Finite Group Databasebeta

← #30637748

Subgroups — #30637748

Order 16256.

288 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 51 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#22550434C211yesmaximal, characteristic
2#22550194C211yesmaximal, characteristic, complemented
3#22549811C211yesmaximal, characteristic, complemented
4#22549611C211yesmaximal, characteristic, complemented
5#22549610C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic, complemented
7#481×C127C211yesmaximal, characteristic, complemented
8C2×D4×D127—21no—
9#16861194—21no—
10#16861191—21no—
11#16861183C2211yescharacteristic
12#16861181—21no—
13#16861179—21no—
14#16861177—21no—
15C2×Dic127⋊C4—21no—
16C2×(C2×D127)⋊C4—21no—
17C2×(C2×D127)⋊C4C2211yescharacteristic
18#16861140—21no—
19#16861129—21no—
20C22⋊(C127⋊C8)C2211yescharacteristic, complemented
21C127⋊#95C2211yescharacteristic, complemented
22#121×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic, complemented
24#106×C127—21no—
25#102×C127—21no—
26C22⋊C8×C127C2211yescharacteristic, complemented
27C23×D127—21no—
28C23×D127D411yescharacteristic
29C2×D508—21no—
30C2×D508D411yescharacteristic
31C2×C22⋊D127—21no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34D4×D127—42no—
35D4×D127—42no—
36C22×Dic127—21no—
37C22×Dic127D411yescharacteristic
38C2×C4×D127—21no—
39C2×C127⋊C8—21no—
40C4⋊Dic127D411yescharacteristic
41Dic127⋊C4—21no—
42Dic127⋊C4—42no—
43C22⋊Dic127—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—21no—
47(C2×D127)⋊C4—21no—
48(C2×D127)⋊C4—42no—
49C2×D4×C127—21no—
50C2×D4×C127D411yescharacteristic
51C22×C4×C127—21no—
52C22×C4×C127C2311yescharacteristic
53C2×C8×C127—21no—
54C4⋊C4×C127—21no—
55C4⋊C4×C127—21no—
56C4⋊C4×C127D411yescharacteristic
57C22⋊C4×C127—21no—
58C22×D127—21no—
59C22×D127—21no—
60C22×D127—21no—
61C22×D127—42no—
62C22×D127—41no—
63C22×D127—41no—
64C22×D127—81no—
65C22×D127—81no—
66C22×D127SD1612yes—
67D508—41no—
68D508—42no—
69C22⋊D127—42no—
70C22⋊D127—41no—
71C22⋊D127—41no—
72C22⋊D127—81no—
73C2×Dic127—22no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—41no—
77C2×Dic127—41no—
78C2×Dic127—41no—
79C4×D127—42no—
80C127⋊C8—41no—
81C23×C127—21no—
82C23×C127C2×D411yescharacteristic
83D4×C127—41no—
84D4×C127—41no—
85D4×C127—41no—
86C2×C4×C127—21no—
87C2×C4×C127—41no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127—41no—
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C8×C127—41no—
94C22×C127—21no—
95C22×C127—21no—
96C22×C127—41no—
97C22×C127—41no—
98C22×C127—41no—
99C22×C127#12011yescharacteristic
100C22×C127C2×SD1611yescharacteristic
101C22×C127AGL(1,ℤ/8)11yescharacteristic
102C2×D127—22no—
103C2×D127—41no—
104C2×D127—41no—
105C2×D127—42no—
106C2×D127—41no—
107C2×D127—41no—
108C2×D127—41no—
109C2×D127—81no—
110C2×D127—84no—
111C4×C127—21no—
112C4×C127—21no—
113C4×C127—41no—
114C4×C127—81no—
115Dic127—42no—
116Dic127—41no—
117Dic127—41no—
118C2×C127—21no—
119C2×C127—21no—
120C2×C127—81no—
121C2×C127#45011yescharacteristic
122C2×C127#45311yescharacteristic
123C2×C127#45811yescharacteristic
124D127—42no—
125D127—81no—
126#1613—1271nomaximal, Sylow
127C127#161311yescharacteristic, complemented, Sylow
128#534—1271no—
129#481—1271no—
130#481D12711yescharacteristic, complemented
131#399—1271no—
132#327—1271no—
133#327—1271no—
134#323—1271no—
135C22×D4—2541no—
136#123—2541no—
137#123—2541no—
138#121—1271no—
139#121—2541no—
140#121C2×D12711yescharacteristic
141#120—2541no—
142#120—2541no—
143C2×C4⋊C4—2541no—
144C2×C4⋊C4C2×D12711yescharacteristic, complemented
145C2×C22⋊C4—1271no—
146C2×C22⋊C4—2541no—
147#106—21no—
148#106—2541no—
149#102—21no—
150#102—2541no—
151C22⋊C8—1271no—
152C22⋊C8C2×D12711yescharacteristic, complemented
153#95—1271no—
154C24—1271no—
155C24—2541no—
156C2×D4—21no—
157C2×D4—1271no—
158C2×D4—5082no—
159C2×D4—5082no—
160C2×D4—2541no—
161C2×D4—2541no—
162C2×D4—2541no—
163C2×D4—2541no—
164C2×D4C22⋊D12711yescharacteristic
165C22×C4—21no—
166C22×C4—1271no—
167C22×C4—2541no—
168C22×C4—2541no—
169C22×C4C22×D12711yescharacteristic
170C2×C8—21no—
171C2×C8—2541no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—1271no—
175C4⋊C4—5082no—
176C4⋊C4—2541no—
177C4⋊C4C22⋊D12711yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—5082no—
180C22⋊C4—2541no—
181C22⋊C4—2541no—
182C22⋊C4—2541no—
183C22⋊C4—2541no—
184C22⋊C4—2541no—
185C23—21no—
186C23—1272no—
187C23—10161no—
188C23—10161no—
189C23—5081no—
190C23—5081no—
191C23—5082no—
192C23—2541no—
193C23—2541no—
194C23—2541no—
195C23D4×D12711yescharacteristic
196D4—41no—
197D4—41no—
198D4—41no—
199D4—10161no—
200D4—5081no—
201D4—5082no—
202D4—5082no—
203D4—5081no—
204D4—5081no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—5082no—
211C2×C4—5081no—
212C2×C4—5081no—
213C2×C4—5081no—
214C2×C4—2542no—
215C2×C4—2541no—
216C2×C4—2541no—
217C2×C4D4×D12711yescharacteristic
218C2×C4C2×C22⋊D12711yescharacteristic
219C8—41no—
220C8—5081no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—10161no—
227C22—10164no—
228C22—5082no—
229C22—5081no—
230C22—5081no—
231C22—5081no—
232C22—5081no—
233C22—5081no—
234C22—2542no—
235C22#1686117711yescharacteristic
236C22SD16×D12711yescharacteristic
237C22#1686123611yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—81no—
242C4—5082no—
243C4—5081no—
244C4—5081no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—10161no—
249C2—5082no—
250C2#2255002511yescharacteristic
251C2#2255004211yescharacteristic
252C2#2255006511yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#481×C127C2
socleC22×C127#120
prime radical#22550434C2
hypercenter#481D127
hypocenterC127#1613
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. #22550194C2semidirect
3. #22549811C2semidirect
4. #22549611C2semidirect
5. #22549610C2semidirect
6. #22549599C2semidirect
7. #481×C127C2semidirect
20. C22⋊(C127⋊C8)C22semidirect
21. C127⋊#95C22semidirect
23. C2×C4⋊C4×C127C22semidirect
26. C22⋊C8×C127C22semidirect
127. C127#1613semidirect
130. #481D127semidirect
144. C2×C4⋊C4C2×D127semidirect
152. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504342
#225501942
#225498112
#225496112
#225496102
#225495992
#481×C1272
#1613127