Finite Group Databasebeta

← #30648412

Subgroups — #30648412

Order 16256.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#16861234C211yesmaximal, characteristic, complemented
2C2×#16861216C211yesmaximal, characteristic, complemented
3#22550440C211yesmaximal, characteristic, complemented
4#22550127C212yesmaximal, complemented
5#22550124C212yesmaximal, complemented
6#22550110C212yesmaximal, complemented
7#22550107C212yesmaximal, complemented
8#22549916C211yesmaximal, characteristic, complemented
9#22549872C211yesmaximal, characteristic, complemented
10C2×#16861129C211yesmaximal, characteristic, complemented
11#537×C127C211yesmaximal, characteristic, complemented
12C22×D508C2211yescharacteristic
13#16861234—24no—
14C2×#14523235C2212yes—
15C2×#14523233C2212yes—
16#16861216—24no—
17C2×#14523232C2211yescharacteristic
18C2×#14523232C2212yescomplemented
19C22×C127⋊C8C2211yescharacteristic
20C22×C127⋊C8C2212yescomplemented
21#16861205—21no—
22#16861198—21no—
23#16861183C2212yes—
24#16861183C2212yes—
25#16861177—22no—
26#16861168C2211yescharacteristic
27#16861142C2212yescomplemented
28#16861140C2212yescomplemented
29#16861133C2211yescharacteristic, complemented
30#16861133C2211yescharacteristic, complemented
31#16861130C2211yescharacteristic, complemented
32#16861130C2211yescharacteristic, complemented
33#16861129C2212yes—
34#16861129C2212yes—
35C2×#41×C127C2211yescharacteristic
36C2×#41×C127C2211yescharacteristic
37C22⋊Q8×C127—21no—
38C22⋊Q8×C127C2212yescomplemented
39#121×C127—21no—
40#121×C127—22no—
41#121×C127C2212yescomplemented
42C2×C4⋊C4×C127C2211yescharacteristic
43C23×D127—21no—
44C2×D508—22no—
45C2×D508—21no—
46C2×D508—22no—
47C2×D508C2312yes—
48C2×C22⋊D127—22no—
49C2×C22⋊D127—22no—
50#14523235—24no—
51#14523233—24no—
52#14523232—24no—
53#14523232D414yescomplemented
54C2×C127⋊C8—24no—
55C2×C127⋊C8—22no—
56C2×C127⋊C8C2311yescharacteristic
57C2×C127⋊C8C2311yescharacteristic
58C2×C127⋊C8C2312yes—
59C4⋊Dic127C2311yescharacteristic
60C4⋊Dic127C2311yescharacteristic
61C22⋊Dic127—21no—
62(C2×D127)⋊C4—22no—
63(C2×D127)⋊C4—22no—
64C4×Dic127—21no—
65#41×C127—22no—
66#41×C127—22no—
67#41×C127—22no—
68#41×C127—22no—
69C2×Q8×C127D411yescharacteristic
70C2×Q8×C127C2311yescharacteristic
71C2×D4×C127—21no—
72C2×D4×C127—22no—
73C2×D4×C127D411yescharacteristic
74C2×D4×C127D412yes—
75C2×D4×C127C2311yescharacteristic
76C22×C4×C127—21no—
77C22×C4×C127—21no—
78C22×C4×C127C2312yes—
79C22×C4×C127C2311yescharacteristic
80C4⋊C4×C127—21no—
81C4⋊C4×C127—22no—
82C4⋊C4×C127C2312yes—
83C22⋊C4×C127—22no—
84C22⋊C4×C127—22no—
85C22×D127—22no—
86C22×D127—42no—
87C22×D127—81no—
88C22×D127—81no—
89D508—22no—
90D508—22no—
91D508—42no—
92C22⋊D127—82no—
93C22⋊D127—82no—
94C2×Dic127—21no—
95C2×Dic127—21no—
96C127⋊C8—21no—
97C127⋊C8—21no—
98C127⋊C8C2×D414yes—
99C23×C127—21no—
100C23×C127C2×D412yes—
101C23×C127C2×D411yescharacteristic
102Q8×C127—22no—
103Q8×C127—21no—
104Q8×C127—21no—
105D4×C127—24no—
106D4×C127—22no—
107D4×C127—21no—
108D4×C127—21no—
109D4×C127—42no—
110D4×C127—82no—
111C2×C4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127—21no—
115C2×C4×C127—42no—
116C2×C4×C127—81no—
117C2×C4×C127C2×D412yes—
118C2×C4×C127C2×D412yes—
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127#4112yes—
121C2×C4×C127C2411yescharacteristic
122C22×C127—21no—
123C22×C127—21no—
124C22×C127—22no—
125C22×C127—22no—
126C22×C127—22no—
127C22×C127—41no—
128C22×C127—41no—
129C22×C127—41no—
130C22×C127#12112yes—
131C22×C127C22×D411yescharacteristic
132C2×D127—42no—
133C2×D127—42no—
134C2×D127—42no—
135C2×D127—84no—
136C4×C127—22no—
137C4×C127—41no—
138C4×C127—42no—
139C4×C127#12112yes—
140C4×C127C22×D411yescharacteristic
141C4×C127C2×#4111yescharacteristic
142Dic127—41no—
143Dic127—41no—
144C2×C127—22no—
145C2×C127—42no—
146C2×C127—81no—
147C2×C127C2×#12111yescharacteristic
148C2×C127#57611yescharacteristic
149C2×C127#57711yescharacteristic
150D127—82no—
151#3062—1271nomaximal, Sylow
152C127#306211yescharacteristic, complemented, Sylow
153C2×AGL(1,ℤ/8)—1271no—
154C2×#131—1271no—
155#537D12711yescharacteristic, complemented
156#535—1271no—
157#469—1272no—
158#468—1272no—
159#466—1272no—
160#465—1272no—
161#428—1271no—
162#416—1271no—
163C2×#102—1271no—
164C2×#41C2×D12711yescharacteristic
165C2×#41C2×D12711yescharacteristic
166C22×D4—1271no—
167AGL(1,ℤ/8)—2544no—
168C2×SD16—1272no—
169C2×D8—1272no—
170#131—2544no—
171C2×M16—1271no—
172C2×M16—1272no—
173C22×C8—1271no—
174C22×C8—1272no—
175#127—2541no—
176#124—2541no—
177C22⋊Q8—21no—
178C22⋊Q8C2×D12712yescomplemented
179#121—21no—
180#121—22no—
181#121—1272no—
182#121—1272no—
183#121C2×D12712yescomplemented
184#120—2542no—
185#117—1271no—
186C2×C4⋊C4C2×D12711yescharacteristic
187#107—1272no—
188#106—1272no—
189#103—1271no—
190#103—1271no—
191#102—1271no—
192#102—1271no—
193#102—1272no—
194#102—1272no—
195C24—2541no—
196#41—22no—
197#41—22no—
198#41—22no—
199#41—22no—
200C2×Q8C22⋊D12711yescharacteristic
201C2×Q8C22×D12711yescharacteristic
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—1272no—
205C2×D4—2542no—
206C2×D4—2542no—
207C2×D4—2542no—
208C2×D4—2541no—
209C2×D4—2542no—
210C2×D4C22⋊D12712yes—
211C2×D4C22⋊D12711yescharacteristic
212C2×D4C22×D12711yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4C22×D12712yes—
216C22×C4C22×D12711yescharacteristic
217SD16—2544no—
218D8—2544no—
219M16—1274no—
220M16—2544no—
221C2×C8—1272no—
222C2×C8—1271no—
223C2×C8—1271no—
224C2×C8—2544no—
225C2×C8—2542no—
226C4⋊C4—21no—
227C4⋊C4—22no—
228C4⋊C4—1271no—
229C4⋊C4—1271no—
230C4⋊C4C22×D12712yes—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—2541no—
234C22⋊C4—2542no—
235C22⋊C4—2542no—
236C42—2541no—
237C23—21no—
238C23—10161no—
239C23—10161no—
240C23—5082no—
241C23—2542no—
242C23C2×C22⋊D12712yes—
243C23C2×C22⋊D12711yescharacteristic
244Q8—22no—
245Q8—21no—
246Q8—21no—
247D4—24no—
248D4—21no—
249D4—21no—
250D4—22no—
251D4—42no—
252D4—82no—
253D4—10162no—
254D4—10162no—
255D4—5082no—
256D4—2542no—
257D4—2542no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—81no—
264C2×C4—2541no—
265C2×C4—2541no—
266C2×C4D4×D12712yes—
267C2×C4C2×C22⋊D12712yes—
268C2×C4C2×C22⋊D12711yescharacteristic
269C2×C4#1452324712yes—
270C2×C4C23×D12711yescharacteristic
271C8—1274no—
272C8—2541no—
273C8—2541no—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—10164no—
283C22—5082no—
284C22—5082no—
285C22—5082no—
286C22#1686118112yes—
287C22C22×C22⋊D12711yescharacteristic
288C4—22no—
289C4—41no—
290C4—42no—
291C4—5081no—
292C4—5081no—
293C4#1686118112yes—
294C4C2×D4×D12711yescharacteristic
295C4C2×#1452324711yescharacteristic
296C2—22no—
297C2—42no—
298C2—81no—
299C2—10162no—
300C2C2×#1686118111yescharacteristic
301C2#2255067911yescharacteristic
302C2#2255068411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#537×C127C2
socleC22×C127C22×D4
hypercenter#537D127
hypocenterC127#3062
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C2×#16861216C2semidirect
3. #22550440C2semidirect
4. #22550127C2semidirect
5. #22550124C2semidirect
6. #22550110C2semidirect
7. #22550107C2semidirect
8. #22549916C2semidirect
9. #22549872C2semidirect
10. C2×#16861129C2semidirect
11. #537×C127C2semidirect
18. C2×#14523232C22semidirect
20. C22×C127⋊C8C22semidirect
27. #16861142C22semidirect
28. #16861140C22semidirect
29. #16861133C22semidirect
30. #16861133C22semidirect
31. #16861130C22semidirect
32. #16861130C22semidirect
38. C22⋊Q8×C127C22semidirect
41. #121×C127C22semidirect
53. #14523232D4semidirect
152. C127#3062semidirect
155. #537D127semidirect
178. C22⋊Q8C2×D127semidirect
183. #121C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C2×#168612162
#225504402
#225501272
#225501242
#225501102
#225501072
#225499162
#225498722
C2×#168611292
#537×C1272
#3062127