Finite Group Databasebeta

← C22×(C3×A4)⋊A4 #13180060

Subgroups — C22×(C3×A4)⋊A4 #13180060

Order 1728.

1302 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 1302 conjugacy classes of subgroups — too many to draw. Only its 75 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×(C3×A4)⋊A4C213yesmaximal, direct product factor
2C22×C3×C22⋊A4C312yesmaximal, complemented
3C24×C3×A4C312yesmaximal, complemented
4C22×C22⋊H(3,𝔽3)—42nomaximal
5(C3×A4)⋊A4C2211yescharacteristic, direct product factor
6C2×C3×C22⋊A4C2×C316yescomplemented
7C23×C3×A4—32no—
8C23×C3×A4—36no—
9C23×C3×A4C2×C316yescomplemented
10C2×C22⋊H(3,𝔽3)—46no—
11C26×C3C3211yescharacteristic
12C22×C22⋊A4—32no—
13C24×A4—32no—
14C24×C32—42no—
15C3×C22⋊A4C22×C312yescomplemented
16C22×C3×A4—34no—
17C22×C3×A4—36no—
18C22×C3×A4—36no—
19C22×C3×A4—36no—
20C22×C3×A4—44no—
21C22×C3×A4—122no—
22C22×C3×A4A412yescomplemented
23C22×C3×A4C22×C312yescomplemented
24C22×H(3,𝔽3)—161no—
25C22⋊H(3,𝔽3)—42no—
26C25×C3—36no—
27C25×C3—32no—
28C25×C3—91no—
29C25×C3—93no—
30C25×C3C2×C3213yes—
31C2×C22⋊A4—36no—
32C23×A4—318no—
33C23×A4—36no—
34C23×A4—36no—
35C23×C32—46no—
36C23×C32—122no—
37C23×C32—126no—
38C2×C3×A4—36no—
39C2×C3×A4—32no—
40C2×C3×A4—412no—
41C2×C3×A4—126no—
42C2×C3×A4C2×A416yescomplemented
43C26H(3,𝔽3)11yescharacteristic, complemented, Sylow
44C2×H(3,𝔽3)—163no—
45C24×C3—34no—
46C24×C3—32no—
47C24×C3—36no—
48C24×C3—36no—
49C24×C3—36no—
50C24×C3—93no—
51C24×C3—91no—
52C24×C3—96no—
53C24×C3—96no—
54C24×C3—92no—
55C24×C3—96no—
56C24×C3—93no—
57C24×C3—93no—
58C24×C3—96no—
59C24×C3—912no—
60C24×C3—912no—
61C24×C3—94no—
62C24×C3—122no—
63C24×C3C3×A412yes—
64C24×C3C22×C3211yescharacteristic
65C22⋊A4—32no—
66C22×A4—312no—
67C22×A4—318no—
68C22×A4—318no—
69C22×A4—318no—
70C22×A4—32no—
71C22×A4—32no—
72C22×A4—126no—
73C22×A4—124no—
74C22×C32—42no—
75C22×C32—42no—
76C22×C32—126no—
77C22×C32—126no—
78C22×C32—126no—
79C22×C32—124no—
80C22×C32—162no—
81C3×A4—44no—
82C3×A4—122no—
83C3×A4C22×A412yescomplemented
84C25—36no—
85C25—32no—
86C25—91no—
87C25—93no—
88C25C2×H(3,𝔽3)13yescomplemented
89H(3,𝔽3)—161noSylow
90C23×C3—36no—
91C23×C3—32no—
92C23×C3—36no—
93C23×C3—36no—
94C23×C3—32no—
95C23×C3—91no—
96C23×C3—93no—
97C23×C3—96no—
98C23×C3—93no—
99C23×C3—96no—
100C23×C3—96no—
101C23×C3—912no—
102C23×C3—96no—
103C23×C3—96no—
104C23×C3—94no—
105C23×C3—912no—
106C23×C3—912no—
107C23×C3—96no—
108C23×C3—93no—
109C23×C3—912no—
110C23×C3—912no—
111C23×C3—96no—
112C23×C3—96no—
113C23×C3—924no—
114C23×C3—91no—
115C23×C3—1218no—
116C23×C3—126no—
117C23×C3—126no—
118C23×C3C2×C3×A416yes—
119C2×A4—36no—
120C2×A4—318no—
121C2×A4—36no—
122C2×A4—1218no—
123C2×A4—1212no—
124C2×C32—46no—
125C2×C32—122no—
126C2×C32—126no—
127C2×C32—166no—
128C24—34no—
129C24—32no—
130C24—36no—
131C24—36no—
132C24—36no—
133C24—93no—
134C24—91no—
135C24—96no—
136C24—96no—
137C24—92no—
138C24—96no—
139C24—93no—
140C24—93no—
141C24—96no—
142C24—912no—
143C24—912no—
144C24—94no—
145C24C22⋊H(3,𝔽3)12yescomplemented
146C24C22×H(3,𝔽3)11yescharacteristic, complemented
147C22×C3—36no—
148C22×C3—34no—
149C22×C3—36no—
150C22×C3—36no—
151C22×C3—32no—
152C22×C3—91no—
153C22×C3—96no—
154C22×C3—93no—
155C22×C3—96no—
156C22×C3—92no—
157C22×C3—96no—
158C22×C3—912no—
159C22×C3—912no—
160C22×C3—96no—
161C22×C3—94no—
162C22×C3—93no—
163C22×C3—93no—
164C22×C3—1218no—
165C22×C3—1212no—
166C22×C3—1218no—
167C22×C3—1218no—
168C22×C3—122no—
169C22×C3—122no—
170C22×C3—482no—
171C22×C3A4211yescharacteristic
172C22×C3C22×C3×A412yes—
173A4—32no—
174A4—126no—
175A4—124no—
176C32—42no—
177C32—162no—
178C23—36no—
179C23—32no—
180C23—36no—
181C23—32no—
182C23—36no—
183C23—91no—
184C23—93no—
185C23—96no—
186C23—93no—
187C23—96no—
188C23—96no—
189C23—912no—
190C23—96no—
191C23—96no—
192C23—94no—
193C23—912no—
194C23—912no—
195C23—96no—
196C23—93no—
197C23—912no—
198C23—912no—
199C23—96no—
200C23—96no—
201C23—924no—
202C23—91no—
203C23C2×C22⋊H(3,𝔽3)16yescomplemented
204C2×C3—32no—
205C2×C3—36no—
206C2×C3—91no—
207C2×C3—93no—
208C2×C3—126no—
209C2×C3—1218no—
210C2×C3—126no—
211C2×C3—486no—
212C2×C3C2×A4213yes—
213C22—36no—
214C22—34no—
215C22—36no—
216C22—36no—
217C22—32no—
218C22—91no—
219C22—96no—
220C22—93no—
221C22—96no—
222C22—92no—
223C22—96no—
224C22—912no—
225C22—912no—
226C22—96no—
227C22—94no—
228C22—93no—
229C22—93no—
230C22(C3×A4)⋊A411yescharacteristic, direct product factor
231C22C22×C22⋊H(3,𝔽3)12yescomplemented
232C3—122no—
233C3—482no—
234C3C22×A4211yescharacteristic
235C2—32no—
236C2—36no—
237C2—91no—
238C2—93no—
239C2C2×(C3×A4)⋊A413yesdirect product factor
subgroupG/H
center, hypercenterC22×C3A42
commutator, cosocleC24×C3C22×C32
FrattiniC3C22×A42
Fitting, socleC26×C3C32
hypocenterC24C22×H(3,𝔽3)
LCSG ▷ C24×C3 ▷ C24
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×(C3×A4)⋊A4C2direct (N × H)
2. C22×C3×C22⋊A4C3semidirect
3. C24×C3×A4C3semidirect
5. (C3×A4)⋊A4C22direct (N × H)
6. C2×C3×C22⋊A4C2×C3semidirect
9. C23×C3×A4C2×C3semidirect
15. C3×C22⋊A4C22×C3semidirect
22. C22×C3×A4A4semidirect
23. C22×C3×A4C22×C3semidirect
42. C2×C3×A4C2×A4semidirect
43. C26H(3,𝔽3)semidirect
83. C3×A4C22×A4semidirect
88. C25C2×H(3,𝔽3)semidirect
145. C24C22⋊H(3,𝔽3)semidirect
146. C24C22×H(3,𝔽3)semidirect
203. C23C2×C22⋊H(3,𝔽3)semidirect
230. C22(C3×A4)⋊A4direct (N × H)
231. C22C22×C22⋊H(3,𝔽3)semidirect
239. C2C2×(C3×A4)⋊A4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×(C3×A4)⋊A42
C22×C3×C22⋊A43
C24×C3×A43
C22×C22⋊H(3,𝔽3)4