Finite Group Databasebeta

← C22×C32×C4×A4 #13181696

Subgroups — C22×C32×C4×A4 #13181696

Order 1728.

2766 conjugacy classes of subgroups; only the 918 normal subgroups are shown.

normal subgroups only

This group has 2766 conjugacy classes of subgroups — too many to draw. Only its 918 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×C32×A4C211yesmaximal, characteristic
2C2×C32×C4×A4C216yesmaximal, direct product factor
3C24×C32×C4C311yesmaximal, characteristic, complemented
4C22×C3×C4×A4C3112yesmaximal, direct product factor
5C22×C32×A4C414yesdirect product factor
6C22×C32×A4C2213yes—
7C22×C33×C4—41nomaximal
8C32×C4×A4C2214yesdirect product factor
9C25×C32C2×C311yescharacteristic
10C23×C3×A4C2×C3112yes—
11C23×C32×C4—31no—
12C23×C32×C4—36no—
13C23×C32×C4—31no—
14C23×C32×C4C2×C316yescomplemented
15C2×C3×C4×A4C2×C3172yesdirect product factor
16C23×C33—41no—
17C2×C32×A4C2×C416yesdirect product factor
18C2×C32×A4C2311yescharacteristic
19C2×C33×C4—46no—
20C24×C3×C4C3214yescomplemented
21C22×C4×A4C3219yesdirect product factor
22C24×C32—31no—
23C24×C32—33no—
24C24×C32—34no—
25C24×C32C3×C414yescomplemented
26C24×C32C22×C313yes—
27C22×C3×A4C3×C4148yesdirect product factor
28C22×C3×A4C22×C3136yes—
29C22×C32×C4—36no—
30C22×C32×C4—38no—
31C22×C32×C4—31no—
32C22×C32×C4—36no—
33C22×C32×C4—36no—
34C22×C32×C4—36no—
35C22×C32×C4—312no—
36C22×C32×C4—412no—
37C22×C32×C4A411yescharacteristic, direct product factor
38C22×C32×C4C22×C314yescomplemented
39C3×C4×A4C22×C3148yesdirect product factor
40C22×C33—44no—
41C22×C33—43no—
42C32×A4C22×C411yescharacteristic, direct product factor
43C33×C4—44no—
44C25×C3C2×C3214yes—
45C23×A4C2×C3219yes—
46C23×C3×C4—34no—
47C23×C3×C4—34no—
48C23×C3×C4—324no—
49C23×C3×C4C2×C32124yescomplemented
50C2×C4×A4C2×C32154yesdirect product factor
51C23×C32—312no—
52C23×C32—36no—
53C23×C32—33no—
54C23×C32—312no—
55C23×C32—32no—
56C23×C32—34no—
57C23×C32—34no—
58C23×C32—33no—
59C23×C32—33no—
60C23×C32—412no—
61C23×C32C2×C3×C416yescomplemented
62C23×C32C2×A411yescharacteristic
63C23×C32C23×C311yescharacteristic
64C2×C3×A4C2×C3×C4172yesdirect product factor
65C2×C3×A4C23×C3112yes—
66C2×C32×C4—312no—
67C2×C32×C4—36no—
68C2×C32×C4—312no—
69C2×C32×C4—34no—
70C2×C32×C4—34no—
71C2×C32×C4—472no—
72C2×C32×C4C2×A416yesdirect product factor
73C24×C4C3311yescharacteristic, complemented, Sylow
74C2×C33—46no—
75C2×C33—41no—
76C24×C3—316no—
77C24×C3—312no—
78C24×C3—34no—
79C24×C3C32×C4116yescomplemented
80C24×C3C22×C32112yes—
81C22×A4C32×C4136yesdirect product factor
82C22×A4C22×C32127yes—
83C22×C3×C4—324no—
84C22×C3×C4—332no—
85C22×C3×C4—34no—
86C22×C3×C4—324no—
87C22×C3×C4—324no—
88C22×C3×C4—324no—
89C22×C3×C4—348no—
90C22×C3×C4—49no—
91C22×C3×C4C3×A414yesdirect product factor
92C22×C3×C4C22×C32116yescomplemented
93C4×A4C22×C32136yesdirect product factor
94C22×C32—36no—
95C22×C32—31no—
96C22×C32—38no—
97C22×C32—36no—
98C22×C32—312no—
99C22×C32—36no—
100C22×C32—36no—
101C22×C32—33no—
102C22×C32—31no—
103C22×C32—436no—
104C22×C32—448no—
105C22×C32C4×A414yesdirect product factor
106C22×C32C22×C3×C411yescharacteristic, complemented
107C22×C32C22×A413yes—
108C3×A4C22×C3×C4112yesdirect product factor
109C32×C4—34no—
110C32×C4—448no—
111C32×C4C22×A414yesdirect product factor
112C25C2×C3311yescharacteristic
113C23×C4—31no—
114C23×C4—31no—
115C23×C4—36no—
116C23×C4C2×C3316yescomplemented
117C33—41noSylow
118C23×C3—312no—
119C23×C3—348no—
120C23×C3—324no—
121C23×C3—38no—
122C23×C3—348no—
123C23×C3—312no—
124C23×C3—312no—
125C23×C3—316no—
126C23×C3—316no—
127C23×C3—49no—
128C23×C3C2×C32×C4124yescomplemented
129C23×C3C2×C3×A414yes—
130C23×C3C23×C3214yes—
131C2×A4C2×C32×C4154yesdirect product factor
132C2×A4C23×C3219yes—
133C2×C3×C4—348no—
134C2×C3×C4—324no—
135C2×C3×C4—348no—
136C2×C3×C4—316no—
137C2×C3×C4—316no—
138C2×C3×C4—454no—
139C2×C3×C4C2×C3×A4124yesdirect product factor
140C2×C32—36no—
141C2×C32—31no—
142C2×C32—31no—
143C2×C32—472no—
144C2×C32—412no—
145C2×C32C2×C4×A416yesdirect product factor
146C2×C32C23×A411yescharacteristic
147C24—33no—
148C24—34no—
149C24—31no—
150C24C33×C414yescomplemented
151C24C22×C3313yes—
152C22×C4—36no—
153C22×C4—38no—
154C22×C4—31no—
155C22×C4—36no—
156C22×C4—36no—
157C22×C4—36no—
158C22×C4—312no—
159C22×C4C32×A411yescharacteristic, direct product factor
160C22×C4C22×C3314yescomplemented
161C22×C3—324no—
162C22×C3—34no—
163C22×C3—324no—
164C22×C3—332no—
165C22×C3—324no—
166C22×C3—348no—
167C22×C3—324no—
168C22×C3—34no—
169C22×C3—312no—
170C22×C3—427no—
171C22×C3—436no—
172C22×C3C3×C4×A4116yesdirect product factor
173C22×C3C22×C32×C414yescomplemented
174C22×C3C22×C3×A4112yes—
175A4C22×C32×C419yesdirect product factor
176C3×C4—316no—
177C3×C4—436no—
178C3×C4C22×C3×A4116yesdirect product factor
179C32—412no—
180C32C22×C4×A411yescharacteristic, direct product factor
181C23—33no—
182C23—312no—
183C23—36no—
184C23—32no—
185C23—312no—
186C23—33no—
187C23—33no—
188C23—34no—
189C23—34no—
190C23C2×C33×C416yescomplemented
191C23C2×C32×A411yescharacteristic
192C23C23×C3311yescharacteristic
193C2×C4—312no—
194C2×C4—36no—
195C2×C4—34no—
196C2×C4—312no—
197C2×C4—34no—
198C2×C4C2×C32×A416yesdirect product factor
199C2×C3—34no—
200C2×C3—324no—
201C2×C3—34no—
202C2×C3—454no—
203C2×C3—49no—
204C2×C3C2×C3×C4×A4124yesdirect product factor
205C2×C3C23×C3×A414yes—
206C22—36no—
207C22—31no—
208C22—36no—
209C22—38no—
210C22—36no—
211C22—312no—
212C22—36no—
213C22—31no—
214C22—33no—
215C22C32×C4×A414yesdirect product factor
216C22C22×C33×C411yescharacteristic, complemented
217C22C22×C32×A413yes—
218C4—34no—
219C4C22×C32×A414yesdirect product factor
220C3—49no—
221C3C22×C3×C4×A414yesdirect product factor
222C2—31no—
223C2—36no—
224C2—31no—
225C2C2×C32×C4×A416yesdirect product factor
226C2C23×C32×A411yescharacteristic
subgroupG/H
center, hypercenterC22×C32×C4A4
commutator, hypocenterC22C22×C33×C4
FrattiniC2C23×C32×A4
FittingC24×C32×C4C3
socleC25×C32C2×C3
prime radicalC23×C32×A4C2
cosocleC23C23×C33
LCSG ▷ C22
UCSC1 ◁ C22×C32×C4

Complements

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

NH ≅ G/Ntype
2. C2×C32×C4×A4C2direct (N × H)
3. C24×C32×C4C3semidirect
4. C22×C3×C4×A4C3direct (N × H)
5. C22×C32×A4C4direct (N × H)
8. C32×C4×A4C22direct (N × H)
14. C23×C32×C4C2×C3semidirect
15. C2×C3×C4×A4C2×C3direct (N × H)
17. C2×C32×A4C2×C4direct (N × H)
20. C24×C3×C4C32semidirect
21. C22×C4×A4C32direct (N × H)
25. C24×C32C3×C4semidirect
27. C22×C3×A4C3×C4direct (N × H)
37. C22×C32×C4A4direct (N × H)
38. C22×C32×C4C22×C3semidirect
39. C3×C4×A4C22×C3direct (N × H)
42. C32×A4C22×C4direct (N × H)
49. C23×C3×C4C2×C32semidirect
50. C2×C4×A4C2×C32direct (N × H)
61. C23×C32C2×C3×C4semidirect
64. C2×C3×A4C2×C3×C4direct (N × H)
72. C2×C32×C4C2×A4direct (N × H)
73. C24×C4C33semidirect
79. C24×C3C32×C4semidirect
81. C22×A4C32×C4direct (N × H)
91. C22×C3×C4C3×A4direct (N × H)
92. C22×C3×C4C22×C32semidirect
93. C4×A4C22×C32direct (N × H)
105. C22×C32C4×A4direct (N × H)
106. C22×C32C22×C3×C4semidirect
108. C3×A4C22×C3×C4direct (N × H)
111. C32×C4C22×A4direct (N × H)
116. C23×C4C2×C33semidirect
128. C23×C3C2×C32×C4semidirect
131. C2×A4C2×C32×C4direct (N × H)
139. C2×C3×C4C2×C3×A4direct (N × H)
145. C2×C32C2×C4×A4direct (N × H)
150. C24C33×C4semidirect
159. C22×C4C32×A4direct (N × H)
160. C22×C4C22×C33semidirect
172. C22×C3C3×C4×A4direct (N × H)
173. C22×C3C22×C32×C4semidirect
175. A4C22×C32×C4direct (N × H)
178. C3×C4C22×C3×A4direct (N × H)
180. C32C22×C4×A4direct (N × H)
190. C23C2×C33×C4semidirect
198. C2×C4C2×C32×A4direct (N × H)
204. C2×C3C2×C3×C4×A4direct (N × H)
215. C22C32×C4×A4direct (N × H)
216. C22C22×C33×C4semidirect
219. C4C22×C32×A4direct (N × H)
221. C3C22×C3×C4×A4direct (N × H)
225. C2C2×C32×C4×A4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×A42
C2×C32×C4×A42
C24×C32×C43
C22×C3×C4×A43
C22×C33×C44