Finite Group Databasebeta

← C3×#91498 #13086646

Subgroups — C3×#91498 #13086646

Order 1440.

294 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 76 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×D4×D15C211yesmaximal, characteristic, complemented
2C3×#6766C211yesmaximal, characteristic, complemented
3C3×C5×#214C211yesmaximal, characteristic, complemented
4C32×#686C211yesmaximal, characteristic, complemented
5C3×#6660C211yesmaximal, characteristic, complemented
6C3×#6653C211yesmaximal, characteristic, complemented
7C3×#6651C211yesmaximal, characteristic, complemented
8C3×#4060—31nomaximal
9#91498C311yesmaximal, characteristic, direct product factor
10C22×C3×D15—21no—
11C32×C5×D4C2211yescharacteristic
12C3×#1220—21no—
13C3×D60C2211yescharacteristic
14C3×C4×D15C2211yescharacteristic
15C3×C5×Dic6C2211yescharacteristic, complemented
16C32×D20C2211yescharacteristic, complemented
17C3×#1202—21no—
18C3×D5×Dic3—21no—
19C3×C5×C3⋊C8C2211yescharacteristic, complemented
20C32×#168C2211yescharacteristic, complemented
21C3×#911—51nomaximal, Hall
22D4×D15C2×C311yescharacteristic, complemented
23C3×#713—31no—
24C3×D4×D5—31no—
25#6766C2×C311yescharacteristic, complemented
26C3×C5×SD16—31no—
27C5×#214C2×C311yescharacteristic, complemented
28C3×#688—31no—
29C3×#686—21no—
30C3×#686D311yescharacteristic, complemented
31C3×#686C2×C311yescharacteristic, complemented
32C3×D40—31no—
33C3×#676—31no—
34#6660C2×C311yescharacteristic, complemented
35#6653C2×C311yescharacteristic, complemented
36#6651C2×C311yescharacteristic, complemented
37C22×C32×C5—21no—
38C2×C3×D15—21no—
39C2×C3×D15—42no—
40C2×C3×D15D411yescharacteristic
41C2×C32×D5—21no—
42C32×C4×C5C2311yescharacteristic
43C3×Dic15D411yescharacteristic
44C3×C5×Dic3—21no—
45#4060—31noHall
46C3×#237—51no—
47C3×D3×D4—51no—
48C32×D8—51no—
49C3×#214D511yescharacteristic, complemented
50C3×#213—51no—
51C3×#204—51no—
52C3×ΓL(1,𝔽25)—51no—
53C22×D15—21no—
54C22×C3×D5—61no—
55C3×C5×Q8—31no—
56C3×C5×D4—21no—
57C3×C5×D4C2×D311yescharacteristic
58C3×C5×D4C22×C311yescharacteristic
59#1220—21no—
60D60C22×C311yescharacteristic
61C4×D15C22×C311yescharacteristic
62C5×Dic6C22×C311yescharacteristic, complemented
63C3×C22⋊D5—61no—
64C3×D20—21no—
65C3×D20—31no—
66C3×D20—61no—
67C3×D20C2×D311yescharacteristic, complemented
68C3×D20C22×C311yescharacteristic, complemented
69C3×C4×D5—31no—
70C3×C4×D5—61no—
71#1202—21no—
72D5×Dic3—21no—
73C3×C5×C8—31no—
74C3×#168—21no—
75C3×#168C2×D311yescharacteristic, complemented
76C3×#168C22×C311yescharacteristic, complemented
77C5×C3⋊C8C22×C311yescharacteristic, complemented
78C3×AGL(1,ℤ/8)—151no—
79#911—51no—
80C2×C32×C5—41no—
81C2×C32×C5C2×D411yescharacteristic
82C3×D15—21no—
83C3×D15—41no—
84C32×D5—41no—
85#713—31no—
86D4×D5—31no—
87C5×SD16—31no—
88#688—31no—
89#686C3×D311yescharacteristic, complemented
90D40—31no—
91#676—31no—
92C22×C3×D3—101no—
93C32×D4—51no—
94C32×D4C2×D511yescharacteristic
95C3×#67—101no—
96C3×#67—101no—
97C2×C3×Dic3—101no—
98C3×D12—51no—
99C3×C4×D3—51no—
100C3×Dic6C2×D511yescharacteristic, complemented
101C32×C8—51no—
102C3×C3⋊C8C2×D511yescharacteristic, complemented
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5—41no—
106C2×D15—21no—
107C2×D15—42no—
108C2×D15C3×D411yescharacteristic
109C2×C3×D5—21no—
110C2×C3×D5—21no—
111C2×C3×D5—31no—
112C2×C3×D5—41no—
113C2×C3×D5—61no—
114C2×C3×D5—122no—
115C3×C4×C5—21no—
116C3×C4×C5—61no—
117C3×C4×C5C22×D311yescharacteristic
118C3×C4×C5C23×C311yescharacteristic
119Dic15C3×D411yescharacteristic
120C3×Dic5—31no—
121C5×Dic3—21no—
122C3×#41—151no—
123C2×C3×D4—151no—
124#237—51no—
125D3×D4—51no—
126C3×SD16—31no—
127C3×SD16—151no—
128C3×D8—51no—
129C3×D8—51no—
130C3×D8—101no—
131C3×D8—151no—
132C3×M16—151no—
133#214C3×D511yescharacteristic, complemented
134#213—51no—
135#204—51no—
136ΓL(1,𝔽25)—51no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C22×D5—61no—
139C5×Q8—31no—
140C5×D4C2×C3×D311yescharacteristic
141C22⋊D5—61no—
142D20—31no—
143D20—61no—
144D20C2×C3×D311yescharacteristic, complemented
145C4×D5—31no—
146C4×D5—61no—
147C5×C8—31no—
148#168C2×C3×D311yescharacteristic, complemented
149C22×C32—21no—
150C22×C32—101no—
151C2×C3×D3—51no—
152C2×C3×D3—101no—
153C2×C3×D3—202no—
154C32×C4C22×D511yescharacteristic
155C3×Dic3—21no—
156C3×Dic3—51no—
157AGL(1,ℤ/8)—151noSylow
158C2×C3×C5—21no—
159C2×C3×C5—41no—
160C2×C3×C5—41no—
161C2×C3×C5—81no—
162C2×C3×C5D3×D411yescharacteristic
163C2×C3×C5C2×C3×D411yescharacteristic
164D15—21no—
165D15—41no—
166C3×D5—41no—
167C3×D5—41no—
168C3×D5—61no—
169C3×D5—81no—
170C3×D5—121no—
171C23×C3—301no—
172C22×D3—101no—
173C3×Q8—31no—
174C3×D4—21no—
175C3×D4—51no—
176C3×D4—51no—
177C3×D4—101no—
178C3×D4—151no—
179C3×D4—301no—
180C3×D4—301no—
181C3×D4D3×D511yescharacteristic
182C3×D4C2×C3×D511yescharacteristic
183C2×C3×C4—151no—
184C2×C3×C4—301no—
185#67—101no—
186#67—101no—
187C2×Dic3—101no—
188D12—51no—
189C4×D3—51no—
190Dic6C2×C3×D511yescharacteristic, complemented
191C3×C8—31no—
192C3×C8—51no—
193C3×C8—51no—
194C3×C8—101no—
195C3⋊C8C2×C3×D511yescharacteristic, complemented
196C22×C5—21no—
197C2×D5—21no—
198C2×D5—31no—
199C2×D5—61no—
200C2×D5—122no—
201C4×C5—61no—
202C4×C5C22×C3×D311yescharacteristic
203Dic5—31no—
204C2×C32—41no—
205C2×C32—201no—
206C2×C32D4×D511yescharacteristic
207C3×D3—101no—
208C3×D3—201no—
209#41—151no—
210C2×D4—151no—
211SD16—31no—
212SD16—151no—
213D8—51no—
214D8—151no—
215M16—151no—
216C3×C5—21no—
217C3×C5#91111yescharacteristic, complemented
218C3×C5C3×AGL(1,ℤ/8)11yescharacteristic, complemented
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—41no—
222C22×C3—101no—
223C22×C3—101no—
224C22×C3—151no—
225C22×C3—201no—
226C22×C3—301no—
227C22×C3—602no—
228C2×D3—51no—
229C2×D3—101no—
230C2×D3—202no—
231C3×C4—21no—
232C3×C4—61no—
233C3×C4—151no—
234C3×C4C2×D3×D511yescharacteristic
235C3×C4C22×C3×D511yescharacteristic
236Dic3—21no—
237Dic3—51no—
238C2×C5—41no—
239C2×C5C3×D3×D411yescharacteristic
240D5—41no—
241D5—61no—
242D5—121no—
243C32#406011yescharacteristic, complemented, Sylow
244C23—301no—
245Q8—31no—
246D4—51no—
247D4—151no—
248D4—301no—
249D4—301no—
250D4C3×D3×D511yescharacteristic
251C2×C4—151no—
252C2×C4—301no—
253C8—31no—
254C8—51no—
255C2×C3—21no—
256C2×C3—41no—
257C2×C3—41no—
258C2×C3—81no—
259C2×C3—201no—
260C2×C3—201no—
261C2×C3—301no—
262C2×C3—401no—
263C2×C3—601no—
264C2×C3#679011yescharacteristic
265C2×C3C3×D4×D511yescharacteristic
266D3—101no—
267D3—201no—
268C5C3×#91111yescharacteristic, complemented, Sylow
269C22—21no—
270C22—101no—
271C22—151no—
272C22—301no—
273C22—602no—
274C4—61no—
275C4—151no—
276C4C2×C3×D3×D511yescharacteristic
277C3—21no—
278C3#9149811yescharacteristic, direct product factor
279C3C3×#406011yescharacteristic, complemented
280C2—41no—
281C2—201no—
282C2—301no—
283C2—601no—
284C2C3×#679011yescharacteristic
subgroupG/H
centerC2×C3#6790
commutator, cosocleC3×C4×C5C23×C3
FrattiniC4C2×C3×D3×D5
FittingC32×C5×D4C22
socleC2×C32×C5C2×D4
hypercenterC3×D4D3×D5
hypocenterC3×C5C3×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×D4×D15C2semidirect
2. C3×#6766C2semidirect
3. C3×C5×#214C2semidirect
4. C32×#686C2semidirect
5. C3×#6660C2semidirect
6. C3×#6653C2semidirect
7. C3×#6651C2semidirect
9. #91498C3direct (N × H)
15. C3×C5×Dic6C22semidirect
16. C32×D20C22semidirect
19. C3×C5×C3⋊C8C22semidirect
20. C32×#168C22semidirect
22. D4×D15C2×C3semidirect
25. #6766C2×C3semidirect
27. C5×#214C2×C3semidirect
30. C3×#686D3semidirect
31. C3×#686C2×C3semidirect
34. #6660C2×C3semidirect
35. #6653C2×C3semidirect
36. #6651C2×C3semidirect
49. C3×#214D5semidirect
62. C5×Dic6C22×C3semidirect
67. C3×D20C2×D3semidirect
68. C3×D20C22×C3semidirect
75. C3×#168C2×D3semidirect
76. C3×#168C22×C3semidirect
77. C5×C3⋊C8C22×C3semidirect
89. #686C3×D3semidirect
100. C3×Dic6C2×D5semidirect
102. C3×C3⋊C8C2×D5semidirect
133. #214C3×D5semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
144. D20C2×C3×D3semidirect
148. #168C2×C3×D3semidirect
190. Dic6C2×C3×D5semidirect
195. C3⋊C8C2×C3×D5semidirect
217. C3×C5#911semidirect
218. C3×C5C3×AGL(1,ℤ/8)semidirect
243. C32#4060semidirect
268. C5C3×#911semidirect
278. C3#91498direct (N × H)
279. C3C3×#4060semidirect

Maximal subgroups

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

subgroupindex
C3×D4×D152
C3×#67662
C3×C5×#2142
C32×#6862
C3×#66602
C3×#66532
C3×#66512
C3×#40603
#914983
C3×#9115