Finite Group Databasebeta

← #30650680

Subgroups — #30650680

Order 16256.

408 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550591C211yesmaximal, characteristic, complemented
2#22550222C211yesmaximal, characteristic, complemented
3#22550197C212yesmaximal, complemented
4#22550171C211yesmaximal, characteristic, complemented
5#22550110C212yesmaximal, complemented
6#22550068C211yesmaximal, characteristic, complemented
7#22550021C212yesmaximal, complemented
8C4×#14523235C211yesmaximal, characteristic, complemented
9D4×C127⋊C8C211yesmaximal, characteristic, complemented
10#22549872C212yesmaximal, complemented
11#550×C127C211yesmaximal, characteristic, complemented
12C2×#14523235C2211yescharacteristic
13C2×#14523233C2212yes—
14C22×C127⋊C8C2212yes—
15#16861209C2211yescharacteristic
16#16861195—22no—
17#16861183C2212yes—
18#16861180—21no—
19C4×D508C2211yescharacteristic
20#16861168C2212yes—
21#16861167—21no—
22#16861142C2212yes—
23#16861140C2211yescharacteristic
24C4⋊(C127⋊C8)C2211yescharacteristic
25#16861133C2212yes—
26#16861133C2211yescharacteristic
27#16861130C2212yes—
28#16861130C2212yes—
29#16861130C2212yes—
30#16861129C2211yescharacteristic
31C22⋊(C127⋊C8)C2212yes—
32C4×C127⋊C8C2211yescharacteristic
33C2×#41×C127C2212yes—
34#127×C127—21no—
35#127×C127C2211yescharacteristic
36#121×C127—22no—
37#121×C127C2212yes—
38C4×Q8×C127C2211yescharacteristic
39C4×D4×C127—21no—
40C4×D4×C127C2211yescharacteristic
41C2×Dic254—21no—
42C2×D508C2311yescharacteristic
43C2×C22⋊D127—22no—
44C2×C4×D127—21no—
45#14523235—22no—
46#14523233—42no—
47C2×C127⋊C8—42no—
48C2×C127⋊C8C2311yescharacteristic
49C2×C127⋊C8C2312yes—
50C2×C127⋊C8C2311yescharacteristic
51C4⋊Dic127—21no—
52C4⋊Dic127C2312yes—
53C4⋊Dic127C2311yescharacteristic
54C22⋊Dic127—22no—
55(C2×D127)⋊C4—21no—
56(C2×D127)⋊C4—22no—
57C4×Dic127—22no—
58#41×C127—24no—
59#41×C127—42no—
60C2×Q8×C127C2311yescharacteristic
61C2×D4×C127—22no—
62C2×D4×C127—22no—
63C2×D4×C127—21no—
64C2×D4×C127—21no—
65C2×D4×C127D411yescharacteristic
66C2×D4×C127C2312yes—
67C22×C4×C127—22no—
68C22×C4×C127C2312yes—
69C4⋊C4×C127—21no—
70C4⋊C4×C127D411yescharacteristic
71C4⋊C4×C127C2311yescharacteristic
72C22⋊C4×C127—22no—
73C22⋊C4×C127D412yes—
74C42×C127—21no—
75C42×C127C2311yescharacteristic
76C22×D127—21no—
77Dic254—81no—
78D508—21no—
79D508—21no—
80C22⋊D127—82no—
81C2×Dic127—21no—
82C2×Dic127—22no—
83C4×D127—42no—
84C127⋊C8—21no—
85C127⋊C8—21no—
86C127⋊C8—22no—
87C23×C127—22no—
88C23×C127C2×D412yes—
89Q8×C127—21no—
90Q8×C127#4112yes—
91D4×C127—22no—
92D4×C127—22no—
93D4×C127—22no—
94D4×C127—44no—
95D4×C127—42no—
96D4×C127—81no—
97D4×C127—82no—
98C2×C4×C127—21no—
99C2×C4×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127—21no—
102C2×C4×C127—44no—
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D412yes—
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2411yescharacteristic
107C22×C127—22no—
108C22×C127—22no—
109C22×C127—22no—
110C22×C127—42no—
111C22×C127—42no—
112C22×C127—42no—
113C22×C127C22×D411yescharacteristic
114C2×D127—41no—
115C2×D127—41no—
116C2×D127—41no—
117C4×C127—22no—
118C4×C127—22no—
119C4×C127—21no—
120C4×C127—41no—
121C4×C127#13512yes—
122C4×C127C2×#4111yescharacteristic
123C4×C127H(5,𝔽2)11yescharacteristic
124Dic127—41no—
125Dic127—42no—
126C2×C127—42no—
127C2×C127—82no—
128C2×C127#54611yescharacteristic
129C2×C127C2×#13511yescharacteristic
130C2×C127#57611yescharacteristic
131D127—81no—
132#3305—1271nomaximal, Sylow
133C127#330511yescharacteristic, complemented, Sylow
134#555—1271no—
135#550D12711yescharacteristic, complemented
136#486—1271no—
137#482—1272no—
138#477—1271no—
139#466—1272no—
140#459—1271no—
141#449—1272no—
142C4×SD16—1271no—
143C8×D4—1271no—
144#416—1272no—
145C2×#41C2×D12712yes—
146C2×SD16—1271no—
147C2×D8—1272no—
148C22×C8—1272no—
149C4⋊Q8—1271no—
150#127—21no—
151#127C2×D12711yescharacteristic
152#124—2542no—
153#121—22no—
154#121—1272no—
155#121—2541no—
156#121C2×D12712yes—
157C4×Q8C2×D12711yescharacteristic
158C4×D4—21no—
159C4×D4—1271no—
160C4×D4C2×D12711yescharacteristic
161#117—1272no—
162#117—2541no—
163#107—1272no—
164#106—1271no—
165C4⋊C8—1271no—
166#103—1271no—
167#103—1272no—
168#102—1272no—
169#102—1272no—
170#102—1272no—
171#102—1271no—
172C22⋊C8—1272no—
173C4×C8—1271no—
174#41—24no—
175#41—42no—
176C2×Q8—2541no—
177C2×Q8C22×D12711yescharacteristic
178C2×D4—22no—
179C2×D4—22no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—1271no—
183C2×D4—2542no—
184C2×D4C22⋊D12711yescharacteristic
185C2×D4C22×D12712yes—
186C22×C4—22no—
187C22×C4—2541no—
188C22×C4C22×D12712yes—
189SD16—2542no—
190D8—5082no—
191C2×C8—1271no—
192C2×C8—1271no—
193C2×C8—1272no—
194C2×C8—5082no—
195C4⋊C4—21no—
196C4⋊C4—1271no—
197C4⋊C4—1272no—
198C4⋊C4—2541no—
199C4⋊C4C22⋊D12711yescharacteristic
200C4⋊C4C22×D12711yescharacteristic
201C22⋊C4—22no—
202C22⋊C4—2542no—
203C22⋊C4—2542no—
204C22⋊C4—2541no—
205C22⋊C4C22⋊D12712yes—
206C42—21no—
207C42—2542no—
208C42C22×D12711yescharacteristic
209C23—22no—
210C23—2541no—
211C23C2×C22⋊D12712yes—
212Q8—21no—
213Q8—10161no—
214Q8#1452324712yes—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—44no—
219D4—42no—
220D4—81no—
221D4—82no—
222D4—10162no—
223D4—2541no—
224D4—2541no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—44no—
230C2×C4—5082no—
231C2×C4—2541no—
232C2×C4—2542no—
233C2×C4C2×C22⋊D12712yes—
234C2×C4C2×C22⋊D12711yescharacteristic
235C2×C4C2×C22⋊D12711yescharacteristic
236C2×C4C23×D12711yescharacteristic
237C8—2541no—
238C8—2541no—
239C8—2542no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—5081no—
247C22—5081no—
248C22—5081no—
249C22C22×C22⋊D12711yescharacteristic
250C4—21no—
251C4—22no—
252C4—22no—
253C4—41no—
254C4—5081no—
255C4—5082no—
256C4#1686122912yes—
257C4C2×#1452324711yescharacteristic
258C4#1686125411yescharacteristic
259C2—42no—
260C2—82no—
261C2—10161no—
262C2#2255052011yescharacteristic
263C2C2×#1686122911yescharacteristic
264C2#2255067911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#550×C127C2
socleC22×C127C22×D4
hypercenter#550D127
hypocenterC127#3305
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #22550591C2semidirect
2. #22550222C2semidirect
3. #22550197C2semidirect
4. #22550171C2semidirect
5. #22550110C2semidirect
6. #22550068C2semidirect
7. #22550021C2semidirect
8. C4×#14523235C2semidirect
9. D4×C127⋊C8C2semidirect
10. #22549872C2semidirect
11. #550×C127C2semidirect
133. C127#3305semidirect
135. #550D127semidirect

Maximal subgroups

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

subgroupindex
#225505912
#225502222
#225501972
#225501712
#225501102
#225500682
#225500212
C4×#145232352
D4×C127⋊C82
#225498722
#550×C1272
#3305127