Finite Group Databasebeta

← #30637662

Subgroups — #30637662

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#22550026C211yesmaximal, characteristic, complemented
3#22550012C211yesmaximal, characteristic
4#22549794C211yesmaximal, characteristic, complemented
5#22549609C211yesmaximal, characteristic, complemented
6#22549601C211yesmaximal, characteristic, complemented
7#328×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523235—21no—
10C2×D1016—21no—
11#16861204—21no—
12#16861200—21no—
13#16861194—21no—
14#16861193—21no—
15#16861185—21no—
16#16861183C2211yescharacteristic
17(C22⋊D127)⋊C4—21no—
18C2×Dic127⋊C4C2211yescharacteristic
19C2×(C2×D127)⋊C4—21no—
20C2×(C2×D127)⋊C4—21no—
21#16861131—21no—
22#16861129—21no—
23C22⋊(C127⋊C8)C2211yescharacteristic
24C22⋊Q8×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C23×D127—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—42no—
31C2×D508D411yescharacteristic
32C2×C22⋊D127—21no—
33C22×Dic127D411yescharacteristic
34C22×Dic127D411yescharacteristic
35C2×C4×D127—21no—
36#14523235—42no—
37D1016—42no—
38C2×C127⋊C8—21no—
39C4⋊Dic127—21no—
40C4⋊Dic127—21no—
41C4⋊Dic127D411yescharacteristic
42Dic127⋊C4—21no—
43Dic127⋊C4—21no—
44Dic127⋊C4—21no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—21no—
47(C2×D127)⋊C4—21no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—42no—
50(C2×D127)⋊C4—42no—
51C4×Dic127—21no—
52C2×Q8×C127D411yescharacteristic
53C22×C4×C127—21no—
54C22×C4×C127C2311yescharacteristic
55C2×C8×C127—21no—
56C4⋊C4×C127—21no—
57C4⋊C4×C127D411yescharacteristic
58C22⋊C4×C127—21no—
59C22×D127—22no—
60C22×D127—21no—
61C22×D127—41no—
62C22×D127—41no—
63C22×D127—41no—
64C22×D127—41no—
65C22×D127—41no—
66C22×D127—41no—
67D508—22no—
68D508—22no—
69D508—41no—
70D508—41no—
71D508—41no—
72C22⋊D127—41no—
73C22⋊D127—41no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C2×Dic127—21no—
78C2×Dic127—21no—
79C2×Dic127—41no—
80C4×D127—81no—
81C127⋊C8—41no—
82C23×C127C2×D411yescharacteristic
83Q8×C127—41no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C8×C127—41no—
93C22×C127—21no—
94C22×C127—21no—
95C22×C127#12011yescharacteristic
96C22×C127#13511yescharacteristic
97C22×C127AGL(1,ℤ/8)11yescharacteristic
98C2×D127—42no—
99C2×D127—42no—
100C2×D127—41no—
101C2×D127—41no—
102C2×D127—41no—
103C2×D127—42no—
104C2×D127—81no—
105C2×D127—81no—
106C2×D127—81no—
107C2×D127—81no—
108C4×C127—21no—
109C4×C127—21no—
110C4×C127—41no—
111C4×C127—41no—
112C4×C127—81no—
113Dic127—41no—
114Dic127—41no—
115Dic127—41no—
116C2×C127—21no—
117C2×C127—21no—
118C2×C127#44911yescharacteristic
119C2×C127#45411yescharacteristic
120C2×C127B(4,𝔽2)11yescharacteristic
121D127—81no—
122D127—82no—
123#1597—1271nomaximal, Sylow
124C127#159711yescharacteristic, complemented, Sylow
125#539—1271no—
126#450—1271no—
127#447—1271no—
128#394—1271no—
129#328D12711yescharacteristic, complemented
130#327—1271no—
131#324—1271no—
132C22×D4—1271no—
133C2×SD16—2541no—
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—2541no—
147#102—2541no—
148C22⋊C8—1271no—
149C22⋊C8C2×D12711yescharacteristic
150#95C2×D12711yescharacteristic
151C24—2541no—
152C2×Q8D50811yescharacteristic
153C2×D4—1271no—
154C2×D4—5082no—
155C2×D4—2541no—
156C2×D4—2541no—
157C2×D4—2541no—
158C22×C4—21no—
159C22×C4—1271no—
160C22×C4—1271no—
161C22×C4—2541no—
162C22×C4C22×D12711yescharacteristic
163SD16—5082no—
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⋊C4D50811yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—5082no—
177C22⋊C4—5082no—
178C22⋊C4—2541no—
179C22⋊C4—2541no—
180C22⋊C4—2541no—
181C22⋊C4—2541no—
182C42—2541no—
183C23—5081no—
184C23—5081no—
185C23—5081no—
186C23—5081no—
187C23—5081no—
188C23—5081no—
189C23—2542no—
190C23—2541no—
191C23D4×D12711yescharacteristic
192Q8—41no—
193D4—5081no—
194D4—5081no—
195D4—5081no—
196D4—5081no—
197D4—5081no—
198D4—2542no—
199D4—2542no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—10161no—
207C2×C4—5081no—
208C2×C4—2541no—
209C2×C4—2541no—
210C2×C4—2541no—
211C2×C4—2541no—
212C2×C4—2541no—
213C2×C4D4×D12711yescharacteristic
214C2×C4C2×D50811yescharacteristic
215C8—41no—
216C8—5081no—
217C22—21no—
218C22—21no—
219C22—10161no—
220C22—10161no—
221C22—10161no—
222C22—10161no—
223C22—5082no—
224C22—5082no—
225C22—5081no—
226C22—5081no—
227C22—5081no—
228C22—5082no—
229C22#1686117911yescharacteristic
230C22#1686123211yescharacteristic
231C22#1686123611yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—81no—
237C4—5081no—
238C4—5081no—
239C4—5081no—
240C2—21no—
241C2—21no—
242C2—10162no—
243C2—10161no—
244C2#2255002411yescharacteristic
245C2#2255005011yescharacteristic
246C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#328×C127C2
socleC22×C127#120
prime radical#22550012C2
hypercenter#328D127
hypocenterC127#1597
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #22550475C2semidirect
2. #22550026C2semidirect
4. #22549794C2semidirect
5. #22549609C2semidirect
6. #22549601C2semidirect
7. #328×C127C2semidirect
124. C127#1597semidirect
129. #328D127semidirect

Maximal subgroups

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

subgroupindex
#225504752
#225500262
#225500122
#225497942
#225496092
#225496012
#328×C1272
#1597127