Finite Group Databasebeta

← #30639857

Subgroups — #30639857

Order 16256.

638 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 75 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×#16861179C212yesmaximal, complemented
3C2×#16861123C212yesmaximal, complemented
4#22549731C211yesmaximal, characteristic, complemented
5#342×C127C211yesmaximal, characteristic, complemented
6C24×D127C2211yescharacteristic
7C22×D508—22no—
8C22×C22⋊D127C2212yes—
9#16861193—22no—
10#16861183—22no—
11#16861179—24no—
12#16861179—22no—
13#16861179—24no—
14C2×C22⋊Dic127C2211yescharacteristic
15C2×(C2×D127)⋊C4—22no—
16C2×(C2×D127)⋊C4—22no—
17#16861123—24no—
18C2×C22⋊C4×C127C2211yescharacteristic
19C2×C22⋊C4×C127C2212yes—
20C23×D127—22no—
21C23×D127—21no—
22C23×D127—22no—
23C23×D127—21no—
24C23×D127—44no—
25C23×D127D412yescomplemented
26C23×D127D412yescomplemented
27C2×D508—28no—
28C2×D508—22no—
29C2×D508—48no—
30C2×C22⋊D127—22no—
31C2×C22⋊D127—22no—
32C2×C22⋊D127—22no—
33C2×C22⋊D127—22no—
34C2×C22⋊D127—22no—
35C2×C22⋊D127—22no—
36C22×Dic127—21no—
37C22×Dic127D412yescomplemented
38C4⋊Dic127—22no—
39C22⋊Dic127—22no—
40(C2×D127)⋊C4—24no—
41(C2×D127)⋊C4—28no—
42(C2×D127)⋊C4—44no—
43C24×C127C2311yescharacteristic
44C22×C4×C127—22no—
45C22×C4×C127D412yescomplemented
46C22⋊C4×C127—22no—
47C22⋊C4×C127—22no—
48C22⋊C4×C127D414yescomplemented
49C22×D127—24no—
50C22×D127—28no—
51C22×D127—24no—
52C22×D127—22no—
53C22×D127—22no—
54C22×D127—22no—
55C22×D127—22no—
56C22×D127—44no—
57C22×D127—48no—
58C22×D127—44no—
59C22×D127—42no—
60C22×D127—44no—
61C22×D127—41no—
62C22×D127—42no—
63C22×D127—41no—
64C22×D127—44no—
65D508—416no—
66D508—82no—
67C22⋊D127—42no—
68C22⋊D127—42no—
69C22⋊D127—42no—
70C22⋊D127—42no—
71C2×Dic127—22no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—42no—
75C2×Dic127—41no—
76C23×C127—42no—
77C23×C127C2×D412yes—
78C23×C127C2×D411yescharacteristic
79C23×C127C2×D411yescharacteristic
80C23×C127C2×D412yes—
81C23×C127#4111yescharacteristic
82C2×C4×C127—22no—
83C2×C4×C127—24no—
84C2×C4×C127—44no—
85C2×C4×C127—42no—
86C22×C127—22no—
87C22×C127—24no—
88C22×C127—41no—
89C22×C127—42no—
90C22×C127—41no—
91C22×C127#12012yes—
92C22×C127#12011yescharacteristic
93C22×C127#12112yes—
94C22×C127#12111yescharacteristic
95C22×C127#12711yescharacteristic
96C2×D127—28no—
97C2×D127—24no—
98C2×D127—44no—
99C2×D127—44no—
100C2×D127—42no—
101C2×D127—48no—
102C2×D127—42no—
103C2×D127—44no—
104C2×D127—44no—
105C2×D127—42no—
106C2×D127—44no—
107C2×D127—82no—
108C4×C127—44no—
109C4×C127—42no—
110Dic127—42no—
111Dic127—81no—
112C2×C127—21no—
113C2×C127—22no—
114C2×C127—21no—
115C2×C127—22no—
116C2×C127#39211yescharacteristic
117C2×C127B(4,𝔽2)12yes—
118D127—44no—
119D127—82no—
120#2021—1271nomaximal, Sylow
121C127#202111yescharacteristic, complemented, Sylow
122#534—1271no—
123C2×#120—1272no—
124C2×#99—1272no—
125#379—1271no—
126#342D12711yescharacteristic, complemented
127C25—1271no—
128C22×D4—1272no—
129C22×D4—2542no—
130#123—2542no—
131#121—2542no—
132#120—2544no—
133#120—2542no—
134#120—2544no—
135C2×C22⋊C4—1271no—
136C2×C22⋊C4—2542no—
137C2×C22⋊C4—2542no—
138C2×C22⋊C4C2×D12711yescharacteristic
139C2×C22⋊C4C2×D12712yes—
140#99—2544no—
141C24—1272no—
142C24—1272no—
143C24—5084no—
144C24—2542no—
145C24—2541no—
146C24—2542no—
147C24—2541no—
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—22no—
159C22×C4—1272no—
160C22×C4—2541no—
161C22×C4D50812yescomplemented
162C4⋊C4—2542no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—5084no—
166C22⋊C4—2548no—
167C22⋊C4—2544no—
168C22⋊C4—2542no—
169C22⋊C4D50814yescomplemented
170C23—42no—
171C23—5084no—
172C23—5082no—
173C23—5084no—
174C23—5088no—
175C23—5084no—
176C23—5084no—
177C23—5082no—
178C23—5081no—
179C23—5081no—
180C23—2544no—
181C23—2548no—
182C23—2542no—
183C23—2542no—
184C23—2542no—
185C23—2542no—
186C23—2544no—
187C23D4×D12712yes—
188C23D4×D12711yescharacteristic
189C23C2×D50811yescharacteristic
190C23C2×D50812yes—
191C23#1452324911yescharacteristic
192D4—10162no—
193D4—50816no—
194D4—5082no—
195D4—5082no—
196D4—5082no—
197D4—5082no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—5082no—
203C2×C4—5081no—
204C2×C4—2542no—
205C2×C4—2542no—
206C2×C4—2542no—
207C22—22no—
208C22—24no—
209C22—41no—
210C22—42no—
211C22—41no—
212C22—10162no—
213C22—5084no—
214C22—5084no—
215C22—5084no—
216C22—5088no—
217C22—5082no—
218C22—5082no—
219C22—5084no—
220C22—5082no—
221C22—5084no—
222C22—2548no—
223C22—2544no—
224C22#1686117912yes—
225C22#1686117911yescharacteristic
226C22#1686118412yes—
227C22#1686118411yescharacteristic
228C22#1686120611yescharacteristic
229C4—44no—
230C4—42no—
231C4—10161no—
232C4—5082no—
233C2—22no—
234C2—21no—
235C2—22no—
236C2—21no—
237C2—10162no—
238C2—5084no—
239C2#2254978611yescharacteristic
240C2#2255006412yes—
subgroupG/H
centerC22#16861179
commutator, cosocleC24×C127C23
FrattiniC24C22×D127
Fitting#342×C127C2
socleC22×C127#120
hypercenter#342D127
hypocenterC127#2021
LCSG ▷ C24×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #22550438C2semidirect
2. C2×#16861179C2semidirect
3. C2×#16861123C2semidirect
4. #22549731C2semidirect
5. #342×C127C2semidirect
25. C23×D127D4semidirect
26. C23×D127D4semidirect
37. C22×Dic127D4semidirect
45. C22×C4×C127D4semidirect
48. C22⋊C4×C127D4semidirect
121. C127#2021semidirect
126. #342D127semidirect
161. C22×C4D508semidirect
169. C22⋊C4D508semidirect

Maximal subgroups

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

subgroupindex
#225504382
C2×#168611792
C2×#168611232
#225497312
#342×C1272
#2021127