Finite Group Databasebeta

← #30637231

Subgroups — #30637231

Order 16256.

334 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 65 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#22550386C211yesmaximal, characteristic
2#22549838C211yesmaximal, characteristic, complemented
3#95×D127C211yesmaximal, characteristic, complemented
4#22549614C211yesmaximal, characteristic, complemented
5#22549612C211yesmaximal, characteristic, complemented
6#22549610C211yesmaximal, characteristic, complemented
7#327×C127C211yesmaximal, characteristic, complemented
8C2×#14523247C411yescharacteristic, complemented
9C22×C4×D127—21no—
10C22×C4×D127C2211yescharacteristic
11C2×#14523231—21no—
12#16861191—21no—
13#16861189C411yescharacteristic, complemented
14#16861188C2211yescharacteristic
15#16861180C411yescharacteristic, complemented
16#16861177—21no—
17D4×Dic127—21no—
18(C22⋊D127)⋊C4—21no—
19#16861167C411yescharacteristic, complemented
20C22⋊(C127⋊C8)C2211yescharacteristic
21(C2×D127)⋊C8—21no—
22C127⋊#95—21no—
23C127⋊#95C2211yescharacteristic
24#121×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28#14523247—42no—
29C2×Dic254C2×C411yescharacteristic, complemented
30C2×C22⋊D127—21no—
31C2×C22⋊D127—21no—
32C22×Dic127—21no—
33C22×Dic127—21no—
34C22×Dic127D411yescharacteristic
35C2×C4×D127—24no—
36C2×C4×D127—42no—
37C2×C4×D127—42no—
38C2×C4×D127—42no—
39C2×C4×D127C2×C411yescharacteristic
40C2×C4×D127C2×C411yescharacteristic
41C2×C4×D127D411yescharacteristic
42C2×C4×D127D411yescharacteristic
43#14523231—42no—
44C2×C127⋊C8—21no—
45C4⋊Dic127—21no—
46C4⋊Dic127C2×C411yescharacteristic, complemented
47Dic127⋊C4—21no—
48C22⋊Dic127—21no—
49C22⋊Dic127—21no—
50(C2×D127)⋊C4—21no—
51(C2×D127)⋊C4—21no—
52C4×Dic127—21no—
53C2×D4×C127—21no—
54C2×D4×C127C2×C411yescharacteristic, complemented
55C22×C4×C127—21no—
56C22×C4×C127C2311yescharacteristic
57C2×C8×C127—21no—
58C4⋊C4×C127C2×C411yescharacteristic, complemented
59C22⋊C4×C127—21no—
60C22×D127—21no—
61C22×D127—21no—
62C22×D127—22no—
63C22×D127—22no—
64C22×D127C22⋊C411yescharacteristic
65C22×D127C22⋊C411yescharacteristic
66Dic254—41no—
67C22⋊D127—42no—
68C22⋊D127—42no—
69C2×Dic127—22no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—41no—
73C2×Dic127—42no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C2×Dic127C22⋊C411yescharacteristic
77C2×Dic127C22⋊C411yescharacteristic
78C4×D127—22no—
79C4×D127—22no—
80C4×D127—41no—
81C4×D127—41no—
82C4×D127—48no—
83C127⋊C8—41no—
84C23×C127—21no—
85C23×C127C2×D411yescharacteristic
86D4×C127—41no—
87D4×C127—41no—
88D4×C127—41no—
89C2×C4×C127—22no—
90C2×C4×C127—21no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127—41no—
94C2×C4×C127C22×C411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C8×C127—41no—
97C22×C127—21no—
98C22×C127—21no—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127—41no—
102C22×C127C2×C22⋊C411yescharacteristic
103C22×C127AGL(1,ℤ/8)11yescharacteristic
104C22×C127#13711yescharacteristic
105C2×D127—24no—
106C2×D127—24no—
107C2×D127—21no—
108C2×D127—21no—
109C2×D127—21no—
110C2×D127—21no—
111C2×D127#9912yes—
112C2×D127#10412yes—
113C4×C127—21no—
114C4×C127—21no—
115C4×C127—42no—
116C4×C127—41no—
117Dic127—21no—
118Dic127—21no—
119Dic127—42no—
120Dic127—41no—
121Dic127—41no—
122C2×C127—21no—
123C2×C127—21no—
124C2×C127—81no—
125C2×C127C2×#9911yescharacteristic
126C2×C127#41711yescharacteristic
127C2×C127C2×#10411yescharacteristic
128D127—22no—
129D127—22no—
130#1504—1271nomaximal, Sylow
131C127#150411yescharacteristic, complemented, Sylow
132#525—1271no—
133#407—1271no—
134C2×#95—1271no—
135#328—1271no—
136#328—1271no—
137#327—1271no—
138#327D12711yescharacteristic, complemented
139C2×#41—1271no—
140C23×C4—1271no—
141C23×C4—2541no—
142C2×M16—2541no—
143#123—2541no—
144C22⋊Q8—1271no—
145C22⋊Q8—1271no—
146#121—1271no—
147#121C2×D12711yescharacteristic
148#120—2541no—
149C4×D4—2541no—
150C4×D4—2541no—
151#117—1271no—
152C22⋊C8—1271no—
153C22⋊C8—2541no—
154C22⋊C8C2×D12711yescharacteristic
155#95—1271no—
156#95—2541no—
157#95C2×D12711yescharacteristic
158C24—1271no—
159#41—5082no—
160C2×Q8—1271no—
161C2×D4—21no—
162C2×D4—2541no—
163C2×D4—2541no—
164C2×D4C4×D12711yescharacteristic, complemented
165C22×C4—21no—
166C22×C4—1271no—
167C22×C4—1271no—
168C22×C4—1271no—
169C22×C4—1271no—
170C22×C4—1271no—
171C22×C4—5082no—
172C22×C4—5082no—
173C22×C4—5082no—
174C22×C4—2544no—
175C22×C4—2541no—
176C22×C4—2541no—
177C22×C4C22×D12711yescharacteristic
178M16—5082no—
179C2×C8—21no—
180C2×C8—2541no—
181C4⋊C4—1271no—
182C4⋊C4—2541no—
183C4⋊C4—2541no—
184C4⋊C4C4×D12711yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—2541no—
187C22⋊C4—2541no—
188C22⋊C4—2541no—
189C22⋊C4—2541no—
190C42—2541no—
191C23—21no—
192C23—1271no—
193C23—1271no—
194C23—2542no—
195C23—2542no—
196C23—2541no—
197C23—2541no—
198C23D4×D12711yescharacteristic
199Q8—5081no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—5082no—
204D4—5082no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—1271no—
211C2×C4—1271no—
212C2×C4—5081no—
213C2×C4—5081no—
214C2×C4—5088no—
215C2×C4—5081no—
216C2×C4—5081no—
217C2×C4—5082no—
218C2×C4—5081no—
219C2×C4—2542no—
220C2×C4—2542no—
221C2×C4—2542no—
222C2×C4—2541no—
223C2×C4—2541no—
224C2×C4C2×C4×D12711yescharacteristic
225C2×C4D4×D12711yescharacteristic
226C8—41no—
227C8—5081no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—1272no—
234C22—1272no—
235C22—2544no—
236C22—2544no—
237C22—2541no—
238C22—2541no—
239C22—2541no—
240C22—2541no—
241C22C22⋊C4×D12711yescharacteristic
242C22#1686123311yescharacteristic
243C22#1686123811yescharacteristic
244C4—21no—
245C4—21no—
246C4—42no—
247C4—41no—
248C4—5082no—
249C4—5081no—
250C4—5081no—
251C4—2541no—
252C4—2541no—
253C2—21no—
254C2—21no—
255C2—81no—
256C2—2542no—
257C2—2542no—
258C2#99×D12711yescharacteristic
259C2#2254987411yescharacteristic
260C2#104×D12711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#327×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550386C2
cosocleC22×C4×C127C23
hypercenter#327D127
hypocenterC127#1504
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #22549838C2semidirect
3. #95×D127C2semidirect
4. #22549614C2semidirect
5. #22549612C2semidirect
6. #22549610C2semidirect
7. #327×C127C2semidirect
8. C2×#14523247C4semidirect
13. #16861189C4semidirect
15. #16861180C4semidirect
19. #16861167C4semidirect
29. C2×Dic254C2×C4semidirect
46. C4⋊Dic127C2×C4semidirect
54. C2×D4×C127C2×C4semidirect
58. C4⋊C4×C127C2×C4semidirect
131. C127#1504semidirect
138. #327D127semidirect
164. C2×D4C4×D127semidirect
184. C4⋊C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
#225503862
#225498382
#95×D1272
#225496142
#225496122
#225496102
#327×C1272
#1504127