Finite Group Databasebeta

← #30647206

Subgroups — #30647206

Order 16256.

756 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 209 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×#16861254C211yesmaximal, characteristic
2#22550642C212yesmaximal, complemented
3#22549858C212yesmaximal, complemented
4#22549857C214yesmaximal, complemented
5C2×#16861126C211yesmaximal, characteristic, complemented
6#100×D127C214yesmaximal, complemented
7C2×#100×C127C211yesmaximal, characteristic, complemented
8#16861254—24no—
9#16861254—24no—
10C2×#14523248C2211yescharacteristic
11C2×#14523248C2211yescharacteristic
12C2×#14523247C412yes—
13C2×#14523247C2212yes—
14C22×C22⋊D127C414yes—
15C2×D4×D127C412yes—
16C2×D4×D127C2212yes—
17#16861217—24no—
18#16861215—24no—
19C2×#14523232C2212yescomplemented
20M16×D127C2218yescomplemented
21#16861127C2214yescomplemented
22#16861126C2214yescomplemented
23#16861125C2214yescomplemented
24C22×D4×C127C2211yescharacteristic
25C2×M16×C127C2212yescomplemented
26#100×C127C2214yescomplemented
27C23×D127C2×C414yes—
28#14523248—22no—
29#14523248—22no—
30#14523248D414yes—
31#14523248D414yes—
32#14523247—28no—
33#14523247—28no—
34C2×Dic254C2×C411yescharacteristic
35C2×Dic254C2311yescharacteristic
36C2×D508C2×C411yescharacteristic
37C2×D508C2311yescharacteristic
38C2×C22⋊D127—28no—
39C2×C22⋊D127—28no—
40C2×C22⋊D127C2×C414yes—
41C2×C22⋊D127C2×C414yes—
42C2×C22⋊D127C2×C412yes—
43C2×C22⋊D127C2×C412yes—
44D4×D127—28no—
45D4×D127—28no—
46C22×Dic127C2×C414yes—
47C2×C4×D127C2×C412yes—
48C2×C4×D127C2312yes—
49#14523232—22no—
50#14523232C2314yes—
51#14523231—28no—
52C2×C127⋊C8—22no—
53C8×D127—28no—
54C24×C127C2×C412yes—
55C2×D4×C127—22no—
56C2×D4×C127—22no—
57C2×D4×C127C2×C412yes—
58C2×D4×C127C2312yes—
59C22×C4×C127C2311yescharacteristic
60M16×C127—22no—
61M16×C127C2314yes—
62C2×C8×C127—22no—
63C22×D127—28no—
64C22×D127—84no—
65C22×D127C22×C412yes—
66C22×D127C22×C412yes—
67Dic254—22no—
68Dic254C22⋊C414yes—
69D508—22no—
70D508C22⋊C414yes—
71C22⋊D127—22no—
72C22⋊D127—22no—
73C22⋊D127—28no—
74C22⋊D127—28no—
75C22⋊D127C22⋊C414yes—
76C22⋊D127C22⋊C414yes—
77C2×Dic127—28no—
78C2×Dic127C22×C412yes—
79C2×Dic127C22×C412yes—
80C4×D127—24no—
81C4×D127C2×D418yes—
82C127⋊C8—24no—
83C23×C127—22no—
84C23×C127—22no—
85C23×C127—82no—
86C23×C127C22×C414yes—
87C23×C127C22×C411yescharacteristic
88D4×C127—24no—
89D4×C127—24no—
90C2×C4×C127C22×C411yescharacteristic
91C2×C4×C127C2×D412yes—
92C2×C4×C127C2×D412yes—
93C2×C4×C127C2411yescharacteristic
94C8×C127—24no—
95C22×C127—24no—
96C22×C127—41no—
97C22×C127—44no—
98C22×C127—82no—
99C22×C127—82no—
100C22×C127C2×C22⋊C411yescharacteristic
101C22×C127C2×C22⋊C411yescharacteristic
102C22×C127C23×C411yescharacteristic
103C2×D127—22no—
104C2×D127—42no—
105C2×D127—42no—
106C2×D127—88no—
107C2×D127C2×C22⋊C414yes—
108C4×C127C2×C22⋊C412yes—
109C4×C127C22×D412yes—
110Dic127—22no—
111Dic127C2×C22⋊C414yes—
112C2×C127—21no—
113C2×C127—21no—
114C2×C127—21no—
115C2×C127—44no—
116C2×C127C22×C22⋊C411yescharacteristic
117D127—24no—
118D127—42no—
119#2888—1271nomaximal, Sylow
120C127#288811yescharacteristic, complemented, Sylow
121C2×H(5,𝔽2)—1271no—
122#568—1272no—
123#413—1272no—
124#413—1274no—
125C2×#100—1271no—
126C2×#100—1274no—
127C2×#100D12711yescharacteristic, complemented
128H(5,𝔽2)—2544no—
129H(5,𝔽2)—2544no—
130C2×#41—1272no—
131C2×#41—1272no—
132C2×#41—1271no—
133C2×#41—1271no—
134C22×D4—1274no—
135C22×D4—1272no—
136C22×D4—1272no—
137C22×D4C2×D12711yescharacteristic
138#131—2544no—
139#131—2544no—
140C2×M16—1272no—
141C2×M16—1278no—
142C2×M16C2×D12712yescomplemented
143#101—1274no—
144#100—1274no—
145#100—1274no—
146#100C2×D12714yescomplemented
147C24—1274no—
148C24C4×D12712yes—
149#41—1274no—
150#41—1274no—
151#41—2548no—
152#41—2548no—
153#41—2542no—
154#41—2542no—
155C2×Q8—1271no—
156C2×Q8—1271no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—1272no—
160C2×D4—1274no—
161C2×D4—1274no—
162C2×D4—1272no—
163C2×D4—1271no—
164C2×D4—1271no—
165C2×D4—2548no—
166C2×D4—2548no—
167C2×D4—2548no—
168C2×D4—2548no—
169C2×D4C4×D12712yes—
170C2×D4C22×D12712yes—
171C22×C4—1272no—
172C22×C4—1274no—
173C22×C4—1272no—
174C22×C4C22×D12711yescharacteristic
175M16—22no—
176M16—1274no—
177M16—2548no—
178M16—2542no—
179M16C22×D12714yes—
180C2×C8—22no—
181C2×C8—2548no—
182C2×C8—2542no—
183C23—22no—
184C23—22no—
185C23—82no—
186C23—1272no—
187C23—1272no—
188C23—10164no—
189C23—2548no—
190C23C2×C4×D12714yes—
191C23C2×C4×D12711yescharacteristic
192Q8—1274no—
193Q8—2542no—
194D4—24no—
195D4—24no—
196D4—1274no—
197D4—1274no—
198D4—1274no—
199D4—2548no—
200D4—2548no—
201D4—2542no—
202D4—2542no—
203D4—2542no—
204C2×C4—1272no—
205C2×C4—1278no—
206C2×C4—1272no—
207C2×C4—2548no—
208C2×C4—2544no—
209C2×C4C2×C4×D12711yescharacteristic
210C2×C4D4×D12712yes—
211C2×C4D4×D12712yes—
212C2×C4C23×D12711yescharacteristic
213C8—24no—
214C8—2544no—
215C22—24no—
216C22—41no—
217C22—44no—
218C22—82no—
219C22—82no—
220C22—1274no—
221C22—10168no—
222C22—5082no—
223C22—5082no—
224C22—2542no—
225C22C22⋊C4×D12711yescharacteristic
226C22C22⋊C4×D12711yescharacteristic
227C22C22×C4×D12711yescharacteristic
228C4—1274no—
229C4—2542no—
230C4C22⋊C4×D12712yes—
231C4C2×D4×D12712yes—
232C2—21no—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—5082no—
237C2—2544no—
238C2C2×C22⋊C4×D12711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D127
commutatorC22×C127C23×C4
FrattiniC2×C4C23×D127
FittingC2×#100×C127C2
socleC2×C127C22×C22⋊C4
prime radicalC2×#16861254C2
cosocleC2×C4×C127C24
hypercenterC2×#100D127
hypocenterC127#2888
LCSG ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #22550642C2semidirect
3. #22549858C2semidirect
4. #22549857C2semidirect
5. C2×#16861126C2semidirect
6. #100×D127C2semidirect
7. C2×#100×C127C2semidirect
19. C2×#14523232C22semidirect
20. M16×D127C22semidirect
21. #16861127C22semidirect
22. #16861126C22semidirect
23. #16861125C22semidirect
25. C2×M16×C127C22semidirect
26. #100×C127C22semidirect
120. C127#2888semidirect
127. C2×#100D127semidirect
142. C2×M16C2×D127semidirect
146. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612542
#225506422
#225498582
#225498572
C2×#168611262
#100×D1272
C2×#100×C1272
#2888127