Finite Group Databasebeta

← D3×#6821 #13089380

Subgroups — D3×#6821 #13089380

Order 1440.

696 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 162 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
1#10624287C211yesmaximal, characteristic, complemented
2C3×#6821C211yesmaximal, characteristic, complemented
3C3×C5×D3×Q8C211yesmaximal, characteristic, complemented
4D3×D60C213yesmaximal, complemented
5C4×D3×D15C213yesmaximal, complemented
6#10624201C213yesmaximal, complemented
7#10624194C213yesmaximal, complemented
8C2×#6821—31nomaximal
9D3×#713—31nomaximal
10C2×D3×D15C2213yes—
11C32×C5×Q8C2211yescharacteristic, complemented
12C4⋊(C3⋊D15)C2213yescomplemented
13C4×C3⋊D15C2213yescomplemented
14C3×D60C2213yescomplemented
15C3×C4×D15C2213yescomplemented
16C3×C4×C5×D3C2213yes—
17C3×C5×Dic6C2213yescomplemented
18#67151C2216yescomplemented
19#67149C2213yescomplemented
20D3×Dic15C2211yescharacteristic
21Dic3×D15C2213yescomplemented
22D3×#239—51nomaximal, Hall
23C2×C3×C5×Q8—31no—
24#6821—31no—
25#6821—31no—
26#6821—33no—
27#6821—33no—
28#6821—61no—
29#6821D311yescharacteristic, direct product factor
30C2×D60—33no—
31C2×C4×D15—33no—
32C5×D3×Q8D311yescharacteristic, complemented
33C3×#713—31no—
34D3×D20—33no—
35C4×D3×D5—33no—
36#6773—33no—
37#6766—33no—
38C2×C3⋊D15C2313yes—
39C2×C3×D15C2313yes—
40C2×C3×C5×D3C2311yescharacteristic
41D3×D15—26no—
42C32×C4×C5C2313yes—
43C3⋊Dic15C2311yescharacteristic
44C3×Dic15C2311yescharacteristic
45C3×C5×Dic3C2313yes—
46C2×#713—91noHall
47#3832—51no—
48C3×#239—51no—
49C3×D3×Q8D511yescharacteristic, complemented
50D3×D12—53no—
51C4×D32—53no—
52#3799—53no—
53#3796—53no—
54C22×D15—33no—
55C2×D3×D5—33no—
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—31no—
62D60—36no—
63D60—33no—
64D60—33no—
65D60—63no—
66D60C2×D313yescomplemented
67C4×D15—33no—
68C4×D15—33no—
69C4×D15—33no—
70C4×D15—33no—
71C4×D15—63no—
72C4×D15C2×D313yescomplemented
73C4×C5×D3C2×D313yes—
74C5×Dic6C2×D313yescomplemented
75C3×D20—33no—
76C3×C4×D5—33no—
77#1202—36no—
78#1200—33no—
79D3×Dic5—31no—
80D5×Dic3—33no—
81D3×#41—151no—
82C2×#239—151no—
83C2×C32×C5C2411yescharacteristic
84C3⋊D15—23no—
85C3×D15—23no—
86C3×C5×D3#4112yescomplemented
87C2×C5×Q8—31no—
88#713—31no—
89#713—93no—
90#713—93no—
91#713—91no—
92C2×D20—93no—
93C2×C4×D5—93no—
94C2×D32—53no—
95C32×Q8C2×D511yescharacteristic, complemented
96C4⋊(C3⋊D3)—53no—
97C4×C3⋊D3—53no—
98C3×D12—53no—
99C3×C4×D3—53no—
100C3×C4×D3C2×D513yes—
101C3×Dic6C2×D513yescomplemented
102#626—56no—
103#624—53no—
104D3×Dic3—51no—
105D3×Dic3—53no—
106C22×C3×C5—31no—
107C2×D15—33no—
108C2×D15—33no—
109C2×D15—63no—
110C2×D15—66no—
111C2×D15C22×D313yes—
112C2×C5×D3C22×D311yescharacteristic
113C2×C3×D5—33no—
114D3×D5—66no—
115C3×C4×C5—23no—
116C3×C4×C5—33no—
117C3×C4×C5C22×D313yes—
118C3×C4×C5C22×D313yes—
119Dic15—31no—
120Dic15—31no—
121Dic15—61no—
122Dic15C22×D311yescharacteristic
123C3×Dic5—31no—
124C5×Dic3C22×D313yes—
125C3×#41—151no—
126C2×C3×Q8—31no—
127#239—51no—
128#239—153no—
129#239—153no—
130#239—151no—
131#239—151no—
132#239—301no—
133D3×Q8D1511yescharacteristic, complemented
134#237—153no—
135D3×D4—153no—
136#235—153no—
137C2×D12—153no—
138C2×C4×D3—153no—
139C2×C4×D3—153no—
140C32×C5C2×#4111yescharacteristic, complemented, Hall
141C22×D5—93no—
142C5×Q8—33no—
143C5×Q8D3211yescharacteristic, complemented
144C2×C4×C5—33no—
145C2×Dic5—91no—
146D20—33no—
147D20—96no—
148D20—93no—
149C4×D5—33no—
150C4×D5—93no—
151C4×D5—93no—
152C4×D5—93no—
153C2×C3⋊D3—53no—
154C2×C3×D3—53no—
155C2×C3×D3C22×D511yescharacteristic
156D32—106no—
157C32×C4C22×D513yes—
158C3⋊Dic3—51no—
159C3×Dic3—51no—
160C3×Dic3C22×D513yes—
161C2×#41—451noSylow
162C2×C3×C5—21no—
163C2×C3×C5—32no—
164C2×C3×C5C23×D311yescharacteristic
165C2×C3×C5C23×D311yescharacteristic
166D15—23no—
167D15—63no—
168D15—63no—
169D15—123no—
170C3×D5—63no—
171C5×D3#23912yescomplemented
172C22×D3—153no—
173C22×D3—153no—
174C3×Q8—21no—
175C3×Q8—33no—
176C3×Q8D3×D511yescharacteristic, complemented
177C3×Q8C2×D1511yescharacteristic, complemented
178C3×D4—153no—
179C2×C3×C4—33no—
180C2×C3×C4—153no—
181#67—156no—
182C2×Dic3—151no—
183C2×Dic3—153no—
184D12—53no—
185D12—156no—
186D12—153no—
187D12—153no—
188D12—303no—
189C4×D3—53no—
190C4×D3—153no—
191C4×D3—153no—
192C4×D3—151no—
193C4×D3—153no—
194C4×D3—153no—
195C4×D3—153no—
196C4×D3—303no—
197C4×D3C2×D1513yes—
198Dic6C2×D1513yescomplemented
199C22×C5—31no—
200C2×D5—33no—
201C2×D5—93no—
202C2×D5—186no—
203C4×C5—33no—
204C4×C5C2×D3213yes—
205Dic5—31no—
206Dic5—91no—
207C2×C32C23×D511yescharacteristic
208C3⋊D3—103no—
209C3×D3—103no—
210C3×D3#71312yescomplemented
211#41—151no—
212#41—453no—
213#41—453no—
214#41—451no—
215C2×Q8—31no—
216C2×D4—453no—
217C22×C4—453no—
218C3×C5—21no—
219C3×C5C2×#23911yescharacteristic, complemented
220C3×C5D3×#4111yescharacteristic, complemented
221C22×C3—31no—
222C22×C3—153no—
223C2×D3—53no—
224C2×D3—153no—
225C2×D3—153no—
226C2×D3—303no—
227C2×D3—306no—
228C2×D3—306no—
229C2×D3C22×D1511yescharacteristic
230C3×C4—23no—
231C3×C4—33no—
232C3×C4—151no—
233C3×C4C2×D3×D513yes—
234C3×C4C22×D1513yes—
235Dic3—51no—
236Dic3—151no—
237Dic3—151no—
238Dic3—301no—
239Dic3C22×D1513yes—
240C2×C5—32no—
241C2×C5C22×D3211yescharacteristic
242D5—63no—
243D5—183no—
244C32C2×#71311yescharacteristic, complemented, Sylow
245C23—453no—
246Q8—33no—
247Q8D3×D1511yescharacteristic, complemented
248D4—153no—
249D4—456no—
250D4—453no—
251C2×C4—33no—
252C2×C4—153no—
253C2×C4—453no—
254C2×C4—453no—
255C2×C4—451no—
256C2×C4—453no—
257C2×C3—21no—
258C2×C3—32no—
259C2×C3—303no—
260C2×C3C22×D3×D511yescharacteristic
261C2×C3C23×D1511yescharacteristic
262D3—103no—
263D3—303no—
264D3—303no—
265D3—603no—
266D3#682112yesdirect product factor
267C5D3×#23911yescharacteristic, complemented, Sylow
268C22—31no—
269C22—153no—
270C22—453no—
271C22—906no—
272C4—33no—
273C4—151no—
274C4—451no—
275C4C2×D3×D1513yes—
276C3—21no—
277C3D3×#71311yescharacteristic, complemented
278C3C2×#682111yescharacteristic, complemented
279C2—32no—
280C2—303no—
281C2—903no—
282C2C22×D3×D1511yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×D15
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C5×Q8C22
hypercenterQ8D3×D15
hypocenterC32×C5C2×#41
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. #10624287C2semidirect
2. C3×#6821C2semidirect
3. C3×C5×D3×Q8C2semidirect
4. D3×D60C2semidirect
5. C4×D3×D15C2semidirect
6. #10624201C2semidirect
7. #10624194C2semidirect
11. C32×C5×Q8C22semidirect
12. C4⋊(C3⋊D15)C22semidirect
13. C4×C3⋊D15C22semidirect
14. C3×D60C22semidirect
15. C3×C4×D15C22semidirect
17. C3×C5×Dic6C22semidirect
18. #67151C22semidirect
19. #67149C22semidirect
21. Dic3×D15C22semidirect
29. #6821D3direct (N × H)
32. C5×D3×Q8D3semidirect
49. C3×D3×Q8D5semidirect
58. C3×C5×Q8C2×D3semidirect
59. C3×C5×Q8C2×D3semidirect
66. D60C2×D3semidirect
72. C4×D15C2×D3semidirect
74. C5×Dic6C2×D3semidirect
86. C3×C5×D3#41semidirect
95. C32×Q8C2×D5semidirect
101. C3×Dic6C2×D5semidirect
133. D3×Q8D15semidirect
140. C32×C5C2×#41semidirect
143. C5×Q8D32semidirect
171. C5×D3#239semidirect
176. C3×Q8D3×D5semidirect
177. C3×Q8C2×D15semidirect
198. Dic6C2×D15semidirect
210. C3×D3#713semidirect
219. C3×C5C2×#239semidirect
220. C3×C5D3×#41semidirect
244. C32C2×#713semidirect
247. Q8D3×D15semidirect
266. D3#6821direct (N × H)
267. C5D3×#239semidirect
277. C3D3×#713semidirect
278. C3C2×#6821semidirect

Maximal subgroups

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

subgroupindex
#106242872
C3×#68212
C3×C5×D3×Q82
D3×D602
C4×D3×D152
#106242012
#106241942
C2×#68213
D3×#7133
D3×#2395