Finite Group Databasebeta

← #29779859

Subgroups — #29779859

Order 15616.

770 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454196C211yesmaximal, characteristic, complemented
2#21448362C211yesmaximal, characteristic
3#21446960C211yesmaximal, characteristic, complemented
4#21446956C211yesmaximal, characteristic
5#21446952C211yesmaximal, characteristic
6#21446627C212yesmaximal
7#21445948C212yesmaximal, complemented
8#21445832C212yesmaximal, complemented
9#21445828C212yesmaximal
10#21445667C211yesmaximal, characteristic, complemented
11C61×#2024C211yesmaximal, characteristic, complemented
12C2×#14519777C2211yescharacteristic
13C2×#14519766C2211yescharacteristic
14C2×Q8×Dic61C2211yescharacteristic
15C2×#14519739C2211yescharacteristic
16C2×#14519706C2212yes—
17C2×#14519701C2212yes—
18C2×#14519699C2212yes—
19C2×#14519697C2212yes—
20C2×C8×Dic61C2212yes—
21#16839669—21no—
22#16839669—21no—
23#16839667—21no—
24#16839667—21no—
25#16839664C2211yescharacteristic
26#16839654—21no—
27#16839654—21no—
28#16839627C2211yescharacteristic
29#16839627C2212yes—
30#16839627C2212yes—
31#16839609—21no—
32#16839609—21no—
33C42×Dic61C2211yescharacteristic
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839517C2211yescharacteristic
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×#103×C61C2212yes—
40C61×#391C2211yescharacteristic
41C61×#340C2212yes—
42C61×#336C2211yescharacteristic
43C22×Dic122—21no—
44C22×D244C2311yescharacteristic
45C22×C4×D61—21no—
46C22×#92418C2312yes—
47#14519777—42no—
48#14519766—22no—
49#14519766Q814yescomplemented
50Q8×Dic61—24no—
51#14519739—24no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2311yescharacteristic
55C2×C4⋊Dic61C2312yes—
56C2×#11758198—21no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61C2×C4×Dic61—21no—
62C2×C4×Dic61D411yescharacteristic
63C2×C4×Dic61D411yescharacteristic
64C2×C4×Dic61D411yescharacteristic
65C2×C4×Dic61D411yescharacteristic
66C2×C4×Dic61C2311yescharacteristic
67C2×C4×Dic61C2311yescharacteristic
68#14519706—24no—
69#14519701—42no—
70#14519699—42no—
71#14519697—42no—
72C8×Dic61—24no—
73#14519684—21no—
74#14519684—21no—
75#14519684—22no—
76#14519684—22no—
77#14519684—22no—
78#14519684—22no—
79C22×Q8×C61—21no—
80C22×Q8×C61C2311yescharacteristic
81C22×C8×C61C2312yes—
82C4⋊Q8×C61—42no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61C2312yes—
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C42×C61C2311yescharacteristic
87#103×C61—42no—
88#95×C61—21no—
89#95×C61—21no—
90C23×D61—21no—
91C2×Q8×C61—21no—
92C2×Q8×C61—22no—
93C2×Q8×C61—22no—
94C2×Q8×C61—42no—
95C2×Q8×C61—42no—
96C2×Q8×C61—42no—
97C2×Q8×C61C2×Q812yes—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—22no—
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2411yescharacteristic
104C22×Dic61—22no—
105C22×Dic61—21no—
106C22×Dic61C2×D411yescharacteristic
107C22×Dic61C2×D411yescharacteristic
108C22×Dic61C2×D411yescharacteristic
109C22×Dic61C2×D411yescharacteristic
110C2×D244—22no—
111C2×D244—22no—
112C2×D244—21no—
113C2×D244C2×Q812yes—
114C2×C4×D61—42no—
115C2×C4×D61—42no—
116C2×C4×D61—42no—
117C2×Dic122—42no—
118C2×Dic122—42no—
119C2×Dic122—42no—
120C2×C8×C61—22no—
121C2×C8×C61—22no—
122C2×C8×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—42no—
127C42×C61—22no—
128#11758198—24no—
129#11758198—24no—
130#11758198—42no—
131#11758198—42no—
132#11758198—42no—
133C4⋊Dic61—22no—
134C4⋊Dic61—22no—
135C4⋊Dic61—22no—
136C4⋊Dic61—42no—
137C4⋊Dic61—42no—
138C4×Dic61—22no—
139C4×Dic61—22no—
140C4×Dic61—41no—
141C4×Dic61—41no—
142C4×Dic61—42no—
143C4×Dic61C2×Q812yes—
144C4×Dic61#4112yes—
145C4×Dic61#4114yes—
146C2×#92418—22no—
147C2×#92418—22no—
148C2×#92418—22no—
149C23×C61C22×D411yescharacteristic
150C22×D61—22no—
151C22×D61—41no—
152C22×D61—41no—
153C22×D61—41no—
154C22×D61—41no—
155C22×D61—41no—
156C22×D61—41no—
157Q8×C61—24no—
158Q8×C61—42no—
159Q8×C61—84no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—22no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—42no—
169C2×C4×C61—42no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—42no—
173C2×C4×C61#13512yes—
174C2×C4×C61#13512yes—
175C2×C4×C61C22×Q811yescharacteristic
176C2×C4×C61C2×#4112yes—
177C2×C4×C61C2×#4111yescharacteristic
178C2×C4×C61H(5,𝔽2)12yes—
179C2×Dic61—21no—
180C2×Dic61—21no—
181C2×Dic61—21no—
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—41no—
188C2×Dic61—42no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61C22⋊Q812yes—
194C2×Dic61C22⋊Q812yes—
195C2×Dic61#12412yes—
196C2×Dic61#12412yes—
197D244—24no—
198D244—42no—
199C4×D61—84no—
200Dic122—84no—
201C8×C61—42no—
202#92418—42no—
203#29142—611nomaximal, Sylow
204C22×C61C2×C22⋊Q811yescharacteristic
205C22×C61C2×#12411yescharacteristic
206C22×C61#53511yescharacteristic
207C22×C61C2×#13511yescharacteristic
208C22×C61C2×#13511yescharacteristic
209C22×C61#57511yescharacteristic
210C22×C61#57511yescharacteristic
211C2×D61—42no—
212C2×D61—42no—
213C2×D61—42no—
214C2×D61—81no—
215C2×D61—81no—
216C2×D61—81no—
217C2×D61—81no—
218C4×C61—22no—
219C4×C61—22no—
220C4×C61—42no—
221C4×C61—81no—
222C4×C61—82no—
223C4×C61#55412yes—
224C4×C61#55512yes—
225Dic61—22no—
226Dic61—22no—
227Dic61—22no—
228Dic61—22no—
229Dic61—81no—
230Dic61—82no—
231#2862—611no—
232C4×#124—611no—
233#2024—611no—
234#2024D6111yescharacteristic, complemented
235#2023—611no—
236#2023—611no—
237#1958—612no—
238#1835—612no—
239#1805—612no—
240#1804—612no—
241C4×C4⋊C8—611no—
242C2×C61#256311yescharacteristic
243C2×C61#307611yescharacteristic
244C2×C61#307611yescharacteristic
245C2×C61#313111yescharacteristic
246C2×C61#314011yescharacteristic
247C2×C61#321111yescharacteristic
248C2×C61#321111yescharacteristic
249D61—82no—
250C2×C4⋊Q8—611no—
251C2×C4⋊Q8C2×D6111yescharacteristic
252C2×#124—611no—
253C2×C4×Q8—611no—
254C2×C4×D4—611no—
255C2×C4⋊C8—612no—
256C2×#103—612no—
257C2×#103C2×D6112yes—
258C2×#102—612no—
259C2×#102—612no—
260C2×C4×C8—612no—
261#391—1221no—
262#391—1221no—
263#391—1221no—
264#391—1221no—
265#391C2×D6111yescharacteristic
266#390—611no—
267#388—1221no—
268#388—1221no—
269#382—612no—
270#382—611no—
271#382—612no—
272C4×C22⋊C4—1221no—
273C4×C22⋊C4—1221no—
274C43—611no—
275#340—612no—
276#340—612no—
277#340—612no—
278#340C2×D6112yes—
279#336—611no—
280#336C2×D6111yescharacteristic
281C61#2914211yescharacteristic, complemented, Sylow
282C22×Q8—21no—
283C22×Q8—1221no—
284C22×Q8C22×D6111yescharacteristic
285C22×D4—611no—
286C23×C4—1221no—
287C22×C8—612no—
288C22×C8C22×D6112yes—
289C4⋊Q8—42no—
290C4⋊Q8—2442no—
291#124—614no—
292#124—1222no—
293C4×Q8—1224no—
294C4×D4—1224no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—611no—
297C2×C4⋊C4—612no—
298C2×C4⋊C4—1221no—
299C2×C4⋊C4—1221no—
300C2×C4⋊C4C22×D6111yescharacteristic
301C2×C4⋊C4C22×D6112yes—
302C2×C22⋊C4—1221no—
303C2×C22⋊C4—1221no—
304C2×C22⋊C4—1221no—
305C2×C22⋊C4—1221no—
306C2×C22⋊C4—1221no—
307C2×C42—611no—
308C2×C42—611no—
309C2×C42—611no—
310C2×C42—611no—
311C2×C42—611no—
312C2×C42—611no—
313C2×C42—1221no—
314C2×C42C22×D6111yescharacteristic
315C4⋊C8—1224no—
316#103—42no—
317#103—2442no—
318#102—2442no—
319#102—2442no—
320C4×C8—1224no—
321#95—21no—
322#95—21no—
323#95—1221no—
324#95—1221no—
325#95—1222no—
326#95—1222no—
327#95—1222no—
328#95—1222no—
329C24—1221no—
330C2×Q8—21no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—2442no—
337C2×Q8—2442no—
338C2×Q8—2442no—
339C2×Q8Q8×D6112yes—
340C2×D4—612no—
341C2×D4—1222no—
342C2×D4—1221no—
343C2×D4—1222no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—611no—
348C22×C4—611no—
349C22×C4—611no—
350C22×C4—611no—
351C22×C4—1222no—
352C22×C4—1221no—
353C22×C4—2442no—
354C22×C4—2442no—
355C22×C4—2442no—
356C22×C4D4×D6111yescharacteristic
357C22×C4D4×D6111yescharacteristic
358C22×C4C23×D6111yescharacteristic
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—1222no—
363C2×C8—1222no—
364C2×C8—1222no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—42no—
369C4⋊C4—1222no—
370C4⋊C4—1222no—
371C4⋊C4—1222no—
372C4⋊C4—2442no—
373C4⋊C4—2442no—
374C22⋊C4—1224no—
375C22⋊C4—1224no—
376C22⋊C4—2442no—
377C22⋊C4—2442no—
378C22⋊C4—2442no—
379C42—22no—
380C42—612no—
381C42—612no—
382C42—614no—
383C42—1222no—
384C42—1222no—
385C42—2441no—
386C42—2441no—
387C42—2442no—
388C23—1222no—
389C23—2441no—
390C23—2441no—
391C23—2441no—
392C23—2441no—
393C23—2441no—
394C23—2441no—
395C23C2×D4×D6111yescharacteristic
396Q8—24no—
397Q8—42no—
398Q8—84no—
399Q8—4884no—
400D4—1224no—
401D4—2442no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—612no—
416C2×C4—612no—
417C2×C4—612no—
418C2×C4—612no—
419C2×C4—1221no—
420C2×C4—1221no—
421C2×C4—1221no—
422C2×C4—1221no—
423C2×C4—1221no—
424C2×C4—1221no—
425C2×C4—1221no—
426C2×C4—1221no—
427C2×C4—4884no—
428C2×C4—2441no—
429C2×C4—2441no—
430C2×C4—2441no—
431C2×C4—2442no—
432C2×C4—2442no—
433C2×C4—2442no—
434C2×C4#1451979812yes—
435C2×C4#1451980012yes—
436C2×C4C2×Q8×D6111yescharacteristic
437C2×C4#41×D6112yes—
438C2×C4#41×D6111yescharacteristic
439C2×C4#1451982212yes—
440C8—42no—
441C8—2442no—
442C22—4881no—
443C22—4881no—
444C22—4881no—
445C22—4881no—
446C22—2442no—
447C22—2442no—
448C22—2442no—
449C22C22⋊Q8×D6111yescharacteristic
450C22#124×D6111yescharacteristic
451C22#1684032711yescharacteristic
452C22C2×#1451979811yescharacteristic
453C22C2×#1451980011yescharacteristic
454C22#1684055811yescharacteristic
455C22#1684056111yescharacteristic
456C4—22no—
457C4—22no—
458C4—42no—
459C4—81no—
460C4—82no—
461C4—1222no—
462C4—1222no—
463C4—1222no—
464C4—1222no—
465C4—4881no—
466C4—4882no—
467C4#1684046612yes—
468C4#1684047412yes—
469C2—4882no—
470C2#2145098711yescharacteristic
471C2#2145560211yescharacteristic
472C2#2145561011yescharacteristic
473C2#2145614311yescharacteristic
474C2#2145620411yescharacteristic
475C2#2145678711yescharacteristic
476C2#2145678911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2024C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2024D61
hypocenterC61#29142
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2024

Complements

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

NH ≅ G/Ntype
1. #21454196C2semidirect
3. #21446960C2semidirect
7. #21445948C2semidirect
8. #21445832C2semidirect
10. #21445667C2semidirect
11. C61×#2024C2semidirect
49. #14519766Q8semidirect
234. #2024D61semidirect
281. C61#29142semidirect

Maximal subgroups

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

subgroupindex
#214541962
#214483622
#214469602
#214469562
#214469522
#214466272
#214459482
#214458322
#214458282
#214456672
C61×#20242
#2914261