Finite Group Databasebeta

← #13157206

Subgroups — #13157206

Order 1728.

396 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 65 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
2C2×C4×C3⋊#157C211yesmaximal, characteristic
3C3×M16×C3⋊D3C211yesmaximal, characteristic
4#10590189—31nomaximal
5C22×C3⋊#157—21no—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7#88994—21no—
8C4⋊(C3⋊#157)C412yescomplemented
9C4×C3⋊#157—21no—
10C4×C3⋊#157—21no—
11C4×C3⋊#157C412yescomplemented
12C33×M16C411yescharacteristic, complemented
13C3×#3748C411yescharacteristic, complemented
14C3×#3743—21no—
15C3×C8×C3⋊D3—21no—
16#64286—31no—
17C2×C4×#157—31no—
18M16×C3⋊D3D311yescharacteristic
19C3×D3×M16—62no—
20C22×C3×C3⋊D3D411yescharacteristic
21C2×C3⋊#157—24no—
22C2×C3⋊#157—22no—
23C2×C3⋊#157—22no—
24C2×C33×C4C2×C411yescharacteristic
25C2×C3×C3⋊Dic3Q811yescharacteristic
26C3×C4×C3⋊D3C2×C411yescharacteristic
27C3×C4×C3⋊D3C2×C411yescharacteristic
28C3×C4×C3⋊D3D411yescharacteristic
29C3×C4×C3⋊D3D411yescharacteristic
30C33×C8—21no—
31C3×C3⋊(C3⋊C8)—21no—
32#4555—91nomaximal
33C22×#157—61no—
34C2×C4×C3⋊D3C2×D311yescharacteristic
35C2×C3×C4×D3—62no—
36C22⋊#157—61no—
37C4⋊#157—32no—
38C4×#157—32no—
39C4×#157—61no—
40C4×#157—61no—
41C32×M16—22no—
42C32×M16—42no—
43C32×M16Dic311yescharacteristic, complemented
44#3748Dic311yescharacteristic, complemented
45#3743—21no—
46C8×C3⋊D3—21no—
47C3×#208—62no—
48C3×ΓL(1,𝔽25)—64no—
49C3×D3×C8—64no—
50C22×C33C4⋊C411yescharacteristic
51C2×C3×C3⋊D3—22no—
52C2×C3×C3⋊D3C22⋊C411yescharacteristic
53C2×C3×C3⋊D3C22⋊C411yescharacteristic
54C3⋊#157—24no—
55C3⋊#157—42no—
56C33×C4C4211yescharacteristic
57C33×C4C22⋊C411yescharacteristic
58C3×C3⋊Dic3C4⋊C411yescharacteristic
59C3×C3⋊Dic3C4⋊C411yescharacteristic
60C2×C3×M16—91no—
61#926—91no—
62C2×C4×Dic3—91no—
63D3×M16—62no—
64C22×C3⋊D3D1211yescharacteristic
65C22×C3×D3—62no—
66C2×#157—64no—
67C2×#157—62no—
68C2×#157—62no—
69C2×C32×C4—22no—
70C2×C32×C4—42no—
71C2×C32×C4C2×Dic311yescharacteristic
72C2×C3⋊Dic3Dic611yescharacteristic
73C4×C3⋊D3C4×D311yescharacteristic
74C4×C3⋊D3C4×D311yescharacteristic
75C4×C3⋊D3#6711yescharacteristic
76C4×C3⋊D3#6711yescharacteristic
77C2×C3×Dic3—62no—
78C3×C4×D3—62no—
79C3×C4×D3—62no—
80C3×C4×D3—62no—
81C3×C4×D3—62no—
82C32×C8—21no—
83C32×C8—24no—
84C32×C8—44no—
85C3⋊(C3⋊C8)—21no—
86C3×C3⋊C8—64no—
87#339—271noSylow
88C2×C33—21no—
89C2×C33#9511yescharacteristic
90C3×C3⋊D3—21no—
91C3×C3⋊D3#10412yes—
92C22×C3×C4—91no—
93C22×Dic3—181no—
94C2×C4×D3—62no—
95C3×M16—22no—
96C3×M16—42no—
97C3×M16—91no—
98C3×M16—181no—
99C3×M16#15711yescharacteristic, complemented
100C2×C3×C8—181no—
101#217—181no—
102C4⋊Dic3—92no—
103C4×Dic3—92no—
104C4×Dic3—181no—
105C4×Dic3—181no—
106#208—62no—
107ΓL(1,𝔽25)—64no—
108D3×C8—64no—
109C22×C32—22no—
110C22×C32—42no—
111C22×C32C4⋊Dic311yescharacteristic
112C2×C3⋊D3—22no—
113C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
114C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
115C2×C3×D3—62no—
116C2×C3×D3—62no—
117C2×C3×D3—124no—
118#157—64no—
119#157—122no—
120C32×C4—22no—
121C32×C4—22no—
122C32×C4—42no—
123C32×C4—42no—
124C32×C4C4×Dic311yescharacteristic
125C32×C4#21711yescharacteristic
126C3⋊Dic3Dic3⋊C411yescharacteristic
127C3⋊Dic3Dic3⋊C411yescharacteristic
128C3×Dic3—62no—
129C3×Dic3—62no—
130C2×M16—91no—
131#117—271no—
132C2×C42—271no—
133C33#33911yescharacteristic, complemented, Sylow
134C23×C3—91no—
135C22×D3—62no—
136C2×C3×C4—22no—
137C2×C3×C4—42no—
138C2×C3×C4—91no—
139C2×C3×C4—91no—
140C2×C3×C4—91no—
141C2×C3×C4—91no—
142C2×C3×C4—91no—
143C2×C3×C4C2×#15711yescharacteristic
144C2×Dic3—62no—
145C2×Dic3—182no—
146C2×Dic3—184no—
147C2×Dic3—182no—
148C4×D3—62no—
149C4×D3—62no—
150C4×D3—62no—
151C4×D3—62no—
152C3×C8—24no—
153C3×C8—21no—
154C3×C8—44no—
155C3×C8—181no—
156C3⋊C8—64no—
157C2×C32—22no—
158C2×C32—21no—
159C2×C32—42no—
160C2×C32—42no—
161C2×C32—82no—
162C2×C32C3⋊#9511yescharacteristic
163C3⋊D3—21no—
164C3⋊D3#82512yes—
165C3×D3—64no—
166C3×D3—122no—
167C22×C4—91no—
168C22×C4—541no—
169M16—91no—
170M16—181no—
171M16C3⋊#15711yescharacteristic, complemented
172C2×C8—181no—
173C4⋊C4—272no—
174C22⋊C4—541no—
175C42—272no—
176C42—541no—
177C42—541no—
178C22×C3—22no—
179C22×C3—42no—
180C22×C3—91no—
181C22×C3—91no—
182C22×C3—182no—
183C22×C3C4⋊#15711yescharacteristic
184C2×D3—62no—
185C2×D3—62no—
186C2×D3—124no—
187C3×C4—22no—
188C3×C4—22no—
189C3×C4—42no—
190C3×C4—42no—
191C3×C4—91no—
192C3×C4—91no—
193C3×C4C4×#15711yescharacteristic
194C3×C4C22⋊#15711yescharacteristic
195Dic3—62no—
196Dic3—62no—
197Dic3—184no—
198Dic3—362no—
199C32—22no—
200C32—42no—
201C32#455511yescharacteristic, complemented
202C23—91no—
203C2×C4—91no—
204C2×C4—91no—
205C2×C4—91no—
206C2×C4—91no—
207C2×C4—91no—
208C2×C4—542no—
209C2×C4—544no—
210C2×C4—542no—
211C2×C4C2×C3⋊#15711yescharacteristic
212C8—21no—
213C8—181no—
214C2×C3—21no—
215C2×C3—22no—
216C2×C3—42no—
217C2×C3—42no—
218C2×C3—82no—
219C2×C3—92no—
220C2×C3—181no—
221C2×C3#6378111yescharacteristic
222D3—64no—
223D3—122no—
224C22—91no—
225C22—91no—
226C22—182no—
227C22C4⋊(C3⋊#157)11yescharacteristic
228C4—91no—
229C4—91no—
230C4—544no—
231C4—1082no—
232C4C4×C3⋊#15711yescharacteristic
233C4#8899411yescharacteristic
234C3—22no—
235C3—42no—
236C3#1059018911yescharacteristic, complemented
237C2—21no—
238C2—92no—
239C2—181no—
240C2#1172310611yescharacteristic
subgroupG/H
centerC4#88994
commutatorC33×C4C42
FrattiniC2×C4C2×C3⋊#157
FittingC33×M16C4
socleC2×C33#95
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C3×C4×C3⋊D3C22
hypercenterM16C3⋊#157
hypocenterC33#339
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
8. C4⋊(C3⋊#157)C4semidirect
11. C4×C3⋊#157C4semidirect
12. C33×M16C4semidirect
13. C3×#3748C4semidirect
43. C32×M16Dic3semidirect
44. #3748Dic3semidirect
99. C3×M16#157semidirect
133. C33#339semidirect
171. M16C3⋊#157semidirect
201. C32#4555semidirect
236. C3#10590189semidirect

Maximal subgroups

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

subgroupindex
#117245142
C2×C4×C3⋊#1572
C3×M16×C3⋊D32
#105901893
#45559