Finite Group Databasebeta

← C23×(C3⋊D3)⋊A4 #13180095

Subgroups — C23×(C3⋊D3)⋊A4 #13180095

Order 1728.

6384 conjugacy classes of subgroups; only the 281 normal subgroups are shown.

normal subgroups only

This group has 6384 conjugacy classes of subgroups — too many to draw. Only its 281 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, complemented
2C22×(C3⋊D3)⋊A4C2114yesmaximal, direct product factor
3C25×C3⋊D3C311yesmaximal, characteristic, complemented
4C23×D3×A4—31nomaximal
5C23×(C3⋊D3)⋊C3—41nomaximal
6C22×C22⋊H(3,𝔽3)C2217yescomplemented
7C2×(C3⋊D3)⋊A4C22128yesdirect product factor
8C25×C32C2×C311yescharacteristic, complemented
9C24×C3⋊D3—31no—
10C24×C3⋊D3—31no—
11C24×C3⋊D3—314no—
12C24×C3⋊D3C2×C3114yescomplemented
13C23×C3×A4—21no—
14C23×C3×A4D311yescharacteristic, complemented
15C22×D3×A4—314no—
16C23×H(3,𝔽3)—41no—
17C22×(C3⋊D3)⋊C3—414no—
18C2×C22⋊H(3,𝔽3)C2317yescomplemented
19(C3⋊D3)⋊A4C2318yesdirect product factor
20C25×D3—31no—
21C25×D3—91no—
22C24×A4—91no—
23C24×C32—31no—
24C24×C32—37no—
25C24×C32C22×C317yescomplemented
26C23×C3⋊D3—314no—
27C23×C3⋊D3—356no—
28C23×C3⋊D3—31no—
29C23×C3⋊D3—314no—
30C23×C3⋊D3—314no—
31C23×C3⋊D3—314no—
32C23×C3⋊D3—384no—
33C23×C3⋊D3A411yescharacteristic, complemented
34C23×C3⋊D3C22×C3128yescomplemented
35C23×C3×D3—121no—
36C22×C3×A4—27no—
37C22×C3×A4C2×D317yescomplemented
38C2×D3×A4—328no—
39C22×H(3,𝔽3)—47no—
40C2×(C3⋊D3)⋊C3—428no—
41C22⋊H(3,𝔽3)C2411yescharacteristic, complemented
42C25×C3—31no—
43C25×C3C3×D311yescharacteristic
44C24×D3—314no—
45C24×D3—914no—
46C24×D3—91no—
47C24×D3—93no—
48C24×D3—942no—
49C24×D3—93no—
50C24×D3—914no—
51C24×D3—91no—
52C23×A4—31no—
53C23×A4—61no—
54C23×A4—914no—
55C23×C32—321no—
56C23×C32—314no—
57C23×C32—37no—
58C23×C32—37no—
59C23×C32—41no—
60C23×C32—81no—
61C23×C32C2×A411yescharacteristic, complemented
62C23×C32C23×C317yescomplemented
63C22×C3⋊D3—384no—
64C22×C3⋊D3—3112no—
65C22×C3⋊D3—314no—
66C22×C3⋊D3—384no—
67C22×C3⋊D3—328no—
68C22×C3⋊D3—328no—
69C22×C3⋊D3—356no—
70C22×C3⋊D3C2×A4114yescomplemented
71C22×C3⋊D3C23×C318yescomplemented
72C22×C3×D3—1214no—
73C2×C3×A4—27no—
74C2×C3×A4C22×D317yescomplemented
75D3×A4—38no—
76C26—91noSylow
77C2×H(3,𝔽3)—47no—
78(C3⋊D3)⋊C3—48no—
79C24×C3—37no—
80C24×C3—321no—
81C24×C3—33no—
82C24×C3—31no—
83C24×C3—37no—
84C24×C3—361no—
85C24×C3C2×C3×D317yes—
86C23×D3—31no—
87C23×D3—328no—
88C23×D3—91no—
89C23×D3—928no—
90C23×D3—914no—
91C23×D3—914no—
92C23×D3—942no—
93C23×D3—942no—
94C23×D3—9252no—
95C23×D3—942no—
96C23×D3—956no—
97C23×D3—9168no—
98C23×D3—942no—
99C23×D3—914no—
100C23×D3—93no—
101C23×D3—91no—
102C23×D3—984no—
103C23×D3—914no—
104C22×A4—37no—
105C22×A4—67no—
106C22×A4—928no—
107C22×C32—37no—
108C22×C32—314no—
109C22×C32—321no—
110C22×C32—37no—
111C22×C32—47no—
112C22×C32—87no—
113C22×C32C22×A417yescomplemented
114C22×C32C24×C311yescharacteristic, complemented
115C2×C3⋊D3—356no—
116C2×C3⋊D3—328no—
117C2×C3⋊D3—356no—
118C2×C3⋊D3—38no—
119C2×C3⋊D3—38no—
120C2×C3⋊D3C22×A4128yescomplemented
121C2×C3×D3—1228no—
122C3×A4—21no—
123C3×A4C23×D311yescharacteristic, complemented
124C25—914no—
125C25—271no—
126C25—2714no—
127C25—271no—
128C25(C3⋊D3)⋊C311yescharacteristic, complemented
129H(3,𝔽3)—41noSylow
130C23×C3—321no—
131C23×C3—363no—
132C23×C3—342no—
133C23×C3—314no—
134C23×C3—321no—
135C23×C3—321no—
136C23×C3—37no—
137C23×C3—37no—
138C23×C3—31no—
139C23×C3—37no—
140C23×C3—121no—
141C23×C3—241no—
142C23×C3—3614no—
143C23×C3D3×A411yescharacteristic
144C23×C3C22×C3×D317yes—
145C22×D3—314no—
146C22×D3—38no—
147C22×D3—914no—
148C22×D3—98no—
149C22×D3—928no—
150C22×D3—984no—
151C22×D3—984no—
152C22×D3—9252no—
153C22×D3—9168no—
154C22×D3—984no—
155C22×D3—9112no—
156C22×D3—9336no—
157C22×D3—9252no—
158C22×D3—984no—
159C22×D3—942no—
160C22×D3—914no—
161C22×D3—956no—
162C22×D3—928no—
163C2×A4—37no—
164C2×A4—67no—
165C2×A4—98no—
166C2×C32—37no—
167C2×C32—31no—
168C2×C32—47no—
169C2×C32—87no—
170C2×C32C23×A417yescomplemented
171C3⋊D3—38no—
172C3⋊D3C23×A418yescomplemented
173C3×D3—128no—
174C24—37no—
175C24—31no—
176C24—91no—
177C24—928no—
178C24—2714no—
179C24—2714no—
180C24—2784no—
181C24—2714no—
182C24—2756no—
183C24—2714no—
184C24—271no—
185C24C2×(C3⋊D3)⋊C317yescomplemented
186C22×C3—37no—
187C22×C3—321no—
188C22×C3—342no—
189C22×C3—314no—
190C22×C3—321no—
191C22×C3—363no—
192C22×C3—37no—
193C22×C3—321no—
194C22×C3—37no—
195C22×C3—31no—
196C22×C3—127no—
197C22×C3—247no—
198C22×C3—3628no—
199C22×C3C2×D3×A417yes—
200C22×C3C23×C3×D311yescharacteristic
201C2×D3—328no—
202C2×D3—928no—
203C2×D3—98no—
204C2×D3—956no—
205C2×D3—924no—
206C2×D3—9168no—
207C2×D3—924no—
208C2×D3—9168no—
209C2×D3—956no—
210C2×D3—984no—
211C2×D3—928no—
212C2×D3—98no—
213A4—31no—
214A4—61no—
215C32—41no—
216C32—81no—
217C32C24×A411yescharacteristic, complemented
218C23—321no—
219C23—314no—
220C23—37no—
221C23—37no—
222C23—914no—
223C23—98no—
224C23—2728no—
225C23—2784no—
226C23—2756no—
227C23—2728no—
228C23—27112no—
229C23—2784no—
230C23—2714no—
231C23(C3⋊D3)⋊A411yescharacteristic, direct product factor
232C23C22×(C3⋊D3)⋊C317yescomplemented
233C2×C3—33no—
234C2×C3—321no—
235C2×C3—31no—
236C2×C3—37no—
237C2×C3—37no—
238C2×C3—127no—
239C2×C3—247no—
240C2×C3—368no—
241C2×C3C22×D3×A417yes—
242D3—38no—
243D3—98no—
244D3—924no—
245D3—98no—
246C22—37no—
247C22—314no—
248C22—37no—
249C22—321no—
250C22—928no—
251C22—278no—
252C22—2756no—
253C22—278no—
254C22—2756no—
255C22—2728no—
256C22C2×(C3⋊D3)⋊A417yesdirect product factor
257C22C23×(C3⋊D3)⋊C311yescharacteristic, complemented
258C3—31no—
259C3—121no—
260C3—241no—
261C3C23×D3×A411yescharacteristic
262C2—31no—
263C2—37no—
264C2—98no—
265C2—278no—
266C2C22×(C3⋊D3)⋊A417yesdirect product factor
subgroupG/H
center, hypercenterC23(C3⋊D3)⋊A4
commutator, cosocle, hypocenterC22×C32C24×C3
FrattiniC3C23×D3×A4
FittingC25×C32C2×C3
socleC25×C3C3×D3
LCSG ▷ C22×C32
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊H(3,𝔽3)C2semidirect
2. C22×(C3⋊D3)⋊A4C2direct (N × H)
3. C25×C3⋊D3C3semidirect
6. C22×C22⋊H(3,𝔽3)C22semidirect
7. C2×(C3⋊D3)⋊A4C22direct (N × H)
8. C25×C32C2×C3semidirect
12. C24×C3⋊D3C2×C3semidirect
14. C23×C3×A4D3semidirect
18. C2×C22⋊H(3,𝔽3)C23semidirect
19. (C3⋊D3)⋊A4C23direct (N × H)
25. C24×C32C22×C3semidirect
33. C23×C3⋊D3A4semidirect
34. C23×C3⋊D3C22×C3semidirect
37. C22×C3×A4C2×D3semidirect
41. C22⋊H(3,𝔽3)C24semidirect
61. C23×C32C2×A4semidirect
62. C23×C32C23×C3semidirect
70. C22×C3⋊D3C2×A4semidirect
71. C22×C3⋊D3C23×C3semidirect
74. C2×C3×A4C22×D3semidirect
113. C22×C32C22×A4semidirect
114. C22×C32C24×C3semidirect
120. C2×C3⋊D3C22×A4semidirect
123. C3×A4C23×D3semidirect
128. C25(C3⋊D3)⋊C3semidirect
170. C2×C32C23×A4semidirect
172. C3⋊D3C23×A4semidirect
185. C24C2×(C3⋊D3)⋊C3semidirect
217. C32C24×A4semidirect
231. C23(C3⋊D3)⋊A4direct (N × H)
232. C23C22×(C3⋊D3)⋊C3semidirect
256. C22C2×(C3⋊D3)⋊A4direct (N × H)
257. C22C23×(C3⋊D3)⋊C3semidirect
266. C2C22×(C3⋊D3)⋊A4direct (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
C22×(C3⋊D3)⋊A42
C25×C3⋊D33
C23×D3×A43
C23×(C3⋊D3)⋊C34