Finite Group Databasebeta

← #29593806

Subgroups — #29593806

Order 15616.

350 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 71 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
1Q16×D244C211yesmaximal, characteristic
2#21448048C211yesmaximal, characteristic, complemented
3#21445205C211yesmaximal, characteristic
4#21443869C211yesmaximal, characteristic, complemented
5#21443783C211yesmaximal, characteristic, complemented
6#21443773C211yesmaximal, characteristic
7C61×#2225C211yesmaximal, characteristic, complemented
8C2×Q16×D61—21no—
9Q8×D244—21no—
10#16840142C2211yescharacteristic
11#16840004—21no—
12#16839972—21no—
13#16839964—21no—
14#16839919—22no—
15C8×D244C2211yescharacteristic
16#16839565—22no—
17C4×#11758185C2211yescharacteristic
18#16839515C2211yescharacteristic
19#16839508—21no—
20C61×#500C2211yescharacteristic
21C4×Q16×C61C2211yescharacteristic
22C61×#367—21no—
23C61×C4⋊C16C2211yescharacteristic
24C61×#358—21no—
25C2×Q8×D61—22no—
26C2×Dic244—21no—
27C2×Dic244D412yes—
28C2×#11758202—22no—
29Q16×D61—42no—
30Q16×D61—42no—
31C2×#11758201—41no—
32C2×C8×D61—21no—
33#14519777—21no—
34#14519777—41no—
35#14519757—22no—
36#14519757—42no—
37C4×D244—41no—
38C4×D244D411yescharacteristic
39C2×#11758185—21no—
40C4⋊#92418—21no—
41C4⋊#92418D411yescharacteristic
42#14519701—41no—
43#14519700—22no—
44#14519689—21no—
45C2×Q16×C61D411yescharacteristic
46C4⋊Q8×C61—21no—
47C4×Q8×C61—21no—
48C2×C16×C61—21no—
49#107×C61—21no—
50#107×C61—21no—
51#107×C61D411yescharacteristic
52#103×C61—21no—
53C4×C8×C61C2311yescharacteristic
54C2×Q8×C61—21no—
55C2×Q8×C61—41no—
56Q8×D61—44no—
57Q8×D61—82no—
58C2×D244D811yescharacteristic
59C2×C4×D61—21no—
60C2×C4×D61—42no—
61C2×Dic122—22no—
62C2×Dic122—42no—
63Q16×C61—21no—
64Q16×C61SD1612yescomplemented
65C2×C8×C61C2×D411yescharacteristic
66C2×C8×C61C2×D411yescharacteristic
67C4⋊C4×C61—21no—
68C4⋊C4×C61—41no—
69C4⋊C4×C61—41no—
70C4⋊C4×C61—41no—
71C42×C61—41no—
72C42×C61C2×D411yescharacteristic
73#11758202—44no—
74#11758201—82no—
75#11758198—21no—
76#11758198—42no—
77C4⋊Dic61—42no—
78C4⋊Dic61—41no—
79C4⋊Dic61—41no—
80C4⋊Dic61D811yescharacteristic
81C2×#92418—21no—
82C2×#92418—41no—
83Dic244—42no—
84Dic244—41no—
85Dic244—41no—
86C8×D61—41no—
87C8×D61—41no—
88C16×C61—41no—
89#11758185—41no—
90C22×D61—21no—
91Q8×C61—22no—
92Q8×C61—41no—
93Q8×C61—81no—
94Q8×C61—81no—
95C2×C4×C61—41no—
96C2×C4×C61—41no—
97C2×C4×C61—81no—
98C2×C4×C61#12011yescharacteristic
99C2×C4×C61C2×D811yescharacteristic
100C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
101C2×Dic61—21no—
102C2×Dic61—42no—
103D244—21no—
104D244SD3212yes—
105C4×D61—41no—
106C4×D61—41no—
107C4×D61—84no—
108Dic122—42no—
109Dic122—42no—
110Dic122—84no—
111C8×C61—21no—
112C8×C61C2×SD1611yescharacteristic
113C8×C61AGL(1,ℤ/8)11yescharacteristic
114#92418—41no—
115#92418—81no—
116#12370—611nomaximal, Sylow
117C22×C61#44711yescharacteristic
118C2×D61—22no—
119C2×D61—41no—
120C2×D61—41no—
121C4×C61—21no—
122C4×C61—42no—
123C4×C61—81no—
124C4×C61—161no—
125C4×C61#45011yescharacteristic
126C4×C61#45511yescharacteristic
127C4×C61C2×SD3211yescharacteristic
128C4×C61#51011yescharacteristic
129Dic61—41no—
130Dic61—82no—
131D4×Q16—611no—
132#2241—611no—
133#2225D6111yescharacteristic, complemented
134#1673—611no—
135#1362—611no—
136#1332—611no—
137D4⋊C16—611no—
138C2×C61#163911yescharacteristic
139C2×C61#218511yescharacteristic
140C2×C61#219511yescharacteristic
141D61—42no—
142C22×Q16—1221no—
143D4×Q8—1221no—
144#500C2×D6111yescharacteristic
145#494—611no—
146#467—1221no—
147#462—1221no—
148#461—1221no—
149#451—1222no—
150C4×Q16C2×D6111yescharacteristic
151C8×D4—611no—
152#367—21no—
153C4⋊C16C2×D6111yescharacteristic
154#358—21no—
155#358—1222no—
156C4×C16—611no—
157#335—611no—
158#332—1221no—
159C61#1237011yescharacteristic, complemented, Sylow
160C22×Q8—1222no—
161C2×Q16—612no—
162C2×Q16—1222no—
163C2×Q16—1221no—
164C2×Q16—2442no—
165C2×Q16—2442no—
166C2×Q16C22⋊D6111yescharacteristic
167C2×SD16—2441no—
168C22×C8—1221no—
169C4⋊Q8—21no—
170C4⋊Q8—1221no—
171C4⋊Q8—2441no—
172C22⋊Q8—1222no—
173C22⋊Q8—2442no—
174C4×Q8—21no—
175C4×D4—611no—
176C4×D4—2441no—
177C2×C16—21no—
178C2×C16—1221no—
179#107—21no—
180#107—21no—
181#107C22⋊D6111yescharacteristic
182C4⋊C8—611no—
183C4⋊C8—1221no—
184#103—21no—
185#103—1222no—
186#103—2441no—
187C22⋊C8—1221no—
188C4×C8C22×D6111yescharacteristic
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—1222no—
192C2×Q8—4882no—
193C2×Q8—2442no—
194C2×Q8—2444no—
195C2×D4—611no—
196C22×C4—1221no—
197C22×C4—2442no—
198Q16—21no—
199Q16—2442no—
200Q16—2444no—
201Q16—2441no—
202Q16—2441no—
203Q16#1175820112yescomplemented
204SD16—4882no—
205C2×C8—1221no—
206C2×C8—2441no—
207C2×C8—2441no—
208C2×C8—2441no—
209C2×C8D4×D6111yescharacteristic
210C2×C8C2×C22⋊D6111yescharacteristic
211C4⋊C4—21no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—611no—
216C4⋊C4—2442no—
217C4⋊C4—2441no—
218C4⋊C4—2441no—
219C22⋊C4—1221no—
220C22⋊C4—2442no—
221C42—41no—
222C42D4×D6111yescharacteristic
223C16—41no—
224C16—2441no—
225C23—1221no—
226Q8—22no—
227Q8—41no—
228Q8—81no—
229Q8—81no—
230Q8—4884no—
231Q8—2442no—
232Q8—2442no—
233D4—612no—
234D4—1221no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—81no—
238C2×C4—1221no—
239C2×C4—4884no—
240C2×C4—2442no—
241C2×C4—2441no—
242C2×C4—2441no—
243C2×C4#1451974511yescharacteristic
244C2×C4D8×D6111yescharacteristic
245C2×C4#1451980111yescharacteristic
246C8—21no—
247C8—4881no—
248C8—2441no—
249C8C2×#1175820111yescharacteristic
250C8#1451980211yescharacteristic
251C22—1222no—
252C22—2441no—
253C22—2441no—
254C22#1683989411yescharacteristic
255C4—21no—
256C4—42no—
257C4—81no—
258C4—161no—
259C4—4882no—
260C4—2441no—
261C4#1683991211yescharacteristic
262C4#1683993811yescharacteristic
263C4SD32×D6111yescharacteristic
264C4#1684020111yescharacteristic
265C2—2442no—
266C2#2144499711yescharacteristic
267C2#2144773411yescharacteristic
268C2#2144779411yescharacteristic
subgroupG/H
centerC22#16839894
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#2225C2
socleC22×C61#447
prime radicalC2×D244D8
hypercenter#2225D61
hypocenterC61#12370
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2225

Complements

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

NH ≅ G/Ntype
2. #21448048C2semidirect
4. #21443869C2semidirect
5. #21443783C2semidirect
7. C61×#2225C2semidirect
64. Q16×C61SD16semidirect
133. #2225D61semidirect
159. C61#12370semidirect
203. Q16#11758201semidirect

Maximal subgroups

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

subgroupindex
Q16×D2442
#214480482
#214452052
#214438692
#214437832
#214437732
C61×#22252
#1237061