Finite Group Databasebeta

← #30642024

Subgroups — #30642024

Order 16256.

504 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 177 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
1C2×C4×Dic254C212yesmaximal
2#22549810C212yesmaximal
3#22549809C212yesmaximal
4#22549797C212yesmaximal
5#22549796C212yesmaximal
6C4×Dic127⋊C4C212yesmaximal
7C4×C4⋊Dic127C211yesmaximal, characteristic
8#22549722C211yesmaximal, characteristic
9C4×C4⋊C4×C127C211yesmaximal, characteristic
10C22×Dic254C2212yes—
11C4×Dic254—28no—
12C2×C4⋊Dic127C2212yes—
13C2×C4⋊Dic127C2212yes—
14C2×Dic127⋊C4C2212yes—
15C2×Dic127⋊C4C2212yes—
16C2×Dic127⋊C4C2212yes—
17C2×Dic127⋊C4C2212yes—
18C2×C4×Dic127C2212yes—
19C2×C4×Dic127C2211yescharacteristic
20C2×C4×Dic127C2211yescharacteristic
21C127⋊#95C2211yescharacteristic
22C127⋊#95C2212yes—
23C127⋊#95C2212yes—
24C127⋊#95C2212yes—
25C127⋊#95C2211yescharacteristic
26C127⋊#95C2212yes—
27C2×C4⋊C4×C127C2212yes—
28C2×C42×C127C2212yes—
29C2×C42×C127C2211yescharacteristic
30#95×C127C2211yescharacteristic
31#95×C127C2211yescharacteristic
32C2×Dic254—24no—
33C2×Dic254—24no—
34C2×Dic254—24no—
35C22×Dic127C2312yes—
36C22×Dic127C2312yes—
37C22×Dic127C2312yes—
38C22×Dic127C2312yes—
39C4⋊Dic127—24no—
40C4⋊Dic127—24no—
41Dic127⋊C4—24no—
42Dic127⋊C4—24no—
43Dic127⋊C4—24no—
44Dic127⋊C4—24no—
45C4×Dic127—24no—
46C4×Dic127D414yes—
47C4×Dic127Q814yescomplemented
48C22×C4×C127C2312yes—
49C22×C4×C127C2311yescharacteristic
50C22×C4×C127C2311yescharacteristic
51C22×C4×C127C2312yes—
52C22×C4×C127C2311yescharacteristic
53C4⋊C4×C127—24no—
54C42×C127—24no—
55C42×C127Q814yescomplemented
56Dic254—216no—
57C2×Dic127—22no—
58C2×Dic127—22no—
59C2×Dic127—22no—
60C2×Dic127—22no—
61C2×Dic127—22no—
62C2×Dic127—22no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C2×Dic127C2×D414yes—
67C2×Dic127C2×Q814yes—
68C2×Dic127#4114yes—
69C23×C127C2411yescharacteristic
70C2×C4×C127—22no—
71C2×C4×C127—22no—
72C2×C4×C127—22no—
73C2×C4×C127—22no—
74C2×C4×C127—22no—
75C2×C4×C127—22no—
76C2×C4×C127—22no—
77C2×C4×C127C2×D412yes—
78C2×C4×C127C2×Q812yes—
79C2×C4×C127C2×Q812yes—
80C2×C4×C127C2×Q812yes—
81C2×C4×C127C2×Q812yes—
82C2×C4×C127#4114yes—
83C22×C127C22×D411yescharacteristic
84C22×C127C22×Q811yescharacteristic
85C22×C127C22×Q811yescharacteristic
86C22×C127C2×#4112yes—
87C22×C127C2×#4112yes—
88C4×C127—24no—
89C4×C127—22no—
90C4×C127—22no—
91C4×C127—42no—
92C4×C127C4⋊Q814yes—
93Dic127—24no—
94Dic127—24no—
95Dic127—24no—
96Dic127—42no—
97C2×C127#52512yes—
98C2×C127C2×C4⋊Q811yescharacteristic
99C2×C127#53311yescharacteristic
100C2×C127#53312yes—
101C2×C127#53311yescharacteristic
102#2399—1271nomaximal, Sylow
103C127#239911yescharacteristic, complemented, Sylow
104C2×C4×Q8—1272no—
105#398—1272no—
106#398—1272no—
107#395—1272no—
108#395—1272no—
109C4×C4⋊C4—1271no—
110C4×C4⋊C4—1272no—
111C4×C4⋊C4D12711yescharacteristic
112#376—1271no—
113C22×Q8—1272no—
114C4×Q8—2548no—
115C2×C4⋊C4—1272no—
116C2×C4⋊C4—1272no—
117C2×C4⋊C4—1272no—
118C2×C4⋊C4—1272no—
119C2×C4⋊C4—1272no—
120C2×C4⋊C4—1272no—
121C2×C4⋊C4C2×D12712yes—
122C2×C42—1271no—
123C2×C42—1271no—
124C2×C42—1272no—
125C2×C42C2×D12712yes—
126C2×C42C2×D12711yescharacteristic
127#95—1272no—
128#95—1272no—
129#95—1272no—
130#95—1271no—
131#95—1272no—
132#95—1271no—
133#95C2×D12711yescharacteristic
134#95C2×D12711yescharacteristic
135C2×Q8—2544no—
136C2×Q8—2544no—
137C2×Q8—2544no—
138C22×C4—1272no—
139C22×C4—1272no—
140C22×C4—1272no—
141C22×C4—1272no—
142C22×C4C22×D12712yes—
143C22×C4C22×D12711yescharacteristic
144C22×C4C22×D12711yescharacteristic
145C22×C4C22×D12712yes—
146C22×C4C22×D12711yescharacteristic
147C4⋊C4—24no—
148C4⋊C4—2544no—
149C4⋊C4—2544no—
150C4⋊C4—2544no—
151C4⋊C4—2544no—
152C4⋊C4—2544no—
153C4⋊C4—2544no—
154C42—24no—
155C42—1274no—
156C42—1274no—
157C42—2544no—
158C42Dic25414yescomplemented
159C23C23×D12711yescharacteristic
160Q8—25416no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—1274no—
169C2×C4—1274no—
170C2×C4—1274no—
171C2×C4—2542no—
172C2×C4—2542no—
173C2×C4—2542no—
174C2×C4—2542no—
175C2×C4—2542no—
176C2×C4—2542no—
177C2×C4—2542no—
178C2×C4—2542no—
179C2×C4—2542no—
180C2×C4D4×D12712yes—
181C2×C4Q8×D12712yes—
182C2×C4C2×Dic25412yes—
183C2×C4C2×Dic25412yes—
184C2×C4C2×Dic25412yes—
185C2×C4#1452324814yes—
186C22C2×D4×D12711yescharacteristic
187C22C2×Q8×D12711yescharacteristic
188C22C22×Dic25411yescharacteristic
189C22#41×D12712yes—
190C22C2×#1452324812yes—
191C4—24no—
192C4—22no—
193C4—22no—
194C4—42no—
195C4—5082no—
196C4—2544no—
197C4—2544no—
198C4—2544no—
199C4#1686120814yes—
200C2#2255038512yes—
201C2C2×#1686120811yescharacteristic
202C2#2255042911yescharacteristic
203C2#2255043012yes—
204C2#2255043311yescharacteristic
subgroupG/H
centerC22×C4C22×D127
commutator, socle, prime radical, cosocleC23×C127C24
FrattiniC23C23×D127
FittingC4×C4⋊C4×C127C2
hypercenterC4×C4⋊C4D127
hypocenterC127#2399
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
47. C4×Dic127Q8semidirect
55. C42×C127Q8semidirect
103. C127#2399semidirect
158. C42Dic254semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic2542
#225498102
#225498092
#225497972
#225497962
C4×Dic127⋊C42
C4×C4⋊Dic1272
#225497222
C4×C4⋊C4×C1272
#2399127