Finite Group Databasebeta

← #30648973

Subgroups — #30648973

Order 16256.

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

normal subgroups only

This group has 418 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#22550481C211yesmaximal, characteristic, complemented
2#22550200C212yesmaximal, complemented
3#22550176C212yesmaximal, complemented
4#22550071C212yesmaximal, complemented
5#22550010C212yesmaximal, complemented
6#22549986C212yesmaximal, complemented
7C4×#14523233C212yesmaximal, complemented
8#22549937C211yesmaximal, characteristic, complemented
9#518×C127C211yesmaximal, characteristic, complemented
10C2×#14523233C2212yes—
11C2×#14523233C2212yes—
12#16861206C2211yescharacteristic, complemented
13#16861201C2211yescharacteristic, complemented
14#16861194—22no—
15#16861183C2212yes—
16#16861182—22no—
17C4×D508C2212yescomplemented
18#16861142C2212yescomplemented
19#16861140C2212yescomplemented
20C4⋊(C127⋊C8)C2212yescomplemented
21#16861130C2212yes—
22#16861130C2212yes—
23#16861129C2212yescomplemented
24#16861129C2212yescomplemented
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×D508—22no—
36C2×D508C2312yes—
37C2×C22⋊D127—22no—
38C2×C4×D127—22no—
39#14523233—24no—
40#14523233—24no—
41C2×C127⋊C8C2312yes—
42C2×C127⋊C8C2312yes—
43C4⋊Dic127C2312yes—
44Dic127⋊C4—22no—
45(C2×D127)⋊C4—22no—
46(C2×D127)⋊C4—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
63C22×D127—22no—
64D508—22no—
65D508—22no—
66D508—82no—
67C22⋊D127—82no—
68C2×Dic127—22no—
69C4×D127—82no—
70C127⋊C8—22no—
71C127⋊C8—22no—
72C23×C127—24no—
73C23×C127—22no—
74C23×C127—82no—
75C23×C127C2×D411yescharacteristic
76D4×C127—24no—
77D4×C127—22no—
78D4×C127#4114yes—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—82no—
82C2×C4×C127—82no—
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D412yes—
85C2×C4×C127C2×D412yes—
86C2×C4×C127C2411yescharacteristic
87C22×C127—21no—
88C22×C127—24no—
89C22×C127—22no—
90C22×C127—42no—
91C22×C127—42no—
92C22×C127—84no—
93C22×C127C22×D411yescharacteristic
94C2×D127—42no—
95C2×D127—42no—
96C2×D127—42no—
97C4×C127—21no—
98C4×C127—22no—
99C4×C127—22no—
100C4×C127—42no—
101C4×C127C2×#4112yes—
102Dic127—42no—
103C2×C127—41no—
104C2×C127—44no—
105C2×C127#52511yescharacteristic
106C2×C127#57612yes—
107D127—82no—
108#3122—1271nomaximal, Sylow
109C127#312211yescharacteristic, complemented, Sylow
110#540—1271no—
111#518D12711yescharacteristic, complemented
112#482—1272no—
113#478—1272no—
114#459—1272no—
115#447—1272no—
116#442—1272no—
117C4×D8—1272no—
118#433—1271no—
119C22×D4C2×D12711yescharacteristic
120C2×D8—1272no—
121C2×D8—1272no—
122#127—1271no—
123#125—1271no—
124#123—2542no—
125#121—1272no—
126#121—2542no—
127C4×D4—22no—
128C4×D4—1272no—
129C4×D4C2×D12712yes—
130C4×D4C2×D12712yes—
131#117C2×D12711yescharacteristic
132#117C2×D12711yescharacteristic
133C2×C22⋊C4—22no—
134#107—1272no—
135#106—1272no—
136C4⋊C8—1272no—
137#102—1272no—
138#102—1272no—
139#102—1272no—
140#102—1272no—
141C22⋊C8—1272no—
142#97—1271no—
143C4×C8—1271no—
144C24—21no—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—1272no—
148C2×D4—2542no—
149C2×D4—2542no—
150C2×D4C22×D12712yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—2542no—
154C22×C4C22×D12711yescharacteristic
155D8—2544no—
156D8—2544no—
157C2×C8—1272no—
158C2×C8—1272no—
159C4⋊C4—1272no—
160C4⋊C4—2542no—
161C4⋊C4C22⋊D12712yes—
162C4⋊C4C22×D12712yes—
163C22⋊C4—21no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—2542no—
167C22⋊C4—2542no—
168C22⋊C4C22⋊D12712yes—
169C42—21no—
170C42C22×D12711yescharacteristic
171C42C22×D12711yescharacteristic
172C23—24no—
173C23—22no—
174C23—82no—
175C23—2542no—
176C23C2×C22⋊D12711yescharacteristic
177D4—24no—
178D4—22no—
179D4—10162no—
180D4—10162no—
181D4—2542no—
182D4—2542no—
183D4#1452324814yes—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—10162no—
189C2×C4—2542no—
190C2×C4C2×C22⋊D12711yescharacteristic
191C2×C4C2×C22⋊D12712yes—
192C2×C4C2×C22⋊D12712yes—
193C2×C4C23×D12711yescharacteristic
194C8—2542no—
195C8—2542no—
196C22—21no—
197C22—24no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22—84no—
202C22—5082no—
203C22—5082no—
204C22—5082no—
205C22C22×C22⋊D12711yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—5082no—
211C4C2×#1452324812yes—
212C2—41no—
213C2—44no—
214C2—10162no—
215C2#2255038711yescharacteristic
216C2#2255067912yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#518×C127C2
socleC22×C127C22×D4
hypercenter#518D127
hypocenterC127#3122
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. #22550481C2semidirect
2. #22550200C2semidirect
3. #22550176C2semidirect
4. #22550071C2semidirect
5. #22550010C2semidirect
6. #22549986C2semidirect
7. C4×#14523233C2semidirect
8. #22549937C2semidirect
9. #518×C127C2semidirect
12. #16861206C22semidirect
13. #16861201C22semidirect
17. C4×D508C22semidirect
18. #16861142C22semidirect
19. #16861140C22semidirect
20. C4⋊(C127⋊C8)C22semidirect
23. #16861129C22semidirect
24. #16861129C22semidirect
26. #16861120C22semidirect
27. C4×C127⋊C8C22semidirect
109. C127#3122semidirect
111. #518D127semidirect

Maximal subgroups

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

subgroupindex
#225504812
#225502002
#225501762
#225500712
#225500102
#225499862
C4×#145232332
#225499372
#518×C1272
#3122127