Finite Group Databasebeta

← C2×C3×D3×C42⋊C3 #13173083

Subgroups — C2×C3×D3×C42⋊C3 #13173083

Order 1728.

556 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C2×C32×C42⋊C3C211yesmaximal, characteristic, complemented
2C3×D3×C42⋊C3C212yesmaximal, direct product factor
3C2×C3×C42×D3C311yesmaximal, characteristic, complemented
4C22×C3×C42⋊C3—31nomaximal
5C2×D3×C42⋊C3C313yesmaximal, direct product factor
6C2×C3×D3×A4—41nomaximal
7C32×C42⋊C3C2211yescharacteristic, complemented
8C22×C3×C4×D3—31no—
9C2×C32×C42C2×C311yescharacteristic, complemented
10C2×C3×C4×Dic3—31no—
11C3×C42×D3—32no—
12C3×C42×D3C2×C312yescomplemented
13C2×C3×C42⋊C3—21no—
14C2×C3×C42⋊C3—23no—
15C2×C3×C42⋊C3—32no—
16C2×C3×C42⋊C3D311yescharacteristic, direct product factor
17C2×C3×C42⋊C3C2×C313yescomplemented
18D3×C42⋊C3C2×C316yesdirect product factor
19C2×C32×A4—41no—
20C3×D3×A4—42no—
21C22×C3×C42—31no—
22C2×C42×D3C3211yescharacteristic, complemented
23C22×C42⋊C3—33no—
24C23×C3×D3A411yescharacteristic
25C22×C3×A4—121no—
26C2×D3×A4—43no—
27C22×C32×C4—31no—
28C22×C3×Dic3—31no—
29C2×C3×C4×D3—32no—
30C2×C3×C4×D3—34no—
31C2×C3×C4×D3—32no—
32C2×C3×C4×D3—32no—
33C2×C3×C4×D3—32no—
34C32×C42—31no—
35C32×C42C22×C311yescharacteristic, complemented
36C3×C4×Dic3—32no—
37C3×C4×Dic3—32no—
38C3×C42⋊C3—21no—
39C3×C42⋊C3—23no—
40C3×C42⋊C3C2×D311yescharacteristic, direct product factor
41C3×C42⋊C3C22×C313yescomplemented
42C2×C32×D3—161no—
43C32×A4—41no—
44C23×C3×C4—91no—
45C22×C4×D3—31no—
46C2×C3×C42—21no—
47C2×C3×C42—32no—
48C2×C3×C42—92no—
49C2×C3×C42—91no—
50C2×C3×C42C3×D311yescharacteristic, complemented
51C2×C3×C42C2×C3211yescharacteristic, complemented
52C2×C4×Dic3—31no—
53C42×D3—32no—
54C42×D3C2×C3212yescomplemented
55C2×C42⋊C3—23no—
56C2×C42⋊C3—36no—
57C2×C42⋊C3C3×D313yesdirect product factor
58C23×C32C2×A411yescharacteristic
59C22×C3×D3—31no—
60C22×C3×D3—31no—
61C22×C3×D3—32no—
62C22×C3×D3C2×A412yes—
63C2×C3×A4—41no—
64C2×C3×A4—43no—
65C2×C3×A4—83no—
66C2×C3×A4—81no—
67C2×C3×A4—122no—
68D3×A4—46no—
69C2×C32×C4—32no—
70C2×C32×C4—32no—
71C2×C32×C4—31no—
72C2×C32×C4—31no—
73C2×C3×Dic3—32no—
74C2×C3×Dic3—32no—
75C2×C3×Dic3—32no—
76C3×C4×D3—34no—
77C3×C4×D3—34no—
78C3×C4×D3—34no—
79C3×C4×D3—34no—
80C22×C42—31noSylow
81C2×C33—161no—
82C32×D3—162no—
83C24×C3—31no—
84C23×D3C3×A411yescharacteristic
85C22×A4—123no—
86C22×C3×C4—31no—
87C22×C3×C4—31no—
88C22×C3×C4—61no—
89C22×C3×C4—92no—
90C22×C3×C4—92no—
91C22×C3×C4—92no—
92C22×C3×C4—94no—
93C22×C3×C4—92no—
94C22×C3×C4—91no—
95C22×Dic3—31no—
96C2×C4×D3—32no—
97C2×C4×D3—34no—
98C2×C4×D3—32no—
99C2×C4×D3—32no—
100C2×C4×D3—32no—
101C3×C42—21no—
102C3×C42—31no—
103C3×C42—31no—
104C3×C42—61no—
105C3×C42—92no—
106C3×C42—92no—
107C3×C42C2×C3×D311yescharacteristic, complemented
108C3×C42C22×C3211yescharacteristic, complemented
109C4×Dic3—32no—
110C4×Dic3—32no—
111C42⋊C3—23no—
112C42⋊C3C2×C3×D313yesdirect product factor
113C22×C32—31no—
114C22×C32—31no—
115C22×C32—481no—
116C22×C32C22×A411yescharacteristic
117C2×C3×D3—32no—
118C2×C3×D3—31no—
119C2×C3×D3—32no—
120C2×C3×D3—32no—
121C2×C3×D3—32no—
122C2×C3×D3—163no—
123C2×C3×D3C42⋊C311yescharacteristic, direct product factor
124C3×A4—41no—
125C3×A4—43no—
126C3×A4—83no—
127C3×A4—81no—
128C32×C4—32no—
129C32×C4—32no—
130C3×Dic3—34no—
131C23×C4—91no—
132C2×C42—32no—
133C2×C42—92no—
134C2×C42—91no—
135C2×C42C32×D311yescharacteristic, complemented
136C33—161noSylow
137C23×C3—21no—
138C23×C3—32no—
139C23×C3—91no—
140C23×C3—92no—
141C23×C3—91no—
142C23×C3D3×A411yescharacteristic
143C23×C3C2×C3×A411yescharacteristic
144C22×D3—31no—
145C22×D3—31no—
146C22×D3—32no—
147C22×D3C2×C3×A412yes—
148C2×A4—43no—
149C2×A4—83no—
150C2×A4—126no—
151C2×C3×C4—32no—
152C2×C3×C4—32no—
153C2×C3×C4—32no—
154C2×C3×C4—32no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—31no—
159C2×C3×C4—61no—
160C2×C3×C4—61no—
161C2×C3×C4—62no—
162C2×C3×C4—62no—
163C2×C3×C4—94no—
164C2×C3×C4—94no—
165C2×C3×C4—94no—
166C2×C3×C4—94no—
167C2×C3×C4—92no—
168C2×C3×C4—92no—
169C2×C3×C4—92no—
170C2×Dic3—32no—
171C2×Dic3—32no—
172C2×Dic3—32no—
173C4×D3—34no—
174C4×D3—34no—
175C4×D3—34no—
176C4×D3—34no—
177C2×C32—31no—
178C2×C32—31no—
179C2×C32—161no—
180C2×C32—163no—
181C2×C32—323no—
182C2×C32—321no—
183C2×C32—482no—
184C2×C32C2×C42⋊C311yescharacteristic, complemented
185C3×D3—32no—
186C3×D3—166no—
187C3×D3C2×C42⋊C312yesdirect product factor
188C24—31no—
189C22×C4—31no—
190C22×C4—92no—
191C22×C4—92no—
192C22×C4—92no—
193C22×C4—94no—
194C22×C4—92no—
195C22×C4—91no—
196C42—31no—
197C42—92no—
198C42—92no—
199C42C2×C32×D311yescharacteristic, complemented
200C22×C3—21no—
201C22×C3—31no—
202C22×C3—31no—
203C22×C3—31no—
204C22×C3—31no—
205C22×C3—31no—
206C22×C3—61no—
207C22×C3—61no—
208C22×C3—92no—
209C22×C3—92no—
210C22×C3—92no—
211C22×C3—92no—
212C22×C3—91no—
213C22×C3—483no—
214C22×C3C2×D3×A411yescharacteristic
215C22×C3C22×C3×A411yescharacteristic
216C2×D3—32no—
217C2×D3—31no—
218C2×D3—32no—
219C2×D3—32no—
220C2×D3—32no—
221C2×D3C3×C42⋊C311yescharacteristic, direct product factor
222A4—43no—
223A4—83no—
224C3×C4—32no—
225C3×C4—32no—
226C3×C4—32no—
227C3×C4—32no—
228C3×C4—62no—
229C3×C4—62no—
230C3×C4—94no—
231Dic3—34no—
232C32—161no—
233C32—163no—
234C32—323no—
235C32—321no—
236C32C22×C42⋊C311yescharacteristic, complemented
237C23—32no—
238C23—91no—
239C23—92no—
240C23—91no—
241C23C3×D3×A411yescharacteristic
242C2×C4—32no—
243C2×C4—32no—
244C2×C4—31no—
245C2×C4—31no—
246C2×C4—94no—
247C2×C4—94no—
248C2×C4—94no—
249C2×C4—94no—
250C2×C4—92no—
251C2×C4—92no—
252C2×C4—92no—
253C2×C3—21no—
254C2×C3—31no—
255C2×C3—31no—
256C2×C3—32no—
257C2×C3—31no—
258C2×C3—31no—
259C2×C3—61no—
260C2×C3—61no—
261C2×C3—92no—
262C2×C3—163no—
263C2×C3—323no—
264C2×C3—486no—
265C2×C3D3×C42⋊C311yescharacteristic, direct product factor
266C2×C3C2×C3×C42⋊C311yescharacteristic, complemented
267D3—32no—
268D3C2×C3×C42⋊C312yesdirect product factor
269C22—31no—
270C22—31no—
271C22—31no—
272C22—92no—
273C22—92no—
274C22—92no—
275C22—92no—
276C22—91no—
277C22C2×C3×D3×A411yescharacteristic
278C4—32no—
279C4—32no—
280C4—94no—
281C3—21no—
282C3—163no—
283C3—323no—
284C3C2×D3×C42⋊C311yescharacteristic, direct product factor
285C3C22×C3×C42⋊C311yescharacteristic, complemented
286C2—31no—
287C2—31no—
288C2—32no—
289C2—92no—
290C2C3×D3×C42⋊C311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3D3×C42⋊C3
commutator, cosocle, hypocenterC3×C42C22×C32
FrattiniC22C2×C3×D3×A4
FittingC2×C32×C42C2×C3
socleC23×C32C2×A4
LCSG ▷ C3×C42
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C32×C42⋊C3C2semidirect
2. C3×D3×C42⋊C3C2direct (N × H)
3. C2×C3×C42×D3C3semidirect
5. C2×D3×C42⋊C3C3direct (N × H)
7. C32×C42⋊C3C22semidirect
9. C2×C32×C42C2×C3semidirect
12. C3×C42×D3C2×C3semidirect
16. C2×C3×C42⋊C3D3direct (N × H)
17. C2×C3×C42⋊C3C2×C3semidirect
18. D3×C42⋊C3C2×C3direct (N × H)
22. C2×C42×D3C32semidirect
35. C32×C42C22×C3semidirect
40. C3×C42⋊C3C2×D3direct (N × H)
41. C3×C42⋊C3C22×C3semidirect
50. C2×C3×C42C3×D3semidirect
51. C2×C3×C42C2×C32semidirect
54. C42×D3C2×C32semidirect
57. C2×C42⋊C3C3×D3direct (N × H)
107. C3×C42C2×C3×D3semidirect
108. C3×C42C22×C32semidirect
112. C42⋊C3C2×C3×D3direct (N × H)
123. C2×C3×D3C42⋊C3direct (N × H)
135. C2×C42C32×D3semidirect
184. C2×C32C2×C42⋊C3semidirect
187. C3×D3C2×C42⋊C3direct (N × H)
199. C42C2×C32×D3semidirect
221. C2×D3C3×C42⋊C3direct (N × H)
236. C32C22×C42⋊C3semidirect
265. C2×C3D3×C42⋊C3direct (N × H)
266. C2×C3C2×C3×C42⋊C3semidirect
268. D3C2×C3×C42⋊C3direct (N × H)
284. C3C2×D3×C42⋊C3direct (N × H)
285. C3C22×C3×C42⋊C3semidirect
290. C2C3×D3×C42⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C42⋊C32
C3×D3×C42⋊C32
C2×C3×C42×D33
C22×C3×C42⋊C33
C2×D3×C42⋊C33
C2×C3×D3×A44