Finite Group Databasebeta

← #30640731

Subgroups — #30640731

Order 16256.

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

normal subgroups only

This group has 270 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#22550062C212yesmaximal, complemented
2#22550053C211yesmaximal, characteristic, complemented
3#22549671C211yesmaximal, characteristic, complemented
4#22549663C212yesmaximal, complemented
5H(3,𝔽4)×C127C211yesmaximal, characteristic, complemented
6#16861254C2211yescharacteristic
7#16861240—21no—
8#16861178—22no—
9#16861137—21no—
10#16861126C2211yescharacteristic
11#16861124C2212yes—
12#16861123—22no—
13#124×C127—21no—
14#124×C127—22no—
15#124×C127—21no—
16#124×C127C2211yescharacteristic
17#120×C127—22no—
18#120×C127C2212yes—
19#14523248—21no—
20#14523247—41no—
21C2×C22⋊D127—22no—
22C2×C22⋊D127—42no—
23D4×D127—41no—
24#14523238—41no—
25#14523235—41no—
26#14523232—21no—
27C22⋊Dic127—22no—
28C22⋊Dic127—42no—
29C24×C127D412yescomplemented
30C2×Q8×C127—21no—
31C2×Q8×C127D411yescharacteristic, complemented
32C2×D4×C127—21no—
33C2×D4×C127—21no—
34C2×D4×C127—22no—
35C2×D4×C127C2311yescharacteristic
36C22⋊C4×C127—22no—
37C22⋊C4×C127—22no—
38C22⋊C4×C127—22no—
39C22⋊C4×C127—22no—
40C22⋊C4×C127D412yescomplemented
41C42×C127—21no—
42C42×C127D411yescharacteristic, complemented
43C22×D127—22no—
44Dic254—21no—
45D508—21no—
46C22⋊D127—42no—
47C22⋊D127—41no—
48C22⋊D127—84no—
49C2×Dic127—22no—
50C2×Dic127—42no—
51C4×D127—41no—
52C127⋊C8—41no—
53C23×C127—22no—
54C23×C127—42no—
55C23×C127—82no—
56C23×C127C2×D412yes—
57Q8×C127—41no—
58Q8×C127—81no—
59D4×C127—41no—
60D4×C127—82no—
61D4×C127—81no—
62D4×C127—81no—
63C2×C4×C127—21no—
64C2×C4×C127—22no—
65C2×C4×C127—21no—
66C2×C4×C127C2×D411yescharacteristic
67C22×C127—22no—
68C22×C127—42no—
69C22×C127—42no—
70C22×C127—42no—
71C22×C127—44no—
72C22×C127—42no—
73C22×C127—82no—
74C22×C127#12011yescharacteristic
75C2×D127—41no—
76C2×D127—82no—
77C4×C127—21no—
78C4×C127—42no—
79C4×C127—41no—
80C4×C127—41no—
81Dic127—41no—
82Dic127—82no—
83C2×C127—21no—
84C2×C127—42no—
85C2×C127—82no—
86C2×C127B(4,𝔽2)11yescharacteristic
87D127—81no—
88#2199—1271nomaximal, Sylow
89C127#219911yescharacteristic, complemented, Sylow
90H(3,𝔽4)D12711yescharacteristic, complemented
91B(4,𝔽2)—1272no—
92#455—1271no—
93#354—1271no—
94AΣL(1,𝔽16)—1272no—
95H(5,𝔽2)—1271no—
96#137—2541no—
97#124—21no—
98#124—22no—
99#124—21no—
100#124C2×D12711yescharacteristic
101#120—22no—
102#120—2542no—
103#120C2×D12712yes—
104#104—2541no—
105#100—1271no—
106#99—1272no—
107#99—2542no—
108C24C22⋊D12712yescomplemented
109#41—5081no—
110#41—2541no—
111C2×Q8—21no—
112C2×Q8C22⋊D12711yescharacteristic, complemented
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—5081no—
117C2×D4—5082no—
118C2×D4—2542no—
119C2×D4C22×D12711yescharacteristic
120Q16—5081no—
121SD16—5081no—
122M16—2541no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—5082no—
128C22⋊C4—2542no—
129C22⋊C4C22⋊D12712yescomplemented
130C42—21no—
131C42C22⋊D12711yescharacteristic, complemented
132C23—22no—
133C23—42no—
134C23—82no—
135C23—2542no—
136C23C2×C22⋊D12712yes—
137Q8—41no—
138Q8—81no—
139Q8—2541no—
140D4—41no—
141D4—82no—
142D4—81no—
143D4—81no—
144D4—10164no—
145D4—5082no—
146D4—5081no—
147D4—2541no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—5081no—
152C2×C4—5082no—
153C2×C4—2542no—
154C2×C4C2×C22⋊D12711yescharacteristic
155C8—5081no—
156C22—22no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22—44no—
161C22—42no—
162C22—82no—
163C22—10162no—
164C22—5081no—
165C22#1686117811yescharacteristic
166C4—21no—
167C4—42no—
168C4—41no—
169C4—41no—
170C4—10162no—
171C4—5081no—
172C2—21no—
173C2—42no—
174C2—82no—
175C2—10161no—
176C2#2255006311yescharacteristic
subgroupG/H
centerC2#22550063
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
FittingH(3,𝔽4)×C127C2
socleC2×C127B(4,𝔽2)
hypercenterH(3,𝔽4)D127
hypocenterC127#2199
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #22550062C2semidirect
2. #22550053C2semidirect
3. #22549671C2semidirect
4. #22549663C2semidirect
5. H(3,𝔽4)×C127C2semidirect
29. C24×C127D4semidirect
31. C2×Q8×C127D4semidirect
40. C22⋊C4×C127D4semidirect
42. C42×C127D4semidirect
89. C127#2199semidirect
90. H(3,𝔽4)D127semidirect
108. C24C22⋊D127semidirect
112. C2×Q8C22⋊D127semidirect
129. C22⋊C4C22⋊D127semidirect
131. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225500622
#225500532
#225496712
#225496632
H(3,𝔽4)×C1272
#2199127