Finite Group Databasebeta

← #30639801

Subgroups — #30639801

Order 16256.

510 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C22×#14523233C212yesmaximal, complemented
2C2×#16861130C212yesmaximal, complemented
3#22549778C211yesmaximal, characteristic, complemented
4#22549631C211yesmaximal, characteristic, complemented
5C2×#127×C127C211yesmaximal, characteristic, complemented
6C22×D508C2211yescharacteristic
7C2×#14523233—24no—
8C2×#14523233—24no—
9C2×#14523233—24no—
10C22×C127⋊C8C2212yes—
11C2×C4⋊Dic127C2211yescharacteristic
12C2×(C2×D127)⋊C4—22no—
13#16861130—24no—
14C22×D4×C127—22no—
15C22×D4×C127C2212yes—
16#127×C127—24no—
17C2×C42×C127C2211yescharacteristic
18C23×D127—21no—
19C2×D508—22no—
20C2×D508—22no—
21C2×D508—21no—
22C2×D508D412yescomplemented
23C22×Dic127—21no—
24#14523233—216no—
25C2×C127⋊C8—22no—
26C2×C127⋊C8—22no—
27C2×C127⋊C8D414yescomplemented
28C4⋊Dic127—22no—
29(C2×D127)⋊C4—44no—
30C24×C127—22no—
31C2×D4×C127—28no—
32C2×D4×C127—22no—
33C2×D4×C127—22no—
34C2×D4×C127—22no—
35C2×D4×C127—22no—
36C2×D4×C127—22no—
37C2×D4×C127—44no—
38C2×D4×C127—44no—
39C2×D4×C127D414yes—
40C22×C4×C127D412yes—
41C22×C4×C127C2311yescharacteristic
42C42×C127—22no—
43C22×D127—22no—
44C22×D127—41no—
45C22×D127—41no—
46C22×D127—42no—
47C22×D127—42no—
48D508—24no—
49D508—42no—
50C2×Dic127—41no—
51C2×Dic127—42no—
52C127⋊C8—24no—
53C23×C127—24no—
54C23×C127—42no—
55C23×C127—42no—
56C23×C127—44no—
57C23×C127—44no—
58C23×C127C2×D411yescharacteristic
59D4×C127—24no—
60D4×C127—24no—
61D4×C127—416no—
62D4×C127—42no—
63D4×C127—42no—
64C2×C4×C127—22no—
65C2×C4×C127—22no—
66C2×C4×C127D814yescomplemented
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D412yes—
69C2×C4×C127C2×D412yes—
70C2×C4×C127#4111yescharacteristic
71C22×C127—44no—
72C22×C127—48no—
73C22×C127—84no—
74C22×C127—82no—
75C22×C127—82no—
76C22×C127#12011yescharacteristic
77C22×C127#12111yescharacteristic
78C22×C127#12711yescharacteristic
79C22×C127C2×D812yes—
80C22×C127AGL(1,ℤ/8)12yes—
81C2×D127—42no—
82C2×D127—44no—
83C2×D127—82no—
84C2×D127—81no—
85C2×D127—81no—
86C4×C127—24no—
87C4×C127#12012yes—
88C4×C127#12112yes—
89Dic127—81no—
90C2×C127—84no—
91C2×C127#39211yescharacteristic
92C2×C127#44712yes—
93C2×C127#45912yes—
94C2×C127#49311yescharacteristic
95C2×C127#49611yescharacteristic
96D127—82no—
97#2011—1271nomaximal, Sylow
98C127#201111yescharacteristic, complemented, Sylow
99C22×D8—1272no—
100C2×#127D12711yescharacteristic, complemented
101C2×#102—1272no—
102#390—1271no—
103#336—1271no—
104C22×D4—22no—
105C22×D4—1271no—
106C22×D4C2×D12712yes—
107C2×D8—2544no—
108C2×D8—2544no—
109C2×D8—2544no—
110C22×C8—1272no—
111#127—24no—
112C2×C4⋊C4—1271no—
113C2×C22⋊C4—2542no—
114C2×C42C2×D12711yescharacteristic
115#102—2544no—
116C24—22no—
117C24—2541no—
118C2×D4—28no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—44no—
125C2×D4—44no—
126C2×D4—1272no—
127C2×D4—2542no—
128C2×D4—2542no—
129C2×D4—2541no—
130C2×D4C22⋊D12714yes—
131C22×C4—2541no—
132C22×C4C22⋊D12712yes—
133C22×C4C22×D12711yescharacteristic
134D8—25416no—
135C2×C8—1274no—
136C2×C8—2542no—
137C2×C8—2542no—
138C4⋊C4—2542no—
139C22⋊C4—5084no—
140C42—22no—
141C23—24no—
142C23—42no—
143C23—44no—
144C23—44no—
145C23—42no—
146C23—5081no—
147C23—5082no—
148C23—5082no—
149C23—5081no—
150C23—2542no—
151C23C2×C22⋊D12711yescharacteristic
152D4—24no—
153D4—24no—
154D4—416no—
155D4—42no—
156D4—42no—
157D4—5082no—
158D4—2544no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—5081no—
162C2×C4—5082no—
163C2×C4#1452323314yescomplemented
164C2×C4D4×D12711yescharacteristic
165C2×C4D4×D12712yes—
166C2×C4C2×C22⋊D12712yes—
167C2×C4#1452324711yescharacteristic
168C8—2544no—
169C22—44no—
170C22—48no—
171C22—84no—
172C22—82no—
173C22—82no—
174C22—10161no—
175C22—10161no—
176C22—10162no—
177C22—5082no—
178C22—5084no—
179C22#1686117811yescharacteristic
180C22#1686118011yescharacteristic
181C22#1686120511yescharacteristic
182C22C2×#1452323312yes—
183C22#1686123512yes—
184C4—24no—
185C4—10161no—
186C4#1686117712yes—
187C4#1686118112yes—
188C2—84no—
189C2—10162no—
190C2#2254978411yescharacteristic
191C2#2255001012yes—
192C2#2255006812yes—
193C2#2255025211yescharacteristic
194C2#2255026311yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#127×C127C2
socleC23×C127C2×D4
hypercenterC2×#127D127
hypocenterC127#2011
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C22×#14523233C2semidirect
2. C2×#16861130C2semidirect
3. #22549778C2semidirect
4. #22549631C2semidirect
5. C2×#127×C127C2semidirect
22. C2×D508D4semidirect
27. C2×C127⋊C8D4semidirect
66. C2×C4×C127D8semidirect
98. C127#2011semidirect
100. C2×#127D127semidirect
163. C2×C4#14523233semidirect

Maximal subgroups

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

subgroupindex
C22×#145232332
C2×#168611302
#225497782
#225496312
C2×#127×C1272
#2011127