Finite Group Databasebeta

← #30637353

Subgroups — #30637353

Order 16256.

248 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 69 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×#16861197C211yesmaximal, characteristic
2#22549933C211yesmaximal, characteristic, complemented
3#22549615C214yesmaximal, complemented
4#432×C127C211yesmaximal, characteristic, complemented
5C22×Dic254C411yescharacteristic
6C22×D508C411yescharacteristic
7#16861197—21no—
8#16861197—21no—
9#16861197C412yes—
10#16861197C2212yes—
11C2×(C2×D127)⋊C4—21no—
12C2×(C2×D127)⋊C4—21no—
13C4⋊(C127⋊C8)—21no—
14C22⋊(C127⋊C8)—21no—
15#16861120C2212yescomplemented
16C2×C42×C127C2211yescharacteristic
17C4⋊C8×C127—21no—
18C22⋊C8×C127—21no—
19#97×C127C2212yescomplemented
20C23×D127—21no—
21C2×Dic254—21no—
22C2×Dic254—42no—
23C2×Dic254C2×C412yes—
24C2×D508—21no—
25C2×D508—42no—
26C2×D508C2×C412yes—
27C22×Dic127—21no—
28C2×C127⋊C8—22no—
29(C2×D127)⋊C4—22no—
30(C2×D127)⋊C4—22no—
31(C2×D127)⋊C4—22no—
32(C2×D127)⋊C4—22no—
33C22×C4×C127C2×C411yescharacteristic
34C22×C4×C127D411yescharacteristic
35C22×C4×C127D411yescharacteristic
36C2×C8×C127—22no—
37C42×C127C2×C411yescharacteristic
38C42×C127D411yescharacteristic
39C42×C127D411yescharacteristic
40C42×C127C2311yescharacteristic
41C22×D127—22no—
42C22×D127—42no—
43C22×D127—42no—
44C22×D127—42no—
45Dic254—42no—
46Dic254—42no—
47D508—42no—
48D508—42no—
49C2×Dic127—22no—
50C2×Dic127—42no—
51C127⋊C8—42no—
52C23×C127C22⋊C411yescharacteristic
53C2×C4×C127—22no—
54C2×C4×C127—41no—
55C2×C4×C127—41no—
56C2×C4×C127C22⋊C411yescharacteristic
57C2×C4×C127C22⋊C411yescharacteristic
58C2×C4×C127C22⋊C411yescharacteristic
59C2×C4×C127C22×C411yescharacteristic
60C2×C4×C127C2×D411yescharacteristic
61C2×C4×C127C2×D411yescharacteristic
62C8×C127—42no—
63C22×C127—21no—
64C22×C127—21no—
65C22×C127#10012yes—
66C22×C127C2×C22⋊C411yescharacteristic
67C2×D127—42no—
68C2×D127—42no—
69C2×D127—42no—
70C2×D127—84no—
71C4×C127—21no—
72C4×C127—21no—
73C4×C127—21no—
74C4×C127—21no—
75C4×C127—21no—
76C4×C127—21no—
77Dic127—42no—
78C2×C127—22no—
79C2×C127C2×#10011yescharacteristic
80C2×C127#42111yescharacteristic
81C2×C127#42111yescharacteristic
82D127—82no—
83#1533—1271nomaximal, Sylow
84C127#153311yescharacteristic, complemented, Sylow
85C2×#124—1271no—
86#432—1271no—
87#432D12711yescharacteristic, complemented
88#329—1274no—
89C22×Q8—1271no—
90C22×D4—1271no—
91#124—1272no—
92#124—1272no—
93#124—2541no—
94#124—2541no—
95C2×C22⋊C4—2541no—
96C2×C22⋊C4—2541no—
97C2×C42C2×D12711yescharacteristic
98C4⋊C8—21no—
99C4⋊C8—2541no—
100C22⋊C8—21no—
101C22⋊C8—2541no—
102#97—1272no—
103#97C2×D12712yescomplemented
104C24—2541no—
105C2×Q8—1272no—
106C2×Q8—5082no—
107C2×Q8—2541no—
108C2×D4—1272no—
109C2×D4—5082no—
110C2×D4—2541no—
111C22×C4—2541no—
112C22×C4C4×D12711yescharacteristic
113C22×C4C22⋊D12711yescharacteristic
114C22×C4D50811yescharacteristic
115C2×C8—22no—
116C2×C8—2542no—
117C22⋊C4—2542no—
118C22⋊C4—2542no—
119C22⋊C4—2542no—
120C22⋊C4—2542no—
121C42C4×D12711yescharacteristic
122C42C22⋊D12711yescharacteristic
123C42D50811yescharacteristic
124C42C22×D12711yescharacteristic
125C23—5082no—
126C23—5082no—
127C23—5082no—
128C23—2542no—
129C23(C2×D127)⋊C411yescharacteristic
130Q8—5082no—
131Q8—5082no—
132D4—5082no—
133D4—5082no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—5082no—
138C2×C4—2542no—
139C2×C4(C2×D127)⋊C411yescharacteristic
140C2×C4(C2×D127)⋊C411yescharacteristic
141C2×C4(C2×D127)⋊C411yescharacteristic
142C2×C4C2×C4×D12711yescharacteristic
143C2×C4C2×C22⋊D12711yescharacteristic
144C2×C4C2×D50811yescharacteristic
145C8—42no—
146C8—5082no—
147C22—21no—
148C22—21no—
149C22—10164no—
150C22—5082no—
151C22—5082no—
152C22—5082no—
153C22#1686112512yes—
154C22C2×(C2×D127)⋊C411yescharacteristic
155C4—21no—
156C4—21no—
157C4—21no—
158C4—21no—
159C4—21no—
160C4—21no—
161C4—5082no—
162C2—22no—
163C2—10162no—
164C2C2×#1686112511yescharacteristic
165C2#2254989211yescharacteristic
166C2#2254989411yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#432×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D508C4
cosocleC42×C127C23
hypercenter#432D127
hypocenterC127#1533
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #22549933C2semidirect
3. #22549615C2semidirect
4. #432×C127C2semidirect
15. #16861120C22semidirect
19. #97×C127C22semidirect
84. C127#1533semidirect
87. #432D127semidirect
103. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611972
#225499332
#225496152
#432×C1272
#1533127