Finite Group Databasebeta

← #30638103

Subgroups — #30638103

Order 16256.

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

normal subgroups only

This group has 166 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#22550272C211yesmaximal, characteristic, complemented
2#22550166C211yesmaximal, characteristic
3D4×C127⋊C8C211yesmaximal, characteristic
4#22549626C211yesmaximal, characteristic, complemented
5#22549624C211yesmaximal, characteristic, complemented
6#22549622C211yesmaximal, characteristic, complemented
7#461×C127C211yesmaximal, characteristic
8C22×C127⋊C8—21no—
9#16861209C2211yescharacteristic
10#16861140—21no—
11#16861140—21no—
12#16861140—21no—
13C4⋊(C127⋊C8)C2211yescharacteristic
14C4⋊(C127⋊C8)C2211yescharacteristic
15#16861130—21no—
16C22⋊(C127⋊C8)—21no—
17C4×C127⋊C8C2211yescharacteristic
18C2×SD16×C127—21no—
19C4⋊Q8×C127C2211yescharacteristic
20C4×D4×C127C2211yescharacteristic
21C4⋊C8×C127C2211yescharacteristic
22#103×C127—21no—
23C2×Dic254—21no—
24C2×C127⋊C8—21no—
25C2×C127⋊C8—21no—
26C2×C127⋊C8—42no—
27C2×C127⋊C8D411yescharacteristic
28C2×C127⋊C8D411yescharacteristic
29C4⋊Dic127—21no—
30C4⋊Dic127—21no—
31C2×Q8×C127—21no—
32C2×D4×C127D411yescharacteristic
33C22×C4×C127—21no—
34SD16×C127—42no—
35C2×C8×C127—21no—
36C4⋊C4×C127—21no—
37C4⋊C4×C127—21no—
38C4⋊C4×C127D411yescharacteristic
39C22⋊C4×C127—21no—
40C42×C127C2311yescharacteristic
41Dic254—81no—
42C2×Dic127—41no—
43C127⋊C8—21no—
44C127⋊C8—41no—
45C127⋊C8—41no—
46C127⋊C8SD1612yescomplemented
47C23×C127—21no—
48Q8×C127—41no—
49Q8×C127—41no—
50D4×C127—21no—
51D4×C127SD1612yes—
52C2×C4×C127—21no—
53C2×C4×C127—41no—
54C2×C4×C127—41no—
55C2×C4×C127—41no—
56C2×C4×C127C2×D411yescharacteristic
57C2×C4×C127C2×D411yescharacteristic
58C2×C4×C127#4111yescharacteristic
59C8×C127—41no—
60C22×C127—22no—
61C22×C127—41no—
62C22×C127—41no—
63C22×C127#12111yescharacteristic
64C4×C127—21no—
65C4×C127—41no—
66C4×C127—81no—
67C4×C127C2×SD1611yescharacteristic
68C4×C127C2×SD1611yescharacteristic
69C4×C127#13511yescharacteristic
70C4×C127#13711yescharacteristic
71Dic127—81no—
72C2×C127—42no—
73C2×C127#46111yescharacteristic
74C2×C127#46511yescharacteristic
75C2×C127#47111yescharacteristic
76#1671—1271nomaximal, Sylow
77C127#167111yescharacteristic, complemented, Sylow
78#498—1271no—
79#476—1271no—
80#461D12711yescharacteristic
81C8×D4—1271no—
82#334—1271no—
83#333—1271no—
84#332—1271no—
85C2×SD16—21no—
86C22×C8—2541no—
87C4⋊Q8—1271no—
88C4⋊Q8C2×D12711yescharacteristic
89C4×D4C2×D12711yescharacteristic
90#106—2541no—
91#106—2541no—
92#106—2541no—
93C4⋊C8—1271no—
94C4⋊C8—1271no—
95C4⋊C8C2×D12711yescharacteristic
96#103—21no—
97#102—2541no—
98C22⋊C8—2541no—
99C4×C8—1271no—
100C2×Q8—21no—
101C2×Q8—2541no—
102C2×D4C22⋊D12711yescharacteristic
103C22×C4—21no—
104SD16—42no—
105C2×C8—21no—
106C2×C8—1271no—
107C2×C8—1271no—
108C2×C8—5082no—
109C2×C8—2541no—
110C2×C8—2541no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—2541no—
114C4⋊C4—2541no—
115C4⋊C4C22⋊D12711yescharacteristic
116C22⋊C4—21no—
117C42C22×D12711yescharacteristic
118C23—21no—
119Q8—41no—
120Q8—41no—
121Q8—10161no—
122D4—21no—
123D4#1452323612yes—
124C2×C4—21no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—5081no—
129C2×C4D4×D12711yescharacteristic
130C2×C4C2×C22⋊D12711yescharacteristic
131C2×C4#1452324711yescharacteristic
132C8—41no—
133C8—1272no—
134C8—5081no—
135C8—5081no—
136C8—2541no—
137C22—22no—
138C22—41no—
139C22—41no—
140C22#1686118111yescharacteristic
141C4—21no—
142C4—41no—
143C4—81no—
144C4—10161no—
145C4SD16×D12711yescharacteristic
146C4C2×#1452323611yescharacteristic
147C4#1686122911yescharacteristic
148C4#1686123811yescharacteristic
149C2—42no—
150C2#2255007911yescharacteristic
151C2#2255010311yescharacteristic
152C2#2255014111yescharacteristic
subgroupG/H
centerC22#16861181
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#461×C127C2
socleC22×C127#121
prime radicalC2×D4×C127D4
hypercenter#461D127
hypocenterC127#1671
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #22550272C2semidirect
4. #22549626C2semidirect
5. #22549624C2semidirect
6. #22549622C2semidirect
46. C127⋊C8SD16semidirect
77. C127#1671semidirect

Maximal subgroups

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

subgroupindex
#225502722
#225501662
D4×C127⋊C82
#225496262
#225496242
#225496222
#461×C1272
#1671127