Finite Group Databasebeta

← #30638020

Subgroups — #30638020

Order 16256.

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

normal subgroups only

This group has 262 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#22550618C211yesmaximal, characteristic, complemented
2#22550266C211yesmaximal, characteristic, complemented
3#22550265C212yesmaximal, complemented
4#22549615C212yesmaximal, complemented
5#329×C127C211yesmaximal, characteristic, complemented
6C2×#14523237—22no—
7C2×#14523235—21no—
8C2×D1016—22no—
9C2×#14523233—21no—
10#16861206C2211yescharacteristic, complemented
11#16861198—21no—
12#16861197C2212yes—
13#16861195—22no—
14#16861179—22no—
15#16861177—21no—
16#16861120C2211yescharacteristic, complemented
17#124×C127C2211yescharacteristic
18#97×C127C2212yescomplemented
19C23×D127—21no—
20C2×Dic254D412yes—
21C2×D508—21no—
22C2×D508—21no—
23C2×D508—22no—
24C2×D508D412yes—
25C2×C22⋊D127—22no—
26#14523237—44no—
27#14523235—42no—
28D1016—44no—
29#14523233—42no—
30C2×C127⋊C8—21no—
31C22⋊Dic127—21no—
32(C2×D127)⋊C4—22no—
33(C2×D127)⋊C4—22no—
34(C2×D127)⋊C4—22no—
35C4×Dic127—21no—
36C2×Q8×C127D411yescharacteristic
37C2×D4×C127D411yescharacteristic
38C2×C8×C127—22no—
39C22⋊C4×C127—22no—
40C42×C127C2311yescharacteristic
41C22×D127—22no—
42C22×D127—41no—
43C22×D127—82no—
44C22×D127—81no—
45Dic254—42no—
46D508—42no—
47D508—41no—
48D508—42no—
49D508—41no—
50D508—42no—
51D508—81no—
52C22⋊D127—82no—
53C2×Dic127—22no—
54C127⋊C8—22no—
55C23×C127—21no—
56Q8×C127—41no—
57D4×C127—41no—
58C2×C4×C127—21no—
59C2×C4×C127C2×D412yes—
60C2×C4×C127C2×D411yescharacteristic
61C8×C127—24no—
62C22×C127—41no—
63C22×C127—42no—
64C22×C127#12011yescharacteristic
65C2×D127—42no—
66C2×D127—42no—
67C2×D127—42no—
68C2×D127—84no—
69C2×D127—82no—
70C2×D127—81no—
71C4×C127—22no—
72C4×C127—21no—
73C4×C127—41no—
74Dic127—42no—
75C2×C127—81no—
76C2×C127#45312yes—
77C2×C127#45311yescharacteristic
78D127—82no—
79D127—161no—
80#1655—1271nomaximal, Sylow
81C127#165511yescharacteristic, complemented, Sylow
82H(3,𝔽4)—1271no—
83#496—1271no—
84#496—1272no—
85#329—1272no—
86#329D12711yescharacteristic, complemented
87C2×SD16—2542no—
88C2×SD16—2541no—
89C2×D8—2542no—
90C2×D8—2541no—
91#127—1271no—
92#124—1272no—
93#124—2542no—
94#124—2541no—
95#124C2×D12711yescharacteristic
96#120—2542no—
97#120—2541no—
98#97—1271no—
99#97C2×D12712yescomplemented
100C24—2541no—
101C2×Q8—1272no—
102C2×Q8D50811yescharacteristic
103C2×D4—1272no—
104C2×D4—2541no—
105C2×D4—2542no—
106C2×D4—2542no—
107C2×D4—2541no—
108C2×D4D50811yescharacteristic
109SD16—5084no—
110SD16—5082no—
111D8—5084no—
112D8—5082no—
113C2×C8—22no—
114C2×C8—2541no—
115C22⋊C4—22no—
116C22⋊C4—2542no—
117C22⋊C4—2541no—
118C22⋊C4—2542no—
119C22⋊C4—2542no—
120C42—2541no—
121C42C22×D12711yescharacteristic
122C23—21no—
123C23—10162no—
124C23—10161no—
125C23—5081no—
126C23—2542no—
127Q8—41no—
128Q8—5082no—
129D4—41no—
130D4—10161no—
131D4—10162no—
132D4—5082no—
133D4—5081no—
134D4—5082no—
135D4—5081no—
136D4—5082no—
137C2×C4—21no—
138C2×C4—2542no—
139C2×C4D4×D12712yes—
140C2×C4C2×D50811yescharacteristic
141C8—24no—
142C8—2542no—
143C22—42no—
144C22—41no—
145C22—10164no—
146C22—10162no—
147C22—10161no—
148C22—5082no—
149C22—5082no—
150C22—5082no—
151C22#1686117911yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—5082no—
156C2—81no—
157C2—20321no—
158C2—10162no—
159C2#2255004512yes—
160C2#2255004611yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#329×C127C2
socleC22×C127#120
hypercenter#329D127
hypocenterC127#1655
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. #22550618C2semidirect
2. #22550266C2semidirect
3. #22550265C2semidirect
4. #22549615C2semidirect
5. #329×C127C2semidirect
10. #16861206C22semidirect
16. #16861120C22semidirect
18. #97×C127C22semidirect
81. C127#1655semidirect
86. #329D127semidirect
99. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506182
#225502662
#225502652
#225496152
#329×C1272
#1655127