Finite Group Databasebeta

← #30651875

Subgroups — #30651875

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#22550689C211yesmaximal, characteristic, complemented
2#22550678C211yesmaximal, characteristic, complemented
3#22550318C212yesmaximal, complemented
4#22550314C211yesmaximal, characteristic, complemented
5#22550313C212yesmaximal, complemented
6#22550309C212yesmaximal, complemented
7#22550306C211yesmaximal, characteristic, complemented
8C2×#16861152C211yesmaximal, characteristic, complemented
9SD32×D127C212yesmaximal, complemented
10#22550294C211yesmaximal, characteristic, complemented
11#510×C127C211yesmaximal, characteristic, complemented
12#16861257—21no—
13#16861255—21no—
14#16861241—22no—
15#16861240—21no—
16#16861236—22no—
17#16861234—21no—
18#16861232C2212yes—
19#16861232C2212yescomplemented
20#16861231C2211yescharacteristic, complemented
21Q16×D127C2212yescomplemented
22C2×D1016C2211yescharacteristic
23D8×D127C2212yes—
24#16861215C2211yescharacteristic
25#16861155C2211yescharacteristic, complemented
26#16861154C2212yescomplemented
27#16861153C2211yescharacteristic, complemented
28#16861152C2212yescomplemented
29#16861152C2212yescomplemented
30#16861152C2211yescharacteristic, complemented
31D2032C2212yescomplemented
32#16861150C2211yescharacteristic, complemented
33#16861148C2212yescomplemented
34C2×C127⋊C16C2211yescharacteristic, complemented
35C16×D127C2212yescomplemented
36#135×C127C2211yescharacteristic
37C2×Q16×C127C2211yescharacteristic, complemented
38Q32×C127C2212yescomplemented
39SD32×C127C2212yescomplemented
40M32×C127C2211yescharacteristic, complemented
41#14523249—22no—
42#14523249—22no—
43#14523249—41no—
44#14523248—21no—
45#14523248—41no—
46#14523248—42no—
47#14523248D411yescharacteristic
48Q8×D127—22no—
49Q8×D127—41no—
50C2×D508—21no—
51C2×D508—41no—
52D4×D127—22no—
53D4×D127—42no—
54Dic508C2311yescharacteristic
55#14523238—22no—
56#14523237—21no—
57#14523235—22no—
58#14523235—22no—
59D1016—21no—
60D1016C2312yes—
61D1016C2311yescharacteristic
62#14523233—22no—
63#14523232D411yescharacteristic
64#14523231D412yes—
65C8×D127C2312yes—
66C127⋊C16C2311yescharacteristic
67C127⋊C16C2311yescharacteristic
68#41×C127—21no—
69C2×Q8×C127—21no—
70Q16×C127—21no—
71Q16×C127C2312yes—
72Q16×C127C2311yescharacteristic
73SD16×C127—21no—
74D8×C127C2311yescharacteristic
75C2×C8×C127C2311yescharacteristic
76C16×C127C2312yes—
77C22×D127—41no—
78C22×D127—42no—
79Dic254—21no—
80Dic254—42no—
81Dic254D811yescharacteristic
82D508—22no—
83D508—21no—
84D508—42no—
85D508—42no—
86D508—41no—
87D508D811yescharacteristic
88C22⋊D127—41no—
89C22⋊D127—42no—
90C22⋊D127D812yes—
91C4×D127—42no—
92C4×D127—41no—
93C4×D127—42no—
94C4×D127—42no—
95C4×D127C2×D412yes—
96C127⋊C8C2×D412yes—
97Q8×C127—21no—
98Q8×C127—22no—
99Q8×C127—41no—
100D4×C127—21no—
101D4×C127—41no—
102C2×C4×C127—41no—
103C2×C4×C127—41no—
104C2×C4×C127C2×D411yescharacteristic
105C8×C127C2×D411yescharacteristic
106C8×C127C2411yescharacteristic
107C22×C127—41no—
108C22×C127C2×D811yescharacteristic
109C2×D127—41no—
110C2×D127—42no—
111C2×D127—162no—
112C2×D127—161no—
113C2×D127C2×D812yes—
114C4×C127—41no—
115C4×C127—42no—
116C4×C127C2×D811yescharacteristic
117C4×C127C22×D411yescharacteristic
118Dic127—41no—
119Dic127C2×D812yes—
120C2×C127—21no—
121C2×C127—81no—
122C2×C127C22×D811yescharacteristic
123D127—22no—
124D127—81no—
125D127—82no—
126#3416—1271nomaximal, Sylow
127C127#341611yescharacteristic, complemented, Sylow
128#578—1271no—
129#576—1271no—
130#510—1272no—
131#510D12711yescharacteristic, complemented
132#509—1272no—
133#509—1271no—
134#508—1271no—
135#508—1272no—
136C2×SD32—1271no—
137C2×SD32—1272no—
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#135—1271no—
148#135C2×D12711yescharacteristic
149C2×Q16—1272no—
150C2×Q16C2×D12711yescharacteristic, complemented
151C2×D8—1272no—
152C2×D8—1271no—
153#131—1271no—
154Q32—1271no—
155Q32C2×D12712yescomplemented
156SD32—1272no—
157SD32—1272no—
158SD32—1271no—
159SD32—1271no—
160SD32—1272no—
161SD32C2×D12712yescomplemented
162D16—1271no—
163D16—1272no—
164M32—1272no—
165M32C2×D12711yescharacteristic, complemented
166C2×C16—1271no—
167C2×C16—1272no—
168#41—21no—
169#41—1271no—
170#41—5082no—
171#41—5081no—
172#41—5081no—
173#41—2542no—
174#41—2542no—
175#41—2541no—
176C2×Q8—21no—
177C2×Q8—5081no—
178C2×Q8—2542no—
179C2×D4—5081no—
180C2×D4—5082no—
181C2×D4—2542no—
182C2×D4—2541no—
183Q16—21no—
184Q16—1271no—
185Q16—2542no—
186Q16C22×D12712yes—
187Q16C22×D12711yescharacteristic
188SD16—21no—
189SD16—2542no—
190SD16—2542no—
191SD16—2541no—
192D8—1271no—
193D8—1272no—
194D8—2542no—
195D8—2541no—
196D8C22×D12711yescharacteristic
197M16—1271no—
198M16—1272no—
199C2×C8—1272no—
200C2×C8C22×D12711yescharacteristic
201C16—1271no—
202C16—1271no—
203C16C22×D12712yes—
204C23—5082no—
205C23—5081no—
206Q8—21no—
207Q8—22no—
208Q8—41no—
209Q8—1271no—
210Q8—5082no—
211Q8—2541no—
212D4—21no—
213D4—41no—
214D4—1272no—
215D4—1271no—
216D4—5082no—
217D4—5082no—
218D4—5082no—
219D4—5081no—
220D4—5081no—
221D4—2542no—
222D4—2541no—
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—1272no—
236C22—20321no—
237C22—20322no—
238C22—5081no—
239C22—5082no—
240C22D8×D12711yescharacteristic
241C4—41no—
242C4—42no—
243C4—1272no—
244C4—5081no—
245C4D8×D12711yescharacteristic
246C4C2×D4×D12711yescharacteristic
247C2—21no—
248C2—81no—
249C2—10162no—
250C2—10161no—
251C2—2542no—
252C2C2×D8×D12711yescharacteristic
subgroupG/H
centerC2C2×D8×D127
commutator, cosocleC8×C127C24
FrattiniC8C23×D127
Fitting#510×C127C2
socleC2×C127C22×D8
hypercenter#510D127
hypocenterC127#3416
LCSG ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. #22550689C2semidirect
2. #22550678C2semidirect
3. #22550318C2semidirect
4. #22550314C2semidirect
5. #22550313C2semidirect
6. #22550309C2semidirect
7. #22550306C2semidirect
8. C2×#16861152C2semidirect
9. SD32×D127C2semidirect
10. #22550294C2semidirect
11. #510×C127C2semidirect
19. #16861232C22semidirect
20. #16861231C22semidirect
21. Q16×D127C22semidirect
25. #16861155C22semidirect
26. #16861154C22semidirect
27. #16861153C22semidirect
28. #16861152C22semidirect
29. #16861152C22semidirect
30. #16861152C22semidirect
31. D2032C22semidirect
32. #16861150C22semidirect
33. #16861148C22semidirect
34. C2×C127⋊C16C22semidirect
35. C16×D127C22semidirect
37. C2×Q16×C127C22semidirect
38. Q32×C127C22semidirect
39. SD32×C127C22semidirect
40. M32×C127C22semidirect
127. C127#3416semidirect
131. #510D127semidirect
150. C2×Q16C2×D127semidirect
155. Q32C2×D127semidirect
161. SD32C2×D127semidirect
165. M32C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506892
#225506782
#225503182
#225503142
#225503132
#225503092
#225503062
C2×#168611522
SD32×D1272
#225502942
#510×C1272
#3416127