Finite Group Databasebeta

← #13086202

Subgroups — #13086202

Order 1440.

310 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C2×#67153C211yesmaximal, characteristic
2C2×#67141C211yesmaximal, characteristic
3#10623831C211yesmaximal, characteristic
4Dic3×Dic15C211yesmaximal, characteristic
5C5×Dic3⋊Dic3C211yesmaximal, characteristic
6C3×Dic3⋊Dic5C211yesmaximal, characteristic
7C3×#6669C211yesmaximal, characteristic
8#91470—31nomaximal
9#91328—31nomaximal
10C2×C3⋊Dic15C2211yescharacteristic
11C2×C5×C3⋊Dic3C2211yescharacteristic
12C2×C3×Dic15C2211yescharacteristic
13C2×C3×Dic15C2211yescharacteristic
14C2×C3×C5×Dic3C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C32×Dic5C2211yescharacteristic
17#67153—22no—
18#67141—22no—
19#63840—51nomaximal, Hall
20C2×Dic30—31no—
21C2×#1204—31no—
22C2×#1204—31no—
23C2×#1204—31no—
24C2×#1204—61no—
25C4×Dic15—31no—
26C5×C4⋊Dic3—31no—
27C5×Dic3⋊C4—31no—
28C3×C4⋊Dic5—31no—
29C3×#683—31no—
30Dic3⋊Dic5D311yescharacteristic
31#6669—31no—
32#6669D311yescharacteristic
33#6668—31no—
34Dic3×Dic5—31no—
35C22×C32×C5C2311yescharacteristic
36C3⋊Dic15D412yes—
37C5×C3⋊Dic3—21no—
38C3×Dic15—21no—
39C3×Dic15Q812yescomplemented
40C3×C5×Dic3—21no—
41C3×C5×Dic3Q812yescomplemented
42C32×Dic5—21no—
43#4034—91noHall
44C2×#634—51no—
45C2×#627—51no—
46C3×Dic3⋊C4—51no—
47C3×Dic3⋊C4—51no—
48Dic3⋊Dic3—51no—
49Dic3⋊Dic3D511yescharacteristic
50Dic32—51no—
51C2×C3×C4×C5—31no—
52C2×C3×C4×C5—31no—
53C2×Dic15—31no—
54C2×Dic15—31no—
55C2×Dic15—61no—
56C2×Dic15C2×D311yescharacteristic
57C2×Dic15C2×D311yescharacteristic
58Dic30—62no—
59C2×C5×Dic3—31no—
60C2×C5×Dic3—31no—
61C2×C5×Dic3—61no—
62C2×C5×Dic3C2×D311yescharacteristic
63C2×C5×Dic3C2×D311yescharacteristic
64C2×C3×Dic5—21no—
65C2×C3×Dic5—31no—
66C2×C3×Dic5—31no—
67C2×C3×Dic5C2×D311yescharacteristic
68C2×C3×Dic5C2×D311yescharacteristic
69#1204—62no—
70#1204—62no—
71#1204—62no—
72#1204—64no—
73#888—151no—
74#888—151no—
75C2×C32×C5C2×D411yescharacteristic
76C2×C32×C5C2×Q811yescharacteristic
77C2×C32×C5C2×Q811yescharacteristic
78C2×Dic10—91no—
79C2×Dic10—91no—
80C5×C4⋊C4—91no—
81C4⋊Dic5—31no—
82#683—31no—
83#683—91no—
84C4×Dic5—91no—
85C2×C32×C4—51no—
86C2×C3⋊Dic3—51no—
87C2×C3⋊Dic3C2×D511yescharacteristic
88#634—102no—
89C2×C3×Dic3—51no—
90C2×C3×Dic3—51no—
91C2×C3×Dic3C2×D511yescharacteristic
92C2×C3×Dic3C2×D511yescharacteristic
93#627—102no—
94C22×C3×C5—21no—
95C22×C3×C5C22×D311yescharacteristic
96C22×C3×C5C22×D311yescharacteristic
97C3×C4×C5—32no—
98C3×C4×C5—61no—
99Dic15—21no—
100Dic15—32no—
101Dic15—32no—
102Dic15—62no—
103Dic15Dic612yescomplemented
104C3×Dic5—22no—
105C3×Dic5—21no—
106C3×Dic5—21no—
107C3×Dic5—32no—
108C3×Dic5—61no—
109C5×Dic3—21no—
110C5×Dic3—62no—
111C5×Dic3—61no—
112C5×Dic3—61no—
113C5×Dic3Dic612yescomplemented
114C2×Dic6—151no—
115C2×Dic6—151no—
116C2×Dic6—151no—
117C2×Dic6—151no—
118C2×Dic6—301no—
119C3×C4⋊C4—151no—
120C3×C4⋊C4—151no—
121C4⋊Dic3—31no—
122C4⋊Dic3—151no—
123Dic3⋊C4—31no—
124Dic3⋊C4—51no—
125Dic3⋊C4—51no—
126Dic3⋊C4—151no—
127C4×Dic3—151no—
128C4×Dic3—151no—
129C32×C5C4⋊Q811yescharacteristic, complemented, Hall
130C2×C4×C5—31no—
131C2×C4×C5—31no—
132C2×C4×C5—91no—
133C2×Dic5—31no—
134C2×Dic5—31no—
135C2×Dic5—91no—
136C2×Dic5D3211yescharacteristic
137Dic10—182no—
138Dic10—182no—
139C22×C32C22×D511yescharacteristic
140C32×C4—101no—
141C3⋊Dic3—21no—
142C3⋊Dic3—52no—
143C3×Dic3—21no—
144C3×Dic3—52no—
145C3×Dic3—101no—
146C3×Dic3Dic1012yescomplemented
147C4⋊Q8—451noSylow
148C2×C3×C5—21no—
149C2×C3×C5—21no—
150C2×C3×C5—21no—
151C2×C3×C5C2×Dic611yescharacteristic
152C2×C3×C5C2×Dic611yescharacteristic
153C2×C3×C5D3×D411yescharacteristic
154C2×C3×C5D3×D411yescharacteristic
155C2×C3×C5D3×Q811yescharacteristic
156C2×C3×C5D3×Q811yescharacteristic
157C2×C3×C4—31no—
158C2×C3×C4—31no—
159C2×C3×C4—51no—
160C2×C3×C4—51no—
161C2×C3×C4—101no—
162C2×C3×C4—151no—
163C2×C3×C4—151no—
164C2×Dic3—31no—
165C2×Dic3—31no—
166C2×Dic3—51no—
167C2×Dic3—51no—
168C2×Dic3—61no—
169C2×Dic3—151no—
170C2×Dic3—151no—
171C2×Dic3—301no—
172C2×Dic3D3×D511yescharacteristic
173C2×Dic3D3×D511yescharacteristic
174Dic6—302no—
175Dic6—302no—
176Dic6—302no—
177Dic6—304no—
178Dic6—302no—
179C22×C5C2×D3211yescharacteristic
180C4×C5—32no—
181C4×C5—61no—
182C4×C5—181no—
183Dic5—21no—
184Dic5—32no—
185Dic5—61no—
186Dic5—92no—
187C2×C32C2×Dic1011yescharacteristic
188C2×C32D4×D511yescharacteristic
189C2×C32Q8×D511yescharacteristic
190C2×Q8—451no—
191C2×Q8—451no—
192C4⋊C4—91no—
193C4⋊C4—151no—
194C4⋊C4—151no—
195C4⋊C4—451no—
196C42—451no—
197C3×C5—21no—
198C3×C5#88811yescharacteristic, complemented
199C3×C5#88811yescharacteristic, complemented
200C22×C3—21no—
201C22×C3C2×D3×D511yescharacteristic
202C22×C3C2×D3×D511yescharacteristic
203C3×C4—32no—
204C3×C4—61no—
205C3×C4—102no—
206C3×C4—101no—
207C3×C4—101no—
208C3×C4—152no—
209C3×C4—301no—
210Dic3—21no—
211Dic3—52no—
212Dic3—62no—
213Dic3—61no—
214Dic3—61no—
215Dic3—101no—
216Dic3—152no—
217Dic3—152no—
218Dic3—302no—
219Dic3#120412yescomplemented
220C2×C5D3×Dic611yescharacteristic
221C2×C5D3×Dic611yescharacteristic
222C2×C5#381111yescharacteristic
223C32#403411yescharacteristic, complemented, Sylow
224Q8—902no—
225Q8—902no—
226C2×C4—31no—
227C2×C4—31no—
228C2×C4—51no—
229C2×C4—91no—
230C2×C4—151no—
231C2×C4—151no—
232C2×C4—451no—
233C2×C3—21no—
234C2×C3—21no—
235C2×C3—21no—
236C2×C3D5×Dic611yescharacteristic
237C2×C3D3×Dic1011yescharacteristic
238C2×C3D15⋊Q811yescharacteristic
239C2×C3#677711yescharacteristic
240C2×C3C2×#120411yescharacteristic
241C2×C3#679011yescharacteristic
242C5#6384011yescharacteristic, complemented, Sylow
243C22D32×D511yescharacteristic
244C4—32no—
245C4—61no—
246C4—101no—
247C4—152no—
248C4—181no—
249C4—301no—
250C4—452no—
251C3—21no—
252C3#9132811yescharacteristic, complemented
253C3#9147011yescharacteristic, complemented
254C2#1062403011yescharacteristic
255C2D3×#120411yescharacteristic
256C2#1062403811yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×D5
commutator, Fitting, socle, prime radical, cosocleC22×C32×C5C23
hypocenterC32×C5C4⋊Q8
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
39. C3×Dic15Q8semidirect
41. C3×C5×Dic3Q8semidirect
103. Dic15Dic6semidirect
113. C5×Dic3Dic6semidirect
129. C32×C5C4⋊Q8semidirect
146. C3×Dic3Dic10semidirect
198. C3×C5#888semidirect
199. C3×C5#888semidirect
219. Dic3#1204semidirect
223. C32#4034semidirect
242. C5#63840semidirect
252. C3#91328semidirect
253. C3#91470semidirect

Maximal subgroups

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

subgroupindex
C2×#671532
C2×#671412
#106238312
Dic3×Dic152
C5×Dic3⋊Dic32
C3×Dic3⋊Dic52
C3×#66692
#914703
#913283
#638405