Finite Group Databasebeta

← #30648108

Subgroups — #30648108

Order 16256.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 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#22550712C211yesmaximal, characteristic
2C2×#16861240C212yesmaximal, complemented
3#22550058C214yesmaximal, complemented
4#22550053C214yesmaximal, complemented
5#22549894C212yesmaximal, complemented
6#22549860C211yesmaximal, characteristic, complemented
7#536×C127C211yesmaximal, characteristic, complemented
8#16861257C2212yes—
9#16861256—22no—
10#16861255—22no—
11#16861254—22no—
12#16861254C2212yes—
13C2×#14523248—22no—
14C2×#14523248C2212yes—
15#41×D127—24no—
16#16861240—24no—
17#16861240C2218yescomplemented
18C2×#14523238—22no—
19C2×#14523235—22no—
20C2×#14523232C2212yescomplemented
21#16861137C2218yescomplemented
22#16861128C2212yescomplemented
23#16861126C2212yescomplemented
24C2×#41×C127C2211yescharacteristic
25C22×Q8×C127C2211yescharacteristic, complemented
26C4⋊Q8×C127C2212yescomplemented
27#124×C127C2212yescomplemented
28#123×C127—22no—
29C22⋊Q8×C127—22no—
30#117×C127C2211yescharacteristic, complemented
31#14523249—24no—
32#14523248—24no—
33#14523248—24no—
34#14523248—22no—
35#14523248—24no—
36#14523248—22no—
37#14523248D414yes—
38#14523248C2314yes—
39#14523247—28no—
40#14523247—24no—
41C2×Dic254—22no—
42C2×Dic254D412yes—
43Q8×D127—24no—
44C2×D508—22no—
45C2×D508D412yes—
46C2×C22⋊D127—24no—
47C2×C22⋊D127—22no—
48D4×D127—28no—
49D4×D127—24no—
50C2×C4×D127—24no—
51C2×C4×D127—22no—
52#14523238—28no—
53#14523235—28no—
54#14523232—22no—
55#14523232C2314yes—
56C2×C127⋊C8—22no—
57#41×C127—22no—
58#41×C127—22no—
59C2×Q8×C127—21no—
60C2×Q8×C127—22no—
61C2×Q8×C127—22no—
62C2×Q8×C127C2312yes—
63C2×Q8×C127C2311yescharacteristic
64C2×D4×C127D412yes—
65C2×D4×C127C2311yescharacteristic
66C22×C4×C127—21no—
67C22×C4×C127D412yes—
68C22×C4×C127C2311yescharacteristic
69C4⋊C4×C127—21no—
70C4⋊C4×C127—24no—
71C22⋊C4×C127—21no—
72C22⋊C4×C127—24no—
73C42×C127C2312yes—
74C22×D127—24no—
75C22×D127—22no—
76Dic254—24no—
77Dic254—22no—
78Dic254C2×D414yes—
79D508—24no—
80D508—22no—
81D508C2×D414yes—
82C22⋊D127—28no—
83C22⋊D127—24no—
84C22⋊D127—24no—
85C2×Dic127—24no—
86C2×Dic127—22no—
87C4×D127—28no—
88C4×D127—24no—
89C4×D127—24no—
90C127⋊C8—24no—
91C23×C127C2×D412yes—
92C23×C127C2×D411yescharacteristic
93Q8×C127—22no—
94Q8×C127—24no—
95Q8×C127—41no—
96Q8×C127—41no—
97D4×C127—24no—
98D4×C127—22no—
99C2×C4×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—42no—
104C2×C4×C127C2×D412yes—
105C2×C4×C127C2×D412yes—
106C2×C4×C127C2×D412yes—
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2411yescharacteristic
109C22×C127—22no—
110C22×C127—41no—
111C22×C127—42no—
112C22×C127#12012yes—
113C22×C127C22×D411yescharacteristic
114C2×D127—24no—
115C2×D127—82no—
116C2×D127—84no—
117C4×C127—22no—
118C4×C127—42no—
119C4×C127—42no—
120C4×C127#12011yescharacteristic
121C4×C127#12011yescharacteristic
122C4×C127C22×D412yes—
123Dic127—24no—
124C2×C127—21no—
125C2×C127—22no—
126C2×C127—42no—
127C2×C127C2×#12011yescharacteristic
128D127—44no—
129#3022—1271nomaximal, Sylow
130C127#302211yescharacteristic, complemented, Sylow
131#585—1271no—
132C2×#137—1272no—
133#536D12711yescharacteristic, complemented
134#456—1274no—
135#455—1274no—
136#421—1272no—
137#413—1271no—
1382−1+4—1272no—
1392−1+4—2542no—
140H(5,𝔽2)—1272no—
141H(5,𝔽2)—2542no—
142H(5,𝔽2)—2542no—
143C2×#41—1272no—
144C2×#41—2544no—
145C2×#41—2542no—
146C2×#41C2×D12711yescharacteristic
147C22×Q8C2×D12711yescharacteristic, complemented
148#137—1278no—
149#137—2544no—
150C2×Q16—2542no—
151C2×SD16—2542no—
152C2×M16—1272no—
153C4⋊Q8C2×D12712yescomplemented
154#124C2×D12712yescomplemented
155#123—22no—
156C22⋊Q8—22no—
157#117C2×D12711yescharacteristic, complemented
158#104—1278no—
159#101—1272no—
160#100—1272no—
161#41—22no—
162#41—22no—
163#41—1274no—
164#41—1274no—
165#41—2544no—
166#41—2544no—
167#41—2544no—
168#41—2548no—
169#41—2544no—
170#41—2542no—
171#41—2542no—
172#41—2544no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—1272no—
177C2×Q8—2542no—
178C2×Q8—2544no—
179C2×Q8C22×D12712yes—
180C2×Q8C22×D12711yescharacteristic
181C2×D4—1272no—
182C2×D4—2542no—
183C2×D4—2544no—
184C2×D4—2548no—
185C2×D4—2542no—
186C2×D4—2544no—
187C2×D4C22⋊D12712yes—
188C2×D4C22×D12711yescharacteristic
189C22×C4—21no—
190C22×C4—2544no—
191C22×C4—2542no—
192C22×C4C22⋊D12712yes—
193C22×C4C22×D12711yescharacteristic
194Q16—2548no—
195SD16—2548no—
196M16—1274no—
197M16—2542no—
198C2×C8—2542no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C22×D12712yes—
204C23—2544no—
205C23—2542no—
206C23C2×C22⋊D12712yes—
207C23C2×C22⋊D12711yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—41no—
211Q8—41no—
212Q8—1274no—
213Q8—2544no—
214Q8—2542no—
215D4—24no—
216D4—22no—
217D4—1274no—
218D4—2544no—
219D4—2548no—
220D4—2544no—
221D4—2544no—
222D4—2542no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—2548no—
229C2×C4—2544no—
230C2×C4—2542no—
231C2×C4—2544no—
232C2×C4—2544no—
233C2×C4D4×D12712yes—
234C2×C4D4×D12712yes—
235C2×C4C2×C22⋊D12712yes—
236C2×C4C2×C22⋊D12711yescharacteristic
237C2×C4C23×D12711yescharacteristic
238C8—2544no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—10162no—
243C22—10164no—
244C22—2544no—
245C22#1686117712yes—
246C22C22×C22⋊D12711yescharacteristic
247C4—22no—
248C4—42no—
249C4—42no—
250C4—2544no—
251C4#1686117711yescharacteristic
252C4#1686117711yescharacteristic
253C4C2×D4×D12712yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—5084no—
258C2C2×#1686117711yescharacteristic
subgroupG/H
centerC2C2×#16861177
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#536×C127C2
socleC2×C127C2×#120
prime radical#22550712C2
hypercenter#536D127
hypocenterC127#3022
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#16861240C2semidirect
3. #22550058C2semidirect
4. #22550053C2semidirect
5. #22549894C2semidirect
6. #22549860C2semidirect
7. #536×C127C2semidirect
17. #16861240C22semidirect
20. C2×#14523232C22semidirect
21. #16861137C22semidirect
22. #16861128C22semidirect
23. #16861126C22semidirect
25. C22×Q8×C127C22semidirect
26. C4⋊Q8×C127C22semidirect
27. #124×C127C22semidirect
30. #117×C127C22semidirect
130. C127#3022semidirect
133. #536D127semidirect
147. C22×Q8C2×D127semidirect
153. C4⋊Q8C2×D127semidirect
154. #124C2×D127semidirect
157. #117C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225507122
C2×#168612402
#225500582
#225500532
#225498942
#225498602
#536×C1272
#3022127