Finite Group Databasebeta

← #30637253

Subgroups — #30637253

Order 16256.

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

normal subgroups only

This group has 242 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
1#22550410C211yesmaximal, characteristic
2#22549933C211yesmaximal, characteristic, complemented
3#22549612C212yesmaximal, complemented
4#22549609C212yesmaximal
5C4×C4⋊C4×C127C211yesmaximal, characteristic, complemented
6#16861203—21no—
7#16861201C411yescharacteristic, complemented
8#16861197C411yescharacteristic, complemented
9#16861192—21no—
10#16861188C2211yescharacteristic
11#16861183C2211yescharacteristic
12C4×Dic254C411yescharacteristic, complemented
13C4×D508C411yescharacteristic, complemented
14C4×C22⋊D127—21no—
15#16861166—21no—
16C4⋊(C127⋊C8)—21no—
17C22⋊(C127⋊C8)C2212yes—
18#16861120—21no—
19C2×C4⋊C4×C127—21no—
20C2×C42×C127—21no—
21C2×C42×C127C2211yescharacteristic
22#95×C127C2212yes—
23C2×Dic254C2×C411yescharacteristic
24C2×D508C2×C411yescharacteristic
25C2×C22⋊D127—21no—
26C2×C4×D127—21no—
27C2×C127⋊C8—22no—
28C4⋊Dic127C2×C411yescharacteristic
29C4⋊Dic127C2×C411yescharacteristic
30Dic127⋊C4—21no—
31Dic127⋊C4—21no—
32Dic127⋊C4—21no—
33Dic127⋊C4—21no—
34C22⋊Dic127—21no—
35(C2×D127)⋊C4—21no—
36(C2×D127)⋊C4—21no—
37(C2×D127)⋊C4—21no—
38(C2×D127)⋊C4—21no—
39C4×Dic127—21no—
40C22×C4×C127—22no—
41C22×C4×C127D411yescharacteristic
42C22×C4×C127D411yescharacteristic
43C22×C4×C127C2311yescharacteristic
44C4⋊C4×C127—24no—
45C42×C127—42no—
46C42×C127C2×C411yescharacteristic
47C42×C127C2×C411yescharacteristic
48C42×C127D411yescharacteristic
49C42×C127D411yescharacteristic
50C22×D127—21no—
51Dic254—41no—
52D508—41no—
53C22⋊D127—41no—
54C22⋊D127—41no—
55C2×Dic127—21no—
56C2×Dic127—21no—
57C2×Dic127—21no—
58C4×D127—42no—
59C127⋊C8—42no—
60C23×C127C2×D411yescharacteristic
61C2×C4×C127—21no—
62C2×C4×C127—21no—
63C2×C4×C127—22no—
64C2×C4×C127—24no—
65C2×C4×C127—41no—
66C2×C4×C127—42no—
67C2×C4×C127—42no—
68C2×C4×C127C22⋊C411yescharacteristic
69C2×C4×C127C22⋊C411yescharacteristic
70C2×C4×C127C22⋊C411yescharacteristic
71C2×C4×C127C22⋊C411yescharacteristic
72C2×C4×C127C22×C411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C22×C127—21no—
75C22×C127—21no—
76C22×C127C2×C22⋊C411yescharacteristic
77C22×C127AGL(1,ℤ/8)11yescharacteristic
78C22×C127#13711yescharacteristic
79C2×D127—41no—
80C2×D127—41no—
81C2×D127—41no—
82C4×C127—21no—
83C4×C127—21no—
84C4×C127—21no—
85C4×C127—21no—
86C4×C127—21no—
87C4×C127—44no—
88C4×C127#10412yes—
89Dic127—41no—
90Dic127—41no—
91Dic127—41no—
92C2×C127—21no—
93C2×C127—21no—
94C2×C127#41011yescharacteristic
95C2×C127#41711yescharacteristic
96C2×C127C2×#10411yescharacteristic
97D127—81no—
98#1509—1271nomaximal, Sylow
99C127#150911yescharacteristic, complemented, Sylow
100#529—1271no—
101#432—1271no—
102C4×C4⋊C4D12711yescharacteristic, complemented
103#328—1272no—
104#327—1272no—
105#126—2541no—
106#125—1271no—
107#124—1271no—
108#123—2541no—
109C22⋊Q8—1271no—
110#121—1271no—
111C4×Q8—1271no—
112C4×D4—1271no—
113C4×D4—2541no—
114#117—2541no—
115C2×C4⋊C4—21no—
116C2×C42—21no—
117C2×C42C2×D12711yescharacteristic
118C4⋊C8—2541no—
119C22⋊C8—1272no—
120#97—2541no—
121#95C2×D12712yes—
122C2×Q8—1271no—
123C2×D4—1271no—
124C2×D4—2541no—
125C22×C4—22no—
126C22×C4—2541no—
127C22×C4C22⋊D12711yescharacteristic
128C22×C4C22⋊D12711yescharacteristic
129C22×C4C22×D12711yescharacteristic
130C2×C8—2542no—
131C4⋊C4—24no—
132C4⋊C4—1271no—
133C4⋊C4—1271no—
134C4⋊C4—2541no—
135C4⋊C4—2541no—
136C4⋊C4—2541no—
137C4⋊C4—2541no—
138C22⋊C4—2541no—
139C22⋊C4—2541no—
140C22⋊C4—2541no—
141C22⋊C4—2541no—
142C22⋊C4—2541no—
143C42—42no—
144C42—2541no—
145C42C4×D12711yescharacteristic
146C42C4×D12711yescharacteristic
147C42D50811yescharacteristic
148C42D50811yescharacteristic
149C23—2541no—
150C23C2×C22⋊D12711yescharacteristic
151Q8—5081no—
152D4—5081no—
153D4—5081no—
154D4—5081no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—5082no—
163C2×C4—2541no—
164C2×C4—2541no—
165C2×C4—2541no—
166C2×C4(C2×D127)⋊C411yescharacteristic
167C2×C4(C2×D127)⋊C411yescharacteristic
168C2×C4(C2×D127)⋊C411yescharacteristic
169C2×C4(C2×D127)⋊C411yescharacteristic
170C2×C4C2×C4×D12711yescharacteristic
171C2×C4C2×D50811yescharacteristic
172C8—5082no—
173C22—21no—
174C22—21no—
175C22—5081no—
176C22—5081no—
177C22—5081no—
178C22C2×(C2×D127)⋊C411yescharacteristic
179C22#1686123511yescharacteristic
180C22#1686124011yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—21no—
186C4—44no—
187C4—5081no—
188C4—5081no—
189C4—5081no—
190C4#1686113712yes—
191C2—21no—
192C2—21no—
193C2—10161no—
194C2#2254984911yescharacteristic
195C2#2254987511yescharacteristic
196C2C2×#1686113711yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC4×C4⋊C4×C127C2
socleC22×C127C2×C22⋊C4
prime radical#16861183C22
cosocleC22×C4×C127C23
hypercenterC4×C4⋊C4D127
hypocenterC127#1509
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22549933C2semidirect
3. #22549612C2semidirect
5. C4×C4⋊C4×C127C2semidirect
7. #16861201C4semidirect
8. #16861197C4semidirect
12. C4×Dic254C4semidirect
13. C4×D508C4semidirect
99. C127#1509semidirect
102. C4×C4⋊C4D127semidirect

Maximal subgroups

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

subgroupindex
#225504102
#225499332
#225496122
#225496092
C4×C4⋊C4×C1272
#1509127