Finite Group Databasebeta

← #13154797

Subgroups — #13154797

Order 1728.

400 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 60 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
1#11724514C211yesmaximal, characteristic
2C3×#64283C211yesmaximal, characteristic
3#11723437C211yesmaximal, characteristic
4#10590152—31nomaximal
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6#88994—21no—
7C4⋊(C3⋊#157)—21no—
8C4×C3⋊#157C412yescomplemented
9C3×#3792—21no—
10C3×#3788—21no—
11C3×#3787C412yescomplemented
12C4×C3⋊(C3⋊Dic3)C411yescharacteristic, complemented
13C32×C4×Dic3C411yescharacteristic, complemented
14#88807—21no—
15#88803—21no—
16#64286—31no—
17#64283D311yescharacteristic
18#64081—31no—
19#63855—62no—
20C22×C3×C3⋊D3D411yescharacteristic
21C2×C3⋊#157—22no—
22C2×C33×C4C2×C411yescharacteristic
23C2×C3⋊(C3⋊Dic3)—21no—
24C2×C3×C3⋊Dic3D411yescharacteristic
25C3×C4×C3⋊D3C2×C411yescharacteristic
26C3×C4×C3⋊D3C2×C411yescharacteristic
27C3×C4×C3⋊D3D411yescharacteristic
28C3×C4×C3⋊D3Q811yescharacteristic
29C2×C32×Dic3—21no—
30C3×#622—22no—
31#4557—91nomaximal
32C2×C4×C3⋊D3C2×D311yescharacteristic
33C2×C3×C4×D3—62no—
34C22⋊#157—61no—
35#3792—21no—
36C4⋊#157—61no—
37C4×#157—32no—
38#3788—21no—
39#3787Dic312yescomplemented
40C32×C42—31no—
41#3752—61no—
42#3750—61no—
43C4×C3⋊Dic3—31no—
44C4×C3⋊Dic3—62no—
45C4×C3⋊Dic3—122no—
46C3×C4×Dic3—22no—
47Dic3⋊Dic3—64no—
48(C2×D3)⋊Dic3—64no—
49C22×C33C22⋊C411yescharacteristic
50C2×C3×C3⋊D3—41no—
51C2×C3×C3⋊D3C22⋊C411yescharacteristic
52C2×C3×C3⋊D3C4⋊C411yescharacteristic
53C3⋊#157—42no—
54C33×C4C4211yescharacteristic
55C33×C4C4⋊C411yescharacteristic
56C3⋊(C3⋊Dic3)—41no—
57C3×C3⋊Dic3C22⋊C411yescharacteristic
58C3×C3⋊Dic3C4⋊C411yescharacteristic
59C32×Dic3—41no—
60C2×C3×M16—91no—
61#926—91no—
62#926—91no—
63#872—182no—
64C22×C3⋊D3D1211yescharacteristic
65C22×C3×D3—62no—
66C2×#157—62no—
67C2×C32×C4—22no—
68C2×C32×C4—42no—
69C2×C32×C4—61no—
70C2×C32×C4C4×D311yescharacteristic
71C2×C3⋊Dic3—61no—
72C2×C3⋊Dic3—64no—
73C2×C3⋊Dic3—124no—
74C2×C3⋊Dic3#6711yescharacteristic
75C4×C3⋊D3Dic611yescharacteristic
76C4×C3⋊D3C4×D311yescharacteristic
77C4×C3⋊D3C2×Dic311yescharacteristic
78C4×C3⋊D3#6711yescharacteristic
79C2×C3×Dic3—24no—
80C2×C3×Dic3—62no—
81C3×C4×D3—62no—
82C3×C4×D3—62no—
83C3×C4×D3—62no—
84C3×C4×D3—62no—
85#622—22no—
86H(3,ℤ/4)—271noSylow
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×C42—62no—
97#217—181no—
98#217—181no—
99(C2×D3)⋊C4—184no—
100C4⋊Dic3—181no—
101C4⋊Dic3—181no—
102Dic3⋊C4—184no—
103C4×Dic3—92no—
104C4×Dic3—91no—
105C4×Dic3—182no—
106C4×Dic3—362no—
107C4×Dic3#15711yescharacteristic, complemented
108C22×C32—22no—
109C22×C32—42no—
110C22×C32(C2×D3)⋊C411yescharacteristic
111C2×C3⋊D3—41no—
112C2×C3⋊D3C4⋊Dic311yescharacteristic
113C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
114C2×C3×D3—62no—
115C2×C3×D3—62no—
116C2×C3×D3—124no—
117#157—122no—
118C32×C4—22no—
119C32×C4—22no—
120C32×C4—42no—
121C32×C4—42no—
122C32×C4—121no—
123C32×C4C4×Dic311yescharacteristic
124C32×C4Dic3⋊C411yescharacteristic
125C3⋊Dic3—121no—
126C3⋊Dic3—124no—
127C3⋊Dic3—128no—
128C3⋊Dic3Dic3⋊C411yescharacteristic
129C3⋊Dic3#21711yescharacteristic
130C3×Dic3—44no—
131C3×Dic3—62no—
132C3×Dic3—62no—
133C2×M16—91no—
134#117—271no—
135#117—271no—
136C33H(3,ℤ/4)11yescharacteristic, complemented, Sylow
137C23×C3—91no—
138C22×D3—62no—
139C2×C3×C4—22no—
140C2×C3×C4—42no—
141C2×C3×C4—64no—
142C2×C3×C4—91no—
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4C2×#15711yescharacteristic
148C2×Dic3—21no—
149C2×Dic3—62no—
150C2×Dic3—184no—
151C2×Dic3—182no—
152C2×Dic3—181no—
153C2×Dic3—364no—
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—541no—
174C4⋊C4—541no—
175C22⋊C4—541no—
176C22⋊C4—541no—
177C42—31no—
178C42—272no—
179C42—271no—
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—42no—
192C3×C4—42no—
193C3×C4—91no—
194C3×C4—91no—
195C3×C4—124no—
196C3×C4C4×#15711yescharacteristic
197C3×C4C4⋊#15711yescharacteristic
198Dic3—41no—
199Dic3—62no—
200Dic3—62no—
201Dic3—364no—
202Dic3—361no—
203Dic3—368no—
204Dic3—362no—
205C32—22no—
206C32—42no—
207C32#455711yescharacteristic, complemented
208C23—91no—
209C2×C4—61no—
210C2×C4—91no—
211C2×C4—91no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—91no—
215C2×C4—542no—
216C2×C4—541no—
217C2×C4D3×#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—
232C22(C2×D3)⋊#15711yescharacteristic
233C4—91no—
234C4—91no—
235C4—121no—
236C4—1081no—
237C4—1082no—
238C4Dic3×#15711yescharacteristic
239C4Dic3⋊#15711yescharacteristic
240C3—22no—
241C3—42no—
242C3#1059015211yescharacteristic, complemented
243C2—21no—
244C2—181no—
245C2—181no—
246C2#1172272211yescharacteristic
subgroupG/H
centerC4Dic3×#157
commutatorC33×C4C42
Frattini, hypercenterC2×C4D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33#95
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C3×C4×C3⋊D3C22
hypocenterC33H(3,ℤ/4)
LCSG ▷ C33×C4 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
8. C4×C3⋊#157C4semidirect
11. C3×#3787C4semidirect
12. C4×C3⋊(C3⋊Dic3)C4semidirect
13. C32×C4×Dic3C4semidirect
39. #3787Dic3semidirect
107. C4×Dic3#157semidirect
136. C33H(3,ℤ/4)semidirect
207. C32#4557semidirect
242. C3#10590152semidirect

Maximal subgroups

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

subgroupindex
#117245142
C3×#642832
#117234372
#105901523
#45579