Finite Group Databasebeta

← #30637224

Subgroups — #30637224

Order 16256.

284 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 89 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#22550426C211yesmaximal, characteristic
2#22549929C211yesmaximal, characteristic, complemented
3#22549609C214yesmaximal, complemented
4C4×C4⋊C4×C127C211yesmaximal, characteristic, complemented
5C2×#14523248—21no—
6#16861209C411yescharacteristic, complemented
7#16861206C411yescharacteristic, complemented
8#16861197—21no—
9#16861183—21no—
10#16861183C2212yes—
11C4×D508—21no—
12C4×D508C412yescomplemented
13C4⋊(C127⋊C8)—21no—
14C22⋊(C127⋊C8)C2212yes—
15C4×C127⋊C8—21no—
16C2×C4⋊C4×C127—21no—
17C2×C42×C127—21no—
18C2×C42×C127C2211yescharacteristic
19#95×C127C2212yes—
20#14523248—42no—
21#14523248—42no—
22C2×Dic254—21no—
23C2×D508—21no—
24C2×D508—21no—
25C2×D508C2×C412yes—
26C2×C22⋊D127—22no—
27C2×C4×D127—22no—
28C2×C127⋊C8—22no—
29C4⋊Dic127—21no—
30C4⋊Dic127C2×C412yes—
31(C2×D127)⋊C4—22no—
32(C2×D127)⋊C4—22no—
33C22×C4×C127—22no—
34C22×C4×C127D411yescharacteristic
35C22×C4×C127D411yescharacteristic
36C22×C4×C127C2311yescharacteristic
37C4⋊C4×C127—24no—
38C42×C127—42no—
39C42×C127C2×C411yescharacteristic
40C42×C127C2×C411yescharacteristic
41C42×C127D411yescharacteristic
42C42×C127D411yescharacteristic
43C22×D127—22no—
44Dic254—42no—
45D508—42no—
46D508—42no—
47D508—42no—
48C22⋊D127—42no—
49C22⋊D127—42no—
50C2×Dic127—22no—
51C4×D127—44no—
52C127⋊C8—42no—
53C23×C127C2×D411yescharacteristic
54C2×C4×C127—22no—
55C2×C4×C127—24no—
56C2×C4×C127—41no—
57C2×C4×C127—42no—
58C2×C4×C127—42no—
59C2×C4×C127C22⋊C411yescharacteristic
60C2×C4×C127C22⋊C411yescharacteristic
61C2×C4×C127C22⋊C411yescharacteristic
62C2×C4×C127C22⋊C411yescharacteristic
63C2×C4×C127D812yes—
64C2×C4×C127SD1612yes—
65C2×C4×C127C22×C411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C22×C127—21no—
68C22×C127—21no—
69C22×C127C2×C22⋊C411yescharacteristic
70C22×C127C2×D811yescharacteristic
71C22×C127C2×SD1611yescharacteristic
72C2×D127—42no—
73C2×D127—42no—
74C2×D127—42no—
75C4×C127—21no—
76C4×C127—21no—
77C4×C127—21no—
78C4×C127—44no—
79C4×C127#10212yes—
80C4×C127#10212yes—
81C4×C127#10412yes—
82Dic127—42no—
83C2×C127—21no—
84C2×C127—21no—
85C2×C127#41011yescharacteristic
86C2×C127C2×#10211yescharacteristic
87C2×C127C2×#10411yescharacteristic
88D127—82no—
89#1502—1271nomaximal, Sylow
90C127#150211yescharacteristic, complemented, Sylow
91#532—1271no—
92#431—1271no—
93C4×C4⋊C4D12711yescharacteristic, complemented
94#327—1274no—
95C2×#41—2541no—
96C4⋊Q8—1271no—
97#127—1271no—
98#124—2541no—
99#121—1272no—
100#121—2541no—
101C4×D4—1272no—
102C4×D4—2541no—
103C2×C4⋊C4—21no—
104C2×C42—21no—
105C2×C42C2×D12711yescharacteristic
106C4⋊C8—2541no—
107C22⋊C8—1272no—
108C4×C8—2541no—
109#95C2×D12712yes—
110#41—5082no—
111#41—5082no—
112C2×Q8—2541no—
113C2×D4—1272no—
114C2×D4—2542no—
115C2×D4—2541no—
116C2×D4—2541no—
117C22×C4—22no—
118C22×C4—2542no—
119C22×C4C22⋊D12711yescharacteristic
120C22×C4C22⋊D12711yescharacteristic
121C22×C4C22×D12711yescharacteristic
122C2×C8—2542no—
123C4⋊C4—24no—
124C4⋊C4—1272no—
125C4⋊C4—2541no—
126C22⋊C4—2542no—
127C22⋊C4—2542no—
128C42—42no—
129C42C4×D12711yescharacteristic
130C42C4×D12711yescharacteristic
131C42D50811yescharacteristic
132C42D50811yescharacteristic
133C23—2542no—
134C23C2×C22⋊D12711yescharacteristic
135Q8—5082no—
136D4—5082no—
137D4—5082no—
138D4—5082no—
139D4—5082no—
140D4—5082no—
141C2×C4—22no—
142C2×C4—24no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—5084no—
147C2×C4—2542no—
148C2×C4(C2×D127)⋊C411yescharacteristic
149C2×C4(C2×D127)⋊C411yescharacteristic
150C2×C4(C2×D127)⋊C411yescharacteristic
151C2×C4(C2×D127)⋊C411yescharacteristic
152C2×C4#1452323312yes—
153C2×C4#1452323512yes—
154C2×C4C2×C4×D12711yescharacteristic
155C2×C4C2×D50811yescharacteristic
156C8—5082no—
157C22—21no—
158C22—21no—
159C22—5082no—
160C22—5082no—
161C22—5082no—
162C22C2×(C2×D127)⋊C411yescharacteristic
163C22C2×#1452323311yescharacteristic
164C22C2×#1452323511yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—44no—
169C4—5082no—
170C4#1686112912yes—
171C4#1686112912yes—
172C4#1686113712yes—
173C2—21no—
174C2—21no—
175C2—10162no—
176C2#2254984911yescharacteristic
177C2C2×#1686112911yescharacteristic
178C2C2×#1686113711yescharacteristic
subgroupG/H
centerC2×C4(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC4×C4⋊C4×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550426C2
cosocleC22×C4×C127C23
hypercenterC4×C4⋊C4D127
hypocenterC127#1502
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22549929C2semidirect
3. #22549609C2semidirect
4. C4×C4⋊C4×C127C2semidirect
6. #16861209C4semidirect
7. #16861206C4semidirect
12. C4×D508C4semidirect
90. C127#1502semidirect
93. C4×C4⋊C4D127semidirect

Maximal subgroups

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

subgroupindex
#225504262
#225499292
#225496092
C4×C4⋊C4×C1272
#1502127