Finite Group Databasebeta

← #30637784

Subgroups — #30637784

Order 16256.

336 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 65 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#22550252C211yesmaximal, characteristic, complemented
2#22550071C212yesmaximal, complemented
3#22549620C211yesmaximal, characteristic, complemented
4D4⋊(C127⋊C8)C212yesmaximal, complemented
5D42×C127C211yesmaximal, characteristic, complemented
6C2×#14523233—22no—
7C2×#14523233—22no—
8#16861206C2211yescharacteristic, complemented
9C4⋊(C127⋊C8)C2212yescomplemented
10#16861129—22no—
11C4×C127⋊C8C2211yescharacteristic, complemented
12C22×D4×C127—22no—
13#127×C127C2211yescharacteristic
14#121×C127—22no—
15#120×C127—22no—
16C4×D4×C127C2212yes—
17C2×D508—21no—
18C2×D508—22no—
19#14523233—44no—
20#14523233—44no—
21C2×C127⋊C8—21no—
22C2×C127⋊C8—22no—
23C24×C127—22no—
24C2×D4×C127—24no—
25C2×D4×C127—22no—
26C2×D4×C127—44no—
27C2×D4×C127—44no—
28C2×D4×C127D412yes—
29C2×D4×C127D412yes—
30C22×C4×C127—22no—
31C4⋊C4×C127D412yes—
32C22⋊C4×C127—22no—
33C22⋊C4×C127—22no—
34C42×C127C2311yescharacteristic
35C22×D127—41no—
36D508—42no—
37D508—42no—
38D508—42no—
39C127⋊C8—22no—
40C127⋊C8—42no—
41C23×C127—22no—
42C23×C127—22no—
43C23×C127—48no—
44C23×C127—82no—
45D4×C127—22no—
46D4×C127—42no—
47D4×C127—48no—
48D4×C127—42no—
49D4×C127—42no—
50D4×C127D814yescomplemented
51C2×C4×C127—22no—
52C2×C4×C127—42no—
53C2×C4×C127—42no—
54C2×C4×C127C2×D411yescharacteristic
55C2×C4×C127C2×D412yes—
56C22×C127—24no—
57C22×C127—42no—
58C22×C127—42no—
59C22×C127—44no—
60C22×C127—42no—
61C22×C127—88no—
62C22×C127#12011yescharacteristic
63C2×D127—82no—
64C2×D127—81no—
65C4×C127—21no—
66C4×C127—42no—
67C4×C127C2×D812yes—
68C4×C127AGL(1,ℤ/8)12yes—
69C2×C127—44no—
70C2×C127—82no—
71C2×C127#44712yes—
72C2×C127#45311yescharacteristic
73D127—161no—
74#1619—1271nomaximal, Sylow
75C127#161911yescharacteristic, complemented, Sylow
76D42D12711yescharacteristic, complemented
77#493—1271no—
78#459—1272no—
79#331—1271no—
80D4⋊C8—1272no—
81C22×D4—22no—
82C2×D8—2542no—
83C2×D8—2542no—
84#127—1271no—
85#127C2×D12711yescharacteristic
86#121—22no—
87#120—22no—
88C4×D4C2×D12712yes—
89C4⋊C8—1272no—
90#102—2542no—
91C4×C8—1271no—
92C24—22no—
93C2×D4—24no—
94C2×D4—22no—
95C2×D4—44no—
96C2×D4—44no—
97C2×D4—2542no—
98C2×D4—2541no—
99C2×D4C22⋊D12712yes—
100C2×D4C22⋊D12712yes—
101C22×C4—22no—
102D8—5084no—
103D8—5084no—
104C2×C8—2542no—
105C2×C8—2541no—
106C4⋊C4C22⋊D12712yes—
107C22⋊C4—22no—
108C22⋊C4—22no—
109C42C22×D12711yescharacteristic
110C23—22no—
111C23—22no—
112C23—48no—
113C23—82no—
114C23—5081no—
115D4—22no—
116D4—42no—
117D4—48no—
118D4—42no—
119D4—42no—
120D4—5082no—
121D4—5082no—
122D4—5082no—
123D4#1452323314yescomplemented
124C2×C4—22no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4C2×C22⋊D12711yescharacteristic
128C2×C4C2×C22⋊D12712yes—
129C8—5082no—
130C8—2542no—
131C22—24no—
132C22—42no—
133C22—42no—
134C22—42no—
135C22—44no—
136C22—88no—
137C22—10161no—
138C22—10162no—
139C22#1686117811yescharacteristic
140C4—21no—
141C4—42no—
142C4C2×#1452323312yes—
143C4#1686123512yes—
144C2—44no—
145C2—82no—
146C2—20321no—
147C2#2255001012yes—
148C2#2255004411yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingD42×C127C2
socleC22×C127#120
hypercenterD42D127
hypocenterC127#1619
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #22550252C2semidirect
2. #22550071C2semidirect
3. #22549620C2semidirect
4. D4⋊(C127⋊C8)C2semidirect
5. D42×C127C2semidirect
8. #16861206C22semidirect
9. C4⋊(C127⋊C8)C22semidirect
11. C4×C127⋊C8C22semidirect
50. D4×C127D8semidirect
75. C127#1619semidirect
76. D42D127semidirect
123. D4#14523233semidirect

Maximal subgroups

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

subgroupindex
#225502522
#225500712
#225496202
D4⋊(C127⋊C8)2
D42×C1272
#1619127