Finite Group Databasebeta

← C2×C3⋊D3×C22⋊A4 #13181862

Subgroups — C2×C3⋊D3×C22⋊A4 #13181862

Order 1728.

6112 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 6112 conjugacy classes of subgroups — too many to draw. Only its 120 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×C22⋊A4C211yesmaximal, characteristic, complemented
2C3⋊D3×C22⋊A4C212yesmaximal, direct product factor
3C25×C3⋊D3C311yesmaximal, characteristic, complemented
4C2×D3×C22⋊A4—34nomaximal
5C32×C22⋊A4C2211yescharacteristic, complemented
6C2×A4×C3⋊D3—45nomaximal
7C25×C32C2×C311yescharacteristic, complemented
8C24×C3⋊D3—35no—
9C24×C3⋊D3—35no—
10C24×C3⋊D3—310no—
11C24×C3⋊D3C2×C312yescomplemented
12C2×C3×C22⋊A4—24no—
13C2×C3×C22⋊A4D314yescomplemented
14D3×C22⋊A4—38no—
15C2×C32×A4—45no—
16A4×C3⋊D3—410no—
17C25×D3—34no—
18C22×C22⋊A4—91no—
19C24×C32—35no—
20C24×C32—35no—
21C24×C32C22×C311yescharacteristic, complemented
22C23×C3⋊D3—360no—
23C23×C3⋊D3—360no—
24C23×C3⋊D3—330no—
25C23×C3⋊D3—310no—
26C23×C3⋊D3—310no—
27C23×C3⋊D3—35no—
28C23×C3⋊D3—310no—
29C23×C3⋊D3—310no—
30C23×C3⋊D3—310no—
31C23×C3⋊D3A415yescomplemented
32C3×C22⋊A4—24no—
33C3×C22⋊A4C2×D314yescomplemented
34C2×D3×A4—1220no—
35C2×C3×C3⋊D3—161no—
36C32×A4—45no—
37C25×C3C3×D314yescomplemented
38C24×D3—38no—
39C24×D3—920no—
40C24×D3—940no—
41C24×D3—920no—
42C2×C22⋊A4—24no—
43C2×C22⋊A4—92no—
44C2×C22⋊A4C3⋊D311yescharacteristic, direct product factor
45C23×C32—330no—
46C23×C32—310no—
47C23×C32—35no—
48C23×C32—35no—
49C23×C32C2×A415yescomplemented
50C22×C3⋊D3—330no—
51C22×C3⋊D3—3120no—
52C22×C3⋊D3—360no—
53C22×C3⋊D3—360no—
54C22×C3⋊D3—330no—
55C22×C3⋊D3—360no—
56C22×C3⋊D3—320no—
57C22×C3⋊D3—310no—
58C22×C3⋊D3—35no—
59C22×C3⋊D3—35no—
60C22×C3⋊D3—310no—
61C22×C3⋊D3C2×A4110yescomplemented
62C2×C3×A4—420no—
63C2×C3×A4—820no—
64D3×A4—1240no—
65C26—91noSylow
66C2×C33—161no—
67C3×C3⋊D3—162no—
68C24×C3—320no—
69C24×C3—320no—
70C24×C3C2×C3×D314yescomplemented
71C23×D3—320no—
72C23×D3—940no—
73C23×D3—940no—
74C23×D3—940no—
75C23×D3—9240no—
76C23×D3—9240no—
77C23×D3—9120no—
78C23×D3—940no—
79C23×D3—920no—
80C23×D3—940no—
81C22⋊A4—24no—
82C22⋊A4C2×C3⋊D311yescharacteristic, direct product factor
83C22×A4—365no—
84C22×C32—35no—
85C22×C32—330no—
86C22×C32—310no—
87C22×C32—35no—
88C22×C32C22×A415yescomplemented
89C2×C3⋊D3—360no—
90C2×C3⋊D3—310no—
91C2×C3⋊D3—360no—
92C2×C3⋊D3—35no—
93C2×C3⋊D3—310no—
94C2×C3⋊D3—310no—
95C2×C3⋊D3—310no—
96C2×C3⋊D3C22⋊A411yescharacteristic, direct product factor
97C2×C3×D3—484no—
98C3×A4—420no—
99C3×A4—820no—
100C25—92no—
101C25—275no—
102C25—2710no—
103C25—275no—
104C25C3×C3⋊D311yescharacteristic, complemented
105C33—161noSylow
106C23×C3—3120no—
107C23×C3—340no—
108C23×C3—320no—
109C23×C3—320no—
110C23×C3D3×A4120yescomplemented
111C22×D3—340no—
112C22×D3—920no—
113C22×D3—980no—
114C22×D3—9240no—
115C22×D3—9120no—
116C22×D3—9480no—
117C22×D3—9240no—
118C22×D3—9240no—
119C22×D3—9120no—
120C22×D3—940no—
121C22×D3—940no—
122C22×D3—920no—
123C2×A4—45no—
124C2×A4—820no—
125C2×A4—3610no—
126C2×C32—35no—
127C2×C32—35no—
128C2×C32—164no—
129C2×C32—324no—
130C2×C32C2×C22⋊A411yescharacteristic, complemented
131C3⋊D3—310no—
132C3⋊D3C2×C22⋊A412yesdirect product factor
133C3×D3—488no—
134C24—35no—
135C24—35no—
136C24—95no—
137C24—2710no—
138C24—2710no—
139C24—2710no—
140C24—2710no—
141C24—2760no—
142C24—2730no—
143C24—2760no—
144C24—2710no—
145C24—275no—
146C24C2×C3×C3⋊D311yescharacteristic, complemented
147C22×C3—320no—
148C22×C3—340no—
149C22×C3—3120no—
150C22×C3—320no—
151C22×C3—1441no—
152C22×C3C2×D3×A4120yescomplemented
153C2×D3—34no—
154C2×D3—940no—
155C2×D3—940no—
156C2×D3—9240no—
157C2×D3—9240no—
158C2×D3—940no—
159C2×D3—920no—
160C2×D3—940no—
161A4—45no—
162A4—820no—
163C32—164no—
164C32—324no—
165C32C22×C22⋊A411yescharacteristic, complemented
166C23—330no—
167C23—310no—
168C23—35no—
169C23—35no—
170C23—910no—
171C23—275no—
172C23—2710no—
173C23—275no—
174C23—2730no—
175C23—2720no—
176C23—2760no—
177C23—2760no—
178C23—2760no—
179C23—27120no—
180C23—2730no—
181C23—2710no—
182C23A4×C3⋊D315yescomplemented
183C2×C3—320no—
184C2×C3—320no—
185C2×C3—161no—
186C2×C3—324no—
187C2×C3—1442no—
188C2×C3D3×C22⋊A414yescomplemented
189D3—38no—
190D3—940no—
191C22—35no—
192C22—310no—
193C22—330no—
194C22—35no—
195C22—91no—
196C22—2710no—
197C22—2710no—
198C22—2710no—
199C22—2760no—
200C22—2710no—
201C22—2760no—
202C22—275no—
203C22C2×A4×C3⋊D315yescomplemented
204C3—161no—
205C3—324no—
206C3C2×D3×C22⋊A414yescomplemented
207C2—35no—
208C2—35no—
209C2—92no—
210C2—2710no—
211C2C3⋊D3×C22⋊A411yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2C3⋊D3×C22⋊A4
commutator, cosocle, hypocenterC24×C32C22×C3
FrattiniC1G
Fitting, socleC25×C32C2×C3
LCSG ▷ C24×C32
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C32×C22⋊A4C2semidirect
2. C3⋊D3×C22⋊A4C2direct (N × H)
3. C25×C3⋊D3C3semidirect
5. C32×C22⋊A4C22semidirect
7. C25×C32C2×C3semidirect
11. C24×C3⋊D3C2×C3semidirect
13. C2×C3×C22⋊A4D3semidirect
21. C24×C32C22×C3semidirect
31. C23×C3⋊D3A4semidirect
33. C3×C22⋊A4C2×D3semidirect
37. C25×C3C3×D3semidirect
44. C2×C22⋊A4C3⋊D3direct (N × H)
49. C23×C32C2×A4semidirect
61. C22×C3⋊D3C2×A4semidirect
70. C24×C3C2×C3×D3semidirect
82. C22⋊A4C2×C3⋊D3direct (N × H)
88. C22×C32C22×A4semidirect
96. C2×C3⋊D3C22⋊A4direct (N × H)
104. C25C3×C3⋊D3semidirect
110. C23×C3D3×A4semidirect
130. C2×C32C2×C22⋊A4semidirect
132. C3⋊D3C2×C22⋊A4direct (N × H)
146. C24C2×C3×C3⋊D3semidirect
152. C22×C3C2×D3×A4semidirect
165. C32C22×C22⋊A4semidirect
182. C23A4×C3⋊D3semidirect
188. C2×C3D3×C22⋊A4semidirect
203. C22C2×A4×C3⋊D3semidirect
206. C3C2×D3×C22⋊A4semidirect
211. C2C3⋊D3×C22⋊A4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C22⋊A42
C3⋊D3×C22⋊A42
C25×C3⋊D33
C2×D3×C22⋊A43
C2×A4×C3⋊D34