Finite Group Databasebeta

← C3×#91489 #13086637

Subgroups — C3×#91489 #13086637

Order 1440.

326 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 80 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
1C32×D4×D5C211yesmaximal, characteristic, complemented
2C3×#6772C211yesmaximal, characteristic, complemented
3C3×#6716C211yesmaximal, characteristic, complemented
4C3×C5×#213C211yesmaximal, characteristic, complemented
5C3×#6657C211yesmaximal, characteristic, complemented
6C3×#6652C211yesmaximal, characteristic, complemented
7C3×#6649C211yesmaximal, characteristic, complemented
8C3×#4057—31nomaximal
9#91489C311yesmaximal, characteristic, direct product factor
10C22×C32×D5—21no—
11C32×C5×D4C2211yescharacteristic
12C3×Dic30C2211yescharacteristic, complemented
13C3×C5×D12C2211yescharacteristic, complemented
14C32×C22⋊D5—21no—
15C32×D20C2211yescharacteristic
16C32×C4×D5C2211yescharacteristic
17C3×#1201—21no—
18C3×D3×Dic5—21no—
19C3×C3⋊#168C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21C3×#932—51nomaximal, Hall
22C3×#711—31no—
23C3×D4×D5—21no—
24C3×D4×D5D311yescharacteristic, complemented
25C3×D4×D5C2×C311yescharacteristic, complemented
26#6772C2×C311yescharacteristic, complemented
27C3×C5×D8—31no—
28#6716C2×C311yescharacteristic, complemented
29C5×#213C2×C311yescharacteristic, complemented
30C3×#687—31no—
31C3×#686—31no—
32C3×#677—31no—
33C3×#676—31no—
34#6657C2×C311yescharacteristic, complemented
35#6652C2×C311yescharacteristic, complemented
36#6649C2×C311yescharacteristic, complemented
37C22×C32×C5—21no—
38C2×C3×C5×D3—21no—
39C2×C32×D5—21no—
40C2×C32×D5—42no—
41C2×C32×D5D411yescharacteristic
42C32×C4×C5C2311yescharacteristic
43C3×Dic15—21no—
44C32×Dic5D411yescharacteristic
45#4057—31noHall
46C2×C32×D4—51no—
47C3×#235—51no—
48C3×#214—51no—
49C3×#214—51no—
50C3×#213—51no—
51C3×#213D511yescharacteristic, complemented
52C3×#208—51no—
53C22×C3×D5—22no—
54C22×C3×D5—21no—
55C22×C3×D5—21no—
56C3×C5×D4—21no—
57C3×C5×D4—31no—
58C3×C5×D4C2×D311yescharacteristic
59C3×C5×D4C22×C311yescharacteristic
60Dic30C22×C311yescharacteristic, complemented
61C5×D12C22×C311yescharacteristic, complemented
62C3×C22⋊D5—22no—
63C3×C22⋊D5—21no—
64C3×C22⋊D5—21no—
65C3×C22⋊D5—61no—
66C2×C3×Dic5—61no—
67C3×D20—21no—
68C3×D20C2×D311yescharacteristic
69C3×D20C22×C311yescharacteristic
70C3×C4×D5—21no—
71C3×C4×D5C2×D311yescharacteristic
72C3×C4×D5C22×C311yescharacteristic
73C3×Dic10—31no—
74#1201—21no—
75D3×Dic5—21no—
76C3×C5×C8—31no—
77C3⋊#168C22×C311yescharacteristic, complemented
78C3×#168—31no—
79C5×C3⋊C8C22×C311yescharacteristic, complemented
80C3×AGL(1,ℤ/8)—151no—
81#932—51no—
82C2×C32×C5—41no—
83C2×C32×C5C2×D411yescharacteristic
84C3×C5×D3—41no—
85C32×D5—21no—
86C32×D5—41no—
87#711—31no—
88D4×D5C3×D311yescharacteristic, complemented
89C5×D8—31no—
90#687—31no—
91#686—31no—
92#677—31no—
93#676—31no—
94C23×C32—101no—
95C32×D4—51no—
96C32×D4—101no—
97C32×D4C2×D511yescharacteristic
98C2×C32×C4—51no—
99C3×#67—101no—
100C3×D12C2×D511yescharacteristic, complemented
101C3×C4×D3—101no—
102C3×Dic6—51no—
103C3×C3⋊C8—51no—
104C3×C3⋊C8C2×D511yescharacteristic, complemented
105C22×C3×C5—22no—
106C22×C3×C5—21no—
107C22×C3×C5—21no—
108C22×C3×C5—61no—
109C2×C5×D3—21no—
110C2×C3×D5—22no—
111C2×C3×D5—21no—
112C2×C3×D5—21no—
113C2×C3×D5—21no—
114C2×C3×D5—44no—
115C2×C3×D5—42no—
116C2×C3×D5—42no—
117C2×C3×D5#6711yescharacteristic
118C2×C3×D5C3×D411yescharacteristic
119C3×C4×C5—21no—
120C3×C4×C5C22×D311yescharacteristic
121C3×C4×C5C23×C311yescharacteristic
122Dic15—21no—
123C3×Dic5—21no—
124C3×Dic5—61no—
125C3×Dic5#6711yescharacteristic
126C3×Dic5C3×D411yescharacteristic
127C3×#41—151no—
128C2×C3×D4—51no—
129C2×C3×D4—51no—
130C2×C3×D4—101no—
131#235—51no—
132C3×SD16—151no—
133C3×SD16—151no—
134C3×D8—31no—
135C3×D8—151no—
136C3×M16—151no—
137#214—51no—
138#214—51no—
139#213—51no—
140#213C3×D511yescharacteristic, complemented
141#208—51no—
142C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
143C22×D5—21no—
144C5×D4—31no—
145C5×D4C2×C3×D311yescharacteristic
146C22⋊D5—21no—
147C22⋊D5—61no—
148C2×Dic5—61no—
149D20C2×C3×D311yescharacteristic
150C4×D5C2×C3×D311yescharacteristic
151Dic10—31no—
152C5×C8—31no—
153#168—31no—
154C22×C32—21no—
155C22×C32—51no—
156C22×C32—101no—
157C22×C32—202no—
158C2×C3×D3—21no—
159C32×C4—51no—
160C32×C4C22×D511yescharacteristic
161C3×Dic3—101no—
162AGL(1,ℤ/8)—151noSylow
163C2×C3×C5—21no—
164C2×C3×C5—42no—
165C2×C3×C5—41no—
166C2×C3×C5—41no—
167C2×C3×C5—121no—
168C2×C3×C5C2×#6711yescharacteristic
169C2×C3×C5C2×C3×D411yescharacteristic
170C3×D5—22no—
171C3×D5—21no—
172C3×D5—21no—
173C3×D5—42no—
174C3×D5—41no—
175C3×D5—41no—
176C5×D3—41no—
177C23×C3—101no—
178C23×C3—101no—
179C23×C3—102no—
180C3×Q8—151no—
181C3×D4—21no—
182C3×D4—31no—
183C3×D4—51no—
184C3×D4—51no—
185C3×D4—101no—
186C3×D4—101no—
187C3×D4—101no—
188C3×D4—102no—
189C3×D4—301no—
190C3×D4D3×D511yescharacteristic
191C3×D4C2×C3×D511yescharacteristic
192C2×C3×C4—51no—
193C2×C3×C4—51no—
194C2×C3×C4—101no—
195C2×C3×C4—301no—
196#67—101no—
197D12C2×C3×D511yescharacteristic, complemented
198C4×D3—101no—
199Dic6—51no—
200C3×C8—31no—
201C3×C8—151no—
202C3⋊C8—51no—
203C3⋊C8C2×C3×D511yescharacteristic, complemented
204C22×C5—21no—
205C22×C5—61no—
206C2×D5—21no—
207C2×D5—42no—
208C2×D5C3×#6711yescharacteristic
209C4×C5C22×C3×D311yescharacteristic
210Dic5—61no—
211Dic5C3×#6711yescharacteristic
212C2×C32—41no—
213C2×C32—101no—
214C2×C32—201no—
215C2×C32D4×D511yescharacteristic
216C3×D3—41no—
217#41—151no—
218C2×D4—51no—
219SD16—151no—
220SD16—151no—
221D8—31no—
222D8—151no—
223M16—151no—
224C3×C5—21no—
225C3×C5#93211yescharacteristic, complemented
226C3×C5C3×AGL(1,ℤ/8)11yescharacteristic, complemented
227C22×C3—22no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—51no—
231C22×C3—51no—
232C22×C3—61no—
233C22×C3—101no—
234C22×C3—102no—
235C22×C3—101no—
236C22×C3—101no—
237C22×C3—204no—
238C22×C3—202no—
239C22×C3—202no—
240C2×D3—21no—
241C3×C4—21no—
242C3×C4—51no—
243C3×C4—51no—
244C3×C4—101no—
245C3×C4—301no—
246C3×C4C2×D3×D511yescharacteristic
247C3×C4C22×C3×D511yescharacteristic
248Dic3—101no—
249C2×C5—41no—
250C2×C5—121no—
251C2×C5C2×C3×#6711yescharacteristic
252D5—21no—
253D5—41no—
254C32#405711yescharacteristic, complemented, Sylow
255C23—101no—
256Q8—151no—
257D4—31no—
258D4—51no—
259D4—101no—
260D4—301no—
261D4C3×D3×D511yescharacteristic
262C2×C4—51no—
263C2×C4—301no—
264C8—31no—
265C8—151no—
266C2×C3—21no—
267C2×C3—42no—
268C2×C3—41no—
269C2×C3—41no—
270C2×C3—102no—
271C2×C3—101no—
272C2×C3—101no—
273C2×C3—121no—
274C2×C3—202no—
275C2×C3—201no—
276C2×C3—201no—
277C2×C3D5×#6711yescharacteristic
278C2×C3C3×D4×D511yescharacteristic
279D3—41no—
280C5C3×#93211yescharacteristic, complemented, Sylow
281C22—21no—
282C22—51no—
283C22—61no—
284C22—101no—
285C22—202no—
286C4—51no—
287C4—301no—
288C4C2×C3×D3×D511yescharacteristic
289C3—21no—
290C3#9148911yescharacteristic, direct product factor
291C3C3×#405711yescharacteristic, complemented
292C2—41no—
293C2—101no—
294C2—121no—
295C2—201no—
296C2C3×D5×#6711yescharacteristic
subgroupG/H
centerC2×C3D5×#67
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. C32×D4×D5C2semidirect
2. C3×#6772C2semidirect
3. C3×#6716C2semidirect
4. C3×C5×#213C2semidirect
5. C3×#6657C2semidirect
6. C3×#6652C2semidirect
7. C3×#6649C2semidirect
9. #91489C3direct (N × H)
12. C3×Dic30C22semidirect
13. C3×C5×D12C22semidirect
19. C3×C3⋊#168C22semidirect
20. C3×C5×C3⋊C8C22semidirect
24. C3×D4×D5D3semidirect
25. C3×D4×D5C2×C3semidirect
26. #6772C2×C3semidirect
28. #6716C2×C3semidirect
29. C5×#213C2×C3semidirect
34. #6657C2×C3semidirect
35. #6652C2×C3semidirect
36. #6649C2×C3semidirect
51. C3×#213D5semidirect
60. Dic30C22×C3semidirect
61. C5×D12C22×C3semidirect
77. C3⋊#168C22×C3semidirect
79. C5×C3⋊C8C22×C3semidirect
88. D4×D5C3×D3semidirect
100. C3×D12C2×D5semidirect
104. C3×C3⋊C8C2×D5semidirect
140. #213C3×D5semidirect
142. C32×C5AGL(1,ℤ/8)semidirect
197. D12C2×C3×D5semidirect
203. C3⋊C8C2×C3×D5semidirect
225. C3×C5#932semidirect
226. C3×C5C3×AGL(1,ℤ/8)semidirect
254. C32#4057semidirect
280. C5C3×#932semidirect
290. C3#91489direct (N × H)
291. C3C3×#4057semidirect

Maximal subgroups

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

subgroupindex
C32×D4×D52
C3×#67722
C3×#67162
C3×C5×#2132
C3×#66572
C3×#66522
C3×#66492
C3×#40573
#914893
C3×#9325