Finite Group Databasebeta

← #29662554

Subgroups — #29662554

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456329C211yesmaximal, characteristic, complemented
2#21455828C211yesmaximal, characteristic
3#21455823C211yesmaximal, characteristic
4C2×#16839839C211yesmaximal, characteristic
5#21445341C212yesmaximal, complemented
6#21445331C212yesmaximal, complemented
7#21445321C212yesmaximal, complemented
8#21445317C212yesmaximal, complemented
9#21444565C211yesmaximal, characteristic, complemented
10#21444477C211yesmaximal, characteristic, complemented
11C61×#1542C211yesmaximal, characteristic, complemented
12C22×#11758189—21no—
13#16840319C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16840165C2211yescharacteristic, complemented
16#16840165C2211yescharacteristic, complemented
17#16840161—21no—
18#16840161—21no—
19#16840158C2211yescharacteristic, complemented
20#16840158C2211yescharacteristic, complemented
21#16840093—21no—
22#16840090—21no—
23#16840053C2212yes—
24#16840049C2212yes—
25#16839968C2212yes—
26#16839964C2212yes—
27#16839915—21no—
28#16839912—21no—
29C4×#11758190—22no—
30C4×#11758201—22no—
31#16839839—21no—
32#16839839—21no—
33#16839839C2212yes—
34#16839839C2212yes—
35#16839815C2211yescharacteristic
36#16839798—21no—
37#16839798—21no—
38C2×#14519689—21no—
39C2×C4×#92418C2211yescharacteristic
40#16839514C2212yescomplemented
41#16839512C2212yescomplemented
42#16839510C2212yescomplemented
43#16839507C2212yescomplemented
44C61×#533C2211yescharacteristic
45C61×#431C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#333C2212yescomplemented
48C61×#332C2212yescomplemented
49C22×D244D411yescharacteristic
50C22×C4×D61—21no—
51C2×#11758190—22no—
52C2×#11758201—22no—
53C2×#11758189—22no—
54C2×#11758189—22no—
55C2×#11758189—44no—
56C22×#92418D411yescharacteristic
57C22×#92418D411yescharacteristic
58#14519777C2311yescharacteristic
59#14519777C2311yescharacteristic
60#14519769—21no—
61#14519756—21no—
62#14519756—21no—
63C4×Dic122—21no—
64C4×Dic122—21no—
65C4×D244—21no—
66C4×D244—21no—
67C4×D244D412yes—
68C4×D244C2312yes—
69#14519736—21no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×#11758198—21no—
72#14519710—21no—
73#14519710—21no—
74#14519710—21no—
75#14519710—21no—
76#14519709—22no—
77#14519708—22no—
78#14519708—21no—
79#14519708—21no—
80#14519708—21no—
81#14519708—21no—
82C4⋊#92418C2312yes—
83#14519702—22no—
84#14519701—22no—
85#14519699—22no—
86#14519697—22no—
87C22⋊#92418—21no—
88#14519689—22no—
89#14519689—22no—
90C4×#92418—21no—
91C4×#92418D411yescharacteristic
92C4×#92418D411yescharacteristic
93C4×#92418C2311yescharacteristic
94C4×#92418C2311yescharacteristic
95C22×C8×C61—21no—
96C4⋊Q8×C61C2311yescharacteristic
97C4⋊Q8×C61C2311yescharacteristic
98#125×C61—21no—
99C22⋊Q8×C61—21no—
100C22⋊Q8×C61—21no—
101C4×Q8×C61—21no—
102C4×Q8×C61—21no—
103#117×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—21no—
106C4⋊C8×C61C2312yes—
107C4⋊C8×C61C2312yes—
108C22⋊C8×C61—21no—
109C4×C8×C61—21no—
110C23×D61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C22×Dic61—21no—
117C2×D244—21no—
118C2×D244—42no—
119C2×D244C2×D412yes—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—42no—
123C2×C4×D61—42no—
124C2×Dic122—21no—
125C2×Dic122—21no—
126C2×C8×C61—22no—
127C2×C8×C61—22no—
128C2×C8×C61—42no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—41no—
134C4⋊C4×C61—41no—
135C22⋊C4×C61—41no—
136C42×C61—41no—
137C42×C61C2×D411yescharacteristic
138C42×C61C2×D411yescharacteristic
139C42×C61#4111yescharacteristic
140C42×C61C2411yescharacteristic
141C22⋊Dic61—41no—
142#11758201—44no—
143#11758198—22no—
144#11758198—22no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61C2×D412yes—
151#11758196—41no—
152#11758196—41no—
153C4×Dic61—41no—
154C2×#92418—22no—
155C2×#92418—22no—
156C2×#92418SD1614yescomplemented
157C2×#92418C2×D411yescharacteristic
158C2×#92418C2×D411yescharacteristic
159C2×#92418C2×D411yescharacteristic
160C2×#92418C2×D411yescharacteristic
161#11758190—44no—
162#11758189—44no—
163#11758189—44no—
164C23×C61#12111yescharacteristic
165C22×D61—22no—
166C22×D61—42no—
167C22×D61—42no—
168C22×D61—42no—
169Q8×C61—42no—
170Q8×C61—81no—
171C2×C4×C61—22no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61#12111yescharacteristic
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61C2×SD1612yes—
179C2×C4×C61#13712yes—
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C2×#4111yescharacteristic
183C2×Dic61—22no—
184C2×Dic61—42no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187D244—22no—
188D244—22no—
189D244—42no—
190C4×D61—42no—
191C4×D61—42no—
192C4×D61—44no—
193Dic122—42no—
194Dic122—81no—
195C8×C61—42no—
196C8×C61—42no—
197#92418—21no—
198#92418—21no—
199#92418—42no—
200#92418C2×SD1612yes—
201#92418C2×SD1612yes—
202#19137—611nomaximal, Sylow
203C22×C61—21no—
204C22×C61—21no—
205C22×C61#46112yes—
206C22×C61C2×#12111yescharacteristic
207C2×D61—42no—
208C2×D61—42no—
209C2×D61—42no—
210C2×D61—84no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—81no—
215C4×C61—81no—
216C4×C61#46112yes—
217C4×C61C22×SD1611yescharacteristic
218C4×C61C2×#13711yescharacteristic
219C4×C61#57512yes—
220Dic61—42no—
221Dic61—81no—
222Dic61—81no—
223#3160—611no—
224#3101—611no—
225#3101—611no—
226C2×#436—611no—
227#1695—612no—
228#1693—612no—
229#1691—612no—
230#1691—612no—
231#1564—611no—
232#1542—611no—
233#1542D6111yescharacteristic, complemented
234C2×C61—22no—
235C2×C61C2×#46111yescharacteristic
236C2×C61#305411yescharacteristic
237C2×C61#306711yescharacteristic
238D61—82no—
239C22×M16—1221no—
240#533—611no—
241#533C2×D6111yescharacteristic
242C2×C4×D4—611no—
243#500—611no—
244#500—611no—
245#499—1221no—
246#499—1221no—
247#498—611no—
248#498—611no—
249#483—1221no—
250#483—1221no—
251#476—612no—
252#476—612no—
253#461—612no—
254#461—612no—
255#450—1221no—
256#450—1221no—
257C4×SD16—1222no—
258C4×SD16—1222no—
259#436—612no—
260#436—612no—
261#436—1221no—
262#436—1221no—
263#431—611no—
264#431C2×D6111yescharacteristic
265#427—1221no—
266#427—1221no—
267C2×C4⋊C8C2×D6111yescharacteristic
268C2×C22⋊C8—1221no—
269C2×C4×C8—611no—
270#335—612no—
271#334—612no—
272#333—612no—
273#333C2×D6112yescomplemented
274#332—612no—
275#332C2×D6112yescomplemented
276C61#1913711yescharacteristic, complemented, Sylow
277C22×D4—611no—
278C23×C4—1221no—
279C2×SD16—1222no—
280C2×SD16—1222no—
281C2×M16—1222no—
282C2×M16—1222no—
283C2×M16—2444no—
284C22×C8—21no—
285C22×C8—611no—
286C22×C8—611no—
287C4⋊Q8—611no—
288C4⋊Q8—611no—
289C4⋊Q8C22×D6111yescharacteristic
290C4⋊Q8C22×D6111yescharacteristic
291#125—21no—
292#125—1221no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C22⋊Q8—1221no—
296C22⋊Q8—1221no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8—1221no—
300C4×Q8—1221no—
301C4×D4—612no—
302C4×D4—612no—
303C4×D4—1221no—
304C4×D4—1221no—
305#117—21no—
306#117—1221no—
307C2×C4⋊C4—611no—
308C2×C22⋊C4—1221no—
309C2×C42C22×D6111yescharacteristic
310#107—1221no—
311#107—1221no—
312#107—1221no—
313#107—1221no—
314#106—1222no—
315#106—1221no—
316#106—1221no—
317#106—1221no—
318#106—1221no—
319#106—1222no—
320C4⋊C8—21no—
321C4⋊C8—612no—
322C4⋊C8C22×D6112yes—
323C4⋊C8C22×D6112yes—
324#103—1222no—
325#103—1222no—
326#102—1222no—
327#102—1222no—
328C22⋊C8—21no—
329C22⋊C8—1222no—
330C22⋊C8—1222no—
331C22⋊C8—1221no—
332C4×C8—21no—
333C4×C8—611no—
334C4×C8—611no—
335C4×C8—611no—
336C4×C8—611no—
337C4×C8—1221no—
338C24—1221no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—1221no—
342C2×Q8—1221no—
343C2×D4—612no—
344C2×D4—1221no—
345C2×D4—2442no—
346C22×C4—1221no—
347C22×C4—1222no—
348C22×C4—1222no—
349C22×C4—2442no—
350C22×C4—2442no—
351C22×C4#1175822211yescharacteristic
352C22×C4D4×D6111yescharacteristic
353C22×C4D4×D6111yescharacteristic
354SD16—2444no—
355SD16—2444no—
356M16—2444no—
357M16—2444no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—42no—
361C2×C8—611no—
362C2×C8—611no—
363C2×C8—614no—
364C2×C8—611no—
365C2×C8—611no—
366C2×C8—1222no—
367C2×C8—1222no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—612no—
375C4⋊C4—1221no—
376C4⋊C4—1221no—
377C4⋊C4—1221no—
378C4⋊C4—1221no—
379C4⋊C4—1221no—
380C4⋊C4—2441no—
381C4⋊C4—2441no—
382C22⋊C4—41no—
383C22⋊C4—1222no—
384C22⋊C4—1222no—
385C22⋊C4—2441no—
386C42—41no—
387C42—2441no—
388C42#1175822211yescharacteristic
389C42D4×D6111yescharacteristic
390C42D4×D6111yescharacteristic
391C42C23×D6111yescharacteristic
392C23—1222no—
393C23—2442no—
394C23—2442no—
395C23—2442no—
396C23#1451975011yescharacteristic
397Q8—42no—
398Q8—81no—
399Q8—4881no—
400Q8—2442no—
401D4—1222no—
402D4—1222no—
403D4—2442no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—1222no—
409C2×C4—2441no—
410C2×C4—2441no—
411C2×C4—2442no—
412C2×C4—2444no—
413C2×C4—2442no—
414C2×C4—2442no—
415C2×C4#1451975011yescharacteristic
416C2×C4#1451975011yescharacteristic
417C2×C4#1451975011yescharacteristic
418C2×C4SD16×D6112yes—
419C2×C4#1451980912yes—
420C2×C4C2×D4×D6111yescharacteristic
421C2×C4C2×D4×D6111yescharacteristic
422C2×C4C2×#1175822211yescharacteristic
423C8—42no—
424C8—42no—
425C8—612no—
426C8—612no—
427C8—1221no—
428C8—1221no—
429C8—2442no—
430C22—21no—
431C22—21no—
432C22—4884no—
433C22—2442no—
434C22—2442no—
435C22—2442no—
436C22#1683996612yes—
437C22C2×#1451975011yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—81no—
442C4—81no—
443C4—4881no—
444C4—4881no—
445C4—2442no—
446C4#1683996612yes—
447C4C2×SD16×D6111yescharacteristic
448C4C2×#1451980911yescharacteristic
449C4#1684055912yes—
450C2—22no—
451C2—4882no—
452C2C2×#1683996611yescharacteristic
453C2#2145542711yescharacteristic
454C2#2145553811yescharacteristic
subgroupG/H
centerC2×C4#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1542C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#1542D61
hypocenterC61#19137
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
1. #21456329C2semidirect
5. #21445341C2semidirect
6. #21445331C2semidirect
7. #21445321C2semidirect
8. #21445317C2semidirect
9. #21444565C2semidirect
10. #21444477C2semidirect
11. C61×#1542C2semidirect
15. #16840165C22semidirect
16. #16840165C22semidirect
19. #16840158C22semidirect
20. #16840158C22semidirect
40. #16839514C22semidirect
41. #16839512C22semidirect
42. #16839510C22semidirect
43. #16839507C22semidirect
47. C61×#333C22semidirect
48. C61×#332C22semidirect
156. C2×#92418SD16semidirect
233. #1542D61semidirect
273. #333C2×D61semidirect
275. #332C2×D61semidirect
276. C61#19137semidirect

Maximal subgroups

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

subgroupindex
#214563292
#214558282
#214558232
C2×#168398392
#214453412
#214453312
#214453212
#214453172
#214445652
#214444772
C61×#15422
#1913761