Finite Group Databasebeta

← C3×#82460 #11805595

Subgroups — C3×#82460 #11805595

Order 1152.

1253 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Q8C211yesmaximal, characteristic
2C2×C3×Q8×Dic3C212yesmaximal
3C3×#4984C216yesmaximal
4C3×#4936C213yesmaximal
5C3×C4×C4⋊Dic3C213yesmaximal
6C3×#2351—31nomaximal
7#82460C311yesmaximal, characteristic, direct product factor
8C22×C32×Q8C2211yescharacteristic
9C32×C4×Q8C418yescomplemented
10C2×C32×C4⋊C4C2213yes—
11C2×C32×C42C2213yes—
12C3×Q8×Dic3—28no—
13C2×C3×C4⋊Dic3C2216yes—
14C2×C3×C4⋊Dic3C2216yes—
15C2×C3×C4×Dic3C2216yes—
16C3×C3⋊#95C2211yescharacteristic
17C3×C3⋊#95C2213yes—
18C3×C3⋊#95C2216yes—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8—32no—
21C2×C3×C4×Q8D311yescharacteristic
22C2×C3×C4×Q8C2×C311yescharacteristic
23C2×Q8×Dic3C2×C312yes—
24C3×#389—36no—
25C3×#382—33no—
26C3×C4×C4⋊C4—33no—
27#4984C2×C316yes—
28#4936C2×C313yes—
29C4×C4⋊Dic3C2×C313yes—
30C2×C32×Q8C2×C414yes—
31C2×C32×Q8D414yes—
32C2×C32×Q8Q814yes—
33C22×C32×C4C2313yes—
34C22×C32×C4C2311yescharacteristic
35C22×C32×C4C2313yes—
36C22×C3×Dic3C2316yes—
37C22×C3×Dic3C2312yes—
38C32×C4⋊C4C2×C4112yes—
39C32×C42C2×C4112yes—
40C3×C4⋊Dic3—212no—
41C3×C4⋊Dic3—212no—
42C3×C4×Dic3—212no—
43#2351—31noSylow
44C22×C3×Q8—21no—
45C22×C3×Q8C2×D311yescharacteristic
46C22×C3×Q8C22×C311yescharacteristic
47C3×C4×Q8—28no—
48C3×C4×Q8—68no—
49C3×C4×Q8Dic318yescomplemented
50C3×C4×Q8C3×C418yescomplemented
51C2×C3×C4⋊C4—23no—
52C2×C3×C4⋊C4—36no—
53C2×C3×C4⋊C4—36no—
54C2×C3×C4⋊C4C2×D313yes—
55C2×C3×C4⋊C4C22×C313yes—
56C2×C3×C42—23no—
57C2×C3×C42—36no—
58C2×C3×C42C2×D313yes—
59C2×C3×C42C22×C313yes—
60Q8×Dic3—28no—
61C2×C4⋊Dic3C22×C316yes—
62C2×C4⋊Dic3C22×C316yes—
63C2×C4×Dic3C22×C316yes—
64C3×#95—31no—
65C3×#95—33no—
66C3×#95—36no—
67C3⋊#95C22×C311yescharacteristic
68C3⋊#95C22×C313yes—
69C3⋊#95C22×C316yes—
70C23×C32C2411yescharacteristic
71C32×Q8C4⋊C4116yescomplemented
72C2×C32×C4—22no—
73C2×C32×C4—26no—
74C2×C32×C4C22×C412yes—
75C2×C32×C4C22×C416yes—
76C2×C32×C4C22×C416yes—
77C2×C32×C4C2×D416yes—
78C2×C32×C4C2×Q816yes—
79C2×C3×Dic3—22no—
80C2×C3×Dic3—22no—
81C2×C3×Dic3—26no—
82C2×C3×Dic3—26no—
83C2×C3×Dic3—26no—
84C2×C3×Dic3#4114yes—
85C2×C4×Q8—32no—
86C2×C4×Q8C3×D311yescharacteristic
87#389—36no—
88#382—33no—
89C4×C4⋊C4—33no—
90C2×C3×Q8—24no—
91C2×C3×Q8—24no—
92C2×C3×Q8—24no—
93C2×C3×Q8Dic614yes—
94C2×C3×Q8D1214yes—
95C2×C3×Q8C2×Dic314yes—
96C2×C3×Q8C2×C3×C414yes—
97C2×C3×Q8C3×D414yes—
98C2×C3×Q8C3×Q814yes—
99C22×C3×C4—21no—
100C22×C3×C4—23no—
101C22×C3×C4—23no—
102C22×C3×C4—32no—
103C22×C3×C4—36no—
104C22×C3×C4C22×D311yescharacteristic
105C22×C3×C4C22×D313yes—
106C22×C3×C4C22×D313yes—
107C22×C3×C4C23×C311yescharacteristic
108C22×C3×C4C23×C313yes—
109C22×C3×C4C23×C313yes—
110C22×Dic3C23×C316yes—
111C22×Dic3C23×C312yes—
112C3×C4⋊C4—212no—
113C3×C4⋊C4—612no—
114C3×C4⋊C4—612no—
115C3×C4⋊C4C2×Dic3112yes—
116C3×C4⋊C4C2×C3×C4112yes—
117C3×C42—212no—
118C3×C42—612no—
119C3×C42C2×Dic3112yes—
120C3×C42C2×C3×C4112yes—
121C4⋊Dic3—212no—
122C4⋊Dic3—212no—
123C4×Dic3—212no—
124C22×C32C23×C411yescharacteristic
125C22×C32C22×D411yescharacteristic
126C22×C32C22×Q811yescharacteristic
127C22×C32C2×#4112yes—
128C22×C322−1+412yes—
129C32×C4—22no—
130C32×C4—26no—
131C32×C4C2×C4⋊C4112yes—
132C3×Dic3—24no—
133C3×Dic3—46no—
134C22×Q8C2×C3×D311yescharacteristic
135C4×Q8—68no—
136C4×Q8C3×Dic318yescomplemented
137C2×C4⋊C4—36no—
138C2×C4⋊C4—36no—
139C2×C4⋊C4C2×C3×D313yes—
140C2×C42—36no—
141C2×C42C2×C3×D313yes—
142#95—31no—
143#95—33no—
144#95—36no—
145C23×C3—21no—
146C23×C3C23×D311yescharacteristic
147C23×C3C24×C311yescharacteristic
148C3×Q8—216no—
149C3×Q8C4⋊Dic3116yescomplemented
150C3×Q8C3×C4⋊C4116yescomplemented
151C2×C3×C4—24no—
152C2×C3×C4—22no—
153C2×C3×C4—26no—
154C2×C3×C4—26no—
155C2×C3×C4—22no—
156C2×C3×C4—26no—
157C2×C3×C4—26no—
158C2×C3×C4—22no—
159C2×C3×C4—26no—
160C2×C3×C4—26no—
161C2×C3×C4—34no—
162C2×C3×C4—46no—
163C2×C3×C4—62no—
164C2×C3×C4—62no—
165C2×C3×C4—66no—
166C2×C3×C4—66no—
167C2×C3×C4—66no—
168C2×C3×C4C2×Dic616yes—
169C2×C3×C4C2×D1216yes—
170C2×C3×C4C22×Dic312yes—
171C2×C3×C4C22×Dic316yes—
172C2×C3×C4C22×Dic316yes—
173C2×C3×C4C22×C3×C412yes—
174C2×C3×C4C22×C3×C416yes—
175C2×C3×C4C22×C3×C416yes—
176C2×C3×C4C2×C3×D416yes—
177C2×C3×C4C2×C3×Q816yes—
178C2×Dic3—22no—
179C2×Dic3—22no—
180C2×Dic3—26no—
181C2×Dic3—26no—
182C2×Dic3—26no—
183C2×Dic3C3×#4114yes—
184C2×C32C22×C4⋊C411yescharacteristic
185C2×C32C4×#4111yescharacteristic
186C2×C32#51911yescharacteristic
187C2×C32#54812yes—
188C2×C32#55712yes—
189C2×Q8C3×Dic614yes—
190C2×Q8C3×D1214yes—
191C2×Q8C2×C3×Dic314yes—
192C22×C4—32no—
193C22×C4—36no—
194C22×C4C22×C3×D311yescharacteristic
195C22×C4C22×C3×D313yes—
196C22×C4C22×C3×D313yes—
197C4⋊C4—612no—
198C4⋊C4—612no—
199C4⋊C4C2×C3×Dic3112yes—
200C42—612no—
201C42C2×C3×Dic3112yes—
202C22×C3—21no—
203C22×C3—21no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—21no—
207C22×C3C22×Dic611yescharacteristic
208C22×C3C22×D1211yescharacteristic
209C22×C3#100712yes—
210C22×C3D3×#4112yes—
211C22×C3C23×Dic311yescharacteristic
212C22×C3C23×C3×C411yescharacteristic
213C22×C3C22×C3×D411yescharacteristic
214C22×C3C22×C3×Q811yescharacteristic
215C22×C3C2×C3×#4112yes—
216C22×C3C3×2−1+412yes—
217C3×C4—24no—
218C3×C4—22no—
219C3×C4—26no—
220C3×C4—26no—
221C3×C4—22no—
222C3×C4—212no—
223C3×C4—46no—
224C3×C4—64no—
225C3×C4—126no—
226C3×C4C2×C4⋊Dic3112yes—
227C3×C4C2×C3×C4⋊C4112yes—
228Dic3—24no—
229Dic3—46no—
230C32#235111yescharacteristic, complemented, Sylow
231C23C23×C3×D311yescharacteristic
232Q8C3×C4⋊Dic3116yescomplemented
233C2×C4—22no—
234C2×C4—26no—
235C2×C4—34no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—66no—
239C2×C4—66no—
240C2×C4—66no—
241C2×C4C2×C3×Dic616yes—
242C2×C4C2×C3×D1216yes—
243C2×C4C22×C3×Dic312yes—
244C2×C4C22×C3×Dic316yes—
245C2×C4C22×C3×Dic316yes—
246C2×C3—21no—
247C2×C3—21no—
248C2×C3—22no—
249C2×C3—21no—
250C2×C3—22no—
251C2×C3#556912yes—
252C2×C3#557612yes—
253C2×C3C22×C4⋊Dic311yescharacteristic
254C2×C3#581211yescharacteristic
255C2×C3Dic3×#4111yescharacteristic
256C2×C3C22×C3×C4⋊C411yescharacteristic
257C2×C3C3×C4×#4111yescharacteristic
258C2×C3C3×#51911yescharacteristic
259C2×C3C3×#54812yes—
260C2×C3C3×#55712yes—
261C22C22×C3×Dic611yescharacteristic
262C22C22×C3×D1211yescharacteristic
263C22C3×#100712yes—
264C22C3×D3×#4112yes—
265C22C23×C3×Dic311yescharacteristic
266C4—22no—
267C4—26no—
268C4—64no—
269C4—126no—
270C4C2×C3×C4⋊Dic3112yes—
271C3—21no—
272C3#8246011yescharacteristic, direct product factor
273C3C3×#235111yescharacteristic, complemented
274C2C3×#556912yes—
275C2C3×#557612yes—
276C2C22×C3×C4⋊Dic311yescharacteristic
277C2C3×#581211yescharacteristic
278C2C3×Dic3×#4111yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC2×C32×C4×Q8C2
socle, prime radicalC23×C32C24
cosocleC23×C3C24×C3
hypercenterC2×C3×C4×Q8D3
hypocenterC3C3×#2351
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8

Complements

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

NH ≅ G/Ntype
7. #82460C3direct (N × H)
9. C32×C4×Q8C4semidirect
49. C3×C4×Q8Dic3semidirect
50. C3×C4×Q8C3×C4semidirect
71. C32×Q8C4⋊C4semidirect
136. C4×Q8C3×Dic3semidirect
149. C3×Q8C4⋊Dic3semidirect
150. C3×Q8C3×C4⋊C4semidirect
230. C32#2351semidirect
232. Q8C3×C4⋊Dic3semidirect
272. C3#82460direct (N × H)
273. C3C3×#2351semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×Q82
C2×C3×Q8×Dic32
C3×#49842
C3×#49362
C3×C4×C4⋊Dic32
C3×#23513
#824603