Finite Group Databasebeta

← C3×C4×#3773 #13166770

Subgroups — C3×C4×#3773 #13166770

Order 1728.

664 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 162 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×C4×#157C211yesmaximal, characteristic
2C2×C3×#3773C212yesmaximal
3C3×C4×#624C212yesmaximal
4C3×#63737C212yesmaximal
5C4×#3773C311yesmaximal, characteristic, direct product factor
6C22×C3×#157C2211yescharacteristic
7C22×C3×#157C2211yescharacteristic
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C3×#624C2214yes—
10C3×C4×#157C414yescomplemented
11C3×#3773C418yesdirect product factor
12C32×C4×Dic3—22no—
13C3×Dic32—22no—
14C2×C4×#157C2×C311yescharacteristic
15C2×#3773C2×C312yes—
16C3×C42×D3—62no—
17C4×#624C2×C312yes—
18#63737C2×C312yes—
19C22×C3×C3⋊D3C2311yescharacteristic
20C2×C3×#157—22no—
21C2×C3×#157C2×C412yes—
22C2×C3×#157C2×C414yes—
23C2×C3×#157C2×C412yes—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3D411yescharacteristic
26C3×C4×C3⋊D3C2×C412yes—
27C3×C4×C3⋊D3Q812yes—
28C2×C32×Dic3—24no—
29C3×#624—24no—
30C3×#624C2×C418yes—
31C3×C4×C4⋊C4—91nomaximal
32C22×#157C22×C311yescharacteristic
33C22×#157C22×C311yescharacteristic
34C2×C4×C3⋊D3C22×C311yescharacteristic
35C2×C3×C4×D3—64no—
36C2×C3×C4×D3—62no—
37C2×#624C22×C314yes—
38C4×#157C3×C414yescomplemented
39#3773C3×C418yesdirect product factor
40C32×C42—62no—
41C3×C4×Dic3—22no—
42C3×C4×Dic3—22no—
43C3×C4×Dic3—42no—
44C3×C4×Dic3—64no—
45Dic32—22no—
46C22×C33C2×D411yescharacteristic
47C2×C3×C3⋊D3C22×C411yescharacteristic
48C2×C3×C3⋊D3C22×C412yes—
49C2×C3×C3⋊D3C2×Q811yescharacteristic
50C2×C3×C3⋊D3#4112yes—
51C3×#157—22no—
52C3×#157C4214yescomplemented
53C33×C4C4⋊C412yescomplemented
54C3×C3⋊Dic3C4⋊C412yescomplemented
55C32×Dic3—28no—
56C2×C3×C4⋊C4—92no—
57C2×C3×C42—92no—
58C2×C3×C42—91no—
59C42×D3—62no—
60C3×#95—92no—
61C22×C3⋊D3C23×C311yescharacteristic
62C22×C3×D3—62no—
63C2×#157—22no—
64C2×#157C2×C3×C412yes—
65C2×#157C2×C3×C414yes—
66C2×#157C2×C3×C412yes—
67C2×C32×C4—22no—
68C2×C32×C4—42no—
69C2×C32×C4—64no—
70C2×C32×C4C3×D411yescharacteristic
71C2×C3⋊Dic3C3×D411yescharacteristic
72C4×C3⋊D3C2×C3×C412yes—
73C4×C3⋊D3C3×Q812yes—
74C2×C3×Dic3—24no—
75C2×C3×Dic3—24no—
76C2×C3×Dic3—44no—
77C2×C3×Dic3—62no—
78C3×C4×D3—68no—
79C3×C4×D3—64no—
80C3×C4×D3—64no—
81C3×C4×D3—68no—
82#624—24no—
83#624C2×C3×C418yes—
84C4×C4⋊C4—91noSylow
85C2×C33C2×C4⋊C411yescharacteristic
86C2×C33C4×D412yes—
87C3×C3⋊D3C2×C4211yescharacteristic
88C3×C3⋊D3#11711yescharacteristic
89C3×C3⋊D3C4×Q812yes—
90C22×C3×C4—94no—
91C22×C3×C4—91no—
92C22×C3×C4—91no—
93C22×C3×C4—91no—
94C2×C4×D3—64no—
95C2×C4×D3—62no—
96C3×C4⋊C4—98no—
97C3×C42—62no—
98C3×C42—62no—
99C3×C42—94no—
100C3×C42—122no—
101C3×C42—182no—
102C4×Dic3—22no—
103C4×Dic3—64no—
104C22×C32—22no—
105C22×C32—42no—
106C22×C32C2×C3×D411yescharacteristic
107C2×C3⋊D3C22×C3×C411yescharacteristic
108C2×C3⋊D3C22×C3×C412yes—
109C2×C3⋊D3C2×C3×Q811yescharacteristic
110C2×C3⋊D3C3×#4112yes—
111C2×C3×D3—64no—
112C2×C3×D3—62no—
113C2×C3×D3—62no—
114C2×C3×D3—64no—
115#157—22no—
116#157C3×C4214yescomplemented
117C32×C4—24no—
118C32×C4—44no—
119C32×C4—68no—
120C32×C4C3×C4⋊C412yescomplemented
121C3⋊Dic3C3×C4⋊C412yescomplemented
122C3×Dic3—28no—
123C3×Dic3—28no—
124C3×Dic3—48no—
125C3×Dic3—64no—
126C2×C4⋊C4—92no—
127C2×C42—92no—
128C2×C42—91no—
129#95—92no—
130C33C4×C4⋊C411yescharacteristic, complemented, Sylow
131C23×C3—91no—
132C22×D3—62no—
133C2×C3×C4—22no—
134C2×C3×C4—42no—
135C2×C3×C4—64no—
136C2×C3×C4—64no—
137C2×C3×C4—98no—
138C2×C3×C4—92no—
139C2×C3×C4—94no—
140C2×C3×C4—91no—
141C2×C3×C4—92no—
142C2×C3×C4—92no—
143C2×C3×C4—92no—
144C2×C3×C4—124no—
145C2×C3×C4—184no—
146C2×C3×C4—182no—
147C2×C3×C4GO+(4,𝔽2)11yescharacteristic
148C2×Dic3—24no—
149C2×Dic3—62no—
150C4×D3—68no—
151C4×D3—64no—
152C4×D3—64no—
153C4×D3—68no—
154C2×C32—24no—
155C2×C32—22no—
156C2×C32—42no—
157C2×C32—44no—
158C2×C32C2×C3×C4⋊C411yescharacteristic
159C2×C32C3×C4×D412yes—
160C3⋊D3C2×C3×C4211yescharacteristic
161C3⋊D3C3×#11711yescharacteristic
162C3⋊D3C3×C4×Q812yes—
163C3×D3—62no—
164C3×D3—62no—
165C3×D3—64no—
166C22×C4—94no—
167C22×C4—91no—
168C22×C4—91no—
169C22×C4—91no—
170C4⋊C4—98no—
171C42—62no—
172C42—94no—
173C42—182no—
174C22×C3—22no—
175C22×C3—42no—
176C22×C3—92no—
177C22×C3—92no—
178C22×C3—91no—
179C22×C3—91no—
180C22×C3C2×GO+(4,𝔽2)11yescharacteristic
181C2×D3—62no—
182C2×D3—64no—
183C2×D3—62no—
184C2×D3—64no—
185C3×C4—24no—
186C3×C4—44no—
187C3×C4—68no—
188C3×C4—68no—
189C3×C4—94no—
190C3×C4—92no—
191C3×C4—128no—
192C3×C4—182no—
193C3×C4#377312yesdirect product factor
194Dic3—28no—
195Dic3—64no—
196C32—22no—
197C32—42no—
198C32C3×C4×C4⋊C411yescharacteristic, complemented
199C23—91no—
200C2×C4—64no—
201C2×C4—98no—
202C2×C4—92no—
203C2×C4—94no—
204C2×C4—91no—
205C2×C4—92no—
206C2×C4—92no—
207C2×C4—92no—
208C2×C4—184no—
209C2×C4—182no—
210C2×C4C3×GO+(4,𝔽2)11yescharacteristic
211C2×C3—22no—
212C2×C3—24no—
213C2×C3—42no—
214C2×C3—44no—
215C2×C3—91no—
216C2×C3—91no—
217C2×C3—92no—
218C2×C3C4×GO+(4,𝔽2)12yes—
219C2×C3C2×#377311yescharacteristic
220D3—62no—
221D3—62no—
222D3—64no—
223C22—91no—
224C22—91no—
225C22—92no—
226C22—92no—
227C22C2×C3×GO+(4,𝔽2)11yescharacteristic
228C4—68no—
229C4—94no—
230C4—92no—
231C4—182no—
232C4C3×#377312yesdirect product factor
233C3—22no—
234C3—42no—
235C3C4×#377311yescharacteristic, direct product factor
236C2—91no—
237C2—91no—
238C2—92no—
239C2C3×C4×GO+(4,𝔽2)12yes—
240C2C2×C3×#377311yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4GO+(4,𝔽2)
commutatorC3⋊D3C2×C3×C42
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3C23
cosocleC22×C3⋊D3C23×C3
hypocenterC32C3×C4×C4⋊C4
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
5. C4×#3773C3direct (N × H)
10. C3×C4×#157C4semidirect
11. C3×#3773C4direct (N × H)
38. C4×#157C3×C4semidirect
39. #3773C3×C4direct (N × H)
52. C3×#157C42semidirect
53. C33×C4C4⋊C4semidirect
54. C3×C3⋊Dic3C4⋊C4semidirect
116. #157C3×C42semidirect
120. C32×C4C3×C4⋊C4semidirect
121. C3⋊Dic3C3×C4⋊C4semidirect
130. C33C4×C4⋊C4semidirect
193. C3×C4#3773direct (N × H)
198. C32C3×C4×C4⋊C4semidirect
232. C4C3×#3773direct (N × H)
235. C3C4×#3773direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×#1572
C2×C3×#37732
C3×C4×#6242
C3×#637372
C4×#37733
C3×C4×C4⋊C49