Finite Group Databasebeta

← #30650511

Subgroups — #30650511

Order 16256.

454 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 117 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#22550693C211yesmaximal, characteristic, complemented
2#22550686C212yesmaximal, complemented
3#22550677C211yesmaximal, characteristic, complemented
4#22550259C211yesmaximal, characteristic, complemented
5#22550152C211yesmaximal, characteristic, complemented
6#22550138C212yesmaximal, complemented
7#22550056C212yesmaximal, complemented
8#22550047C212yesmaximal, complemented
9#22549992C211yesmaximal, characteristic, complemented
10#22549991C211yesmaximal, characteristic, complemented
11#472×C127C211yesmaximal, characteristic, complemented
12#16861256C2212yescomplemented
13#16861254C2212yes—
14#16861242—22no—
15#16861242C2212yes—
16#16861242C2212yescomplemented
17#16861238—22no—
18#16861235C2212yes—
19#16861233—22no—
20#16861233—22no—
21#16861231—21no—
22#16861231C2211yescharacteristic
23#16861229C2211yescharacteristic
24#16861228—21no—
25#16861228—22no—
26C2×Dic508—21no—
27C2×Dic508C2211yescharacteristic, complemented
28C2×#14523236C2212yescomplemented
29SD16×D127—22no—
30D8×D127—21no—
31#16861217C2212yescomplemented
32#16861215C2211yescharacteristic
33#16861196C2212yescomplemented
34#16861144C2211yescharacteristic, complemented
35#16861136C2211yescharacteristic, complemented
36#16861135C2211yescharacteristic
37#16861135C2212yescomplemented
38#16861127C2212yescomplemented
39C8×Dic127C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C127C2212yescomplemented
41C2×D8×C127C2211yescharacteristic
42#131×C127C2211yescharacteristic, complemented
43#108×C127C2211yescharacteristic, complemented
44#100×C127C2212yescomplemented
45#14523248—22no—
46#14523248C2311yescharacteristic
47#14523248C2312yes—
48#14523247—22no—
49#14523247—22no—
50#14523247—22no—
51#14523247—42no—
52C2×Dic254—22no—
53C2×Dic254C2312yes—
54Q8×D127—22no—
55C2×C22⋊D127—42no—
56D4×D127—22no—
57D4×D127—22no—
58Dic508—22no—
59Dic508—22no—
60Dic508—21no—
61Dic508—21no—
62Dic508D411yescharacteristic
63#14523238D411yescharacteristic
64#14523237—22no—
65#14523237—22no—
66#14523237D412yes—
67#14523236—22no—
68#14523236—22no—
69#14523236—22no—
70#14523236D411yescharacteristic
71#14523235D411yescharacteristic
72D1016D411yescharacteristic
73#14523233—22no—
74#14523233D411yescharacteristic
75#14523232C2311yescharacteristic
76#14523231—22no—
77#14523231—21no—
78C2×C127⋊C8C2311yescharacteristic
79C8×D127—22no—
80C8×D127—21no—
81C22⋊Dic127—42no—
82C4×Dic127C2311yescharacteristic
83#41×C127C2311yescharacteristic
84C2×D4×C127C2312yes—
85SD16×C127—22no—
86D8×C127—22no—
87D8×C127—21no—
88D8×C127—21no—
89M16×C127—21no—
90M16×C127C2312yes—
91M16×C127C2311yescharacteristic
92C2×C8×C127—21no—
93C2×C8×C127C2311yescharacteristic
94C22×D127—42no—
95Dic254—22no—
96Dic254—22no—
97Dic254—42no—
98Dic254C2×D412yes—
99Dic254C2×D411yescharacteristic
100D508C2×D412yes—
101D508C2×D411yescharacteristic
102C22⋊D127—21no—
103C22⋊D127—22no—
104C22⋊D127—42no—
105C22⋊D127—42no—
106C22⋊D127—42no—
107C2×Dic127—21no—
108C2×Dic127—22no—
109C2×Dic127—42no—
110C2×Dic127—42no—
111C4×D127—21no—
112C4×D127—22no—
113C4×D127—42no—
114C127⋊C8—21no—
115C127⋊C8C2×D411yescharacteristic
116C127⋊C8C2×D411yescharacteristic
117C23×C127—22no—
118Q8×C127C2×D411yescharacteristic
119D4×C127—21no—
120D4×C127—22no—
121D4×C127—22no—
122D4×C127C2×D411yescharacteristic
123C2×C4×C127—21no—
124C2×C4×C127C2411yescharacteristic
125C8×C127—21no—
126C8×C127—22no—
127C8×C127C2×D411yescharacteristic
128C8×C127C2×D411yescharacteristic
129C22×C127—21no—
130C22×C127—42no—
131C22×C127—82no—
132C22×C127H(5,𝔽2)11yescharacteristic
133C2×D127—22no—
134C2×D127—21no—
135C2×D127—162no—
136C4×C127—21no—
137C4×C127C22×D411yescharacteristic
138C4×C127C22×D411yescharacteristic
139Dic127—21no—
140Dic127—22no—
141Dic127—41no—
142Dic127—42no—
143C2×C127—21no—
144C2×C127—41no—
145C2×C127—82no—
146C2×C127D4211yescharacteristic
147D127—42no—
148D127—41no—
149#3290—1271nomaximal, Sylow
150C127#329011yescharacteristic, complemented, Sylow
151#578—1271no—
152#577—1272no—
153#576—1271no—
154#495—1271no—
155#473—1271no—
156#472D12711yescharacteristic, complemented
157#471—1272no—
158#455—1272no—
159#454—1272no—
160#443—1271no—
161#443—1271no—
1622−1+4—1272no—
163H(5,𝔽2)—1272no—
164#137—1272no—
165#137—1272no—
166#137—2542no—
167#137—2542no—
168AGL(1,ℤ/8)—1272no—
169AGL(1,ℤ/8)—2542no—
170AGL(1,ℤ/8)—2542no—
171AGL(1,ℤ/8)C2×D12712yescomplemented
172#135—1271no—
173#135—1271no—
174#135—2541no—
175#135—2542no—
176#135—2541no—
177C2×Q16—1271no—
178C2×Q16—2541no—
179C2×SD16—1272no—
180C2×SD16—2542no—
181C2×D8—2541no—
182C2×D8C2×D12711yescharacteristic
183#131—1272no—
184#131—1271no—
185#131C2×D12711yescharacteristic, complemented
186#124—1272no—
187#108—1271no—
188#108C2×D12711yescharacteristic, complemented
189#104—1271no—
190#104—1272no—
191#104—1271no—
192#101—1272no—
193#100C2×D12712yescomplemented
194C4×C8—1271no—
195#41—1272no—
196#41—1271no—
197#41—5082no—
198#41—2542no—
199#41—2542no—
200#41—2542no—
201#41—2542no—
202#41C22×D12711yescharacteristic
203C2×Q8—1272no—
204C2×Q8—2542no—
205C2×Q8—2542no—
206C2×D4—5082no—
207C2×D4—2542no—
208C2×D4—2542no—
209C2×D4C22×D12712yes—
210Q16—1271no—
211Q16—1271no—
212Q16—2542no—
213Q16—2542no—
214Q16—2541no—
215Q16—2541no—
216SD16—22no—
217SD16—1271no—
218SD16—1271no—
219SD16—1272no—
220SD16—2542no—
221SD16—2542no—
222SD16—2542no—
223SD16—2542no—
224SD16—2542no—
225D8—22no—
226D8—21no—
227D8—21no—
228D8—1271no—
229D8—1271no—
230D8—2542no—
231M16—21no—
232M16—1271no—
233M16—2542no—
234M16—2541no—
235M16C22×D12711yescharacteristic
236M16C22×D12712yes—
237C2×C8—21no—
238C2×C8—1271no—
239C2×C8—2542no—
240C2×C8—2541no—
241C2×C8C22×D12711yescharacteristic
242C22⋊C4—5082no—
243C42—1271no—
244C23—22no—
245C23—5082no—
246Q8—1272no—
247Q8—1271no—
248Q8—5082no—
249Q8—2542no—
250Q8—2542no—
251Q8D4×D12711yescharacteristic
252D4—21no—
253D4—22no—
254D4—22no—
255D4—1272no—
256D4—1271no—
257D4—5082no—
258D4—5082no—
259D4—5082no—
260D4—2541no—
261D4—2542no—
262D4D4×D12711yescharacteristic
263C2×C4—21no—
264C2×C4—5082no—
265C2×C4—5082no—
266C2×C4—5082no—
267C2×C4—2541no—
268C2×C4—2542no—
269C2×C4—2541no—
270C2×C4—2542no—
271C2×C4C23×D12711yescharacteristic
272C8—21no—
273C8—22no—
274C8—1271no—
275C8—1271no—
276C8—2541no—
277C8D4×D12711yescharacteristic
278C8D4×D12711yescharacteristic
279C22—21no—
280C22—42no—
281C22—82no—
282C22—20322no—
283C22—2542no—
284C22—2541no—
285C22#1686125511yescharacteristic
286C4—21no—
287C4—5081no—
288C4—5082no—
289C4—2541no—
290C4—2542no—
291C4C2×D4×D12711yescharacteristic
292C4C2×D4×D12711yescharacteristic
293C2—21no—
294C2—41no—
295C2—82no—
296C2—5081no—
297C2—5082no—
298C2#2255051011yescharacteristic
subgroupG/H
centerC2#22550510
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#472×C127C2
socleC2×C127D42
hypercenter#472D127
hypocenterC127#3290
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #22550693C2semidirect
2. #22550686C2semidirect
3. #22550677C2semidirect
4. #22550259C2semidirect
5. #22550152C2semidirect
6. #22550138C2semidirect
7. #22550056C2semidirect
8. #22550047C2semidirect
9. #22549992C2semidirect
10. #22549991C2semidirect
11. #472×C127C2semidirect
12. #16861256C22semidirect
16. #16861242C22semidirect
27. C2×Dic508C22semidirect
28. C2×#14523236C22semidirect
31. #16861217C22semidirect
33. #16861196C22semidirect
34. #16861144C22semidirect
35. #16861136C22semidirect
37. #16861135C22semidirect
38. #16861127C22semidirect
39. C8×Dic127C22semidirect
40. AGL(1,ℤ/8)×C127C22semidirect
42. #131×C127C22semidirect
43. #108×C127C22semidirect
44. #100×C127C22semidirect
150. C127#3290semidirect
156. #472D127semidirect
171. AGL(1,ℤ/8)C2×D127semidirect
185. #131C2×D127semidirect
188. #108C2×D127semidirect
193. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506932
#225506862
#225506772
#225502592
#225501522
#225501382
#225500562
#225500472
#225499922
#225499912
#472×C1272
#3290127