Finite Group Databasebeta

← #30640858

Subgroups — #30640858

Order 16256.

144 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 47 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
1C2×#16861153C211yesmaximal, characteristic, complemented
2#22550293C211yesmaximal, characteristic, complemented
3#22550151C211yesmaximal, characteristic, complemented
4#22549705C211yesmaximal, characteristic, complemented
5#22549691C211yesmaximal, characteristic, complemented
6#22549689C211yesmaximal, characteristic, complemented
7#472×C127C211yesmaximal, characteristic
8#16861239—21no—
9C2×Dic508C2211yescharacteristic
10#16861153—21no—
11#16861153—21no—
12#16861149—21no—
13#16861149C2211yescharacteristic
14C2×C127⋊C16—21no—
15C2×C127⋊C16C2211yescharacteristic
16#16861145C2211yescharacteristic
17#16861127—21no—
18AGL(1,ℤ/8)×C127—21no—
19C2×D8×C127C2211yescharacteristic
20#131×C127C2211yescharacteristic
21#108×C127C2211yescharacteristic
22#100×C127—21no—
23C2×Dic254—21no—
24Dic508—21no—
25Dic508—21no—
26#14523238—41no—
27#14523236—41no—
28#14523232—21no—
29C127⋊C16—21no—
30C127⋊C16D411yescharacteristic
31C127⋊C16D411yescharacteristic
32#41×C127D411yescharacteristic
33C2×D4×C127—21no—
34SD16×C127—41no—
35D8×C127—21no—
36D8×C127—21no—
37D8×C127—41no—
38M16×C127—21no—
39M16×C127—21no—
40M16×C127D411yescharacteristic
41C2×C8×C127—21no—
42C2×C8×C127C2311yescharacteristic
43Dic254—41no—
44Dic254—41no—
45C2×Dic127—41no—
46C127⋊C8—41no—
47C23×C127—41no—
48Q8×C127D811yescharacteristic
49D4×C127—21no—
50D4×C127—41no—
51D4×C127—41no—
52D4×C127D811yescharacteristic
53C2×C4×C127—21no—
54C2×C4×C127C2×D411yescharacteristic
55C8×C127—21no—
56C8×C127—41no—
57C8×C127C2×D411yescharacteristic
58C8×C127#4111yescharacteristic
59C22×C127—21no—
60C22×C127—81no—
61C22×C127—161no—
62C22×C127#13511yescharacteristic
63C4×C127—21no—
64C4×C127#12111yescharacteristic
65C4×C127C2×D811yescharacteristic
66Dic127—81no—
67C2×C127—21no—
68C2×C127—41no—
69C2×C127—161no—
70C2×C127#46611yescharacteristic
71#2223—1271nomaximal, Sylow
72C127#222311yescharacteristic, complemented, Sylow
73C2×SD32—1271no—
74#504—1271no—
75#473—1271no—
76#472D12711yescharacteristic
77#368—1271no—
78#362—1271no—
79#361—1271no—
80#137—2541no—
81AGL(1,ℤ/8)—21no—
82C2×Q16—1271no—
83C2×D8C2×D12711yescharacteristic
84#131C2×D12711yescharacteristic
85SD32—2541no—
86SD32—2541no—
87M32—1271no—
88M32—2541no—
89C2×C16—1271no—
90C2×C16—2541no—
91#108—1271no—
92#108C2×D12711yescharacteristic
93#101—2541no—
94#100—21no—
95#41C22⋊D12711yescharacteristic
96C2×Q8—2541no—
97C2×D4—21no—
98Q16—5081no—
99Q16—2541no—
100Q16—2541no—
101SD16—41no—
102SD16—5081no—
103D8—21no—
104D8—21no—
105D8—41no—
106M16—21no—
107M16—21no—
108M16—2541no—
109M16C22⋊D12711yescharacteristic
110C2×C8—21no—
111C2×C8C22×D12711yescharacteristic
112C16—1271no—
113C16—1271no—
114C16—2541no—
115C23—41no—
116Q8—5081no—
117Q8—5081no—
118Q8#1452323311yescharacteristic
119D4—21no—
120D4—41no—
121D4—41no—
122D4#1452323311yescharacteristic
123C2×C4—21no—
124C2×C4—5081no—
125C2×C4C2×C22⋊D12711yescharacteristic
126C8—21no—
127C8—41no—
128C8—5081no—
129C8D4×D12711yescharacteristic
130C8#1452324711yescharacteristic
131C22—21no—
132C22—81no—
133C22—161no—
134C22#1686122911yescharacteristic
135C4—21no—
136C4—10161no—
137C4#1686118111yescharacteristic
138C4C2×#1452323311yescharacteristic
139C2—21no—
140C2—41no—
141C2—161no—
142C2#2255011011yescharacteristic
subgroupG/H
centerC2#22550110
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting, prime radical#472×C127C2
socleC2×C127#466
hypercenter#472D127
hypocenterC127#2223
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #472

Complements

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

NH ≅ G/Ntype
1. C2×#16861153C2semidirect
2. #22550293C2semidirect
3. #22550151C2semidirect
4. #22549705C2semidirect
5. #22549691C2semidirect
6. #22549689C2semidirect
72. C127#2223semidirect

Maximal subgroups

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

subgroupindex
C2×#168611532
#225502932
#225501512
#225497052
#225496912
#225496892
#472×C1272
#2223127