Finite Group Databasebeta

← #30638403

Subgroups — #30638403

Order 16256.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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#22550281C211yesmaximal, characteristic, complemented
2#22550279C211yesmaximal, characteristic, complemented
3#22550255C211yesmaximal, characteristic, complemented
4#22550248C211yesmaximal, characteristic
5#22550222C211yesmaximal, characteristic
6#22549598C211yesmaximal, characteristic, complemented
7#486×C127C211yesmaximal, characteristic
8C2×#14523238—22no—
9C2×#14523236—22no—
10#16861209C2211yescharacteristic
11#16861209C2211yescharacteristic
12#16861143—22no—
13#16861142—22no—
14#16861130—22no—
15C4×C127⋊C8C2211yescharacteristic
16C4×C127⋊C8C2211yescharacteristic
17C4⋊Q8×C127C2211yescharacteristic
18#127×C127C2211yescharacteristic
19#102×C127—22no—
20C4×C8×C127C2211yescharacteristic
21C2×Dic254—21no—
22C2×Dic254—21no—
23#14523238—44no—
24#14523236—44no—
25C2×C127⋊C8D412yes—
26C2×C127⋊C8D412yes—
27C4⋊Dic127—21no—
28C4⋊Dic127—21no—
29C4⋊Dic127—21no—
30C4⋊Dic127—21no—
31C2×Q8×C127—21no—
32C2×D4×C127—21no—
33C2×D4×C127—21no—
34C2×D4×C127—21no—
35C2×C8×C127D412yes—
36C4⋊C4×C127—21no—
37C4⋊C4×C127—21no—
38C42×C127C2311yescharacteristic
39Dic254—42no—
40Dic254—42no—
41C2×Dic127—41no—
42C2×Dic127—41no—
43C127⋊C8—22no—
44C127⋊C8D814yes—
45C23×C127—41no—
46Q8×C127—42no—
47D4×C127—42no—
48D4×C127—42no—
49D4×C127—42no—
50C2×C4×C127—41no—
51C2×C4×C127C2×D411yescharacteristic
52C2×C4×C127C2×D411yescharacteristic
53C2×C4×C127C2×D411yescharacteristic
54C8×C127—22no—
55C22×C127—81no—
56C22×C127—81no—
57C22×C127—81no—
58C22×C127#12711yescharacteristic
59C4×C127—81no—
60C4×C127C2×D812yes—
61C4×C127#13712yes—
62C4×C127#13712yes—
63Dic127—81no—
64Dic127—81no—
65C2×C127—161no—
66C2×C127#49311yescharacteristic
67C2×C127#49711yescharacteristic
68C2×C127#49711yescharacteristic
69#1722—1271nomaximal, Sylow
70C127#172211yescharacteristic, complemented, Sylow
71#500—1271no—
72#500—1271no—
73#494—1271no—
74#492—1271no—
75#486—1271no—
76#486D12711yescharacteristic
77#322—1271no—
78C2×Q16—2542no—
79C2×SD16—2542no—
80C4⋊Q8—1271no—
81C4⋊Q8—1271no—
82C4⋊Q8C2×D12711yescharacteristic
83#127C2×D12711yescharacteristic
84#107—2542no—
85#107—2542no—
86#102—22no—
87#102—2542no—
88C4×C8—1271no—
89C4×C8—1271no—
90C4×C8C2×D12711yescharacteristic
91C2×Q8—21no—
92C2×Q8—2541no—
93C2×Q8—2541no—
94C2×D4—21no—
95C2×D4—21no—
96C2×D4—21no—
97Q16—5084no—
98SD16—5084no—
99C2×C8—1272no—
100C2×C8—1272no—
101C2×C8C22⋊D12712yes—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—2541no—
105C4⋊C4—2541no—
106C4⋊C4—2541no—
107C4⋊C4—2541no—
108C42C22×D12711yescharacteristic
109C23—41no—
110Q8—42no—
111Q8—5082no—
112Q8—5082no—
113D4—42no—
114D4—42no—
115D4—42no—
116C2×C4—41no—
117C2×C4—5081no—
118C2×C4—5081no—
119C2×C4D4×D12711yescharacteristic
120C2×C4D4×D12711yescharacteristic
121C2×C4C2×C22⋊D12711yescharacteristic
122C8—22no—
123C8—1274no—
124C8—2542no—
125C22—81no—
126C22—81no—
127C22—81no—
128C22#1686120511yescharacteristic
129C4—81no—
130C4—10161no—
131C4—10161no—
132C4D8×D12712yes—
133C4#1686123812yes—
134C4#1686123912yes—
135C2—161no—
136C2#2255025111yescharacteristic
137C2#2255026711yescharacteristic
138C2#2255027111yescharacteristic
subgroupG/H
centerC22#16861205
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#486×C127C2
socleC22×C127#127
prime radical#127×C127C22
hypercenter#486D127
hypocenterC127#1722
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #22550281C2semidirect
2. #22550279C2semidirect
3. #22550255C2semidirect
6. #22549598C2semidirect
70. C127#1722semidirect

Maximal subgroups

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

subgroupindex
#225502812
#225502792
#225502552
#225502482
#225502222
#225495982
#486×C1272
#1722127