Finite Group Databasebeta

← #13145736

Subgroups — #13145736

Order 1728.

626 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3748C212yesmaximal
2C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
3#10589166—32nomaximal
4#10589165—31nomaximal
5C22×C33×C4C2211yescharacteristic
6C3×#3748—22no—
7C3×#3748C414yes—
8C2×C3×C3⋊(C3⋊C8)—22no—
9C2×C3×C3⋊(C3⋊C8)—22no—
10C2×C3×C3⋊(C3⋊C8)C412yes—
11C2×#3748D312yes—
12C22×C3⋊(C3⋊C8)D311yescharacteristic
13C2×C3×#208—32no—
14C2×C3×#208—32no—
15C2×C3×#208—62no—
16C22×C3×C3⋊C8—32no—
17C22×C3×C3⋊C8—61no—
18C23×C33Q811yescharacteristic
19C2×C33×C4C2×C411yescharacteristic
20C2×C33×C4C2×C412yes—
21C2×C33×C4D411yescharacteristic
22C2×C33×C4D412yes—
23C3×C3⋊(C3⋊C8)—22no—
24C3×C3⋊(C3⋊C8)—24no—
25#4558—92no—
26#4552—91no—
27C22×C32×C4—21no—
28C22×C32×C4—22no—
29C22×C32×C4—41no—
30C22×C32×C4C2×D312yes—
31C22×C32×C4C2×D311yescharacteristic
32#3748—22no—
33#3748Dic314yes—
34C2×C3⋊(C3⋊C8)—22no—
35C2×C3⋊(C3⋊C8)—22no—
36C2×C3⋊(C3⋊C8)Dic312yes—
37C3×#208—34no—
38C3×#208—34no—
39C3×#208—64no—
40C3×#208—64no—
41C3×#208—62no—
42C3×#208—62no—
43C2×C3×C3⋊C8—34no—
44C2×C3×C3⋊C8—62no—
45C2×C3×C3⋊C8—64no—
46C2×C3×C3⋊C8—64no—
47C2×C3×C3⋊C8—62no—
48C2×C3×C3⋊C8—62no—
49C2×C3×C3⋊C8—62no—
50C2×C3×C3⋊C8—62no—
51C22×C33—22no—
52C22×C33C4⋊C412yes—
53C22×C33C4⋊C411yescharacteristic
54C33×C4C4211yescharacteristic
55C33×C4C22⋊C411yescharacteristic
56C33×C4C22⋊C412yes—
57C2×C3×M16—92no—
58C22×C3×C8—91no—
59C2×#208—32no—
60C2×#208—32no—
61C2×#208—62no—
62C22×C3⋊C8—32no—
63C22×C3⋊C8—61no—
64C23×C32—21no—
65C23×C32—22no—
66C23×C32—41no—
67C23×C32Dic611yescharacteristic
68C23×C32Dic612yes—
69C2×C32×C4—22no—
70C2×C32×C4—22no—
71C2×C32×C4—22no—
72C2×C32×C4—21no—
73C2×C32×C4—22no—
74C2×C32×C4—22no—
75C2×C32×C4—21no—
76C2×C32×C4—22no—
77C2×C32×C4—22no—
78C2×C32×C4—22no—
79C2×C32×C4—41no—
80C2×C32×C4—42no—
81C2×C32×C4—42no—
82C2×C32×C4—41no—
83C2×C32×C4C4×D312yes—
84C2×C32×C4C4×D312yes—
85C2×C32×C4C4×D312yes—
86C2×C32×C4D1211yescharacteristic
87C2×C32×C4D1212yes—
88C2×C32×C4C2×Dic311yescharacteristic
89C2×C32×C4C2×Dic312yes—
90C2×C32×C4#6712yes—
91C2×C32×C4#6712yes—
92C2×C32×C4#6712yes—
93C3⋊(C3⋊C8)—22no—
94C3⋊(C3⋊C8)—24no—
95C3×C3⋊C8—64no—
96C3×C3⋊C8—68no—
97C3×C3⋊C8—64no—
98C3×C3⋊C8—64no—
99C3×C3⋊C8—64no—
100#341—271noSylow
101C2×C33—22no—
102C2×C33#9511yescharacteristic
103C2×C33#10812yes—
104C22×C3×C4—21no—
105C22×C3×C4—22no—
106C22×C3×C4—41no—
107C22×C3×C4D3212yes—
108C22×C3×C4D3211yescharacteristic
109C3×M16—94no—
110C3×M16—182no—
111C2×C3×C8—92no—
112C2×C3×C8—182no—
113C2×C3×C8—182no—
114#208—34no—
115#208—34no—
116#208—64no—
117#208—64no—
118#208—62no—
119#208—62no—
120C2×C3⋊C8—34no—
121C2×C3⋊C8—62no—
122C2×C3⋊C8—64no—
123C2×C3⋊C8—64no—
124C2×C3⋊C8—62no—
125C2×C3⋊C8—62no—
126C2×C3⋊C8—62no—
127C2×C3⋊C8—62no—
128C22×C32—22no—
129C22×C32—24no—
130C22×C32—24no—
131C22×C32—22no—
132C22×C32—22no—
133C22×C32—22no—
134C22×C32—21no—
135C22×C32—22no—
136C22×C32—41no—
137C22×C32—42no—
138C22×C32—44no—
139C22×C32—44no—
140C22×C32Dic3⋊C412yes—
141C22×C32Dic3⋊C412yes—
142C22×C32Dic3⋊C412yes—
143C22×C32C4⋊Dic312yes—
144C22×C32C4⋊Dic311yescharacteristic
145C32×C4—21no—
146C32×C4—22no—
147C32×C4—22no—
148C32×C4—22no—
149C32×C4—22no—
150C32×C4—21no—
151C32×C4—22no—
152C32×C4—41no—
153C32×C4—41no—
154C32×C4—42no—
155C32×C4C4×Dic311yescharacteristic
156C32×C4C4×Dic312yes—
157C32×C4(C2×D3)⋊C412yes—
158C32×C4(C2×D3)⋊C412yes—
159C32×C4(C2×D3)⋊C412yes—
160C32×C4#21711yescharacteristic
161C32×C4#21712yes—
162C2×M16—92no—
163C22×C8—91no—
164C33#34111yescharacteristic, complemented, Sylow
165C23×C3—21no—
166C23×C3—22no—
167C23×C3—41no—
168C23×C3#62712yes—
169C23×C3#62711yescharacteristic
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—21no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—21no—
179C2×C3×C4—22no—
180C2×C3×C4—41no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—41no—
184C2×C3×C4D3×Dic312yes—
185C2×C3×C4D3×Dic312yes—
186C2×C3×C4D3×Dic312yes—
187C2×C3×C4#62411yescharacteristic
188C2×C3×C4#62412yes—
189C2×C3×C4#62512yes—
190C2×C3×C4#62511yescharacteristic
191C2×C3×C4#62612yes—
192C2×C3×C4#62612yes—
193C2×C3×C4#62612yes—
194C3×C8—182no—
195C3×C8—184no—
196C3⋊C8—64no—
197C3⋊C8—68no—
198C3⋊C8—64no—
199C3⋊C8—64no—
200C3⋊C8—64no—
201C2×C32—22no—
202C2×C32—24no—
203C2×C32—22no—
204C2×C32—22no—
205C2×C32—24no—
206C2×C32—22no—
207C2×C32—21no—
208C2×C32—22no—
209C2×C32—41no—
210C2×C32—42no—
211C2×C32—42no—
212C2×C32—42no—
213C2×C32—44no—
214C2×C32#81912yes—
215C2×C32#82214yes—
216C2×C32C3⋊#9511yescharacteristic
217C2×C32C3⋊#9512yes—
218C22×C4#111311yescharacteristic
219M16—94no—
220M16—182no—
221C2×C8—92no—
222C2×C8—182no—
223C2×C8—182no—
224C22×C3—24no—
225C22×C3—24no—
226C22×C3—22no—
227C22×C3—22no—
228C22×C3—22no—
229C22×C3—21no—
230C22×C3—22no—
231C22×C3—22no—
232C22×C3—44no—
233C22×C3—44no—
234C22×C3—42no—
235C22×C3—41no—
236C22×C3Dic3⋊Dic312yes—
237C22×C3Dic3⋊Dic312yes—
238C22×C3Dic3⋊Dic312yes—
239C22×C3#372412yes—
240C22×C3#372411yescharacteristic
241C3×C4—21no—
242C3×C4—22no—
243C3×C4—21no—
244C3×C4—22no—
245C3×C4—22no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—41no—
249C3×C4—41no—
250C3×C4—42no—
251C3×C4Dic3211yescharacteristic
252C3×C4Dic3212yes—
253C3×C4(C2×D3)⋊Dic312yes—
254C3×C4(C2×D3)⋊Dic312yes—
255C3×C4(C2×D3)⋊Dic312yes—
256C3×C4#372211yescharacteristic
257C3×C4#372212yes—
258C32—21no—
259C32—22no—
260C32—41no—
261C32#455211yescharacteristic, complemented
262C32#455812yescomplemented
263C23#629611yescharacteristic
264C2×C4#629411yescharacteristic
265C2×C4#629412yes—
266C2×C4#629512yes—
267C2×C4#629511yescharacteristic
268C8—182no—
269C8—184no—
270C2×C3—24no—
271C2×C3—22no—
272C2×C3—22no—
273C2×C3—22no—
274C2×C3—22no—
275C2×C3—24no—
276C2×C3—21no—
277C2×C3—22no—
278C2×C3—42no—
279C2×C3—42no—
280C2×C3—44no—
281C2×C3—42no—
282C2×C3—41no—
283C2×C3#6357714yes—
284C2×C3#6357812yes—
285C2×C3#6357911yescharacteristic
286C2×C3#6357912yes—
287C22—22no—
288C22#8881512yes—
289C22#8881511yescharacteristic
290C4#8881311yescharacteristic
291C4#8881411yescharacteristic
292C4#8881412yes—
293C3—21no—
294C3—22no—
295C3—41no—
296C3#1058916511yescharacteristic, complemented
297C3#1058916612yescomplemented
298C2—22no—
299C2#1172208112yes—
300C2#1172208211yescharacteristic
subgroupG/H
centerC2×C4#6295
commutatorC33×C4C42
Frattini, hypercenterC22×C4#1113
Fitting, cosocleC22×C33×C4C22
socleC22×C33C4⋊C4
prime radicalC23×C33Q8
hypocenterC33#341
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
164. C33#341semidirect
261. C32#4552semidirect
262. C32#4558semidirect
296. C3#10589165semidirect
297. C3#10589166semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37482
C22×C3×C3⋊(C3⋊C8)2
#105891663
#105891653