Finite Group Databasebeta

← #30639897

Subgroups — #30639897

Order 16256.

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

normal subgroups only

This group has 290 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
1#22550502C211yesmaximal, characteristic
2#22549739C212yesmaximal
3#22549643C211yesmaximal, characteristic
4#22549642C212yesmaximal
5#381×C127C211yesmaximal, characteristic
6#16861188—22no—
7#16861186—24no—
8C2×C22⋊Dic127C2212yes—
9C2×C22⋊Dic127C2212yes—
10C127⋊#95—24no—
11C23×C4×C127C2211yescharacteristic
12C2×C22⋊C4×C127C2211yescharacteristic
13C2×C22⋊C4×C127C2211yescharacteristic
14#95×C127—22no—
15C2×Dic254—22no—
16C22×Dic127—22no—
17C22×Dic127D414yes—
18C4⋊Dic127—22no—
19Dic127⋊C4—24no—
20C22⋊Dic127—22no—
21C22⋊Dic127—22no—
22C22⋊Dic127—22no—
23C22⋊Dic127—22no—
24C24×C127C2311yescharacteristic
25C22×C4×C127—21no—
26C22×C4×C127—22no—
27C22×C4×C127—22no—
28C22×C4×C127—21no—
29C22×C4×C127—21no—
30C22×C4×C127D412yes—
31C22×C4×C127Q812yescomplemented
32C22⋊C4×C127—22no—
33C22⋊C4×C127—22no—
34Dic254—82no—
35C2×Dic127—24no—
36C2×Dic127—44no—
37C2×Dic127—44no—
38C2×Dic127—42no—
39C23×C127—42no—
40C23×C127C2×D412yes—
41C23×C127C2×D411yescharacteristic
42C23×C127C2×Q811yescharacteristic
43C23×C127#4112yes—
44C23×C127#4111yescharacteristic
45C2×C4×C127—22no—
46C2×C4×C127—24no—
47C2×C4×C127—22no—
48C2×C4×C127—41no—
49C2×C4×C127—42no—
50C2×C4×C127—41no—
51C2×C4×C127—42no—
52C2×C4×C127—42no—
53C2×C4×C127—41no—
54C22×C127—22no—
55C22×C127—24no—
56C22×C127—41no—
57C22×C127—42no—
58C22×C127—41no—
59C22×C127#12011yescharacteristic
60C22×C127C22⋊Q812yes—
61C22×C127C22⋊Q811yescharacteristic
62C22×C127#12412yes—
63C22×C127#12411yescharacteristic
64C4×C127—42no—
65C4×C127—42no—
66C4×C127—81no—
67Dic127—44no—
68Dic127—82no—
69C2×C127—22no—
70C2×C127—21no—
71C2×C127—22no—
72C2×C127—21no—
73C2×C127#39311yescharacteristic
74C2×C127#45812yes—
75#2030—1271nomaximal, Sylow
76C127#203011yescharacteristic, complemented, Sylow
77#543—1271no—
78#381—1272no—
79#381D12711yescharacteristic
80#342—1272no—
81#342—1271no—
82C23×C4C2×D12711yescharacteristic
83C22⋊Q8—2544no—
84C22⋊Q8—2542no—
85C2×C22⋊C4—1272no—
86C2×C22⋊C4—1272no—
87C2×C22⋊C4C2×D12711yescharacteristic
88C2×C22⋊C4C2×D12711yescharacteristic
89#95—22no—
90#95—2544no—
91C24C22×D12711yescharacteristic
92C2×Q8—2542no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—1274no—
99C22×C4—2542no—
100C22×C4C22⋊D12712yes—
101C22×C4Dic25412yescomplemented
102C4⋊C4—2544no—
103C4⋊C4—2542no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—2542no—
107C22⋊C4—2542no—
108C22⋊C4—2542no—
109C22⋊C4—2542no—
110C23—42no—
111C23D4×D12712yes—
112C23C2×C22⋊D12711yescharacteristic
113C23C2×Dic25411yescharacteristic
114C23#1452324712yes—
115C23#1452324811yescharacteristic
116Q8—10162no—
117C2×C4—24no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—42no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—41no—
126C2×C4—5084no—
127C2×C4—5082no—
128C2×C4—5084no—
129C2×C4—2544no—
130C22—22no—
131C22—24no—
132C22—41no—
133C22—42no—
134C22—41no—
135C22#1686117711yescharacteristic
136C22#1686118612yes—
137C22#1686118811yescharacteristic
138C22#1686119512yes—
139C22#1686119611yescharacteristic
140C4—42no—
141C4—42no—
142C4—81no—
143C4—10162no—
144C4—5084no—
145C2—21no—
146C2—22no—
147C2—22no—
148C2—21no—
149C2#2254978811yescharacteristic
150C2#2255006512yes—
subgroupG/H
centerC22#16861177
commutator, prime radical, cosocleC24×C127C23
FrattiniC24C22×D127
Fitting#381×C127C2
socleC22×C127#120
hypercenter#381D127
hypocenterC127#2030
LCSG ▷ C24×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
31. C22×C4×C127Q8semidirect
76. C127#2030semidirect
101. C22×C4Dic254semidirect

Maximal subgroups

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

subgroupindex
#225505022
#225497392
#225496432
#225496422
#381×C1272
#2030127