Finite Group Databasebeta

← #30640283

Subgroups — #30640283

Order 16256.

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

normal subgroups only

This group has 210 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#22549925C211yesmaximal, characteristic, complemented
2#22549902C212yesmaximal, complemented
3#22549893C212yesmaximal
4#22549633C211yesmaximal, characteristic, complemented
5#408×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7#16861168C2212yes—
8#16861145—22no—
9Dic127⋊C8—24no—
10#16861136—24no—
11C2×#41×C127C2211yescharacteristic
12C2×M16×C127C2211yescharacteristic
13C22×C8×C127C2211yescharacteristic
14C22⋊C8×C127—22no—
15#14523232—22no—
16#14523232—22no—
17C2×C127⋊C8—22no—
18C4⋊Dic127—22no—
19C22⋊Dic127—22no—
20C4×Dic127D414yes—
21#41×C127—22no—
22#41×C127—22no—
23C2×Q8×C127Q811yescharacteristic
24C2×D4×C127—21no—
25C2×D4×C127Q811yescharacteristic
26C22×C4×C127—21no—
27C22×C4×C127C2311yescharacteristic
28M16×C127—22no—
29C2×C8×C127—21no—
30C2×C8×C127—22no—
31C2×C8×C127D412yes—
32C2×Dic127—24no—
33C127⋊C8—42no—
34C23×C127—21no—
35C23×C127#4111yescharacteristic
36Q8×C127—22no—
37D4×C127—22no—
38D4×C127—42no—
39C2×C4×C127—21no—
40C2×C4×C127—41no—
41C2×C4×C127—41no—
42C2×C4×C127C2×D412yes—
43C2×C4×C127C2×D411yescharacteristic
44C2×C4×C127C2×Q811yescharacteristic
45C2×C4×C127#4112yes—
46C8×C127—22no—
47C8×C127—41no—
48C22×C127—41no—
49C22×C127—42no—
50C22×C127—41no—
51C22×C127C22⋊Q811yescharacteristic
52C22×C127#12312yes—
53C4×C127—41no—
54C4×C127C22⋊Q812yes—
55C4×C127#12311yescharacteristic
56C4×C127#12711yescharacteristic
57Dic127—44no—
58C2×C127—22no—
59C2×C127—21no—
60C2×C127—81no—
61C2×C127#39911yescharacteristic
62#2098—1271nomaximal, Sylow
63C127#209811yescharacteristic, complemented, Sylow
64#430—1271no—
65#424—1272no—
66#421—1272no—
67#408D12711yescharacteristic
68H(3,ℤ/4)—1271no—
69C2×#41C2×D12711yescharacteristic
70C2×M16—1272no—
71C2×M16C2×D12711yescharacteristic
72C22×C8C2×D12711yescharacteristic
73#117—1272no—
74#108—2542no—
75C4⋊C8—2544no—
76#104—2544no—
77C22⋊C8—22no—
78#41—22no—
79#41—22no—
80C2×Q8Dic25411yescharacteristic
81C2×D4—21no—
82C2×D4Dic25411yescharacteristic
83C22×C4—21no—
84C22×C4C22×D12711yescharacteristic
85M16—22no—
86M16—2542no—
87M16—2542no—
88C2×C8—22no—
89C2×C8—21no—
90C2×C8—2542no—
91C2×C8C22⋊D12712yes—
92C4⋊C4—2542no—
93C22⋊C4—2542no—
94C42—1274no—
95C23—21no—
96C23#1452324811yescharacteristic
97Q8—22no—
98D4—22no—
99D4—42no—
100C2×C4—21no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—2544no—
104C2×C4D4×D12712yes—
105C2×C4C2×C22⋊D12711yescharacteristic
106C2×C4C2×Dic25411yescharacteristic
107C2×C4#1452324712yes—
108C8—22no—
109C8—41no—
110C8—5082no—
111C22—41no—
112C22—42no—
113C22—41no—
114C22#1686118811yescharacteristic
115C22#1686119112yes—
116C4—41no—
117C4—5084no—
118C4#1686118612yes—
119C4#1686119011yescharacteristic
120C4#1686120511yescharacteristic
121C2—22no—
122C2—21no—
123C2—81no—
124C2#2254981211yescharacteristic
subgroupG/H
centerC4#16861205
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#408×C127C2
socleC2×C127#399
prime radicalC2×#41×C127C22
hypercenter#408D127
hypocenterC127#2098
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #22549925C2semidirect
2. #22549902C2semidirect
4. #22549633C2semidirect
63. C127#2098semidirect

Maximal subgroups

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

subgroupindex
#225499252
#225499022
#225498932
#225496332
#408×C1272
#2098127