Finite Group Databasebeta

← #30648133

Subgroups — #30648133

Order 16256.

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

normal subgroups only

This group has 772 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, complemented
2#22550445C212yesmaximal, complemented
3#22550440C211yesmaximal, characteristic, complemented
4#22550064C218yesmaximal, complemented
5#22549849C212yesmaximal, complemented
6#410×C127C211yesmaximal, characteristic, complemented
7#16861257—21no—
8#16861256—22no—
9#16861255—22no—
10#16861254—21no—
11#16861254C2214yes—
12C2×#14523248—22no—
13C2×#14523248C2212yes—
14#41×D127—24no—
15C22×D508C2211yescharacteristic, complemented
16#16861206—22no—
17#16861205—21no—
18#16861198—21no—
19#16861197—22no—
20#16861184—24no—
21#16861183—22no—
22#16861179C2218yescomplemented
23#16861177C2214yescomplemented
24#16861168C2211yescharacteristic, complemented
25#16861123C2218yescomplemented
26C2×#41×C127C2211yescharacteristic
27#117×C127C2212yescomplemented
28#99×C127C2214yescomplemented
29C23×D127C2312yes—
30#14523249—24no—
31#14523248—24no—
32#14523248—24no—
33#14523248—22no—
34#14523248—22no—
35#14523248—24no—
36#14523248—22no—
37#14523248—22no—
38#14523247—24no—
39#14523247—28no—
40C2×Dic254—22no—
41C2×Dic254D412yescomplemented
42Q8×D127—24no—
43C2×D508—22no—
44C2×D508—28no—
45C2×D508—22no—
46C2×D508—22no—
47C2×D508—21no—
48C2×D508—21no—
49C2×D508D412yescomplemented
50C2×C22⋊D127—24no—
51C2×C22⋊D127—24no—
52C2×C22⋊D127C2314yes—
53D4×D127—24no—
54D4×D127—28no—
55C2×C4×D127—24no—
56C2×C4×D127D414yescomplemented
57C4⋊Dic127—21no—
58C22⋊Dic127C2312yes—
59(C2×D127)⋊C4—28no—
60(C2×D127)⋊C4—24no—
61C4×Dic127—21no—
62#41×C127—22no—
63#41×C127—22no—
64C2×Q8×C127D411yescharacteristic, complemented
65C2×D4×C127D411yescharacteristic, complemented
66C2×D4×C127C2312yes—
67C22×C4×C127D412yescomplemented
68C22×C4×C127C2311yescharacteristic
69C4⋊C4×C127—22no—
70C22⋊C4×C127C2314yes—
71C42×C127—22no—
72C22×D127—24no—
73C22×D127—22no—
74C22×D127—24no—
75C22×D127—82no—
76C22×D127C2×D414yes—
77Dic254—24no—
78Dic254—22no—
79Dic254—22no—
80D508—24no—
81D508—22no—
82D508—22no—
83D508—41no—
84D508—41no—
85D508—42no—
86D508—88no—
87C22⋊D127—28no—
88C22⋊D127—24no—
89C22⋊D127—24no—
90C22⋊D127—84no—
91C2×Dic127—24no—
92C2×Dic127—22no—
93C2×Dic127C2×D414yes—
94C4×D127—28no—
95C4×D127—24no—
96C4×D127—24no—
97C23×C127C2×D412yes—
98C23×C127C2411yescharacteristic
99Q8×C127—22no—
100D4×C127—24no—
101D4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—22no—
104C2×C4×C127—24no—
105C2×C4×C127C2×D412yes—
106C2×C4×C127C2×D412yes—
107C2×C4×C127C2×D412yes—
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2×D411yescharacteristic
110C22×C127—22no—
111C22×C127—41no—
112C22×C127—42no—
113C22×C127C22×D412yes—
114C22×C127C22×D411yescharacteristic
115C2×D127—24no—
116C2×D127—44no—
117C2×D127—44no—
118C2×D127—48no—
119C2×D127—42no—
120C2×D127—84no—
121C4×C127—22no—
122C4×C127—44no—
123C4×C127#12011yescharacteristic
124C4×C127#12011yescharacteristic
125C4×C127#12012yes—
126Dic127—24no—
127Dic127—42no—
128C2×C127—21no—
129C2×C127—22no—
130C2×C127—42no—
131C2×C127C2×#12011yescharacteristic
132D127—44no—
133D127—82no—
134#3027—1271nomaximal, Sylow
135C127#302711yescharacteristic, complemented, Sylow
136#585—1271no—
137#535—1271no—
138#535—1272no—
139B(4,𝔽2)—1278no—
140#410—1272no—
141#410D12711yescharacteristic, complemented
1422−1+4—2542no—
1432−1+4—2541no—
144H(5,𝔽2)—1274no—
145H(5,𝔽2)—2542no—
146H(5,𝔽2)—2541no—
147C2×#41—1272no—
148C2×#41—2544no—
149C2×#41—2542no—
150C2×#41C2×D12711yescharacteristic
151C22×D4—1271no—
152#127—2542no—
153#127—2541no—
154#124—2542no—
155#124—2541no—
156#121—2544no—
157#121—2542no—
158#120—1274no—
159#120—1278no—
160#117—1271no—
161#117C2×D12712yescomplemented
162#99—1278no—
163#99C2×D12714yescomplemented
164C24—1272no—
165#41—22no—
166#41—22no—
167#41—2544no—
168#41—2544no—
169#41—2548no—
170#41—2544no—
171#41—2544no—
172#41—2542no—
173#41—2542no—
174#41—2542no—
175#41—2542no—
176#41—2544no—
177C2×Q8—1272no—
178C2×Q8—2542no—
179C2×Q8—2544no—
180C2×Q8D50811yescharacteristic, complemented
181C2×D4—1274no—
182C2×D4—1272no—
183C2×D4—2548no—
184C2×D4—2542no—
185C2×D4—2548no—
186C2×D4—2544no—
187C2×D4—2544no—
188C2×D4—2544no—
189C2×D4—2542no—
190C2×D4—2542no—
191C2×D4—2541no—
192C2×D4—2541no—
193C2×D4D50811yescharacteristic, complemented
194C2×D4C22×D12712yes—
195C22×C4—1274no—
196C22×C4—2544no—
197C22×C4D50812yescomplemented
198C22×C4C22×D12711yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—2541no—
201C22⋊C4—1272no—
202C22⋊C4—2548no—
203C22⋊C4—2544no—
204C22⋊C4C22×D12714yes—
205C42—22no—
206C42—2541no—
207C23—1274no—
208C23—10162no—
209C23—2544no—
210C23—2544no—
211C23—2542no—
212C23C2×D50812yes—
213C23C23×D12711yescharacteristic
214Q8—22no—
215Q8—2544no—
216Q8—2542no—
217Q8—2542no—
218D4—24no—
219D4—22no—
220D4—10168no—
221D4—10164no—
222D4—5081no—
223D4—5081no—
224D4—5082no—
225D4—2544no—
226D4—2548no—
227D4—2544no—
228D4—2544no—
229D4—2542no—
230D4—2542no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—1274no—
235C2×C4—2548no—
236C2×C4—2542no—
237C2×C4—2544no—
238C2×C4—2544no—
239C2×C4—2544no—
240C2×C4D4×D12712yes—
241C2×C4D4×D12712yes—
242C2×C4C2×D50812yes—
243C2×C4C2×D50811yescharacteristic
244C2×C4C2×D50811yescharacteristic
245C22—22no—
246C22—41no—
247C22—42no—
248C22—10164no—
249C22—5088no—
250C22—5084no—
251C22—5084no—
252C22—5082no—
253C22—2544no—
254C22C2×D4×D12712yes—
255C22C22×D50811yescharacteristic
256C4—22no—
257C4—44no—
258C4—5082no—
259C4—2544no—
260C4#1686117911yescharacteristic
261C4#1686117911yescharacteristic
262C4#1686117912yes—
263C2—22no—
264C2—21no—
265C2—42no—
266C2—10162no—
267C2—5084no—
268C2C2×#1686117911yescharacteristic
subgroupG/H
centerC2C2×#16861179
commutator, cosocleC23×C127C24
FrattiniC23C23×D127
Fitting#410×C127C2
socleC2×C127C2×#120
hypercenter#410D127
hypocenterC127#3027
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #22550712C2semidirect
2. #22550445C2semidirect
3. #22550440C2semidirect
4. #22550064C2semidirect
5. #22549849C2semidirect
6. #410×C127C2semidirect
15. C22×D508C22semidirect
22. #16861179C22semidirect
23. #16861177C22semidirect
24. #16861168C22semidirect
25. #16861123C22semidirect
27. #117×C127C22semidirect
28. #99×C127C22semidirect
41. C2×Dic254D4semidirect
49. C2×D508D4semidirect
56. C2×C4×D127D4semidirect
64. C2×Q8×C127D4semidirect
65. C2×D4×C127D4semidirect
67. C22×C4×C127D4semidirect
135. C127#3027semidirect
141. #410D127semidirect
161. #117C2×D127semidirect
163. #99C2×D127semidirect
180. C2×Q8D508semidirect
193. C2×D4D508semidirect
197. C22×C4D508semidirect

Maximal subgroups

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

subgroupindex
#225507122
#225504452
#225504402
#225500642
#225498492
#410×C1272
#3027127