Finite Group Databasebeta

← #13145732

Subgroups — #13145732

Order 1728.

650 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 155 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×#64135C212yesmaximal
2C2×C3×#3748C211yesmaximal, characteristic
3#10589158—32nomaximal
4#10589157—31nomaximal
5C22×C33×C4C2211yescharacteristic
6C3×#3757—22no—
7C3×C4⋊(C3⋊Dic3)—22no—
8C3×C4×C3⋊Dic3C414yescomplemented
9C3×#3748—22no—
10C2×C3×C3⋊(C3⋊C8)C412yescomplemented
11#64135D312yes—
12C2×#3748D311yescharacteristic
13C3×#926—32no—
14C3×#926—32no—
15C3×#926—62no—
16C2×C3×#208—32no—
17C2×C3×#208—61no—
18C23×C33D411yescharacteristic
19C2×C33×C4C2×C411yescharacteristic
20C2×C33×C4C2×C412yes—
21C2×C33×C4D412yes—
22C2×C33×C4Q811yescharacteristic
23C2×C3×C3⋊Dic3—24no—
24C3×C3⋊(C3⋊C8)—22no—
25#4557—91no—
26#4531—92no—
27C22×C32×C4—21no—
28C22×C32×C4—22no—
29C22×C32×C4—41no—
30C22×C32×C4C2×D312yes—
31C22×C32×C4C2×D311yescharacteristic
32#3757—22no—
33C4⋊(C3⋊Dic3)—22no—
34C4×C3⋊Dic3Dic314yescomplemented
35#3748—22no—
36C2×C3⋊(C3⋊C8)Dic312yescomplemented
37C3×#217—64no—
38C3×#217—62no—
39C3×#217—62no—
40C3×C4⋊Dic3—64no—
41C3×C4⋊Dic3—62no—
42C3×C4⋊Dic3—62no—
43C3×C4×Dic3—34no—
44C3×C4×Dic3—34no—
45C3×C4×Dic3—64no—
46C3×#208—64no—
47C3×#208—62no—
48C3×#208—62no—
49C2×C3×C3⋊C8—34no—
50C2×C3×C3⋊C8—62no—
51C22×C33—41no—
52C22×C33C22⋊C412yes—
53C22×C33C4⋊C411yescharacteristic
54C33×C4C4211yescharacteristic
55C33×C4C22⋊C411yescharacteristic
56C33×C4C4⋊C412yes—
57C3×C3⋊Dic3—44no—
58C2×C3×M16—91no—
59C3×#117—92no—
60#926—32no—
61#926—32no—
62#926—62no—
63C2×#208—32no—
64C2×#208—61no—
65C23×C32—21no—
66C23×C32—22no—
67C23×C32—41no—
68C23×C32D1211yescharacteristic
69C23×C32#6712yes—
70C2×C32×C4—22no—
71C2×C32×C4—22no—
72C2×C32×C4—22no—
73C2×C32×C4—21no—
74C2×C32×C4—22no—
75C2×C32×C4—22no—
76C2×C32×C4—22no—
77C2×C32×C4—22no—
78C2×C32×C4—21no—
79C2×C32×C4—22no—
80C2×C32×C4—41no—
81C2×C32×C4—42no—
82C2×C32×C4—42no—
83C2×C32×C4—41no—
84C2×C32×C4Dic611yescharacteristic
85C2×C32×C4Dic612yes—
86C2×C32×C4C4×D312yes—
87C2×C32×C4C4×D312yes—
88C2×C32×C4C4×D312yes—
89C2×C32×C4D1212yes—
90C2×C32×C4C2×Dic312yes—
91C2×C32×C4C2×Dic311yescharacteristic
92C2×C32×C4#6712yes—
93C2×C32×C4#6712yes—
94C2×C3⋊Dic3—24no—
95C2×C3×Dic3—68no—
96C2×C3×Dic3—64no—
97C2×C3×Dic3—64no—
98C3⋊(C3⋊C8)—22no—
99C3×C3⋊C8—64no—
100C3×C3⋊C8—64no—
101H(3,ℤ/4)—271noSylow
102C2×C33—22no—
103C2×C33—21no—
104C2×C33#9511yescharacteristic
105C22×C3×C4—21no—
106C22×C3×C4—22no—
107C22×C3×C4—41no—
108C22×C3×C4D3212yes—
109C22×C3×C4D3211yescharacteristic
110C3×M16—182no—
111C2×C3×C8—92no—
112C3×C4⋊C4—182no—
113C3×C22⋊C4—182no—
114C3×C42—94no—
115#217—64no—
116#217—62no—
117#217—62no—
118C4⋊Dic3—64no—
119C4⋊Dic3—62no—
120C4⋊Dic3—62no—
121C4×Dic3—34no—
122C4×Dic3—34no—
123C4×Dic3—64no—
124#208—64no—
125#208—62no—
126#208—62no—
127C2×C3⋊C8—34no—
128C2×C3⋊C8—62no—
129C22×C32—22no—
130C22×C32—22no—
131C22×C32—22no—
132C22×C32—22no—
133C22×C32—21no—
134C22×C32—42no—
135C22×C32—41no—
136C22×C32—41no—
137C22×C32—42no—
138C22×C32—42no—
139C22×C32—44no—
140C22×C32—44no—
141C22×C32Dic3⋊C412yes—
142C22×C32C4⋊Dic311yescharacteristic
143C22×C32(C2×D3)⋊C412yes—
144C22×C32(C2×D3)⋊C412yes—
145C22×C32#21712yes—
146C32×C4—21no—
147C32×C4—22no—
148C32×C4—22no—
149C32×C4—21no—
150C32×C4—22no—
151C32×C4—22no—
152C32×C4—22no—
153C32×C4—41no—
154C32×C4—41no—
155C32×C4—42no—
156C32×C4C4×Dic312yes—
157C32×C4C4×Dic311yescharacteristic
158C32×C4Dic3⋊C412yes—
159C32×C4Dic3⋊C412yes—
160C32×C4C4⋊Dic312yes—
161C32×C4(C2×D3)⋊C412yes—
162C32×C4#21711yescharacteristic
163C3⋊Dic3—44no—
164C3×Dic3—128no—
165C3×Dic3—124no—
166C3×Dic3—124no—
167C2×M16—91no—
168#117—92no—
169C33H(3,ℤ/4)11yescharacteristic, complemented, Sylow
170C23×C3—21no—
171C23×C3—22no—
172C23×C3—41no—
173C23×C3#62511yescharacteristic
174C23×C3#62612yes—
175C2×C3×C4—22no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—21no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—21no—
184C2×C3×C4—22no—
185C2×C3×C4—41no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—41no—
189C2×C3×C4—184no—
190C2×C3×C4D3×Dic312yes—
191C2×C3×C4D3×Dic312yes—
192C2×C3×C4D3×Dic312yes—
193C2×C3×C4#62411yescharacteristic
194C2×C3×C4#62412yes—
195C2×C3×C4#62512yes—
196C2×C3×C4#62612yes—
197C2×C3×C4#62612yes—
198C2×C3×C4#62711yescharacteristic
199C2×C3×C4#62712yes—
200C2×Dic3—68no—
201C2×Dic3—64no—
202C2×Dic3—64no—
203C3×C8—182no—
204C3⋊C8—64no—
205C3⋊C8—64no—
206C2×C32—22no—
207C2×C32—21no—
208C2×C32—24no—
209C2×C32—22no—
210C2×C32—22no—
211C2×C32—22no—
212C2×C32—22no—
213C2×C32—21no—
214C2×C32—22no—
215C2×C32—41no—
216C2×C32—42no—
217C2×C32—42no—
218C2×C32—42no—
219C2×C32—44no—
220C2×C32—42no—
221C2×C32C3⋊#9512yes—
222C2×C32C3⋊#9511yescharacteristic
223C22×C4#111311yescharacteristic
224M16—182no—
225C2×C8—92no—
226C4⋊C4—182no—
227C22⋊C4—182no—
228C42—94no—
229C22×C3—22no—
230C22×C3—22no—
231C22×C3—22no—
232C22×C3—21no—
233C22×C3—22no—
234C22×C3—41no—
235C22×C3—42no—
236C22×C3—44no—
237C22×C3—42no—
238C22×C3—44no—
239C22×C3—42no—
240C22×C3—41no—
241C22×C3(C2×D3)⋊Dic312yes—
242C22×C3(C2×D3)⋊Dic312yes—
243C22×C3#372212yes—
244C22×C3Dic3⋊Dic312yes—
245C22×C3#372411yescharacteristic
246C3×C4—21no—
247C3×C4—22no—
248C3×C4—21no—
249C3×C4—22no—
250C3×C4—22no—
251C3×C4—22no—
252C3×C4—22no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—42no—
256C3×C4—364no—
257C3×C4Dic3211yescharacteristic
258C3×C4Dic3212yes—
259C3×C4(C2×D3)⋊Dic312yes—
260C3×C4#372211yescharacteristic
261C3×C4Dic3⋊Dic312yes—
262C3×C4Dic3⋊Dic312yes—
263C3×C4#372412yes—
264Dic3—128no—
265Dic3—124no—
266Dic3—124no—
267C32—21no—
268C32—22no—
269C32—41no—
270C32#453112yescomplemented
271C32#455711yescharacteristic, complemented
272C23#629511yescharacteristic
273C2×C4—184no—
274C2×C4#629411yescharacteristic
275C2×C4#629412yes—
276C2×C4#629512yes—
277C2×C4#629611yescharacteristic
278C8—182no—
279C2×C3—24no—
280C2×C3—22no—
281C2×C3—22no—
282C2×C3—21no—
283C2×C3—22no—
284C2×C3—22no—
285C2×C3—22no—
286C2×C3—22no—
287C2×C3—21no—
288C2×C3—42no—
289C2×C3—42no—
290C2×C3—42no—
291C2×C3—44no—
292C2×C3—42no—
293C2×C3—41no—
294C2×C3#6357911yescharacteristic
295C2×C3#6357912yes—
296C22—41no—
297C22#8881412yes—
298C22#8881511yescharacteristic
299C4—364no—
300C4#8881311yescharacteristic
301C4#8881411yescharacteristic
302C4#8881512yes—
303C3—22no—
304C3—21no—
305C3—41no—
306C3#1058915711yescharacteristic, complemented
307C3#1058915812yescomplemented
308C2—22no—
309C2—21no—
310C2#1172208211yescharacteristic
subgroupG/H
centerC4#88813
commutatorC33×C4C42
Frattini, hypercenterC22×C4#1113
Fitting, cosocleC22×C33×C4C22
socleC2×C33#95
prime radicalC23×C33D4
hypocenterC33H(3,ℤ/4)
LCSG ▷ C33×C4 ▷ C33
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
8. C3×C4×C3⋊Dic3C4semidirect
10. C2×C3×C3⋊(C3⋊C8)C4semidirect
34. C4×C3⋊Dic3Dic3semidirect
36. C2×C3⋊(C3⋊C8)Dic3semidirect
169. C33H(3,ℤ/4)semidirect
270. C32#4531semidirect
271. C32#4557semidirect
306. C3#10589157semidirect
307. C3#10589158semidirect

Maximal subgroups

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

subgroupindex
C3×#641352
C2×C3×#37482
#105891583
#105891573