Finite Group Databasebeta

← #30637424

Subgroups — #30637424

Order 16256.

252 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 67 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#22550427C211yesmaximal, characteristic
2#22549932C211yesmaximal, characteristic, complemented
3C4⋊C8×D127C211yesmaximal, characteristic, complemented
4#22549625C211yesmaximal, characteristic, complemented
5#22549619C211yesmaximal, characteristic, complemented
6#22549619C211yesmaximal, characteristic, complemented
7#333×C127C211yesmaximal, characteristic, complemented
8C2×#14523249C412yes—
9C2×C8×D127—21no—
10#16861206C2211yescharacteristic
11#16861198C412yes—
12#16861184—21no—
13#16861184—21no—
14#16861171—21no—
15#16861171—21no—
16C42×D127C2211yescharacteristic
17C4⋊(C127⋊C8)C2211yescharacteristic
18C4⋊(C127⋊C8)C2211yescharacteristic
19Dic127⋊C8—21no—
20(C2×D127)⋊C8—21no—
21#16861119—21no—
22C4⋊Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4⋊C8×C127C2211yescharacteristic
25#14523249—44no—
26C2×D508—22no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508C2×C412yes—
30C2×C4×D127—22no—
31C2×C4×D127C2×C411yescharacteristic
32C2×C4×D127D411yescharacteristic
33C2×C4×D127D411yescharacteristic
34C2×C127⋊C8—21no—
35C2×C127⋊C8—21no—
36C8×D127—42no—
37(C2×D127)⋊C4—22no—
38(C2×D127)⋊C4—22no—
39C4×Dic127C2×C411yescharacteristic
40C4×Dic127D411yescharacteristic
41C4×Dic127D411yescharacteristic
42C2×Q8×C127C2×C412yes—
43C2×C8×C127—21no—
44C2×C8×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127—21no—
47C42×C127C2311yescharacteristic
48C22×D127—22no—
49C22×D127C22⋊C411yescharacteristic
50D508—42no—
51D508—44no—
52D508—41no—
53D508—41no—
54D508—41no—
55D508—41no—
56C2×Dic127C22⋊C411yescharacteristic
57C2×Dic127C22⋊C411yescharacteristic
58C2×Dic127C22⋊C411yescharacteristic
59C4×D127—22no—
60C4×D127—21no—
61C4×D127—21no—
62C4×D127—21no—
63C4×D127—21no—
64C4×D127—44no—
65C127⋊C8—41no—
66C127⋊C8—41no—
67Q8×C127—42no—
68C2×C4×C127—22no—
69C2×C4×C127C22×C411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C8×C127—41no—
73C8×C127—41no—
74C22×C127C2×C22⋊C411yescharacteristic
75C2×D127—21no—
76C2×D127—21no—
77C2×D127—42no—
78C2×D127—42no—
79C2×D127—42no—
80C2×D127#10012yes—
81C4×C127—21no—
82C4×C127—42no—
83C4×C127#13511yescharacteristic
84C4×C127#13511yescharacteristic
85C4×C127AGL(1,ℤ/8)11yescharacteristic
86C4×C127#13711yescharacteristic
87Dic127—21no—
88Dic127—21no—
89Dic127—21no—
90C2×C127C2×#10011yescharacteristic
91C2×C127#41611yescharacteristic
92C2×C127#41711yescharacteristic
93D127—22no—
94D127—82no—
95#1551—1271nomaximal, Sylow
96C127#155111yescharacteristic, complemented, Sylow
97#532—1271no—
98#432—1271no—
99C2×C4⋊C8—1271no—
100#333—1271no—
101#333D12711yescharacteristic, complemented
102#331—1271no—
103#331—1271no—
104C2×#41—1272no—
105C22×C8—2541no—
106C4⋊Q8C2×D12711yescharacteristic
107#127—1271no—
108#124—1272no—
109#121—2541no—
110#121—2541no—
111C4×D4—2541no—
112C4×D4—2541no—
113C2×C42—1271no—
114C4⋊C8—1271no—
115C4⋊C8—1271no—
116C4⋊C8—2541no—
117C4⋊C8C2×D12711yescharacteristic
118C4⋊C8C2×D12711yescharacteristic
119C22⋊C8—2541no—
120#97—2541no—
121#41—5084no—
122C2×Q8C4×D12712yes—
123C2×D4—1272no—
124C2×D4—2542no—
125C2×D4—2541no—
126C2×D4—2541no—
127C22×C4—1271no—
128C22×C4—1271no—
129C22×C4—1271no—
130C22×C4—2542no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—5082no—
134C2×C8—2541no—
135C2×C8—2541no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C22⋊C4—2542no—
139C22⋊C4—2542no—
140C42—1271no—
141C42—1271no—
142C42—1271no—
143C42C22×D12711yescharacteristic
144C23—1271no—
145C23—2542no—
146Q8—42no—
147D4—5082no—
148D4—5084no—
149D4—5081no—
150D4—5081no—
151D4—5081no—
152D4—5081no—
153C2×C4—22no—
154C2×C4—1271no—
155C2×C4—1271no—
156C2×C4—1271no—
157C2×C4—5084no—
158C2×C4—2542no—
159C2×C4—2541no—
160C2×C4—2541no—
161C2×C4—2541no—
162C2×C4—2541no—
163C2×C4C2×C4×D12711yescharacteristic
164C2×C4D4×D12711yescharacteristic
165C2×C4D4×D12711yescharacteristic
166C8—41no—
167C8—41no—
168C8—5081no—
169C8—5081no—
170C22—1272no—
171C22—5082no—
172C22—5082no—
173C22—5082no—
174C22—2541no—
175C22—2541no—
176C22C22⋊C4×D12711yescharacteristic
177C4—21no—
178C4—42no—
179C4—2541no—
180C4—2541no—
181C4—2541no—
182C4#1686123011yescharacteristic
183C4#1686123211yescharacteristic
184C4#1686123611yescharacteristic
185C4#1686124111yescharacteristic
186C2—10162no—
187C2—2542no—
188C2#100×D12711yescharacteristic
189C2#2254987111yescharacteristic
190C2#2254987611yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#333×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550427C2
cosocleC42×C127C23
hypercenter#333D127
hypocenterC127#1551
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #22549932C2semidirect
3. C4⋊C8×D127C2semidirect
4. #22549625C2semidirect
5. #22549619C2semidirect
6. #22549619C2semidirect
7. #333×C127C2semidirect
96. C127#1551semidirect
101. #333D127semidirect

Maximal subgroups

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

subgroupindex
#225504272
#225499322
C4⋊C8×D1272
#225496252
#225496192
#333×C1272
#1551127