Finite Group Databasebeta

← C32×A4×C22⋊A4 #72252822

Subgroups — C32×A4×C22⋊A4 #72252822

Order 5184.

6584 conjugacy classes of subgroups; only the 444 normal subgroups are shown.

normal subgroups only

This group has 6584 conjugacy classes of subgroups — too many to draw. Only its 444 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
1C24×C32×A4C311yesmaximal, characteristic, complemented
2C22×C32×C22⋊A4C311yesmaximal, characteristic, complemented
3C32×C22⋊(C22⋊A4)C312yesmaximal, complemented
4C3×A4×C22⋊A4C3136yesmaximal, direct product factor
5C33×C22⋊A4—41nomaximal
6C32×A42—45nomaximal
7C2×C32×C22⋊A4—31no—
8C23×C32×A4—35no—
9C26×C32C3211yescharacteristic, complemented
10C3×C22⋊(C22⋊A4)C32124yescomplemented
11C22×C3×C22⋊A4C32112yescomplemented
12C24×C3×A4C32112yescomplemented
13A4×C22⋊A4C32181yesdirect product factor
14C24×C33—41no—
15C32×C22⋊A4—42no—
16C32×C22⋊A4—410no—
17C32×C22⋊A4—436no—
18C32×C22⋊A4—1210no—
19C32×C22⋊A4A411yescharacteristic, direct product factor
20C22×C32×A4—310no—
21C22×C32×A4—45no—
22C22×C32×A4A415yescomplemented
23C3×A42—4180no—
24C33×A4—165no—
25C33×A4—161no—
26C25×C32—35no—
27C25×C32—31no—
28C25×C32—95no—
29C2×C3×C22⋊A4—312no—
30C23×C3×A4—360no—
31C23×C33—125no—
32C2×C32×A4—35no—
33C2×C32×A4—125no—
34C26×C3C3314yescomplemented
35C22⋊(C22⋊A4)C33118yescomplemented
36C22×C22⋊A4C3319yescomplemented
37C24×A4C3319yescomplemented
38C24×C32—310no—
39C24×C32—310no—
40C24×C32—412no—
41C24×C32—95no—
42C24×C32—95no—
43C24×C32—95no—
44C24×C32—920no—
45C24×C32—930no—
46C24×C32C3×A411yescharacteristic, complemented
47C24×C32C3×A415yescomplemented
48C3×C22⋊A4—424no—
49C3×C22⋊A4—481no—
50C3×C22⋊A4—4120no—
51C3×C22⋊A4—12120no—
52C3×C22⋊A4C3×A4112yesdirect product factor
53C22×C3×A4—3120no—
54C22×C3×A4—460no—
55C22×C3×A4C3×A4160yescomplemented
56A42—4405no—
57C22×C33—45no—
58C22×C33—1210no—
59C22×C33—161no—
60C32×A4—45no—
61C32×A4—16180no—
62C32×A4—1610no—
63C32×A4—162no—
64C32×A4—1636no—
65C32×A4—4810no—
66C32×A4C22⋊A411yescharacteristic, direct product factor
67C25×C3—320no—
68C25×C3—34no—
69C25×C3—920no—
70C2×C22⋊A4—39no—
71C23×A4—345no—
72C34—641noSylow
73C23×C32—35no—
74C23×C32—35no—
75C23×C32—35no—
76C23×C32—95no—
77C23×C32—910no—
78C23×C32—940no—
79C23×C32—930no—
80C23×C32—930no—
81C23×C32—930no—
82C23×C32—95no—
83C23×C32—1260no—
84C2×C3×A4—360no—
85C2×C3×A4—1260no—
86C26C3411yescharacteristic, complemented, Sylow
87C2×C33—125no—
88C2×C33—481no—
89C24×C3—340no—
90C24×C3—340no—
91C24×C3—49no—
92C24×C3—920no—
93C24×C3—920no—
94C24×C3—920no—
95C24×C3—9120no—
96C24×C3—980no—
97C24×C3C32×A414yescomplemented
98C24×C3C32×A4120yescomplemented
99C22⋊A4—418no—
100C22⋊A4—490no—
101C22⋊A4—1290no—
102C22⋊A4C32×A419yesdirect product factor
103C22×A4—390no—
104C22×A4—445no—
105C22×A4C32×A4145yescomplemented
106C22×C32—310no—
107C22×C32—310no—
108C22×C32—460no—
109C22×C32—95no—
110C22×C32—95no—
111C22×C32—95no—
112C22×C32—930no—
113C22×C32—920no—
114C22×C32—12120no—
115C22×C32—1612no—
116C22×C32A4215yescomplemented
117C22×C32C3×C22⋊A411yescharacteristic, complemented
118C3×A4—460no—
119C3×A4—16405no—
120C3×A4—16120no—
121C3×A4—1624no—
122C3×A4—1681no—
123C3×A4—48120no—
124C3×A4C3×C22⋊A4112yesdirect product factor
125C25—35no—
126C25—31no—
127C25—95no—
128C33—41no—
129C33—161no—
130C33—6436no—
131C33—642no—
132C23×C3—320no—
133C23×C3—320no—
134C23×C3—320no—
135C23×C3—920no—
136C23×C3—940no—
137C23×C3—9160no—
138C23×C3—9120no—
139C23×C3—9120no—
140C23×C3—9120no—
141C23×C3—920no—
142C23×C3—1245no—
143C2×A4—345no—
144C2×A4—1245no—
145C2×C32—35no—
146C2×C32—31no—
147C2×C32—95no—
148C2×C32—1260no—
149C2×C32—4812no—
150C24—310no—
151C24—310no—
152C24—95no—
153C24—95no—
154C24—95no—
155C24—930no—
156C24—920no—
157C24C33×A411yescharacteristic, complemented
158C24C33×A415yescomplemented
159C22×C3—340no—
160C22×C3—340no—
161C22×C3—445no—
162C22×C3—920no—
163C22×C3—920no—
164C22×C3—920no—
165C22×C3—9120no—
166C22×C3—980no—
167C22×C3—1290no—
168C22×C3—169no—
169C22×C3C3×A42120yescomplemented
170C22×C3C32×C22⋊A414yescomplemented
171A4—445no—
172A4—1618no—
173A4—1690no—
174A4—4890no—
175A4C32×C22⋊A419yesdirect product factor
176C32—412no—
177C32—1612no—
178C32—6481no—
179C32—6424no—
180C32A4×C22⋊A411yescharacteristic, direct product factor
181C23—35no—
182C23—35no—
183C23—35no—
184C23—95no—
185C23—910no—
186C23—940no—
187C23—930no—
188C23—930no—
189C23—930no—
190C23—95no—
191C2×C3—320no—
192C2×C3—34no—
193C2×C3—920no—
194C2×C3—1245no—
195C2×C3—489no—
196C22—310no—
197C22—310no—
198C22—95no—
199C22—95no—
200C22—95no—
201C22—930no—
202C22—920no—
203C22C32×A4215yescomplemented
204C22C33×C22⋊A411yescharacteristic, complemented
205C3—49no—
206C3—169no—
207C3—6418no—
208C3C3×A4×C22⋊A414yesdirect product factor
209C2—35no—
210C2—31no—
211C2—95no—
subgroupG/H
center, hypercenterC32A4×C22⋊A4
commutator, cosocle, hypocenterC26C34
FrattiniC1G
Fitting, socleC26×C32C32
LCSG ▷ C26
UCSC1 ◁ C32

Complements

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

NH ≅ G/Ntype
1. C24×C32×A4C3semidirect
2. C22×C32×C22⋊A4C3semidirect
3. C32×C22⋊(C22⋊A4)C3semidirect
4. C3×A4×C22⋊A4C3direct (N × H)
9. C26×C32C32semidirect
10. C3×C22⋊(C22⋊A4)C32semidirect
11. C22×C3×C22⋊A4C32semidirect
12. C24×C3×A4C32semidirect
13. A4×C22⋊A4C32direct (N × H)
19. C32×C22⋊A4A4direct (N × H)
22. C22×C32×A4A4semidirect
34. C26×C3C33semidirect
35. C22⋊(C22⋊A4)C33semidirect
36. C22×C22⋊A4C33semidirect
37. C24×A4C33semidirect
46. C24×C32C3×A4semidirect
47. C24×C32C3×A4semidirect
52. C3×C22⋊A4C3×A4direct (N × H)
55. C22×C3×A4C3×A4semidirect
66. C32×A4C22⋊A4direct (N × H)
86. C26C34semidirect
97. C24×C3C32×A4semidirect
98. C24×C3C32×A4semidirect
102. C22⋊A4C32×A4direct (N × H)
105. C22×A4C32×A4semidirect
116. C22×C32A42semidirect
117. C22×C32C3×C22⋊A4semidirect
124. C3×A4C3×C22⋊A4direct (N × H)
157. C24C33×A4semidirect
158. C24C33×A4semidirect
169. C22×C3C3×A42semidirect
170. C22×C3C32×C22⋊A4semidirect
175. A4C32×C22⋊A4direct (N × H)
180. C32A4×C22⋊A4direct (N × H)
203. C22C32×A42semidirect
204. C22C33×C22⋊A4semidirect
208. C3C3×A4×C22⋊A4direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C32×A43
C22×C32×C22⋊A43
C32×C22⋊(C22⋊A4)3
C3×A4×C22⋊A43
C33×C22⋊A44
C32×A424