Finite Group Databasebeta

← #30637243

Subgroups — #30637243

Order 16256.

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

normal subgroups only

This group has 228 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#22550433C211yesmaximal, characteristic
2#22549722C211yesmaximal, characteristic
3#22549614C214yesmaximal
4#432×C127C211yesmaximal, characteristic
5#16861209C411yescharacteristic, complemented
6#16861209C411yescharacteristic, complemented
7#16861201—21no—
8#16861188—21no—
9#16861188C2212yes—
10C4×Dic254—21no—
11C4×Dic254C412yescomplemented
12#16861168—21no—
13C2×C4×Dic127—21no—
14C127⋊#95—21no—
15C127⋊#95C2212yes—
16C2×C42×C127C2211yescharacteristic
17C4⋊C8×C127—21no—
18C22⋊C8×C127C2212yes—
19#97×C127—21no—
20C2×Dic254—21no—
21C2×Dic254C2×C412yes—
22C22×Dic127—22no—
23C4⋊Dic127—21no—
24C4⋊Dic127—21no—
25C4⋊Dic127—21no—
26C4⋊Dic127C2×C412yes—
27Dic127⋊C4—22no—
28Dic127⋊C4—22no—
29C22⋊Dic127—22no—
30C4×Dic127—24no—
31C4×Dic127—22no—
32C22×C4×C127D411yescharacteristic
33C22×C4×C127D411yescharacteristic
34C22×C4×C127C2311yescharacteristic
35C2×C8×C127—22no—
36C42×C127C2×C411yescharacteristic
37C42×C127C2×C411yescharacteristic
38C42×C127D411yescharacteristic
39C42×C127D411yescharacteristic
40Dic254—42no—
41Dic254—42no—
42C2×Dic127—24no—
43C2×Dic127—22no—
44C2×Dic127—22no—
45C2×Dic127—42no—
46C2×Dic127—42no—
47C23×C127C2×D411yescharacteristic
48C2×C4×C127—22no—
49C2×C4×C127—22no—
50C2×C4×C127—41no—
51C2×C4×C127C22⋊C411yescharacteristic
52C2×C4×C127C22⋊C411yescharacteristic
53C2×C4×C127C22⋊C411yescharacteristic
54C2×C4×C127C22⋊C411yescharacteristic
55C2×C4×C127C22×C411yescharacteristic
56C2×C4×C127C2×D411yescharacteristic
57C8×C127—42no—
58C22×C127—21no—
59C22×C127—21no—
60C22×C127C2×C22⋊C411yescharacteristic
61C22×C127#13712yes—
62C4×C127—21no—
63C4×C127—21no—
64C4×C127—21no—
65C4×C127—21no—
66C4×C127—21no—
67C4×C127#10412yes—
68Dic127—44no—
69Dic127—42no—
70Dic127—42no—
71C2×C127—22no—
72C2×C127#41011yescharacteristic
73C2×C127#41911yescharacteristic
74C2×C127C2×#10411yescharacteristic
75#1506—1271nomaximal, Sylow
76C127#150611yescharacteristic, complemented, Sylow
77#533—1271no—
78#432D12711yescharacteristic
79#376—1271no—
80#328—1274no—
81C4⋊Q8—1271no—
82C4⋊Q8—1271no—
83#125—2541no—
84C22⋊Q8—1272no—
85C22⋊Q8—2541no—
86C4×Q8—1272no—
87C4×Q8—2541no—
88#117—2541no—
89C2×C42—2541no—
90C2×C42C2×D12711yescharacteristic
91C4⋊C8—21no—
92C22⋊C8C2×D12712yes—
93#97—21no—
94#95—1272no—
95#95—2541no—
96C2×Q8—1272no—
97C2×Q8—2541no—
98C22×C4—2542no—
99C22×C4D50811yescharacteristic
100C22×C4D50811yescharacteristic
101C22×C4C22×D12711yescharacteristic
102C2×C8—22no—
103C4⋊C4—1272no—
104C4⋊C4—2542no—
105C4⋊C4—2541no—
106C4⋊C4—2541no—
107C4⋊C4—2541no—
108C4⋊C4—2542no—
109C22⋊C4—2542no—
110C42—2544no—
111C42—2542no—
112C42C4×D12711yescharacteristic
113C42C4×D12711yescharacteristic
114C42C22⋊D12711yescharacteristic
115C42C22⋊D12711yescharacteristic
116C23C2×D50811yescharacteristic
117Q8—5082no—
118Q8—5082no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—41no—
122C2×C4—5082no—
123C2×C4—5082no—
124C2×C4—2544no—
125C2×C4—2542no—
126C2×C4—2542no—
127C2×C4(C2×D127)⋊C411yescharacteristic
128C2×C4(C2×D127)⋊C411yescharacteristic
129C2×C4(C2×D127)⋊C411yescharacteristic
130C2×C4(C2×D127)⋊C411yescharacteristic
131C2×C4C2×C4×D12711yescharacteristic
132C2×C4C2×C22⋊D12711yescharacteristic
133C8—42no—
134C22—21no—
135C22—21no—
136C22C2×(C2×D127)⋊C411yescharacteristic
137C22#1686124212yes—
138C4—21no—
139C4—21no—
140C4—21no—
141C4—21no—
142C4—21no—
143C4—5084no—
144C4—5082no—
145C4—5082no—
146C4#1686113512yes—
147C2—22no—
148C2#2254984911yescharacteristic
149C2#2254988611yescharacteristic
150C2C2×#1686113511yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#432×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenter#432D127
hypocenterC127#1506
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
5. #16861209C4semidirect
6. #16861209C4semidirect
11. C4×Dic254C4semidirect
76. C127#1506semidirect

Maximal subgroups

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

subgroupindex
#225504332
#225497222
#225496142
#432×C1272
#1506127