Finite Group Databasebeta

← #30638639

Subgroups — #30638639

Order 16256.

852 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C23×C4×D127C211yesmaximal, characteristic
2C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
3C2×(C2×D127)⋊C8C214yesmaximal, complemented
4C2×C22⋊C8×C127C211yesmaximal, characteristic, complemented
5C24×D127C411yescharacteristic
6C23×Dic127C411yescharacteristic
7C22×C4×D127—28no—
8C22×C4×D127—22no—
9C22×C4×D127—22no—
10C22×C4×D127C412yes—
11C22×C4×D127C2212yes—
12C22×C127⋊C8C2212yescomplemented
13C22⋊(C127⋊C8)—22no—
14(C2×D127)⋊C8—28no—
15C23×C4×C127C2211yescharacteristic
16C22×C8×C127C2212yescomplemented
17C22⋊C8×C127—22no—
18C23×D127—24no—
19C23×D127—24no—
20C23×D127—22no—
21C23×D127—22no—
22C23×D127C814yescomplemented
23C23×D127C2×C412yes—
24C22×Dic127—24no—
25C22×Dic127—22no—
26C22×Dic127C2×C412yes—
27C2×C4×D127—216no—
28C2×C4×D127—216no—
29C2×C4×D127—24no—
30C2×C4×D127—24no—
31C2×C4×D127—24no—
32C2×C4×D127—24no—
33C2×C4×D127—22no—
34C2×C4×D127—22no—
35C2×C4×D127D418yes—
36C2×C127⋊C8—24no—
37C2×C127⋊C8—22no—
38C24×C127C2×C411yescharacteristic
39C22×C4×C127—24no—
40C22×C4×C127C2×C411yescharacteristic
41C22×C4×C127D412yes—
42C22×C4×C127D412yes—
43C22×C4×C127C2311yescharacteristic
44C2×C8×C127—22no—
45C2×C8×C127—24no—
46C22×D127—28no—
47C22×D127—216no—
48C22×D127—28no—
49C22×D127—28no—
50C22×D127—22no—
51C22×D127—22no—
52C22×D127—22no—
53C22×D127—22no—
54C22×D127—216no—
55C22×D127C22⋊C414yes—
56C22×D127C2×C814yes—
57C22×D127M1614yes—
58C2×Dic127—28no—
59C2×Dic127—22no—
60C2×Dic127—22no—
61C2×Dic127C22⋊C414yes—
62C4×D127—216no—
63C4×D127—28no—
64C4×D127—28no—
65C127⋊C8—42no—
66C23×C127—22no—
67C23×C127—22no—
68C23×C127C22⋊C412yes—
69C23×C127C2×C812yes—
70C23×C127M1612yes—
71C23×C127C22×C411yescharacteristic
72C2×C4×C127—22no—
73C2×C4×C127—24no—
74C2×C4×C127—24no—
75C2×C4×C127C22⋊C412yes—
76C2×C4×C127C2×D414yes—
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127C2×D411yescharacteristic
79C8×C127—42no—
80C22×C127—22no—
81C22×C127—24no—
82C22×C127—22no—
83C22×C127—24no—
84C22×C127C22⋊C814yescomplemented
85C22×C127C2×C22⋊C412yes—
86C22×C127C2×C22⋊C411yescharacteristic
87C22×C127C22×C811yescharacteristic
88C22×C127C2×M1612yes—
89C22×C127C2×M1611yescharacteristic
90C2×D127—232no—
91C2×D127—24no—
92C2×D127—24no—
93C2×D127—24no—
94C2×D127—24no—
95C2×D127—24no—
96C2×D127—24no—
97C2×D127C22⋊C818yescomplemented
98C4×C127—22no—
99C4×C127#12012yes—
100C4×C127#12012yes—
101Dic127—24no—
102C2×C127—24no—
103C2×C127#37911yescharacteristic
104C2×C127C2×C22⋊C812yes—
105C2×C127C2×C22⋊C811yescharacteristic
106C2×C127#40712yes—
107C2×C127#40711yescharacteristic
108D127—28no—
109#1779—1271nomaximal, Sylow
110C127#177911yescharacteristic, complemented, Sylow
111C24×C4—1271no—
112C2×C22⋊C8—1271no—
113C2×C22⋊C8—1274no—
114C2×C22⋊C8D12711yescharacteristic, complemented
115C25—1271no—
116C23×C4—1272no—
117C23×C4—1271no—
118C23×C4—1272no—
119C23×C4—2548no—
120C23×C4—2542no—
121C23×C4—2542no—
122C23×C4C2×D12711yescharacteristic
123C22×C8—1272no—
124C22×C8C2×D12712yescomplemented
125C22⋊C8—22no—
126C22⋊C8—2548no—
127C22⋊C8—2542no—
128C24—1272no—
129C24—1274no—
130C24—2544no—
131C24—2544no—
132C24—2542no—
133C24—2542no—
134C24C4×D12711yescharacteristic
135C22×C4—24no—
136C22×C4—1272no—
137C22×C4—1278no—
138C22×C4—2544no—
139C22×C4—2544no—
140C22×C4—25416no—
141C22×C4—25416no—
142C22×C4—2544no—
143C22×C4—2542no—
144C22×C4—2544no—
145C22×C4—2544no—
146C22×C4—2542no—
147C22×C4—2542no—
148C22×C4C4×D12711yescharacteristic
149C22×C4C22⋊D12712yes—
150C22×C4D50812yes—
151C22×C4C22×D12711yescharacteristic
152C2×C8—24no—
153C2×C8—22no—
154C2×C8—2544no—
155C2×C8—2542no—
156C23—22no—
157C23—22no—
158C23—1274no—
159C23—1274no—
160C23—1274no—
161C23—25416no—
162C23—2548no—
163C23—25416no—
164C23—2548no—
165C23—2542no—
166C23—2542no—
167C23—2548no—
168C23—2542no—
169C23—2542no—
170C23(C2×D127)⋊C412yes—
171C23C8×D12712yes—
172C23#1452323112yes—
173C23C2×C4×D12711yescharacteristic
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—1274no—
178C2×C4—25416no—
179C2×C4—2548no—
180C2×C4—2542no—
181C2×C4—2542no—
182C2×C4—2548no—
183C2×C4—2548no—
184C2×C4(C2×D127)⋊C412yes—
185C2×C4D4×D12714yes—
186C2×C4C2×C22⋊D12711yescharacteristic
187C2×C4C2×D50811yescharacteristic
188C8—42no—
189C8—5082no—
190C22—24no—
191C22—24no—
192C22—22no—
193C22—22no—
194C22—1278no—
195C22—25432no—
196C22—2544no—
197C22—2544no—
198C22—2544no—
199C22—2544no—
200C22—2544no—
201C22—2544no—
202C22(C2×D127)⋊C814yescomplemented
203C22C22⋊C4×D12712yes—
204C22C2×(C2×D127)⋊C411yescharacteristic
205C22C2×C8×D12711yescharacteristic
206C22M16×D12712yes—
207C22C2×#1452323111yescharacteristic
208C4—22no—
209C4—2544no—
210C4#1686117712yes—
211C4#1686117912yes—
212C2—24no—
213C2—2548no—
214C2#2254973111yescharacteristic
215C2C22⋊C8×D12712yes—
216C2C2×(C2×D127)⋊C811yescharacteristic
217C2#2254983812yes—
218C2#2254983911yescharacteristic
subgroupG/H
center, FrattiniC22×C4C22×D127
commutatorC22×C127C22×C8
FittingC2×C22⋊C8×C127C2
socleC23×C127C22×C4
prime radicalC24×D127C4
cosocleC22×C4×C127C23
hypercenterC2×C22⋊C8D127
hypocenterC127#1779
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊(C127⋊C8)C2semidirect
3. C2×(C2×D127)⋊C8C2semidirect
4. C2×C22⋊C8×C127C2semidirect
12. C22×C127⋊C8C22semidirect
16. C22×C8×C127C22semidirect
22. C23×D127C8semidirect
84. C22×C127C22⋊C8semidirect
97. C2×D127C22⋊C8semidirect
110. C127#1779semidirect
114. C2×C22⋊C8D127semidirect
124. C22×C8C2×D127semidirect
202. C22(C2×D127)⋊C8semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1272
C2×C22⋊(C127⋊C8)2
C2×(C2×D127)⋊C82
C2×C22⋊C8×C1272
#1779127