Finite Group Databasebeta

← #30648335

Subgroups — #30648335

Order 16256.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14523231C211yesmaximal, characteristic, complemented
2#22550383C212yesmaximal, complemented
3#22550131C212yesmaximal, complemented
4#22550126C212yesmaximal, complemented
5#22550119C214yesmaximal, complemented
6#22549914C211yesmaximal, characteristic, complemented
7#22549872C212yesmaximal, complemented
8C2×#135×C127C211yesmaximal, characteristic, complemented
9C22×C4×D127C2211yescharacteristic
10C2×#14523231—24no—
11C2×#14523231C2212yes—
12C2×#14523231C2212yes—
13C22×C127⋊C8C2211yescharacteristic
14#16861190—22no—
15#16861185C2214yes—
16#16861180C2214yes—
17#16861177—22no—
18C4×C22⋊D127—24no—
19#16861168C2212yes—
20#16861143C2212yescomplemented
21#16861141C2212yescomplemented
22#16861133C2214yescomplemented
23#16861130C2214yescomplemented
24C2×#41×C127C2212yes—
25#135×C127—24no—
26C2×Q16×C127C2211yescharacteristic, complemented
27C2×SD16×C127C2212yescomplemented
28C2×D8×C127C2211yescharacteristic, complemented
29C22×C8×C127C2211yescharacteristic
30C23×D127D411yescharacteristic
31C2×C22⋊D127—24no—
32C22×Dic127D411yescharacteristic
33C2×C4×D127—22no—
34C2×C4×D127—22no—
35C2×C4×D127D412yes—
36C2×C4×D127C2312yes—
37#14523231—24no—
38#14523231—24no—
39C2×C127⋊C8—22no—
40C2×C127⋊C8C2312yes—
41C4⋊Dic127C2314yes—
42Dic127⋊C4—24no—
43C22⋊Dic127—24no—
44C22⋊Dic127—22no—
45(C2×D127)⋊C4—24no—
46C4×Dic127—22no—
47#41×C127—24no—
48#41×C127—24no—
49C2×Q8×C127C2312yes—
50C2×D4×C127—22no—
51C2×D4×C127C2312yes—
52C22×C4×C127—22no—
53C22×C4×C127C2311yescharacteristic
54Q16×C127—22no—
55SD16×C127—24no—
56D8×C127—22no—
57C2×C8×C127D414yescomplemented
58C2×C8×C127C2311yescharacteristic
59C2×C8×C127C2311yescharacteristic
60C22×D127—22no—
61C22×D127—42no—
62C22×D127C2×D412yes—
63C22⋊D127—48no—
64C2×Dic127—22no—
65C2×Dic127—24no—
66C2×Dic127C2×D412yes—
67C4×D127—24no—
68C4×D127—22no—
69C4×D127—22no—
70C127⋊C8—22no—
71C23×C127—22no—
72C23×C127C2×D411yescharacteristic
73Q8×C127—22no—
74Q8×C127—22no—
75D4×C127—22no—
76D4×C127—22no—
77D4×C127—44no—
78C2×C4×C127—22no—
79C2×C4×C127—44no—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D412yes—
82C2×C4×C127#4112yes—
83C2×C4×C127C2411yescharacteristic
84C8×C127C2×D412yes—
85C8×C127C2×D412yes—
86C22×C127—22no—
87C22×C127—42no—
88C22×C127—44no—
89C22×C127#12112yes—
90C22×C127C22×D411yescharacteristic
91C2×D127—22no—
92C2×D127—24no—
93C2×D127—44no—
94C4×C127—42no—
95C4×C127#12112yes—
96C4×C127C22×D411yescharacteristic
97C4×C127C2×#4111yescharacteristic
98Dic127—22no—
99Dic127—44no—
100C2×C127—22no—
101C2×C127—82no—
102C2×C127C2×#12111yescharacteristic
103C2×C127#57512yes—
104D127—42no—
105#3054—1271nomaximal, Sylow
106C127#305411yescharacteristic, complemented, Sylow
107C2×#135D12711yescharacteristic, complemented
108C22×M16—1271no—
109#525—1272no—
110#470—1272no—
111#469—1272no—
112#468—1274no—
113#427—1271no—
114#416—1272no—
115C2×#41C2×D12712yes—
116C23×C4—1271no—
117#135—24no—
118C2×Q16C2×D12711yescharacteristic, complemented
119C2×SD16C2×D12712yescomplemented
120C2×D8C2×D12711yescharacteristic, complemented
121C2×M16—1272no—
122C2×M16—1272no—
123C2×M16—2544no—
124C22×C8—1271no—
125C22×C8C2×D12711yescharacteristic
126#123—2542no—
127C22⋊Q8—1274no—
128#121—1274no—
129#120—2542no—
130C4×D4—2544no—
131#117—1272no—
132#107—1272no—
133#106—1272no—
134#103—1274no—
135#102—1274no—
136C24—1271no—
137#41—24no—
138#41—24no—
139C2×Q8C22×D12712yes—
140C2×D4—22no—
141C2×D4—2544no—
142C2×D4C22×D12712yes—
143C22×C4—22no—
144C22×C4—1271no—
145C22×C4—1272no—
146C22×C4—1272no—
147C22×C4—2542no—
148C22×C4—2542no—
149C22×C4C22×D12711yescharacteristic
150Q16—22no—
151SD16—24no—
152D8—22no—
153M16—2544no—
154M16—2544no—
155C2×C8—1272no—
156C2×C8—2542no—
157C2×C8C22⋊D12714yescomplemented
158C2×C8C22×D12711yescharacteristic
159C2×C8C22×D12711yescharacteristic
160C4⋊C4—1274no—
161C4⋊C4—2544no—
162C22⋊C4—2544no—
163C22⋊C4—2544no—
164C22⋊C4—2542no—
165C42—2542no—
166C23—22no—
167C23—1272no—
168C23—5082no—
169C23—2542no—
170C23D4×D12711yescharacteristic
171Q8—22no—
172Q8—22no—
173D4—22no—
174D4—22no—
175D4—44no—
176D4—5088no—
177C2×C4—22no—
178C2×C4—44no—
179C2×C4—1272no—
180C2×C4—2544no—
181C2×C4—2542no—
182C2×C4—2542no—
183C2×C4—2542no—
184C2×C4—2544no—
185C2×C4D4×D12711yescharacteristic
186C2×C4C2×C22⋊D12712yes—
187C2×C4#1452324712yes—
188C2×C4C23×D12711yescharacteristic
189C8—2542no—
190C8C2×C22⋊D12712yes—
191C8C2×C22⋊D12712yes—
192C22—22no—
193C22—42no—
194C22—44no—
195C22—5084no—
196C22—2542no—
197C22—2544no—
198C22#1686118012yes—
199C22C2×D4×D12711yescharacteristic
200C4—42no—
201C4—5084no—
202C4—2542no—
203C4#1686118012yes—
204C4C22×C22⋊D12711yescharacteristic
205C4C2×#1452324711yescharacteristic
206C2—22no—
207C2—82no—
208C2—5082no—
209C2C2×#1686118011yescharacteristic
210C2#2255067312yes—
subgroupG/H
centerC2×C4D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#135×C127C2
socleC22×C127C22×D4
hypercenterC2×#135D127
hypocenterC127#3054
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C22×#14523231C2semidirect
2. #22550383C2semidirect
3. #22550131C2semidirect
4. #22550126C2semidirect
5. #22550119C2semidirect
6. #22549914C2semidirect
7. #22549872C2semidirect
8. C2×#135×C127C2semidirect
20. #16861143C22semidirect
21. #16861141C22semidirect
22. #16861133C22semidirect
23. #16861130C22semidirect
26. C2×Q16×C127C22semidirect
27. C2×SD16×C127C22semidirect
28. C2×D8×C127C22semidirect
57. C2×C8×C127D4semidirect
106. C127#3054semidirect
107. C2×#135D127semidirect
118. C2×Q16C2×D127semidirect
119. C2×SD16C2×D127semidirect
120. C2×D8C2×D127semidirect
157. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232312
#225503832
#225501312
#225501262
#225501192
#225499142
#225498722
C2×#135×C1272
#3054127