Finite Group Databasebeta

← #30638026

Subgroups — #30638026

Order 16256.

206 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 45 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#22550626C211yesmaximal, characteristic, complemented
2#22550240C212yesmaximal, complemented
3#22549617C212yesmaximal, complemented
4#22549616C211yesmaximal, characteristic, complemented
5#496×C127C211yesmaximal, characteristic
6#16861207—21no—
7#16861202—22no—
8#16861201C2212yes—
9#16861190—21no—
10#16861186—22no—
11#16861130—22no—
12#16861130—22no—
13#16861120C2212yes—
14C2×SD16×C127—21no—
15C2×D8×C127—21no—
16#127×C127C2211yescharacteristic
17#124×C127C2211yescharacteristic
18#97×C127C2211yescharacteristic
19C2×Dic254—21no—
20C22×Dic127—21no—
21C2×C127⋊C8—22no—
22C4⋊Dic127D412yes—
23C4⋊Dic127D412yes—
24Dic127⋊C4—22no—
25Dic127⋊C4—22no—
26Dic127⋊C4—22no—
27C22⋊Dic127—22no—
28C22⋊Dic127—21no—
29C4×Dic127—21no—
30C2×Q8×C127D411yescharacteristic
31C2×D4×C127—21no—
32C2×D4×C127—22no—
33C2×D4×C127D411yescharacteristic
34SD16×C127—42no—
35D8×C127—42no—
36C2×C8×C127—21no—
37C22⋊C4×C127—22no—
38C42×C127C2311yescharacteristic
39Dic254—81no—
40C2×Dic127—22no—
41C2×Dic127—22no—
42C2×Dic127—81no—
43C127⋊C8—42no—
44C23×C127—21no—
45C23×C127—41no—
46Q8×C127—41no—
47D4×C127—41no—
48D4×C127—41no—
49D4×C127—41no—
50D4×C127—44no—
51C2×C4×C127—21no—
52C2×C4×C127C2×D412yes—
53C2×C4×C127C2×D411yescharacteristic
54C8×C127—22no—
55C22×C127—41no—
56C22×C127—42no—
57C22×C127—81no—
58C22×C127—82no—
59C22×C127#12011yescharacteristic
60C4×C127—21no—
61C4×C127—22no—
62C4×C127—41no—
63Dic127—42no—
64Dic127—42no—
65C2×C127—81no—
66C2×C127—161no—
67C2×C127#45311yescharacteristic
68C2×C127#45412yes—
69#1657—1271nomaximal, Sylow
70C127#165711yescharacteristic, complemented, Sylow
71#563—1271no—
72#496D12711yescharacteristic
73#490—1272no—
74#330—1272no—
75#329—1271no—
76C2×SD16—21no—
77C2×D8—21no—
78C4⋊Q8—2541no—
79#127C2×D12711yescharacteristic
80#126—2542no—
81#125—1272no—
82#124C2×D12711yescharacteristic
83#123—2541no—
84C22⋊Q8—2542no—
85#102—2542no—
86#102—2542no—
87#97—1272no—
88#97C2×D12711yescharacteristic
89C2×Q8—2541no—
90C2×Q8C22⋊D12711yescharacteristic
91C2×D4—22no—
92C2×D4—21no—
93C2×D4C22⋊D12711yescharacteristic
94C22×C4—2541no—
95SD16—42no—
96D8—42no—
97C2×C8—21no—
98C2×C8—2542no—
99C4⋊C4—1272no—
100C4⋊C4—1272no—
101C4⋊C4—2542no—
102C4⋊C4—2542no—
103C4⋊C4—2542no—
104C22⋊C4—22no—
105C22⋊C4—2542no—
106C22⋊C4—2541no—
107C42—2541no—
108C42C22×D12711yescharacteristic
109C23—21no—
110C23—41no—
111Q8—41no—
112Q8—10161no—
113D4—41no—
114D4—41no—
115D4—41no—
116D4—44no—
117C2×C4—21no—
118C2×C4—10161no—
119C2×C4—2542no—
120C2×C4—2542no—
121C2×C4D4×D12712yes—
122C2×C4C2×C22⋊D12711yescharacteristic
123C8—22no—
124C8—5082no—
125C22—41no—
126C22—42no—
127C22—81no—
128C22—82no—
129C22#1686117711yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—5082no—
134C4—5082no—
135C2—81no—
136C2—161no—
137C2#2255004311yescharacteristic
138C2#2255004712yes—
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting, prime radical#496×C127C2
socleC22×C127#120
hypercenter#496D127
hypocenterC127#1657
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #22550626C2semidirect
2. #22550240C2semidirect
3. #22549617C2semidirect
4. #22549616C2semidirect
70. C127#1657semidirect

Maximal subgroups

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

subgroupindex
#225506262
#225502402
#225496172
#225496162
#496×C1272
#1657127