Finite Group Databasebeta

← #30637422

Subgroups — #30637422

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#22550424C211yesmaximal, characteristic
2#22549932C211yesmaximal, characteristic, complemented
3C4⋊C8×D127C211yesmaximal, characteristic, complemented
4#22549624C211yesmaximal, characteristic, complemented
5#22549624C211yesmaximal, characteristic, complemented
6#22549620C211yesmaximal, characteristic, complemented
7#331×C127C211yesmaximal, characteristic, complemented
8C2×#14523247C412yes—
9C2×C8×D127—21no—
10#16861209C2211yescharacteristic
11#16861196C412yes—
12#16861180—21no—
13#16861180—21no—
14D4×Dic127—21no—
15D4×Dic127—21no—
16C42×D127C2211yescharacteristic
17C4⋊(C127⋊C8)C2211yescharacteristic
18C4⋊(C127⋊C8)C2211yescharacteristic
19Dic127⋊C8—21no—
20(C2×D127)⋊C8—21no—
21#16861119—21no—
22#127×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4⋊C8×C127C2211yescharacteristic
25#14523247—44no—
26C2×Dic254C2×C412yes—
27C2×C22⋊D127—22no—
28C22×Dic127—22no—
29C2×C4×D127C2×C411yescharacteristic
30C2×C4×D127D411yescharacteristic
31C2×C4×D127D411yescharacteristic
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C8×D127—42no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C22⋊Dic127—22no—
38C22⋊Dic127—22no—
39C4×Dic127C2×C411yescharacteristic
40C4×Dic127D411yescharacteristic
41C4×Dic127D411yescharacteristic
42C2×D4×C127—21no—
43C2×D4×C127—21no—
44C2×D4×C127C2×C412yes—
45C2×C8×C127—21no—
46C2×C8×C127—21no—
47C42×C127C2311yescharacteristic
48C22×D127C22⋊C411yescharacteristic
49Dic254—42no—
50C22⋊D127—44no—
51C2×Dic127—22no—
52C2×Dic127—44no—
53C2×Dic127C22⋊C411yescharacteristic
54C2×Dic127C22⋊C411yescharacteristic
55C2×Dic127C22⋊C411yescharacteristic
56C4×D127—22no—
57C4×D127—21no—
58C4×D127—21no—
59C4×D127—21no—
60C4×D127—21no—
61C127⋊C8—41no—
62C127⋊C8—41no—
63C23×C127—22no—
64D4×C127—42no—
65D4×C127—41no—
66D4×C127—41no—
67D4×C127—41no—
68D4×C127—41no—
69C2×C4×C127C22×C411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C8×C127—41no—
73C8×C127—41no—
74C22×C127—42no—
75C22×C127—42no—
76C22×C127—42no—
77C22×C127C2×C22⋊C411yescharacteristic
78C2×D127—21no—
79C2×D127—21no—
80C2×D127#10012yes—
81C4×C127—21no—
82C4×C127#13511yescharacteristic
83C4×C127#13511yescharacteristic
84C4×C127AGL(1,ℤ/8)11yescharacteristic
85C4×C127#13711yescharacteristic
86Dic127—21no—
87Dic127—21no—
88Dic127—21no—
89Dic127—42no—
90C2×C127—82no—
91C2×C127C2×#10011yescharacteristic
92C2×C127#41611yescharacteristic
93C2×C127#41711yescharacteristic
94D127—22no—
95#1551—1271nomaximal, Sylow
96C127#155111yescharacteristic, complemented, Sylow
97#532—1271no—
98#432—1271no—
99C2×C4⋊C8—1271no—
100#333—1271no—
101#333—1271no—
102#331—1271no—
103#331D12711yescharacteristic, complemented
104C2×#41—1272no—
105C22×C8—2541no—
106C4⋊Q8—1271no—
107#127C2×D12711yescharacteristic
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×Q8—1272no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—2542no—
126C2×D4C4×D12712yes—
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—2541no—
137C4⋊C4—2541no—
138C22⋊C4—2542no—
139C22⋊C4—2542no—
140C42—1271no—
141C42—1271no—
142C42—1271no—
143C42C22×D12711yescharacteristic
144C23—22no—
145C23—1271no—
146Q8—5082no—
147D4—42no—
148D4—41no—
149D4—41no—
150D4—41no—
151D4—41no—
152D4—5084no—
153C2×C4—1271no—
154C2×C4—1271no—
155C2×C4—1271no—
156C2×C4—5084no—
157C2×C4—2542no—
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—42no—
171C22—42no—
172C22—42no—
173C22—1272no—
174C22—2541no—
175C22—2541no—
176C22C22⋊C4×D12711yescharacteristic
177C4—21no—
178C4—5082no—
179C4—2541no—
180C4—2541no—
181C4—2541no—
182C4#1686122811yescharacteristic
183C4#1686123011yescharacteristic
184C4#1686123311yescharacteristic
185C4#1686123811yescharacteristic
186C2—82no—
187C2—2542no—
188C2#100×D12711yescharacteristic
189C2#2254986911yescharacteristic
190C2#2254987411yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#331×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550424C2
cosocleC42×C127C23
hypercenter#331D127
hypocenterC127#1551
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #22549932C2semidirect
3. C4⋊C8×D127C2semidirect
4. #22549624C2semidirect
5. #22549624C2semidirect
6. #22549620C2semidirect
7. #331×C127C2semidirect
96. C127#1551semidirect
103. #331D127semidirect

Maximal subgroups

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

subgroupindex
#225504242
#225499322
C4⋊C8×D1272
#225496242
#225496202
#331×C1272
#1551127