Finite Group Databasebeta

← #29713963

Subgroups — #29713963

Order 15616.

1154 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840478C211yesmaximal, characteristic
2C2×C4×#11758201C211yesmaximal, characteristic, complemented
3#21450737C211yesmaximal, characteristic, complemented
4#21447003C212yesmaximal, complemented
5#21446821C212yesmaximal, complemented
6#21446781C211yesmaximal, characteristic, complemented
7#21446612C211yesmaximal, characteristic, complemented
8#21446128C212yesmaximal, complemented
9#21446015C212yesmaximal, complemented
10#21445820C211yesmaximal, characteristic, complemented
11C2×C61×#477C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13#16840478—24no—
14#16840478—24no—
15C2×#14519766—22no—
16C2×#14519753C2212yes—
17C2×#14519752—21no—
18C2×#14519752C2212yes—
19C2×C4×D244C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519735—21no—
22C2×#14519735C2212yes—
23C4×#11758201—24no—
24C2×#14519708C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519699C2212yes—
27C2×#14519697C2211yescharacteristic
28C2×#14519689C2212yes—
29C2×#14519689C2212yes—
30C2×C4×#92418C2211yescharacteristic
31#16839684—22no—
32#16839664—21no—
33#16839660—21no—
34#16839632C2211yescharacteristic
35#16839619C2212yes—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4×Q8×C61C2211yescharacteristic
41C61×#477—24no—
42C2×#107×C61C2212yes—
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#103×C61C2212yes—
45C22×D244—22no—
46C22×D244D411yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49C22×C4×D61C2312yes—
50C2×#11758201—22no—
51C2×#11758201—22no—
52C2×#11758201—22no—
53C22×#92418C2311yescharacteristic
54C22×#92418C2311yescharacteristic
55#14519766—216no—
56#14519753—24no—
57#14519753—42no—
58#14519752—28no—
59#14519752—24no—
60#14519752—24no—
61C4×D244—22no—
62C4×D244—22no—
63C4×D244—22no—
64C4×D244D414yes—
65#14519735—28no—
66#14519735—24no—
67#14519735—24no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—22no—
74C2×#11758198—22no—
75C2×#11758198—21no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×#11758198—22no—
79C2×#11758198D412yes—
80C2×C4×Dic61—22no—
81#14519708—22no—
82#14519701—22no—
83#14519699—42no—
84#14519697—22no—
85#14519689—24no—
86#14519689—42no—
87C4×#92418—22no—
88#14519684—22no—
89C22×Q8×C61—21no—
90C22×Q8×C61C2311yescharacteristic
91C22×C8×C61C2312yes—
92C4⋊Q8×C61—22no—
93C4⋊Q8×C61D414yes—
94C4×Q8×C61—22no—
95C4×Q8×C61—22no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C4⋊C4×C61C2312yes—
100C2×C42×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#107×C61—24no—
103C4⋊C8×C61—22no—
104#103×C61—24no—
105C23×D61—21no—
106C23×D61C2×D412yes—
107C2×Q8×C61—24no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—81no—
114C2×Q8×C61—81no—
115C2×Q8×C61#4112yes—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61C2×D412yes—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—21no—
133C2×D244—28no—
134C2×D244—82no—
135C2×D244—82no—
136C2×D244C2×D412yes—
137C2×C4×D61—24no—
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—42no—
145C2×C4×D61—42no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—42no—
154C4⋊C4×C61C2×D414yes—
155C42×C61—42no—
156C42×C61C2×D412yes—
157C42×C61C2×D412yes—
158#11758201—28no—
159#11758198—28no—
160#11758198—28no—
161#11758198—24no—
162#11758198—24no—
163#11758198—44no—
164#11758198—82no—
165C4⋊Dic61—22no—
166C4⋊Dic61—21no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—42no—
170C4⋊Dic61—81no—
171C4⋊Dic61C2×D412yes—
172#11758196—44no—
173C4×Dic61—28no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418#4112yes—
180C23×C61C22×D411yescharacteristic
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195Q8×C61—22no—
196Q8×C61—22no—
197Q8×C61—41no—
198Q8×C61—41no—
199Q8×C61—84no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—24no—
203C2×C4×C61—22no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61#12012yes—
212C2×C4×C61#12012yes—
213C2×C4×C61#13512yes—
214C2×C4×C61#13712yes—
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61H(5,𝔽2)11yescharacteristic
220C2×C4×C612−1+411yescharacteristic
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230D244—22no—
231D244—22no—
232D244—41no—
233D244—41no—
234D244—44no—
235D244—88no—
236C4×D61—42no—
237C4×D61—42no—
238C4×D61—42no—
239C4×D61—42no—
240C4×D61—48no—
241C8×C61—42no—
242#92418—22no—
243#92418—41no—
244#24121—611nomaximal, Sylow
245C22×C61C2×#12011yescharacteristic
246C22×C61#54611yescharacteristic
247C22×C61#54711yescharacteristic
248C22×C61C2×#13511yescharacteristic
249C22×C61C2×#13711yescharacteristic
250C22×C61#57611yescharacteristic
251C22×C61#57711yescharacteristic
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—81no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C4×C61—22no—
267C4×C61—42no—
268C4×C61—44no—
269C4×C61—81no—
270C4×C61#45214yes—
271C4×C61#52511yescharacteristic
272C4×C61#53711yescharacteristic
273C4×C61#54811yescharacteristic
274C4×C61#55011yescharacteristic
275Dic61—42no—
276Dic61—42no—
277Dic61—82no—
278C2×#555—611no—
279C2×#477D6111yescharacteristic, complemented
280C2×C4×SD16—611no—
281#2538—611no—
282#2033—612no—
283#2000—612no—
284#1990—611no—
285#1954—611no—
286#1873—612no—
287#1854—612no—
288#1802—611no—
289C2×C61#240711yescharacteristic
290C2×C61C2×#45211yescharacteristic
291C2×C61#301211yescharacteristic
292C2×C61#330511yescharacteristic
293C2×C61#332911yescharacteristic
294C2×C61#334011yescharacteristic
295C2×C61#335711yescharacteristic
296D61—82no—
297D61—82no—
298C22×SD16—611no—
299#555—1224no—
300#555—1224no—
301C2×C4⋊Q8C2×D6111yescharacteristic
302C2×#124—1222no—
303C2×C22⋊Q8—612no—
304C2×#121—612no—
305C2×#121—1221no—
306C2×C4×Q8C2×D6111yescharacteristic
307C2×C4×D4—611no—
308C2×C4×D4—611no—
309C2×#117—612no—
310C2×#117—1221no—
311#477—24no—
312C4×SD16—1224no—
313C2×#107C2×D6112yes—
314C2×#106—611no—
315C2×C4⋊C8C2×D6111yescharacteristic
316C2×#103—611no—
317C2×#103C2×D6112yes—
318C2×#102—611no—
319C2×#102—612no—
320C2×C22⋊C8—612no—
321C2×C22⋊C8—612no—
322C2×C4×C8—611no—
323#396—1222no—
324#390—1221no—
325#389—1221no—
326#384—611no—
327#380—612no—
328#340—612no—
329#340—612no—
330#340—612no—
331C61#2412111yescharacteristic, complemented, Sylow
332C22×Q8—21no—
333C22×Q8C22×D6111yescharacteristic
334C22×D4—611no—
335C22×D4—611no—
336C22×D4—1222no—
337C23×C4—612no—
338C23×C4—1221no—
339C2×SD16—1222no—
340C2×SD16—1222no—
341C2×SD16—1222no—
342C22×C8—611no—
343C22×C8—611no—
344C22×C8C22×D6112yes—
345C4⋊Q8—22no—
346C4⋊Q8C22⋊D6114yes—
347#124—12216no—
348C22⋊Q8—1224no—
349C22⋊Q8—2442no—
350#121—1224no—
351#121—1228no—
352#121—1224no—
353C4×Q8—22no—
354C4×Q8—22no—
355C4×D4—614no—
356C4×D4—1222no—
357C4×D4—1222no—
358C4×D4—1222no—
359#117—1224no—
360#117—1228no—
361#117—1224no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—612no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4C22×D6111yescharacteristic
371C2×C4⋊C4C22×D6112yes—
372C2×C22⋊C4—612no—
373C2×C22⋊C4—1222no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1222no—
378C2×C42—21no—
379C2×C42—1222no—
380C2×C42C22×D6111yescharacteristic
381#107—24no—
382#106—1222no—
383C4⋊C8—22no—
384#103—24no—
385#103—1222no—
386#102—1222no—
387#102—2442no—
388C22⋊C8—1224no—
389C22⋊C8—2442no—
390C4×C8—1222no—
391#95—1222no—
392C24—612no—
393C24—1221no—
394C2×Q8—21no—
395C2×Q8—24no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×Q8#1175822612yes—
403C2×D4—612no—
404C2×D4—1228no—
405C2×D4—1222no—
406C2×D4—1221no—
407C2×D4—1221no—
408C2×D4—1221no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1222no—
412C2×D4—1222no—
413C2×D4—4882no—
414C2×D4—4882no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—612no—
419C22×C4—1221no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1224no—
427C22×C4—1222no—
428C22×C4—2442no—
429C22×C4—2442no—
430C22×C4D4×D6111yescharacteristic
431C22×C4D4×D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433SD16—1228no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—612no—
438C2×C8—1221no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—42no—
448C4⋊C4—612no—
449C4⋊C4—1221no—
450C4⋊C4—1222no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—4881no—
454C4⋊C4—2444no—
455C4⋊C4—2442no—
456C4⋊C4C2×C22⋊D6114yes—
457C22⋊C4—1228no—
458C22⋊C4—1228no—
459C22⋊C4—1224no—
460C22⋊C4—1224no—
461C22⋊C4—4882no—
462C22⋊C4—2444no—
463C42—42no—
464C42—1228no—
465C42D4×D6112yes—
466C42C2×C22⋊D6112yes—
467C23—1222no—
468C23—1222no—
469C23—1222no—
470C23—1222no—
471C23—1222no—
472C23—1222no—
473C23—1222no—
474C23—1222no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23—2441no—
480C23—2441no—
481C23C2×D4×D6111yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—41no—
485Q8—41no—
486Q8—84no—
487D4—1222no—
488D4—1222no—
489D4—4888no—
490D4—2441no—
491D4—2441no—
492D4—2444no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2448no—
518C2×C4#1451974512yes—
519C2×C4#1451974512yes—
520C2×C4#1451980012yes—
521C2×C4#1451980912yes—
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C22×C22⋊D6111yescharacteristic
524C2×C4#41×D6111yescharacteristic
525C2×C4C2×#1175822611yescharacteristic
526C2×C4#1451982311yescharacteristic
527C2×C4#1451982511yescharacteristic
528C8—42no—
529C8—1222no—
530C8—2441no—
531C22—4881no—
532C22—4881no—
533C22—4881no—
534C22—4881no—
535C22—2442no—
536C22—2442no—
537C22—2442no—
538C22—2442no—
539C22—2442no—
540C22—2442no—
541C22—2442no—
542C22—2442no—
543C22—2442no—
544C22—2442no—
545C22C2×#1451974511yescharacteristic
546C22#1684040311yescharacteristic
547C22#1684041411yescharacteristic
548C22C2×#1451980011yescharacteristic
549C22C2×#1451980911yescharacteristic
550C22#1684056611yescharacteristic
551C22#1684057211yescharacteristic
552C4—22no—
553C4—44no—
554C4—42no—
555C4—81no—
556C4—4882no—
557C4—2442no—
558C4—2442no—
559C4#1683992314yes—
560C4#1684027511yescharacteristic
561C4#1684034211yescharacteristic
562C4#1684042311yescharacteristic
563C4#1684043111yescharacteristic
564C2—4882no—
565C2—4882no—
566C2#2144916611yescharacteristic
567C2C2×#1683992311yescharacteristic
568C2#2145512211yescharacteristic
569C2#2145776611yescharacteristic
570C2#2145811011yescharacteristic
571C2#2145823511yescharacteristic
572C2#2145844211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#477C2
socleC23×C61C22×D4
prime radicalC2×#16840478C2
hypercenterC2×#477D61
hypocenterC61#24121
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758201C2semidirect
3. #21450737C2semidirect
4. #21447003C2semidirect
5. #21446821C2semidirect
6. #21446781C2semidirect
7. #21446612C2semidirect
8. #21446128C2semidirect
9. #21446015C2semidirect
10. #21445820C2semidirect
11. C2×C61×#477C2semidirect
279. C2×#477D61semidirect
331. C61#24121semidirect

Maximal subgroups

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

subgroupindex
C2×#168404782
C2×C4×#117582012
#214507372
#214470032
#214468212
#214467812
#214466122
#214461282
#214460152
#214458202
C2×C61×#4772
#2412161