Finite Group Databasebeta

← #508×D127 #30651844

Subgroups — #508×D127 #30651844

Order 16256.

368 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 115 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#135×D127C212yesmaximal, complemented
2#22550309C211yesmaximal, characteristic, complemented
3#22550308C211yesmaximal, characteristic, complemented
4#22550307C212yesmaximal, complemented
5#22550306C212yesmaximal, complemented
6#22550305C211yesmaximal, characteristic, complemented
7Q32×D127C211yesmaximal, characteristic, complemented
8SD32×D127C212yesmaximal, complemented
9D16×D127C211yesmaximal, characteristic, complemented
10C2×C16×D127C211yesmaximal, characteristic, complemented
11#508×C127C211yesmaximal, characteristic, complemented
12#41×D127—22no—
13#16861232C2212yescomplemented
14#16861231C2212yescomplemented
15#16861230—22no—
16#16861229—22no—
17#16861228C2212yescomplemented
18Q16×D127C2212yes—
19SD16×D127—22no—
20D8×D127C2212yes—
21C2×C8×D127C2211yescharacteristic
22Dic1016C2211yescharacteristic, complemented
23#16861155C2212yescomplemented
24#16861154C2212yescomplemented
25#16861153C2212yescomplemented
26#16861152C2212yescomplemented
27D2032C2211yescharacteristic, complemented
28#16861150C2212yescomplemented
29C2×C127⋊C16C2211yescharacteristic, complemented
30C16×D127C2211yescharacteristic
31C16×D127C2211yescharacteristic
32C16×D127C2211yescharacteristic, complemented
33C16×D127C2211yescharacteristic, complemented
34#135×C127C2212yescomplemented
35Q32×C127C2211yescharacteristic, complemented
36SD32×C127C2212yescomplemented
37D16×C127C2211yescharacteristic, complemented
38C2×C16×C127C2211yescharacteristic, complemented
39#14523249—22no—
40#14523248—22no—
41#14523248—42no—
42#14523247—22no—
43#14523247—42no—
44Q8×D127—22no—
45D4×D127—22no—
46D4×D127—42no—
47C2×C4×D127—42no—
48C2×C4×D127D411yescharacteristic
49Dic508C2312yes—
50#14523238—22no—
51#14523237—22no—
52#14523236—22no—
53#14523235—22no—
54D1016C2312yes—
55#14523233—22no—
56C2×C127⋊C8D411yescharacteristic
57C8×D127D411yescharacteristic
58C8×D127D411yescharacteristic
59C8×D127C2311yescharacteristic
60C8×D127C2311yescharacteristic
61C127⋊C16C2311yescharacteristic
62C127⋊C16C2311yescharacteristic
63#41×C127—22no—
64Q16×C127C2312yes—
65SD16×C127—22no—
66D8×C127C2312yes—
67C2×C8×C127C2311yescharacteristic
68C16×C127C2311yescharacteristic
69C16×C127C2311yescharacteristic
70C22×D127—42no—
71C22×D127D811yescharacteristic
72Dic254—22no—
73Dic254—42no—
74D508—22no—
75D508—42no—
76C22⋊D127—42no—
77C22⋊D127—42no—
78C22⋊D127—42no—
79C2×Dic127—42no—
80C2×Dic127D811yescharacteristic
81C4×D127—42no—
82C4×D127—42no—
83C4×D127—42no—
84C4×D127D811yescharacteristic
85C4×D127D811yescharacteristic
86C4×D127C2×D411yescharacteristic
87C4×D127C2×D411yescharacteristic
88C127⋊C8C2×D411yescharacteristic
89C127⋊C8C2×D411yescharacteristic
90Q8×C127—22no—
91D4×C127—22no—
92D4×C127—42no—
93C2×C4×C127—42no—
94C2×C4×C127C2×D411yescharacteristic
95C8×C127C2×D411yescharacteristic
96C8×C127C2411yescharacteristic
97C22×C127—42no—
98C22×C127C2×D811yescharacteristic
99C2×D127—22no—
100C2×D127—42no—
101C2×D127—84no—
102C2×D127C2×D811yescharacteristic
103C2×D127C2×D811yescharacteristic
104C4×C127—42no—
105C4×C127C2×D811yescharacteristic
106C4×C127C22×D411yescharacteristic
107Dic127—42no—
108Dic127C2×D811yescharacteristic
109Dic127C2×D811yescharacteristic
110C2×C127—21no—
111C2×C127—82no—
112C2×C127C22×D811yescharacteristic
113D127—21no—
114D127—82no—
115D127#50812yesdirect product factor
116C2×#508—1271nomaximal, Sylow
117C127C2×#50811yescharacteristic, complemented, Sylow
118C2×#135—1272no—
119#508—1272no—
120#508—1271no—
121#508—1272no—
122#508—1271no—
123#508—1271no—
124#508D12711yescharacteristic, direct product factor
125C2×Q32—1271no—
126C2×SD32—1272no—
127C2×D16—1271no—
128C22×C16—1271no—
129C2×#41—2542no—
130#135—1272no—
131#135—1272no—
132#135—1272no—
133#135—2542no—
134#135—2542no—
135#135C2×D12712yescomplemented
136C2×Q16—1272no—
137C2×SD16—2542no—
138C2×D8—1272no—
139C22×C8—1271no—
140Q32—1272no—
141Q32—1271no—
142Q32C2×D12711yescharacteristic, complemented
143SD32—1272no—
144SD32—1272no—
145SD32—1272no—
146SD32C2×D12712yescomplemented
147D16—1272no—
148D16—1271no—
149D16C2×D12711yescharacteristic, complemented
150C2×C16—1271no—
151C2×C16—1271no—
152C2×C16—1271no—
153C2×C16—1271no—
154C2×C16—1271no—
155C2×C16C2×D12711yescharacteristic, complemented
156#41—22no—
157#41—5082no—
158#41—5082no—
159#41—2542no—
160#41—2542no—
161#41—2542no—
162C2×Q8—2542no—
163C2×D4—5082no—
164C2×D4—2542no—
165C22×C4—1271no—
166C22×C4—5082no—
167Q16—1272no—
168Q16—2542no—
169Q16C22×D12712yes—
170SD16—22no—
171SD16—2542no—
172SD16—2542no—
173SD16—2542no—
174D8—1272no—
175D8—2542no—
176D8C22×D12712yes—
177C2×C8—1271no—
178C2×C8—1271no—
179C2×C8—1271no—
180C2×C8—1271no—
181C2×C8—1271no—
182C2×C8C22×D12711yescharacteristic
183C16—1271no—
184C16—1271no—
185C16C22×D12711yescharacteristic
186C16C22×D12711yescharacteristic
187C23—1271no—
188C23—5082no—
189Q8—22no—
190Q8—5082no—
191Q8—2542no—
192D4—22no—
193D4—42no—
194D4—5082no—
195D4—5082no—
196D4—5082no—
197D4—5082no—
198D4—2542no—
199C2×C4—42no—
200C2×C4—1271no—
201C2×C4—1271no—
202C2×C4—1271no—
203C2×C4—1271no—
204C2×C4—1271no—
205C2×C4—5082no—
206C2×C4—5082no—
207C2×C4—5082no—
208C2×C4—5082no—
209C2×C4D4×D12711yescharacteristic
210C8—1271no—
211C8—1271no—
212C8D4×D12711yescharacteristic
213C8C23×D12711yescharacteristic
214C22—42no—
215C22—1271no—
216C22—1271no—
217C22—10164no—
218C22—5082no—
219C22—2542no—
220C22D8×D12711yescharacteristic
221C4—42no—
222C4—1271no—
223C4—1271no—
224C4—5082no—
225C4D8×D12711yescharacteristic
226C4C2×D4×D12711yescharacteristic
227C2—21no—
228C2—82no—
229C2—1272no—
230C2—10162no—
231C2—2541no—
232C2C2×D8×D12711yescharacteristic
subgroupG/H
centerC4D8×D127
commutator, cosocleC8×C127C24
FrattiniC8C23×D127
Fitting#508×C127C2
socleC2×C127C22×D8
hypercenter#508D127
hypocenterC127C2×#508
LCSG ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D127C2semidirect
2. #22550309C2semidirect
3. #22550308C2semidirect
4. #22550307C2semidirect
5. #22550306C2semidirect
6. #22550305C2semidirect
7. Q32×D127C2semidirect
8. SD32×D127C2semidirect
9. D16×D127C2semidirect
10. C2×C16×D127C2semidirect
11. #508×C127C2semidirect
13. #16861232C22semidirect
14. #16861231C22semidirect
17. #16861228C22semidirect
22. Dic1016C22semidirect
23. #16861155C22semidirect
24. #16861154C22semidirect
25. #16861153C22semidirect
26. #16861152C22semidirect
27. D2032C22semidirect
28. #16861150C22semidirect
29. C2×C127⋊C16C22semidirect
32. C16×D127C22semidirect
33. C16×D127C22semidirect
34. #135×C127C22semidirect
35. Q32×C127C22semidirect
36. SD32×C127C22semidirect
37. D16×C127C22semidirect
38. C2×C16×C127C22semidirect
115. D127#508direct (N × H)
117. C127C2×#508semidirect
124. #508D127direct (N × H)
135. #135C2×D127semidirect
142. Q32C2×D127semidirect
146. SD32C2×D127semidirect
149. D16C2×D127semidirect
155. C2×C16C2×D127semidirect

Maximal subgroups

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

subgroupindex
#135×D1272
#225503092
#225503082
#225503072
#225503062
#225503052
Q32×D1272
SD32×D1272
D16×D1272
C2×C16×D1272
#508×C1272
C2×#508127