Finite Group Databasebeta

← #30638042

Subgroups — #30638042

Order 16256.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#22550626C211yesmaximal, characteristic
2#22550285C212yesmaximal, complemented
3#22550271C211yesmaximal, characteristic
4#22549617C212yesmaximal, complemented
5#329×C127C211yesmaximal, characteristic
6C2×#14523238—21no—
7C2×#14523236—21no—
8#16861209C2211yescharacteristic
9#16861207—21no—
10#16861202—22no—
11#16861201C2212yes—
12#16861190—21no—
13#16861186—22no—
14#16861143—22no—
15#16861141—22no—
16#16861120C2211yescharacteristic
17#124×C127C2211yescharacteristic
18#97×C127C2212yes—
19C2×Dic254—21no—
20C2×Dic254—21no—
21C22×Dic127—21no—
22#14523238—42no—
23#14523236—42no—
24C2×C127⋊C8—21no—
25C4⋊Dic127—22no—
26C4⋊Dic127D412yes—
27C4⋊Dic127D412yes—
28Dic127⋊C4—22no—
29Dic127⋊C4—22no—
30Dic127⋊C4—22no—
31C22⋊Dic127—22no—
32C22⋊Dic127—21no—
33C4×Dic127—21no—
34C2×Q8×C127D411yescharacteristic
35C2×D4×C127D411yescharacteristic
36C2×C8×C127—22no—
37C22⋊C4×C127—22no—
38C42×C127C2311yescharacteristic
39Dic254—41no—
40Dic254—41no—
41Dic254—81no—
42C2×Dic127—22no—
43C2×Dic127—22no—
44C2×Dic127—41no—
45C2×Dic127—81no—
46C127⋊C8—22no—
47C23×C127—21no—
48Q8×C127—41no—
49D4×C127—41no—
50C2×C4×C127—21no—
51C2×C4×C127C2×D412yes—
52C2×C4×C127C2×D411yescharacteristic
53C8×C127—24no—
54C22×C127—41no—
55C22×C127—42no—
56C22×C127#12011yescharacteristic
57C4×C127—21no—
58C4×C127—22no—
59C4×C127—41no—
60Dic127—42no—
61Dic127—42no—
62Dic127—81no—
63C2×C127—81no—
64C2×C127#45311yescharacteristic
65C2×C127#45612yes—
66#1660—1271nomaximal, Sylow
67C127#166011yescharacteristic, complemented, Sylow
68#563—1271no—
69#501—1272no—
70#497—1271no—
71#330—1272no—
72#329D12711yescharacteristic
73C2×Q16—2541no—
74C2×SD16—2541no—
75C4⋊Q8—1271no—
76C4⋊Q8—2541no—
77#126—2542no—
78#125—1272no—
79#124C2×D12711yescharacteristic
80#123—2541no—
81C22⋊Q8—2542no—
82#107—2542no—
83#106—2542no—
84#97—1271no—
85#97C2×D12712yes—
86C2×Q8—2541no—
87C2×Q8—2541no—
88C2×Q8D50811yescharacteristic
89C2×D4D50811yescharacteristic
90C22×C4—2541no—
91Q16—5082no—
92SD16—5082no—
93C2×C8—22no—
94C2×C8—2541no—
95C4⋊C4—1272no—
96C4⋊C4—1272no—
97C4⋊C4—2542no—
98C4⋊C4—2542no—
99C4⋊C4—2542no—
100C4⋊C4—2542no—
101C22⋊C4—22no—
102C22⋊C4—2542no—
103C22⋊C4—2541no—
104C42—2541no—
105C42C22×D12711yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—10161no—
109Q8—5081no—
110Q8—5081no—
111D4—41no—
112C2×C4—21no—
113C2×C4—10161no—
114C2×C4—5081no—
115C2×C4—2542no—
116C2×C4—2542no—
117C2×C4D4×D12712yes—
118C2×C4C2×D50811yescharacteristic
119C8—24no—
120C8—2542no—
121C22—41no—
122C22—42no—
123C22#1686117911yescharacteristic
124C4—21no—
125C4—22no—
126C4—41no—
127C4—10161no—
128C4—5082no—
129C4—5082no—
130C2—81no—
131C2#2255004611yescharacteristic
132C2#2255006012yes—
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#329×C127C2
socleC22×C127#120
prime radicalC2×D4×C127D4
hypercenter#329D127
hypocenterC127#1660
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #22550285C2semidirect
4. #22549617C2semidirect
67. C127#1660semidirect

Maximal subgroups

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

subgroupindex
#225506262
#225502852
#225502712
#225496172
#329×C1272
#1660127