Finite Group Databasebeta

← #30637674

Subgroups — #30637674

Order 16256.

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

normal subgroups only

This group has 320 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#22550464C211yesmaximal, characteristic, complemented
2#22550027C211yesmaximal, characteristic, complemented
3#22550010C211yesmaximal, characteristic, complemented
4#22549612C211yesmaximal, characteristic, complemented
5#22549609C211yesmaximal, characteristic, complemented
6#22549600C211yesmaximal, characteristic, complemented
7#392×C127C211yesmaximal, characteristic, complemented
8C2×#14523236—21no—
9C2×#14523233—21no—
10#16861202—21no—
11#16861195—21no—
12#16861191—21no—
13#16861188C2211yescharacteristic
14#16861183C2211yescharacteristic
15#16861182—21no—
16#16861178—21no—
17C4×C22⋊D127—21no—
18#16861130—21no—
19#16861130—21no—
20C22⋊(C127⋊C8)C2211yescharacteristic
21C22⋊(C127⋊C8)C2211yescharacteristic
22C22×D4×C127—21no—
23C22×D4×C127C2211yescharacteristic
24C2×C22⋊C4×C127—21no—
25C2×C22⋊C4×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C2×Dic254D411yescharacteristic, complemented
28C2×D508D411yescharacteristic, complemented
29C2×C22⋊D127—21no—
30C2×C22⋊D127—21no—
31C2×C22⋊D127—21no—
32C2×C4×D127—21no—
33#14523236—42no—
34#14523233—42no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C4⋊Dic127D411yescharacteristic, complemented
38C4⋊Dic127D411yescharacteristic, complemented
39Dic127⋊C4—21no—
40Dic127⋊C4—21no—
41C22⋊Dic127—21no—
42C22⋊Dic127—21no—
43C22⋊Dic127—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46C4×Dic127—21no—
47C24×C127—21no—
48C24×C127D411yescharacteristic
49C2×D4×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127—42no—
52C2×D4×C127—42no—
53C2×D4×C127—42no—
54C2×D4×C127—42no—
55C22×C4×C127—21no—
56C22×C4×C127D411yescharacteristic
57C22×C4×C127C2311yescharacteristic
58C22⋊C4×C127—21no—
59C22⋊C4×C127—21no—
60C22⋊C4×C127—42no—
61C22×D127—21no—
62Dic254—41no—
63D508—41no—
64C22⋊D127—41no—
65C22⋊D127—41no—
66C22⋊D127—81no—
67C22⋊D127—81no—
68C2×Dic127—21no—
69C2×Dic127—21no—
70C2×Dic127—21no—
71C4×D127—81no—
72C127⋊C8—41no—
73C127⋊C8—41no—
74C23×C127—22no—
75C23×C127—21no—
76C23×C127—21no—
77C23×C127—21no—
78C23×C127—41no—
79C23×C127—41no—
80C23×C127—41no—
81C23×C127—41no—
82C23×C127—41no—
83C23×C127—41no—
84C23×C127—41no—
85C23×C127—41no—
86C23×C127C2×D411yescharacteristic
87D4×C127—22no—
88D4×C127—22no—
89D4×C127—41no—
90D4×C127—41no—
91D4×C127—48no—
92C2×C4×C127—22no—
93C2×C4×C127—21no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127—41no—
97C2×C4×C127—41no—
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C22×C127—21no—
101C22×C127—21no—
102C22×C127—42no—
103C22×C127—42no—
104C22×C127—42no—
105C22×C127—41no—
106C22×C127—41no—
107C22×C127—42no—
108C22×C127—41no—
109C22×C127—81no—
110C22×C127—81no—
111C22×C127—81no—
112C22×C127—81no—
113C22×C127—81no—
114C22×C127#12011yescharacteristic
115C22×C127#13511yescharacteristic
116C22×C127AGL(1,ℤ/8)11yescharacteristic
117C2×D127—41no—
118C2×D127—41no—
119C2×D127—41no—
120C4×C127—21no—
121C4×C127—21no—
122C4×C127—42no—
123C4×C127—41no—
124Dic127—41no—
125Dic127—41no—
126Dic127—41no—
127C2×C127—21no—
128C2×C127—21no—
129C2×C127—81no—
130C2×C127—82no—
131C2×C127#44911yescharacteristic
132C2×C127#45311yescharacteristic
133C2×C127B(4,𝔽2)11yescharacteristic
134D127—81no—
135#1600—1271nomaximal, Sylow
136C127#160011yescharacteristic, complemented, Sylow
137#538—1271no—
138#450—1271no—
139#447—1271no—
140#392D12711yescharacteristic, complemented
141#328—1271no—
142#327—1271no—
143#323—1271no—
144C22×D4—21no—
145C22×D4C2×D12711yescharacteristic
146C2×SD16—2541no—
147C2×D8—2541no—
148#126—2541no—
149#124—2541no—
150#123—2541no—
151C22⋊Q8—1271no—
152#121—1271no—
153#121—2541no—
154#120—2541no—
155C4×D4—2541no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4C2×D12711yescharacteristic
158#102—2541no—
159#102—2541no—
160C22⋊C8—1271no—
161C22⋊C8—1271no—
162#95C2×D12711yescharacteristic
163C24—21no—
164C24C22⋊D12711yescharacteristic
165C2×Q8—1271no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—1271no—
173C2×D4—2541no—
174C2×D4—2541no—
175C2×D4—2541no—
176C22×C4—21no—
177C22×C4—2541no—
178C22×C4C22⋊D12711yescharacteristic
179C22×C4C22×D12711yescharacteristic
180SD16—5082no—
181D8—5082no—
182C2×C8—2541no—
183C2×C8—2541no—
184C4⋊C4—1271no—
185C4⋊C4—1271no—
186C4⋊C4—2541no—
187C4⋊C4—2541no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—42no—
191C22⋊C4—2541no—
192C22⋊C4—2541no—
193C22⋊C4—2541no—
194C22⋊C4—2541no—
195C22⋊C4—2541no—
196C42—2541no—
197C23—22no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—2541no—
210C23C2×C22⋊D12711yescharacteristic
211Q8—5081no—
212D4—22no—
213D4—22no—
214D4—41no—
215D4—41no—
216D4—48no—
217D4—10161no—
218D4—10161no—
219D4—5081no—
220D4—5081no—
221D4—5081no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—10161no—
229C2×C4—2541no—
230C2×C4—2541no—
231C2×C4—2541no—
232C2×C4D4×D12711yescharacteristic
233C2×C4D4×D12711yescharacteristic
234C8—5081no—
235C8—5081no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—42no—
244C22—41no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22—5081no—
251C22—5081no—
252C22—5081no—
253C22#1686117711yescharacteristic
254C22#1686122911yescharacteristic
255C22#1686123511yescharacteristic
256C4—21no—
257C4—21no—
258C4—42no—
259C4—41no—
260C4—5081no—
261C4—5081no—
262C4—5081no—
263C2—21no—
264C2—21no—
265C2—81no—
266C2—82no—
267C2—10161no—
268C2#2255001911yescharacteristic
269C2#2255004311yescharacteristic
270C2#2255006211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#392×C127C2
socleC22×C127#120
hypercenter#392D127
hypocenterC127#1600
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #22550464C2semidirect
2. #22550027C2semidirect
3. #22550010C2semidirect
4. #22549612C2semidirect
5. #22549609C2semidirect
6. #22549600C2semidirect
7. #392×C127C2semidirect
27. C2×Dic254D4semidirect
28. C2×D508D4semidirect
37. C4⋊Dic127D4semidirect
38. C4⋊Dic127D4semidirect
136. C127#1600semidirect
140. #392D127semidirect

Maximal subgroups

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

subgroupindex
#225504642
#225500272
#225500102
#225496122
#225496092
#225496002
#392×C1272
#1600127