Finite Group Databasebeta

← #30637849

Subgroups — #30637849

Order 16256.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550612C211yesmaximal, characteristic
2#22550168C212yesmaximal, complemented
3#22549620C211yesmaximal, characteristic
4#22549608C212yesmaximal, complemented
5#492×C127C211yesmaximal, characteristic
6#16861196—22no—
7#16861190—22no—
8C4×Dic254C2212yes—
9D4×Dic127—22no—
10#16861142—22no—
11C4⋊(C127⋊C8)C2212yes—
12#16861132—22no—
13C2×SD16×C127—22no—
14C4⋊Q8×C127C2211yescharacteristic
15#127×C127C2211yescharacteristic
16C4×C8×C127C2211yescharacteristic
17C2×Dic254D412yes—
18C22×Dic127—22no—
19C2×C127⋊C8—22no—
20C4⋊Dic127D412yes—
21Dic127⋊C4—22no—
22C22⋊Dic127—22no—
23C22⋊Dic127—24no—
24C4×Dic127—22no—
25C2×Q8×C127—21no—
26C2×D4×C127—22no—
27C2×D4×C127D412yes—
28SD16×C127—44no—
29C2×C8×C127—21no—
30C4⋊C4×C127—22no—
31C42×C127C2311yescharacteristic
32Dic254—22no—
33Dic254—22no—
34C2×Dic127—22no—
35C2×Dic127—22no—
36C2×Dic127—82no—
37C127⋊C8—42no—
38C23×C127—22no—
39Q8×C127—42no—
40D4×C127—42no—
41D4×C127—42no—
42D4×C127—42no—
43C2×C4×C127—41no—
44C2×C4×C127C2×D412yes—
45C2×C4×C127C2×D411yescharacteristic
46C8×C127—22no—
47C22×C127—44no—
48C22×C127—42no—
49C22×C127#12011yescharacteristic
50C4×C127—21no—
51C4×C127—81no—
52C4×C127#13512yes—
53C4×C127#13712yes—
54Dic127—42no—
55Dic127—42no—
56C2×C127—82no—
57C2×C127#45212yes—
58C2×C127#45311yescharacteristic
59#1630—1271nomaximal, Sylow
60C127#163011yescharacteristic, complemented, Sylow
61#559—1271no—
62#492D12711yescharacteristic
63#477—1272no—
64#331—1271no—
65Q8⋊C8—1272no—
66C2×SD16—22no—
67C4⋊Q8C2×D12711yescharacteristic
68#127C2×D12711yescharacteristic
69#124—2542no—
70#123—2542no—
71C4×Q8—1272no—
72C4×D4—2542no—
73#107—2542no—
74C4⋊C8—1272no—
75#103—2542no—
76C4×C8C2×D12711yescharacteristic
77C2×Q8—21no—
78C2×Q8—1272no—
79C2×D4—22no—
80C2×D4C22⋊D12712yes—
81C22×C4—2542no—
82SD16—44no—
83C2×C8—21no—
84C2×C8—2542no—
85C4⋊C4—22no—
86C4⋊C4—1272no—
87C4⋊C4—2542no—
88C22⋊C4—2544no—
89C22⋊C4—2542no—
90C42—2542no—
91C42C22×D12711yescharacteristic
92C23—22no—
93Q8—42no—
94Q8—2542no—
95Q8—2542no—
96D4—42no—
97D4—42no—
98D4—42no—
99C2×C4—41no—
100C2×C4—10162no—
101C2×C4—2542no—
102C2×C4—2542no—
103C2×C4D4×D12712yes—
104C2×C4C2×C22⋊D12711yescharacteristic
105C8—22no—
106C8—5082no—
107C22—44no—
108C22—42no—
109C22#1686117711yescharacteristic
110C4—21no—
111C4—81no—
112C4—5082no—
113C4—5082no—
114C4#1686123012yes—
115C4#1686123812yes—
116C2—82no—
117C2#2255003512yes—
118C2#2255004311yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#492×C127C2
socleC22×C127#120
prime radical#127×C127C22
hypercenter#492D127
hypocenterC127#1630
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. #22550168C2semidirect
4. #22549608C2semidirect
60. C127#1630semidirect

Maximal subgroups

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

subgroupindex
#225506122
#225501682
#225496202
#225496082
#492×C1272
#1630127