Finite Group Databasebeta

← #30639566

Subgroups — #30639566

Order 16256.

408 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16861183C211yesmaximal, characteristic
2C2×#16861123C214yesmaximal, complemented
3#22549736C211yesmaximal, characteristic, complemented
4C4×C22⋊C4×C127C211yesmaximal, characteristic, complemented
5C22×D508C411yescharacteristic, complemented
6C22×C22⋊D127C2212yes—
7#16861183—21no—
8#16861183—21no—
9#16861183—21no—
10#16861183—21no—
11C2×C4⋊Dic127—21no—
12C2×C4⋊Dic127C411yescharacteristic, complemented
13C2×C22⋊Dic127C2212yes—
14C2×(C2×D127)⋊C4C412yescomplemented
15#16861123—28no—
16C127⋊#95—21no—
17C23×C4×C127C2211yescharacteristic
18C2×C22⋊C4×C127C2212yes—
19C2×C42×C127—21no—
20#95×C127—21no—
21C23×D127C2×C412yes—
22C2×D508—21no—
23C2×D508—21no—
24C2×D508—42no—
25C2×C22⋊D127—24no—
26C2×C22⋊D127—22no—
27C2×C22⋊D127—22no—
28C2×C22⋊D127—24no—
29C22×Dic127—22no—
30C22×Dic127C2×C412yes—
31C4⋊Dic127—21no—
32C4⋊Dic127—21no—
33C4⋊Dic127—42no—
34C22⋊Dic127—24no—
35(C2×D127)⋊C4—22no—
36(C2×D127)⋊C4—22no—
37C24×C127C2311yescharacteristic
38C22×C4×C127—22no—
39C22×C4×C127—42no—
40C22×C4×C127C2×C411yescharacteristic
41C22×C4×C127C2×C411yescharacteristic
42C22×C4×C127D411yescharacteristic
43C22×C4×C127D411yescharacteristic
44C22×C4×C127D411yescharacteristic
45C22×C4×C127D411yescharacteristic
46C22⋊C4×C127—22no—
47C22⋊C4×C127D414yescomplemented
48C42×C127—42no—
49C22×D127—22no—
50C22×D127—24no—
51C22×D127—42no—
52C22×D127—42no—
53D508—42no—
54D508—42no—
55C22⋊D127—44no—
56C22⋊D127—42no—
57C22⋊D127—42no—
58C2×Dic127—22no—
59C2×Dic127—24no—
60C2×Dic127—42no—
61C2×Dic127—42no—
62C2×Dic127—42no—
63C23×C127—41no—
64C23×C127—41no—
65C23×C127C22×C411yescharacteristic
66C23×C127C2×D411yescharacteristic
67C23×C127C2×D411yescharacteristic
68C23×C127C2×D412yes—
69C23×C127#4112yes—
70C2×C4×C127—22no—
71C2×C4×C127—22no—
72C2×C4×C127—22no—
73C2×C4×C127—24no—
74C2×C4×C127—41no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127—42no—
78C2×C4×C127—42no—
79C2×C4×C127C22⋊C411yescharacteristic
80C2×C4×C127C22⋊C411yescharacteristic
81C2×C4×C127C22⋊C411yescharacteristic
82C2×C4×C127C22⋊C411yescharacteristic
83C22×C127—21no—
84C22×C127—21no—
85C22×C127—21no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—42no—
90C22×C127—42no—
91C22×C127C2×C22⋊C411yescharacteristic
92C22×C127C4×D412yes—
93C22×C127#12112yes—
94C22×C127#12411yescharacteristic
95C22×C127#12711yescharacteristic
96C2×D127—42no—
97C2×D127—42no—
98C2×D127—42no—
99C2×D127—84no—
100C4×C127—21no—
101C4×C127—21no—
102C4×C127—21no—
103C4×C127—44no—
104C4×C127#9912yescomplemented
105Dic127—42no—
106Dic127—82no—
107C2×C127—21no—
108C2×C127—22no—
109C2×C127—22no—
110C2×C127—21no—
111C2×C127#39011yescharacteristic
112C2×C127C2×#9911yescharacteristic
113C2×C127#41011yescharacteristic
114D127—82no—
115#1964—1271nomaximal, Sylow
116C127#196411yescharacteristic, complemented, Sylow
117C2×#121—1271no—
118C2×#99—1274no—
119#380—1271no—
120C4×C22⋊C4D12711yescharacteristic, complemented
121C22×D4—1272no—
122C22×D4—1271no—
123C23×C4C2×D12711yescharacteristic
124#121—2541no—
125#121—2541no—
126#121—2541no—
127#121—2541no—
128C2×C4⋊C4—1271no—
129C2×C4⋊C4—2541no—
130C2×C22⋊C4—1272no—
131C2×C22⋊C4—1272no—
132C2×C22⋊C4C2×D12712yes—
133C2×C42—21no—
134#99—2548no—
135#95—21no—
136#95—2541no—
137C24—1272no—
138C24C22×D12711yescharacteristic
139C2×D4—5082no—
140C2×D4—2542no—
141C2×D4—2544no—
142C2×D4—2542no—
143C2×D4—2544no—
144C2×D4—2541no—
145C2×D4—2541no—
146C22×C4—22no—
147C22×C4—42no—
148C22×C4—1272no—
149C22×C4—2542no—
150C22×C4C4×D12711yescharacteristic
151C22×C4C4×D12711yescharacteristic
152C22×C4C22⋊D12711yescharacteristic
153C22×C4C22⋊D12711yescharacteristic
154C22×C4D50811yescharacteristic
155C22×C4D50811yescharacteristic
156C4⋊C4—5082no—
157C4⋊C4—2541no—
158C4⋊C4—2541no—
159C22⋊C4—22no—
160C22⋊C4—2544no—
161C22⋊C4—2542no—
162C22⋊C4—2542no—
163C22⋊C4D50814yescomplemented
164C42—42no—
165C23—41no—
166C23—41no—
167C23—5082no—
168C23—5082no—
169C23—2544no—
170C23—2542no—
171C23C2×C4×D12711yescharacteristic
172C23C2×C22⋊D12711yescharacteristic
173C23C2×D50811yescharacteristic
174C23C2×D50812yes—
175C23#1452324812yes—
176D4—5082no—
177D4—5082no—
178D4—5082no—
179D4—5084no—
180D4—5082no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—5082no—
191C2×C4—5082no—
192C2×C4—5082no—
193C2×C4—2542no—
194C2×C4—2544no—
195C2×C4(C2×D127)⋊C411yescharacteristic
196C2×C4(C2×D127)⋊C411yescharacteristic
197C2×C4(C2×D127)⋊C411yescharacteristic
198C2×C4(C2×D127)⋊C411yescharacteristic
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—42no—
206C22—42no—
207C22—10164no—
208C22—5082no—
209C22—5082no—
210C22—5082no—
211C22C2×(C2×D127)⋊C411yescharacteristic
212C22C4×D50812yes—
213C22#1686118312yes—
214C22#1686119711yescharacteristic
215C22#1686120611yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—44no—
220C4—10162no—
221C4—5082no—
222C4#1686112312yescomplemented
223C2—22no—
224C2—21no—
225C2—22no—
226C2—21no—
227C2—10162no—
228C2#2254977811yescharacteristic
229C2C2×#1686112311yescharacteristic
230C2#2254984911yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC23×C127C22×C4
FrattiniC24C22×D127
FittingC4×C22⋊C4×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC2×#16861183C2
cosocleC24×C127C23
hypercenterC4×C22⋊C4D127
hypocenterC127#1964
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16861123C2semidirect
3. #22549736C2semidirect
4. C4×C22⋊C4×C127C2semidirect
5. C22×D508C4semidirect
12. C2×C4⋊Dic127C4semidirect
14. C2×(C2×D127)⋊C4C4semidirect
47. C22⋊C4×C127D4semidirect
104. C4×C127#99semidirect
116. C127#1964semidirect
120. C4×C22⋊C4D127semidirect
163. C22⋊C4D508semidirect
222. C4#16861123semidirect

Maximal subgroups

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

subgroupindex
C2×#168611832
C2×#168611232
#225497362
C4×C22⋊C4×C1272
#1964127