Finite Group Databasebeta

← C3×#10590152 #13157077

Subgroups — C3×#10590152 #13157077

Order 1728.

400 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 68 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×#64286C211yesmaximal, characteristic
2C3×#64283C211yesmaximal, characteristic
3C3×#64081C211yesmaximal, characteristic
4#10590152C311yesmaximal, characteristic, direct product factor
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6C3×C22⋊#157—21no—
7C3×#3792—21no—
8C3×C4⋊#157—21no—
9C3×C4×#157C412yescomplemented
10C3×#3788—21no—
11C3×#3787C412yescomplemented
12C33×C42C411yescharacteristic, complemented
13C3×#3752—21no—
14C3×#3750—21no—
15C3×C4×C3⋊Dic3C411yescharacteristic, complemented
16#64286C2×C311yescharacteristic
17#64283C2×C311yescharacteristic
18#64081C2×C311yescharacteristic
19C3×#872—62no—
20C22×C3×C3⋊D3D411yescharacteristic
21C2×C3×#157—22no—
22C2×C33×C4—21no—
23C2×C33×C4C2×C411yescharacteristic
24C2×C3×C3⋊Dic3—21no—
25C2×C3×C3⋊Dic3D411yescharacteristic
26C3×C4×C3⋊D3C2×C411yescharacteristic
27C3×C4×C3⋊D3C2×C411yescharacteristic
28C3×C4×C3⋊D3D411yescharacteristic
29C3×C4×C3⋊D3Q811yescharacteristic
30C3×#622—22no—
31C3×H(3,ℤ/4)—91nomaximal
32C2×C4×C3⋊D3C22×C311yescharacteristic
33C2×C3×C4×D3—62no—
34C22⋊#157—21no—
35#3792—21no—
36C4⋊#157—21no—
37C4×#157C3×C412yescomplemented
38#3788—21no—
39#3787C3×C412yescomplemented
40C32×C42—22no—
41C32×C42—42no—
42C32×C42C3×C411yescharacteristic, complemented
43#3752—21no—
44#3750—21no—
45C4×C3⋊Dic3C3×C411yescharacteristic, complemented
46C3×(C2×D3)⋊C4—64no—
47C3×Dic3⋊C4—64no—
48C3×C4×Dic3—62no—
49C22×C33C22⋊C411yescharacteristic
50C2×C3×C3⋊D3—41no—
51C2×C3×C3⋊D3C22⋊C411yescharacteristic
52C2×C3×C3⋊D3C4⋊C411yescharacteristic
53C3×#157—42no—
54C33×C4—41no—
55C33×C4C4211yescharacteristic
56C33×C4C4⋊C411yescharacteristic
57C3×C3⋊Dic3—41no—
58C3×C3⋊Dic3C22⋊C411yescharacteristic
59C3×C3⋊Dic3C4⋊C411yescharacteristic
60C2×C3×M16—91no—
61C3×#117—91no—
62C3×#117—91no—
63#872—62no—
64C22×C3⋊D3C3×D411yescharacteristic
65C22×C3×D3—62no—
66C2×#157—22no—
67C2×C32×C4—21no—
68C2×C32×C4—24no—
69C2×C32×C4—22no—
70C2×C32×C4—44no—
71C2×C32×C4—42no—
72C2×C32×C4C2×C3×C411yescharacteristic
73C2×C3⋊Dic3—21no—
74C2×C3⋊Dic3C3×D411yescharacteristic
75C4×C3⋊D3C2×C3×C411yescharacteristic
76C4×C3⋊D3C2×C3×C411yescharacteristic
77C4×C3⋊D3C3×D411yescharacteristic
78C4×C3⋊D3C3×Q811yescharacteristic
79C2×C3×Dic3—62no—
80C2×C3×Dic3—64no—
81C3×C4×D3—62no—
82C3×C4×D3—62no—
83C3×C4×D3—62no—
84C3×C4×D3—62no—
85#622—22no—
86H(3,ℤ/4)—91noSylow
87C2×C33—21no—
88C2×C33#9511yescharacteristic
89C3×C3⋊D3—21no—
90C3×C3⋊D3—21no—
91C22×C3×C4—91no—
92C2×C4×D3—62no—
93C3×M16—181no—
94C3×M16—181no—
95C2×C3×C8—92no—
96C3×C4⋊C4—181no—
97C3×C4⋊C4—181no—
98C3×C22⋊C4—181no—
99C3×C22⋊C4—181no—
100C3×C42—22no—
101C3×C42—42no—
102C3×C42—92no—
103C3×C42—91no—
104C3×C42#15711yescharacteristic, complemented
105(C2×D3)⋊C4—64no—
106Dic3⋊C4—64no—
107C4×Dic3—62no—
108C22×C32—22no—
109C22×C32—42no—
110C22×C32C3×C22⋊C411yescharacteristic
111C2×C3⋊D3—41no—
112C2×C3⋊D3C3×C22⋊C411yescharacteristic
113C2×C3⋊D3C3×C4⋊C411yescharacteristic
114C2×C3×D3—62no—
115C2×C3×D3—62no—
116C2×C3×D3—124no—
117#157—42no—
118C32×C4—22no—
119C32×C4—22no—
120C32×C4—42no—
121C32×C4—42no—
122C32×C4—41no—
123C32×C4—44no—
124C32×C4—48no—
125C32×C4C3×C4211yescharacteristic
126C32×C4C3×C4⋊C411yescharacteristic
127C3⋊Dic3—41no—
128C3⋊Dic3C3×C22⋊C411yescharacteristic
129C3⋊Dic3C3×C4⋊C411yescharacteristic
130C3×Dic3—62no—
131C3×Dic3—62no—
132C3×Dic3—124no—
133C2×M16—91no—
134#117—91no—
135#117—91no—
136C33H(3,ℤ/4)11yescharacteristic, complemented, Sylow
137C23×C3—91no—
138C22×D3—62no—
139C2×C3×C4—24no—
140C2×C3×C4—21no—
141C2×C3×C4—22no—
142C2×C3×C4—44no—
143C2×C3×C4—42no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4—91no—
148C2×C3×C4—91no—
149C2×C3×C4—182no—
150C2×C3×C4—181no—
151C2×C3×C4C2×#15711yescharacteristic
152C2×Dic3—62no—
153C2×Dic3—64no—
154C4×D3—62no—
155C4×D3—62no—
156C4×D3—62no—
157C4×D3—62no—
158C3×C8—182no—
159C2×C32—21no—
160C2×C32—24no—
161C2×C32—22no—
162C2×C32—44no—
163C2×C32—42no—
164C2×C32C3×#9511yescharacteristic
165C3⋊D3—21no—
166C3⋊D3—21no—
167C3×D3—122no—
168C3×D3—122no—
169C22×C4—91no—
170M16—181no—
171M16—181no—
172C2×C8—92no—
173C4⋊C4—181no—
174C4⋊C4—181no—
175C22⋊C4—181no—
176C22⋊C4—181no—
177C42—92no—
178C42—91no—
179C42C3×#15711yescharacteristic, complemented
180C22×C3—22no—
181C22×C3—42no—
182C22×C3—91no—
183C22×C3—91no—
184C22×C3—361no—
185C22×C3C22⋊#15711yescharacteristic
186C2×D3—62no—
187C2×D3—62no—
188C2×D3—124no—
189C3×C4—22no—
190C3×C4—22no—
191C3×C4—44no—
192C3×C4—41no—
193C3×C4—48no—
194C3×C4—42no—
195C3×C4—42no—
196C3×C4—91no—
197C3×C4—91no—
198C3×C4—361no—
199C3×C4—362no—
200C3×C4C4×#15711yescharacteristic
201C3×C4C4⋊#15711yescharacteristic
202Dic3—62no—
203Dic3—62no—
204Dic3—124no—
205C32—22no—
206C32—42no—
207C32C3×H(3,ℤ/4)11yescharacteristic, complemented
208C23—91no—
209C2×C4—21no—
210C2×C4—91no—
211C2×C4—91no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—91no—
215C2×C4—182no—
216C2×C4—181no—
217C2×C4C2×C3×#15711yescharacteristic
218C8—182no—
219C2×C3—24no—
220C2×C3—21no—
221C2×C3—22no—
222C2×C3—44no—
223C2×C3—42no—
224C2×C3—181no—
225C2×C3—181no—
226C2×C3#6378111yescharacteristic
227D3—122no—
228D3—122no—
229C22—91no—
230C22—91no—
231C22—361no—
232C22C3×C22⋊#15711yescharacteristic
233C4—41no—
234C4—91no—
235C4—91no—
236C4—361no—
237C4—362no—
238C4C3×C4×#15711yescharacteristic
239C4C3×C4⋊#15711yescharacteristic
240C3—22no—
241C3—42no—
242C3#1059015211yescharacteristic, direct product factor
243C2—21no—
244C2—181no—
245C2—181no—
246C2C3×#6378111yescharacteristic
subgroupG/H
centerC3×C4C4×#157
commutatorC32×C4C3×C42
FrattiniC2×C4C2×C3×#157
FittingC33×C42C4
socleC2×C33#95
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C4×C3⋊D3C22×C3
hypercenterC3×C42#157
hypocenterC32C3×H(3,ℤ/4)
LCSG ▷ C32×C4 ▷ C32
UCSC1 ◁ C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
4. #10590152C3direct (N × H)
9. C3×C4×#157C4semidirect
11. C3×#3787C4semidirect
12. C33×C42C4semidirect
15. C3×C4×C3⋊Dic3C4semidirect
37. C4×#157C3×C4semidirect
39. #3787C3×C4semidirect
42. C32×C42C3×C4semidirect
45. C4×C3⋊Dic3C3×C4semidirect
104. C3×C42#157semidirect
136. C33H(3,ℤ/4)semidirect
179. C42C3×#157semidirect
207. C32C3×H(3,ℤ/4)semidirect
242. C3#10590152direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642862
C3×#642832
C3×#640812
#105901523
C3×H(3,ℤ/4)9