Finite Group Databasebeta

← #30651873

Subgroups — #30651873

Order 16256.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#22550690C211yesmaximal, characteristic, complemented
2#22550678C211yesmaximal, characteristic, complemented
3#22550318C212yesmaximal, complemented
4#22550316C211yesmaximal, characteristic, complemented
5#22550311C211yesmaximal, characteristic, complemented
6#22550310C212yesmaximal, complemented
7#22550307C212yesmaximal, complemented
8C2×#16861154C211yesmaximal, characteristic, complemented
9SD32×D127C212yesmaximal, complemented
10#22550292C211yesmaximal, characteristic, complemented
11#508×C127C211yesmaximal, characteristic, complemented
12#16861256—21no—
13#16861255—21no—
14#16861239—21no—
15#16861238—22no—
16#16861236—22no—
17#16861234—21no—
18#16861232C2212yes—
19#16861228C2212yescomplemented
20C2×Dic508C2211yescharacteristic, complemented
21Q16×D127C2212yescomplemented
22C2×D1016C2211yescharacteristic
23D8×D127C2212yes—
24#16861215C2211yescharacteristic
25#16861155C2212yescomplemented
26#16861154C2212yescomplemented
27#16861154C2212yescomplemented
28#16861153C2212yescomplemented
29#16861152C2212yescomplemented
30#16861150C2212yescomplemented
31#16861149C2211yescharacteristic, complemented
32#16861148C2212yescomplemented
33C16×D127C2212yescomplemented
34#135×C127C2211yescharacteristic
35#135×C127C2211yescharacteristic, complemented
36Q32×C127C2211yescharacteristic, complemented
37SD32×C127C2211yescharacteristic, complemented
38SD32×C127C2211yescharacteristic, complemented
39D16×C127C2211yescharacteristic, complemented
40C2×C16×C127C2211yescharacteristic, complemented
41#14523249—22no—
42#14523248—41no—
43#14523248—41no—
44#14523248D411yescharacteristic
45#14523247—22no—
46#14523247—42no—
47C2×Dic254—21no—
48C2×Dic254—41no—
49Q8×D127—22no—
50C2×D508—21no—
51C2×D508—41no—
52D4×D127—22no—
53D4×D127—42no—
54Dic508—21no—
55Dic508C2312yes—
56#14523238—22no—
57#14523236—22no—
58#14523235—22no—
59D1016—21no—
60D1016C2312yes—
61#14523233—22no—
62#14523232D411yescharacteristic
63#14523231D412yes—
64C8×D127C2312yes—
65C127⋊C16C2312yes—
66#41×C127—21no—
67#41×C127—21no—
68Q16×C127C2311yescharacteristic
69Q16×C127C2311yescharacteristic
70SD16×C127—21no—
71SD16×C127—21no—
72D8×C127C2311yescharacteristic
73D8×C127C2311yescharacteristic
74C2×C8×C127C2311yescharacteristic
75C16×C127C2311yescharacteristic
76C16×C127C2311yescharacteristic
77C22×D127—41no—
78C22×D127—42no—
79Dic254—22no—
80Dic254—41no—
81Dic254—42no—
82Dic254D811yescharacteristic
83D508—22no—
84D508—42no—
85D508—41no—
86D508D811yescharacteristic
87C22⋊D127—42no—
88C22⋊D127—42no—
89C22⋊D127D812yes—
90C2×Dic127—41no—
91C2×Dic127—42no—
92C4×D127—42no—
93C4×D127—42no—
94C4×D127C2×D412yes—
95C127⋊C8C2×D412yes—
96Q8×C127—21no—
97Q8×C127—21no—
98D4×C127—21no—
99D4×C127—21no—
100D4×C127—41no—
101D4×C127—41no—
102C2×C4×C127—41no—
103C2×C4×C127—41no—
104C2×C4×C127C2×D411yescharacteristic
105C8×C127C2×D411yescharacteristic
106C8×C127C2411yescharacteristic
107C22×C127—41no—
108C22×C127—41no—
109C22×C127C2×D811yescharacteristic
110C2×D127—42no—
111C2×D127—162no—
112C2×D127—161no—
113C2×D127C2×D812yes—
114C4×C127—41no—
115C4×C127—41no—
116C4×C127C2×D811yescharacteristic
117C4×C127C22×D411yescharacteristic
118Dic127—42no—
119Dic127C2×D812yes—
120C2×C127—21no—
121C2×C127—81no—
122C2×C127—81no—
123C2×C127C22×D811yescharacteristic
124D127—22no—
125D127—82no—
126#3416—1271nomaximal, Sylow
127C127#341611yescharacteristic, complemented, Sylow
128#578—1271no—
129#576—1271no—
130#510—1271no—
131#510—1272no—
132#509—1271no—
133#509—1272no—
134#508—1272no—
135#508D12711yescharacteristic, complemented
136C2×SD32—1272no—
137C2×SD32—1271no—
138#504—1271no—
1392−1+4—2541no—
140H(5,𝔽2)—2541no—
141#137—2542no—
142#137—2541no—
143AGL(1,ℤ/8)—2542no—
144AGL(1,ℤ/8)—2541no—
145#135—1272no—
146#135—1272no—
147#135C2×D12711yescharacteristic
148#135C2×D12711yescharacteristic, complemented
149C2×Q16—1272no—
150C2×Q16—1271no—
151C2×D8—1272no—
152C2×D8—1271no—
153#131—1271no—
154Q32—1272no—
155Q32C2×D12711yescharacteristic, complemented
156SD32—1272no—
157SD32—1272no—
158SD32—1272no—
159SD32—1272no—
160SD32C2×D12711yescharacteristic, complemented
161SD32C2×D12711yescharacteristic, complemented
162D16—1272no—
163D16C2×D12711yescharacteristic, complemented
164M32—1271no—
165M32—1272no—
166C2×C16—1272no—
167C2×C16C2×D12711yescharacteristic, complemented
168#41—21no—
169#41—21no—
170#41—1271no—
171#41—5081no—
172#41—5081no—
173#41—5082no—
174#41—2542no—
175#41—2542no—
176C2×Q8—5081no—
177C2×Q8—2542no—
178C2×Q8—2541no—
179C2×D4—5081no—
180C2×D4—5082no—
181C2×D4—2542no—
182C2×D4—2541no—
183Q16—1272no—
184Q16—2542no—
185Q16—2541no—
186Q16C22×D12711yescharacteristic
187Q16C22×D12711yescharacteristic
188SD16—21no—
189SD16—21no—
190SD16—2542no—
191SD16—2542no—
192D8—1272no—
193D8—2542no—
194D8—2541no—
195D8C22×D12711yescharacteristic
196D8C22×D12711yescharacteristic
197M16—1271no—
198M16—1272no—
199C2×C8—1272no—
200C2×C8C22×D12711yescharacteristic
201C16—1272no—
202C16C22×D12711yescharacteristic
203C16C22×D12711yescharacteristic
204C23—5082no—
205C23—5081no—
206Q8—21no—
207Q8—21no—
208Q8—1271no—
209Q8—5082no—
210Q8—5081no—
211Q8—2542no—
212D4—21no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—1272no—
217D4—1271no—
218D4—5082no—
219D4—5082no—
220D4—5082no—
221D4—5081no—
222D4—2542no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—1272no—
226C2×C4—5082no—
227C2×C4—5082no—
228C2×C4—5082no—
229C2×C4—5081no—
230C2×C4D4×D12711yescharacteristic
231C8—1272no—
232C8D4×D12711yescharacteristic
233C8C23×D12711yescharacteristic
234C22—41no—
235C22—41no—
236C22—1272no—
237C22—20321no—
238C22—20322no—
239C22—5082no—
240C22D8×D12711yescharacteristic
241C4—41no—
242C4—41no—
243C4—1272no—
244C4—5082no—
245C4D8×D12711yescharacteristic
246C4C2×D4×D12711yescharacteristic
247C2—21no—
248C2—81no—
249C2—81no—
250C2—10162no—
251C2—2542no—
252C2C2×D8×D12711yescharacteristic
subgroupG/H
centerC2C2×D8×D127
commutator, cosocleC8×C127C24
FrattiniC8C23×D127
Fitting#508×C127C2
socleC2×C127C22×D8
hypercenter#508D127
hypocenterC127#3416
LCSG ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #22550690C2semidirect
2. #22550678C2semidirect
3. #22550318C2semidirect
4. #22550316C2semidirect
5. #22550311C2semidirect
6. #22550310C2semidirect
7. #22550307C2semidirect
8. C2×#16861154C2semidirect
9. SD32×D127C2semidirect
10. #22550292C2semidirect
11. #508×C127C2semidirect
19. #16861228C22semidirect
20. C2×Dic508C22semidirect
21. Q16×D127C22semidirect
25. #16861155C22semidirect
26. #16861154C22semidirect
27. #16861154C22semidirect
28. #16861153C22semidirect
29. #16861152C22semidirect
30. #16861150C22semidirect
31. #16861149C22semidirect
32. #16861148C22semidirect
33. C16×D127C22semidirect
35. #135×C127C22semidirect
36. Q32×C127C22semidirect
37. SD32×C127C22semidirect
38. SD32×C127C22semidirect
39. D16×C127C22semidirect
40. C2×C16×C127C22semidirect
127. C127#3416semidirect
135. #508D127semidirect
148. #135C2×D127semidirect
155. Q32C2×D127semidirect
160. SD32C2×D127semidirect
161. SD32C2×D127semidirect
163. D16C2×D127semidirect
167. C2×C16C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506902
#225506782
#225503182
#225503162
#225503112
#225503102
#225503072
C2×#168611542
SD32×D1272
#225502922
#508×C1272
#3416127