Finite Group Databasebeta

← #30648339

Subgroups — #30648339

Order 16256.

500 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861231C211yesmaximal, characteristic, complemented
2#22550387C212yesmaximal, complemented
3#22550134C212yesmaximal, complemented
4#22550129C212yesmaximal, complemented
5#22550122C214yesmaximal, complemented
6#22549914C211yesmaximal, characteristic, complemented
7#22549873C212yesmaximal, complemented
8C22×M16×C127C211yesmaximal, characteristic, complemented
9C2×#14523248C2212yes—
10#16861231—24no—
11C2×Dic508C2211yescharacteristic, complemented
12C2×#14523237C2212yescomplemented
13C2×D1016C2211yescharacteristic, complemented
14#16861192—22no—
15#16861188C2214yes—
16#16861183C2214yes—
17#16861178—22no—
18C4×C22⋊D127—24no—
19#16861168C2212yes—
20#16861143C2212yescomplemented
21#16861141C2212yescomplemented
22#16861134C2214yescomplemented
23#16861131C2214yescomplemented
24C23×C4×C127C2211yescharacteristic
25C2×M16×C127—24no—
26C2×M16×C127C2212yes—
27C2×M16×C127C2212yes—
28C22×C8×C127C2211yescharacteristic
29C22×C8×C127C2211yescharacteristic
30#14523248—24no—
31#14523248—24no—
32C2×Dic254C2312yes—
33C2×D508C2312yes—
34C2×C22⋊D127—22no—
35C2×C22⋊D127—24no—
36C2×C4×D127—22no—
37Dic508—22no—
38#14523237—24no—
39D1016—22no—
40C4⋊Dic127C2314yes—
41Dic127⋊C4—24no—
42C22⋊Dic127—24no—
43C22⋊Dic127—22no—
44(C2×D127)⋊C4—24no—
45C4×Dic127—22no—
46C24×C127D411yescharacteristic
47C22×C4×C127—22no—
48C22×C4×C127—22no—
49C22×C4×C127D411yescharacteristic
50C22×C4×C127D412yes—
51C22×C4×C127C2312yes—
52C22×C4×C127C2311yescharacteristic
53M16×C127—24no—
54M16×C127—24no—
55C2×C8×C127—22no—
56C2×C8×C127D414yescomplemented
57C2×C8×C127C2312yes—
58C2×C8×C127C2311yescharacteristic
59C2×C8×C127C2311yescharacteristic
60C22×D127—22no—
61Dic254—22no—
62Dic254—22no—
63D508—22no—
64D508—22no—
65C22⋊D127—48no—
66C22⋊D127—44no—
67C2×Dic127—24no—
68C2×Dic127—22no—
69C4×D127—44no—
70C23×C127—22no—
71C23×C127—42no—
72C23×C127C2×D412yes—
73C23×C127C2×D411yescharacteristic
74C2×C4×C127—22no—
75C2×C4×C127—24no—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127C2×D412yes—
79C2×C4×C127C2×D412yes—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127#4112yes—
82C2×C4×C127C2411yescharacteristic
83C8×C127—22no—
84C8×C127C2×D412yes—
85C8×C127C2×D412yes—
86C22×C127—22no—
87C22×C127—22no—
88C22×C127—24no—
89C22×C127—44no—
90C22×C127#12112yes—
91C22×C127C22×D411yescharacteristic
92C2×D127—42no—
93C2×D127—44no—
94C4×C127—22no—
95C4×C127#12112yes—
96C4×C127C22×D411yescharacteristic
97C4×C127C2×#4111yescharacteristic
98Dic127—42no—
99Dic127—44no—
100C2×C127—22no—
101C2×C127—42no—
102C2×C127C2×#12111yescharacteristic
103C2×C127#57512yes—
104D127—82no—
105#3054—1271nomaximal, Sylow
106C127#305411yescharacteristic, complemented, Sylow
107C2×#135—1271no—
108C22×M16D12711yescharacteristic, complemented
109#525—1272no—
110#470—1272no—
111#469—1272no—
112#468—1274no—
113#427—1271no—
114#416—1272no—
115C2×#41—1272no—
116C23×C4C2×D12711yescharacteristic
117#135—2544no—
118C2×Q16—1271no—
119C2×SD16—1272no—
120C2×D8—1271no—
121C2×M16—24no—
122C2×M16C2×D12712yes—
123C2×M16C2×D12712yes—
124C22×C8C2×D12711yescharacteristic
125C22×C8C2×D12711yescharacteristic
126#123—2542no—
127C22⋊Q8—1274no—
128#121—1274no—
129#120—2542no—
130C4×D4—2544no—
131#117—1272no—
132#107—1272no—
133#106—1272no—
134#103—1274no—
135#102—1274no—
136C24D50811yescharacteristic
137#41—2544no—
138#41—2544no—
139C2×Q8—1272no—
140C2×D4—1272no—
141C2×D4—2544no—
142C2×D4—2542no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—2542no—
146C22×C4D50811yescharacteristic
147C22×C4D50812yes—
148C22×C4C22×D12712yes—
149C22×C4C22×D12711yescharacteristic
150Q16—2542no—
151SD16—2544no—
152D8—2542no—
153M16—24no—
154M16—24no—
155C2×C8—22no—
156C2×C8C22⋊D12714yescomplemented
157C2×C8C22×D12712yes—
158C2×C8C22×D12711yescharacteristic
159C2×C8C22×D12711yescharacteristic
160C4⋊C4—1274no—
161C4⋊C4—2544no—
162C22⋊C4—2542no—
163C22⋊C4—2544no—
164C22⋊C4—2544no—
165C42—2542no—
166C23—22no—
167C23—42no—
168C23—2542no—
169C23C2×D50812yes—
170C23C2×D50811yescharacteristic
171Q8—2542no—
172Q8—2542no—
173D4—5088no—
174D4—5084no—
175D4—2542no—
176D4—2542no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—5084no—
182C2×C4—2544no—
183C2×C4—2542no—
184C2×C4C2×C22⋊D12712yes—
185C2×C4C2×D50812yes—
186C2×C4C2×D50811yescharacteristic
187C2×C4#1452324812yes—
188C2×C4C23×D12711yescharacteristic
189C8—22no—
190C8C2×C22⋊D12712yes—
191C8C2×C22⋊D12712yes—
192C22—22no—
193C22—22no—
194C22—24no—
195C22—44no—
196C22—5082no—
197C22—5084no—
198C22#1686118312yes—
199C22C22×D50811yescharacteristic
200C4—22no—
201C4—5082no—
202C4—5084no—
203C4#1686118312yes—
204C4C22×C22⋊D12711yescharacteristic
205C4C2×#1452324811yescharacteristic
206C2—22no—
207C2—42no—
208C2—10162no—
209C2C2×#1686118311yescharacteristic
210C2#2255067612yes—
subgroupG/H
centerC2×C4C2×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×M16×C127C2
socleC22×C127C22×D4
hypercenterC22×M16D127
hypocenterC127#3054
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C2×#16861231C2semidirect
2. #22550387C2semidirect
3. #22550134C2semidirect
4. #22550129C2semidirect
5. #22550122C2semidirect
6. #22549914C2semidirect
7. #22549873C2semidirect
8. C22×M16×C127C2semidirect
11. C2×Dic508C22semidirect
12. C2×#14523237C22semidirect
13. C2×D1016C22semidirect
20. #16861143C22semidirect
21. #16861141C22semidirect
22. #16861134C22semidirect
23. #16861131C22semidirect
56. C2×C8×C127D4semidirect
106. C127#3054semidirect
108. C22×M16D127semidirect
156. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612312
#225503872
#225501342
#225501292
#225501222
#225499142
#225498732
C22×M16×C1272
#3054127