Finite Group Databasebeta

← #30637171

Subgroups — #30637171

Order 16256.

316 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C4×D4×D127C211yesmaximal, characteristic
2#22549932C211yesmaximal, characteristic, complemented
3C4×#14523231C211yesmaximal, characteristic, complemented
4#22549605C211yesmaximal, characteristic, complemented
5D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7D4⋊C8×C127C211yesmaximal, characteristic, complemented
8C2×D4×D127C411yescharacteristic
9C22×C4×D127—21no—
10C2×#14523231—21no—
11C4×D508C2211yescharacteristic
12D4×Dic127C411yescharacteristic
13#16861171C411yescharacteristic
14C4×C22⋊D127—21no—
15(C22⋊D127)⋊C4—21no—
16C4⋊C4×D127C411yescharacteristic
17C22⋊C4×D127—21no—
18C42×D127C2211yescharacteristic
19C4⋊(C127⋊C8)C2211yescharacteristic, complemented
20(C2×D127)⋊C8—21no—
21#16861119—21no—
22#16861119—21no—
23C4×C127⋊C8C2211yescharacteristic, complemented
24C4×D4×C127C2211yescharacteristic
25C4⋊C8×C127C2211yescharacteristic, complemented
26C4×C8×C127C2211yescharacteristic, complemented
27C23×D127—21no—
28C2×D508C2×C411yescharacteristic
29C2×C22⋊D127—21no—
30D4×D127—22no—
31D4×D127—22no—
32C22×Dic127—21no—
33C2×C4×D127—21no—
34C2×C4×D127—21no—
35C2×C4×D127—21no—
36C2×C4×D127—42no—
37C2×C4×D127—42no—
38C2×C4×D127C2×C411yescharacteristic
39C2×C4×D127D411yescharacteristic
40C2×C4×D127D411yescharacteristic
41#14523231—24no—
42C2×C127⋊C8—21no—
43C2×C127⋊C8—21no—
44C4⋊Dic127C2×C411yescharacteristic
45Dic127⋊C4—21no—
46C22⋊Dic127—21no—
47(C2×D127)⋊C4—21no—
48(C2×D127)⋊C4—21no—
49C4×Dic127C2×C411yescharacteristic
50C4×Dic127D411yescharacteristic
51C4×Dic127D411yescharacteristic
52C2×D4×C127C2×C411yescharacteristic
53C22×C4×C127—21no—
54C2×C8×C127—21no—
55C2×C8×C127—21no—
56C4⋊C4×C127C2×C411yescharacteristic
57C22⋊C4×C127—21no—
58C42×C127C2311yescharacteristic
59C22×D127—24no—
60C22×D127—21no—
61C22×D127—42no—
62C22×D127—42no—
63C22×D127C22⋊C411yescharacteristic
64D508—21no—
65D508M1612yescomplemented
66C22⋊D127—24no—
67C2×Dic127—21no—
68C2×Dic127—42no—
69C2×Dic127C22⋊C411yescharacteristic
70C2×Dic127C22⋊C411yescharacteristic
71C2×Dic127C22⋊C411yescharacteristic
72C4×D127—22no—
73C4×D127—21no—
74C4×D127—21no—
75C4×D127—21no—
76C4×D127—21no—
77C4×D127—42no—
78C4×D127—42no—
79C4×D127—41no—
80C4×D127—41no—
81C127⋊C8—22no—
82C127⋊C8—41no—
83C23×C127—21no—
84D4×C127—21no—
85D4×C127M1612yescomplemented
86C2×C4×C127—21no—
87C2×C4×C127—41no—
88C2×C4×C127—41no—
89C2×C4×C127C22×C411yescharacteristic
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C8×C127—22no—
93C8×C127—41no—
94C22×C127—22no—
95C22×C127—41no—
96C22×C127—41no—
97C22×C127C2×C22⋊C411yescharacteristic
98C2×D127—21no—
99C2×D127—21no—
100C2×D127—22no—
101C2×D127—48no—
102C2×D127—41no—
103C2×D127—41no—
104C2×D127#10412yes—
105C4×C127—21no—
106C4×C127—41no—
107C4×C127C2×M1611yescharacteristic
108C4×C127C2×M1611yescharacteristic
109C4×C127AGL(1,ℤ/8)11yescharacteristic
110C4×C127AGL(1,ℤ/8)11yescharacteristic
111Dic127—21no—
112Dic127—21no—
113Dic127—21no—
114Dic127—41no—
115C2×C127—42no—
116C2×C127#40711yescharacteristic
117C2×C127#41811yescharacteristic
118C2×C127C2×#10411yescharacteristic
119D127—22no—
120D127—42no—
121#1490—1271nomaximal, Sylow
122C127#149011yescharacteristic, complemented, Sylow
123C2×C4×D4—1271no—
124#432—1271no—
125C4×M16—1271no—
126D4⋊C8—1271no—
127D4⋊C8—1271no—
128D4⋊C8—1271no—
129D4⋊C8D12711yescharacteristic, complemented
130C22×D4—1271no—
131C23×C4—2541no—
132C2×M16—2541no—
133C4×D4—1271no—
134C4×D4—1271no—
135C4×D4—1271no—
136C4×D4—2541no—
137C4×D4—2541no—
138C4×D4C2×D12711yescharacteristic
139C2×C4⋊C4—1271no—
140C2×C22⋊C4—2541no—
141C2×C42—1271no—
142C4⋊C8—1271no—
143C4⋊C8C2×D12711yescharacteristic, complemented
144C22⋊C8—2541no—
145#97—2541no—
146#97—2541no—
147C4×C8—1271no—
148C4×C8C2×D12711yescharacteristic, complemented
149C24—2541no—
150C2×D4—1271no—
151C2×D4—2541no—
152C2×D4—2542no—
153C2×D4—2542no—
154C2×D4C4×D12711yescharacteristic
155C22×C4—21no—
156C22×C4—1271no—
157C22×C4—1271no—
158C22×C4—1271no—
159C22×C4—5082no—
160C22×C4—5082no—
161C22×C4—2541no—
162C22×C4—2541no—
163C22×C4—2541no—
164C22×C4—2541no—
165M16—2544no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—2541no—
169C2×C8—2541no—
170C4⋊C4—1271no—
171C4⋊C4—2541no—
172C4⋊C4C4×D12711yescharacteristic
173C22⋊C4—21no—
174C22⋊C4—2541no—
175C22⋊C4—2541no—
176C22⋊C4—2541no—
177C42—1271no—
178C42—1271no—
179C42—1271no—
180C42C22×D12711yescharacteristic
181C23—21no—
182C23—1271no—
183C23—5082no—
184C23—5082no—
185C23—2544no—
186C23—2541no—
187D4—21no—
188D4—1272no—
189D4—2544no—
190D4—2541no—
191D4#1452323112yescomplemented
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—1271no—
196C2×C4—1271no—
197C2×C4—1271no—
198C2×C4—5082no—
199C2×C4—5082no—
200C2×C4—5082no—
201C2×C4—5081no—
202C2×C4—5081no—
203C2×C4—2542no—
204C2×C4—2541no—
205C2×C4—2541no—
206C2×C4—2541no—
207C2×C4—2541no—
208C2×C4—2541no—
209C2×C4C2×C4×D12711yescharacteristic
210C2×C4D4×D12711yescharacteristic
211C2×C4D4×D12711yescharacteristic
212C8—22no—
213C8—41no—
214C8—5081no—
215C8—2542no—
216C22—22no—
217C22—41no—
218C22—41no—
219C22—1272no—
220C22—5088no—
221C22—5081no—
222C22—5081no—
223C22—2542no—
224C22—2541no—
225C22—2541no—
226C22C22⋊C4×D12711yescharacteristic
227C4—21no—
228C4—41no—
229C4—5081no—
230C4—2541no—
231C4—2541no—
232C4—2541no—
233C4M16×D12711yescharacteristic
234C4C2×#1452323111yescharacteristic
235C4#1686123311yescharacteristic
236C4#1686123611yescharacteristic
237C2—42no—
238C2—5082no—
239C2—2542no—
240C2#2254983811yescharacteristic
241C2#2254987911yescharacteristic
242C2#104×D12711yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
FittingD4⋊C8×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC2×D4×D127C4
cosocleC42×C127C23
hypercenterD4⋊C8D127
hypocenterC127#1490
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #22549932C2semidirect
3. C4×#14523231C2semidirect
4. #22549605C2semidirect
5. D4⋊(C127⋊C8)C2semidirect
6. #22549603C2semidirect
7. D4⋊C8×C127C2semidirect
19. C4⋊(C127⋊C8)C22semidirect
23. C4×C127⋊C8C22semidirect
25. C4⋊C8×C127C22semidirect
26. C4×C8×C127C22semidirect
65. D508M16semidirect
85. D4×C127M16semidirect
122. C127#1490semidirect
129. D4⋊C8D127semidirect
143. C4⋊C8C2×D127semidirect
148. C4×C8C2×D127semidirect
191. D4#14523231semidirect

Maximal subgroups

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

subgroupindex
C4×D4×D1272
#225499322
C4×#145232312
#225496052
D4⋊(C127⋊C8)2
#225496032
D4⋊C8×C1272
#1490127