Finite Group Databasebeta

← #30639052

Subgroups — #30639052

Order 16256.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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×#16861129C212yesmaximal, complemented
3C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
4#22549638C212yesmaximal, complemented
5C2×#117×C127C211yesmaximal, characteristic, complemented
6C22×D508C2211yescharacteristic
7C22×C22⋊D127—21no—
8C22×C127⋊C8C2212yes—
9#16861183—22no—
10#16861183C414yescomplemented
11C2×C4⋊Dic127C2211yescharacteristic
12C2×(C2×D127)⋊C4—21no—
13#16861129—24no—
14C22⋊(C127⋊C8)—22no—
15C23×C4×C127C2211yescharacteristic
16#117×C127—24no—
17C2×C4⋊C4×C127C2212yes—
18C2×C22⋊C4×C127—21no—
19C2×C42×C127—21no—
20C23×D127—21no—
21C2×D508—22no—
22C2×D508—21no—
23C2×D508—22no—
24C2×D508C2×C412yes—
25C2×C22⋊D127—24no—
26C2×C22⋊D127—44no—
27C22×Dic127—21no—
28C2×C127⋊C8—22no—
29C2×C127⋊C8—22no—
30C2×C127⋊C8—22no—
31C4⋊Dic127—21no—
32C4⋊Dic127C2×C412yes—
33(C2×D127)⋊C4—24no—
34C24×C127D411yescharacteristic
35C22×C4×C127—22no—
36C22×C4×C127—22no—
37C22×C4×C127—22no—
38C22×C4×C127C2×C412yes—
39C22×C4×C127D411yescharacteristic
40C22×C4×C127D412yes—
41C22×C4×C127C2311yescharacteristic
42C4⋊C4×C127—22no—
43C4⋊C4×C127D414yescomplemented
44C22⋊C4×C127—24no—
45C42×C127—24no—
46C22×D127—22no—
47C22×D127—41no—
48C22×D127—41no—
49C22×D127—42no—
50C22×D127—42no—
51D508—24no—
52D508—42no—
53C22⋊D127—48no—
54C2×Dic127—22no—
55C2×Dic127—42no—
56C127⋊C8—42no—
57C23×C127—21no—
58C23×C127—21no—
59C23×C127—22no—
60C23×C127—22no—
61C23×C127C22⋊C412yes—
62C23×C127C2×D411yescharacteristic
63C2×C4×C127—21no—
64C2×C4×C127—21no—
65C2×C4×C127—24no—
66C2×C4×C127—24no—
67C2×C4×C127—24no—
68C2×C4×C127—42no—
69C2×C4×C127—42no—
70C2×C4×C127C22⋊C412yes—
71C2×C4×C127C22×C411yescharacteristic
72C2×C4×C127C2×D412yes—
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127#4112yes—
75C22×C127—22no—
76C22×C127—22no—
77C22×C127—22no—
78C22×C127—42no—
79C22×C127—42no—
80C22×C127C2×C22⋊C411yescharacteristic
81C22×C127#12011yescharacteristic
82C22×C127#12311yescharacteristic
83C22×C127#13512yes—
84C22×C127AGL(1,ℤ/8)12yes—
85C2×D127—42no—
86C2×D127—42no—
87C2×D127—42no—
88C2×D127—82no—
89C2×D127—82no—
90C4×C127—22no—
91C4×C127—44no—
92C4×C127C4×D412yes—
93C4×C127#12112yes—
94Dic127—42no—
95C2×C127—42no—
96C2×C127#38611yescharacteristic
97C2×C127#41611yescharacteristic
98C2×C127#41811yescharacteristic
99C2×C127#44912yes—
100C2×C127#48212yes—
101D127—82no—
102#1875—1271nomaximal, Sylow
103C127#187511yescharacteristic, complemented, Sylow
104C2×#121—1271no—
105C2×#117D12711yescharacteristic, complemented
106C2×#102—1272no—
107C2×C22⋊C8—1271no—
108#340—1272no—
109C22×D4—1271no—
110C22×D4—2541no—
111C23×C4C2×D12711yescharacteristic
112C22×C8—1272no—
113#121—1274no—
114#121—2542no—
115#117—24no—
116C2×C4⋊C4—1271no—
117C2×C4⋊C4C2×D12712yes—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—2541no—
120C2×C42—21no—
121#102—2544no—
122C22⋊C8—2542no—
123C24—2541no—
124C24C22⋊D12711yescharacteristic
125C2×D4—1272no—
126C2×D4—5084no—
127C2×D4—2544no—
128C2×D4—2542no—
129C2×D4—2541no—
130C2×D4—2542no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—2541no—
135C22×C4C4×D12712yes—
136C22×C4C22⋊D12711yescharacteristic
137C22×C4D50812yes—
138C22×C4C22×D12711yescharacteristic
139C2×C8—2542no—
140C2×C8—2542no—
141C2×C8—2542no—
142C4⋊C4—22no—
143C4⋊C4—1272no—
144C4⋊C4—2541no—
145C4⋊C4C22⋊D12714yescomplemented
146C22⋊C4—24no—
147C22⋊C4—2544no—
148C42—24no—
149C23—21no—
150C23—22no—
151C23—22no—
152C23—21no—
153C23—5081no—
154C23—5082no—
155C23—5082no—
156C23—5081no—
157C23—2542no—
158C23(C2×D127)⋊C412yes—
159C23C2×C22⋊D12711yescharacteristic
160D4—5088no—
161D4—5082no—
162D4—2544no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—5082no—
171C2×C4—2542no—
172C2×C4(C2×D127)⋊C412yes—
173C2×C4C2×C4×D12711yescharacteristic
174C2×C4C2×C22⋊D12712yes—
175C2×C4C2×D50811yescharacteristic
176C2×C4#1452324812yes—
177C8—5082no—
178C22—22no—
179C22—22no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—10162no—
184C22—10162no—
185C22—5082no—
186C22—5082no—
187C22—5082no—
188C22C2×(C2×D127)⋊C411yescharacteristic
189C22#1686117811yescharacteristic
190C22#1686119211yescharacteristic
191C22#1686122912yes—
192C22#1686123412yes—
193C4—22no—
194C4—44no—
195C4—5082no—
196C4C4×C22⋊D12712yes—
197C4#1686118312yes—
198C2—42no—
199C2—10162no—
200C2#2254975511yescharacteristic
201C2#2254987011yescharacteristic
202C2#2254988011yescharacteristic
203C2#2255002112yes—
204C2#2255020012yes—
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#117×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861183C2
cosocleC22×C4×C127C23
hypercenterC2×#117D127
hypocenterC127#1875
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#16861129C2semidirect
3. C2×C22⋊(C127⋊C8)C2semidirect
4. #22549638C2semidirect
5. C2×#117×C127C2semidirect
10. #16861183C4semidirect
43. C4⋊C4×C127D4semidirect
103. C127#1875semidirect
105. C2×#117D127semidirect
145. C4⋊C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611832
C2×#168611292
C2×C22⋊(C127⋊C8)2
#225496382
C2×#117×C1272
#1875127