Finite Group Databasebeta

← #30637658

Subgroups — #30637658

Order 16256.

272 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550475C211yesmaximal, characteristic, complemented
2#22550010C211yesmaximal, characteristic
3#22549790C211yesmaximal, characteristic, complemented
4#22549613C211yesmaximal, characteristic, complemented
5#22549611C211yesmaximal, characteristic, complemented
6#22549601C211yesmaximal, characteristic, complemented
7#450×C127C211yesmaximal, characteristic, complemented
8C2×#14523233—21no—
9#16861204—21no—
10#16861200—21no—
11#16861194—21no—
12#16861193—21no—
13#16861185—21no—
14#16861183C2211yescharacteristic
15(C22⋊D127)⋊C4—21no—
16C2×Dic127⋊C4C2211yescharacteristic
17C2×C22⋊Dic127—21no—
18C2×C22⋊Dic127—21no—
19#16861130—21no—
20C22⋊(C127⋊C8)C2211yescharacteristic
21C127⋊#95C2211yescharacteristic
22C22×D4×C127C2211yescharacteristic
23C2×SD16×C127—21no—
24C22⋊Q8×C127C2211yescharacteristic
25#102×C127—21no—
26C22⋊C8×C127C2211yescharacteristic
27C2×D508D411yescharacteristic
28C2×C22⋊D127—21no—
29C22×Dic127—21no—
30C22×Dic127D411yescharacteristic
31C22×Dic127D411yescharacteristic
32C2×C4×D127—21no—
33#14523233—42no—
34C2×C127⋊C8—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127D411yescharacteristic
38Dic127⋊C4—21no—
39Dic127⋊C4—21no—
40Dic127⋊C4—21no—
41C22⋊Dic127—42no—
42C22⋊Dic127—42no—
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—21no—
47C4×Dic127—21no—
48C24×C127—21no—
49C2×Q8×C127D411yescharacteristic
50C2×D4×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—42no—
53C22×C4×C127C2311yescharacteristic
54SD16×C127—42no—
55C2×C8×C127—21no—
56C4⋊C4×C127—21no—
57C4⋊C4×C127D411yescharacteristic
58C22⋊C4×C127—21no—
59C22×D127—21no—
60D508—41no—
61C22⋊D127—41no—
62C22⋊D127—41no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—21no—
68C2×Dic127—41no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C4×D127—81no—
73C127⋊C8—41no—
74C23×C127—22no—
75C23×C127—41no—
76C23×C127—41no—
77C23×C127—41no—
78C23×C127—41no—
79C23×C127—41no—
80C23×C127—41no—
81C23×C127C2×D411yescharacteristic
82Q8×C127—41no—
83D4×C127—22no—
84D4×C127—22no—
85D4×C127—41no—
86D4×C127—41no—
87C2×C4×C127—21no—
88C2×C4×C127—21no—
89C2×C4×C127—41no—
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C8×C127—41no—
93C22×C127—21no—
94C22×C127—21no—
95C22×C127—42no—
96C22×C127—42no—
97C22×C127—42no—
98C22×C127—81no—
99C22×C127—81no—
100C22×C127—81no—
101C22×C127—81no—
102C22×C127#12011yescharacteristic
103C22×C127#13511yescharacteristic
104C22×C127AGL(1,ℤ/8)11yescharacteristic
105C2×D127—41no—
106C2×D127—41no—
107C2×D127—41no—
108C4×C127—21no—
109C4×C127—21no—
110C4×C127—41no—
111C4×C127—41no—
112Dic127—41no—
113Dic127—41no—
114Dic127—41no—
115Dic127—81no—
116C2×C127—21no—
117C2×C127—21no—
118C2×C127—82no—
119C2×C127#44911yescharacteristic
120C2×C127#45411yescharacteristic
121C2×C127B(4,𝔽2)11yescharacteristic
122D127—81no—
123#1597—1271nomaximal, Sylow
124C127#159711yescharacteristic, complemented, Sylow
125#539—1271no—
126#450D12711yescharacteristic, complemented
127#447—1271no—
128#394—1271no—
129#328—1271no—
130#327—1271no—
131#324—1271no—
132C22×D4C2×D12711yescharacteristic
133C2×SD16—21no—
134C2×D8—2541no—
135#126—2541no—
136#125—2541no—
137#123—2541no—
138#123—2541no—
139C22⋊Q8—2541no—
140C22⋊Q8C2×D12711yescharacteristic
141#121—1271no—
142C4×D4—2541no—
143C2×C4⋊C4—1271no—
144C2×C22⋊C4—2541no—
145C2×C22⋊C4—2541no—
146#102—21no—
147#102—2541no—
148C22⋊C8—1271no—
149C22⋊C8C2×D12711yescharacteristic
150#95—1271no—
151C24—21no—
152C2×Q8C22⋊D12711yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—42no—
156C2×D4—1271no—
157C2×D4—2541no—
158C22×C4—1271no—
159C22×C4—1271no—
160C22×C4—2541no—
161C22×C4—2541no—
162C22×C4C22×D12711yescharacteristic
163SD16—42no—
164D8—5082no—
165C2×C8—21no—
166C2×C8—2541no—
167C4⋊C4—21no—
168C4⋊C4—1271no—
169C4⋊C4—2541no—
170C4⋊C4—2541no—
171C4⋊C4—2541no—
172C4⋊C4—2541no—
173C4⋊C4—2541no—
174C4⋊C4C22⋊D12711yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—5082no—
177C22⋊C4—5082no—
178C22⋊C4—2541no—
179C22⋊C4—2541no—
180C22⋊C4—2541no—
181C22⋊C4—2541no—
182C42—2541no—
183C23—22no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—2541no—
191C23D4×D12711yescharacteristic
192Q8—41no—
193D4—22no—
194D4—22no—
195D4—41no—
196D4—41no—
197D4—5081no—
198D4—5081no—
199D4—5081no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—10161no—
204C2×C4—5081no—
205C2×C4—5081no—
206C2×C4—5081no—
207C2×C4—5081no—
208C2×C4—2541no—
209C2×C4—2541no—
210C2×C4—2541no—
211C2×C4—2541no—
212C2×C4—2541no—
213C2×C4D4×D12711yescharacteristic
214C2×C4C2×C22⋊D12711yescharacteristic
215C8—41no—
216C8—5081no—
217C22—21no—
218C22—21no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—81no—
223C22—81no—
224C22—81no—
225C22—81no—
226C22—5081no—
227C22—5081no—
228C22—5081no—
229C22#1686117711yescharacteristic
230C22#1686123011yescharacteristic
231C22#1686123611yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—10161no—
237C4—5081no—
238C4—5081no—
239C4—5081no—
240C2—21no—
241C2—21no—
242C2—82no—
243C2—10161no—
244C2#2255002011yescharacteristic
245C2#2255004711yescharacteristic
246C2#2255006211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#450×C127C2
socleC22×C127#120
prime radical#22550010C2
hypercenter#450D127
hypocenterC127#1597
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #22550475C2semidirect
3. #22549790C2semidirect
4. #22549613C2semidirect
5. #22549611C2semidirect
6. #22549601C2semidirect
7. #450×C127C2semidirect
124. C127#1597semidirect
126. #450D127semidirect

Maximal subgroups

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

subgroupindex
#225504752
#225500102
#225497902
#225496132
#225496112
#225496012
#450×C1272
#1597127