Finite Group Databasebeta

← #30648372

Subgroups — #30648372

Order 16256.

492 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C22×#14523235C211yesmaximal, characteristic, complemented
2C2×#16861217C211yesmaximal, characteristic, complemented
3#22550450C211yesmaximal, characteristic, complemented
4#22550445C211yesmaximal, characteristic
5#22550125C214yesmaximal, complemented
6#22550108C214yesmaximal, complemented
7#22549912C211yesmaximal, characteristic, complemented
8#22549882C211yesmaximal, characteristic, complemented
9#419×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C22×D508C2211yescharacteristic
12C2×#14523235—24no—
13C2×#14523235C2212yes—
14C2×#14523235C2212yes—
15#16861217—24no—
16C2×#14523231C2212yescomplemented
17C22×C127⋊C8C2211yescharacteristic
18C2×C8×D127C2212yescomplemented
19#16861207—21no—
20#16861206—21no—
21#16861198—21no—
22#16861197—21no—
23#16861192—22no—
24#16861185C2214yescomplemented
25#16861184C2214yes—
26#16861179—22no—
27#16861168C2211yescharacteristic
28#16861142C2212yescomplemented
29#16861140C2212yescomplemented
30#16861131C2214yes—
31C22×Q8×C127C2211yescharacteristic
32C2×M16×C127C2211yescharacteristic
33#117×C127C2211yescharacteristic
34#103×C127C2214yescomplemented
35C23×D127—21no—
36#14523248—22no—
37#14523248—22no—
38C2×Dic254D411yescharacteristic
39C2×D508—24no—
40C2×D508—22no—
41C2×D508—21no—
42C2×D508—22no—
43C2×D508D411yescharacteristic
44C2×D508C2312yes—
45C2×C22⋊D127D412yes—
46C2×C4×D127C2312yes—
47#14523235—24no—
48#14523235—24no—
49#14523231—24no—
50C2×C127⋊C8D414yescomplemented
51C2×C127⋊C8C2311yescharacteristic
52C2×C127⋊C8C2311yescharacteristic
53C8×D127—24no—
54C4⋊Dic127C2312yes—
55Dic127⋊C4—24no—
56C22⋊Dic127—21no—
57(C2×D127)⋊C4—24no—
58(C2×D127)⋊C4—24no—
59C4×Dic127—21no—
60C2×Q8×C127—21no—
61C2×Q8×C127—22no—
62C2×Q8×C127—22no—
63C2×Q8×C127C2312yes—
64C22×C4×C127—21no—
65C22×C4×C127C2311yescharacteristic
66M16×C127—22no—
67C2×C8×C127C2312yes—
68C4⋊C4×C127C2312yes—
69C22⋊C4×C127—21no—
70C42×C127—21no—
71C22×D127—22no—
72C22×D127—42no—
73C22×D127—82no—
74C22×D127C2×D412yes—
75Dic254—22no—
76D508—22no—
77D508—22no—
78D508—22no—
79D508—42no—
80D508—84no—
81C22⋊D127—24no—
82C2×Dic127—22no—
83C2×Dic127C2×D412yes—
84C4×D127—22no—
85C4×D127—22no—
86C127⋊C8C2×D412yes—
87C127⋊C8C2×D412yes—
88C23×C127C2×D411yescharacteristic
89Q8×C127—22no—
90Q8×C127—22no—
91Q8×C127—42no—
92C2×C4×C127—22no—
93C2×C4×C127—22no—
94C2×C4×C127—42no—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D412yes—
97C2×C4×C127#4112yes—
98C2×C4×C127C2411yescharacteristic
99C8×C127—22no—
100C22×C127—22no—
101C22×C127#12112yes—
102C22×C127C22×D411yescharacteristic
103C2×D127—22no—
104C2×D127—24no—
105C2×D127—42no—
106C2×D127—44no—
107C2×D127—84no—
108C4×C127—42no—
109C4×C127—42no—
110C4×C127#12112yes—
111C4×C127C22×D411yescharacteristic
112C4×C127C2×#4111yescharacteristic
113Dic127—22no—
114Dic127—42no—
115C2×C127—22no—
116C2×C127C2×#12111yescharacteristic
117C2×C127#57712yes—
118D127—42no—
119D127—82no—
120#3057—1271nomaximal, Sylow
121C127#305711yescharacteristic, complemented, Sylow
122C22×SD16—1271no—
123C2×#131—1271no—
124#536—1271no—
125#535—1271no—
126#468—1274no—
127#465—1274no—
128#427—1271no—
129#419D12711yescharacteristic, complemented
130#418—1271no—
131C2×#41—1271no—
132C22×Q8C2×D12711yescharacteristic
133C22×D4—1271no—
134C2×SD16—1272no—
135C2×SD16—1272no—
136C2×SD16—2544no—
137#131—2544no—
138C2×M16—1272no—
139C2×M16C2×D12711yescharacteristic
140C22×C8—1271no—
141C22×C8—1272no—
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#117C2×D12711yescharacteristic
152#107—1272no—
153#106—1272no—
154#103C2×D12714yescomplemented
155#102—1274no—
156C24—2541no—
157#41—2542no—
158#41—2542no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—1271no—
163C2×Q8C22×D12712yes—
164C2×D4—1272no—
165C2×D4—1271no—
166C2×D4—1272no—
167C2×D4—2544no—
168C2×D4—2541no—
169C2×D4—2542no—
170C2×D4—2542no—
171C22×C4—21no—
172C22×C4—1272no—
173C22×C4C22×D12711yescharacteristic
174SD16—2544no—
175SD16—2544no—
176M16—22no—
177M16—2544no—
178C2×C8—1274no—
179C2×C8—1271no—
180C2×C8—1271no—
181C2×C8—2544no—
182C2×C8C22×D12712yes—
183C4⋊C4—1272no—
184C4⋊C4—2544no—
185C4⋊C4C22×D12712yes—
186C22⋊C4—21no—
187C22⋊C4—2544no—
188C22⋊C4—2544no—
189C22⋊C4—2541no—
190C42—21no—
191C42—2541no—
192C23—1272no—
193C23—10162no—
194C23—5082no—
195C23—2542no—
196C23D4×D12711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—42no—
200Q8—2542no—
201D4—10164no—
202D4—5082no—
203D4—2544no—
204D4—2542no—
205D4—2542no—
206D4—2542no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—1272no—
211C2×C4—2542no—
212C2×C4—2542no—
213C2×C4—2542no—
214C2×C4D4×D12712yes—
215C2×C4D4×D12711yescharacteristic
216C2×C4#1452324812yes—
217C2×C4C23×D12711yescharacteristic
218C8—22no—
219C8—1272no—
220C8—1272no—
221C22—22no—
222C22—10164no—
223C22—5082no—
224C22—5084no—
225C22—2542no—
226C22—2544no—
227C22#1686118212yes—
228C22C2×D4×D12711yescharacteristic
229C4—42no—
230C4—42no—
231C4—5082no—
232C4—2542no—
233C4#1686118212yes—
234C4C2×D4×D12711yescharacteristic
235C4C2×#1452324811yescharacteristic
236C2—22no—
237C2—10162no—
238C2—5082no—
239C2C2×#1686118211yescharacteristic
240C2#2255068512yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#419×C127C2
socleC22×C127C22×D4
prime radical#22550445C2
hypercenter#419D127
hypocenterC127#3057
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#14523235C2semidirect
2. C2×#16861217C2semidirect
3. #22550450C2semidirect
5. #22550125C2semidirect
6. #22550108C2semidirect
7. #22549912C2semidirect
8. #22549882C2semidirect
9. #419×C127C2semidirect
16. C2×#14523231C22semidirect
18. C2×C8×D127C22semidirect
24. #16861185C22semidirect
28. #16861142C22semidirect
29. #16861140C22semidirect
34. #103×C127C22semidirect
50. C2×C127⋊C8D4semidirect
121. C127#3057semidirect
129. #419D127semidirect
154. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232352
C2×#168612172
#225504502
#225504452
#225501252
#225501082
#225499122
#225498822
#419×C1272
#3057127