Finite Group Databasebeta

← C2×#22550027 #30647912

Subgroups — C2×#22550027 #30647912

Order 16256.

996 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 197 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
1C22×#14523236C212yesmaximal, complemented
2C2×#16861188C211yesmaximal, characteristic, complemented
3#22550027C218yesmaximal, direct product factor
4C2×#16861130C212yesmaximal, complemented
5C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6C23×D4×C127C211yesmaximal, characteristic
7C22×Dic254C2211yescharacteristic, complemented
8C2×#14523236—28no—
9C2×#14523236C2218yescomplemented
10C22×C127⋊C8C2212yescomplemented
11#16861188—22no—
12#16861188C2214yescomplemented
13C2×C4⋊Dic127C2211yescharacteristic, complemented
14C2×Dic127⋊C4—21no—
15C2×C22⋊Dic127—21no—
16#16861130C2218yescomplemented
17C22⋊(C127⋊C8)C2214yescomplemented
18C25×C127—21no—
19C22×D4×C127—28no—
20C22×D4×C127—22no—
21C22×D4×C127—21no—
22C22×D4×C127C2214yes—
23C22×D4×C127C2212yes—
24C23×C4×C127C2211yescharacteristic
25C2×Dic254—21no—
26C2×Dic254—24no—
27C2×Dic254C2312yescomplemented
28C22×Dic127—21no—
29C22×Dic127—21no—
30#14523236—216no—
31C2×C127⋊C8—24no—
32C2×C127⋊C8C2314yescomplemented
33C4⋊Dic127—21no—
34C4⋊Dic127C2312yescomplemented
35Dic127⋊C4—24no—
36C22⋊Dic127—24no—
37C24×C127—28no—
38C24×C127—24no—
39C24×C127—22no—
40C24×C127—44no—
41C24×C127—44no—
42C24×C127D411yescharacteristic
43C2×D4×C127—28no—
44C2×D4×C127—216no—
45C2×D4×C127—216no—
46C2×D4×C127—24no—
47C2×D4×C127—22no—
48C2×D4×C127—24no—
49C2×D4×C127D418yes—
50C2×D4×C127C2314yes—
51C22×C4×C127—24no—
52C22×C4×C127D412yes—
53C22×C4×C127D411yescharacteristic
54C22×C4×C127C2312yes—
55C22×C4×C127C2311yescharacteristic
56Dic254—24no—
57Dic254—42no—
58C2×Dic127—22no—
59C2×Dic127—22no—
60C2×Dic127—42no—
61C2×Dic127—42no—
62C127⋊C8—24no—
63C23×C127—22no—
64C23×C127—22no—
65C23×C127—216no—
66C23×C127—28no—
67C23×C127—24no—
68C23×C127—48no—
69C23×C127—432no—
70C23×C127—44no—
71C23×C127—48no—
72C23×C127—44no—
73C23×C127SD1614yescomplemented
74C23×C127C2×D412yes—
75C23×C127C2×D411yescharacteristic
76D4×C127—216no—
77D4×C127—28no—
78D4×C127—24no—
79D4×C127C2×D418yes—
80C2×C4×C127—22no—
81C2×C4×C127—24no—
82C2×C4×C127—24no—
83C2×C4×C127C2×D412yes—
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D414yes—
86C2×C4×C127C2411yescharacteristic
87C22×C127—22no—
88C22×C127—22no—
89C22×C127—28no—
90C22×C127—28no—
91C22×C127—432no—
92C22×C127—44no—
93C22×C127—44no—
94C22×C127—416no—
95C22×C127#12012yes—
96C22×C127C2×SD1612yes—
97C22×C127C2×SD1614yescomplemented
98C22×C127AGL(1,ℤ/8)12yes—
99C22×C127C22×D411yescharacteristic
100C4×C127—22no—
101C4×C127#12012yes—
102C4×C127C22×D412yes—
103Dic127—42no—
104Dic127—42no—
105C2×C127—24no—
106C2×C127—48no—
107C2×C127#45014yescomplemented
108C2×C127C2×#12011yescharacteristic
109C2×C127C22×SD1611yescharacteristic
110C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
111C2×#450—1271nomaximal, Sylow
112C127C2×#45011yescharacteristic, complemented, Sylow
113C23×D4D12711yescharacteristic
114C22×SD16—1272no—
115C2×C22⋊Q8—1271no—
116#450—1278no—
117C2×#102—1272no—
118C2×C22⋊C8—1271no—
119C25—21no—
120C22×Q8—1271no—
121C22×D4—28no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4C2×D12714yes—
125C22×D4C2×D12712yes—
126C23×C4C2×D12711yescharacteristic
127C2×SD16—1278no—
128C2×SD16—2548no—
129C22×C8—1272no—
130C22⋊Q8—1274no—
131C22⋊Q8—2542no—
132C2×C4⋊C4—1271no—
133C2×C4⋊C4—2541no—
134C2×C22⋊C4—2541no—
135#102—1278no—
136C22⋊C8—1274no—
137C24—28no—
138C24—24no—
139C24—22no—
140C24—44no—
141C24—44no—
142C24C22⋊D12711yescharacteristic
143C2×Q8—1272no—
144C2×Q8—2544no—
145C2×Q8—2541no—
146C2×D4—28no—
147C2×D4—216no—
148C2×D4—216no—
149C2×D4—24no—
150C2×D4—24no—
151C2×D4—22no—
152C2×D4C22⋊D12718yes—
153C2×D4C22×D12714yes—
154C22×C4—24no—
155C22×C4—2541no—
156C22×C4—2541no—
157C22×C4C22⋊D12712yes—
158C22×C4C22⋊D12711yescharacteristic
159C22×C4C22×D12712yes—
160C22×C4C22×D12711yescharacteristic
161SD16—25416no—
162C2×C8—1274no—
163C2×C8—2544no—
164C4⋊C4—1272no—
165C4⋊C4—2544no—
166C4⋊C4—2541no—
167C22⋊C4—2544no—
168C23—22no—
169C23—22no—
170C23—216no—
171C23—24no—
172C23—28no—
173C23—432no—
174C23—48no—
175C23—44no—
176C23—48no—
177C23—44no—
178C23#1452323614yescomplemented
179C23C2×C22⋊D12712yes—
180C23C2×C22⋊D12711yescharacteristic
181Q8—5082no—
182Q8—2544no—
183D4—216no—
184D4—24no—
185D4—28no—
186D4C2×C22⋊D12718yes—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—5082no—
191C2×C4—5082no—
192C2×C4—2542no—
193C2×C4—2542no—
194C2×C4C2×C22⋊D12712yes—
195C2×C4C2×C22⋊D12714yes—
196C2×C4C2×C22⋊D12711yescharacteristic
197C2×C4C23×D12711yescharacteristic
198C8—2544no—
199C22—28no—
200C22—22no—
201C22—22no—
202C22—28no—
203C22—432no—
204C22—416no—
205C22—44no—
206C22—44no—
207C22#1686117812yes—
208C22C2×#1452323612yes—
209C22C2×#1452323614yescomplemented
210C22#1686123512yes—
211C22C22×C22⋊D12711yescharacteristic
212C4—22no—
213C4—5082no—
214C4—5082no—
215C4#1686117812yes—
216C4C22×C22⋊D12712yes—
217C2—24no—
218C2—48no—
219C2#2255002714yesdirect product factor
220C2C2×#1686117811yescharacteristic
221C2C22×#1452323611yescharacteristic
222C2C2×#1686123511yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radicalC23×D4×C127C2
socleC23×C127C2×D4
hypercenterC23×D4D127
hypocenterC127C2×#450
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C22×#14523236C2semidirect
2. C2×#16861188C2semidirect
3. #22550027C2direct (N × H)
4. C2×#16861130C2semidirect
5. C2×C22⋊(C127⋊C8)C2semidirect
7. C22×Dic254C22semidirect
9. C2×#14523236C22semidirect
10. C22×C127⋊C8C22semidirect
12. #16861188C22semidirect
13. C2×C4⋊Dic127C22semidirect
16. #16861130C22semidirect
17. C22⋊(C127⋊C8)C22semidirect
27. C2×Dic254C23semidirect
32. C2×C127⋊C8C23semidirect
34. C4⋊Dic127C23semidirect
73. C23×C127SD16semidirect
97. C22×C127C2×SD16semidirect
107. C2×C127#450semidirect
112. C127C2×#450semidirect
178. C23#14523236semidirect
209. C22C2×#14523236semidirect
219. C2#22550027direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232362
C2×#168611882
#225500272
C2×#168611302
C2×C22⋊(C127⋊C8)2
C23×D4×C1272
C2×#450127