Finite Group Databasebeta

← #30638036

Subgroups — #30638036

Order 16256.

228 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550271C211yesmaximal, characteristic, complemented
2#22550235C212yesmaximal, complemented
3#22549616C212yesmaximal, complemented
4#22549616C211yesmaximal, characteristic, complemented
5H(3,𝔽4)×C127C211yesmaximal, characteristic
6C2×#14523238—21no—
7C2×#14523236—21no—
8#16861209C2211yescharacteristic, complemented
9#16861133—22no—
10#16861130—22no—
11#16861120C2212yescomplemented
12#16861120C2211yescharacteristic, complemented
13#124×C127—21no—
14#124×C127—22no—
15#124×C127C2212yes—
16#124×C127C2211yescharacteristic
17#120×C127—21no—
18#120×C127—22no—
19C2×Dic254—21no—
20#14523238—42no—
21#14523236—42no—
22C2×C127⋊C8—21no—
23C2×C127⋊C8—22no—
24C4⋊Dic127—22no—
25C24×C127—21no—
26C2×Q8×C127D411yescharacteristic
27C2×Q8×C127D412yes—
28C2×D4×C127—21no—
29C2×D4×C127—22no—
30C2×D4×C127D411yescharacteristic
31C2×D4×C127D412yes—
32C22⋊C4×C127—22no—
33C22⋊C4×C127—21no—
34C22⋊C4×C127—22no—
35C22⋊C4×C127—22no—
36C22⋊C4×C127—22no—
37C42×C127—21no—
38C42×C127C2311yescharacteristic
39Dic254—41no—
40Dic254—41no—
41C2×Dic127—41no—
42C127⋊C8—22no—
43C127⋊C8—42no—
44C23×C127—21no—
45C23×C127—22no—
46C23×C127—81no—
47C23×C127—82no—
48Q8×C127—42no—
49Q8×C127—41no—
50D4×C127—42no—
51D4×C127—41no—
52D4×C127—81no—
53D4×C127—82no—
54C2×C4×C127—21no—
55C2×C4×C127—22no—
56C2×C4×C127C2×D412yes—
57C2×C4×C127C2×D411yescharacteristic
58C22×C127—41no—
59C22×C127—42no—
60C22×C127—42no—
61C22×C127—42no—
62C22×C127—42no—
63C22×C127—84no—
64C22×C127#12011yescharacteristic
65C4×C127—21no—
66C4×C127—22no—
67C4×C127—41no—
68C4×C127—42no—
69Dic127—81no—
70C2×C127—81no—
71C2×C127—82no—
72C2×C127#45311yescharacteristic
73C2×C127#45512yes—
74#1659—1271nomaximal, Sylow
75C127#165911yescharacteristic, complemented, Sylow
76H(3,𝔽4)D12711yescharacteristic
77#497—1271no—
78#489—1272no—
79#329—1271no—
80#329—1272no—
81C2×Q16—2541no—
82C2×SD16—2541no—
83C4⋊Q8—1271no—
84#124—22no—
85#124—21no—
86#124C2×D12712yes—
87#124C2×D12711yescharacteristic
88#120—22no—
89#120—21no—
90#103—2542no—
91#102—2542no—
92#97—1272no—
93#97—1271no—
94C24—21no—
95C2×Q8—2541no—
96C2×Q8C22⋊D12712yes—
97C2×Q8C22⋊D12711yescharacteristic
98C2×D4—21no—
99C2×D4—22no—
100C2×D4C22⋊D12712yes—
101C2×D4C22⋊D12711yescharacteristic
102Q16—5082no—
103SD16—5082no—
104C2×C8—2542no—
105C2×C8—2541no—
106C4⋊C4—2542no—
107C22⋊C4—22no—
108C22⋊C4—21no—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C42—21no—
113C42C22×D12711yescharacteristic
114C23—21no—
115C23—22no—
116C23—81no—
117C23—82no—
118Q8—41no—
119Q8—42no—
120Q8—5081no—
121Q8—5081no—
122D4—41no—
123D4—42no—
124D4—81no—
125D4—82no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—5081no—
129C2×C4C2×C22⋊D12712yes—
130C2×C4C2×C22⋊D12711yescharacteristic
131C8—5082no—
132C8—2542no—
133C22—41no—
134C22—42no—
135C22—42no—
136C22—42no—
137C22—42no—
138C22—84no—
139C22#1686117811yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—42no—
144C4—10161no—
145C2—81no—
146C2—82no—
147C2#2255004411yescharacteristic
148C2#2255005412yes—
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting, prime radicalH(3,𝔽4)×C127C2
socleC22×C127#120
hypercenterH(3,𝔽4)D127
hypocenterC127#1659
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #22550271C2semidirect
2. #22550235C2semidirect
3. #22549616C2semidirect
4. #22549616C2semidirect
8. #16861209C22semidirect
11. #16861120C22semidirect
12. #16861120C22semidirect
75. C127#1659semidirect

Maximal subgroups

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

subgroupindex
#225502712
#225502352
#225496162
H(3,𝔽4)×C1272
#1659127