Finite Group Databasebeta

← D3×Q8×D15 #13089378

Subgroups — D3×Q8×D15 #13089378

Order 1440.

662 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 172 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
1Q8×C3⋊D15C211yesmaximal, characteristic, complemented
2C3×Q8×D15C211yesmaximal, characteristic, complemented
3C3×C5×D3×Q8C211yesmaximal, characteristic, complemented
4C4×D3×D15C213yesmaximal
5#10624198C213yesmaximal, complemented
6D3×Dic30C213yesmaximal, complemented
7Dic6×D15C213yesmaximal, complemented
8C2×Q8×D15—31nomaximal
9D3×Q8×D5—31nomaximal
10C2×D3×D15C2211yescharacteristic
11C32×C5×Q8C2211yescharacteristic, complemented
12C4×C3⋊D15C2213yes—
13#67180C2213yescomplemented
14C3×C4×D15C2213yes—
15C3×Dic30C2213yescomplemented
16C3×C4×C5×D3C2213yes—
17C3×C5×Dic6C2213yescomplemented
18#67153C2216yescomplemented
19#67149C2213yes—
20D3×Dic15C2213yes—
21Dic3×D15C2213yes—
22D32×Q8—51nomaximal, Hall
23C2×C3×C5×Q8—31no—
24Q8×D15—31no—
25Q8×D15—31no—
26Q8×D15—33no—
27Q8×D15—33no—
28Q8×D15—61no—
29Q8×D15D311yescharacteristic, direct product factor
30C2×C4×D15—33no—
31C2×Dic30—33no—
32C5×D3×Q8D311yescharacteristic, complemented
33C3×Q8×D5—31no—
34C4×D3×D5—33no—
35D15⋊Q8—33no—
36D3×Dic10—33no—
37D5×Dic6—33no—
38C2×C3⋊D15C2311yescharacteristic
39C2×C3×D15C2311yescharacteristic
40C2×C3×C5×D3C2311yescharacteristic
41D3×D15Q814yesdirect product factor
42C32×C4×C5C2313yes—
43C3⋊Dic15C2313yes—
44C3×Dic15C2313yes—
45C3×C5×Dic3C2313yes—
46C2×Q8×D5—91noHall
47Q8×C3⋊D3—51no—
48C3×D3×Q8—51no—
49C3×D3×Q8D511yescharacteristic, complemented
50C4×D32—53no—
51#3797—53no—
52D3×Dic6—53no—
53D3×Dic6—53no—
54C22×D15—31no—
55C2×D3×D5—31no—
56C3×C5×Q8—21no—
57C3×C5×Q8—33no—
58C3×C5×Q8C2×D311yescharacteristic, complemented
59C3×C5×Q8C2×D311yescharacteristic, complemented
60C2×C3×C4×C5—33no—
61C2×Dic15—33no—
62C4×D15—33no—
63C4×D15—33no—
64C4×D15—33no—
65C4×D15—33no—
66C4×D15—63no—
67C4×D15C2×D313yes—
68Dic30—33no—
69Dic30—33no—
70Dic30—36no—
71Dic30—63no—
72Dic30C2×D313yescomplemented
73C4×C5×D3C2×D313yes—
74C5×Dic6C2×D313yescomplemented
75C3×C4×D5—33no—
76C3×Dic10—33no—
77#1204—36no—
78#1200—33no—
79D3×Dic5—33no—
80D5×Dic3—33no—
81C2×D3×Q8—151no—
82C2×D3×Q8—151no—
83C2×C32×C5C2411yescharacteristic
84C3⋊D15C2×Q812yescomplemented
85C3×D15C2×Q812yescomplemented
86C3×C5×D3C2×Q812yescomplemented
87C2×C5×Q8—31no—
88Q8×D5—31no—
89Q8×D5—93no—
90Q8×D5—93no—
91Q8×D5—91no—
92C2×C4×D5—93no—
93C2×Dic10—93no—
94C2×D32—51no—
95C32×Q8C2×D511yescharacteristic, complemented
96C4×C3⋊D3—53no—
97#634—53no—
98C3×C4×D3—53no—
99C3×C4×D3C2×D513yes—
100C3×Dic6—53no—
101C3×Dic6C2×D513yescomplemented
102#627—56no—
103#624—53no—
104D3×Dic3—53no—
105D3×Dic3—53no—
106C22×C3×C5—31no—
107C2×D15—31no—
108C2×D15—31no—
109C2×D15—34no—
110C2×D15—61no—
111C2×D15C22×D311yescharacteristic
112C2×C5×D3C22×D311yescharacteristic
113C2×C3×D5—31no—
114D3×D5—34no—
115C3×C4×C5—23no—
116C3×C4×C5—33no—
117C3×C4×C5C22×D313yes—
118C3×C4×C5C22×D313yes—
119Dic15—33no—
120Dic15—33no—
121Dic15—63no—
122Dic15C22×D313yes—
123C3×Dic5—33no—
124C5×Dic3C22×D313yes—
125C2×C3×Q8—31no—
126C2×C3×Q8—151no—
127D3×Q8—51no—
128D3×Q8—153no—
129D3×Q8—153no—
130D3×Q8—151no—
131D3×Q8—151no—
132D3×Q8—153no—
133D3×Q8—153no—
134D3×Q8—301no—
135D3×Q8D1511yescharacteristic, direct product factor
136C2×C4×D3—153no—
137C2×C4×D3—153no—
138C2×Dic6—153no—
139C2×Dic6—153no—
140C32×C5C22×Q811yescharacteristic, complemented, Hall
141C22×D5—91no—
142C5×Q8—33no—
143C5×Q8D3211yescharacteristic, complemented
144C2×C4×C5—33no—
145C2×Dic5—93no—
146C4×D5—33no—
147C4×D5—93no—
148C4×D5—93no—
149C4×D5—93no—
150Dic10—33no—
151Dic10—96no—
152Dic10—93no—
153C2×C3⋊D3—51no—
154C2×C3×D3—51no—
155C2×C3×D3C22×D511yescharacteristic
156D32—54no—
157C32×C4C22×D513yes—
158C3⋊Dic3—53no—
159C3×Dic3—53no—
160C3×Dic3C22×D513yes—
161C22×Q8—451noSylow
162C2×C3×C5—21no—
163C2×C3×C5—32no—
164C2×C3×C5C23×D311yescharacteristic
165C2×C3×C5C23×D311yescharacteristic
166D15—32no—
167D15—32no—
168D15—62no—
169D15D3×Q812yesdirect product factor
170C3×D5—32no—
171C5×D3D3×Q812yescomplemented
172C22×D3—151no—
173C22×D3—151no—
174C3×Q8—21no—
175C3×Q8—33no—
176C3×Q8—153no—
177C3×Q8D3×D511yescharacteristic, complemented
178C3×Q8C2×D1511yescharacteristic, complemented
179C2×C3×C4—33no—
180C2×C3×C4—153no—
181C2×Dic3—153no—
182C2×Dic3—153no—
183C4×D3—53no—
184C4×D3—153no—
185C4×D3—153no—
186C4×D3—153no—
187C4×D3—153no—
188C4×D3—153no—
189C4×D3—153no—
190C4×D3—303no—
191C4×D3C2×D1513yes—
192Dic6—53no—
193Dic6—156no—
194Dic6—153no—
195Dic6—153no—
196Dic6—156no—
197Dic6—303no—
198Dic6C2×D1513yescomplemented
199C22×C5—31no—
200C2×D5—31no—
201C2×D5—94no—
202C2×D5—91no—
203C4×C5—33no—
204C4×C5C2×D3213yes—
205Dic5—33no—
206Dic5—93no—
207C2×C32C23×D511yescharacteristic
208C3⋊D3—52no—
209C3×D3—52no—
210C3×D3Q8×D512yescomplemented
211C2×Q8—31no—
212C2×Q8—151no—
213C2×Q8—453no—
214C2×Q8—453no—
215C2×Q8—453no—
216C2×Q8—451no—
217C22×C4—453no—
218C3×C5—21no—
219C3×C5C2×D3×Q811yescharacteristic, complemented
220C3×C5C2×D3×Q811yescharacteristic, complemented
221C22×C3—31no—
222C22×C3—151no—
223C2×D3—51no—
224C2×D3—154no—
225C2×D3—151no—
226C2×D3—151no—
227C2×D3—154no—
228C2×D3—301no—
229C2×D3C22×D1511yescharacteristic
230C3×C4—23no—
231C3×C4—33no—
232C3×C4—153no—
233C3×C4C2×D3×D513yes—
234C3×C4C22×D1513yes—
235Dic3—53no—
236Dic3—153no—
237Dic3—153no—
238Dic3—303no—
239Dic3C22×D1513yes—
240C2×C5—32no—
241C2×C5C22×D3211yescharacteristic
242D5—32no—
243D5—92no—
244C32C2×Q8×D511yescharacteristic, complemented, Sylow
245C23—451no—
246Q8—33no—
247Q8—153no—
248Q8—456no—
249Q8—453no—
250Q8D3×D1511yescharacteristic, direct product factor
251C2×C4—33no—
252C2×C4—153no—
253C2×C4—453no—
254C2×C4—453no—
255C2×C4—453no—
256C2×C4—453no—
257C2×C3—21no—
258C2×C3—32no—
259C2×C3—152no—
260C2×C3C22×D3×D511yescharacteristic
261C2×C3C23×D1511yescharacteristic
262D3—52no—
263D3—152no—
264D3—152no—
265D3—302no—
266D3Q8×D1512yesdirect product factor
267C5D32×Q811yescharacteristic, complemented, Sylow
268C22—31no—
269C22—151no—
270C22—454no—
271C22—451no—
272C4—33no—
273C4—153no—
274C4—453no—
275C4C2×D3×D1513yes—
276C3—21no—
277C3D3×Q8×D511yescharacteristic, complemented
278C3C2×Q8×D1511yescharacteristic, complemented
279C2—32no—
280C2—152no—
281C2—452no—
282C2C22×D3×D1511yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×D15
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C5×Q8C22
prime radicalC2×D3×D15C22
hypercenterQ8D3×D15
hypocenterC32×C5C22×Q8
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C3⋊D15C2semidirect
2. C3×Q8×D15C2semidirect
3. C3×C5×D3×Q8C2semidirect
5. #10624198C2semidirect
6. D3×Dic30C2semidirect
7. Dic6×D15C2semidirect
11. C32×C5×Q8C22semidirect
13. #67180C22semidirect
15. C3×Dic30C22semidirect
17. C3×C5×Dic6C22semidirect
18. #67153C22semidirect
29. Q8×D15D3direct (N × H)
32. C5×D3×Q8D3semidirect
41. D3×D15Q8direct (N × H)
49. C3×D3×Q8D5semidirect
58. C3×C5×Q8C2×D3semidirect
59. C3×C5×Q8C2×D3semidirect
72. Dic30C2×D3semidirect
74. C5×Dic6C2×D3semidirect
84. C3⋊D15C2×Q8semidirect
85. C3×D15C2×Q8semidirect
86. C3×C5×D3C2×Q8semidirect
95. C32×Q8C2×D5semidirect
101. C3×Dic6C2×D5semidirect
135. D3×Q8D15direct (N × H)
140. C32×C5C22×Q8semidirect
143. C5×Q8D32semidirect
169. D15D3×Q8direct (N × H)
171. C5×D3D3×Q8semidirect
177. C3×Q8D3×D5semidirect
178. C3×Q8C2×D15semidirect
198. Dic6C2×D15semidirect
210. C3×D3Q8×D5semidirect
219. C3×C5C2×D3×Q8semidirect
220. C3×C5C2×D3×Q8semidirect
244. C32C2×Q8×D5semidirect
250. Q8D3×D15direct (N × H)
266. D3Q8×D15direct (N × H)
267. C5D32×Q8semidirect
277. C3D3×Q8×D5semidirect
278. C3C2×Q8×D15semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊D152
C3×Q8×D152
C3×C5×D3×Q82
C4×D3×D152
#106241982
D3×Dic302
Dic6×D152
C2×Q8×D153
D3×Q8×D53
D32×Q85