Finite Group Databasebeta

← #30641452

Subgroups — #30641452

Order 16256.

740 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 205 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×#16861205C211yesmaximal, characteristic
2C2×C4×D508C211yesmaximal, characteristic, complemented
3#22549779C211yesmaximal, characteristic, complemented
4#22549777C211yesmaximal, characteristic, complemented
5#22549755C214yesmaximal, complemented
6#22549752C214yesmaximal, complemented
7#22549741C211yesmaximal, characteristic, complemented
8#22549722C211yesmaximal, characteristic, complemented
9C2×C4×D4×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C22×C22⋊D127C2214yes—
12C22×C4×D127C2212yes—
13#16861205C418yescomplemented
14C4×D508—24no—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×C22⋊Dic127C2214yes—
17C2×(C2×D127)⋊C4C2212yes—
18C2×(C2×D127)⋊C4C2214yes—
19C2×C4×Dic127C2211yescharacteristic
20C2×C4×Dic127C2211yescharacteristic
21C127⋊#95C2212yes—
22C127⋊#95C2212yes—
23C127⋊#95C2214yes—
24C22×D4×C127C2211yescharacteristic
25C23×C4×C127C2212yes—
26C4×D4×C127—24no—
27C2×C4⋊C4×C127C2211yescharacteristic
28C2×C22⋊C4×C127C2212yes—
29C2×C42×C127C2211yescharacteristic
30C23×D127C2312yes—
31C2×D508—22no—
32C2×D508—22no—
33C2×D508C2×C414yescomplemented
34C2×C22⋊D127—44no—
35C2×C22⋊D127—44no—
36C2×C22⋊D127C2×C4116yescomplemented
37C22×Dic127C2312yes—
38C22×Dic127C2312yes—
39C22×Dic127C2312yes—
40C2×C4×D127—24no—
41C2×C4×D127—44no—
42C4⋊Dic127—22no—
43C22⋊Dic127—44no—
44(C2×D127)⋊C4—24no—
45(C2×D127)⋊C4—44no—
46C4×Dic127—22no—
47C4×Dic127C2×C414yescomplemented
48C24×C127C2312yes—
49C2×D4×C127—22no—
50C2×D4×C127—22no—
51C2×D4×C127C2×C414yescomplemented
52C22×C4×C127—24no—
53C22×C4×C127—44no—
54C22×C4×C127C2312yes—
55C22×C4×C127C2311yescharacteristic
56C22×C4×C127C2311yescharacteristic
57C22×C4×C127C2311yescharacteristic
58C4⋊C4×C127—22no—
59C22⋊C4×C127—24no—
60C42×C127D414yescomplemented
61C22×D127—22no—
62C22×D127—22no—
63C22×D127—24no—
64C22×D127—24no—
65C22×D127C22×C414yes—
66D508—28no—
67C22⋊D127—416no—
68C2×Dic127—22no—
69C2×Dic127—22no—
70C2×Dic127—22no—
71C2×Dic127—22no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—24no—
75C2×Dic127C22×C414yes—
76C4×D127—48no—
77C23×C127—22no—
78C23×C127—22no—
79C23×C127—24no—
80C23×C127—24no—
81C23×C127C22×C414yes—
82C23×C127C2411yescharacteristic
83D4×C127—28no—
84C2×C4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127—22no—
87C2×C4×C127—24no—
88C2×C4×C127—48no—
89C2×C4×C127C22×C412yes—
90C2×C4×C127C2×D412yes—
91C2×C4×C127C2×D412yes—
92C2×C4×C127C2×D412yes—
93C2×C4×C127#4112yes—
94C22×C127—24no—
95C22×C127—28no—
96C22×C127—42no—
97C22×C127—42no—
98C22×C127—44no—
99C22×C127C23×C411yescharacteristic
100C22×C127C22×D411yescharacteristic
101C22×C127C2×#4111yescharacteristic
102C22×C127H(5,𝔽2)12yes—
103C22×C127H(5,𝔽2)12yes—
104C2×D127—24no—
105C2×D127—28no—
106C2×D127—42no—
107C2×D127—42no—
108C2×D127—44no—
109C4×C127—22no—
110C4×C127—22no—
111C4×C127—42no—
112C4×C127C4×D414yes—
113Dic127—24no—
114Dic127—42no—
115Dic127—42no—
116C2×C127—44no—
117C2×C127C2×C4×D411yescharacteristic
118C2×C127#51812yes—
119C2×C127#53511yescharacteristic
120C2×C127#53511yescharacteristic
121C2×C127#54012yes—
122D127—44no—
123#2328—1271nomaximal, Sylow
124C127#232811yescharacteristic, complemented, Sylow
125C2×#127—1271no—
126C2×C4×D4—1271no—
127C2×C4×D4D12711yescharacteristic, complemented
128#390—1271no—
129#390—1271no—
130#386—1274no—
131#386—1274no—
132#382—1271no—
133#376—1271no—
134C22×D4—1274no—
135C22×D4—1271no—
136C22×D4C2×D12711yescharacteristic
137C23×C4—1272no—
138C23×C4C2×D12712yes—
139#127—1278no—
140C4×D4—24no—
141C4×D4—2544no—
142C2×C4⋊C4—1271no—
143C2×C4⋊C4C2×D12711yescharacteristic
144C2×C22⋊C4—1274no—
145C2×C22⋊C4—1274no—
146C2×C22⋊C4—1272no—
147C2×C22⋊C4C2×D12712yes—
148C2×C42—1271no—
149C2×C42—1271no—
150C2×C42C2×D12711yescharacteristic
151#95—1272no—
152#95—1272no—
153#95—1274no—
154C24—1272no—
155C24C22×D12712yes—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—12716no—
159C2×D4—1274no—
160C2×D4—5084no—
161C2×D4—5084no—
162C2×D4—2542no—
163C2×D4—2542no—
164C2×D4C4×D12714yescomplemented
165C22×C4—24no—
166C22×C4—44no—
167C22×C4—1272no—
168C22×C4—1272no—
169C22×C4—1272no—
170C22×C4—5084no—
171C22×C4—2544no—
172C22×C4C22×D12712yes—
173C22×C4C22×D12711yescharacteristic
174C22×C4C22×D12711yescharacteristic
175C22×C4C22×D12711yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—2542no—
178C22⋊C4—24no—
179C22⋊C4—5084no—
180C22⋊C4—5084no—
181C22⋊C4—2544no—
182C42—1274no—
183C42—2542no—
184C42C22⋊D12714yescomplemented
185C23—22no—
186C23—22no—
187C23—24no—
188C23—24no—
189C23—1274no—
190C23—2542no—
191C23—2542no—
192C23—2544no—
193C23—2544no—
194C23C2×C4×D12714yes—
195C23C23×D12711yescharacteristic
196D4—28no—
197D4—50816no—
198D4—2548no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—48no—
204C2×C4—1274no—
205C2×C4—5088no—
206C2×C4—2542no—
207C2×C4—2542no—
208C2×C4—2542no—
209C2×C4—2542no—
210C2×C4—2544no—
211C2×C4—2542no—
212C2×C4—2542no—
213C2×C4C2×C4×D12712yes—
214C2×C4C2×C22⋊D12712yes—
215C2×C4C2×C22⋊D12712yes—
216C2×C4C2×C22⋊D12712yes—
217C2×C4#1452324812yes—
218C22—24no—
219C22—28no—
220C22—42no—
221C22—42no—
222C22—44no—
223C22—5082no—
224C22—5082no—
225C22—5084no—
226C22—2544no—
227C22—2548no—
228C22C22×C4×D12711yescharacteristic
229C22C22×C22⋊D12711yescharacteristic
230C22C2×#1452324811yescharacteristic
231C22#1686125412yes—
232C22#1686125512yes—
233C4—22no—
234C4—22no—
235C4—42no—
236C4—5082no—
237C4—5082no—
238C4—2544no—
239C4C4×C22⋊D12714yes—
240C2—44no—
241C2—5084no—
242C2C2×C4×C22⋊D12711yescharacteristic
243C2#2255034812yes—
244C2#2255044011yescharacteristic
245C2#2255044311yescharacteristic
246C2#2255047712yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC2×C4×D4×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861205C2
hypercenterC2×C4×D4D127
hypocenterC127#2328
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4×D508C2semidirect
3. #22549779C2semidirect
4. #22549777C2semidirect
5. #22549755C2semidirect
6. #22549752C2semidirect
7. #22549741C2semidirect
8. #22549722C2semidirect
9. C2×C4×D4×C127C2semidirect
13. #16861205C4semidirect
33. C2×D508C2×C4semidirect
36. C2×C22⋊D127C2×C4semidirect
47. C4×Dic127C2×C4semidirect
51. C2×D4×C127C2×C4semidirect
60. C42×C127D4semidirect
124. C127#2328semidirect
127. C2×C4×D4D127semidirect
164. C2×D4C4×D127semidirect
184. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612052
C2×C4×D5082
#225497792
#225497772
#225497552
#225497522
#225497412
#225497222
C2×C4×D4×C1272
#2328127