Finite Group Databasebeta

← C3×C4×#3712 #13157350

Subgroups — C3×C4×#3712 #13157350

Order 1728.

794 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 240 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
1C3×C42×C3⋊D3C211yesmaximal, characteristic
2C2×C3×#3712C212yesmaximal
3C32×C4×C3⋊C8C212yesmaximal, complemented
4C3×#63555C212yesmaximal, complemented
5C3×C4×ΓL(1,𝔽25)—32nomaximal
6C4×#3712C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C412yes—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C33×C42C2211yescharacteristic
10C3×C4×C3⋊Dic3C412yes—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C32×C3⋊C8C2214yes—
13C3×#3712C418yesdirect product factor
14C42×C3⋊D3C2×C311yescharacteristic
15C2×C3×ΓL(1,𝔽25)—34no—
16C3×C42×D3—32no—
17C3×C42×D3—61no—
18C2×#3712C2×C312yes—
19C32×C4×C8—32no—
20C3×#815—34no—
21C3×C4×C3⋊C8—22no—
22C3×C4×C3⋊C8D312yescomplemented
23C3×C4×C3⋊C8C2×C312yescomplemented
24#63555C2×C312yescomplemented
25C22×C3×C3⋊D3C2×C411yescharacteristic
26C2×C33×C4C2×C412yes—
27C2×C33×C4C2311yescharacteristic
28C2×C3×C3⋊Dic3C2×C412yes—
29C2×C3×C3⋊Dic3C2×C411yescharacteristic
30C3×C4×C3⋊D3—24no—
31C3×C4×C3⋊D3C2×C414yes—
32C32×C3⋊C8C2×C418yescomplemented
33C3×C4×M16—91no—
34C4×ΓL(1,𝔽25)—32no—
35C2×C4×C3⋊D3C3×C412yes—
36C2×C4×C3⋊D3C22×C311yescharacteristic
37C2×C3×C4×D3—32no—
38C2×C3×C4×D3—34no—
39C2×C3×C4×D3—62no—
40C2×C3×C4×D3—61no—
41C2×C32×C8—34no—
42C32×C42—21no—
43C32×C42—22no—
44C32×C42—41no—
45C32×C42C2×D312yes—
46C32×C42C22×C311yescharacteristic
47C4×C3⋊Dic3C3×C412yes—
48C4×C3⋊Dic3C22×C311yescharacteristic
49C3×C4×Dic3—34no—
50C3×C4×Dic3—32no—
51C3×C4×Dic3—62no—
52C3×C4×Dic3—61no—
53C2×C3×C3⋊C8—24no—
54C2×C3×C3⋊C8C2×D314yes—
55C2×C3×C3⋊C8C22×C314yes—
56C3×ΓL(1,𝔽25)—316no—
57#3712C3×C418yesdirect product factor
58C22×C33C22×C411yescharacteristic
59C2×C3×C3⋊D3—22no—
60C2×C3×C3⋊D3C4212yes—
61C33×C4M1614yescomplemented
62C33×C4C22×C412yes—
63C3×C3⋊Dic3—22no—
64C3×C3⋊Dic3C4212yes—
65C2×C3×M16—92no—
66C2×C3×C42—91no—
67C2×ΓL(1,𝔽25)—34no—
68C42×D3—32no—
69C42×D3—61no—
70C3×#97—92no—
71C3×C4×C8—32no—
72C3×C4×C8—32no—
73C3×C4×C8—62no—
74#815—34no—
75C4×C3⋊C8C3×D312yescomplemented
76C22×C3⋊D3C2×C3×C411yescharacteristic
77C22×C3×D3—32no—
78C22×C3×D3—61no—
79C2×C32×C4—22no—
80C2×C32×C4—24no—
81C2×C32×C4—21no—
82C2×C32×C4—22no—
83C2×C32×C4—42no—
84C2×C32×C4—41no—
85C2×C32×C4C4×D314yes—
86C2×C32×C4C2×C3×C412yes—
87C2×C32×C4C22×D312yes—
88C2×C32×C4C23×C311yescharacteristic
89C2×C3⋊Dic3C2×C3×C412yes—
90C2×C3⋊Dic3C2×C3×C411yescharacteristic
91C4×C3⋊D3—24no—
92C4×C3⋊D3C2×C3×C414yes—
93C2×C3×Dic3—32no—
94C2×C3×Dic3—34no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—62no—
97C3×C4×D3—38no—
98C3×C4×D3—64no—
99C3×C4×D3—68no—
100C3×C4×D3—68no—
101C32×C8—38no—
102C3×C3⋊C8—28no—
103C3×C3⋊C8C4×D318yescomplemented
104C3×C3⋊C8C2×C3×C418yescomplemented
105C4×M16—91noSylow
106C2×C33C2×C4211yescharacteristic
107C2×C33C2×M1612yes—
108C3×C3⋊D3—22no—
109C22×C3×C4—91no—
110C22×C3×C4—92no—
111C2×C4×D3—32no—
112C2×C4×D3—34no—
113C2×C4×D3—62no—
114C2×C4×D3—61no—
115C3×M16—98no—
116C2×C3×C8—34no—
117C2×C3×C8—34no—
118C2×C3×C8—64no—
119C3×C42—22no—
120C3×C42—21no—
121C3×C42—41no—
122C3×C42—92no—
123C3×C42—91no—
124C3×C42D3211yescharacteristic
125C3×C42C2×C3×D312yes—
126C4×Dic3—34no—
127C4×Dic3—32no—
128C4×Dic3—62no—
129C4×Dic3—61no—
130C2×C3⋊C8C2×C3×D314yes—
131ΓL(1,𝔽25)—316no—
132C22×C32—21no—
133C22×C32—22no—
134C22×C32—41no—
135C22×C32C2×C4×D312yes—
136C22×C32C22×C3×C411yescharacteristic
137C2×C3⋊D3—22no—
138C2×C3⋊D3C3×C4212yes—
139C2×C3×D3—34no—
140C2×C3×D3—62no—
141C2×C3×D3—64no—
142C2×C3×D3—64no—
143C32×C4—24no—
144C32×C4—28no—
145C32×C4—22no—
146C32×C4—24no—
147C32×C4—44no—
148C32×C4—42no—
149C32×C4ΓL(1,𝔽25)18yescomplemented
150C32×C4C3×M1614yescomplemented
151C32×C4C2×C4×D314yes—
152C32×C4C22×C3×C412yes—
153C3⋊Dic3—22no—
154C3⋊Dic3C3×C4212yes—
155C3×Dic3—34no—
156C3×Dic3—62no—
157C3×Dic3—64no—
158C3×Dic3—64no—
159C2×M16—92no—
160C2×C42—91no—
161#97—92no—
162C4×C8—32no—
163C33C4×M1611yescharacteristic, complemented, Sylow
164C23×C3—91no—
165C22×D3—32no—
166C22×D3—61no—
167C2×C3×C4—24no—
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—42no—
172C2×C3×C4—41no—
173C2×C3×C4—91no—
174C2×C3×C4—94no—
175C2×C3×C4—92no—
176C2×C3×C4—184no—
177C2×C3×C4#62412yes—
178C2×C3×C4C3×C4×D314yes—
179C2×C3×C4C2×D3211yescharacteristic
180C2×C3×C4C22×C3×D312yes—
181C2×Dic3—32no—
182C2×Dic3—34no—
183C2×Dic3—61no—
184C2×Dic3—62no—
185C4×D3—38no—
186C4×D3—64no—
187C4×D3—68no—
188C4×D3—68no—
189C3×C8—38no—
190C3×C8—38no—
191C3×C8—68no—
192C3⋊C8C3×C4×D318yescomplemented
193C2×C32—24no—
194C2×C32—22no—
195C2×C32—21no—
196C2×C32—22no—
197C2×C32—41no—
198C2×C32—42no—
199C2×C32C42×D312yes—
200C2×C32C2×ΓL(1,𝔽25)14yes—
201C2×C32C2×C3×C4211yescharacteristic
202C2×C32C2×C3×M1612yes—
203C3⋊D3—22no—
204C3×D3—64no—
205C3×D3—64no—
206C22×C4—91no—
207C22×C4—92no—
208M16—98no—
209C2×C8—34no—
210C42—92no—
211C42—91no—
212C42C3×D3211yescharacteristic
213C22×C3—21no—
214C22×C3—22no—
215C22×C3—41no—
216C22×C3—92no—
217C22×C3—182no—
218C22×C3C2×#62411yescharacteristic
219C22×C3C2×C3×C4×D312yes—
220C2×D3—34no—
221C2×D3—62no—
222C2×D3—64no—
223C2×D3—64no—
224C3×C4—24no—
225C3×C4—22no—
226C3×C4—24no—
227C3×C4—28no—
228C3×C4—42no—
229C3×C4—44no—
230C3×C4—92no—
231C3×C4—182no—
232C3×C4#371214yesdirect product factor
233C3×C4C3×ΓL(1,𝔽25)18yescomplemented
234C3×C4C4×D3212yes—
235C3×C4C2×C3×C4×D314yes—
236Dic3—34no—
237Dic3—62no—
238Dic3—64no—
239Dic3—64no—
240C32—22no—
241C32—21no—
242C32—41no—
243C32C4×ΓL(1,𝔽25)12yescomplemented
244C32C3×C4×M1611yescharacteristic, complemented
245C23—91no—
246C2×C4—91no—
247C2×C4—94no—
248C2×C4—92no—
249C2×C4—184no—
250C2×C4C3×#62412yes—
251C2×C4C2×C3×D3211yescharacteristic
252C8—38no—
253C2×C3—22no—
254C2×C3—24no—
255C2×C3—21no—
256C2×C3—22no—
257C2×C3—42no—
258C2×C3—41no—
259C2×C3—182no—
260C2×C3C2×#371212yes—
261C2×C3C4×#62411yescharacteristic
262C2×C3C3×C42×D312yes—
263C2×C3C2×C3×ΓL(1,𝔽25)14yes—
264D3—64no—
265D3—64no—
266C22—92no—
267C22—182no—
268C22C2×C3×#62411yescharacteristic
269C4—92no—
270C4—182no—
271C4C3×#371214yesdirect product factor
272C4C3×C4×D3212yes—
273C3—21no—
274C3—22no—
275C3—41no—
276C3C4×#371211yescharacteristic, direct product factor
277C3C3×C4×ΓL(1,𝔽25)12yescomplemented
278C2—182no—
279C2C2×C3×#371212yes—
280C2C3×C4×#62411yescharacteristic
subgroupG/H
center, hypercenterC3×C42D32
commutatorC2×C32C2×C3×C42
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C32×C4C23×C3
hypocenterC32C3×C4×M16
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
3. C32×C4×C3⋊C8C2semidirect
4. C3×#63555C2semidirect
6. C4×#3712C3direct (N × H)
13. C3×#3712C4direct (N × H)
22. C3×C4×C3⋊C8D3semidirect
23. C3×C4×C3⋊C8C2×C3semidirect
24. #63555C2×C3semidirect
32. C32×C3⋊C8C2×C4semidirect
57. #3712C3×C4direct (N × H)
61. C33×C4M16semidirect
75. C4×C3⋊C8C3×D3semidirect
103. C3×C3⋊C8C4×D3semidirect
104. C3×C3⋊C8C2×C3×C4semidirect
149. C32×C4ΓL(1,𝔽25)semidirect
150. C32×C4C3×M16semidirect
163. C33C4×M16semidirect
192. C3⋊C8C3×C4×D3semidirect
232. C3×C4#3712direct (N × H)
233. C3×C4C3×ΓL(1,𝔽25)semidirect
243. C32C4×ΓL(1,𝔽25)semidirect
244. C32C3×C4×M16semidirect
271. C4C3×#3712direct (N × H)
276. C3C4×#3712direct (N × H)
277. C3C3×C4×ΓL(1,𝔽25)semidirect

Maximal subgroups

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

subgroupindex
C3×C42×C3⋊D32
C2×C3×#37122
C32×C4×C3⋊C82
C3×#635552
C3×C4×ΓL(1,𝔽25)3
C4×#37123