Finite Group Databasebeta

← Dic3×#1220 #13087562

Subgroups — Dic3×#1220 #13087562

Order 1440.

410 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 114 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×C3×#1220C211yesmaximal, characteristic
2C22×C3×C5×Dic3C211yesmaximal, characteristic, complemented
3C2×Dic3×D15C211yesmaximal, characteristic, complemented
4#10623913C211yesmaximal, characteristic, complemented
5#10623814C211yesmaximal, characteristic, complemented
6#10623811C211yesmaximal, characteristic, complemented
7Dic3×Dic15C211yesmaximal, characteristic, complemented
8C4×#1220—31nomaximal
9Dic3×C22⋊D5—31nomaximal
10C23×C32×C5C2211yescharacteristic
11C22×C3×D15C2211yescharacteristic
12C2×C3⋊Dic15C2211yescharacteristic
13C2×C3⋊Dic15C2211yescharacteristic
14C3×#1220C414yescomplemented
15C2×C3×Dic15C2211yescharacteristic
16C2×C3×C5×Dic3—22no—
17C2×C3×C5×Dic3C2211yescharacteristic
18C2×C3×C5×Dic3C2211yescharacteristic
19Dic3×D15—22no—
20Dic3×#67—51nomaximal, Hall
21C22×C3×C4×C5—31no—
22C2×#1220D311yescharacteristic
23C2×C4×D15—31no—
24C22×C5×Dic3D311yescharacteristic, complemented
25C2×C3×C22⋊D5—31no—
26C2×D5×Dic3—31no—
27#6719—31no—
28#6719—31no—
29#6719—61no—
30#6714—31no—
31#6712—31no—
32C4×Dic15—31no—
33#6668—31no—
34#6665—31no—
35Dic3×Dic5—31no—
36C22×C32×C5—21no—
37C22×C32×C5—21no—
38C22×C32×C5C2×C412yescomplemented
39C22×C32×C5C2311yescharacteristic
40C2×C3×D15—21no—
41C2×C3×D15—21no—
42C2×C3×D15C2×C412yescomplemented
43C3⋊Dic15—21no—
44C3⋊Dic15—21no—
45C3×Dic15C2×C412yescomplemented
46C3×C5×Dic3—21no—
47C3×C5×Dic3D412yescomplemented
48C4×C22⋊D5—91noHall
49C2×C3×#67—51no—
50C22×C3×Dic3D511yescharacteristic, complemented
51C2×D3×Dic3—51no—
52#3757—51no—
53Dic3⋊Dic3—51no—
54(C2×D3)⋊Dic3—51no—
55Dic32—51no—
56C23×C3×C5—21no—
57C23×C3×C5C2×D311yescharacteristic
58C23×C3×C5C2×D311yescharacteristic
59C22×D15C2×D311yescharacteristic
60C22×C3×D5—31no—
61C2×C3×C4×C5—31no—
62C2×C3×C4×C5—31no—
63C2×C3×C4×C5—62no—
64#1220Dic314yesdirect product factor
65C2×Dic15—31no—
66C2×Dic15—31no—
67C2×Dic15—31no—
68C2×Dic15—31no—
69C2×Dic15—61no—
70C2×Dic15—61no—
71C2×Dic15C2×D311yescharacteristic
72C4×D15—62no—
73C2×C5×Dic3—22no—
74C2×C5×Dic3C2×D311yescharacteristic
75C2×C5×Dic3C2×D311yescharacteristic
76C3×C22⋊D5—34no—
77C2×C3×Dic5—31no—
78D5×Dic3—62no—
79D4×Dic3—151no—
80C4×#67—151no—
81C2×C32×C5—22no—
82C2×C32×C5C22×C411yescharacteristic
83C2×C32×C5C2×D411yescharacteristic
84C2×C32×C5#4111yescharacteristic
85C3×D15—22no—
86C22×C4×C5—31no—
87C2×C22⋊D5—31no—
88C2×C4×D5—91no—
89C22⋊Dic5—91no—
90#685—91no—
91#683—91no—
92C4×Dic5—91no—
93C23×C32C2×D511yescharacteristic
94C22×C3×D3—51no—
95C2×C3⋊Dic3—51no—
96C2×C3⋊Dic3—51no—
97C3×#67—54no—
98C2×C3×Dic3—22no—
99C2×C3×Dic3—51no—
100C2×C3×Dic3C2×D511yescharacteristic
101C2×C3×Dic3C2×D511yescharacteristic
102D3×Dic3—102no—
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5—21no—
106C22×C3×C5—21no—
107C22×C3×C5—22no—
108C22×C3×C5—21no—
109C22×C3×C5—41no—
110C22×C3×C5—41no—
111C22×C3×C5C4×D312yescomplemented
112C22×C3×C5C2×Dic312yescomplemented
113C22×C3×C5C22×D311yescharacteristic
114C22×C3×C5C22×D311yescharacteristic
115C2×D15—21no—
116C2×D15—21no—
117C2×D15C2×Dic312yescomplemented
118C2×C3×D5—32no—
119C2×C3×D5—61no—
120C2×C3×D5—61no—
121C3×C4×C5—32no—
122C3×C4×C5—61no—
123Dic15—61no—
124Dic15—61no—
125Dic15—61no—
126Dic15—61no—
127Dic15—121no—
128Dic15—121no—
129Dic15C2×Dic312yescomplemented
130C3×Dic5—32no—
131C5×Dic3—21no—
132C5×Dic3#6712yescomplemented
133C2×C3×D4—151no—
134C22×C3×C4—31no—
135C2×#67—51no—
136C22×Dic3—151no—
137C22×Dic3D1511yescharacteristic, complemented
138C2×C4×D3—151no—
139#217—151no—
140#217—151no—
141#217—151no—
142#217—301no—
143(C2×D3)⋊C4—151no—
144C4⋊Dic3—151no—
145Dic3⋊C4—151no—
146C4×Dic3—151no—
147C4×Dic3—151no—
148C32×C5C4×D411yescharacteristic, complemented, Hall
149C23×C5D3211yescharacteristic
150C22×D5—31no—
151C2×C4×C5—31no—
152C2×C4×C5—31no—
153C2×C4×C5—62no—
154C22⋊D5—34no—
155C2×Dic5—31no—
156C2×Dic5—91no—
157C2×Dic5—91no—
158C4×D5—182no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32C4×D512yescomplemented
162C22×C32C22×D511yescharacteristic
163C2×C3×D3—52no—
164C2×C3×D3—101no—
165C2×C3×D3—101no—
166C3⋊Dic3—101no—
167C3⋊Dic3—101no—
168C3×Dic3—21no—
169C3×Dic3—52no—
170C3×Dic3C22⋊D512yescomplemented
171C4×D4—451noSylow
172C2×C3×C5—22no—
173C2×C3×C5—22no—
174C2×C3×C5—21no—
175C2×C3×C5—21no—
176C2×C3×C5—21no—
177C2×C3×C5—42no—
178C2×C3×C5C2×C4×D311yescharacteristic
179C2×C3×C5#23511yescharacteristic
180C2×C3×C5D3×D411yescharacteristic
181C2×C3×C5#23711yescharacteristic
182C2×C3×C5C22×Dic311yescharacteristic
183C2×C3×C5C2×#6711yescharacteristic
184D15—22no—
185C3×D5—62no—
186C23×C3—21no—
187C23×C3—151no—
188C23×C3D3×D511yescharacteristic
189C23×C3C2×D1511yescharacteristic
190C22×D3—51no—
191C3×D4—154no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—62no—
195C2×C3×C4—151no—
196#67—54no—
197C2×Dic3—22no—
198C2×Dic3—51no—
199C2×Dic3—151no—
200C2×Dic3—151no—
201C2×Dic3—151no—
202C2×Dic3—151no—
203C2×Dic3—301no—
204C2×Dic3—301no—
205C2×Dic3—302no—
206C2×Dic3C2×D1511yescharacteristic
207C2×Dic3C2×D1511yescharacteristic
208C4×D3—302no—
209C22×C5—21no—
210C22×C5—21no—
211C22×C5D3×Dic312yescomplemented
212C22×C5C2×D3211yescharacteristic
213C2×D5—32no—
214C2×D5—61no—
215C2×D5—61no—
216C4×C5—32no—
217C4×C5—61no—
218Dic5—32no—
219Dic5—181no—
220Dic5—181no—
221C2×C32—22no—
222C2×C32C2×C4×D511yescharacteristic
223C2×C32#70911yescharacteristic
224C2×C32C2×C22⋊D511yescharacteristic
225C3×D3—102no—
226C2×D4—151no—
227C22×C4—31no—
228C22×C4—451no—
229C4⋊C4—451no—
230C22⋊C4—451no—
231C22⋊C4—451no—
232C42—451no—
233C3×C5—21no—
234C3×C5C4×#6711yescharacteristic, complemented
235C3×C5D4×Dic311yescharacteristic, complemented
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—22no—
241C22×C3—21no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—152no—
245C22×C3—301no—
246C22×C3—301no—
247C22×C3D5×Dic312yescomplemented
248C22×C3C4×D1512yescomplemented
249C22×C3C2×D3×D511yescharacteristic
250C22×C3C22×D1511yescharacteristic
251C2×D3—52no—
252C2×D3—101no—
253C2×D3—101no—
254C3×C4—32no—
255C3×C4—61no—
256C3×C4—152no—
257Dic3—21no—
258Dic3—52no—
259Dic3—301no—
260Dic3—301no—
261Dic3—301no—
262Dic3—301no—
263Dic3—601no—
264Dic3—601no—
265Dic3#122012yesdirect product factor
266C2×C5—22no—
267C2×C5C2×D3×Dic311yescharacteristic
268C2×C5#380411yescharacteristic
269C2×C5D3×#6711yescharacteristic
270D5—62no—
271C32C4×C22⋊D511yescharacteristic, complemented, Sylow
272C23—151no—
273C23D3×D1511yescharacteristic
274D4—154no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—62no—
278C2×C4—151no—
279C2×C4—451no—
280C2×C4—451no—
281C2×C4—902no—
282C2×C3—22no—
283C2×C3—22no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—42no—
288C2×C3—302no—
289C2×C3C2×D5×Dic311yescharacteristic
290C2×C3#677911yescharacteristic
291C2×C3D3×C22⋊D511yescharacteristic
292C2×C3C2×C4×D1511yescharacteristic
293C2×C3#681711yescharacteristic
294C2×C3C2×#122011yescharacteristic
295D3—102no—
296C5Dic3×#6711yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22—152no—
300C22—301no—
301C22—301no—
302C22Dic3×D1512yescomplemented
303C22C2×D3×D1511yescharacteristic
304C4—32no—
305C4—61no—
306C4—152no—
307C4—901no—
308C4—901no—
309C3—21no—
310C3Dic3×C22⋊D511yescharacteristic, complemented
311C3C4×#122011yescharacteristic, complemented
312C2—22no—
313C2—302no—
314C2C2×Dic3×D1511yescharacteristic
315C2#1062420711yescharacteristic
316C2D3×#122011yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×D15
commutatorC2×C32×C5C22×C4
FittingC23×C32×C5C22
socle, cosocleC22×C32×C5C23
prime radicalC2×C3×#1220C2
hypercenterC23D3×D15
hypocenterC32×C5C4×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C3×C5×Dic3C2semidirect
3. C2×Dic3×D15C2semidirect
4. #10623913C2semidirect
5. #10623814C2semidirect
6. #10623811C2semidirect
7. Dic3×Dic15C2semidirect
14. C3×#1220C4semidirect
24. C22×C5×Dic3D3semidirect
38. C22×C32×C5C2×C4semidirect
42. C2×C3×D15C2×C4semidirect
45. C3×Dic15C2×C4semidirect
47. C3×C5×Dic3D4semidirect
50. C22×C3×Dic3D5semidirect
64. #1220Dic3direct (N × H)
111. C22×C3×C5C4×D3semidirect
112. C22×C3×C5C2×Dic3semidirect
117. C2×D15C2×Dic3semidirect
129. Dic15C2×Dic3semidirect
132. C5×Dic3#67semidirect
137. C22×Dic3D15semidirect
148. C32×C5C4×D4semidirect
161. C22×C32C4×D5semidirect
170. C3×Dic3C22⋊D5semidirect
211. C22×C5D3×Dic3semidirect
234. C3×C5C4×#67semidirect
235. C3×C5D4×Dic3semidirect
247. C22×C3D5×Dic3semidirect
248. C22×C3C4×D15semidirect
265. Dic3#1220direct (N × H)
271. C32C4×C22⋊D5semidirect
296. C5Dic3×#67semidirect
302. C22Dic3×D15semidirect
310. C3Dic3×C22⋊D5semidirect
311. C3C4×#1220semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#12202
C22×C3×C5×Dic32
C2×Dic3×D152
#106239132
#106238142
#106238112
Dic3×Dic152
C4×#12203
Dic3×C22⋊D53
Dic3×#675