Finite Group Databasebeta

← #30652388

Subgroups — #30652388

Order 16256.

1536 conjugacy classes of subgroups; only the 549 normal subgroups are shown.

normal subgroups only

This group has 1536 conjugacy classes of subgroups — too many to draw. Only its 549 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
1C2×#16861257C211yesmaximal, characteristic, complemented
2#22550545C218yesmaximal, complemented
3#22550533C218yesmaximal, complemented
4#22550443C216yesmaximal, complemented
5#22550387C216yesmaximal, complemented
6#22550354C211yesmaximal, characteristic, complemented
7C22×#41×C127C211yesmaximal, characteristic, complemented
8#16861257—28no—
9C2×#14523249C2214yescomplemented
10C2×#14523248C2216yes—
11C2×Q8×D127C2214yescomplemented
12#16861205C22112yescomplemented
13#16861196C22112yescomplemented
14#16861192C2214yes—
15#16861188C22112yescomplemented
16#16861183C22112yescomplemented
17#16861180C22124yescomplemented
18#16861178C22112yes—
19Q8×Dic127C2218yescomplemented
20C4×C22⋊D127C22124yescomplemented
21#16861168C2216yes—
22C2×#41×C127—28no—
23C2×#41×C127C2218yes—
24C22×Q8×C127C2211yescharacteristic
25C22×D4×C127C2213yes—
26C23×C4×C127C2213yes—
27#14523249—216no—
28#14523248—224no—
29C2×Dic254C2316yes—
30Q8×D127—216no—
31C2×D508C2316yes—
32C2×C22⋊D127C2314yes—
33C2×C22⋊D127C23124yes—
34C2×C4×D127C23112yes—
35C4⋊Dic127C23112yes—
36Dic127⋊C4C2318yes—
37C22⋊Dic127C23124yes—
38C22⋊Dic127C2314yes—
39(C2×D127)⋊C4C2318yes—
40C4×Dic127C23112yes—
41C24×C127C2313yes—
42#41×C127—232no—
43C2×Q8×C127D418yescomplemented
44C2×Q8×C127C2314yes—
45C2×D4×C127—212no—
46C2×D4×C127C23112yes—
47C22×C4×C127—212no—
48C22×C4×C127C23112yes—
49C22×C4×C127C2311yescharacteristic
50C22×C4×C127C2313yes—
51C22×D127C2414yes—
52Dic254—212no—
53D508—212no—
54C22⋊D127—28no—
55C22⋊D127—424no—
56C2×Dic127C2418yes—
57C2×Dic127C2414yes—
58C4×D127—224no—
59C23×C127—26no—
60C23×C127—46no—
61C23×C127C2416yes—
62C23×C127C2411yescharacteristic
63Q8×C127C2×D4116yes—
64D4×C127—224no—
65C2×C4×C127—22no—
66C2×C4×C127—224no—
67C2×C4×C127C2×D4112yes—
68C2×C4×C127C2412yes—
69C2×C4×C127C2416yes—
70C22×C127—22no—
71C22×C127—26no—
72C22×C127—212no—
73C22×C127—412no—
74C22×C127C22×D412yes—
75C22×C127C2511yescharacteristic
76C2×D127—24no—
77C2×D127—28no—
78C4×C127—22no—
79C4×C127C22×D4112yes—
80Dic127—24no—
81Dic127—28no—
82C2×C127—22no—
83C2×C127—46no—
84C2×C127C23×D411yescharacteristic
85C2×C127#58512yes—
86D127—44no—
87#3487—1271nomaximal, Sylow
88C127#348711yescharacteristic, complemented, Sylow
89C2×2−1+4—1271no—
90C22×#41D12711yescharacteristic, complemented
91#550—1278no—
92#548—1278no—
93#535—1276no—
94#525—1276no—
95#519—1271no—
962−1+4—2548no—
97C2×#41—28no—
98C2×#41—1274no—
99C2×#41—1276no—
100C2×#41C2×D12718yes—
101C22×Q8—1274no—
102C22×Q8C2×D12711yescharacteristic
103C22×D4C2×D12713yes—
104C23×C4C2×D12713yes—
105#127—12712no—
106#124—12712no—
107#123—1274no—
108C22⋊Q8—12712no—
109#121—12724no—
110#121—12712no—
111#120—12712no—
112C4×Q8—1278no—
113C4×D4—12724no—
114#117—1276no—
115C24C22×D12713yes—
116#41—232no—
117#41—25416no—
118#41—25424no—
119C2×Q8—1276no—
120C2×Q8—25416no—
121C2×Q8C22⋊D12718yescomplemented
122C2×Q8C22×D12714yes—
123C2×D4—212no—
124C2×D4—12724no—
125C2×D4—1274no—
126C2×D4—1276no—
127C2×D4C22×D127112yes—
128C22×C4—212no—
129C22×C4—12712no—
130C22×C4C22×D127112yes—
131C22×C4C22×D12711yescharacteristic
132C22×C4C22×D12713yes—
133C4⋊C4—1278no—
134C4⋊C4—12712no—
135C22⋊C4—1274no—
136C22⋊C4—12724no—
137C22⋊C4—1278no—
138C42—12712no—
139C23—26no—
140C23—46no—
141C23—1274no—
142C23C23×D12716yes—
143C23C23×D12711yescharacteristic
144Q8—25412no—
145Q8C2×C22⋊D127116yes—
146D4—224no—
147D4—50824no—
148D4—2548no—
149D4—25412no—
150C2×C4—22no—
151C2×C4—224no—
152C2×C4—1278no—
153C2×C4—1274no—
154C2×C4—25424no—
155C2×C4C2×C22⋊D127112yes—
156C2×C4C23×D12712yes—
157C2×C4C23×D12716yes—
158C22—22no—
159C22—26no—
160C22—212no—
161C22—412no—
162C22—2544no—
163C22—2548no—
164C22C22×C22⋊D12712yes—
165C22C24×D12711yescharacteristic
166C4—22no—
167C4—2544no—
168C4—2548no—
169C4C22×C22⋊D127112yes—
170C2—22no—
171C2—46no—
172C2—5084no—
173C2C23×C22⋊D12711yescharacteristic
174C2#2255071212yes—
subgroupG/H
center, FrattiniC22C24×D127
commutator, socle, cosocleC22×C127C25
FittingC22×#41×C127C2
hypercenterC22×#41D127
hypocenterC127#3487
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16861257C2semidirect
2. #22550545C2semidirect
3. #22550533C2semidirect
4. #22550443C2semidirect
5. #22550387C2semidirect
6. #22550354C2semidirect
7. C22×#41×C127C2semidirect
9. C2×#14523249C22semidirect
11. C2×Q8×D127C22semidirect
12. #16861205C22semidirect
13. #16861196C22semidirect
15. #16861188C22semidirect
16. #16861183C22semidirect
17. #16861180C22semidirect
19. Q8×Dic127C22semidirect
20. C4×C22⋊D127C22semidirect
43. C2×Q8×C127D4semidirect
88. C127#3487semidirect
90. C22×#41D127semidirect
121. C2×Q8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612572
#225505452
#225505332
#225504432
#225503872
#225503542
C22×#41×C1272
#3487127