Finite Group Databasebeta

← #30638181

Subgroups — #30638181

Order 16256.

160 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550227C211yesmaximal, characteristic, complemented
2#22550166C211yesmaximal, characteristic
3#22550156C211yesmaximal, characteristic
4#22549629C211yesmaximal, characteristic, complemented
5#22549624C211yesmaximal, characteristic, complemented
6#22549623C211yesmaximal, characteristic, complemented
7C8×D4×C127C211yesmaximal, characteristic
8#16861209C2211yescharacteristic
9#16861209C2211yescharacteristic
10#16861142—21no—
11#16861140—21no—
12C4⋊(C127⋊C8)C2211yescharacteristic
13C4⋊(C127⋊C8)C2211yescharacteristic
14#16861134—21no—
15#16861134—21no—
16#16861130—21no—
17#16861130—21no—
18C22×C8×C127—21no—
19C4×D4×C127C2211yescharacteristic
20C4⋊C8×C127C2211yescharacteristic
21C22⋊C8×C127—21no—
22C4×C8×C127C2211yescharacteristic
23C2×Dic254—21no—
24C2×Dic254—21no—
25C2×C127⋊C8—21no—
26C2×C127⋊C8—21no—
27C4⋊Dic127—21no—
28C4⋊Dic127—21no—
29C4⋊Dic127—21no—
30C4⋊Dic127—21no—
31C2×D4×C127D411yescharacteristic
32C22×C4×C127—21no—
33C2×C8×C127—21no—
34C2×C8×C127—42no—
35C2×C8×C127D411yescharacteristic
36C2×C8×C127D411yescharacteristic
37C4⋊C4×C127D411yescharacteristic
38C22⋊C4×C127—21no—
39C42×C127C2311yescharacteristic
40Dic254—41no—
41Dic254—41no—
42Dic254—81no—
43C2×Dic127—41no—
44C2×Dic127—41no—
45C127⋊C8—41no—
46C127⋊C8—41no—
47C23×C127—21no—
48D4×C127—21no—
49D4×C127Q1612yes—
50C2×C4×C127—21no—
51C2×C4×C127—41no—
52C2×C4×C127—41no—
53C2×C4×C127C2×D411yescharacteristic
54C2×C4×C127C2×D411yescharacteristic
55C2×C4×C127#4111yescharacteristic
56C8×C127—21no—
57C8×C127—21no—
58C8×C127—41no—
59C22×C127—22no—
60C22×C127—41no—
61C22×C127—41no—
62C22×C127#12111yescharacteristic
63C4×C127—21no—
64C4×C127—41no—
65C4×C127C2×Q1611yescharacteristic
66C4×C127#13511yescharacteristic
67C4×C127#13511yescharacteristic
68C4×C127AGL(1,ℤ/8)11yescharacteristic
69Dic127—81no—
70Dic127—81no—
71C2×C127—42no—
72C2×C127#46311yescharacteristic
73C2×C127#46711yescharacteristic
74C2×C127#47111yescharacteristic
75#1683—1271nomaximal, Sylow
76C127#168311yescharacteristic, complemented, Sylow
77#487—1271no—
78#476—1271no—
79#474—1271no—
80C8×D4D12711yescharacteristic
81#335—1271no—
82#333—1271no—
83#332—1271no—
84C22×C8—21no—
85C4⋊Q8—1271no—
86C4⋊Q8—1271no—
87C4×D4C2×D12711yescharacteristic
88#107—2541no—
89#106—2541no—
90C4⋊C8—1271no—
91C4⋊C8—1271no—
92C4⋊C8C2×D12711yescharacteristic
93#103—2541no—
94#103—2541no—
95#102—2541no—
96#102—2541no—
97C22⋊C8—21no—
98C4×C8C2×D12711yescharacteristic
99C2×Q8—2541no—
100C2×Q8—2541no—
101C2×D4D50811yescharacteristic
102C22×C4—21no—
103C2×C8—21no—
104C2×C8—42no—
105C2×C8—2541no—
106C2×C8—2541no—
107C2×C8C22⋊D12711yescharacteristic
108C2×C8C22⋊D12711yescharacteristic
109C4⋊C4—2541no—
110C4⋊C4—2541no—
111C4⋊C4—2541no—
112C4⋊C4—2541no—
113C4⋊C4D50811yescharacteristic
114C22⋊C4—21no—
115C42C22×D12711yescharacteristic
116C23—21no—
117Q8—10161no—
118Q8—5081no—
119Q8—5081no—
120D4—21no—
121D4Dic50812yes—
122C2×C4—21no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—5081no—
126C2×C4—5081no—
127C2×C4C2×C22⋊D12711yescharacteristic
128C2×C4C2×D50811yescharacteristic
129C2×C4#1452324811yescharacteristic
130C8—21no—
131C8—21no—
132C8—41no—
133C8—5081no—
134C8—5081no—
135C22—22no—
136C22—41no—
137C22—41no—
138C22#1686118311yescharacteristic
139C4—21no—
140C4—41no—
141C4—10161no—
142C4—10161no—
143C4C2×Dic50811yescharacteristic
144C4#1686122911yescharacteristic
145C4#1686123111yescharacteristic
146C4#1686123511yescharacteristic
147C2—42no—
148C2#2255009111yescharacteristic
149C2#2255011711yescharacteristic
150C2#2255013911yescharacteristic
subgroupG/H
centerC22#16861183
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingC8×D4×C127C2
socleC22×C127#121
prime radicalC2×D4×C127D4
hypercenterC8×D4D127
hypocenterC127#1683
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #22550227C2semidirect
4. #22549629C2semidirect
5. #22549624C2semidirect
6. #22549623C2semidirect
76. C127#1683semidirect

Maximal subgroups

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

subgroupindex
#225502272
#225501662
#225501562
#225496292
#225496242
#225496232
C8×D4×C1272
#1683127