Finite Group Databasebeta

← #29834985

Subgroups — #29834985

Order 15616.

616 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456390C211yesmaximal, characteristic, complemented
2C2×D8×Dic61C211yesmaximal, characteristic, complemented
3#21448019C211yesmaximal, characteristic, complemented
4#21448005C211yesmaximal, characteristic, complemented
5#21447978C212yesmaximal, complemented
6#21447876C212yesmaximal, complemented
7#21447731C211yesmaximal, characteristic
8SD32×Dic61C212yesmaximal, complemented
9#21447663C212yesmaximal, complemented
10#21447615C211yesmaximal, characteristic, complemented
11C61×#2187C211yesmaximal, characteristic, complemented
12#16840298—21no—
13C2×D4×Dic61—21no—
14C2×#14519718C2212yes—
15#16840163C2211yescharacteristic, complemented
16#16840147C2211yescharacteristic, complemented
17#16840118—21no—
18#16839998C2211yescharacteristic
19#16839993—21no—
20Q16×Dic61C2211yescharacteristic, complemented
21#16839847—21no—
22D8×Dic61—21no—
23D8×Dic61—21no—
24D8×Dic61C2212yes—
25D8×Dic61C2212yes—
26#16839844C2211yescharacteristic, complemented
27C2×#14519711C2211yescharacteristic
28C2×#14519698—21no—
29C2×C8×Dic61C2211yescharacteristic
30#16839590C2212yescomplemented
31#16839587C2212yescomplemented
32#16839580C2212yescomplemented
33#16839566C2212yescomplemented
34#16839564C2212yescomplemented
35#16839563C2212yes—
36C22⋊#11758185C2211yescharacteristic
37C16×Dic61C2212yescomplemented
38C22×D8×C61C2211yescharacteristic
39C2×SD32×C61C2212yes—
40C61×#467C2211yescharacteristic
41C61×#357C2212yes—
42C61×C22⋊C16C2211yescharacteristic
43C23×Dic61—41no—
44C2×#11758201—21no—
45C2×D488C2311yescharacteristic
46C22×#92418D411yescharacteristic
47#14519772—41no—
48#14519768—21no—
49#14519766—21no—
50#14519761—41no—
51#14519751—21no—
52Q8×Dic61—21no—
53D4×Dic61—22no—
54D4×Dic61—22no—
55D4×Dic61—41no—
56D4×Dic61—41no—
57#14519739—21no—
58#14519737—41no—
59#14519735—41no—
60C2×C4⋊Dic61—21no—
61C2×C22⋊Dic61—41no—
62C2×C4×Dic61D411yescharacteristic
63#14519718—24no—
64C2×#11758185C2312yes—
65#14519711—21no—
66#14519711C2311yescharacteristic
67#14519711C2312yes—
68#14519708—21no—
69#14519701—21no—
70#14519699—21no—
71#14519698—22no—
72#14519698—22no—
73#14519697—21no—
74C8×Dic61D411yescharacteristic
75C8×Dic61D411yescharacteristic
76C8×Dic61C2311yescharacteristic
77C8×Dic61C2311yescharacteristic
78C22×D4×C61—21no—
79C2×Q16×C61C2311yescharacteristic
80C2×D8×C61—24no—
81C2×D8×C61—21no—
82C2×D8×C61C2312yes—
83C22×C8×C61C2311yescharacteristic
84C22⋊Q8×C61—21no—
85SD32×C61—24no—
86C2×C16×C61C2312yes—
87#107×C61C2311yescharacteristic
88#103×C61—21no—
89C24×C61—41no—
90C2×Q8×C61—21no—
91C2×D4×C61—24no—
92C2×D4×C61—22no—
93C2×D4×C61—42no—
94C2×D4×C61—41no—
95C22×C4×C61C2×D411yescharacteristic
96C2×C22⋊D61—41no—
97C22×Dic61—41no—
98C22×Dic61—42no—
99C22×Dic61—42no—
100C22×Dic61—84no—
101C22×Dic61D811yescharacteristic
102C22×Dic61D811yescharacteristic
103C2×D244—21no—
104C2×C4×D61—41no—
105Q16×C61—22no—
106D8×C61—24no—
107D8×C61—22no—
108D8×C61#4114yes—
109C2×C8×C61—22no—
110C2×C8×C61C2×D411yescharacteristic
111C2×C8×C61C2411yescharacteristic
112C4⋊C4×C61—21no—
113C4⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C22⋊Dic61—42no—
116C22⋊Dic61—42no—
117#11758201—42no—
118#11758198—41no—
119#11758198—41no—
120#11758198—41no—
121#11758198—41no—
122C4⋊Dic61—22no—
123C4⋊Dic61—21no—
124C4⋊Dic61—41no—
125C4⋊Dic61—41no—
126C4⋊Dic61—41no—
127#11758196—41no—
128C4×Dic61—41no—
129C4×Dic61D811yescharacteristic
130C4×Dic61D811yescharacteristic
131C4×Dic61C2×D411yescharacteristic
132C4×Dic61C2×D411yescharacteristic
133C2×#92418—41no—
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136D488—22no—
137C16×C61—22no—
138#11758185—22no—
139C23×C61—44no—
140C23×C61—42no—
141C23×C61—82no—
142C23×C61—82no—
143C23×C61C2×D811yescharacteristic
144C22×D61—41no—
145Q8×C61—41no—
146Q8×C61—41no—
147D4×C61—24no—
148D4×C61—44no—
149D4×C61—42no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61C2×D811yescharacteristic
155C2×C4×C61C22×D411yescharacteristic
156C22⋊D61—82no—
157C2×Dic61—22no—
158C2×Dic61—22no—
159C2×Dic61—41no—
160C2×Dic61—42no—
161C2×Dic61—84no—
162C2×Dic61—82no—
163C2×Dic61—84no—
164C2×Dic61C2×D811yescharacteristic
165C2×Dic61C2×D811yescharacteristic
166C2×Dic61C2×D811yescharacteristic
167C2×Dic61C2×D811yescharacteristic
168D244—41no—
169D244—41no—
170C4×D61—82no—
171C8×C61—21no—
172C8×C61C2×#4112yes—
173#92418—21no—
174#92418—21no—
175#33665—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—44no—
179C22×C61—88no—
180C22×C61—82no—
181C22×C61—82no—
182C22×C61#50812yes—
183C22×C61C22×D811yescharacteristic
184C2×D61—81no—
185C2×D61—81no—
186C2×D61—81no—
187C4×C61—21no—
188C4×C61—81no—
189C4×C61—81no—
190C4×C61#52511yescharacteristic
191C4×C61#57511yescharacteristic
192Dic61—21no—
193Dic61—21no—
194Dic61—21no—
195Dic61—81no—
196Dic61—82no—
197Dic61#50912yes—
198#3168—611no—
199C2×C4×D8—611no—
200#2236—611no—
201#2233—611no—
202#2228—612no—
203#2210—612no—
204#2187D6111yescharacteristic, complemented
205#2184—611no—
206C4×SD32—612no—
207C4×D16—612no—
208#2162—611no—
209C2×C61—22no—
210C2×C61—84no—
211C2×C61#310011yescharacteristic
212C2×C61C2×#50811yescharacteristic
213C2×C61C2×#50911yescharacteristic
214D61—161no—
215C22×D8C2×D6111yescharacteristic
216#529—1221no—
217C2×C4×D4—1221no—
218C2×SD32C2×D6112yes—
219C2×D16—612no—
220#499—611no—
221#495—611no—
222#488—1221no—
223#467C2×D6111yescharacteristic
224#466—611no—
225#465—1221no—
226C4×Q16—611no—
227C4×SD16—1221no—
228C4×D8—612no—
229C4×D8—612no—
230C4×D8—611no—
231C4×D8—1221no—
232C4×D8—1221no—
233C2×#107—611no—
234C2×#102—1221no—
235C2×C4×C8—611no—
236#367—612no—
237#366—612no—
238C4⋊C16—612no—
239#358—612no—
240#357—612no—
241#357—612no—
242#357C2×D6112yes—
243C22⋊C16—611no—
244C22⋊C16C2×D6111yescharacteristic
245C4×C16—612no—
246C61#3366511yescharacteristic, complemented, Sylow
247C22×D4—21no—
248C23×C4—2441no—
249C2×Q16C22×D6111yescharacteristic
250C2×SD16—1221no—
251C2×D8—24no—
252C2×D8—21no—
253C2×D8—611no—
254C2×D8C22×D6112yes—
255C22×C8—611no—
256C22×C8C22×D6111yescharacteristic
257#126—2441no—
258#125—1221no—
259#124—1221no—
260#123—2441no—
261C22⋊Q8—21no—
262#121—1221no—
263C4×Q8—1221no—
264C4×D4—1221no—
265C4×D4—1222no—
266C4×D4—1222no—
267C4×D4—2441no—
268C4×D4—2441no—
269C4×D4—2441no—
270#117—2441no—
271C2×C4⋊C4—1221no—
272C2×C22⋊C4—2441no—
273C2×C42—611no—
274SD32—24no—
275D16—1224no—
276C2×C16—612no—
277C2×C16C22×D6112yes—
278#107—611no—
279#107—612no—
280#107—1221no—
281#107C22×D6111yescharacteristic
282#106—1221no—
283#103—21no—
284#103—1221no—
285#102—1221no—
286#102—1222no—
287#102—1222no—
288#102—1221no—
289C4×C8—611no—
290C4×C8—611no—
291C4×C8—611no—
292C4×C8—611no—
293C24—41no—
294C2×Q8—21no—
295C2×D4—24no—
296C2×D4—22no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—1221no—
300C2×D4—2441no—
301C22×C4—611no—
302C22×C4—611no—
303C22×C4—4884no—
304C22×C4—2441no—
305C22×C4—2442no—
306C22×C4—2442no—
307C22×C4—2441no—
308C22×C4D4×D6111yescharacteristic
309Q16—22no—
310SD16—2442no—
311D8—24no—
312D8—22no—
313D8—1222no—
314D8#1175822414yes—
315C2×C8—22no—
316C2×C8—611no—
317C2×C8—611no—
318C2×C8—2441no—
319C2×C8D4×D6111yescharacteristic
320C2×C8C23×D6111yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—1222no—
324C4⋊C4—1221no—
325C4⋊C4—2441no—
326C4⋊C4—2441no—
327C4⋊C4—2441no—
328C4⋊C4—2441no—
329C22⋊C4—41no—
330C22⋊C4—2441no—
331C22⋊C4—2441no—
332C22⋊C4—2441no—
333C22⋊C4—2442no—
334C22⋊C4—2442no—
335C22⋊C4—2441no—
336C42—611no—
337C42—611no—
338C42—611no—
339C42—611no—
340C42—2441no—
341C16—22no—
342C16—1222no—
343C23—44no—
344C23—42no—
345C23—82no—
346C23—82no—
347C23—2441no—
348C23D8×D6111yescharacteristic
349Q8—41no—
350Q8—41no—
351D4—24no—
352D4—44no—
353D4—42no—
354D4—4882no—
355D4—2441no—
356D4—2441no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—611no—
362C2×C4—611no—
363C2×C4—611no—
364C2×C4—611no—
365C2×C4—1222no—
366C2×C4—1222no—
367C2×C4—4882no—
368C2×C4—4884no—
369C2×C4—4884no—
370C2×C4—4882no—
371C2×C4—2441no—
372C2×C4—2442no—
373C2×C4D8×D6111yescharacteristic
374C2×C4C2×D4×D6111yescharacteristic
375C8—21no—
376C8—1221no—
377C8—1221no—
378C8C2×#1175822412yes—
379C22—21no—
380C22—21no—
381C22—44no—
382C22—88no—
383C22—82no—
384C22—82no—
385C22—4881no—
386C22—4881no—
387C22—4881no—
388C22#1684019312yes—
389C22C2×D8×D6111yescharacteristic
390C4—21no—
391C4—81no—
392C4—81no—
393C4—612no—
394C4—1221no—
395C4—1221no—
396C4—1221no—
397C4—4881no—
398C4—4882no—
399C4#1684026811yescharacteristic
400C4#1684055811yescharacteristic
401C2—22no—
402C2—84no—
403C2—9761no—
404C2#2145581411yescharacteristic
405C2C2×#1684019311yescharacteristic
406C2#509×D6111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2187C2
socleC22×C61C22×D8
prime radical#21447731C2
hypercenter#2187D61
hypocenterC61#33665
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2187

Complements

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

NH ≅ G/Ntype
1. #21456390C2semidirect
2. C2×D8×Dic61C2semidirect
3. #21448019C2semidirect
4. #21448005C2semidirect
5. #21447978C2semidirect
6. #21447876C2semidirect
8. SD32×Dic61C2semidirect
9. #21447663C2semidirect
10. #21447615C2semidirect
11. C61×#2187C2semidirect
15. #16840163C22semidirect
16. #16840147C22semidirect
20. Q16×Dic61C22semidirect
26. #16839844C22semidirect
30. #16839590C22semidirect
31. #16839587C22semidirect
32. #16839580C22semidirect
33. #16839566C22semidirect
34. #16839564C22semidirect
37. C16×Dic61C22semidirect
204. #2187D61semidirect
246. C61#33665semidirect

Maximal subgroups

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

subgroupindex
#214563902
C2×D8×Dic612
#214480192
#214480052
#214479782
#214478762
#214477312
SD32×Dic612
#214476632
#214476152
C61×#21872
#3366561