Finite Group Databasebeta

← #30638200

Subgroups — #30638200

Order 16256.

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

normal subgroups only

This group has 168 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#22550161C211yesmaximal, characteristic, complemented
3#22550007C211yesmaximal, characteristic, complemented
4#22549628C211yesmaximal, characteristic, complemented
5#22549621C211yesmaximal, characteristic, complemented
6#22549620C211yesmaximal, characteristic
7#460×C127C211yesmaximal, characteristic
8C2×#14523236—21no—
9C2×#14523236—21no—
10#16861209C2211yescharacteristic
11#16861140—21no—
12C4⋊(C127⋊C8)—21no—
13C4⋊(C127⋊C8)C2211yescharacteristic
14C4⋊(C127⋊C8)C2211yescharacteristic
15#16861133—21no—
16#16861120—21no—
17C4×C127⋊C8C2211yescharacteristic
18C2×SD16×C127—21no—
19#127×C127C2211yescharacteristic
20C4×Q8×C127C2211yescharacteristic
21C4⋊C8×C127C2211yescharacteristic
22#102×C127—21no—
23C2×Dic254—21no—
24#14523236—42no—
25#14523236—42no—
26C2×C127⋊C8—21no—
27C2×C127⋊C8—21no—
28C2×C127⋊C8D411yescharacteristic
29C2×C127⋊C8D411yescharacteristic
30C4⋊Dic127—21no—
31C4⋊Dic127—21no—
32C2×Q8×C127D411yescharacteristic
33C2×D4×C127—21no—
34C2×D4×C127—21no—
35C2×D4×C127—21no—
36SD16×C127—42no—
37C2×C8×C127—21no—
38C4⋊C4×C127—21no—
39C4⋊C4×C127D411yescharacteristic
40C42×C127—21no—
41C42×C127C2311yescharacteristic
42Dic254—41no—
43Dic254—41no—
44C2×Dic127—41no—
45C127⋊C8—21no—
46C127⋊C8—41no—
47C127⋊C8—41no—
48C127⋊C8SD1612yescomplemented
49C23×C127—41no—
50Q8×C127—21no—
51Q8×C127—21no—
52D4×C127—41no—
53D4×C127—41no—
54D4×C127—41no—
55D4×C127—41no—
56D4×C127—81no—
57C2×C4×C127—21no—
58C2×C4×C127—21no—
59C2×C4×C127C2×D411yescharacteristic
60C2×C4×C127C2×D411yescharacteristic
61C2×C4×C127#4111yescharacteristic
62C8×C127—41no—
63C22×C127—81no—
64C22×C127—81no—
65C22×C127—81no—
66C22×C127#12111yescharacteristic
67C4×C127—21no—
68C4×C127—41no—
69C4×C127—41no—
70C4×C127C2×SD1611yescharacteristic
71C4×C127AGL(1,ℤ/8)11yescharacteristic
72C4×C127AGL(1,ℤ/8)11yescharacteristic
73C4×C127#13711yescharacteristic
74Dic127—81no—
75C2×C127—161no—
76C2×C127#46011yescharacteristic
77C2×C127#46811yescharacteristic
78C2×C127#47111yescharacteristic
79#1686—1271nomaximal, Sylow
80C127#168611yescharacteristic, complemented, Sylow
81#492—1271no—
82#475—1271no—
83#460D12711yescharacteristic
84#446—1271no—
85#335—1271no—
86#332—1271no—
87#331—1271no—
88C2×SD16—21no—
89C2×SD16—2541no—
90C2×SD16—2541no—
91C4⋊Q8—1271no—
92#127C2×D12711yescharacteristic
93C4×Q8C2×D12711yescharacteristic
94#106—2541no—
95C4⋊C8—1271no—
96C4⋊C8—1271no—
97C4⋊C8—2541no—
98C4⋊C8C2×D12711yescharacteristic
99#103—2541no—
100#102—21no—
101#97—2541no—
102C4×C8—1271no—
103C2×Q8—2541no—
104C2×Q8C22⋊D12711yescharacteristic
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—21no—
108SD16—42no—
109SD16—5082no—
110SD16—5082no—
111C2×C8—21no—
112C2×C8—1271no—
113C2×C8—1271no—
114C2×C8—2541no—
115C2×C8—2541no—
116C4⋊C4—21no—
117C4⋊C4—2541no—
118C4⋊C4—2541no—
119C4⋊C4C22⋊D12711yescharacteristic
120C42—21no—
121C42C22×D12711yescharacteristic
122C23—41no—
123Q8—21no—
124Q8—21no—
125Q8—5081no—
126Q8—5081no—
127D4—41no—
128D4—41no—
129D4—41no—
130D4—41no—
131D4—81no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—5081no—
135C2×C4D4×D12711yescharacteristic
136C2×C4C2×C22⋊D12711yescharacteristic
137C2×C4#1452324711yescharacteristic
138C8—41no—
139C8—1272no—
140C8—5081no—
141C8—5081no—
142C8—2541no—
143C22—81no—
144C22—81no—
145C22—81no—
146C22#1686118111yescharacteristic
147C4—21no—
148C4—41no—
149C4—41no—
150C4—10161no—
151C4SD16×D12711yescharacteristic
152C4#1686123511yescharacteristic
153C4#1686123611yescharacteristic
154C4#1686123911yescharacteristic
155C2—161no—
156C2#2255007411yescharacteristic
157C2#2255012011yescharacteristic
158C2#2255013711yescharacteristic
subgroupG/H
centerC22#16861181
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#460×C127C2
socleC22×C127#121
prime radical#127×C127C22
hypercenter#460D127
hypocenterC127#1686
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. #22550161C2semidirect
3. #22550007C2semidirect
4. #22549628C2semidirect
5. #22549621C2semidirect
48. C127⋊C8SD16semidirect
80. C127#1686semidirect

Maximal subgroups

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

subgroupindex
#225502482
#225501612
#225500072
#225496282
#225496212
#225496202
#460×C1272
#1686127