Finite Group Databasebeta

← #30647967

Subgroups — #30647967

Order 16256.

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

normal subgroups only

This group has 738 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
1C23×Dic254C211yesmaximal, characteristic, complemented
2C2×#16861242C212yesmaximal, complemented
3#22550387C211yesmaximal, characteristic
4#22550034C214yesmaximal, complemented
5#22550017C214yesmaximal, complemented
6#22549886C212yesmaximal, complemented
7#407×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×Dic254—28no—
10C22×Dic254—21no—
11C22×Dic254—22no—
12C22×Dic254C2214yes—
13C22×Dic254C2212yes—
14C23×Dic127—21no—
15#16861242—28no—
16C2×Dic508C2214yescomplemented
17C2×#14523237C2214yescomplemented
18#16861192—21no—
19#16861188C2212yes—
20#16861183C2212yes—
21#16861178—21no—
22C4×C22⋊D127—22no—
23#16861168C2211yescharacteristic
24#16861134C2218yescomplemented
25C23×C4×C127C2211yescharacteristic
26C2×M16×C127C2212yes—
27C22⋊C8×C127C2214yes—
28#14523248—24no—
29C2×Dic254—28no—
30C2×Dic254—216no—
31C2×Dic254—216no—
32C2×Dic254—24no—
33C2×Dic254—24no—
34C2×Dic254—22no—
35C2×Dic254D418yescomplemented
36C2×Dic254C2311yescharacteristic
37C2×Dic254C2314yes—
38C2×D508C2311yescharacteristic
39C2×C22⋊D127—21no—
40C2×C22⋊D127—22no—
41C22×Dic127—28no—
42C22×Dic127—22no—
43C22×Dic127—24no—
44C2×C4×D127—21no—
45Dic508—28no—
46#14523237—28no—
47C4⋊Dic127C2312yes—
48Dic127⋊C4—22no—
49C22⋊Dic127—22no—
50C22⋊Dic127—21no—
51(C2×D127)⋊C4—22no—
52C4×Dic127—21no—
53C24×C127D411yescharacteristic
54C22×C4×C127—24no—
55C22×C4×C127D411yescharacteristic
56C22×C4×C127D412yes—
57C22×C4×C127C2312yes—
58C22×C4×C127C2311yescharacteristic
59M16×C127—24no—
60C2×C8×C127C2314yes—
61C22×D127—21no—
62Dic254—216no—
63Dic254—28no—
64Dic254—24no—
65Dic254—22no—
66Dic254C2×D418yes—
67D508—22no—
68C22⋊D127—42no—
69C22⋊D127—44no—
70C2×Dic127—216no—
71C2×Dic127—28no—
72C2×Dic127—24no—
73C2×Dic127—22no—
74C2×Dic127—21no—
75C4×D127—42no—
76C23×C127—22no—
77C23×C127—24no—
78C23×C127C2×D411yescharacteristic
79C23×C127C2×D412yes—
80C2×C4×C127—22no—
81C2×C4×C127—24no—
82C2×C4×C127—24no—
83C2×C4×C127C2×D414yes—
84C2×C4×C127C2×D412yes—
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2411yescharacteristic
87C8×C127—24no—
88C22×C127—28no—
89C22×C127—22no—
90C22×C127—22no—
91C22×C127—22no—
92C22×C127#12012yes—
93C22×C127#13714yescomplemented
94C22×C127C22×D411yescharacteristic
95C2×D127—41no—
96C2×D127—42no—
97C4×C127—22no—
98C4×C127#12012yes—
99C4×C127C22×D412yes—
100Dic127—28no—
101Dic127—41no—
102Dic127—42no—
103C2×C127—22no—
104C2×C127—24no—
105C2×C127C2×#12011yescharacteristic
106C2×C127C2×#13712yes—
107D127—81no—
108#3004—1271nomaximal, Sylow
109C127#300411yescharacteristic, complemented, Sylow
110C23×Q8—1271no—
111C2×#137—1272no—
112#525—1271no—
113#451—1274no—
114#448—1274no—
115#419—1272no—
116#407D12711yescharacteristic, complemented
117C2×#41—1271no—
118C22×Q8—1272no—
119C22×Q8—1274no—
120C22×Q8—2548no—
121C22×Q8—2542no—
122C22×Q8—2541no—
123C23×C4—2541no—
124C23×C4C2×D12711yescharacteristic
125#137—2548no—
126C2×Q16—1274no—
127C2×SD16—1274no—
128C2×M16C2×D12712yes—
129#123—2541no—
130C22⋊Q8—1272no—
131#121—1272no—
132#120—2541no—
133C4×D4—2542no—
134#117—1271no—
135#103—1278no—
136C22⋊C8C2×D12714yes—
137C24D50811yescharacteristic
138#41—2544no—
139C2×Q8—1278no—
140C2×Q8—1274no—
141C2×Q8—1271no—
142C2×Q8—25416no—
143C2×Q8—2548no—
144C2×Q8—25416no—
145C2×Q8—2544no—
146C2×Q8—2544no—
147C2×Q8—2542no—
148C2×D4—1271no—
149C2×D4—2542no—
150C2×D4—2541no—
151C22×C4—24no—
152C22×C4—2541no—
153C22×C4—2548no—
154C22×C4—2544no—
155C22×C4—2542no—
156C22×C4D50812yes—
157C22×C4D50811yescharacteristic
158C22×C4C22×D12712yes—
159C22×C4C22×D12711yescharacteristic
160Q16—2548no—
161SD16—2548no—
162M16—24no—
163C2×C8C22×D12714yes—
164C4⋊C4—1272no—
165C4⋊C4—2542no—
166C22⋊C4—2541no—
167C22⋊C4—2542no—
168C22⋊C4—2542no—
169C42—2541no—
170C23—24no—
171C23—22no—
172C23—2541no—
173C23C2×D50812yes—
174C23C2×D50811yescharacteristic
175Q8—1278no—
176Q8—25416no—
177Q8—2542no—
178Q8—2544no—
179Q8—2548no—
180D4—5084no—
181D4—5082no—
182D4—2542no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—5082no—
187C2×C4—2542no—
188C2×C4—2541no—
189C2×C4—25416no—
190C2×C4—2544no—
191C2×C4—2548no—
192C2×C4D4×D12714yes—
193C2×C4C2×D50812yes—
194C2×C4C2×D50811yescharacteristic
195C2×C4C23×D12711yescharacteristic
196C8—24no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—5081no—
202C22—5082no—
203C22#1686117912yes—
204C22#1686124214yescomplemented
205C22C22×D50811yescharacteristic
206C4—22no—
207C4—5081no—
208C4—5082no—
209C4—2548no—
210C4#1686117912yes—
211C4C2×D4×D12712yes—
212C2—24no—
213C2—22no—
214C2—10161no—
215C2C2×#1686117911yescharacteristic
216C2C2×#1686124212yes—
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#407×C127C2
socleC22×C127C22×D4
prime radical#22550387C2
hypercenter#407D127
hypocenterC127#3004
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C23×Dic254C2semidirect
2. C2×#16861242C2semidirect
4. #22550034C2semidirect
5. #22550017C2semidirect
6. #22549886C2semidirect
7. #407×C127C2semidirect
16. C2×Dic508C22semidirect
17. C2×#14523237C22semidirect
24. #16861134C22semidirect
35. C2×Dic254D4semidirect
93. C22×C127#137semidirect
109. C127#3004semidirect
116. #407D127semidirect
204. C22#16861242semidirect

Maximal subgroups

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

subgroupindex
C23×Dic2542
C2×#168612422
#225503872
#225500342
#225500172
#225498862
#407×C1272
#3004127