Finite Group Databasebeta

← #13145742

Subgroups — #13145742

Order 1728.

514 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 67 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×#64145C211yesmaximal, characteristic
2C3×#63661C212yesmaximal
3#10589181—31nomaximal
4#10589180—32nomaximal
5C2×C33×D4C2211yescharacteristic
6C2×C3×#634C411yescharacteristic, complemented
7C3×#3757—22no—
8C3×#3757—22no—
9C3×C4×C3⋊Dic3C411yescharacteristic, complemented
10#64145D311yescharacteristic
11C3×#936—32no—
12C3×#936—61no—
13#63661D312yes—
14C3×#834—32no—
15C3×#834—32no—
16C3×#834—62no—
17C23×C33D412yes—
18C33×D4—41no—
19C2×C33×C4C2×C411yescharacteristic
20C2×C3×C3⋊Dic3—21no—
21C2×C3×C3⋊Dic3—21no—
22C2×C3×C3⋊Dic3—42no—
23C3×#634—41no—
24#4541—91no—
25#4473—92no—
26C2×C32×D4—21no—
27C2×C32×D4—22no—
28C2×C32×D4—41no—
29C2×C32×D4C2×D312yes—
30C2×C32×D4C2×D311yescharacteristic
31C2×#634Dic311yescharacteristic, complemented
32C2×C3×Dic6—32no—
33C2×C3×Dic6—61no—
34#3757—22no—
35#3757—22no—
36C4×C3⋊Dic3Dic311yescharacteristic, complemented
37C3×#217—64no—
38C3×#217—62no—
39C3×#217—62no—
40C3×#217—64no—
41C3×#217—62no—
42C3×#217—62no—
43C3×C4×Dic3—32no—
44C3×C4×Dic3—61no—
45C22×C33—22no—
46C22×C33—42no—
47C22×C33C22⋊C411yescharacteristic
48C33×C4—21no—
49C3×C3⋊Dic3—41no—
50C3×C3⋊Dic3—41no—
51C3×C3⋊Dic3—82no—
52C3×#124—91no—
53#936—32no—
54#936—61no—
55C3×#99—92no—
56#834—32no—
57#834—32no—
58#834—62no—
59C23×C32—22no—
60C23×C32—22no—
61C23×C32—22no—
62C23×C32—42no—
63C23×C32D1212yes—
64C23×C32#6712yes—
65C23×C32#6712yes—
66C32×D4—41no—
67C32×D4—42no—
68C32×D4—42no—
69C32×D4—44no—
70C32×D4—44no—
71C2×C32×C4—22no—
72C2×C32×C4—21no—
73C2×C32×C4—41no—
74C2×C32×C4C4×D312yes—
75C2×C32×C4C2×Dic311yescharacteristic
76C2×C3⋊Dic3—21no—
77C2×C3⋊Dic3—21no—
78C2×C3⋊Dic3—42no—
79#634—41no—
80C2×C3×Dic3—62no—
81C2×C3×Dic3—62no—
82C2×C3×Dic3—62no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—124no—
85C2×C3×Dic3—122no—
86C2×C3×Dic3—122no—
87C3×Dic6—122no—
88C3×Dic6—122no—
89#354—271noSylow
90C2×C33—21no—
91C2×C33—42no—
92C2×C33#9911yescharacteristic
93C2×C3×Q8—91no—
94C2×C3×D4—22no—
95C2×C3×D4—21no—
96C2×C3×D4—41no—
97C2×C3×D4D3212yes—
98C2×C3×D4D3211yescharacteristic
99C2×Dic6—32no—
100C2×Dic6—61no—
101C3×C22⋊C4—182no—
102C3×C22⋊C4—182no—
103C3×C42—91no—
104#217—64no—
105#217—62no—
106#217—62no—
107#217—64no—
108#217—62no—
109#217—62no—
110C4×Dic3—32no—
111C4×Dic3—61no—
112C22×C32—22no—
113C22×C32—24no—
114C22×C32—22no—
115C22×C32—22no—
116C22×C32—22no—
117C22×C32—21no—
118C22×C32—41no—
119C22×C32—42no—
120C22×C32—42no—
121C22×C32—44no—
122C22×C32—42no—
123C22×C32—42no—
124C22×C32—42no—
125C22×C32—44no—
126C22×C32—82no—
127C22×C32—82no—
128C22×C32—84no—
129C22×C32(C2×D3)⋊C412yes—
130C22×C32#21711yescharacteristic
131C32×C4—21no—
132C32×C4—22no—
133C32×C4—22no—
134C32×C4—42no—
135C32×C4—42no—
136C3⋊Dic3—41no—
137C3⋊Dic3—41no—
138C3⋊Dic3—82no—
139C3×Dic3—122no—
140C3×Dic3—122no—
141C3×Dic3—122no—
142C3×Dic3—122no—
143C3×Dic3—244no—
144C3×Dic3—242no—
145C3×Dic3—242no—
146#124—91no—
147#99—92no—
148C33#35411yescharacteristic, complemented, Sylow
149C23×C3—22no—
150C23×C3—22no—
151C23×C3—22no—
152C23×C3—42no—
153C23×C3#62512yes—
154C23×C3#62612yes—
155C23×C3#62612yes—
156C3×Q8—361no—
157C3×D4—42no—
158C3×D4—41no—
159C3×D4—42no—
160C3×D4—44no—
161C3×D4—44no—
162C2×C3×C4—21no—
163C2×C3×C4—22no—
164C2×C3×C4—41no—
165C2×C3×C4—181no—
166C2×C3×C4—181no—
167C2×C3×C4—362no—
168C2×C3×C4D3×Dic312yes—
169C2×C3×C4#62411yescharacteristic
170C2×Dic3—62no—
171C2×Dic3—62no—
172C2×Dic3—62no—
173C2×Dic3—62no—
174C2×Dic3—124no—
175C2×Dic3—122no—
176C2×Dic3—122no—
177Dic6—122no—
178Dic6—122no—
179C2×C32—21no—
180C2×C32—22no—
181C2×C32—22no—
182C2×C32—22no—
183C2×C32—21no—
184C2×C32—41no—
185C2×C32—42no—
186C2×C32—42no—
187C2×C32—42no—
188C2×C32—44no—
189C2×C32—42no—
190C2×C32—42no—
191C2×C32—82no—
192C2×C32—82no—
193C2×C32—84no—
194C2×C32#80612yes—
195C2×C32#83411yescharacteristic
196C2×Q8—91no—
197C2×D4#111311yescharacteristic
198C22⋊C4—182no—
199C22⋊C4—182no—
200C42—91no—
201C22×C3—24no—
202C22×C3—22no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3—21no—
206C22×C3—22no—
207C22×C3—42no—
208C22×C3—42no—
209C22×C3—44no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3—44no—
214C22×C3—41no—
215C22×C3—82no—
216C22×C3—82no—
217C22×C3—84no—
218C22×C3(C2×D3)⋊Dic312yes—
219C22×C3#372211yescharacteristic
220C3×C4—22no—
221C3×C4—22no—
222C3×C4—21no—
223C3×C4—42no—
224C3×C4—42no—
225C3×C4—361no—
226C3×C4—361no—
227C3×C4—722no—
228Dic3—122no—
229Dic3—122no—
230Dic3—122no—
231Dic3—122no—
232Dic3—244no—
233Dic3—242no—
234Dic3—242no—
235C32—22no—
236C32—21no—
237C32—41no—
238C32#447312yescomplemented
239C32#454111yescharacteristic, complemented
240C23#629512yes—
241Q8—361no—
242D4—41no—
243C2×C4—181no—
244C2×C4—181no—
245C2×C4—362no—
246C2×C4#629411yescharacteristic
247C2×C3—22no—
248C2×C3—22no—
249C2×C3—21no—
250C2×C3—21no—
251C2×C3—22no—
252C2×C3—42no—
253C2×C3—42no—
254C2×C3—44no—
255C2×C3—42no—
256C2×C3—42no—
257C2×C3—42no—
258C2×C3—41no—
259C2×C3—82no—
260C2×C3—82no—
261C2×C3—84no—
262C2×C3#6358012yes—
263C2×C3#6358111yescharacteristic
264C22—22no—
265C22—42no—
266C22#8881411yescharacteristic
267C4—21no—
268C4—361no—
269C4—361no—
270C4—722no—
271C3—21no—
272C3—22no—
273C3—41no—
274C3#1058918012yescomplemented
275C3#1058918111yescharacteristic, complemented
276C2—21no—
277C2—42no—
278C2#1172208311yescharacteristic
subgroupG/H
centerC2#11722083
commutatorC2×C33×C4C2×C4
Frattini, hypercenterC2×D4#1113
Fitting, prime radical, cosocleC2×C33×D4C22
socleC2×C33#99
hypocenterC33#354
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
6. C2×C3×#634C4semidirect
9. C3×C4×C3⋊Dic3C4semidirect
31. C2×#634Dic3semidirect
36. C4×C3⋊Dic3Dic3semidirect
148. C33#354semidirect
238. C32#4473semidirect
239. C32#4541semidirect
274. C3#10589180semidirect
275. C3#10589181semidirect

Maximal subgroups

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

subgroupindex
C3×#641452
C3×#636612
#105891813
#105891803