Finite Group Databasebeta

← #30639856

Subgroups — #30639856

Order 16256.

638 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 73 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#22550438C211yesmaximal, characteristic, complemented
2C2×#16861178C212yesmaximal, complemented
3C2×#16861123C212yesmaximal, complemented
4#22549643C211yesmaximal, characteristic, complemented
5#379×C127C211yesmaximal, characteristic, complemented
6C22×C22⋊D127—22no—
7C22×C22⋊D127C2212yes—
8#16861193—22no—
9#16861183—22no—
10#16861179—22no—
11#16861178—24no—
12#16861178—24no—
13C2×C22⋊Dic127—22no—
14C2×C22⋊Dic127C2212yes—
15#16861123—24no—
16C25×C127C2211yescharacteristic
17C2×C22⋊C4×C127—22no—
18C2×C22⋊C4×C127C2211yescharacteristic
19C2×C22⋊C4×C127C2211yescharacteristic
20C23×D127D412yescomplemented
21C2×D508—22no—
22C2×C22⋊D127—28no—
23C2×C22⋊D127—22no—
24C2×C22⋊D127—22no—
25C2×C22⋊D127—22no—
26C2×C22⋊D127—22no—
27C2×C22⋊D127—22no—
28C2×C22⋊D127—22no—
29C2×C22⋊D127—48no—
30C22×Dic127—22no—
31C22×Dic127D412yescomplemented
32C4⋊Dic127—22no—
33C22⋊Dic127—28no—
34C22⋊Dic127—22no—
35C22⋊Dic127D414yescomplemented
36(C2×D127)⋊C4—24no—
37C24×C127—22no—
38C24×C127—21no—
39C24×C127—22no—
40C24×C127—21no—
41C24×C127—44no—
42C24×C127D412yescomplemented
43C24×C127C2311yescharacteristic
44C22×C4×C127—21no—
45C22×C4×C127D412yescomplemented
46C22⋊C4×C127—22no—
47C22⋊C4×C127—22no—
48C22⋊C4×C127—44no—
49C22×D127—22no—
50C22×D127—22no—
51C22×D127—22no—
52C22×D127—44no—
53D508—82no—
54C22⋊D127—416no—
55C22⋊D127—42no—
56C22⋊D127—42no—
57C22⋊D127—42no—
58C22⋊D127—42no—
59C2×Dic127—22no—
60C2×Dic127—22no—
61C2×Dic127—24no—
62C2×Dic127—22no—
63C2×Dic127—44no—
64C23×C127—24no—
65C23×C127—28no—
66C23×C127—24no—
67C23×C127—22no—
68C23×C127—44no—
69C23×C127—42no—
70C23×C127—42no—
71C23×C127—44no—
72C23×C127—44no—
73C23×C127—48no—
74C23×C127—41no—
75C23×C127—42no—
76C23×C127—41no—
77C23×C127C2×D412yes—
78C23×C127C2×D412yes—
79C23×C127C2×D411yescharacteristic
80C23×C127C2×D411yescharacteristic
81C23×C127#4111yescharacteristic
82C2×C4×C127—22no—
83C2×C4×C127—42no—
84C2×C4×C127—42no—
85C2×C4×C127—41no—
86C22×C127—22no—
87C22×C127—24no—
88C22×C127—24no—
89C22×C127—28no—
90C22×C127—42no—
91C22×C127—41no—
92C22×C127—41no—
93C22×C127—44no—
94C22×C127—44no—
95C22×C127—48no—
96C22×C127—42no—
97C22×C127—42no—
98C22×C127—44no—
99C22×C127#12011yescharacteristic
100C22×C127#12012yes—
101C22×C127#12112yes—
102C22×C127#12111yescharacteristic
103C22×C127#12711yescharacteristic
104C2×D127—44no—
105C2×D127—42no—
106C2×D127—44no—
107C2×D127—82no—
108C4×C127—42no—
109C4×C127—81no—
110Dic127—44no—
111Dic127—42no—
112C2×C127—22no—
113C2×C127—21no—
114C2×C127—22no—
115C2×C127—21no—
116C2×C127—44no—
117C2×C127#39211yescharacteristic
118C2×C127B(4,𝔽2)12yes—
119D127—82no—
120#2021—1271nomaximal, Sylow
121C127#202111yescharacteristic, complemented, Sylow
122#534—1271no—
123C2×#120—1272no—
124C2×#99—1272no—
125#379D12711yescharacteristic, complemented
126#342—1271no—
127C25C2×D12711yescharacteristic
128C22×D4—1272no—
129C22×D4—2542no—
130#123—2542no—
131#121—2542no—
132#120—2542no—
133#120—2544no—
134#120—2544no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—1272no—
137C2×C22⋊C4—2542no—
138C2×C22⋊C4C2×D12711yescharacteristic
139C2×C22⋊C4C2×D12711yescharacteristic
140#99—2544no—
141C24—22no—
142C24—21no—
143C24—22no—
144C24—21no—
145C24—44no—
146C24—1272no—
147C24D50812yescomplemented
148C24C22×D12711yescharacteristic
149C2×D4—5088no—
150C2×D4—2542no—
151C2×D4—2548no—
152C2×D4—2542no—
153C2×D4—2542no—
154C2×D4—2542no—
155C2×D4—2542no—
156C2×D4—2542no—
157C2×D4—2542no—
158C22×C4—21no—
159C22×C4—1272no—
160C22×C4—2542no—
161C22×C4C22⋊D12712yescomplemented
162C4⋊C4—2542no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—44no—
166C22⋊C4—1274no—
167C22⋊C4—2544no—
168C22⋊C4—2548no—
169C22⋊C4—2542no—
170C23—24no—
171C23—28no—
172C23—24no—
173C23—22no—
174C23—44no—
175C23—42no—
176C23—42no—
177C23—44no—
178C23—48no—
179C23—44no—
180C23—41no—
181C23—41no—
182C23—42no—
183C23—5084no—
184C23—2542no—
185C23—2542no—
186C23—2542no—
187C23D4×D12712yes—
188C23D4×D12712yes—
189C23C2×C22⋊D12711yescharacteristic
190C23C2×D50811yescharacteristic
191C23#1452324811yescharacteristic
192D4—10162no—
193D4—50816no—
194D4—5082no—
195D4—5082no—
196D4—5082no—
197D4—5082no—
198C2×C4—22no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—5084no—
203C2×C4—2542no—
204C2×C4—2544no—
205C2×C4—2542no—
206C2×C4—2542no—
207C22—22no—
208C22—24no—
209C22—28no—
210C22—24no—
211C22—42no—
212C22—41no—
213C22—41no—
214C22—44no—
215C22—44no—
216C22—42no—
217C22—48no—
218C22—42no—
219C22—44no—
220C22—10162no—
221C22—5084no—
222C22—5082no—
223C22—5084no—
224C22#1686117711yescharacteristic
225C22#1686117912yes—
226C22#1686118212yes—
227C22#1686118311yescharacteristic
228C22#1686120511yescharacteristic
229C4—42no—
230C4—81no—
231C4—5084no—
232C4—5082no—
233C2—21no—
234C2—22no—
235C2—21no—
236C2—22no—
237C2—44no—
238C2—10162no—
239C2#2254978511yescharacteristic
240C2#2255006212yes—
subgroupG/H
centerC22#16861177
commutator, cosocleC24×C127C23
FrattiniC24C22×D127
Fitting#379×C127C2
socleC22×C127#120
hypercenter#379D127
hypocenterC127#2021
LCSG ▷ C24×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #379

Complements

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

NH ≅ G/Ntype
1. #22550438C2semidirect
2. C2×#16861178C2semidirect
3. C2×#16861123C2semidirect
4. #22549643C2semidirect
5. #379×C127C2semidirect
20. C23×D127D4semidirect
31. C22×Dic127D4semidirect
35. C22⋊Dic127D4semidirect
42. C24×C127D4semidirect
45. C22×C4×C127D4semidirect
121. C127#2021semidirect
125. #379D127semidirect
147. C24D508semidirect
161. C22×C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225504382
C2×#168611782
C2×#168611232
#225496432
#379×C1272
#2021127