Finite Group Databasebeta

← #29713302

Subgroups — #29713302

Order 15616.

1144 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2C2×#16840091C212yesmaximal
3#389×D61C211yesmaximal, characteristic
4#21447153C211yesmaximal, characteristic, complemented
5#21447152C211yesmaximal, characteristic, complemented
6#21447062C211yesmaximal, characteristic, complemented
7#21446123C212yesmaximal
8#21446089C212yesmaximal, complemented
9#21446085C212yesmaximal, complemented
10#21445886C211yesmaximal, characteristic
11C61×#2043C211yesmaximal, characteristic, complemented
12#16840317—28no—
13C2×#14519777C2211yescharacteristic
14C2×#14519768—21no—
15C2×#14519757—21no—
16C2×#14519753C2212yes—
17C2×Q8×Dic61C2212yes—
18C2×#14519742—21no—
19C2×#14519735—21no—
20C2×C4⋊C4×D61—21no—
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23C2×C42×D61C2211yescharacteristic
24#16840091—28no—
25C2×#14519709C2212yes—
26C2×#14519708C2212yes—
27C2×#14519701C2212yes—
28C2×#14519701C2212yes—
29C2×#14519689C2212yes—
30#16839661C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839659—21no—
33#16839658C2211yescharacteristic
34#16839657—21no—
35#95×D61—21no—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839517C2211yescharacteristic
39#16839516C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#103×C61C2212yes—
42C2×#103×C61C2212yes—
43C61×#389C2211yescharacteristic
44C61×#336C2211yescharacteristic
45C22×Dic122—21no—
46C22×C4×D61—21no—
47C22×C4×D61—21no—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52#14519777—24no—
53#14519768—28no—
54#14519757—28no—
55#14519753—24no—
56#14519753D418yes—
57Q8×Dic61—24no—
58Q8×Dic61—24no—
59#14519742—28no—
60#14519735—28no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61C2311yescharacteristic
65C2×C4⋊Dic61C2312yes—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×#11758196—21no—
68C2×#11758196—22no—
69C2×#11758196—21no—
70C2×#11758196—21no—
71C2×#11758196—21no—
72C4⋊C4×D61—22no—
73C4⋊C4×D61—22no—
74C4⋊C4×D61—24no—
75C4⋊C4×D61—44no—
76C2×#11758198—22no—
77C2×C4×Dic61—22no—
78C2×C4×Dic61D411yescharacteristic
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61C2311yescharacteristic
81C42×D61—24no—
82#14519709—24no—
83#14519708—24no—
84#14519701—24no—
85#14519701—24no—
86#14519689—24no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90C22×Q8×C61C2312yes—
91C22×C8×C61C2312yes—
92C4⋊Q8×C61—24no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C42×C61C2311yescharacteristic
99#103×C61—24no—
100#103×C61—24no—
101#95×C61—21no—
102C23×D61C2×D411yescharacteristic
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61C2×D414yes—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2411yescharacteristic
115C22×Dic61—21no—
116C22×Dic61—22no—
117C22×Dic61—21no—
118C22×Dic61C2×D411yescharacteristic
119C22×Dic61C2×D411yescharacteristic
120C22×Dic61C2×D411yescharacteristic
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—21no—
124C2×C4×D61—21no—
125C2×C4×D61—22no—
126C2×C4×D61—24no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—41no—
130C2×C4×D61—42no—
131C2×C4×D61—42no—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134C2×C4×D61—42no—
135C2×C4×D61SD1614yes—
136C2×C4×D61C2×D414yes—
137C2×Dic122—24no—
138C2×Dic122—42no—
139C2×Dic122—42no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—42no—
148C4⋊C4×C61#4112yes—
149C42×C61—22no—
150#11758198—28no—
151C4⋊Dic61—24no—
152C4⋊Dic61—28no—
153C4⋊Dic61—22no—
154C4⋊Dic61—21no—
155C4⋊Dic61—22no—
156C4⋊Dic61—42no—
157C4⋊Dic61C2×D414yes—
158C4⋊Dic61#4112yes—
159#11758196—24no—
160#11758196—28no—
161#11758196—42no—
162#11758196—42no—
163#11758196—42no—
164C4×Dic61—28no—
165C4×Dic61—22no—
166C4×Dic61—22no—
167C4×Dic61—22no—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C23×C61C22×D411yescharacteristic
172C22×D61—21no—
173C22×D61—21no—
174C22×D61—21no—
175C22×D61—21no—
176C22×D61#12012yes—
177C22×D61C2×SD1612yes—
178C22×D61#13712yes—
179Q8×C61—24no—
180Q8×C61—24no—
181Q8×C61—42no—
182Q8×C61—42no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—24no—
188C2×C4×C61—22no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61C2×SD1612yes—
197C2×C4×C61#13712yes—
198C2×C4×C61C22×D412yes—
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C61C2×#4111yescharacteristic
201C2×C4×C612−1+412yes—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—24no—
208C2×Dic61—22no—
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—41no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61#12012yes—
219C2×Dic61C2×SD1612yes—
220C4×D61—28no—
221C4×D61—24no—
222C4×D61—24no—
223C4×D61—44no—
224C4×D61—48no—
225C4×D61—84no—
226Dic122—48no—
227C8×C61—42no—
228#92418—42no—
229#24063—611nomaximal, Sylow
230C22×C61C2×#12011yescharacteristic
231C22×C61#52511yescharacteristic
232C22×C61#53611yescharacteristic
233C22×C61C22×SD1611yescharacteristic
234C22×C61C2×#13711yescharacteristic
235C22×C61C2×#13711yescharacteristic
236C22×C61#57511yescharacteristic
237C2×D61—22no—
238C2×D61—22no—
239C2×D61—22no—
240C2×D61—22no—
241C2×D61—22no—
242C2×D61—22no—
243C2×D61#44814yes—
244C4×C61—22no—
245C4×C61—22no—
246C4×C61—42no—
247C4×C61—44no—
248C4×C61—81no—
249C4×C61#54712yes—
250C4×C61#54712yes—
251Dic61—22no—
252Dic61—22no—
253Dic61—42no—
254Dic61—44no—
255Dic61—41no—
256Dic61—81no—
257C2×#533—611no—
258C2×#483—612no—
259C2×#389—611no—
260#2058—611no—
261#2058—611no—
262#2043—611no—
263#2043D6111yescharacteristic, complemented
264#1872—612no—
265#1865—612no—
266#1865—612no—
267#1817—611no—
268C2×C61#240611yescharacteristic
269C2×C61C2×#44811yescharacteristic
270C2×C61#300711yescharacteristic
271C2×C61#310111yescharacteristic
272C2×C61#311311yescharacteristic
273C2×C61#322911yescharacteristic
274C2×C61#323611yescharacteristic
275D61—24no—
276#533—1228no—
277C2×C4⋊Q8—611no—
278C2×C4⋊Q8C2×D6111yescharacteristic
279C2×#125—1221no—
280C2×C22⋊Q8—612no—
281C2×C22⋊Q8—1221no—
282C2×C4×Q8—612no—
283C2×C4×Q8—1221no—
284C2×#117—1221no—
285C22×C4⋊C4—611no—
286C22×C4⋊C4—611no—
287C22×C4⋊C4—1221no—
288C22×C42—611no—
289#483—1228no—
290C2×#106—612no—
291C2×#106—612no—
292C2×#103—612no—
293C2×#103—612no—
294C2×#103C2×D6112yes—
295C2×#103C2×D6112yes—
296C2×C22⋊C8—612no—
297#389—611no—
298#389—611no—
299#389—611no—
300#389—1221no—
301#389—1221no—
302#389C2×D6111yescharacteristic
303C2×#95—1221no—
304#340—612no—
305#340—612no—
306#336—612no—
307#336—611no—
308#336C2×D6111yescharacteristic
309C61#2406311yescharacteristic, complemented, Sylow
310C22×Q8—1221no—
311C22×Q8C22×D6112yes—
312C23×C4—611no—
313C23×C4—611no—
314C23×C4—611no—
315C23×C4—1221no—
316C23×C4—1221no—
317C22×C8—612no—
318C22×C8C22×D6112yes—
319C4⋊Q8—24no—
320C4⋊Q8—1224no—
321#125—1228no—
322C22⋊Q8—618no—
323C22⋊Q8—1228no—
324C22⋊Q8—1224no—
325C4×Q8—1228no—
326C4×Q8—1224no—
327C4×Q8—1224no—
328#117—1228no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—612no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—1222no—
336C2×C4⋊C4—1224no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—1222no—
339C2×C4⋊C4—1222no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—2444no—
347C2×C4⋊C4C22×D6111yescharacteristic
348C2×C4⋊C4C22×D6111yescharacteristic
349C2×C22⋊C4—1222no—
350C2×C42—611no—
351C2×C42—611no—
352C2×C42—611no—
353C2×C42—1224no—
354C2×C42—1222no—
355C2×C42C22×D6111yescharacteristic
356#106—1224no—
357#106—1224no—
358#103—24no—
359#103—24no—
360#103—1224no—
361#103—1224no—
362C22⋊C8—1224no—
363#95—21no—
364#95—1221no—
365#95—1221no—
366#95—1221no—
367C24—611no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—1224no—
374C2×Q8—2442no—
375C2×Q8—2442no—
376C2×Q8D4×D6114yes—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—611no—
381C22×C4—611no—
382C22×C4—614no—
383C22×C4—614no—
384C22×C4—611no—
385C22×C4—1222no—
386C22×C4—1222no—
387C22×C4—1222no—
388C22×C4—1224no—
389C22×C4—1221no—
390C22×C4—1221no—
391C22×C4—1222no—
392C22×C4—1221no—
393C22×C4—1222no—
394C22×C4—1222no—
395C22×C4—1221no—
396C22×C4—2441no—
397C22×C4—2442no—
398C22×C4—2442no—
399C22×C4—2442no—
400C22×C4—2442no—
401C22×C4—2442no—
402C22×C4D4×D6111yescharacteristic
403C22×C4D4×D6111yescharacteristic
404C22×C4C23×D6111yescharacteristic
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—1222no—
409C2×C8—1222no—
410C2×C8—1222no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—42no—
416C4⋊C4—614no—
417C4⋊C4—612no—
418C4⋊C4—1228no—
419C4⋊C4—1224no—
420C4⋊C4—1224no—
421C4⋊C4—1228no—
422C4⋊C4—1222no—
423C4⋊C4—1221no—
424C4⋊C4—1222no—
425C4⋊C4—2442no—
426C4⋊C4—2442no—
427C4⋊C4—2442no—
428C4⋊C4—2442no—
429C4⋊C4#1175822412yes—
430C22⋊C4—1228no—
431C42—22no—
432C42—1222no—
433C42—1222no—
434C42—1228no—
435C42—1222no—
436C23—612no—
437C23—612no—
438C23—612no—
439C23—1221no—
440C23—1221no—
441C23—1221no—
442C23—1221no—
443C23C2×D4×D6111yescharacteristic
444Q8—24no—
445Q8—24no—
446Q8—42no—
447Q8—42no—
448Q8—2448no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—612no—
463C2×C4—612no—
464C2×C4—1228no—
465C2×C4—1221no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—1221no—
469C2×C4—1224no—
470C2×C4—1224no—
471C2×C4—1221no—
472C2×C4—1224no—
473C2×C4—1222no—
474C2×C4—1221no—
475C2×C4—1221no—
476C2×C4—4884no—
477C2×C4—2444no—
478C2×C4—2448no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2442no—
484C2×C4—2441no—
485C2×C4—2442no—
486C2×C4SD16×D6112yes—
487C2×C4#1451980912yes—
488C2×C4C2×D4×D6112yes—
489C2×C4#41×D6111yescharacteristic
490C2×C4C2×#1175822411yescharacteristic
491C2×C4#1451982412yes—
492C8—42no—
493C8—2442no—
494C22—614no—
495C22—1222no—
496C22—1222no—
497C22—1222no—
498C22—1222no—
499C22—1222no—
500C22—1222no—
501C22#120×D6111yescharacteristic
502C22#1684027211yescharacteristic
503C22#1684033211yescharacteristic
504C22C2×SD16×D6111yescharacteristic
505C22#137×D6111yescharacteristic
506C22C2×#1451980911yescharacteristic
507C22#1684056111yescharacteristic
508C4—22no—
509C4—22no—
510C4—44no—
511C4—42no—
512C4—81no—
513C4—1222no—
514C4—1222no—
515C4—4881no—
516C4—2442no—
517C4—2441no—
518C4—2444no—
519C4#1684041012yes—
520C4#1684041212yes—
521C2—1224no—
522C2#2144914911yescharacteristic
523C2#448×D6111yescharacteristic
524C2#2145507811yescharacteristic
525C2#2145582611yescharacteristic
526C2#2145594811yescharacteristic
527C2#2145695911yescharacteristic
528C2#2145702011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2043C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2043D61
hypocenterC61#24063
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2043

Complements

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

NH ≅ G/Ntype
4. #21447153C2semidirect
5. #21447152C2semidirect
6. #21447062C2semidirect
8. #21446089C2semidirect
9. #21446085C2semidirect
11. C61×#2043C2semidirect
263. #2043D61semidirect
309. C61#24063semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168400912
#389×D612
#214471532
#214471522
#214470622
#214461232
#214460892
#214460852
#214458862
C61×#20432
#2406361