
#214469532
#214467932
#214467922
#214461332
#214461232
#214455822
C2×C61×#4892
#2590861