Finite Group Databasebeta

← #30653517

Subgroups — #30653517

Order 16256.

1412 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1412 conjugacy classes of subgroups — too many to draw. Only its 545 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#22550712C211yesmaximal, characteristic, complemented
2#22550691C218yesmaximal, complemented
3#22550684C218yesmaximal, complemented
4#22550674C216yesmaximal, complemented
5C2×#16861240C216yesmaximal, complemented
6#22550643C211yesmaximal, characteristic, complemented
7C2×2−1+4×C127C211yesmaximal, characteristic, complemented
8#16861257—21no—
9#16861256C2214yescomplemented
10#16861255C2214yescomplemented
11#16861254—23no—
12C2×#14523248C2216yes—
13#41×D127—24no—
14#16861240C22112yescomplemented
15#16861240C22124yescomplemented
16#16861239C22112yescomplemented
17#16861235C2214yescomplemented
18#16861234C22112yescomplemented
19#16861229C22124yescomplemented
20C2×#14523238C22112yescomplemented
21C2×#14523235C22112yescomplemented
22#16861216C2218yescomplemented
23C2×#14523232C2216yescomplemented
242−1+4×C127—24no—
252−1+4×C127C2218yescomplemented
26C2×#41×C127—23no—
27C2×#41×C127C2211yescharacteristic
28C2×#41×C127C2213yescomplemented
29C22×Q8×C127—21no—
30C22×Q8×C127C2213yescomplemented
31#14523249—24no—
32#14523248—26no—
33#14523248—26no—
34#14523248C23112yes—
35#14523248C2314yes—
36#14523247—212no—
37C2×Dic254C2316yes—
38Q8×D127—24no—
39C2×D508C2316yes—
40C2×C22⋊D127—26no—
41D4×D127—212no—
42C2×C4×D127—26no—
43#14523238C23124yes—
44#14523236C2318yes—
45#14523235C23124yes—
46#14523233C2318yes—
47#14523232C2314yes—
48#14523232C23112yes—
49C2×C127⋊C8C23112yes—
50#41×C127—224no—
51#41×C127—26no—
52#41×C127D414yes—
53#41×C127C23112yes—
54#41×C127C2314yes—
55C2×Q8×C127—28no—
56C2×Q8×C127—23no—
57C2×Q8×C127—26no—
58C2×Q8×C127D411yescharacteristic
59C2×Q8×C127C23112yes—
60C2×Q8×C127C2313yes—
61C2×D4×C127—23no—
62C2×D4×C127D413yes—
63C2×D4×C127C2311yescharacteristic
64C22×C4×C127—23no—
65C22×C4×C127—23no—
66C22×C4×C127C2313yes—
67C22×D127—26no—
68Dic254—26no—
69Dic254C2414yes—
70D508—26no—
71D508C2414yes—
72C22⋊D127—212no—
73C22⋊D127—24no—
74C2×Dic127—26no—
75C4×D127—24no—
76C4×D127—212no—
77C127⋊C8C2418yes—
78C23×C127—21no—
79C23×C127C2×D413yes—
80Q8×C127—212no—
81Q8×C127—23no—
82Q8×C127C2×D414yes—
83Q8×C127C2416yes—
84D4×C127—212no—
85D4×C127—21no—
86D4×C127C2×D4112yes—
87D4×C127C2412yes—
88C2×C4×C127—23no—
89C2×C4×C127—212no—
90C2×C4×C127—212no—
91C2×C4×C127C2×D413yes—
92C2×C4×C127C2×D416yes—
93C2×C4×C127C2411yescharacteristic
94C2×C4×C127C2416yes—
95C22×C127—22no—
96C22×C127—43no—
97C22×C127—42no—
98C22×C127C22×D411yescharacteristic
99C22×C127C22×D416yes—
100C2×D127—24no—
101C2×D127—86no—
102C4×C127—26no—
103C4×C127C22×D416yes—
104C4×C127C22×D411yescharacteristic
105C4×C127C2511yescharacteristic
106Dic127—24no—
107C2×C127—26no—
108C2×C127—21no—
109C2×C127—42no—
110C2×C127C23×D411yescharacteristic
111D127—44no—
112#3586—1271nomaximal, Sylow
113C127#358611yescharacteristic, complemented, Sylow
114#585—1271no—
115C2×2−1+4D12711yescharacteristic, complemented
116#578—1278no—
117#577—1278no—
118#575—1276no—
119C2×#137—1276no—
120#568—1271no—
1212−1+4—24no—
1222−1+4—1274no—
1232−1+4—2541no—
1242−1+4C2×D12718yescomplemented
125H(5,𝔽2)—1274no—
126H(5,𝔽2)—2543no—
127C2×#41—23no—
128C2×#41—1276no—
129C2×#41—2544no—
130C2×#41C2×D12711yescharacteristic
131C2×#41C2×D12713yescomplemented
132C22×Q8—21no—
133C22×Q8C2×D12713yescomplemented
134#137—12712no—
135#137—12712no—
136#137—12724no—
137AGL(1,ℤ/8)—12712no—
138AGL(1,ℤ/8)—1274no—
139#135—12724no—
140C2×Q16—12712no—
141C2×SD16—12712no—
142#131—1278no—
143C2×M16—1276no—
144#41—224no—
145#41—26no—
146#41—12712no—
147#41—1274no—
148#41—2544no—
149#41—25412no—
150#41—2546no—
151#41—2546no—
152#41C22⋊D12714yes—
153#41C22×D127112yes—
154#41C22×D12714yes—
155C2×Q8—28no—
156C2×Q8—23no—
157C2×Q8—26no—
158C2×Q8—1276no—
159C2×Q8—2544no—
160C2×Q8C22⋊D12711yescharacteristic
161C2×Q8C22×D127112yes—
162C2×Q8C22×D12713yes—
163C2×D4—23no—
164C2×D4—1276no—
165C2×D4—2546no—
166C2×D4—25412no—
167C2×D4C22⋊D12713yes—
168C2×D4C22×D12711yescharacteristic
169C22×C4—23no—
170C22×C4—23no—
171C22×C4—2546no—
172C22×C4C22×D12713yes—
173Q16—12724no—
174SD16—12724no—
175SD16—1278no—
176D8—1278no—
177M16—1274no—
178M16—12712no—
179C2×C8—12712no—
180C23—21no—
181C23—2546no—
182C23C2×C22⋊D12713yes—
183Q8—212no—
184Q8—23no—
185Q8—1274no—
186Q8—2546no—
187Q8C2×C22⋊D12714yes—
188Q8C23×D12716yes—
189D4—212no—
190D4—21no—
191D4—1274no—
192D4—2544no—
193D4—25412no—
194D4—2546no—
195D4C2×C22⋊D127112yes—
196D4C23×D12712yes—
197C2×C4—23no—
198C2×C4—212no—
199C2×C4—212no—
200C2×C4—2546no—
201C2×C4—2544no—
202C2×C4—25412no—
203C2×C4C2×C22⋊D12713yes—
204C2×C4C2×C22⋊D12716yes—
205C2×C4C23×D12716yes—
206C2×C4C23×D12711yescharacteristic
207C8—1278no—
208C22—22no—
209C22—43no—
210C22—42no—
211C22—10166no—
212C22—2544no—
213C22C22×C22⋊D12711yescharacteristic
214C22C22×C22⋊D12716yes—
215C4—26no—
216C4—2544no—
217C4C22×C22⋊D12716yes—
218C4C22×C22⋊D12711yescharacteristic
219C4C24×D12711yescharacteristic
220C2—21no—
221C2—26no—
222C2—42no—
223C2—5084no—
224C2C23×C22⋊D12711yescharacteristic
subgroupG/H
centerC2C23×C22⋊D127
commutator, cosocleC4×C127C25
FrattiniC4C24×D127
FittingC2×2−1+4×C127C2
socleC2×C127C23×D4
hypercenterC2×2−1+4D127
hypocenterC127#3586
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×#41 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #22550712C2semidirect
2. #22550691C2semidirect
3. #22550684C2semidirect
4. #22550674C2semidirect
5. C2×#16861240C2semidirect
6. #22550643C2semidirect
7. C2×2−1+4×C127C2semidirect
9. #16861256C22semidirect
10. #16861255C22semidirect
14. #16861240C22semidirect
15. #16861240C22semidirect
16. #16861239C22semidirect
17. #16861235C22semidirect
18. #16861234C22semidirect
19. #16861229C22semidirect
20. C2×#14523238C22semidirect
21. C2×#14523235C22semidirect
22. #16861216C22semidirect
23. C2×#14523232C22semidirect
25. 2−1+4×C127C22semidirect
28. C2×#41×C127C22semidirect
30. C22×Q8×C127C22semidirect
113. C127#3586semidirect
115. C2×2−1+4D127semidirect
124. 2−1+4C2×D127semidirect
131. C2×#41C2×D127semidirect
133. C22×Q8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225507122
#225506912
#225506842
#225506742
C2×#168612402
#225506432
C2×2−1+4×C1272
#3586127