Finite Group Databasebeta

← #30638077

Subgroups — #30638077

Order 16256.

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

normal subgroups only

This group has 172 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#22550248C211yesmaximal, characteristic
2#22550162C211yesmaximal, characteristic, complemented
3#22550073C211yesmaximal, characteristic
4#22550000C211yesmaximal, characteristic, complemented
5#22549626C211yesmaximal, characteristic, complemented
6#22549623C211yesmaximal, characteristic, complemented
7#331×C127C211yesmaximal, characteristic
8C2×#14523236—21no—
9C2×#14523236—21no—
10C2×#14523236—21no—
11#16861209C2211yescharacteristic
12C4×Dic254C2211yescharacteristic
13#16861141—21no—
14C4⋊(C127⋊C8)C2211yescharacteristic
15Dic127⋊C8—21no—
16#16861134—21no—
17#16861130—21no—
18C4×C127⋊C8C2211yescharacteristic
19C8×Dic127—21no—
20#127×C127C2211yescharacteristic
21C4⋊C8×C127C2211yescharacteristic
22C4⋊C8×C127C2211yescharacteristic
23C2×Dic254—21no—
24C2×Dic254D411yescharacteristic
25#14523236—42no—
26#14523236—42no—
27#14523236—42no—
28C2×C127⋊C8—21no—
29C2×C127⋊C8D411yescharacteristic
30C2×C127⋊C8D411yescharacteristic
31C4⋊Dic127—21no—
32C4⋊Dic127—21no—
33C4⋊Dic127D411yescharacteristic
34Dic127⋊C4—21no—
35C4×Dic127—21no—
36C2×D4×C127—21no—
37C2×D4×C127—21no—
38C2×D4×C127—21no—
39C2×C8×C127—21no—
40C2×C8×C127—21no—
41C42×C127C2311yescharacteristic
42Dic254—21no—
43Dic254—41no—
44Dic254—41no—
45Dic254SD1612yescomplemented
46C2×Dic127—21no—
47C2×Dic127—21no—
48C2×Dic127—41no—
49C127⋊C8—21no—
50C127⋊C8—41no—
51C127⋊C8SD1612yescomplemented
52C23×C127—41no—
53D4×C127—41no—
54D4×C127—41no—
55D4×C127—41no—
56D4×C127—41no—
57D4×C127—81no—
58C2×C4×C127C2×D411yescharacteristic
59C2×C4×C127C2×D411yescharacteristic
60C2×C4×C127#4111yescharacteristic
61C8×C127—41no—
62C8×C127—41no—
63C22×C127—81no—
64C22×C127—81no—
65C22×C127—81no—
66C22×C127#12111yescharacteristic
67C4×C127—21no—
68C4×C127C2×SD1611yescharacteristic
69C4×C127C2×SD1611yescharacteristic
70C4×C127#13511yescharacteristic
71C4×C127AGL(1,ℤ/8)11yescharacteristic
72Dic127—22no—
73Dic127—41no—
74Dic127—81no—
75C2×C127—161no—
76C2×C127#46011yescharacteristic
77C2×C127#46511yescharacteristic
78C2×C127#47111yescharacteristic
79#1666—1271nomaximal, Sylow
80C127#166611yescharacteristic, complemented, Sylow
81#492—1271no—
82#475—1271no—
83#460—1271no—
84C8×Q8—1271no—
85#334—1271no—
86#332—1271no—
87#331D12711yescharacteristic
88C2×SD16—2541no—
89C2×SD16—2541no—
90C2×SD16—2541no—
91C4⋊Q8—1271no—
92#127C2×D12711yescharacteristic
93C4×Q8—1271no—
94#106—2541no—
95C4⋊C8—1271no—
96C4⋊C8—2541no—
97C4⋊C8C2×D12711yescharacteristic
98C4⋊C8C2×D12711yescharacteristic
99#103—2541no—
100#102—2541no—
101C4×C8—1271no—
102C4×C8—2541no—
103C2×Q8—1271no—
104C2×Q8—2541no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—21no—
108SD16—5082no—
109SD16—5082no—
110SD16—5082no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—1271no—
114C2×C8—1271no—
115C2×C8—2541no—
116C4⋊C4—1271no—
117C4⋊C4—2541no—
118C4⋊C4—2541no—
119C4⋊C4—2541no—
120C42—2541no—
121C42C22×D12711yescharacteristic
122C23—41no—
123Q8—1272no—
124Q8—5081no—
125Q8—5081no—
126Q8—2541no—
127D4—41no—
128D4—41no—
129D4—41no—
130D4—41no—
131D4—81no—
132C2×C4—5081no—
133C2×C4—2541no—
134C2×C4—2541no—
135C2×C4D4×D12711yescharacteristic
136C2×C4D4×D12711yescharacteristic
137C2×C4#1452324811yescharacteristic
138C8—41no—
139C8—41no—
140C8—1272no—
141C8—5081no—
142C8—2541no—
143C22—81no—
144C22—81no—
145C22—81no—
146C22#1686118211yescharacteristic
147C4—21no—
148C4—10161no—
149C4—5081no—
150C4—2542no—
151C4SD16×D12711yescharacteristic
152C4SD16×D12711yescharacteristic
153C4#1686122811yescharacteristic
154C4#1686123311yescharacteristic
155C2—161no—
156C2#2255007511yescharacteristic
157C2#2255010411yescharacteristic
158C2#2255013811yescharacteristic
subgroupG/H
centerC22#16861182
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#331×C127C2
socleC22×C127#121
prime radical#127×C127C22
hypercenter#331D127
hypocenterC127#1666
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #22550162C2semidirect
4. #22550000C2semidirect
5. #22549626C2semidirect
6. #22549623C2semidirect
45. Dic254SD16semidirect
51. C127⋊C8SD16semidirect
80. C127#1666semidirect

Maximal subgroups

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

subgroupindex
#225502482
#225501622
#225500732
#225500002
#225496262
#225496232
#331×C1272
#1666127