Finite Group Databasebeta

← #30638268

Subgroups — #30638268

Order 16256.

156 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 51 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#22550222C211yesmaximal, characteristic
2#22550171C211yesmaximal, characteristic, complemented
3#22550007C211yesmaximal, characteristic, complemented
4#22549628C211yesmaximal, characteristic, complemented
5#22549624C211yesmaximal, characteristic, complemented
6#22549620C211yesmaximal, characteristic
7#460×C127C211yesmaximal, characteristic
8#16861209C2211yescharacteristic
9#16861142—21no—
10C4⋊(C127⋊C8)—21no—
11C4⋊(C127⋊C8)C2211yescharacteristic
12C4⋊(C127⋊C8)C2211yescharacteristic
13#16861133—21no—
14#16861130—21no—
15#16861130—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—
24C2×C127⋊C8—21no—
25C2×C127⋊C8—21no—
26C2×C127⋊C8D411yescharacteristic
27C2×C127⋊C8D411yescharacteristic
28C4⋊Dic127—21no—
29C4⋊Dic127—21no—
30C2×Q8×C127D411yescharacteristic
31C2×D4×C127—21no—
32C2×D4×C127—21no—
33C2×D4×C127—21no—
34SD16×C127—42no—
35C2×C8×C127—21no—
36C4⋊C4×C127—21no—
37C4⋊C4×C127D411yescharacteristic
38C42×C127—21no—
39C42×C127C2311yescharacteristic
40Dic254—81no—
41C2×Dic127—41no—
42C127⋊C8—21no—
43C127⋊C8—21no—
44C127⋊C8—41no—
45C127⋊C8—41no—
46C23×C127—41no—
47Q8×C127—21no—
48Q8×C127—21no—
49D4×C127—41no—
50D4×C127—41no—
51D4×C127—41no—
52D4×C127—41no—
53D4×C127—81no—
54C2×C4×C127—21no—
55C2×C4×C127—21no—
56C2×C4×C127C2×D411yescharacteristic
57C2×C4×C127C2×D411yescharacteristic
58C2×C4×C127#4111yescharacteristic
59C8×C127—41no—
60C22×C127—81no—
61C22×C127—81no—
62C22×C127—81no—
63C22×C127#12111yescharacteristic
64C4×C127—21no—
65C4×C127—41no—
66C4×C127—41no—
67C4×C127#13511yescharacteristic
68C4×C127AGL(1,ℤ/8)11yescharacteristic
69C4×C127#13711yescharacteristic
70C4×C127#13711yescharacteristic
71Dic127—81no—
72C2×C127—161no—
73C2×C127#46411yescharacteristic
74C2×C127#46811yescharacteristic
75C2×C127#47211yescharacteristic
76#1698—1271nomaximal, Sylow
77C127#169811yescharacteristic, complemented, Sylow
78#486—1271no—
79#477—1271no—
80#460D12711yescharacteristic
81#446—1271no—
82#335—1271no—
83#333—1271no—
84#331—1271no—
85C2×SD16—21no—
86C4⋊Q8—1271no—
87#127C2×D12711yescharacteristic
88C4×Q8C2×D12711yescharacteristic
89#107—2541no—
90C4⋊C8—1271no—
91C4⋊C8—1271no—
92C4⋊C8—2541no—
93C4⋊C8C2×D12711yescharacteristic
94#103—2541no—
95#102—21no—
96#102—2541no—
97#102—2541no—
98#97—2541no—
99C4×C8—1271no—
100C2×Q8—2541no—
101C2×Q8C22⋊D12711yescharacteristic
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—21no—
105SD16—42no—
106C2×C8—21no—
107C2×C8—1271no—
108C2×C8—1271no—
109C2×C8—2541no—
110C2×C8—2541no—
111C4⋊C4—21no—
112C4⋊C4—2541no—
113C4⋊C4—2541no—
114C4⋊C4C22⋊D12711yescharacteristic
115C42—21no—
116C42C22×D12711yescharacteristic
117C23—41no—
118Q8—21no—
119Q8—21no—
120Q8—10161no—
121D4—41no—
122D4—41no—
123D4—41no—
124D4—41no—
125D4—81no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—5081no—
129C2×C4D4×D12711yescharacteristic
130C2×C4C2×C22⋊D12711yescharacteristic
131C2×C4#1452324711yescharacteristic
132C8—41no—
133C8—5081no—
134C8—5081no—
135C8—2541no—
136C8—2541no—
137C22—81no—
138C22—81no—
139C22—81no—
140C22#1686118111yescharacteristic
141C4—21no—
142C4—41no—
143C4—41no—
144C4—10161no—
145C4#1686123011yescharacteristic
146C4#1686123511yescharacteristic
147C4#1686123811yescharacteristic
148C4#1686123911yescharacteristic
149C2—161no—
150C2#2255009611yescharacteristic
151C2#2255012011yescharacteristic
152C2#2255014411yescharacteristic
subgroupG/H
centerC22#16861181
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#460×C127C2
socleC22×C127#121
prime radical#127×C127C22
hypercenter#460D127
hypocenterC127#1698
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. #22550171C2semidirect
3. #22550007C2semidirect
4. #22549628C2semidirect
5. #22549624C2semidirect
77. C127#1698semidirect

Maximal subgroups

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

subgroupindex
#225502222
#225501712
#225500072
#225496282
#225496242
#225496202
#460×C1272
#1698127