Finite Group Databasebeta

← Dic3×#1203 #13085912

Subgroups — Dic3×#1203 #13085912

Order 1440.

402 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 106 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×Dic3×D15C211yesmaximal, characteristic, complemented
2C2×C5×D3×Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#1203C211yesmaximal, characteristic
4#10623829C211yesmaximal, characteristic, complemented
5(C2×D3)⋊Dic15C211yesmaximal, characteristic, complemented
6#10623773C211yesmaximal, characteristic, complemented
7C3×Dic3×Dic5C211yesmaximal, characteristic, complemented
8Dic3×C22⋊D5—31nomaximal
9C4×#1203—31nomaximal
10C22×C3×D15C2211yescharacteristic
11C22×C3×C5×D3C2211yescharacteristic
12C2×C3⋊Dic15C2211yescharacteristic
13C2×C5×C3⋊Dic3C2211yescharacteristic
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C32×Dic5C2211yescharacteristic
17Dic3×D15—22no—
18C5×D3×Dic3—22no—
19C3×#1203C414yescomplemented
20Dic3×D12—51nomaximal, Hall
21C2×C4×D15—31no—
22C22×C5×Dic3—31no—
23C2×C4×C5×D3—31no—
24C2×C3×C22⋊D5—31no—
25C2×#1203D311yescharacteristic
26C2×D5×Dic3—31no—
27#6719—31no—
28C3×C4×Dic5—31no—
29#6668—31no—
30#6668—31no—
31#6668—61no—
32#6667—31no—
33(C2×D3)⋊Dic5—31no—
34#6665—31no—
35Dic3×Dic5D311yescharacteristic, complemented
36C22×C32×C5C2311yescharacteristic
37C2×C3×D15—21no—
38C2×C3×D15—21no—
39C2×C3×D15C2×C412yescomplemented
40C2×C3×C5×D3—21no—
41C2×C3×C5×D3—21no—
42C2×C3×C5×D3C2×C412yescomplemented
43C3⋊Dic15—21no—
44C5×C3⋊Dic3—21no—
45C3×Dic15—21no—
46C3×C5×Dic3D412yescomplemented
47C32×Dic5C2×C412yescomplemented
48C4×C22⋊D5—91noHall
49C2×C3×D12—51no—
50C2×D3×Dic3—51no—
51C2×D3×Dic3D511yescharacteristic, complemented
52C4⋊(C3⋊Dic3)—51no—
53C3×C4×Dic3—51no—
54(C2×D3)⋊Dic3—51no—
55(C2×D3)⋊Dic3—51no—
56C23×C3×C5—31no—
57C22×D15C2×D311yescharacteristic
58C22×C5×D3C2×D311yescharacteristic
59C22×C3×D5—31no—
60C2×C3×C4×C5—31no—
61C2×Dic15—31no—
62C2×Dic15—31no—
63C2×Dic15—61no—
64C2×Dic15C2×D311yescharacteristic
65C4×D15—62no—
66C2×C5×Dic3—31no—
67C2×C5×Dic3—31no—
68C2×C5×Dic3—62no—
69C2×C5×Dic3—61no—
70C2×C5×Dic3C2×D311yescharacteristic
71C4×C5×D3—62no—
72C3×C22⋊D5—34no—
73C2×C3×Dic5—21no—
74C2×C3×Dic5—31no—
75C2×C3×Dic5C2×D311yescharacteristic
76C2×C3×Dic5C2×D311yescharacteristic
77#1203Dic314yesdirect product factor
78D5×Dic3—62no—
79D4×Dic3—151no—
80C4×D12—151no—
81C2×C32×C5C22×C411yescharacteristic
82C2×C32×C5C2×D411yescharacteristic
83C2×C32×C5#4111yescharacteristic
84C3×D15—22no—
85C3×C5×D3—22no—
86C22×C4×C5—91no—
87C2×C22⋊D5—31no—
88C2×C4×D5—91no—
89C22⋊Dic5—91no—
90#685—91no—
91#683—91no—
92C4×Dic5—31no—
93C22×C3×D3—51no—
94C22×C3×D3C2×D511yescharacteristic
95C2×C32×C4—51no—
96C2×C3⋊Dic3—51no—
97C2×C3⋊Dic3C2×D511yescharacteristic
98C2×C3×Dic3—51no—
99C2×C3×Dic3C2×D511yescharacteristic
100C3×D12—54no—
101D3×Dic3—22no—
102D3×Dic3—102no—
103C22×C3×C5—21no—
104C22×C3×C5—32no—
105C22×C3×C5—61no—
106C22×C3×C5—61no—
107C22×C3×C5C22×D311yescharacteristic
108C22×C3×C5C22×D311yescharacteristic
109C2×D15—21no—
110C2×D15—21no—
111C2×D15C2×Dic312yescomplemented
112C2×C5×D3—21no—
113C2×C5×D3—21no—
114C2×C5×D3C2×Dic312yescomplemented
115C2×C3×D5—32no—
116C2×C3×D5—61no—
117C2×C3×D5—61no—
118C3×C4×C5—32no—
119Dic15—21no—
120Dic15—61no—
121Dic15—61no—
122Dic15—121no—
123C3×Dic5—22no—
124C3×Dic5—61no—
125C3×Dic5C4×D312yescomplemented
126C3×Dic5C2×Dic312yescomplemented
127C5×Dic3—61no—
128C5×Dic3—61no—
129C5×Dic3—121no—
130C5×Dic3D1212yescomplemented
131C2×C3×D4—151no—
132C22×Dic3—31no—
133C22×Dic3—151no—
134C2×D12—51no—
135C2×C4×D3—31no—
136C2×C4×D3—151no—
137C3×C42—151no—
138#217—151no—
139#217—151no—
140(C2×D3)⋊C4—151no—
141(C2×D3)⋊C4—151no—
142C4⋊Dic3—151no—
143C4⋊Dic3—151no—
144C4⋊Dic3—301no—
145C4×Dic3—51no—
146C32×C5C4×D411yescharacteristic, complemented, Hall
147C23×C5—31no—
148C22×D5—31no—
149C2×C4×C5—31no—
150C2×C4×C5—91no—
151C2×C4×C5—182no—
152C22⋊D5—34no—
153C2×Dic5—31no—
154C2×Dic5—91no—
155C2×Dic5D3211yescharacteristic
156C4×D5—182no—
157C22×C32C22×D511yescharacteristic
158C2×C3×D3—21no—
159C2×C3×D3—21no—
160C2×C3×D3—52no—
161C2×C3×D3—101no—
162C2×C3×D3—101no—
163C2×C3×D3C4×D512yescomplemented
164C32×C4—52no—
165C3⋊Dic3—21no—
166C3⋊Dic3—101no—
167C3×Dic3—101no—
168C3×Dic3C22⋊D512yescomplemented
169C4×D4—451noSylow
170C2×C3×C5—21no—
171C2×C3×C5—21no—
172C2×C3×C5—21no—
173C2×C3×C5—62no—
174C2×C3×C5C2×C4×D311yescharacteristic
175C2×C3×C5C2×D1211yescharacteristic
176C2×C3×C5#23511yescharacteristic
177C2×C3×C5D3×D411yescharacteristic
178C2×C3×C5#23711yescharacteristic
179C2×C3×C5C22×Dic311yescharacteristic
180D15—22no—
181C3×D5—62no—
182C5×D3—22no—
183C23×C3—31no—
184C23×C3—151no—
185C22×D3—51no—
186C22×D3D3×D511yescharacteristic
187C3×D4—154no—
188C2×C3×C4—31no—
189C2×C3×C4—51no—
190C2×C3×C4—51no—
191C2×C3×C4—101no—
192C2×C3×C4—151no—
193C2×Dic3—31no—
194C2×Dic3—31no—
195C2×Dic3—51no—
196C2×Dic3—61no—
197C2×Dic3—62no—
198C2×Dic3—151no—
199C2×Dic3—151no—
200C2×Dic3—301no—
201C2×Dic3—302no—
202C2×Dic3D3×D511yescharacteristic
203D12—54no—
204C4×D3—62no—
205C4×D3—302no—
206C22×C5—32no—
207C22×C5—61no—
208C22×C5—61no—
209C22×C5C2×D3211yescharacteristic
210C2×D5—32no—
211C2×D5—61no—
212C2×D5—61no—
213C4×C5—32no—
214C4×C5—181no—
215Dic5—61no—
216Dic5—181no—
217Dic5D3×Dic312yescomplemented
218C2×C32C2×C4×D511yescharacteristic
219C2×C32#70911yescharacteristic
220C2×C32C2×C22⋊D511yescharacteristic
221C3×D3—22no—
222C3×D3—102no—
223C2×D4—151no—
224C22×C4—91no—
225C22×C4—451no—
226C4⋊C4—451no—
227C22⋊C4—451no—
228C22⋊C4—451no—
229C42—151no—
230C3×C5—21no—
231C3×C5C4×D1211yescharacteristic, complemented
232C3×C5D4×Dic311yescharacteristic, complemented
233C22×C3—21no—
234C22×C3—32no—
235C22×C3—61no—
236C22×C3—61no—
237C22×C3—152no—
238C22×C3—301no—
239C22×C3—301no—
240C22×C3C2×D3×D511yescharacteristic
241C22×C3C2×D3×D511yescharacteristic
242C2×D3—21no—
243C2×D3—21no—
244C2×D3—52no—
245C2×D3—101no—
246C2×D3—101no—
247C2×D3D5×Dic312yescomplemented
248C3×C4—32no—
249C3×C4—52no—
250C3×C4—52no—
251C3×C4—102no—
252C3×C4—301no—
253Dic3—61no—
254Dic3—61no—
255Dic3—101no—
256Dic3—121no—
257Dic3—301no—
258Dic3—301no—
259Dic3—601no—
260Dic3#120312yesdirect product factor
261C2×C5—62no—
262C2×C5#379511yescharacteristic
263C2×C5D3×D1211yescharacteristic
264C2×C5C2×D3×Dic311yescharacteristic
265D5—62no—
266C32C4×C22⋊D511yescharacteristic, complemented, Sylow
267C23—31no—
268C23—151no—
269D4—154no—
270C2×C4—31no—
271C2×C4—51no—
272C2×C4—91no—
273C2×C4—151no—
274C2×C4—182no—
275C2×C4—451no—
276C2×C4—902no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—62no—
281C2×C3—302no—
282C2×C3#677111yescharacteristic
283C2×C3C4×D3×D511yescharacteristic
284C2×C3C2×D5×Dic311yescharacteristic
285C2×C3#677911yescharacteristic
286C2×C3C2×#120311yescharacteristic
287C2×C3D3×C22⋊D511yescharacteristic
288D3—22no—
289D3—102no—
290C5Dic3×D1211yescharacteristic, complemented, Sylow
291C22—32no—
292C22—61no—
293C22—61no—
294C22—152no—
295C22—301no—
296C22—301no—
297C22D32×D511yescharacteristic
298C4—32no—
299C4—52no—
300C4—181no—
301C4—301no—
302C4—901no—
303C3—21no—
304C3C4×#120311yescharacteristic, complemented
305C3Dic3×C22⋊D511yescharacteristic, complemented
306C2—62no—
307C2—302no—
308C2D3×D5×Dic311yescharacteristic
309C2D3×#120311yescharacteristic
310C2#1062405311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×D5
commutatorC2×C32×C5C22×C4
Fitting, socle, cosocleC22×C32×C5C23
prime radicalC2×C3×#1203C2
hypocenterC32×C5C4×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×Dic3×D15C2semidirect
2. C2×C5×D3×Dic3C2semidirect
4. #10623829C2semidirect
5. (C2×D3)⋊Dic15C2semidirect
6. #10623773C2semidirect
7. C3×Dic3×Dic5C2semidirect
19. C3×#1203C4semidirect
35. Dic3×Dic5D3semidirect
39. C2×C3×D15C2×C4semidirect
42. C2×C3×C5×D3C2×C4semidirect
46. C3×C5×Dic3D4semidirect
47. C32×Dic5C2×C4semidirect
51. C2×D3×Dic3D5semidirect
77. #1203Dic3direct (N × H)
111. C2×D15C2×Dic3semidirect
114. C2×C5×D3C2×Dic3semidirect
125. C3×Dic5C4×D3semidirect
126. C3×Dic5C2×Dic3semidirect
130. C5×Dic3D12semidirect
146. C32×C5C4×D4semidirect
163. C2×C3×D3C4×D5semidirect
168. C3×Dic3C22⋊D5semidirect
217. Dic5D3×Dic3semidirect
231. C3×C5C4×D12semidirect
232. C3×C5D4×Dic3semidirect
247. C2×D3D5×Dic3semidirect
260. Dic3#1203direct (N × H)
266. C32C4×C22⋊D5semidirect
290. C5Dic3×D12semidirect
304. C3C4×#1203semidirect
305. C3Dic3×C22⋊D5semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×D152
C2×C5×D3×Dic32
C2×C3×#12032
#106238292
(C2×D3)⋊Dic152
#106237732
C3×Dic3×Dic52
Dic3×C22⋊D53
C4×#12033
Dic3×D125