Finite Group Databasebeta

← #30648083

Subgroups — #30648083

Order 16256.

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

normal subgroups only

This group has 724 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#22550676C212yesmaximal, complemented
3#22550422C211yesmaximal, characteristic, complemented
4#22550061C212yesmaximal, complemented
5#22550056C212yesmaximal, complemented
6#22550051C212yesmaximal, complemented
7#22550046C212yesmaximal, complemented
8C2×#16861135C212yesmaximal, complemented
9#413×C127C211yesmaximal, characteristic, complemented
10#16861257C2212yes—
11#16861256—22no—
12#16861255—22no—
13#16861254—22no—
14#16861254C2212yes—
15C2×#14523248—22no—
16C2×#14523248C2212yes—
17C2×#14523248C2211yescharacteristic, complemented
18#41×D127—24no—
19#16861242—22no—
20#16861242C2214yescomplemented
21#16861237—22no—
22#16861237C2214yescomplemented
23#16861231—24no—
24#16861207C2211yescharacteristic, complemented
25#16861205C2211yescharacteristic, complemented
26#16861198C2211yescharacteristic, complemented
27#16861196C2211yescharacteristic, complemented
28#16861181—22no—
29C4×C22⋊D127—22no—
30C2×C4×Dic127C2211yescharacteristic, complemented
31#16861135C2214yescomplemented
32#16861135C2214yescomplemented
33C2×#41×C127C2211yescharacteristic
34C2×M16×C127C2212yescomplemented
35#101×C127C2212yescomplemented
36#100×C127C2212yescomplemented
37#14523249—24no—
38#14523248—24no—
39#14523248—24no—
40#14523248—22no—
41#14523248—24no—
42#14523248—22no—
43#14523248—22no—
44#14523248—22no—
45#14523248D414yescomplemented
46#14523248C2314yes—
47#14523247—28no—
48#14523247—24no—
49C2×Dic254—22no—
50C2×Dic254D412yescomplemented
51C2×Dic254C2311yescharacteristic
52Q8×D127—24no—
53C2×D508—22no—
54C2×D508D412yescomplemented
55C2×D508C2311yescharacteristic
56C2×C22⋊D127—24no—
57C2×C22⋊D127—22no—
58C2×C22⋊D127—22no—
59C2×C22⋊D127—21no—
60D4×D127—28no—
61D4×D127—24no—
62C22×Dic127—21no—
63C2×C4×D127—24no—
64C2×C4×D127—22no—
65C2×C4×D127—21no—
66Dic508—24no—
67#14523237—24no—
68#14523237—24no—
69D1016—24no—
70Dic127⋊C4—22no—
71C22⋊Dic127—22no—
72(C2×D127)⋊C4—22no—
73C4×Dic127—21no—
74C4×Dic127C2311yescharacteristic
75C4×Dic127C2311yescharacteristic
76#41×C127—22no—
77#41×C127—22no—
78C2×Q8×C127C2311yescharacteristic
79C2×D4×C127D412yes—
80C2×D4×C127C2311yescharacteristic
81C22×C4×C127D412yes—
82C22×C4×C127C2311yescharacteristic
83M16×C127—22no—
84M16×C127C2314yes—
85C2×C8×C127—22no—
86C22×D127—24no—
87C22×D127—22no—
88C22×D127—21no—
89Dic254—24no—
90Dic254—22no—
91Dic254—22no—
92Dic254C2×D414yes—
93D508—24no—
94D508—22no—
95D508—22no—
96D508C2×D414yes—
97C22⋊D127—28no—
98C22⋊D127—24no—
99C22⋊D127—24no—
100C22⋊D127—44no—
101C22⋊D127—42no—
102C2×Dic127—24no—
103C2×Dic127—22no—
104C2×Dic127—21no—
105C2×Dic127—21no—
106C2×Dic127—21no—
107C2×Dic127—42no—
108C4×D127—28no—
109C4×D127—24no—
110C4×D127—24no—
111C4×D127—41no—
112C4×D127—41no—
113C23×C127C2×D412yes—
114C23×C127C2×D411yescharacteristic
115Q8×C127—22no—
116D4×C127—24no—
117D4×C127—22no—
118C2×C4×C127—22no—
119C2×C4×C127—22no—
120C2×C4×C127C2×D412yes—
121C2×C4×C127C2×D412yes—
122C2×C4×C127C2×D412yes—
123C2×C4×C127C2×D411yescharacteristic
124C2×C4×C127C2411yescharacteristic
125C8×C127—24no—
126C22×C127—22no—
127C22×C127—41no—
128C22×C127—42no—
129C22×C127#12012yes—
130C22×C127C22×D411yescharacteristic
131C2×D127—24no—
132C2×D127—42no—
133C2×D127—41no—
134C2×D127—82no—
135C2×D127—84no—
136C4×C127—22no—
137C4×C127#12011yescharacteristic
138C4×C127#12011yescharacteristic
139C4×C127C22×D412yes—
140Dic127—22no—
141Dic127—24no—
142Dic127—41no—
143Dic127—41no—
144C2×C127—21no—
145C2×C127—22no—
146C2×C127—42no—
147C2×C127C2×#12011yescharacteristic
148D127—44no—
149D127—81no—
150#3018—1271nomaximal, Sylow
151C127#301811yescharacteristic, complemented, Sylow
152#585—1271no—
153#575—1272no—
154#532—1271no—
155#456—1272no—
156#455—1272no—
157#454—1272no—
158#453—1272no—
159C2×#104—1272no—
160#413D12711yescharacteristic, complemented
1612−1+4—1272no—
1622−1+4—2542no—
163H(5,𝔽2)—1272no—
164H(5,𝔽2)—2542no—
165H(5,𝔽2)—2542no—
166C2×#41—1272no—
167C2×#41—1271no—
168C2×#41—2544no—
169C2×#41—2542no—
170C2×#41C2×D12711yescharacteristic
171#137—1274no—
172#137—2542no—
173AGL(1,ℤ/8)—1274no—
174AGL(1,ℤ/8)—2542no—
175#135—2544no—
176C2×M16C2×D12712yescomplemented
177C4⋊Q8—1271no—
178#127—1271no—
179#124—1271no—
180#124—1271no—
181#121—2542no—
182C4×D4—2542no—
183C2×C42—1271no—
184#104—1274no—
185#104—1274no—
186#101C2×D12712yescomplemented
187#100C2×D12712yescomplemented
188#41—22no—
189#41—22no—
190#41—1274no—
191#41—1274no—
192#41—2544no—
193#41—2544no—
194#41—2542no—
195#41—2542no—
196#41—2544no—
197#41—2548no—
198#41—2544no—
199#41—2542no—
200#41—2542no—
201#41—2544no—
202C2×Q8—1272no—
203C2×Q8—1271no—
204C2×Q8—2542no—
205C2×Q8—2544no—
206C2×Q8C22×D12711yescharacteristic
207C2×D4—1272no—
208C2×D4—1271no—
209C2×D4—2542no—
210C2×D4—2544no—
211C2×D4—2548no—
212C2×D4—2542no—
213C2×D4—2541no—
214C2×D4—2542no—
215C2×D4—2544no—
216C2×D4D50812yes—
217C2×D4C22×D12711yescharacteristic
218C22×C4—2544no—
219C22×C4—2541no—
220C22×C4—2541no—
221C22×C4—2542no—
222C22×C4D50812yes—
223C22×C4C22×D12711yescharacteristic
224Q16—2544no—
225SD16—2544no—
226SD16—2544no—
227D8—2544no—
228M16—22no—
229M16C22×D12714yes—
230C2×C8—22no—
231C4⋊C4—2542no—
232C22⋊C4—2542no—
233C22⋊C4—2542no—
234C42—1271no—
235C42—1271no—
236C42—2541no—
237C23—2544no—
238C23—2542no—
239C23—2541no—
240C23C2×D50812yes—
241C23C2×D50811yescharacteristic
242Q8—22no—
243Q8—1274no—
244Q8—2544no—
245Q8—2542no—
246Q8—2542no—
247D4—24no—
248D4—22no—
249D4—1274no—
250D4—5082no—
251D4—5084no—
252D4—2544no—
253D4—2548no—
254D4—2544no—
255D4—2544no—
256D4—2542no—
257D4—2542no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—5082no—
261C2×C4—5081no—
262C2×C4—5081no—
263C2×C4—2548no—
264C2×C4—2541no—
265C2×C4—2541no—
266C2×C4—2544no—
267C2×C4—2541no—
268C2×C4—2542no—
269C2×C4—2544no—
270C2×C4—2544no—
271C2×C4D4×D12712yes—
272C2×C4D4×D12712yes—
273C2×C4C2×D50812yes—
274C2×C4C2×D50811yescharacteristic
275C2×C4C23×D12711yescharacteristic
276C8—24no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—10162no—
281C22—10164no—
282C22—5082no—
283C22—5081no—
284C22—2544no—
285C22#1686117912yes—
286C22C22×D50811yescharacteristic
287C4—22no—
288C4—5081no—
289C4—5081no—
290C4—2542no—
291C4—2544no—
292C4#1686117911yescharacteristic
293C4#1686117911yescharacteristic
294C4C2×D4×D12712yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—10161no—
299C2—5084no—
300C2C2×#1686117911yescharacteristic
subgroupG/H
centerC4#16861179
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#413×C127C2
socleC2×C127C2×#120
hypercenter#413D127
hypocenterC127#3018
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #22550712C2semidirect
2. #22550676C2semidirect
3. #22550422C2semidirect
4. #22550061C2semidirect
5. #22550056C2semidirect
6. #22550051C2semidirect
7. #22550046C2semidirect
8. C2×#16861135C2semidirect
9. #413×C127C2semidirect
17. C2×#14523248C22semidirect
20. #16861242C22semidirect
22. #16861237C22semidirect
24. #16861207C22semidirect
25. #16861205C22semidirect
26. #16861198C22semidirect
27. #16861196C22semidirect
30. C2×C4×Dic127C22semidirect
31. #16861135C22semidirect
32. #16861135C22semidirect
34. C2×M16×C127C22semidirect
35. #101×C127C22semidirect
36. #100×C127C22semidirect
45. #14523248D4semidirect
50. C2×Dic254D4semidirect
54. C2×D508D4semidirect
151. C127#3018semidirect
160. #413D127semidirect
176. C2×M16C2×D127semidirect
186. #101C2×D127semidirect
187. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225507122
#225506762
#225504222
#225500612
#225500562
#225500512
#225500462
C2×#168611352
#413×C1272
#3018127