Finite Group Databasebeta

← #29733302

Subgroups — #29733302

Order 15616.

1422 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840417C211yesmaximal, characteristic, complemented
2C2×#16840121C211yesmaximal, characteristic, complemented
3#21450759C211yesmaximal, characteristic, complemented
4#21446819C212yesmaximal, complemented
5#21446814C212yesmaximal, complemented
6#21446766C211yesmaximal, characteristic, complemented
7#21446616C211yesmaximal, characteristic, complemented
8#21446133C212yesmaximal, complemented
9#21446123C212yesmaximal, complemented
10#21445644C211yesmaximal, characteristic, complemented
11C2×C61×#463C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13#16840417—24no—
14#16840417—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519759—22no—
17C2×#14519756—21no—
18C2×#14519753C2212yes—
19C2×#14519748C2212yes—
20C2×#14519748C2212yes—
21C2×C4×D244C2211yescharacteristic
22C2×C4⋊C4×D61C2212yes—
23#16840121—24no—
24C2×#14519701C2212yes—
25C2×#14519698C2212yes—
26C2×#14519698C2212yes—
27C2×#14519689C2212yes—
28C2×#14519689C2212yes—
29C2×#14519688C2211yescharacteristic
30#16839698—21no—
31#16839671—21no—
32#16839662—22no—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839525C2212yes—
35#16839525C2212yes—
36#16839524C2212yes—
37C22×SD16×C61C2211yescharacteristic
38C22×D8×C61C2211yescharacteristic
39C2×#124×C61C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C61×#463—24no—
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2211yescharacteristic
44C2×#102×C61C2211yescharacteristic
45C2×#11758222—28no—
46C2×#11758222—48no—
47C22×Dic122—21no—
48C22×D244D411yescharacteristic
49C22×C22⋊D61—22no—
50C22×C22⋊D61—22no—
51C22×C4×D61—22no—
52C22×C4×D61C2312yes—
53C22×#92418C2312yes—
54#14519777—22no—
55#14519777D414yescomplemented
56#14519759—216no—
57#14519756—28no—
58#14519753—24no—
59#14519753—42no—
60#14519748—24no—
61#14519748—24no—
62#14519748—24no—
63#14519748—42no—
64C4×D244—22no—
65C4×D244—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×C4⋊Dic61C2312yes—
70C2×#11758196—22no—
71C2×#11758196—22no—
72C4⋊C4×D61—24no—
73C4⋊C4×D61—24no—
74C2×C22⋊Dic61—22no—
75C2×C22⋊Dic61—22no—
76C2×#11758198—22no—
77C2×#11758198D412yes—
78C2×C4×Dic61—21no—
79#14519701—24no—
80#14519698—24no—
81#14519698—24no—
82#14519689—42no—
83#14519689—42no—
84#14519688—22no—
85#14519684—21no—
86C22×Q8×C61C2311yescharacteristic
87C22×D4×C61C2311yescharacteristic
88C22×D4×C61C2311yescharacteristic
89C23×C4×C61—21no—
90C2×SD16×C61—22no—
91C2×SD16×C61—42no—
92C2×D8×C61—22no—
93C2×D8×C61—42no—
94C22×C8×C61C2311yescharacteristic
95C22×C8×C61C2311yescharacteristic
96#124×C61—22no—
97#124×C61—22no—
98C4×D4×C61—22no—
99C4×D4×C61D414yescomplemented
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C22⋊C4×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—22no—
106#103×C61—22no—
107#102×C61—22no—
108C24×C61—21no—
109C24×C61—21no—
110C23×D61C2×D412yes—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61C2×D412yes—
123C22×C4×C61—21no—
124C22×C4×C61—24no—
125C22×C4×C61—21no—
126C22×C4×C61—44no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—28no—
131C2×C22⋊D61—44no—
132C2×C22⋊D61—44no—
133C2×C22⋊D61—44no—
134C2×C22⋊D61—44no—
135C2×C22⋊D61—82no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61C2×D412yes—
139#11758222—432no—
140C2×D244—24no—
141C2×D244—22no—
142C2×C4×D61—28no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—44no—
150C2×C4×D61—44no—
151C2×Dic122—24no—
152C2×Dic122—44no—
153SD16×C61—44no—
154D8×C61—44no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61C2×D412yes—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—42no—
163C22⋊C4×C61—42no—
164C42×C61C2×D412yes—
165C42×C61C2×D412yes—
166C22⋊Dic61—28no—
167C22⋊Dic61—44no—
168#11758198—24no—
169#11758198—44no—
170C4⋊Dic61—24no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61C2×D414yes—
175C4⋊Dic61#4114yescomplemented
176#11758196—28no—
177#11758196—44no—
178C4×Dic61—81no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C23×C61—22no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61C22×D411yescharacteristic
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—22no—
199C22×D61AGL(1,ℤ/8)14yes—
200Q8×C61—41no—
201Q8×C61—41no—
202Q8×C61—42no—
203D4×C61—22no—
204D4×C61—22no—
205D4×C61—41no—
206D4×C61—41no—
207D4×C61—42no—
208D4×C61—42no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—48no—
214C2×C4×C61—42no—
215C2×C4×C61—41no—
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×C61H(5,𝔽2)12yes—
223C22⋊D61—416no—
224C22⋊D61—88no—
225C2×Dic61—22no—
226C2×Dic61—24no—
227C2×Dic61—22no—
228C2×Dic61—24no—
229C2×Dic61—22no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234D244—28no—
235C4×D61—416no—
236C4×D61—42no—
237C4×D61—42no—
238C4×D61—42no—
239C4×D61—42no—
240Dic122—48no—
241C8×C61—41no—
242C8×C61—41no—
243#92418—42no—
244#25577—611nomaximal, Sylow
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—82no—
249C22×C61—82no—
250C22×C61—81no—
251C22×C61—81no—
252C22×C61—81no—
253C22×C61—82no—
254C22×C61—82no—
255C22×C61C2×#12011yescharacteristic
256C22×C61#54612yes—
257C22×C61C2×AGL(1,ℤ/8)12yes—
258C22×C61#57711yescharacteristic
259C22×C61#57811yescharacteristic
260C2×D61—24no—
261C2×D61—24no—
262C2×D61—24no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C4×C61—22no—
268C4×C61—22no—
269C4×C61—42no—
270C4×C61—81no—
271C4×C61#53211yescharacteristic
272C4×C61#53511yescharacteristic
273C4×C61#55012yes—
274Dic61—42no—
275Dic61—44no—
276Dic61—44no—
277C2×#547—611no—
278C2×#489—611no—
279C2×#463D6111yescharacteristic, complemented
280#2540—611no—
281#2000—612no—
282#1999—612no—
283#1987—611no—
284#1955—611no—
285#1874—612no—
286#1872—612no—
287#1762—611no—
288C2×C61—82no—
289C2×C61—161no—
290C2×C61#243111yescharacteristic
291C2×C61#300311yescharacteristic
292C2×C61#301311yescharacteristic
293C2×C61#330612yes—
294C2×C61#330812yes—
295D61—44no—
296C22×#41—1221no—
297C22×SD16C2×D6111yescharacteristic
298C22×D8C2×D6111yescharacteristic
299#547—1224no—
300#547—1224no—
301C2×C4⋊Q8—611no—
302C2×#124C2×D6111yescharacteristic
303C2×#123—1222no—
304C2×C22⋊Q8—612no—
305C2×C22⋊Q8—1221no—
306C2×#121—612no—
307C2×#121—612no—
308C2×C4×D4—611no—
309C2×C4×D4C2×D6111yescharacteristic
310C22×C4⋊C4—612no—
311#489—1224no—
312#463—24no—
313C2×C4⋊C8C2×D6111yescharacteristic
314C2×#103—612no—
315C2×#103C2×D6111yescharacteristic
316C2×#102—612no—
317C2×#102—612no—
318C2×#102C2×D6111yescharacteristic
319C2×C22⋊C8—612no—
320C2×C22⋊C8—612no—
321C2×#97—611no—
322#399—1221no—
323#392—1221no—
324#390—1222no—
325C4×C4⋊C4—611no—
326#340—612no—
327#340—612no—
328#340—612no—
329C61#2557711yescharacteristic, complemented, Sylow
330C2×#41—1228no—
331C2×#41—2448no—
332C22×Q8—1221no—
333C22×Q8C22×D6111yescharacteristic
334C22×D4—611no—
335C22×D4—1222no—
336C22×D4—1222no—
337C22×D4C22×D6111yescharacteristic
338C22×D4C22×D6111yescharacteristic
339C23×C4—21no—
340C23×C4—612no—
341C23×C4—1222no—
342C2×SD16—22no—
343C2×SD16—42no—
344C2×D8—22no—
345C2×D8—42no—
346C22×C8—612no—
347C22×C8C22×D6111yescharacteristic
348C22×C8C22×D6111yescharacteristic
349C4⋊Q8—614no—
350C4⋊Q8—1222no—
351#124—22no—
352#124—22no—
353#123—12216no—
354C22⋊Q8—1228no—
355C22⋊Q8—1224no—
356C22⋊Q8—2442no—
357#121—1224no—
358#121—1224no—
359#121—1224no—
360#121—2442no—
361C4×D4—22no—
362C4×D4—1222no—
363C4×D4—1222no—
364C4×D4C22⋊D6114yescomplemented
365C2×C4⋊C4—612no—
366C2×C4⋊C4—612no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—1224no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1224no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4C22×D6111yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—612no—
378C2×C22⋊C4—1222no—
379C2×C22⋊C4—1222no—
380C2×C22⋊C4—1222no—
381C2×C42—1221no—
382C2×C42C22×D6111yescharacteristic
383C4⋊C8—22no—
384#103—22no—
385#103—1224no—
386#102—22no—
387#102—1224no—
388#102—1224no—
389C22⋊C8—2442no—
390C22⋊C8—2442no—
391#97—1222no—
392#95—1221no—
393C24—21no—
394C24—21no—
395C24—612no—
396#41—24432no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—1224no—
402C2×Q8—2444no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—1224no—
411C2×D4—1228no—
412C2×D4—1222no—
413C2×D4—4882no—
414C2×D4—2444no—
415C2×D4—2444no—
416C2×D4—2444no—
417C2×D4—2444no—
418C2×D4C2×C22⋊D6112yes—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—44no—
423C22×C4—612no—
424C22×C4—1228no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—2444no—
434C22×C4—2444no—
435C22×C4D4×D6111yescharacteristic
436C22×C4D4×D6111yescharacteristic
437C22×C4C23×D6111yescharacteristic
438SD16—44no—
439D8—44no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—1222no—
445C2×C8—1222no—
446C2×C8—1222no—
447C4⋊C4—21no—
448C4⋊C4—614no—
449C4⋊C4—614no—
450C4⋊C4—1228no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—1222no—
454C4⋊C4—1224no—
455C4⋊C4—2444no—
456C4⋊C4C2×C22⋊D6112yes—
457C22⋊C4—24no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—1228no—
461C22⋊C4—1224no—
462C22⋊C4—2444no—
463C22⋊C4—2444no—
464C42—4881no—
465C42D4×D6112yes—
466C42C2×C22⋊D6112yes—
467C23—22no—
468C23—42no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—614no—
478C23—1222no—
479C23—1222no—
480C23—1222no—
481C23—1222no—
482C23—1222no—
483C23—1222no—
484C23C2×D4×D6111yescharacteristic
485Q8—41no—
486Q8—41no—
487Q8—42no—
488Q8—2448no—
489D4—22no—
490D4—22no—
491D4—41no—
492D4—41no—
493D4—42no—
494D4—42no—
495D4—1228no—
496D4—4888no—
497D4—24416no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—1222no—
507C2×C4—1224no—
508C2×C4—1224no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—24416no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4#1451974512yes—
521C2×C4#1451974512yes—
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C22×C22⋊D6111yescharacteristic
524C2×C4#41×D6112yes—
525C2×C4#1451982212yes—
526C8—41no—
527C8—41no—
528C8—2442no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—82no—
533C22—82no—
534C22—81no—
535C22—82no—
536C22—82no—
537C22—81no—
538C22—81no—
539C22—1224no—
540C22—1224no—
541C22—1224no—
542C22—2442no—
543C22—2442no—
544C22—2442no—
545C22—2442no—
546C22C2×#1451974511yescharacteristic
547C22#1684040912yes—
548C22AGL(1,ℤ/8)×D6112yes—
549C22#1684056911yescharacteristic
550C22#1684057611yescharacteristic
551C4—22no—
552C4—22no—
553C4—42no—
554C4—81no—
555C4—2444no—
556C4—2444no—
557C4—2442no—
558C4#1684030811yescharacteristic
559C4#1684032911yescharacteristic
560C4#1684043012yes—
561C2—82no—
562C2—161no—
563C2—2444no—
564C2#2144943911yescharacteristic
565C2#2145504411yescharacteristic
566C2#2145513011yescharacteristic
567C2#2145778712yes—
568C2#2145781712yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#463C2
socleC23×C61C22×D4
hypercenterC2×#463D61
hypocenterC61#25577
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#463

Complements

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

NH ≅ G/Ntype
1. C2×#16840417C2semidirect
2. C2×#16840121C2semidirect
3. #21450759C2semidirect
4. #21446819C2semidirect
5. #21446814C2semidirect
6. #21446766C2semidirect
7. #21446616C2semidirect
8. #21446133C2semidirect
9. #21446123C2semidirect
10. #21445644C2semidirect
11. C2×C61×#463C2semidirect
55. #14519777D4semidirect
99. C4×D4×C61D4semidirect
175. C4⋊Dic61#41semidirect
279. C2×#463D61semidirect
329. C61#25577semidirect
364. C4×D4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404172
C2×#168401212
#214507592
#214468192
#214468142
#214467662
#214466162
#214461332
#214461232
#214456442
C2×C61×#4632
#2557761