Finite Group Databasebeta

← #30638028

Subgroups — #30638028

Order 16256.

206 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550266C211yesmaximal, characteristic
2#22550242C212yesmaximal, complemented
3#22549618C212yesmaximal, complemented
4#22549616C211yesmaximal, characteristic, complemented
5#563×C127C211yesmaximal, characteristic, complemented
6C2×#14523235—21no—
7C2×#14523233—21no—
8#16861206C2211yescharacteristic
9#16861129—22no—
10#16861129—22no—
11#16861120C2212yes—
12#16861120C2211yescharacteristic
13C4⋊Q8×C127—21no—
14#126×C127—22no—
15#125×C127C2212yes—
16#124×C127C2211yescharacteristic
17#123×C127—21no—
18C22⋊Q8×C127—22no—
19C2×D508—21no—
20C2×D508—22no—
21#14523235—42no—
22#14523233—42no—
23C2×C127⋊C8—21no—
24C2×C127⋊C8—22no—
25C2×Q8×C127—21no—
26C2×Q8×C127D411yescharacteristic
27C2×D4×C127D411yescharacteristic
28C22×C4×C127—21no—
29C4⋊C4×C127—22no—
30C4⋊C4×C127—22no—
31C4⋊C4×C127—22no—
32C4⋊C4×C127D412yes—
33C4⋊C4×C127D412yes—
34C22⋊C4×C127—22no—
35C22⋊C4×C127—21no—
36C22⋊C4×C127—22no—
37C42×C127—21no—
38C42×C127C2311yescharacteristic
39C22×D127—41no—
40D508—41no—
41D508—41no—
42D508—44no—
43C127⋊C8—22no—
44C127⋊C8—42no—
45C23×C127—21no—
46Q8×C127—41no—
47Q8×C127—81no—
48D4×C127—41no—
49C2×C4×C127—22no—
50C2×C4×C127—21no—
51C2×C4×C127—22no—
52C2×C4×C127—81no—
53C2×C4×C127C2×D412yes—
54C2×C4×C127C2×D411yescharacteristic
55C22×C127—41no—
56C22×C127—42no—
57C22×C127#12011yescharacteristic
58C2×D127—81no—
59C2×D127—82no—
60C4×C127—21no—
61C4×C127—22no—
62C4×C127—42no—
63C4×C127—41no—
64C4×C127—42no—
65C2×C127—81no—
66C2×C127#45311yescharacteristic
67C2×C127#45412yes—
68D127—161no—
69#1657—1271nomaximal, Sylow
70C127#165711yescharacteristic, complemented, Sylow
71#563D12711yescharacteristic, complemented
72#496—1271no—
73#490—1272no—
74#330—1272no—
75#329—1271no—
76C2×SD16—2541no—
77C2×D8—2541no—
78C4⋊Q8—21no—
79#127—1271no—
80#126—22no—
81#125C2×D12712yes—
82#124C2×D12711yescharacteristic
83#123—21no—
84C22⋊Q8—22no—
85#102—2542no—
86#102—2542no—
87#97—1272no—
88#97—1271no—
89C2×Q8—21no—
90C2×Q8C22⋊D12711yescharacteristic
91C2×D4—2542no—
92C2×D4—2541no—
93C2×D4C22⋊D12711yescharacteristic
94C22×C4—21no—
95SD16—5082no—
96D8—5082no—
97C2×C8—2542no—
98C2×C8—2541no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4C22⋊D12712yes—
103C4⋊C4C22⋊D12712yes—
104C22⋊C4—22no—
105C22⋊C4—21no—
106C22⋊C4—22no—
107C42—21no—
108C42C22×D12711yescharacteristic
109C23—21no—
110C23—5081no—
111Q8—41no—
112Q8—81no—
113D4—41no—
114D4—5081no—
115D4—5081no—
116D4—5084no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—81no—
121C2×C4C2×C22⋊D12712yes—
122C2×C4C2×C22⋊D12711yescharacteristic
123C8—5082no—
124C8—2542no—
125C22—41no—
126C22—42no—
127C22—10161no—
128C22—10162no—
129C22#1686117811yescharacteristic
130C4—21no—
131C4—22no—
132C4—42no—
133C4—41no—
134C4—42no—
135C2—81no—
136C2—20321no—
137C2#2255004411yescharacteristic
138C2#2255004912yes—
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#563×C127C2
socleC22×C127#120
prime radical#22550266C2
hypercenter#563D127
hypocenterC127#1657
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
2. #22550242C2semidirect
3. #22549618C2semidirect
4. #22549616C2semidirect
5. #563×C127C2semidirect
70. C127#1657semidirect
71. #563D127semidirect

Maximal subgroups

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

subgroupindex
#225502662
#225502422
#225496182
#225496162
#563×C1272
#1657127