Finite Group Databasebeta

← C2×C32×#857 #13172939

Subgroups — C2×C32×#857 #13172939

Order 1728.

552 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 66 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×C32×C42⋊C3C211yesmaximal, characteristic, complemented
2C32×#857C212yesmaximal, direct product factor
3C2×C3×#857C314yesmaximal, direct product factor
4C2×C32×#104—31nomaximal
5C2×C32×S4—41nomaximal
6C32×C42⋊C3C2211yescharacteristic, complemented
7C2×C32×#41—31no—
8C2×C32×M16—31no—
9C2×C32×C42D311yescharacteristic, complemented
10C2×C3×C42⋊C3—24no—
11C2×C3×C42⋊C3C2×C314yescomplemented
12C3×#857C2×C318yesdirect product factor
13C32×#104—32no—
14C32×#104—32no—
15C2×C32×A4—41no—
16C32×S4—42no—
17C2×#857C3211yescharacteristic, direct product factor
18C2×C3×#104—34no—
19C2×C3×S4—44no—
20C32×#41—32no—
21C32×#41—32no—
22C32×#41—61no—
23C32×#41—61no—
24C2×C32×Q8—31no—
25C2×C32×D4—31no—
26C2×C32×D4—61no—
27C22×C32×C4—31no—
28C22×C32×C4—61no—
29C32×M16—32no—
30C32×M16—61no—
31C2×C32×C8—61no—
32C32×C42—31no—
33C32×C42C2×D311yescharacteristic, complemented
34C3×C42⋊C3—24no—
35C3×C42⋊C3C22×C314yescomplemented
36C2×C32×D3—161no—
37C32×A4—41no—
38C2×C3×#41—34no—
39C2×C3×M16—34no—
40C2×C3×C42C3×D314yescomplemented
41C2×C42⋊C3—24no—
42C2×C42⋊C3C2×C3211yescharacteristic, complemented
43#857C2×C3212yesdirect product factor
44C3×#104—38no—
45C3×#104—38no—
46C23×C32—61no—
47C23×C32S411yescharacteristic
48C2×C3×A4—44no—
49C2×C3×A4—84no—
50C3×S4—48no—
51C32×Q8—32no—
52C32×Q8—61no—
53C32×D4—32no—
54C32×D4—61no—
55C32×D4—62no—
56C32×D4—62no—
57C2×C32×C4—31no—
58C2×C32×C4—31no—
59C2×C32×C4—31no—
60C2×C32×C4—31no—
61C2×C32×C4—31no—
62C2×C32×C4—31no—
63C2×C32×C4—62no—
64C2×C32×C4—62no—
65C2×C32×C4—61no—
66C32×C8—62no—
67C2×#104—31noSylow
68C2×C33—161no—
69C32×D3—162no—
70C2×S4—41no—
71C3×#41—38no—
72C3×#41—38no—
73C3×#41—64no—
74C3×#41—64no—
75C2×C3×Q8—34no—
76C2×C3×D4—34no—
77C2×C3×D4—64no—
78C22×C3×C4—34no—
79C22×C3×C4—64no—
80C3×M16—38no—
81C3×M16—64no—
82C2×C3×C8—64no—
83C3×C42—34no—
84C3×C42C2×C3×D314yescomplemented
85C42⋊C3—24no—
86C42⋊C3C22×C3211yescharacteristic, complemented
87C22×C32—31no—
88C22×C32—31no—
89C22×C32—62no—
90C22×C32—121no—
91C22×C32—121no—
92C22×C32C2×S411yescharacteristic
93C2×C3×D3—164no—
94C3×A4—44no—
95C3×A4—84no—
96C32×C4—31no—
97C32×C4—31no—
98C32×C4—31no—
99C32×C4—31no—
100C32×C4—62no—
101C2×#41—31no—
102C2×M16—31no—
103C2×C42C32×D311yescharacteristic, complemented
104#104—32no—
105#104—32no—
106C33—161noSylow
107C23×C3—64no—
108C23×C3C3×S414yes—
109C2×A4—41no—
110C2×A4—84no—
111S4—42no—
112C3×Q8—38no—
113C3×Q8—64no—
114C3×D4—38no—
115C3×D4—64no—
116C3×D4—68no—
117C3×D4—68no—
118C2×C3×C4—34no—
119C2×C3×C4—34no—
120C2×C3×C4—34no—
121C2×C3×C4—34no—
122C2×C3×C4—34no—
123C2×C3×C4—34no—
124C2×C3×C4—68no—
125C2×C3×C4—68no—
126C2×C3×C4—64no—
127C3×C8—68no—
128C2×C32—31no—
129C2×C32—31no—
130C2×C32—122no—
131C2×C32—164no—
132C2×C32—324no—
133C2×C32#85711yescharacteristic, direct product factor
134C3×D3—168no—
135#41—32no—
136#41—32no—
137#41—61no—
138#41—61no—
139C2×Q8—31no—
140C2×D4—31no—
141C2×D4—61no—
142C22×C4—31no—
143C22×C4—61no—
144M16—32no—
145M16—61no—
146C2×C8—61no—
147C42—31no—
148C42C2×C32×D311yescharacteristic, complemented
149C22×C3—34no—
150C22×C3—34no—
151C22×C3—68no—
152C22×C3—124no—
153C22×C3—124no—
154C22×C3C2×C3×S414yes—
155C2×D3—161no—
156A4—41no—
157A4—84no—
158C3×C4—34no—
159C3×C4—34no—
160C3×C4—34no—
161C3×C4—34no—
162C3×C4—68no—
163C32—164no—
164C32—324no—
165C32C2×#85711yescharacteristic, direct product factor
166C23—61no—
167C23C32×S411yescharacteristic
168Q8—32no—
169Q8—61no—
170D4—32no—
171D4—61no—
172D4—62no—
173D4—62no—
174C2×C4—31no—
175C2×C4—31no—
176C2×C4—31no—
177C2×C4—31no—
178C2×C4—31no—
179C2×C4—31no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—61no—
183C8—62no—
184C2×C3—34no—
185C2×C3—34no—
186C2×C3—128no—
187C2×C3—161no—
188C2×C3—324no—
189C2×C3C3×#85714yesdirect product factor
190D3—162no—
191C22—31no—
192C22—31no—
193C22—62no—
194C22—121no—
195C22—121no—
196C22C2×C32×S411yescharacteristic
197C4—31no—
198C4—31no—
199C4—31no—
200C4—31no—
201C4—62no—
202C3—161no—
203C3—324no—
204C3C2×C3×#85714yesdirect product factor
205C2—31no—
206C2—31no—
207C2—122no—
208C2C32×#85711yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C32#857
commutator, cosocle, hypocenterC42⋊C3C22×C32
FrattiniC22C2×C32×S4
FittingC2×C32×C42D3
socleC23×C32S4
LCSG ▷ C42⋊C3
UCSC1 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
1. C2×C32×C42⋊C3C2semidirect
2. C32×#857C2direct (N × H)
3. C2×C3×#857C3direct (N × H)
6. C32×C42⋊C3C22semidirect
9. C2×C32×C42D3semidirect
11. C2×C3×C42⋊C3C2×C3semidirect
12. C3×#857C2×C3direct (N × H)
17. C2×#857C32direct (N × H)
33. C32×C42C2×D3semidirect
35. C3×C42⋊C3C22×C3semidirect
40. C2×C3×C42C3×D3semidirect
42. C2×C42⋊C3C2×C32semidirect
43. #857C2×C32direct (N × H)
84. C3×C42C2×C3×D3semidirect
86. C42⋊C3C22×C32semidirect
103. C2×C42C32×D3semidirect
133. C2×C32#857direct (N × H)
148. C42C2×C32×D3semidirect
165. C32C2×#857direct (N × H)
189. C2×C3C3×#857direct (N × H)
204. C3C2×C3×#857direct (N × H)
208. C2C32×#857direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C42⋊C32
C32×#8572
C2×C3×#8573
C2×C32×#1043
C2×C32×S44