Finite Group Databasebeta

← #30637785

Subgroups — #30637785

Order 16256.

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

normal subgroups only

This group has 336 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#22550511C211yesmaximal, characteristic, complemented
2#22550252C211yesmaximal, characteristic, complemented
3#22550072C212yesmaximal, complemented
4#22549603C212yesmaximal, complemented
5#331×C127C211yesmaximal, characteristic, complemented
6C2×D4×D127—22no—
7C2×D1016—22no—
8C2×#14523233—22no—
9#16861206C2211yescharacteristic, complemented
10#16861180—22no—
11#16861177—22no—
12C4×D508C2212yes—
13#16861131—22no—
14C4×C127⋊C8C2211yescharacteristic, complemented
15#127×C127C2211yescharacteristic
16C4⋊C8×C127C2212yescomplemented
17C23×D127—22no—
18C2×D508—21no—
19C2×D508—22no—
20C2×D508D412yes—
21C2×C22⋊D127—24no—
22D4×D127—44no—
23D4×D127—44no—
24C2×C4×D127—22no—
25D1016—44no—
26#14523233—44no—
27C2×C127⋊C8—21no—
28C4⋊Dic127D412yes—
29C22⋊Dic127—22no—
30(C2×D127)⋊C4—22no—
31C2×D4×C127—22no—
32C2×D4×C127D412yes—
33C2×C8×C127—22no—
34C42×C127C2311yescharacteristic
35C22×D127—22no—
36C22×D127—48no—
37C22×D127—41no—
38C22×D127—82no—
39D508—22no—
40D508—42no—
41D508—42no—
42D508—42no—
43D508D814yescomplemented
44C22⋊D127—48no—
45C2×Dic127—22no—
46C4×D127—42no—
47C4×D127—42no—
48C127⋊C8—22no—
49C23×C127—22no—
50D4×C127—42no—
51D4×C127—42no—
52D4×C127—42no—
53C2×C4×C127C2×D412yes—
54C2×C4×C127C2×D411yescharacteristic
55C8×C127—42no—
56C22×C127—44no—
57C22×C127—42no—
58C22×C127#12011yescharacteristic
59C2×D127—24no—
60C2×D127—42no—
61C2×D127—42no—
62C2×D127—88no—
63C2×D127—82no—
64C2×D127—81no—
65C4×C127—21no—
66C4×C127C2×D812yes—
67C4×C127AGL(1,ℤ/8)12yes—
68Dic127—42no—
69C2×C127—82no—
70C2×C127#44712yes—
71C2×C127#45311yescharacteristic
72D127—44no—
73D127—161no—
74#1619—1271nomaximal, Sylow
75C127#161911yescharacteristic, complemented, Sylow
76D42—1271no—
77#493—1271no—
78#459—1272no—
79#331D12711yescharacteristic, complemented
80D4⋊C8—1272no—
81C22×D4—2542no—
82C2×D8—2542no—
83C2×D8—2542no—
84#127—1271no—
85#127C2×D12711yescharacteristic
86#121—2542no—
87#120—2542no—
88C4×D4—1272no—
89C4⋊C8C2×D12712yescomplemented
90#102—2542no—
91C4×C8—1271no—
92C24—2542no—
93C2×D4—22no—
94C2×D4—1272no—
95C2×D4—5084no—
96C2×D4—5084no—
97C2×D4—2544no—
98C2×D4—2542no—
99C2×D4—2541no—
100C2×D4D50812yes—
101C22×C4—2542no—
102D8—5084no—
103D8—5084no—
104C2×C8—22no—
105C2×C8—2541no—
106C4⋊C4—1272no—
107C22⋊C4—2542no—
108C22⋊C4—2542no—
109C42C22×D12711yescharacteristic
110C23—22no—
111C23—10162no—
112C23—5088no—
113C23—5081no—
114C23—2542no—
115D4—42no—
116D4—42no—
117D4—42no—
118D4—1274no—
119D4—5082no—
120D4—5088no—
121D4—5082no—
122D4—5082no—
123D4—2542no—
124C2×C4—5082no—
125C2×C4—5082no—
126C2×C4—2542no—
127C2×C4D4×D12712yes—
128C2×C4C2×D50811yescharacteristic
129C8—42no—
130C8—2542no—
131C22—42no—
132C22—44no—
133C22—10168no—
134C22—10161no—
135C22—10162no—
136C22—5082no—
137C22—5082no—
138C22—2544no—
139C22#1686117911yescharacteristic
140C4—21no—
141C4—5082no—
142C4D8×D12712yes—
143C4#1686123612yes—
144C2—82no—
145C2—20321no—
146C2—5084no—
147C2#2255001112yes—
148C2#2255004611yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#331×C127C2
socleC22×C127#120
hypercenter#331D127
hypocenterC127#1619
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
1. #22550511C2semidirect
2. #22550252C2semidirect
3. #22550072C2semidirect
4. #22549603C2semidirect
5. #331×C127C2semidirect
9. #16861206C22semidirect
14. C4×C127⋊C8C22semidirect
16. C4⋊C8×C127C22semidirect
43. D508D8semidirect
75. C127#1619semidirect
79. #331D127semidirect
89. C4⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505112
#225502522
#225500722
#225496032
#331×C1272
#1619127