Finite Group Databasebeta

← #30642009

Subgroups — #30642009

Order 16256.

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

normal subgroups only

This group has 840 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×D508C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D127C214yesmaximal, complemented
3#22549812C212yesmaximal, complemented
4#22549811C212yesmaximal, complemented
5#22549785C212yesmaximal, complemented
6#22549784C212yesmaximal, complemented
7#22549722C211yesmaximal, characteristic, complemented
8C2×C4×D4×C127C211yesmaximal, characteristic, complemented
9C22×D508C2211yescharacteristic
10C22×C22⋊D127C2214yes—
11C22×C4×D127C2212yes—
12C4×D508—24no—
13C4×C22⋊D127—216no—
14C2×C4⋊Dic127C2211yescharacteristic
15C2×Dic127⋊C4C2214yes—
16C2×C22⋊Dic127C2214yes—
17C2×(C2×D127)⋊C4C2212yes—
18C2×(C2×D127)⋊C4C2214yes—
19C2×C4×Dic127C2212yes—
20C127⋊#95C2212yes—
21C127⋊#95C2212yes—
22C22×D4×C127C2211yescharacteristic
23C23×C4×C127C2212yes—
24C4×D4×C127—24no—
25C2×C4⋊C4×C127C2211yescharacteristic
26C2×C22⋊C4×C127C2212yes—
27C2×C42×C127C2211yescharacteristic
28C23×D127C2312yes—
29C2×D508—24no—
30C2×D508—22no—
31C2×C22⋊D127—28no—
32C2×C22⋊D127—28no—
33C2×C22⋊D127—28no—
34C22×Dic127C2312yes—
35C22×Dic127C2314yes—
36C2×C4×D127—28no—
37C2×C4×D127—24no—
38C4⋊Dic127—22no—
39Dic127⋊C4—28no—
40C22⋊Dic127—28no—
41(C2×D127)⋊C4—28no—
42(C2×D127)⋊C4—24no—
43C4×Dic127D418yescomplemented
44C24×C127C2312yes—
45C2×D4×C127—24no—
46C2×D4×C127—22no—
47C22×C4×C127—24no—
48C22×C4×C127—28no—
49C22×C4×C127C2312yes—
50C22×C4×C127C2311yescharacteristic
51C22×C4×C127C2311yescharacteristic
52C22×C4×C127C2311yescharacteristic
53C4⋊C4×C127—22no—
54C22⋊C4×C127—24no—
55C42×C127D414yescomplemented
56C22×D127—24no—
57C22×D127—22no—
58C22×D127—24no—
59C22×D127—22no—
60C22×D127#4114yes—
61D508—28no—
62C22⋊D127—232no—
63C2×Dic127—24no—
64C2×Dic127—24no—
65C2×Dic127—22no—
66C2×Dic127—24no—
67C2×Dic127C2×D418yes—
68C4×D127—48no—
69C23×C127—22no—
70C23×C127—24no—
71C23×C127—22no—
72C23×C127—24no—
73C23×C127#4114yes—
74C23×C127C2411yescharacteristic
75D4×C127—28no—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—24no—
80C2×C4×C127—48no—
81C2×C4×C127C2×D414yes—
82C2×C4×C127C2×D412yes—
83C2×C4×C127C2×D412yes—
84C2×C4×C127C2×D412yes—
85C22×C127—24no—
86C22×C127—28no—
87C22×C127—42no—
88C22×C127—44no—
89C22×C127—42no—
90C22×C127C22×D412yes—
91C22×C127C22×D411yescharacteristic
92C22×C127C2×#4112yes—
93C22×C127C2×#4112yes—
94C2×D127—28no—
95C2×D127—24no—
96C2×D127—44no—
97C2×D127—42no—
98C2×D127—42no—
99C4×C127—22no—
100C4×C127—22no—
101C4×C127—42no—
102C4×C127#12714yes—
103Dic127—28no—
104Dic127—42no—
105C2×C127—44no—
106C2×C127#52511yescharacteristic
107C2×C127#52514yes—
108C2×C127#52511yescharacteristic
109C2×C127C2×#12711yescharacteristic
110D127—44no—
111#2397—1271nomaximal, Sylow
112C127#239711yescharacteristic, complemented, Sylow
113C2×C4×D4—1274no—
114C2×C4×D4—1271no—
115C2×C4×D4D12711yescharacteristic, complemented
116#399—1272no—
117#399—1272no—
118#392—1272no—
119#392—1272no—
120#376—1271no—
121C22×D4—1274no—
122C22×D4—1271no—
123C22×D4C2×D12711yescharacteristic
124C23×C4—1272no—
125C23×C4C2×D12712yes—
126C4×D4—24no—
127C4×D4—25416no—
128C4×D4—2544no—
129C2×C4⋊C4—1274no—
130C2×C4⋊C4—1271no—
131C2×C4⋊C4C2×D12711yescharacteristic
132C2×C22⋊C4—1274no—
133C2×C22⋊C4—1274no—
134C2×C22⋊C4—1272no—
135C2×C22⋊C4C2×D12712yes—
136C2×C42—1272no—
137C2×C42C2×D12711yescharacteristic
138#95—1272no—
139#95—1272no—
140C24—1272no—
141C24C22×D12712yes—
142C2×D4—24no—
143C2×D4—22no—
144C2×D4—2544no—
145C2×D4—2548no—
146C2×D4—2548no—
147C2×D4—2548no—
148C2×D4—2542no—
149C22×C4—28no—
150C22×C4—24no—
151C22×C4—1274no—
152C22×C4—1272no—
153C22×C4—2548no—
154C22×C4—2544no—
155C22×C4C22×D12712yes—
156C22×C4C22×D12711yescharacteristic
157C22×C4C22×D12711yescharacteristic
158C22×C4C22×D12711yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—2548no—
161C4⋊C4—2542no—
162C22⋊C4—24no—
163C22⋊C4—2548no—
164C22⋊C4—2548no—
165C22⋊C4—2544no—
166C42—1278no—
167C42C22⋊D12714yescomplemented
168C23—24no—
169C23—22no—
170C23—24no—
171C23—22no—
172C23—1274no—
173C23—2544no—
174C23—2542no—
175C23—2542no—
176C23—2544no—
177C23#1452324814yes—
178C23C23×D12711yescharacteristic
179D4—28no—
180D4—2548no—
181D4—25432no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—48no—
187C2×C4—1278no—
188C2×C4—5088no—
189C2×C4—2544no—
190C2×C4—2542no—
191C2×C4—2544no—
192C2×C4—2544no—
193C2×C4D4×D12714yes—
194C2×C4C2×C22⋊D12712yes—
195C2×C4C2×C22⋊D12712yes—
196C2×C4C2×C22⋊D12712yes—
197C22—28no—
198C22—24no—
199C22—42no—
200C22—44no—
201C22—42no—
202C22—5084no—
203C22—5082no—
204C22—5082no—
205C22—2544no—
206C22—2548no—
207C22C2×D4×D12712yes—
208C22C22×C22⋊D12711yescharacteristic
209C22#41×D12712yes—
210C22C2×#1452324812yes—
211C4—22no—
212C4—22no—
213C4—42no—
214C4—5082no—
215C4—2548no—
216C4#1686120514yes—
217C2—44no—
218C2—5084no—
219C2#2255038311yescharacteristic
220C2#2255038514yes—
221C2#2255038711yescharacteristic
222C2C2×#1686120511yescharacteristic
subgroupG/H
centerC22×C4C22×D127
commutator, socle, cosocleC23×C127C24
FrattiniC23C23×D127
FittingC2×C4×D4×C127C2
hypercenterC2×C4×D4D127
hypocenterC127#2397
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4×D508C2semidirect
2. C2×C4×C22⋊D127C2semidirect
3. #22549812C2semidirect
4. #22549811C2semidirect
5. #22549785C2semidirect
6. #22549784C2semidirect
7. #22549722C2semidirect
8. C2×C4×D4×C127C2semidirect
43. C4×Dic127D4semidirect
55. C42×C127D4semidirect
112. C127#2397semidirect
115. C2×C4×D4D127semidirect
167. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D5082
C2×C4×C22⋊D1272
#225498122
#225498112
#225497852
#225497842
#225497222
C2×C4×D4×C1272
#2397127