Finite Group Databasebeta

← #30648982

Subgroups — #30648982

Order 16256.

394 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550488C211yesmaximal, characteristic, complemented
2#22550217C212yesmaximal, complemented
3#22550176C212yesmaximal, complemented
4#22550081C212yesmaximal, complemented
5#22550027C212yesmaximal
6#22549976C212yesmaximal, complemented
7C4×#14523236C212yesmaximal, complemented
8#22549937C211yesmaximal, characteristic, complemented
9#518×C127C211yesmaximal, characteristic
10C2×#14523236C2212yes—
11C2×#14523236C2212yes—
12#16861209C2211yescharacteristic, complemented
13#16861202—22no—
14#16861201C2211yescharacteristic, complemented
15#16861199—22no—
16#16861188C2212yes—
17C4×Dic254C2212yescomplemented
18#16861142C2212yescomplemented
19#16861140C2212yescomplemented
20C4⋊(C127⋊C8)C2212yescomplemented
21#16861132C2212yescomplemented
22#16861132C2212yescomplemented
23#16861130C2212yes—
24#16861130C2212yes—
25C22⋊(C127⋊C8)C2212yes—
26#16861120C2211yescharacteristic, complemented
27C4×C127⋊C8C2211yescharacteristic, complemented
28C22×D4×C127C2211yescharacteristic
29C4×D4×C127—22no—
30C4×D4×C127C2212yes—
31C4×D4×C127C2212yes—
32#117×C127C2211yescharacteristic
33#117×C127C2211yescharacteristic
34C2×C22⋊C4×C127—22no—
35C2×Dic254C2312yes—
36#14523236—24no—
37#14523236—24no—
38C2×C127⋊C8C2312yes—
39C2×C127⋊C8C2312yes—
40C4⋊Dic127—22no—
41C4⋊Dic127C2312yes—
42Dic127⋊C4—22no—
43Dic127⋊C4—22no—
44Dic127⋊C4—22no—
45C22⋊Dic127—22no—
46C4×Dic127—22no—
47C24×C127—21no—
48C2×D4×C127—24no—
49C2×D4×C127—21no—
50C2×D4×C127C2312yes—
51C22×C4×C127—22no—
52C22×C4×C127—22no—
53C22×C4×C127C2311yescharacteristic
54C4⋊C4×C127D412yes—
55C4⋊C4×C127C2312yes—
56C22⋊C4×C127—21no—
57C22⋊C4×C127—22no—
58C22⋊C4×C127—22no—
59C22⋊C4×C127D412yes—
60C42×C127—21no—
61C42×C127C2311yescharacteristic
62C42×C127C2311yescharacteristic
63Dic254—22no—
64Dic254—22no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C127⋊C8—22no—
68C127⋊C8—22no—
69C23×C127—24no—
70C23×C127—22no—
71C23×C127—82no—
72C23×C127C2×D411yescharacteristic
73D4×C127—24no—
74D4×C127—22no—
75D4×C127#4114yes—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127—82no—
79C2×C4×C127—82no—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D412yes—
82C2×C4×C127C2×D412yes—
83C2×C4×C127C2411yescharacteristic
84C22×C127—21no—
85C22×C127—24no—
86C22×C127—22no—
87C22×C127—42no—
88C22×C127—42no—
89C22×C127—84no—
90C22×C127C22×D411yescharacteristic
91C4×C127—21no—
92C4×C127—22no—
93C4×C127—22no—
94C4×C127—42no—
95C4×C127C2×#4112yes—
96Dic127—42no—
97Dic127—42no—
98C2×C127—41no—
99C2×C127—44no—
100C2×C127#52511yescharacteristic
101C2×C127#57712yes—
102#3123—1271nomaximal, Sylow
103C127#312311yescharacteristic, complemented, Sylow
104#541—1271no—
105#518D12711yescharacteristic
106#485—1272no—
107#478—1272no—
108#461—1272no—
109#450—1272no—
110#440—1272no—
111C4×SD16—1272no—
112#433—1271no—
113C22×D4C2×D12711yescharacteristic
114C2×SD16—1272no—
115C2×SD16—1272no—
116C4⋊Q8—1271no—
117#126—2542no—
118#125—1271no—
119#125—2542no—
120C22⋊Q8—1272no—
121C4×Q8—1272no—
122C4×D4—22no—
123C4×D4C2×D12712yes—
124C4×D4C2×D12712yes—
125#117C2×D12711yescharacteristic
126#117C2×D12711yescharacteristic
127C2×C22⋊C4—22no—
128#107—1272no—
129#106—1272no—
130C4⋊C8—1272no—
131#103—1272no—
132#103—1272no—
133#102—1272no—
134#102—1272no—
135C22⋊C8—1272no—
136#97—1271no—
137C4×C8—1271no—
138C24—21no—
139C2×Q8—1272no—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4C22×D12712yes—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4C22×D12711yescharacteristic
146SD16—2544no—
147SD16—2544no—
148C2×C8—1272no—
149C2×C8—1272no—
150C4⋊C4—1272no—
151C4⋊C4—2542no—
152C4⋊C4—2542no—
153C4⋊C4—2542no—
154C4⋊C4—2542no—
155C4⋊C4C22⋊D12712yes—
156C4⋊C4C22×D12712yes—
157C22⋊C4—21no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—2542no—
161C22⋊C4C22⋊D12712yes—
162C42—21no—
163C42—2542no—
164C42C22×D12711yescharacteristic
165C42C22×D12711yescharacteristic
166C23—24no—
167C23—22no—
168C23—82no—
169C23C2×C22⋊D12711yescharacteristic
170Q8—2542no—
171Q8—2542no—
172D4—24no—
173D4—22no—
174D4#1452324814yes—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—2542no—
180C2×C4—2542no—
181C2×C4C2×C22⋊D12711yescharacteristic
182C2×C4C2×C22⋊D12712yes—
183C2×C4C2×C22⋊D12712yes—
184C2×C4C23×D12711yescharacteristic
185C8—2542no—
186C8—2542no—
187C22—21no—
188C22—24no—
189C22—22no—
190C22—42no—
191C22—42no—
192C22—84no—
193C22C22×C22⋊D12711yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—5082no—
199C4—5082no—
200C4C2×#1452324812yes—
201C2—41no—
202C2—44no—
203C2#2255038711yescharacteristic
204C2#2255068712yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#518×C127C2
socleC22×C127C22×D4
prime radicalC22×D4×C127C22
hypercenter#518D127
hypocenterC127#3123
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #22550488C2semidirect
2. #22550217C2semidirect
3. #22550176C2semidirect
4. #22550081C2semidirect
6. #22549976C2semidirect
7. C4×#14523236C2semidirect
8. #22549937C2semidirect
12. #16861209C22semidirect
14. #16861201C22semidirect
17. C4×Dic254C22semidirect
18. #16861142C22semidirect
19. #16861140C22semidirect
20. C4⋊(C127⋊C8)C22semidirect
21. #16861132C22semidirect
22. #16861132C22semidirect
26. #16861120C22semidirect
27. C4×C127⋊C8C22semidirect
103. C127#3123semidirect

Maximal subgroups

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

subgroupindex
#225504882
#225502172
#225501762
#225500812
#225500272
#225499762
C4×#145232362
#225499372
#518×C1272
#3123127