Finite Group Databasebeta

← #29736667

Subgroups — #29736667

Order 15616.

1320 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519776C211yesmaximal, characteristic, complemented
2C2×#16840123C212yesmaximal, complemented
3#21454122C211yesmaximal, characteristic
4#21446958C212yesmaximal, complemented
5#21446954C212yesmaximal, complemented
6#21446790C212yesmaximal, complemented
7#21446136C211yesmaximal, characteristic
8#21446134C211yesmaximal, characteristic
9#21446127C211yesmaximal, characteristic, complemented
10#21445583C211yesmaximal, characteristic, complemented
11C61×#1872C211yesmaximal, characteristic, complemented
12C23×Dic122—21no—
13C2×#14519776—28no—
14C2×#14519776—21no—
15C2×#14519776—21no—
16C2×#14519776C2212yes—
17C2×#14519776C2212yes—
18C2×#14519766C2211yescharacteristic
19C2×#14519751C2211yescharacteristic
20C22×C4⋊Dic61C2211yescharacteristic
21C22×#11758196—21no—
22C22×C4×Dic61C2211yescharacteristic
23#16840123—28no—
24C2×#14519701C2212yes—
25C2×#14519699C2212yes—
26C2×#14519697C2212yes—
27C2×C22⋊#92418C2211yescharacteristic
28C2×#14519687C2212yes—
29#16839669C2211yescharacteristic
30#16839663C2211yescharacteristic
31#16839652—21no—
32#16839652—21no—
33#16839639—21no—
34#16839639—21no—
35#16839627C2211yescharacteristic
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839516C2212yes—
40C2×C22⋊Q8×C61C2211yescharacteristic
41C22×C4⋊C4×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C2×C22⋊C8×C61C2211yescharacteristic
44C61×#340C2212yes—
45C22×Dic122—28no—
46C22×Dic122—22no—
47C22×Dic122—22no—
48C22×Dic122—44no—
49C22×Dic122—44no—
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C23×Dic61—21no—
53C23×Dic61D411yescharacteristic
54C23×Dic61D411yescharacteristic
55C22×#92418C2312yes—
56#14519776—28no—
57#14519776—28no—
58#14519776—24no—
59#14519776—24no—
60#14519776—24no—
61#14519776D418yescomplemented
62#14519766—24no—
63#14519751—22no—
64#14519751—22no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—24no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×#11758196—28no—
70C2×#11758196—22no—
71C2×#11758196—22no—
72C2×#11758198—21no—
73C2×#11758198—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×C4×Dic61—24no—
78C2×C4×Dic61D411yescharacteristic
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61C2311yescharacteristic
81C2×C4×Dic61C2311yescharacteristic
82#14519701—24no—
83#14519699—24no—
84#14519697—24no—
85C22⋊#92418—22no—
86#14519687—24no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91C22×Q8×C61C2311yescharacteristic
92C23×C4×C61—21no—
93C23×C4×C61C2311yescharacteristic
94C22×C8×C61C2312yes—
95C22⋊Q8×C61—22no—
96C22⋊Q8×C61—22no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—24no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2312yes—
102C2×C22⋊C4×C61—21no—
103#103×C61—24no—
104C22⋊C8×C61—22no—
105C24×C61C2×D411yescharacteristic
106C23×D61—21no—
107C2×Q8×C61—22no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—24no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—42no—
118C22×C4×C61—42no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61#4112yes—
121C22×C4×C61#4112yes—
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—42no—
124C2×C22⋊D61—42no—
125C2×C22⋊D61—42no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—24no—
133C22×Dic61—22no—
134C22×Dic61—21no—
135C22×Dic61—42no—
136C22×Dic61—42no—
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C2×D244—22no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—22no—
145C2×Dic122—28no—
146C2×Dic122—28no—
147C2×Dic122—432no—
148C2×Dic122—44no—
149C2×Dic122—44no—
150C2×Dic122—44no—
151C2×Dic122—44no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61C2×D414yes—
161C22⋊C4×C61—42no—
162#11758198—42no—
163#11758198—42no—
164#11758198—42no—
165#11758198—42no—
166#11758198—42no—
167C4⋊Dic61—24no—
168C4⋊Dic61—22no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61C2×D414yes—
172#11758196—28no—
173#11758196—28no—
174C4×Dic61—22no—
175C4×Dic61—22no—
176C4×Dic61—22no—
177C4×Dic61C2×D414yes—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C23×C61—21no—
182C23×C61—21no—
183C23×C61—21no—
184C23×C61—21no—
185C23×C61#12412yes—
186C23×C61AGL(1,ℤ/8)12yes—
187C23×C61#13712yes—
188C23×C61C22×D411yescharacteristic
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196Q8×C61—42no—
197Q8×C61—42no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—24no—
201C2×C4×C61—24no—
202C2×C4×C61—24no—
203C2×C4×C61—48no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61#12412yes—
213C2×C4×C61C22×D412yes—
214C2×C4×C61C2×#4111yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C612−1+412yes—
217C22⋊D61—84no—
218C2×Dic61—28no—
219C2×Dic61—28no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—24no—
229C2×Dic61—48no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61#12012yes—
236C2×Dic61#12012yes—
237C2×Dic61AGL(1,ℤ/8)12yes—
238C2×Dic61#13712yes—
239D244—42no—
240D244—42no—
241Dic122—416no—
242Dic122—416no—
243C8×C61—42no—
244#92418—42no—
245#25908—611nomaximal, Sylow
246C22×C61—22no—
247C22×C61—22no—
248C22×C61—22no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61#48914yescomplemented
253C22×C61C2×#12011yescharacteristic
254C22×C61C2×#12411yescharacteristic
255C22×C61#53611yescharacteristic
256C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
257C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C61C2×#13711yescharacteristic
259C22×C61C2×#13711yescharacteristic
260C2×D61—81no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C2×D61—81no—
267C4×C61—22no—
268C4×C61—44no—
269C4×C61—81no—
270C4×C61—81no—
271C4×C61#54812yes—
272C4×C61#54812yes—
273Dic61—22no—
274Dic61—22no—
275Dic61—22no—
276Dic61—22no—
277Dic61—44no—
278Dic61—81no—
279C22×C4⋊Q8—611no—
280C2×#489—612no—
281#2850—611no—
282#2024—612no—
283#2023—612no—
284#1993—612no—
285#1874—611no—
286#1874—611no—
287#1872—611no—
288#1872D6111yescharacteristic, complemented
289#1749—611no—
290C2×C61—24no—
291C2×C61#243611yescharacteristic
292C2×C61#300311yescharacteristic
293C2×C61#300411yescharacteristic
294C2×C61C2×#48911yescharacteristic
295C2×C61#314111yescharacteristic
296C2×C61#323811yescharacteristic
297C2×C61#323911yescharacteristic
298D61—161no—
299C23×Q8—1221no—
300C2×C4⋊Q8—612no—
301C2×C4⋊Q8—612no—
302C2×C4⋊Q8—1221no—
303C2×C4⋊Q8—1221no—
304C2×C4⋊Q8—1228no—
305C2×#124—611no—
306C2×C22⋊Q8C2×D6111yescharacteristic
307C2×#121—611no—
308C22×C4⋊C4—611no—
309C22×C4⋊C4—1221no—
310C22×C4⋊C4C2×D6111yescharacteristic
311C22×C42—611no—
312#489—1228no—
313C2×#103—612no—
314C2×#103C2×D6112yes—
315C2×#102—612no—
316C2×#102—612no—
317C2×C22⋊C8—611no—
318C2×C22⋊C8C2×D6111yescharacteristic
319C2×#97—612no—
320#391—611no—
321#390—611no—
322#388—1221no—
323#388—1221no—
324#386—1221no—
325#386—1221no—
326#382—611no—
327#340—612no—
328#340—612no—
329#340—612no—
330#340C2×D6112yes—
331#336—612no—
332C61#2590811yescharacteristic, complemented, Sylow
333C22×Q8—1222no—
334C22×Q8—1222no—
335C22×Q8—1228no—
336C22×Q8—2444no—
337C22×Q8—2444no—
338C22×Q8C22×D6111yescharacteristic
339C22×D4—611no—
340C22×D4—1221no—
341C23×C4—21no—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—1221no—
345C23×C4C22×D6111yescharacteristic
346C22×C8—612no—
347C22×C8C22×D6112yes—
348C4⋊Q8—618no—
349C4⋊Q8—1224no—
350C4⋊Q8—1224no—
351C4⋊Q8—1228no—
352C4⋊Q8—1228no—
353C4⋊Q8—1224no—
354#124—1224no—
355C22⋊Q8—22no—
356C22⋊Q8—22no—
357#121—1222no—
358#121—1222no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—612no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—1224no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1228no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4C22×D6111yescharacteristic
370C2×C4⋊C4C22×D6112yes—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—1221no—
373C2×C22⋊C4—1221no—
374C2×C22⋊C4—1221no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1221no—
377C2×C42—611no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—611no—
381C2×C42—1224no—
382#103—24no—
383#103—1224no—
384#102—1224no—
385#102—1224no—
386C22⋊C8—22no—
387C22⋊C8—1222no—
388#97—1224no—
389#95—1221no—
390#95—1221no—
391#95—1221no—
392#95—1221no—
393C24—1221no—
394C24D4×D6111yescharacteristic
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—1228no—
400C2×Q8—1228no—
401C2×Q8—2444no—
402C2×Q8—24432no—
403C2×Q8—2444no—
404C2×Q8—2444no—
405C2×Q8—2444no—
406C2×D4—1222no—
407C2×D4—1221no—
408C2×D4—1221no—
409C2×D4—1222no—
410C2×D4—2442no—
411C2×D4—2442no—
412C2×D4—2442no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1224no—
432C22×C4—1222no—
433C22×C4—1221no—
434C22×C4—2442no—
435C22×C4—2442no—
436C22×C4#1175822212yes—
437C22×C4D4×D6111yescharacteristic
438C22×C4#1175822412yes—
439C22×C4C23×D6111yescharacteristic
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—1222no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—42no—
451C4⋊C4—614no—
452C4⋊C4—1228no—
453C4⋊C4—1228no—
454C4⋊C4—1224no—
455C4⋊C4—1222no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4D4×D6114yes—
459C22⋊C4—42no—
460C22⋊C4—2442no—
461C22⋊C4—2442no—
462C22⋊C4—2442no—
463C22⋊C4—2442no—
464C22⋊C4—2442no—
465C42—614no—
466C42—1222no—
467C42—1222no—
468C42—1222no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—2441no—
474C23—2441no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23—2441no—
480C23#1451976312yes—
481C23#1451980412yes—
482C23#1451980912yes—
483C23C2×D4×D6111yescharacteristic
484Q8—42no—
485Q8—42no—
486Q8—24416no—
487Q8—24416no—
488D4—4884no—
489D4—2442no—
490D4—2442no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—48no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—612no—
506C2×C4—612no—
507C2×C4—612no—
508C2×C4—612no—
509C2×C4—1228no—
510C2×C4—1228no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1224no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2448no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4#1451976312yes—
527C2×C4C2×D4×D6112yes—
528C2×C4C2×#1175822411yescharacteristic
529C2×C4C2×#1175822211yescharacteristic
530C2×C4#1451982412yes—
531C8—42no—
532C8—2442no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—4881no—
540C22—4881no—
541C22—4881no—
542C22—4881no—
543C22—4881no—
544C22—4881no—
545C22—4881no—
546C22#1684012314yescomplemented
547C22#120×D6111yescharacteristic
548C22C2×#1451976311yescharacteristic
549C22#1684033411yescharacteristic
550C22AGL(1,ℤ/8)×D6111yescharacteristic
551C22C2×#1451980411yescharacteristic
552C22#137×D6111yescharacteristic
553C22C2×#1451980911yescharacteristic
554C4—22no—
555C4—44no—
556C4—81no—
557C4—81no—
558C4—1222no—
559C4—1222no—
560C4—1222no—
561C4—1222no—
562C4—4881no—
563C4—2444no—
564C4#1684041812yes—
565C4#1684042112yes—
566C2—24no—
567C2—9761no—
568C2#2144947511yescharacteristic
569C2#2145504711yescharacteristic
570C2#2145505411yescharacteristic
571C2C2×#1684012311yescharacteristic
572C2#2145621011yescharacteristic
573C2#2145704311yescharacteristic
574C2#2145705211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1872C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#1872D61
hypocenterC61#25908
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
1. C22×#14519776C2semidirect
2. C2×#16840123C2semidirect
4. #21446958C2semidirect
5. #21446954C2semidirect
6. #21446790C2semidirect
9. #21446127C2semidirect
10. #21445583C2semidirect
11. C61×#1872C2semidirect
61. #14519776D4semidirect
252. C22×C61#489semidirect
288. #1872D61semidirect
332. C61#25908semidirect
546. C22#16840123semidirect

Maximal subgroups

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

subgroupindex
C22×#145197762
C2×#168401232
#214541222
#214469582
#214469542
#214467902
#214461362
#214461342
#214461272
#214455832
C61×#18722
#2590861