Finite Group Databasebeta

← #30638029

Subgroups — #30638029

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#22550266C211yesmaximal, characteristic
3#22550241C212yesmaximal, complemented
4#22549617C212yesmaximal, complemented
5#329×C127C211yesmaximal, characteristic, complemented
6C2×#14523235—21no—
7C2×#14523233—21no—
8#16861207—21no—
9#16861206C2211yescharacteristic
10#16861202—22no—
11#16861201C2212yes—
12#16861190—21no—
13#16861186—22no—
14#16861131—22no—
15#16861131—22no—
16#16861120C2211yescharacteristic
17#124×C127C2211yescharacteristic
18#97×C127C2212yes—
19C2×Dic254—21no—
20C2×D508—21no—
21C2×D508—22no—
22C22×Dic127—21no—
23#14523235—42no—
24#14523233—42no—
25C2×C127⋊C8—21no—
26C4⋊Dic127D412yes—
27C4⋊Dic127D412yes—
28Dic127⋊C4—22no—
29Dic127⋊C4—22no—
30Dic127⋊C4—22no—
31C22⋊Dic127—22no—
32C22⋊Dic127—21no—
33C4×Dic127—21no—
34C2×Q8×C127D411yescharacteristic
35C2×D4×C127D411yescharacteristic
36C2×C8×C127—22no—
37C22⋊C4×C127—22no—
38C42×C127C2311yescharacteristic
39C22×D127—41no—
40Dic254—81no—
41D508—41no—
42D508—41no—
43D508—44no—
44C2×Dic127—22no—
45C2×Dic127—22no—
46C2×Dic127—81no—
47C127⋊C8—22no—
48C23×C127—21no—
49Q8×C127—41no—
50D4×C127—41no—
51C2×C4×C127—21no—
52C2×C4×C127C2×D412yes—
53C2×C4×C127C2×D411yescharacteristic
54C8×C127—42no—
55C22×C127—41no—
56C22×C127—42no—
57C22×C127#12011yescharacteristic
58C2×D127—81no—
59C2×D127—82no—
60C4×C127—21no—
61C4×C127—22no—
62C4×C127—41no—
63Dic127—42no—
64Dic127—42no—
65C2×C127—81no—
66C2×C127#45311yescharacteristic
67C2×C127#45412yes—
68D127—161no—
69#1657—1271nomaximal, Sylow
70C127#165711yescharacteristic, complemented, Sylow
71#563—1271no—
72#496—1271no—
73#490—1272no—
74#330—1272no—
75#329D12711yescharacteristic, complemented
76C2×SD16—2541no—
77C2×D8—2541no—
78C4⋊Q8—2541no—
79#127—1271no—
80#126—2542no—
81#125—1272no—
82#124C2×D12711yescharacteristic
83#123—2541no—
84C22⋊Q8—2542no—
85#102—2542no—
86#102—2542no—
87#97—1271no—
88#97C2×D12712yes—
89C2×Q8—2541no—
90C2×Q8D50811yescharacteristic
91C2×D4—2542no—
92C2×D4—2541no—
93C2×D4D50811yescharacteristic
94C22×C4—2541no—
95SD16—5082no—
96D8—5082no—
97C2×C8—22no—
98C2×C8—2541no—
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—5081no—
111Q8—41no—
112Q8—10161no—
113D4—41no—
114D4—5081no—
115D4—5081no—
116D4—5084no—
117C2×C4—21no—
118C2×C4—10161no—
119C2×C4—2542no—
120C2×C4—2542no—
121C2×C4D4×D12712yes—
122C2×C4C2×D50811yescharacteristic
123C8—42no—
124C8—2542no—
125C22—41no—
126C22—42no—
127C22—10161no—
128C22—10162no—
129C22#1686117911yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—5082no—
134C4—5082no—
135C2—81no—
136C2—20321no—
137C2#2255004611yescharacteristic
138C2#2255005012yes—
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#329×C127C2
socleC22×C127#120
prime radical#22550266C2
hypercenter#329D127
hypocenterC127#1657
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #22550626C2semidirect
3. #22550241C2semidirect
4. #22549617C2semidirect
5. #329×C127C2semidirect
70. C127#1657semidirect
75. #329D127semidirect

Maximal subgroups

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

subgroupindex
#225506262
#225502662
#225502412
#225496172
#329×C1272
#1657127