Finite Group Databasebeta

← D3×#6717 #13087516

Subgroups — D3×#6717 #13087516

Order 1440.

292 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C3×C5×D3×Q8C211yesmaximal, characteristic, complemented
2D3×D60C211yesmaximal, characteristic
3#10623911C211yesmaximal, characteristic, complemented
4C3×#6717C211yesmaximal, characteristic, complemented
5#10623806C211yesmaximal, characteristic, complemented
6#10623801C211yesmaximal, characteristic, complemented
7D3×C3⋊#168C211yesmaximal, characteristic, complemented
8C2×#6717—31nomaximal
9D3×#688—31nomaximal
10C2×D3×D15—21no—
11C32×C5×Q8C2211yescharacteristic, complemented
12C4⋊(C3⋊D15)C2211yescharacteristic
13C3×D60C2211yescharacteristic
14C3×C4×C5×D3—21no—
15C3×C4×C5×D3C2211yescharacteristic
16C3×C5×Dic6—21no—
17C3×C5×Dic6C2211yescharacteristic, complemented
18#67151—21no—
19C3⋊(C3⋊#168)C2211yescharacteristic, complemented
20C3×C3⋊#168C2211yescharacteristic, complemented
21D3×#215—51nomaximal, Hall
22C2×C3×C5×Q8—31no—
23C2×D60—31no—
24C5×D3×Q8D311yescharacteristic, complemented
25D3×D20—31no—
26#6717—31no—
27#6717—31no—
28#6717—31no—
29#6717—31no—
30#6717—61no—
31#6717D311yescharacteristic, direct product factor
32C2×C3⋊#168—31no—
33C3×#688—31no—
34#6660—31no—
35#6655—31no—
36D3×#168—31no—
37C2×C3⋊D15—21no—
38C2×C3×D15—21no—
39C2×C3×C5×D3D411yescharacteristic
40D3×D15—42no—
41C32×C4×C5—21no—
42C32×C4×C5C2311yescharacteristic
43C3×C5×Dic3—21no—
44C3×C5×Dic3D411yescharacteristic
45C2×#688—91noHall
46C3×D3×Q8D511yescharacteristic, complemented
47D3×D12—51no—
48#3755—51no—
49C3×#215—51no—
50#3717—51no—
51#3715—51no—
52D3×C3⋊C8—51no—
53C22×D15—61no—
54C2×D3×D5—61no—
55C3×C5×Q8—21no—
56C3×C5×Q8—31no—
57C3×C5×Q8—61no—
58C3×C5×Q8C2×D311yescharacteristic, complemented
59C3×C5×Q8C2×D311yescharacteristic, complemented
60C2×C3×C4×C5—31no—
61C2×C3×C4×C5—61no—
62D60—31no—
63D60—31no—
64D60—61no—
65D60—61no—
66D60C2×D311yescharacteristic
67C4×C5×D3—21no—
68C4×C5×D3C2×D311yescharacteristic
69C5×Dic6—21no—
70C5×Dic6C2×D311yescharacteristic, complemented
71C3×D20—31no—
72#1202—61no—
73C3⋊#168—31no—
74C3⋊#168—31no—
75C3⋊#168—61no—
76C3⋊#168C2×D311yescharacteristic, complemented
77C3×#168—31no—
78C2×#215—151no—
79D3×SD16—151no—
80C2×C32×C5C2×D411yescharacteristic
81C3⋊D15—41no—
82C3×D15—41no—
83C3×C5×D3SD1612yescomplemented
84C2×C5×Q8—31no—
85C2×D20—91no—
86#688—31no—
87#688—91no—
88#688—91no—
89#688—91no—
90C2×#168—91no—
91C2×D32—101no—
92C32×Q8C2×D511yescharacteristic, complemented
93C4⋊(C3⋊D3)—51no—
94C3×D12—51no—
95C3×C4×D3—21no—
96C3×C4×D3C2×D511yescharacteristic
97C3×Dic6—21no—
98C3×Dic6C2×D511yescharacteristic, complemented
99#626—101no—
100C3⋊(C3⋊C8)—51no—
101C3×C3⋊C8—51no—
102C22×C3×C5—31no—
103C2×D15—21no—
104C2×D15—61no—
105C2×D15—61no—
106C2×D15—121no—
107C2×D15—122no—
108C2×C5×D3#6711yescharacteristic
109C2×C3×D5—61no—
110D3×D5—122no—
111C3×C4×C5—21no—
112C3×C4×C5—21no—
113C3×C4×C5—21no—
114C3×C4×C5—31no—
115C3×C4×C5—41no—
116C3×C4×C5—61no—
117C3×C4×C5C22×D311yescharacteristic
118C3×C4×C5C22×D311yescharacteristic
119C5×Dic3—21no—
120C5×Dic3#6711yescharacteristic
121C2×C3×Q8—31no—
122D3×Q8D1511yescharacteristic, complemented
123D3×D4—151no—
124C2×D12—151no—
125C3×SD16—151no—
126#215—51no—
127#215—151no—
128#215—151no—
129#215—151no—
130#215—151no—
131#215—301no—
132#214—151no—
133C2×C3⋊C8—151no—
134#204—151no—
135D3×C8—151no—
136C32×C5C2×SD1611yescharacteristic, complemented, Hall
137C22×D5—181no—
138C5×Q8—31no—
139C5×Q8—61no—
140C5×Q8D3211yescharacteristic, complemented
141C2×C4×C5—31no—
142C2×C4×C5—61no—
143D20—31no—
144D20—91no—
145D20—181no—
146#168—31no—
147#168—91no—
148C2×C3⋊D3—101no—
149C2×C3×D3—101no—
150C2×C3×D3C22⋊D511yescharacteristic
151D32—202no—
152C32×C4—21no—
153C32×C4C22×D511yescharacteristic
154C3×Dic3—21no—
155C3×Dic3C22⋊D511yescharacteristic
156C2×SD16—451noSylow
157C2×C3×C5—21no—
158C2×C3×C5—32no—
159C2×C3×C5D3×D411yescharacteristic
160C2×C3×C5C2×#6711yescharacteristic
161D15—41no—
162D15—121no—
163D15—121no—
164D15—241no—
165C3×D5—121no—
166C5×D3#21512yescomplemented
167C22×D3—301no—
168C22×D3—301no—
169C3×Q8—21no—
170C3×Q8—31no—
171C3×Q8—61no—
172C3×Q8D3×D511yescharacteristic, complemented
173C3×Q8C2×D1511yescharacteristic, complemented
174C3×D4—151no—
175C2×C3×C4—31no—
176C2×C3×C4—61no—
177#67—301no—
178D12—51no—
179D12—151no—
180D12—151no—
181D12—301no—
182D12—301no—
183C4×D3—21no—
184C4×D3C2×D1511yescharacteristic
185Dic6—21no—
186Dic6C2×D1511yescharacteristic, complemented
187C3×C8—151no—
188C3⋊C8—51no—
189C3⋊C8—151no—
190C3⋊C8—151no—
191C3⋊C8—301no—
192C22×C5—31no—
193C2×D5—61no—
194C2×D5—181no—
195C2×D5—362no—
196C4×C5—21no—
197C4×C5—31no—
198C4×C5—61no—
199C4×C5C2×D3211yescharacteristic
200C2×C32C2×C22⋊D511yescharacteristic
201C3⋊D3—201no—
202C3×D3—201no—
203C3×D3#68812yescomplemented
204C2×Q8—31no—
205C2×D4—451no—
206SD16—151no—
207SD16—451no—
208SD16—451no—
209SD16—451no—
210C2×C8—451no—
211C3×C5—21no—
212C3×C5D3×SD1611yescharacteristic, complemented
213C3×C5C2×#21511yescharacteristic, complemented
214C22×C3—31no—
215C22×C3—301no—
216C2×D3—101no—
217C2×D3—301no—
218C2×D3—301no—
219C2×D3—601no—
220C2×D3—602no—
221C2×D3—602no—
222C2×D3#122011yescharacteristic
223C3×C4—21no—
224C3×C4—21no—
225C3×C4—21no—
226C3×C4—31no—
227C3×C4—41no—
228C3×C4—61no—
229C3×C4C2×D3×D511yescharacteristic
230C3×C4C22×D1511yescharacteristic
231Dic3—21no—
232Dic3#122011yescharacteristic
233C2×C5—32no—
234C2×C5D3×#6711yescharacteristic
235D5—121no—
236D5—361no—
237C32C2×#68811yescharacteristic, complemented, Sylow
238C23—901no—
239Q8—31no—
240Q8—61no—
241Q8D3×D1511yescharacteristic, complemented
242D4—151no—
243D4—451no—
244D4—901no—
245C2×C4—31no—
246C2×C4—61no—
247C8—151no—
248C8—451no—
249C2×C3—21no—
250C2×C3—32no—
251C2×C3—601no—
252C2×C3D3×C22⋊D511yescharacteristic
253C2×C3C2×#122011yescharacteristic
254D3—201no—
255D3—601no—
256D3—601no—
257D3—1201no—
258D3#671712yesdirect product factor
259C5D3×#21511yescharacteristic, complemented, Sylow
260C22—31no—
261C22—301no—
262C22—901no—
263C22—1802no—
264C4—21no—
265C4—31no—
266C4—61no—
267C4C2×D3×D1511yescharacteristic
268C3—21no—
269C3D3×#68811yescharacteristic, complemented
270C3C2×#671711yescharacteristic, complemented
271C2—32no—
272C2—601no—
273C2—1801no—
274C2D3×#122011yescharacteristic
subgroupG/H
centerC2D3×#1220
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×D3×D15
FittingC32×C5×Q8C22
socleC2×C32×C5C2×D4
prime radicalD3×D60C2
hypercenterQ8D3×D15
hypocenterC32×C5C2×SD16
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C3×C5×D3×Q8C2semidirect
3. #10623911C2semidirect
4. C3×#6717C2semidirect
5. #10623806C2semidirect
6. #10623801C2semidirect
7. D3×C3⋊#168C2semidirect
11. C32×C5×Q8C22semidirect
17. C3×C5×Dic6C22semidirect
19. C3⋊(C3⋊#168)C22semidirect
20. C3×C3⋊#168C22semidirect
24. C5×D3×Q8D3semidirect
31. #6717D3direct (N × H)
46. C3×D3×Q8D5semidirect
58. C3×C5×Q8C2×D3semidirect
59. C3×C5×Q8C2×D3semidirect
70. C5×Dic6C2×D3semidirect
76. C3⋊#168C2×D3semidirect
83. C3×C5×D3SD16semidirect
92. C32×Q8C2×D5semidirect
98. C3×Dic6C2×D5semidirect
122. D3×Q8D15semidirect
136. C32×C5C2×SD16semidirect
140. C5×Q8D32semidirect
166. C5×D3#215semidirect
172. C3×Q8D3×D5semidirect
173. C3×Q8C2×D15semidirect
186. Dic6C2×D15semidirect
203. C3×D3#688semidirect
212. C3×C5D3×SD16semidirect
213. C3×C5C2×#215semidirect
237. C32C2×#688semidirect
241. Q8D3×D15semidirect
258. D3#6717direct (N × H)
259. C5D3×#215semidirect
269. C3D3×#688semidirect
270. C3C2×#6717semidirect

Maximal subgroups

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

subgroupindex
C3×C5×D3×Q82
D3×D602
#106239112
C3×#67172
#106238062
#106238012
D3×C3⋊#1682
C2×#67173
D3×#6883
D3×#2155