Finite Group Databasebeta

← #30637806

Subgroups — #30637806

Order 16256.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550606C211yesmaximal, characteristic, complemented
2#22550073C212yesmaximal
3#22549625C211yesmaximal, characteristic, complemented
4#22549608C212yesmaximal, complemented
5#492×C127C211yesmaximal, characteristic
6C2×#14523236—22no—
7#16861208—22no—
8#16861207—22no—
9C4×Dic254C2212yes—
10#16861174—22no—
11C4⋊(C127⋊C8)C2212yes—
12#16861130—22no—
13C2×SD16×C127—22no—
14C4⋊Q8×C127C2211yescharacteristic
15#127×C127C2211yescharacteristic
16C4×C8×C127C2211yescharacteristic
17C2×Dic254—22no—
18C2×Dic254D412yes—
19#14523236—44no—
20C2×C127⋊C8—22no—
21C4⋊Dic127D412yes—
22Dic127⋊C4—24no—
23Dic127⋊C4—22no—
24C4×Dic127—22no—
25C4×Dic127—22no—
26C2×Q8×C127D412yes—
27C2×D4×C127—22no—
28C2×D4×C127—21no—
29SD16×C127—44no—
30C2×C8×C127—21no—
31C4⋊C4×C127—22no—
32C42×C127C2311yescharacteristic
33Dic254—22no—
34Dic254—82no—
35Dic254SD1614yescomplemented
36C2×Dic127—22no—
37C2×Dic127—22no—
38C127⋊C8—42no—
39C23×C127—41no—
40Q8×C127—42no—
41D4×C127—42no—
42D4×C127—42no—
43D4×C127—42no—
44C2×C4×C127—22no—
45C2×C4×C127C2×D412yes—
46C2×C4×C127C2×D411yescharacteristic
47C8×C127—22no—
48C22×C127—82no—
49C22×C127—81no—
50C22×C127#12011yescharacteristic
51C4×C127—21no—
52C4×C127—42no—
53C4×C127C2×SD1612yes—
54C4×C127#13712yes—
55Dic127—24no—
56Dic127—42no—
57C2×C127—161no—
58C2×C127#44812yes—
59C2×C127#45311yescharacteristic
60#1623—1271nomaximal, Sylow
61C127#162311yescharacteristic, complemented, Sylow
62Q82—1271no—
63#492D12711yescharacteristic
64#460—1272no—
65#333—1271no—
66Q8⋊C8—1272no—
67C2×SD16—22no—
68C2×SD16—2542no—
69C4⋊Q8—2542no—
70C4⋊Q8—2542no—
71C4⋊Q8C2×D12711yescharacteristic
72#127C2×D12711yescharacteristic
73C4×Q8—1272no—
74C4×Q8—2542no—
75C4⋊C8—1272no—
76#102—2542no—
77C4×C8C2×D12711yescharacteristic
78C2×Q8—1272no—
79C2×Q8—2542no—
80C2×Q8C22⋊D12712yes—
81C2×D4—22no—
82C2×D4—21no—
83SD16—44no—
84SD16—5084no—
85C2×C8—21no—
86C2×C8—2542no—
87C4⋊C4—22no—
88C4⋊C4—1272no—
89C4⋊C4—2544no—
90C4⋊C4—2542no—
91C42—2542no—
92C42—2542no—
93C42C22×D12711yescharacteristic
94C23—41no—
95Q8—42no—
96Q8—1274no—
97Q8—10162no—
98Q8—2542no—
99D4—42no—
100D4—42no—
101D4—42no—
102C2×C4—22no—
103C2×C4—2542no—
104C2×C4—2542no—
105C2×C4D4×D12712yes—
106C2×C4C2×C22⋊D12711yescharacteristic
107C8—22no—
108C8—5082no—
109C22—81no—
110C22—82no—
111C22#1686117711yescharacteristic
112C4—21no—
113C4—42no—
114C4—5082no—
115C4—2544no—
116C4SD16×D12712yes—
117C4#1686123812yes—
118C2—161no—
119C2#2255001312yes—
120C2#2255004311yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#492×C127C2
socleC22×C127#120
prime radical#127×C127C22
hypercenter#492D127
hypocenterC127#1623
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #22550606C2semidirect
3. #22549625C2semidirect
4. #22549608C2semidirect
35. Dic254SD16semidirect
61. C127#1623semidirect

Maximal subgroups

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

subgroupindex
#225506062
#225500732
#225496252
#225496082
#492×C1272
#1623127