Finite Group Databasebeta

← C22×C4×C22⋊H(3,𝔽3) #13164841

Subgroups — C22×C4×C22⋊H(3,𝔽3) #13164841

Order 1728.

1848 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 1848 conjugacy classes of subgroups — too many to draw. Only its 270 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×C22⋊H(3,𝔽3)C211yesmaximal, characteristic
2C2×C4×C22⋊H(3,𝔽3)C216yesmaximal, direct product factor
3C24×C32×C4C311yesmaximal, characteristic, complemented
4C22×C3×C4×A4C313yesmaximal, complemented
5C22×C22⋊H(3,𝔽3)C414yesdirect product factor
6C22×C22⋊H(3,𝔽3)C2213yes—
7C22×C4×H(3,𝔽3)—41nomaximal
8C4×C22⋊H(3,𝔽3)C2214yesdirect product factor
9C25×C32C2×C311yescharacteristic
10C23×C3×A4C2×C313yes—
11C23×C32×C4—31no—
12C23×C32×C4—36no—
13C23×C32×C4—31no—
14C23×C32×C4C2×C316yescomplemented
15C2×C3×C4×A4C2×C3118yescomplemented
16C23×H(3,𝔽3)—41no—
17C2×C22⋊H(3,𝔽3)C2×C416yesdirect product factor
18C2×C22⋊H(3,𝔽3)C2311yescharacteristic
19C2×C4×H(3,𝔽3)—46no—
20C24×C3×C4—31no—
21C24×C3×C4C3211yescharacteristic
22C22×C4×A4—33no—
23C24×C32—31no—
24C24×C32—33no—
25C24×C32—34no—
26C24×C32C3×C414yescomplemented
27C24×C32C22×C313yes—
28C22×C3×A4C3×C4112yescomplemented
29C22×C3×A4C22×C319yes—
30C22×C32×C4—36no—
31C22×C32×C4—38no—
32C22×C32×C4—31no—
33C22×C32×C4—36no—
34C22×C32×C4—36no—
35C22×C32×C4—36no—
36C22×C32×C4—312no—
37C22×C32×C4—43no—
38C22×C32×C4A411yescharacteristic, complemented
39C22×C32×C4C22×C314yescomplemented
40C3×C4×A4C22×C3112yescomplemented
41C22×H(3,𝔽3)—44no—
42C22×H(3,𝔽3)—43no—
43C22⋊H(3,𝔽3)C22×C411yescharacteristic, direct product factor
44C4×H(3,𝔽3)—44no—
45C25×C3—31no—
46C25×C3C2×C3211yescharacteristic
47C23×A4—33no—
48C23×C3×C4—33no—
49C23×C3×C4—33no—
50C23×C3×C4—318no—
51C23×C3×C4—31no—
52C23×C3×C4—31no—
53C23×C3×C4—36no—
54C23×C3×C4—36no—
55C23×C3×C4C2×C3216yes—
56C2×C4×A4—318no—
57C23×C32—312no—
58C23×C32—36no—
59C23×C32—33no—
60C23×C32—312no—
61C23×C32—32no—
62C23×C32—34no—
63C23×C32—34no—
64C23×C32—33no—
65C23×C32—33no—
66C23×C32—43no—
67C23×C32C2×C3×C416yescomplemented
68C23×C32C2×A411yescharacteristic
69C23×C32C23×C311yescharacteristic
70C2×C3×A4C2×C3×C4118yescomplemented
71C2×C3×A4C23×C313yes—
72C2×C32×C4—312no—
73C2×C32×C4—36no—
74C2×C32×C4—312no—
75C2×C32×C4—34no—
76C2×C32×C4—34no—
77C2×C32×C4—418no—
78C2×C32×C4C2×A416yescomplemented
79C24×C4H(3,𝔽3)11yescharacteristic, complemented, Sylow
80C2×H(3,𝔽3)—46no—
81C2×H(3,𝔽3)—41no—
82C24×C3—34no—
83C24×C3—33no—
84C24×C3—312no—
85C24×C3—39no—
86C24×C3—33no—
87C24×C3—31no—
88C24×C3—34no—
89C24×C3—33no—
90C24×C3C32×C414yes—
91C24×C3C22×C3213yes—
92C22×A4—312no—
93C22×A4—39no—
94C22×C3×C4—318no—
95C22×C3×C4—324no—
96C22×C3×C4—36no—
97C22×C3×C4—38no—
98C22×C3×C4—33no—
99C22×C3×C4—318no—
100C22×C3×C4—318no—
101C22×C3×C4—318no—
102C22×C3×C4—336no—
103C22×C3×C4—31no—
104C22×C3×C4—36no—
105C22×C3×C4—36no—
106C22×C3×C4—36no—
107C22×C3×C4—312no—
108C22×C3×C4—31no—
109C22×C3×C4—34no—
110C22×C3×C4—123no—
111C22×C3×C4C3×A411yescharacteristic
112C22×C3×C4C22×C3214yes—
113C4×A4—312no—
114C22×C32—36no—
115C22×C32—31no—
116C22×C32—38no—
117C22×C32—36no—
118C22×C32—312no—
119C22×C32—36no—
120C22×C32—36no—
121C22×C32—33no—
122C22×C32—31no—
123C22×C32—49no—
124C22×C32—412no—
125C22×C32C4×A414yescomplemented
126C22×C32C22×C3×C411yescharacteristic, complemented
127C22×C32C22×A413yes—
128C3×A4C22×C3×C413yescomplemented
129C32×C4—34no—
130C32×C4—412no—
131C32×C4C22×A414yescomplemented
132C25C2×H(3,𝔽3)11yescharacteristic
133C23×C4—31no—
134C23×C4—31no—
135C23×C4—36no—
136C23×C4C2×H(3,𝔽3)16yescomplemented
137H(3,𝔽3)—41noSylow
138C23×C3—33no—
139C23×C3—312no—
140C23×C3—36no—
141C23×C3—39no—
142C23×C3—36no—
143C23×C3—32no—
144C23×C3—39no—
145C23×C3—39no—
146C23×C3—336no—
147C23×C3—318no—
148C23×C3—336no—
149C23×C3—312no—
150C23×C3—312no—
151C23×C3—312no—
152C23×C3—33no—
153C23×C3—33no—
154C23×C3—34no—
155C23×C3—34no—
156C23×C3—36no—
157C23×C3—31no—
158C23×C3—31no—
159C23×C3—123no—
160C23×C3C2×C32×C416yes—
161C23×C3C2×C3×A411yescharacteristic
162C23×C3C23×C3211yescharacteristic
163C2×A4—318no—
164C2×A4—33no—
165C2×C3×C4—336no—
166C2×C3×C4—312no—
167C2×C3×C4—318no—
168C2×C3×C4—312no—
169C2×C3×C4—336no—
170C2×C3×C4—312no—
171C2×C3×C4—36no—
172C2×C3×C4—312no—
173C2×C3×C4—34no—
174C2×C3×C4—34no—
175C2×C3×C4—36no—
176C2×C3×C4—1218no—
177C2×C3×C4C2×C3×A416yes—
178C2×C32—36no—
179C2×C32—31no—
180C2×C32—31no—
181C2×C32—418no—
182C2×C32—43no—
183C2×C32C2×C4×A416yescomplemented
184C2×C32C23×A411yescharacteristic
185C24—33no—
186C24—34no—
187C24—31no—
188C24C4×H(3,𝔽3)14yescomplemented
189C24C22×H(3,𝔽3)13yes—
190C22×C4—36no—
191C22×C4—38no—
192C22×C4—31no—
193C22×C4—36no—
194C22×C4—36no—
195C22×C4—36no—
196C22×C4—312no—
197C22×C4C22⋊H(3,𝔽3)11yescharacteristic, direct product factor
198C22×C4C22×H(3,𝔽3)14yescomplemented
199C22×C3—36no—
200C22×C3—31no—
201C22×C3—33no—
202C22×C3—39no—
203C22×C3—318no—
204C22×C3—33no—
205C22×C3—318no—
206C22×C3—324no—
207C22×C3—36no—
208C22×C3—38no—
209C22×C3—318no—
210C22×C3—336no—
211C22×C3—318no—
212C22×C3—36no—
213C22×C3—312no—
214C22×C3—36no—
215C22×C3—31no—
216C22×C3—33no—
217C22×C3—33no—
218C22×C3—34no—
219C22×C3—31no—
220C22×C3—129no—
221C22×C3—1212no—
222C22×C3C3×C4×A414yes—
223C22×C3C22×C32×C411yescharacteristic
224C22×C3C22×C3×A413yes—
225A4—33no—
226C3×C4—312no—
227C3×C4—34no—
228C3×C4—34no—
229C3×C4—1212no—
230C3×C4C22×C3×A414yes—
231C32—43no—
232C32C22×C4×A411yescharacteristic, complemented
233C23—33no—
234C23—312no—
235C23—36no—
236C23—32no—
237C23—312no—
238C23—33no—
239C23—33no—
240C23—34no—
241C23—34no—
242C23C2×C4×H(3,𝔽3)16yescomplemented
243C23C2×C22⋊H(3,𝔽3)11yescharacteristic
244C23C23×H(3,𝔽3)11yescharacteristic
245C2×C4—312no—
246C2×C4—36no—
247C2×C4—34no—
248C2×C4—312no—
249C2×C4—34no—
250C2×C4C2×C22⋊H(3,𝔽3)16yesdirect product factor
251C2×C3—33no—
252C2×C3—318no—
253C2×C3—33no—
254C2×C3—31no—
255C2×C3—36no—
256C2×C3—31no—
257C2×C3—31no—
258C2×C3—36no—
259C2×C3—123no—
260C2×C3—1218no—
261C2×C3C2×C3×C4×A416yes—
262C2×C3C23×C3×A411yescharacteristic
263C22—36no—
264C22—31no—
265C22—36no—
266C22—38no—
267C22—31no—
268C22—33no—
269C22—36no—
270C22—312no—
271C22—36no—
272C22C4×C22⋊H(3,𝔽3)14yesdirect product factor
273C22C22×C4×H(3,𝔽3)11yescharacteristic, complemented
274C22C22×C22⋊H(3,𝔽3)13yes—
275C4—34no—
276C4C22×C22⋊H(3,𝔽3)14yesdirect product factor
277C3—31no—
278C3—123no—
279C3C22×C3×C4×A411yescharacteristic
280C2—31no—
281C2—36no—
282C2—31no—
283C2C2×C4×C22⋊H(3,𝔽3)16yesdirect product factor
284C2C23×C22⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC22×C3×C4C3×A4
commutatorC22×C3C22×C32×C4
FrattiniC2×C3C23×C3×A4
FittingC24×C32×C4C3
socleC25×C3C2×C32
prime radicalC23×C22⋊H(3,𝔽3)C2
cosocleC23×C3C23×C32
hypercenterC22×C32×C4A4
hypocenterC22C22×C4×H(3,𝔽3)
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C22×C3×C4 ◁ C22×C32×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊H(3,𝔽3)C2direct (N × H)
3. C24×C32×C4C3semidirect
4. C22×C3×C4×A4C3semidirect
5. C22×C22⋊H(3,𝔽3)C4direct (N × H)
8. C4×C22⋊H(3,𝔽3)C22direct (N × H)
14. C23×C32×C4C2×C3semidirect
15. C2×C3×C4×A4C2×C3semidirect
17. C2×C22⋊H(3,𝔽3)C2×C4direct (N × H)
26. C24×C32C3×C4semidirect
28. C22×C3×A4C3×C4semidirect
38. C22×C32×C4A4semidirect
39. C22×C32×C4C22×C3semidirect
40. C3×C4×A4C22×C3semidirect
43. C22⋊H(3,𝔽3)C22×C4direct (N × H)
67. C23×C32C2×C3×C4semidirect
70. C2×C3×A4C2×C3×C4semidirect
78. C2×C32×C4C2×A4semidirect
79. C24×C4H(3,𝔽3)semidirect
125. C22×C32C4×A4semidirect
126. C22×C32C22×C3×C4semidirect
128. C3×A4C22×C3×C4semidirect
131. C32×C4C22×A4semidirect
136. C23×C4C2×H(3,𝔽3)semidirect
183. C2×C32C2×C4×A4semidirect
188. C24C4×H(3,𝔽3)semidirect
197. C22×C4C22⋊H(3,𝔽3)direct (N × H)
198. C22×C4C22×H(3,𝔽3)semidirect
232. C32C22×C4×A4semidirect
242. C23C2×C4×H(3,𝔽3)semidirect
250. C2×C4C2×C22⋊H(3,𝔽3)direct (N × H)
272. C22C4×C22⋊H(3,𝔽3)direct (N × H)
273. C22C22×C4×H(3,𝔽3)semidirect
276. C4C22×C22⋊H(3,𝔽3)direct (N × H)
283. C2C2×C4×C22⋊H(3,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊H(3,𝔽3)2
C2×C4×C22⋊H(3,𝔽3)2
C24×C32×C43
C22×C3×C4×A43
C22×C4×H(3,𝔽3)4