Finite Group Databasebeta

← Dic3×#1200 #13085851

Subgroups — Dic3×#1200 #13085851

Order 1440.

462 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 140 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×#67154C211yesmaximal, characteristic
2C2×Dic3×D15C211yesmaximal, characteristic
3C2×C3×#1200C211yesmaximal, characteristic
4Dic3×Dic15C211yesmaximal, characteristic, complemented
5C5×Dic32C211yesmaximal, characteristic, complemented
6Dic5×C3⋊Dic3C211yesmaximal, characteristic, complemented
7C3×Dic3×Dic5C211yesmaximal, characteristic, complemented
8C4×#1200—31nomaximal
9C4×D5×Dic3—31nomaximal
10C22×C3×D15C2211yescharacteristic
11C2×C3⋊Dic15C2211yescharacteristic
12C2×C5×C3⋊Dic3C2211yescharacteristic
13C2×C3×Dic15C2211yescharacteristic
14C2×C3×C5×Dic3C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C32×Dic5C2211yescharacteristic
17#67154C414yescomplemented
18Dic3×D15C414yescomplemented
19C3×#1200C414yescomplemented
20C4×D3×Dic3—51nomaximal, Hall
21C2×C4×D15—31no—
22C2×C3×C4×D5—31no—
23C2×#1200—31no—
24C2×#1200D311yescharacteristic
25C2×D5×Dic3—31no—
26C2×D5×Dic3—31no—
27C4×Dic15—31no—
28C4×C5×Dic3—31no—
29C4×C5×Dic3—31no—
30C3×C4×Dic5—31no—
31Dic3×Dic5—31no—
32Dic3×Dic5—31no—
33Dic3×Dic5—31no—
34Dic3×Dic5—61no—
35Dic3×Dic5D311yescharacteristic, complemented
36C22×C32×C5C2311yescharacteristic
37C2×C3×D15C2×C412yes—
38C2×C3×D15C2×C412yes—
39C2×C3×D15C2×C412yes—
40C3⋊Dic15C2×C412yescomplemented
41C5×C3⋊Dic3C2×C412yescomplemented
42C3×Dic15C2×C412yescomplemented
43C3×C5×Dic3C2×C412yescomplemented
44C3×C5×Dic3C2×C412yescomplemented
45C32×Dic5C2×C412yescomplemented
46C42×D5—91noHall
47C2×C3×C4×D3—51no—
48C2×D3×Dic3—51no—
49C2×D3×Dic3—51no—
50C4×C3⋊Dic3—51no—
51C3×C4×Dic3—51no—
52Dic32—51no—
53Dic32D511yescharacteristic, complemented
54C22×D15C2×D311yescharacteristic
55C22×C3×D5—31no—
56C2×C3×C4×C5—31no—
57C2×C3×C4×C5—31no—
58C2×Dic15—31no—
59C2×Dic15—31no—
60C2×Dic15—61no—
61C2×Dic15C2×D311yescharacteristic
62C4×D15—34no—
63C2×C5×Dic3—31no—
64C2×C5×Dic3—31no—
65C2×C5×Dic3—61no—
66C2×C5×Dic3C2×D311yescharacteristic
67C2×C5×Dic3C2×D311yescharacteristic
68C2×C3×Dic5—21no—
69C2×C3×Dic5—31no—
70C2×C3×Dic5C2×D311yescharacteristic
71C2×C3×Dic5C2×D311yescharacteristic
72C3×C4×D5—34no—
73#1200—34no—
74#1200Dic314yesdirect product factor
75D5×Dic3—34no—
76D5×Dic3—34no—
77C2×C4×Dic3—151no—
78C42×D3—151no—
79C2×C32×C5C22×C411yescharacteristic
80C2×C32×C5C22×C411yescharacteristic
81C2×C32×C5C22×C411yescharacteristic
82C3×D15C4214yescomplemented
83C2×C4×D5—31no—
84C2×C4×D5—91no—
85C2×C4×D5—91no—
86C42×C5—91no—
87C4×Dic5—31no—
88C4×Dic5—91no—
89C4×Dic5—91no—
90C22×C3×D3—51no—
91C2×C32×C4—51no—
92C2×C3⋊Dic3—51no—
93C2×C3⋊Dic3C2×D511yescharacteristic
94C2×C3×Dic3—51no—
95C2×C3×Dic3C2×D511yescharacteristic
96C2×C3×Dic3C2×D511yescharacteristic
97C3×C4×D3—54no—
98D3×Dic3—54no—
99D3×Dic3—54no—
100C22×C3×C5—21no—
101C22×C3×C5C22×D311yescharacteristic
102C22×C3×C5C22×D311yescharacteristic
103C2×D15C4×D312yes—
104C2×D15C4×D312yes—
105C2×D15C2×Dic312yes—
106C2×C3×D5—32no—
107C2×C3×D5—32no—
108C2×C3×D5—32no—
109C3×C4×C5—32no—
110C3×C4×C5—32no—
111Dic15—32no—
112Dic15—32no—
113Dic15—62no—
114Dic15C4×D312yescomplemented
115C3×Dic5—22no—
116C3×Dic5—32no—
117C3×Dic5C4×D312yescomplemented
118C3×Dic5C2×Dic312yescomplemented
119C5×Dic3—32no—
120C5×Dic3—32no—
121C5×Dic3—62no—
122C5×Dic3C4×D312yescomplemented
123C5×Dic3C2×Dic312yescomplemented
124C22×C3×C4—151no—
125C22×Dic3—151no—
126C22×Dic3—151no—
127C2×C4×D3—51no—
128C2×C4×D3—151no—
129C2×C4×D3—151no—
130C3×C42—151no—
131C4×Dic3—31no—
132C4×Dic3—31no—
133C4×Dic3—51no—
134C4×Dic3—151no—
135C4×Dic3—151no—
136C4×Dic3—151no—
137C4×Dic3—151no—
138C4×Dic3—301no—
139C32×C5C2×C4211yescharacteristic, complemented, Hall
140C22×D5—31no—
141C2×C4×C5—31no—
142C2×C4×C5—31no—
143C2×C4×C5—91no—
144C2×Dic5—31no—
145C2×Dic5—91no—
146C2×Dic5D3211yescharacteristic
147C4×D5—34no—
148C4×D5—94no—
149C4×D5—94no—
150C22×C32C22×D511yescharacteristic
151C2×C3×D3—52no—
152C2×C3×D3—52no—
153C2×C3×D3—52no—
154C32×C4—52no—
155C3⋊Dic3—52no—
156C3⋊Dic3C4×D512yescomplemented
157C3×Dic3—52no—
158C3×Dic3C4×D512yescomplemented
159C3×Dic3C4×D512yescomplemented
160C2×C42—451noSylow
161C2×C3×C5—21no—
162C2×C3×C5—21no—
163C2×C3×C5—21no—
164C2×C3×C5C2×C4×D311yescharacteristic
165C2×C3×C5C2×C4×D311yescharacteristic
166C2×C3×C5C2×C4×D311yescharacteristic
167C2×C3×C5C2×C4×D311yescharacteristic
168C2×C3×C5C2×C4×D311yescharacteristic
169C2×C3×C5C22×Dic311yescharacteristic
170D15C4×Dic314yescomplemented
171C3×D5—34no—
172C23×C3—151no—
173C22×D3—51no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—51no—
177C2×C3×C4—51no—
178C2×C3×C4—101no—
179C2×C3×C4—154no—
180C2×C3×C4—151no—
181C2×Dic3—31no—
182C2×Dic3—31no—
183C2×Dic3—51no—
184C2×Dic3—61no—
185C2×Dic3—154no—
186C2×Dic3—151no—
187C2×Dic3—151no—
188C2×Dic3—154no—
189C2×Dic3—301no—
190C2×Dic3D3×D511yescharacteristic
191C2×Dic3D3×D511yescharacteristic
192C4×D3—54no—
193C4×D3—154no—
194C4×D3—154no—
195C22×C5C2×D3211yescharacteristic
196C2×D5—32no—
197C2×D5—32no—
198C2×D5—32no—
199C4×C5—32no—
200C4×C5—32no—
201C4×C5—92no—
202Dic5—32no—
203Dic5—92no—
204Dic5D3×Dic312yescomplemented
205C2×C32C2×C4×D511yescharacteristic
206C2×C32C2×C4×D511yescharacteristic
207C2×C32C2×C4×D511yescharacteristic
208C3×D3—54no—
209C22×C4—151no—
210C22×C4—451no—
211C22×C4—451no—
212C42—91no—
213C42—151no—
214C42—451no—
215C42—451no—
216C3×C5—21no—
217C3×C5C42×D311yescharacteristic, complemented
218C3×C5C2×C4×Dic311yescharacteristic, complemented
219C22×C3—21no—
220C22×C3—152no—
221C22×C3—152no—
222C22×C3—152no—
223C22×C3C2×D3×D511yescharacteristic
224C22×C3C2×D3×D511yescharacteristic
225C2×D3—52no—
226C2×D3—52no—
227C2×D3—52no—
228C3×C4—32no—
229C3×C4—32no—
230C3×C4—52no—
231C3×C4—52no—
232C3×C4—102no—
233C3×C4—152no—
234Dic3—32no—
235Dic3—32no—
236Dic3—52no—
237Dic3—62no—
238Dic3—152no—
239Dic3—152no—
240Dic3—302no—
241Dic3D5×Dic312yescomplemented
242Dic3#120012yesdirect product factor
243C2×C5C4×D3211yescharacteristic
244C2×C5C4×D3211yescharacteristic
245C2×C5C2×D3×Dic311yescharacteristic
246D5—34no—
247C32C42×D511yescharacteristic, complemented, Sylow
248C23—151no—
249C2×C4—31no—
250C2×C4—31no—
251C2×C4—51no—
252C2×C4—91no—
253C2×C4—154no—
254C2×C4—151no—
255C2×C4—454no—
256C2×C4—454no—
257C2×C4—451no—
258C2×C3—21no—
259C2×C3—21no—
260C2×C3—21no—
261C2×C3—154no—
262C2×C3C4×D3×D511yescharacteristic
263C2×C3C4×D3×D511yescharacteristic
264C2×C3C4×D3×D511yescharacteristic
265C2×C3C4×D3×D511yescharacteristic
266C2×C3C2×D5×Dic311yescharacteristic
267C2×C3C2×#120011yescharacteristic
268D3—54no—
269C5C4×D3×Dic311yescharacteristic, complemented, Sylow
270C22—152no—
271C22—152no—
272C22—152no—
273C22D32×D511yescharacteristic
274C4—32no—
275C4—32no—
276C4—52no—
277C4—92no—
278C4—152no—
279C4—452no—
280C3—21no—
281C3C4×D5×Dic311yescharacteristic, complemented
282C3C4×#120011yescharacteristic, complemented
283C2—154no—
284C2D3×D5×Dic311yescharacteristic
285C2D3×#120011yescharacteristic
286C2#1062402211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×D5
commutator, hypocenterC32×C5C2×C42
Fitting, socle, cosocleC22×C32×C5C23
prime radicalC22×C3×D15C22
LCSG ▷ C32×C5
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
4. Dic3×Dic15C2semidirect
5. C5×Dic32C2semidirect
6. Dic5×C3⋊Dic3C2semidirect
7. C3×Dic3×Dic5C2semidirect
17. #67154C4semidirect
18. Dic3×D15C4semidirect
19. C3×#1200C4semidirect
35. Dic3×Dic5D3semidirect
40. C3⋊Dic15C2×C4semidirect
41. C5×C3⋊Dic3C2×C4semidirect
42. C3×Dic15C2×C4semidirect
43. C3×C5×Dic3C2×C4semidirect
44. C3×C5×Dic3C2×C4semidirect
45. C32×Dic5C2×C4semidirect
53. Dic32D5semidirect
74. #1200Dic3direct (N × H)
82. C3×D15C42semidirect
114. Dic15C4×D3semidirect
117. C3×Dic5C4×D3semidirect
118. C3×Dic5C2×Dic3semidirect
122. C5×Dic3C4×D3semidirect
123. C5×Dic3C2×Dic3semidirect
139. C32×C5C2×C42semidirect
156. C3⋊Dic3C4×D5semidirect
158. C3×Dic3C4×D5semidirect
159. C3×Dic3C4×D5semidirect
170. D15C4×Dic3semidirect
204. Dic5D3×Dic3semidirect
217. C3×C5C42×D3semidirect
218. C3×C5C2×C4×Dic3semidirect
241. Dic3D5×Dic3semidirect
242. Dic3#1200direct (N × H)
247. C32C42×D5semidirect
269. C5C4×D3×Dic3semidirect
281. C3C4×D5×Dic3semidirect
282. C3C4×#1200semidirect

Maximal subgroups

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

subgroupindex
C2×#671542
C2×Dic3×D152
C2×C3×#12002
Dic3×Dic152
C5×Dic322
Dic5×C3⋊Dic32
C3×Dic3×Dic52
C4×#12003
C4×D5×Dic33
C4×D3×Dic35