Finite Group Databasebeta

← #30648365

Subgroups — #30648365

Order 16256.

492 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#16861215C211yesmaximal, characteristic, complemented
2#22550450C211yesmaximal, characteristic, complemented
3#22550443C211yesmaximal, characteristic
4#22550119C214yesmaximal, complemented
5#22550102C214yesmaximal, complemented
6#22549914C211yesmaximal, characteristic, complemented
7#22549885C211yesmaximal, characteristic, complemented
8#22549881C211yesmaximal, characteristic, complemented
9C22×SD16×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11#16861215—24no—
12C2×#14523232C2211yescharacteristic
13C2×#14523231C2212yescomplemented
14C2×C8×D127C2212yescomplemented
15#16861207—21no—
16#16861205—21no—
17#16861198—21no—
18#16861196—21no—
19#16861192—22no—
20#16861185C2214yescomplemented
21#16861180C2214yes—
22#16861178—22no—
23#16861168C2211yescharacteristic
24#16861168C2211yescharacteristic
25#16861143C2212yescomplemented
26#16861141C2212yescomplemented
27#16861133C2214yescomplemented
28#16861130C2214yes—
29C22×Q8×C127C2211yescharacteristic
30C22×D4×C127C2211yescharacteristic
31C2×SD16×C127—24no—
32C2×SD16×C127C2212yes—
33C2×SD16×C127C2212yes—
34C22×C8×C127C2211yescharacteristic
35#14523248—22no—
36#14523248—22no—
37C2×Dic254D411yescharacteristic
38C2×D508D411yescharacteristic
39C2×C22⋊D127—24no—
40C2×C22⋊D127D412yes—
41C2×C4×D127C2312yes—
42#14523232—22no—
43#14523231—24no—
44C2×C127⋊C8C2312yes—
45C8×D127—24no—
46C4⋊Dic127C2312yes—
47C4⋊Dic127C2312yes—
48Dic127⋊C4—24no—
49C22⋊Dic127—24no—
50C22⋊Dic127—21no—
51C22⋊Dic127—21no—
52(C2×D127)⋊C4—24no—
53C4×Dic127—21no—
54C4×Dic127—21no—
55C24×C127—21no—
56C2×Q8×C127—21no—
57C2×Q8×C127—22no—
58C2×Q8×C127—22no—
59C2×Q8×C127C2312yes—
60C2×D4×C127—22no—
61C2×D4×C127—21no—
62C2×D4×C127—22no—
63C2×D4×C127C2312yes—
64C22×C4×C127—21no—
65C22×C4×C127C2311yescharacteristic
66SD16×C127—24no—
67SD16×C127—24no—
68C2×C8×C127D414yescomplemented
69C2×C8×C127C2311yescharacteristic
70C2×C8×C127C2311yescharacteristic
71C22×D127C2×D412yes—
72Dic254—22no—
73D508—22no—
74C22⋊D127—24no—
75C22⋊D127—84no—
76C2×Dic127—22no—
77C2×Dic127—22no—
78C2×Dic127C2×D412yes—
79C4×D127—22no—
80C4×D127—22no—
81C127⋊C8—22no—
82C23×C127—22no—
83C23×C127—42no—
84C23×C127—82no—
85C23×C127C2×D411yescharacteristic
86Q8×C127—22no—
87Q8×C127—22no—
88Q8×C127—42no—
89D4×C127—22no—
90D4×C127—22no—
91D4×C127—42no—
92C2×C4×C127—22no—
93C2×C4×C127—42no—
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D412yes—
96C2×C4×C127#4112yes—
97C2×C4×C127C2411yescharacteristic
98C8×C127C2×D412yes—
99C8×C127C2×D412yes—
100C22×C127—22no—
101C22×C127—42no—
102C22×C127—44no—
103C22×C127—84no—
104C22×C127#12112yes—
105C22×C127C22×D411yescharacteristic
106C2×D127—22no—
107C2×D127—24no—
108C4×C127—42no—
109C4×C127#12112yes—
110C4×C127C22×D411yescharacteristic
111C4×C127C2×#4111yescharacteristic
112Dic127—22no—
113Dic127—42no—
114Dic127—42no—
115C2×C127—22no—
116C2×C127—82no—
117C2×C127C2×#12111yescharacteristic
118C2×C127#57712yes—
119D127—42no—
120#3057—1271nomaximal, Sylow
121C127#305711yescharacteristic, complemented, Sylow
122C22×SD16D12711yescharacteristic, complemented
123C2×#131—1271no—
124#536—1271no—
125#535—1271no—
126#468—1274no—
127#465—1274no—
128#427—1271no—
129#419—1271no—
130#418—1271no—
131C2×#41—1271no—
132C22×Q8C2×D12711yescharacteristic
133C22×D4C2×D12711yescharacteristic
134C2×SD16—24no—
135C2×SD16C2×D12712yes—
136C2×SD16C2×D12712yes—
137#131—2544no—
138C2×M16—1271no—
139C2×M16—1272no—
140C22×C8—1272no—
141C22×C8C2×D12711yescharacteristic
142C4⋊Q8—2541no—
143#127—2541no—
144#124—2541no—
145#124—2541no—
146#123—2542no—
147C22⋊Q8—1274no—
148#121—1274no—
149#120—2542no—
150#117—1271no—
151#117—1271no—
152#107—1272no—
153#106—1272no—
154#103—1274no—
155#102—1274no—
156C24—21no—
157#41—2542no—
158#41—2542no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—1271no—
163C2×Q8C22×D12712yes—
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—1272no—
168C2×D4—1271no—
169C2×D4—2544no—
170C2×D4C22×D12712yes—
171C22×C4—21no—
172C22×C4—1272no—
173C22×C4C22×D12711yescharacteristic
174SD16—24no—
175SD16—24no—
176M16—2542no—
177M16—2544no—
178C2×C8—1272no—
179C2×C8—2544no—
180C2×C8C22⋊D12714yescomplemented
181C2×C8C22×D12711yescharacteristic
182C2×C8C22×D12711yescharacteristic
183C4⋊C4—1272no—
184C4⋊C4—1272no—
185C4⋊C4—2544no—
186C22⋊C4—2544no—
187C22⋊C4—2544no—
188C22⋊C4—2541no—
189C22⋊C4—2541no—
190C42—2541no—
191C42—2541no—
192C23—22no—
193C23—42no—
194C23—82no—
195C23—1272no—
196C23D4×D12711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—42no—
200Q8—2542no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—10164no—
205D4—2544no—
206D4—2542no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—1272no—
210C2×C4—2542no—
211C2×C4—2542no—
212C2×C4—2542no—
213C2×C4—2542no—
214C2×C4D4×D12711yescharacteristic
215C2×C4C2×C22⋊D12712yes—
216C2×C4#1452324712yes—
217C2×C4C23×D12711yescharacteristic
218C8—2542no—
219C8C2×C22⋊D12712yes—
220C8C2×C22⋊D12712yes—
221C22—22no—
222C22—42no—
223C22—44no—
224C22—84no—
225C22—2542no—
226C22—2544no—
227C22#1686118012yes—
228C22C2×D4×D12711yescharacteristic
229C4—42no—
230C4—5082no—
231C4—5082no—
232C4—2542no—
233C4#1686118012yes—
234C4C22×C22⋊D12711yescharacteristic
235C4C2×#1452324711yescharacteristic
236C2—22no—
237C2—82no—
238C2—5082no—
239C2C2×#1686118011yescharacteristic
240C2#2255068212yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×SD16×C127C2
socleC22×C127C22×D4
prime radical#22550443C2
hypercenterC22×SD16D127
hypocenterC127#3057
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16861215C2semidirect
2. #22550450C2semidirect
4. #22550119C2semidirect
5. #22550102C2semidirect
6. #22549914C2semidirect
7. #22549885C2semidirect
8. #22549881C2semidirect
9. C22×SD16×C127C2semidirect
13. C2×#14523231C22semidirect
14. C2×C8×D127C22semidirect
20. #16861185C22semidirect
25. #16861143C22semidirect
26. #16861141C22semidirect
27. #16861133C22semidirect
68. C2×C8×C127D4semidirect
121. C127#3057semidirect
122. C22×SD16D127semidirect
180. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612152
#225504502
#225504432
#225501192
#225501022
#225499142
#225498852
#225498812
C22×SD16×C1272
#3057127