Finite Group Databasebeta

← #30649156

Subgroups — #30649156

Order 16256.

572 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 157 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×#16861229C214yesmaximal, complemented
2#22550426C211yesmaximal, characteristic, complemented
3#22550258C214yesmaximal, complemented
4#22550255C211yesmaximal, characteristic, complemented
5#22550252C211yesmaximal, characteristic, complemented
6#22550250C211yesmaximal, characteristic, complemented
7#22550248C211yesmaximal, characteristic, complemented
8C2×C4×C127⋊C8C211yesmaximal, characteristic, complemented
9#532×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2212yes—
11#16861229—216no—
12C2×#14523238C2214yes—
13C2×#14523236C2214yes—
14C2×#14523235C2214yes—
15C2×#14523233C2214yes—
16C22×C127⋊C8C2212yes—
17#16861209C2211yescharacteristic
18#16861206C2211yescharacteristic
19#16861197C2212yes—
20#16861183—22no—
21C4×D508—22no—
22C4×C127⋊C8C2211yescharacteristic
23C4×C127⋊C8C2211yescharacteristic
24C4×C127⋊C8C2212yes—
25C2×#41×C127C2212yes—
26C4⋊Q8×C127C2211yescharacteristic
27#127×C127C2211yescharacteristic
28#124×C127C2212yes—
29#121×C127—22no—
30C4×D4×C127—22no—
31C2×C42×C127C2211yescharacteristic
32#14523248—28no—
33C2×Dic254C2312yes—
34C2×D508—22no—
35C2×D508C2312yes—
36C2×C22⋊D127—22no—
37C2×C4×D127—22no—
38#14523238—28no—
39#14523236—28no—
40#14523235—28no—
41#14523233—28no—
42C2×C127⋊C8D418yescomplemented
43C2×C127⋊C8C2312yes—
44C2×C127⋊C8C2312yes—
45C4⋊Dic127—22no—
46(C2×D127)⋊C4—24no—
47#41×C127—28no—
48C2×Q8×C127C2312yes—
49C2×D4×C127—22no—
50C2×D4×C127—22no—
51C2×D4×C127C2312yes—
52C22×C4×C127—22no—
53C22×C4×C127D412yes—
54C22×C4×C127C2311yescharacteristic
55C4⋊C4×C127—22no—
56C22⋊C4×C127—24no—
57C42×C127D412yes—
58C42×C127C2311yescharacteristic
59C42×C127C2311yescharacteristic
60C22×D127—22no—
61Dic254—24no—
62D508—24no—
63D508—44no—
64C22⋊D127—44no—
65C2×Dic127—22no—
66C4×D127—44no—
67C127⋊C8C2×D414yes—
68C127⋊C8C2×D414yes—
69C23×C127—22no—
70C23×C127C2×D411yescharacteristic
71Q8×C127—24no—
72D4×C127—24no—
73D4×C127—44no—
74D4×C127—44no—
75C2×C4×C127—24no—
76C2×C4×C127—22no—
77C2×C4×C127—44no—
78C2×C4×C127C2×D414yes—
79C2×C4×C127C2×D412yes—
80C2×C4×C127C2×D412yes—
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2411yescharacteristic
83C22×C127—22no—
84C22×C127—42no—
85C22×C127—44no—
86C22×C127#12712yes—
87C22×C127C22×D411yescharacteristic
88C2×D127—42no—
89C2×D127—44no—
90C4×C127—22no—
91C4×C127—42no—
92C4×C127#12712yes—
93C4×C127#13514yescomplemented
94C4×C127C22×D412yes—
95Dic127—42no—
96C2×C127—22no—
97C2×C127—82no—
98C2×C127C2×#12711yescharacteristic
99C2×C127C2×#13512yes—
100D127—82no—
101#3147—1271nomaximal, Sylow
102C127#314711yescharacteristic, complemented, Sylow
103C2×#135—1274no—
104#532—1271no—
105#532D12711yescharacteristic, complemented
106#495—1274no—
107#494—1271no—
108#493—1271no—
109#492—1271no—
110#492—1271no—
111C2×C4×C8—1271no—
112C2×#41—1272no—
113C2×#41C2×D12712yes—
114#135—25416no—
115C2×Q16—1274no—
116C2×SD16—1274no—
117C2×SD16—1274no—
118C2×D8—1274no—
119C22×C8—1272no—
120C4⋊Q8—1271no—
121C4⋊Q8C2×D12711yescharacteristic
122#127—1271no—
123#127C2×D12711yescharacteristic
124#124—1272no—
125#124C2×D12712yes—
126#121—22no—
127#121—2542no—
128C4×D4—22no—
129C4×D4—2542no—
130C2×C42C2×D12711yescharacteristic
131C4×C8—1271no—
132C4×C8—1271no—
133C4×C8—1272no—
134#41—28no—
135#41—2548no—
136C2×Q8—1272no—
137C2×Q8C22×D12712yes—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—1272no—
141C2×D4—2542no—
142C2×D4—2542no—
143C2×D4C22×D12712yes—
144C22×C4—22no—
145C22×C4—2542no—
146C22×C4C22⋊D12712yes—
147C22×C4C22×D12711yescharacteristic
148Q16—2548no—
149SD16—2548no—
150SD16—2548no—
151D8—2548no—
152C2×C8—1278no—
153C2×C8—1272no—
154C2×C8—1272no—
155C4⋊C4—22no—
156C4⋊C4—2542no—
157C22⋊C4—24no—
158C22⋊C4—2544no—
159C42C22⋊D12712yes—
160C42C22×D12711yescharacteristic
161C42C22×D12711yescharacteristic
162C23—22no—
163C23—2542no—
164C23C2×C22⋊D12711yescharacteristic
165Q8—24no—
166Q8—2544no—
167D4—24no—
168D4—44no—
169D4—44no—
170D4—5084no—
171D4—5084no—
172D4—2544no—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—44no—
176C2×C4—5084no—
177C2×C4—2542no—
178C2×C4D4×D12714yes—
179C2×C4C2×C22⋊D12712yes—
180C2×C4C2×C22⋊D12712yes—
181C2×C4C2×C22⋊D12711yescharacteristic
182C2×C4C23×D12711yescharacteristic
183C8—1274no—
184C8—1274no—
185C22—22no—
186C22—44no—
187C22—42no—
188C22—5084no—
189C22—5082no—
190C22#1686120512yes—
191C22C22×C22⋊D12711yescharacteristic
192C4—22no—
193C4—42no—
194C4—5082no—
195C4#1686120512yes—
196C4#1686122914yescomplemented
197C4C2×D4×D12712yes—
198C2—22no—
199C2—82no—
200C2—10162no—
201C2C2×#1686120511yescharacteristic
202C2C2×#1686122912yes—
subgroupG/H
centerC2×C4C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#532×C127C2
socleC22×C127C22×D4
hypercenter#532D127
hypocenterC127#3147
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#16861229C2semidirect
2. #22550426C2semidirect
3. #22550258C2semidirect
4. #22550255C2semidirect
5. #22550252C2semidirect
6. #22550250C2semidirect
7. #22550248C2semidirect
8. C2×C4×C127⋊C8C2semidirect
9. #532×C127C2semidirect
42. C2×C127⋊C8D4semidirect
93. C4×C127#135semidirect
102. C127#3147semidirect
105. #532D127semidirect
196. C4#16861229semidirect

Maximal subgroups

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

subgroupindex
C2×#168612292
#225504262
#225502582
#225502552
#225502522
#225502502
#225502482
C2×C4×C127⋊C82
#532×C1272
#3147127