Finite Group Databasebeta

← #13087529

Subgroups — #13087529

Order 1440.

268 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 62 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×#6821C211yesmaximal, characteristic, complemented
2#10624201C211yesmaximal, characteristic, complemented
3#10623911C211yesmaximal, characteristic, complemented
4C3×C5×#216C211yesmaximal, characteristic, complemented
5#10623801C211yesmaximal, characteristic, complemented
6#10623799C211yesmaximal, characteristic, complemented
7C3⋊C8×D15C211yesmaximal, characteristic, complemented
8#91808—31nomaximal
9#91516—31nomaximal
10C32×C5×Q8C2211yescharacteristic
11C4⋊(C3⋊D15)C2211yescharacteristic
12C3×D60—21no—
13C3×D60C2211yescharacteristic
14C3×C4×D15—21no—
15C3×C4×D15C2211yescharacteristic
16C3×C5×Dic6C2211yescharacteristic
17#67151—21no—
18#67149—21no—
19C3⋊(C3⋊#168)C2211yescharacteristic
20C3×C5×C3⋊C8C2211yescharacteristic
21#63949—51nomaximal, Hall
22#6821—31no—
23#6821D311yescharacteristic, complemented
24C3×#713—31no—
25#6773—31no—
26C3×C5×Q16—31no—
27#6717—31no—
28#6717—31no—
29#6717—31no—
30#6717—61no—
31D120—31no—
32C8×D15—31no—
33C5×#216D311yescharacteristic, complemented
34#6655—31no—
35#6653—31no—
36D5×C3⋊C8—31no—
37C2×C3⋊D15—21no—
38C2×C3×D15—21no—
39C2×C3×D15D411yescharacteristic
40C32×C4×C5—21no—
41C32×C4×C5C2311yescharacteristic
42C3×Dic15D411yescharacteristic
43C3×C5×Dic3—21no—
44#4065—91noHall
45C3×#239—51no—
46#3799—51no—
47#3755—51no—
48C3×#216D511yescharacteristic, complemented
49#3715—51no—
50#3714—51no—
51D3×C3⋊C8—51no—
52C3×C5×Q8—21no—
53C3×C5×Q8—31no—
54C3×C5×Q8C2×D311yescharacteristic
55C3×C5×Q8C2×D311yescharacteristic
56D60—21no—
57D60—31no—
58D60—31no—
59D60—61no—
60D60—61no—
61D60C2×D311yescharacteristic
62C4×D15—21no—
63C4×D15—61no—
64C4×D15C2×D311yescharacteristic
65C5×Dic6C2×D311yescharacteristic
66C3×D20—31no—
67C3×D20—61no—
68C3×C4×D5—31no—
69C3×C4×D5—61no—
70#1202—61no—
71#1200—61no—
72C3×C5×C8—31no—
73C3⋊#168—31no—
74C3⋊#168—31no—
75C3⋊#168—61no—
76C5×C3⋊C8C2×D311yescharacteristic
77#950—151no—
78#919—151no—
79C2×C32×C5C2×D411yescharacteristic
80C3⋊D15—41no—
81C3×D15—21no—
82C3×D15—41no—
83#713—31no—
84#713—91no—
85C5×Q16—31no—
86#688—91no—
87#688—91no—
88D40—91no—
89C8×D5—91no—
90C32×Q8C2×D511yescharacteristic
91C4⋊(C3⋊D3)—51no—
92C3×D12—51no—
93C3×D12—101no—
94C3×C4×D3—51no—
95C3×C4×D3—101no—
96C3×Dic6C2×D511yescharacteristic
97#626—101no—
98#624—101no—
99C3⋊(C3⋊C8)—51no—
100C3×C3⋊C8C2×D511yescharacteristic
101C2×D15—21no—
102C2×D15—61no—
103C2×D15—61no—
104C2×D15—121no—
105C2×D15#6711yescharacteristic
106C2×C3×D5—31no—
107C2×C3×D5—61no—
108C3×C4×C5—21no—
109C3×C4×C5—21no—
110C3×C4×C5—21no—
111C3×C4×C5—41no—
112C3×C4×C5—61no—
113C3×C4×C5C22×D311yescharacteristic
114C3×C4×C5C22×D311yescharacteristic
115Dic15#6711yescharacteristic
116C3×Dic5—31no—
117C5×Dic3—21no—
118C3×#41—151no—
119#239—51no—
120#239—151no—
121#235—151no—
122C3×Q16—31no—
123#216D1511yescharacteristic, complemented
124#215—151no—
125#215—151no—
126#215—151no—
127#215—301no—
128#214—151no—
129#213—151no—
130C2×C3⋊C8—151no—
131D24—151no—
132D3×C8—151no—
133C32×C5#13511yescharacteristic, complemented, Hall
134C5×Q8—31no—
135C5×Q8D3211yescharacteristic
136D20—31no—
137D20—61no—
138D20—91no—
139D20—181no—
140C4×D5—31no—
141C4×D5—61no—
142C4×D5—181no—
143C5×C8—31no—
144#168—91no—
145C2×C3⋊D3—101no—
146C2×C3×D3—51no—
147C2×C3×D3—101no—
148C32×C4—21no—
149C32×C4C22×D511yescharacteristic
150C3×Dic3—21no—
151C3×Dic3—51no—
152#135—451noSylow
153C2×C3×C5—21no—
154C2×C3×C5D3×D411yescharacteristic
155C2×C3×C5C2×#6711yescharacteristic
156D15—21no—
157D15—41no—
158D15—121no—
159D15—121no—
160D15—241no—
161C3×D5—61no—
162C3×D5—121no—
163C3×Q8—21no—
164C3×Q8—31no—
165C3×Q8D3×D511yescharacteristic
166C3×Q8C2×D1511yescharacteristic
167C3×D4—151no—
168C3×D4—301no—
169C2×C3×C4—151no—
170C2×C3×C4—301no—
171#67—301no—
172D12—51no—
173D12—101no—
174D12—151no—
175D12—151no—
176D12—301no—
177D12—301no—
178C4×D3—51no—
179C4×D3—101no—
180C4×D3—301no—
181C4×D3—301no—
182Dic6C2×D1511yescharacteristic
183C3×C8—31no—
184C3⋊C8—151no—
185C3⋊C8—151no—
186C3⋊C8—301no—
187C3⋊C8C2×D1511yescharacteristic
188C2×D5—31no—
189C2×D5—61no—
190C2×D5—181no—
191C4×C5—21no—
192C4×C5—61no—
193C4×C5C2×D3211yescharacteristic
194Dic5—31no—
195C2×C32D4×D511yescharacteristic
196C3⋊D3—201no—
197C3×D3—101no—
198C3×D3—201no—
199#41—151no—
200#41—451no—
201Q16—31no—
202SD16—451no—
203SD16—451no—
204D8—451no—
205C2×C8—451no—
206C3×C5—21no—
207C3×C5#91911yescharacteristic, complemented
208C3×C5#95011yescharacteristic, complemented
209C22×C3—151no—
210C22×C3—301no—
211C2×D3—51no—
212C2×D3—101no—
213C2×D3—301no—
214C2×D3—301no—
215C2×D3—601no—
216C3×C4—21no—
217C3×C4—21no—
218C3×C4—21no—
219C3×C4—41no—
220C3×C4—61no—
221C3×C4—151no—
222C3×C4C2×D3×D511yescharacteristic
223C3×C4C22×D1511yescharacteristic
224Dic3—21no—
225Dic3—51no—
226C2×C5D3×#6711yescharacteristic
227D5—61no—
228D5—121no—
229D5—361no—
230C32#406511yescharacteristic, complemented, Sylow
231Q8—31no—
232Q8D3×D1511yescharacteristic
233D4—151no—
234D4—301no—
235D4—451no—
236D4—901no—
237C2×C4—151no—
238C2×C4—301no—
239C2×C4—901no—
240C8—31no—
241C8—451no—
242C2×C3—21no—
243C2×C3—301no—
244C2×C3—601no—
245C2×C3D5×#6711yescharacteristic
246C2×C3D4×D1511yescharacteristic
247D3—101no—
248D3—201no—
249D3—601no—
250D3—601no—
251D3—1201no—
252C5#6394911yescharacteristic, complemented, Sylow
253C22—151no—
254C22—301no—
255C22—901no—
256C4—21no—
257C4—61no—
258C4—151no—
259C4C2×D3×D1511yescharacteristic
260C3—21no—
261C3#9151611yescharacteristic, complemented
262C3#9180811yescharacteristic, complemented
263C2—301no—
264C2—601no—
265C2—1801no—
266C2#67×D1511yescharacteristic
subgroupG/H
centerC2#67×D15
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×D3×D15
FittingC32×C5×Q8C22
socleC2×C32×C5C2×D4
hypercenterQ8D3×D15
hypocenterC32×C5#135
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×#6821C2semidirect
2. #10624201C2semidirect
3. #10623911C2semidirect
4. C3×C5×#216C2semidirect
5. #10623801C2semidirect
6. #10623799C2semidirect
7. C3⋊C8×D15C2semidirect
23. #6821D3semidirect
33. C5×#216D3semidirect
48. C3×#216D5semidirect
123. #216D15semidirect
133. C32×C5#135semidirect
207. C3×C5#919semidirect
208. C3×C5#950semidirect
230. C32#4065semidirect
252. C5#63949semidirect
261. C3#91516semidirect
262. C3#91808semidirect

Maximal subgroups

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

subgroupindex
C3×#68212
#106242012
#106239112
C3×C5×#2162
#106238012
#106237992
C3⋊C8×D152
#918083
#915163
#639495