Finite Group Databasebeta

← #30601936

Subgroups — #30601936

Order 15616.

396 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461537C211yesmaximal, characteristic, complemented
2#21461380C211yesmaximal, characteristic, complemented
3#21461292C211yesmaximal, characteristic, complemented
4#21461290C211yesmaximal, characteristic
5#21461211C211yesmaximal, characteristic, complemented
6#21461210C211yesmaximal, characteristic
7#21456381C211yesmaximal, characteristic
8#16840784C2211yescharacteristic
9#16840784C2211yescharacteristic
10#16840721C2211yescharacteristic
11#16840698C2211yescharacteristic
12(C2×#6923)⋊C8—21no—
13(C2×#6923)⋊C8—21no—
14#16840625—21no—
15#16840623—21no—
16#16840619—22no—
17#16840618—22no—
18#16840615—22no—
19#16840614—22no—
20#16840317C2211yescharacteristic
21#16840313C2211yescharacteristic
22#16840158C411yescharacteristic, complemented
23#16840136C411yescharacteristic, complemented
24#16840109C411yescharacteristic, complemented
25#16839994—21no—
26#16839994—21no—
27#16839847—21no—
28#16839847—21no—
29C4×C8×D61C2211yescharacteristic
30C61×#487C411yescharacteristic, complemented
31C2×D61⋊C8—21no—
32C2×C4×#6923—21no—
33C4⋊#92420—21no—
34C4⋊#92420—21no—
35#14519834—21no—
36#14519834—21no—
37#14519832—21no—
38#14519832—21no—
39#14519832—21no—
40#14519832D412yes—
41(C2×#6923)⋊C4—21no—
42(C2×#6923)⋊C4D412yes—
43C2×#11758226—21no—
44C2×#11758201—21no—
45C2×#11758201—21no—
46C2×C8×D61D411yescharacteristic
47C2×C8×D61D411yescharacteristic
48#14519777C2×C411yescharacteristic
49#14519774C2×C411yescharacteristic
50#14519768—21no—
51#14519766—21no—
52#14519757—21no—
53#14519753—21no—
54#14519752—21no—
55#14519752—21no—
56Q8×Dic61—21no—
57#14519742—21no—
58#14519739—21no—
59#14519739—21no—
60#14519735—21no—
61C42×D61C2311yescharacteristic
62#14519708—21no—
63#14519708—21no—
64#14519699—21no—
65#14519699—21no—
66C4×#92418C2×C411yescharacteristic
67C8×Dic61D411yescharacteristic
68C8×Dic61D411yescharacteristic
69C4⋊Q8×C61C2×C411yescharacteristic
70C4⋊Q8×C61C2×C411yescharacteristic
71#103×C61—21no—
72#103×C61—21no—
73C4×C8×C61C2×C411yescharacteristic
74C22×#6923—22no—
75C2×Q8×C61—21no—
76C2×Q8×C61—21no—
77#11758226—82no—
78C2×D244—21no—
79C2×D244—21no—
80C2×D244—21no—
81C2×D244—41no—
82C2×C4×D61—41no—
83C2×C4×D61C2×D411yescharacteristic
84C2×C4×D61C2×D411yescharacteristic
85C2×C4×D61#4111yescharacteristic
86C2×Dic122—21no—
87C2×#92420—22no—
88C2×#92420—41no—
89C2×#92420—41no—
90C4×#6923—41no—
91C4×#6923—41no—
92D61⋊C8—41no—
93D61⋊C8—41no—
94C2×C8×C61C22⋊C411yescharacteristic
95C2×C8×C61C22⋊C411yescharacteristic
96C4⋊C4×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—21no—
100C42×C61C22×C411yescharacteristic
101#11758201—42no—
102#11758201—42no—
103#11758198—41no—
104#11758198—41no—
105#11758198—41no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—41no—
109#11758196—41no—
110C4×Dic61—41no—
111C4×Dic61C2×D411yescharacteristic
112C4×Dic61C2×D411yescharacteristic
113C4×Dic61#4111yescharacteristic
114C2×#92418C22⋊C411yescharacteristic
115C2×#92418C22⋊C411yescharacteristic
116C8×D61—41no—
117C8×D61—41no—
118C22×D61—41no—
119C22×D61#12111yescharacteristic
120C2×#6923—42no—
121C2×#6923—42no—
122C2×#6923—42no—
123Q8×C61—41no—
124Q8×C61—41no—
125Q8×C61—81no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61C2×C22⋊C411yescharacteristic
129C2×C4×C61C4×D411yescharacteristic
130C2×C4×C61C4×D411yescharacteristic
131C2×Dic61—41no—
132C2×Dic61#12111yescharacteristic
133C2×Dic61#12411yescharacteristic
134C2×Dic61#12711yescharacteristic
135D244—41no—
136D244—41no—
137D244—81no—
138D244—81no—
139D244—82no—
140C4×D61—22no—
141C4×D61—21no—
142C4×D61—21no—
143C4×D61—41no—
144C4×D61—82no—
145Dic122—81no—
146#92420—42no—
147#92420—81no—
148#92420—81no—
149C8×C61—21no—
150C8×C61—21no—
151#92418—21no—
152#92418—21no—
153#9766—611nomaximal, Sylow
154C22×C61#39011yescharacteristic
155C2×D61—21no—
156C2×D61—21no—
157C2×D61—81no—
158C2×D61—81no—
159C2×D61—81no—
160C2×D61#47112yes—
161#6923—82no—
162C4×C61—21no—
163C4×C61—21no—
164C4×C61—81no—
165C4×C61—81no—
166C4×C61#42111yescharacteristic
167C4×C61#42111yescharacteristic
168Dic61—21no—
169Dic61—21no—
170Dic61—21no—
171Dic61—81no—
172#3167—611no—
173#1746—611no—
174#1568—611no—
175#1531—611no—
176#1530—611no—
177#1506—611no—
178#1505—611no—
179C2×C61#196311yescharacteristic
180C2×C61#197611yescharacteristic
181C2×C61#197711yescharacteristic
182D61—22no—
183D61—161no—
184#533—611no—
185#532—611no—
186#498—611no—
187#492—611no—
188#487#692311yescharacteristic, complemented
189#486—611no—
190#465—1221no—
191#465—1221no—
192C4×SD16—1221no—
193C4×SD16—1221no—
194#432—611no—
195#432—611no—
196C2×#97—611no—
197C2×C4×C8—611no—
198#376—611no—
199#336—1221no—
200#336—1221no—
201#335—1221no—
202#334—1221no—
203#330—1222no—
204#329—1222no—
205#328—1222no—
206#327—1222no—
207C61#976611yescharacteristic, complemented, Sylow
208C2×#41—1221no—
209C2×SD16—1221no—
210C2×SD16—1221no—
211C22×C8—611no—
212C22×C8—611no—
213C22×C8—1221no—
214C4⋊Q8—611no—
215C4⋊Q8C2×#692311yescharacteristic
216C4⋊Q8C2×#692311yescharacteristic
217#127—611no—
218#125—1221no—
219#124—1221no—
220C22⋊Q8—1221no—
221C22⋊Q8—1221no—
222#121—1221no—
223#121—1221no—
224C4×Q8—1221no—
225C4×Q8—1221no—
226C4×D4—1221no—
227C4×D4—1221no—
228#117—1221no—
229C2×C42—611no—
230C2×C42—1221no—
231#106—1221no—
232#106—1221no—
233C4⋊C8—1221no—
234C4⋊C8—1221no—
235#103—21no—
236#103—21no—
237#102—1221no—
238#102—1221no—
239C22⋊C8—1221no—
240C22⋊C8—1221no—
241#97—612no—
242#97—1221no—
243#97—1221no—
244#97—1221no—
245C4×C8—611no—
246C4×C8—611no—
247C4×C8—611no—
248C4×C8C2×#692311yescharacteristic
249#95—612no—
250#95—1221no—
251#41—4882no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—1221no—
255C2×D4—1221no—
256C2×D4—1221no—
257C2×D4—1221no—
258C2×D4—2441no—
259C22×C4—611no—
260C22×C4—611no—
261C22×C4—611no—
262C22×C4—1222no—
263C22×C4—2441no—
264SD16—2442no—
265SD16—2442no—
266C2×C8—611no—
267C2×C8—611no—
268C2×C8—1222no—
269C2×C8—2441no—
270C2×C8—2441no—
271C2×C8—2441no—
272C2×C8—2441no—
273C2×C8—2441no—
274C2×C8—2441no—
275C2×C8C22⋊#692311yescharacteristic
276C2×C8C22⋊#692311yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—1221no—
282C4⋊C4—1221no—
283C4⋊C4—2441no—
284C4⋊C4—2441no—
285C22⋊C4—2441no—
286C22⋊C4—2441no—
287C22⋊C4—2441no—
288C42—611no—
289C42—611no—
290C42—611no—
291C42—2441no—
292C42—2441no—
293C42—2441no—
294C42C22×#692311yescharacteristic
295C23—611no—
296C23—2441no—
297Q8—41no—
298Q8—41no—
299Q8—81no—
300Q8—4881no—
301D4—4881no—
302D4—4881no—
303D4—4882no—
304D4—2441no—
305D4—2441no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—611no—
309C2×C4—611no—
310C2×C4—611no—
311C2×C4—1222no—
312C2×C4—1221no—
313C2×C4—1221no—
314C2×C4—4882no—
315C2×C4—2441no—
316C2×C4—2442no—
317C2×C4—2442no—
318C2×C4—2442no—
319C2×C4—2441no—
320C2×C4C2×C22⋊#692311yescharacteristic
321C2×C4D4×#692311yescharacteristic
322C2×C4D4×#692311yescharacteristic
323C8—21no—
324C8—21no—
325C8—1221no—
326C8—1221no—
327C8—4881no—
328C8—4881no—
329C8—2442no—
330C22—612no—
331C22—1221no—
332C22—1221no—
333C22—4881no—
334C22—4881no—
335C22—4881no—
336C22#1684071411yescharacteristic
337C4—21no—
338C4—21no—
339C4—81no—
340C4—81no—
341C4—1221no—
342C4—1221no—
343C4—1221no—
344C4—4882no—
345C4—4881no—
346C4#1684075911yescharacteristic
347C4#1684075911yescharacteristic
348C2—1222no—
349C2—9761no—
350C2#2146195611yescharacteristic
351C2#2146197811yescharacteristic
352C2#2146197911yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#487C4
socleC22×C61#390
prime radical#16840313C22
cosocleC42×D61C23
hypercenter#487#6923
hypocenterC61#9766
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #21461537C2semidirect
2. #21461380C2semidirect
3. #21461292C2semidirect
5. #21461211C2semidirect
22. #16840158C4semidirect
23. #16840136C4semidirect
24. #16840109C4semidirect
30. C61×#487C4semidirect
188. #487#6923semidirect
207. C61#9766semidirect

Maximal subgroups

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

subgroupindex
#214615372
#214613802
#214612922
#214612902
#214612112
#214612102
#214563812
#976661