Finite Group Databasebeta

← #10725610

Subgroups — #10725610

Order 768.

1290 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#395C211yesmaximal, characteristic
2#72134C212yesmaximal
3#71565C213yesmaximal
4#71563C213yesmaximal
5#69407C216yesmaximal
6#14601—31nomaximal, Sylow
7C23×C3×Q8C2211yescharacteristic
8C22×C3×C4⋊C4C2213yes—
9C22×C4⋊Dic3C2211yescharacteristic
10C22×C4⋊Dic3C2213yes—
11C22×C4×Dic3C2213yes—
12C3×#395C418yescomplemented
13C2×C3×#95C2213yes—
14C2×C3⋊#95C2216yes—
15C2×C3⋊#95C2216yes—
16C2×C3⋊#95C2216yes—
17C2×C3⋊#95C2213yes—
18C2×#395D311yescharacteristic
19#1446—32no—
20#1441—33no—
21#1436—36no—
22C4⋊#95—33no—
23C22×C3×Q8—43no—
24C22×C3×Q8—43no—
25C22×C3×Q8C2×C414yes—
26C23×C3×C4C2313yes—
27C23×C3×C4C2311yescharacteristic
28C23×C3×C4C2313yes—
29C23×Dic3C2312yes—
30C23×Dic3C2316yes—
31C2×C3×C4⋊C4—43no—
32C2×C3×C4⋊C4—43no—
33C2×C3×C4⋊C4C2×C4112yes—
34C2×C4⋊Dic3—43no—
35C2×C4⋊Dic3—43no—
36C2×C4⋊Dic3—43no—
37C2×C4⋊Dic3Q8112yescomplemented
38C2×C4×Dic3—26no—
39C2×C4×Dic3—43no—
40C2×C4×Dic3—43no—
41C3×#95C2×C4112yes—
42C3⋊#95—212no—
43C3⋊#95—43no—
44C3⋊#95—46no—
45C3⋊#95—46no—
46C23×Q8C2×D311yescharacteristic
47C22×C4⋊C4—33no—
48C22×C4⋊C4—31no—
49C22×C4⋊C4C2×D313yes—
50C22×C42—33no—
51#395Dic318yescomplemented
52C2×#95—36no—
53C2×#95—33no—
54C2×#95—36no—
55C2×#95—36no—
56C2×#95C2×D313yes—
57C24×C3C2411yescharacteristic
58C2×C3×Q8—412no—
59C2×C3×Q8—412no—
60C2×C3×Q8—44no—
61C22×C3×C4—23no—
62C22×C3×C4—23no—
63C22×C3×C4—23no—
64C22×C3×C4—23no—
65C22×C3×C4—23no—
66C22×C3×C4—26no—
67C22×C3×C4—26no—
68C22×C3×C4—23no—
69C22×C3×C4—23no—
70C22×C3×C4—23no—
71C22×C3×C4C22×C416yes—
72C22×C3×C4C22×C412yes—
73C22×C3×C4C22×C416yes—
74C22×C3×C4C2×Q816yes—
75C22×C3×C4#4116yes—
76C22×Dic3—26no—
77C22×Dic3—26no—
78C22×Dic3—26no—
79C22×Dic3—26no—
80C22×Dic3—26no—
81C22×Dic3—26no—
82C22×Dic3—26no—
83C22×Dic3—26no—
84C22×Dic3—22no—
85C22×Dic3C2×Q8112yes—
86C3×C4⋊C4—412no—
87C4⋊Dic3—412no—
88C4⋊Dic3—44no—
89C4×Dic3—412no—
90C22×Q8—43no—
91C22×Q8—43no—
92C22×Q8C2×Dic314yes—
93C23×C4—36no—
94C23×C4—32no—
95C23×C4C22×D311yescharacteristic
96C23×C4C22×D313yes—
97C23×C4C22×D313yes—
98C2×C4⋊C4—312no—
99C2×C4⋊C4—43no—
100C2×C4⋊C4—43no—
101C2×C4⋊C4—123no—
102C2×C4⋊C4—123no—
103C2×C4⋊C4—123no—
104C2×C4⋊C4C2×Dic3112yes—
105C2×C42—66no—
106C2×C42—123no—
107C2×C42—123no—
108#95—612no—
109#95—123no—
110#95—126no—
111#95—126no—
112#95C2×Dic3112yes—
113C23×C3C23×C411yescharacteristic
114C23×C3C22×Q813yes—
115C23×C3C2×#4113yes—
116C23×C3H(5,𝔽2)13yes—
117C23×C3H(5,𝔽2)13yes—
118C23×C3H(5,𝔽2)11yescharacteristic
119C23×C32−1+411yescharacteristic
120C3×Q8—416no—
121C2×C3×C4—26no—
122C2×C3×C4—26no—
123C2×C3×C4—26no—
124C2×C3×C4—26no—
125C2×C3×C4—26no—
126C2×C3×C4—26no—
127C2×C3×C4—26no—
128C2×C3×C4—26no—
129C2×C3×C4—212no—
130C2×C3×C4—43no—
131C2×C3×C4—43no—
132C2×C3×C4—46no—
133C2×C3×C4—41no—
134C2×C3×C4—43no—
135C2×C3×C4C4×Q8112yes—
136C2×Dic3—212no—
137C2×Dic3—212no—
138C2×Dic3—212no—
139C2×Dic3—46no—
140C2×Dic3—46no—
141C2×Dic3—46no—
142C2×Dic3—46no—
143C2×Dic3—46no—
144C2×Dic3—46no—
145C2×Dic3—42no—
146C24C23×D311yescharacteristic
147C2×Q8—412no—
148C2×Q8—44no—
149C2×Q8—412no—
150C22×C4—23no—
151C22×C4—23no—
152C22×C4—23no—
153C22×C4—23no—
154C22×C4—23no—
155C22×C4—26no—
156C22×C4—26no—
157C22×C4—23no—
158C22×C4—23no—
159C22×C4—23no—
160C22×C4—312no—
161C22×C4—66no—
162C22×C4—66no—
163C22×C4—66no—
164C22×C4—66no—
165C22×C4—66no—
166C22×C4—66no—
167C22×C4—66no—
168C22×C4—66no—
169C22×C4—62no—
170C22×C4D3×Q816yes—
171C22×C4#23916yes—
172C22×C4C22×Dic316yes—
173C22×C4C22×Dic312yes—
174C22×C4C22×Dic316yes—
175C4⋊C4—412no—
176C4⋊C4—1212no—
177C4⋊C4—124no—
178C42—1212no—
179C22×C3C2×C4×Q813yes—
180C22×C3#51813yes—
181C22×C3#52011yescharacteristic
182C22×C3#53813yes—
183C22×C3#53913yes—
184C22×C3#54313yes—
185C22×C3#54413yes—
186C22×C3#55416yes—
187C22×C3#55513yes—
188C22×C3Q8213yes—
189C22×C3#56013yes—
190C22×C3H(3,𝔽4)11yescharacteristic
191C3×C4—212no—
192C3×C4—42no—
193C3×C4—46no—
194Dic3—412no—
195Dic3—82no—
196C23#100413yes—
197C23#100413yes—
198C23C2×D3×Q813yes—
199C23C2×#23913yes—
200C23#100911yescharacteristic
201C23#101011yescharacteristic
202C23C23×Dic311yescharacteristic
203Q8—416no—
204C2×C4—26no—
205C2×C4—26no—
206C2×C4—26no—
207C2×C4—26no—
208C2×C4—26no—
209C2×C4—26no—
210C2×C4—26no—
211C2×C4—26no—
212C2×C4—212no—
213C2×C4—41no—
214C2×C4—43no—
215C2×C4—46no—
216C2×C4—43no—
217C2×C4—43no—
218C2×C4—612no—
219C2×C4—612no—
220C2×C4—612no—
221C2×C4—126no—
222C2×C4—126no—
223C2×C4—126no—
224C2×C4—126no—
225C2×C4—126no—
226C2×C4—126no—
227C2×C4—122no—
228C2×C4Q8×Dic3112yes—
229C2×C3#232913yes—
230C2×C3#236913yes—
231C2×C3#237511yescharacteristic
232C2×C3#278413yes—
233C2×C3#278511yescharacteristic
234C2×C3#279213yes—
235C2×C3#281111yescharacteristic
236C22#559411yescharacteristic
237C22#559513yes—
238C22#562313yes—
239C22#563413yes—
240C22#564413yes—
241C22#564513yes—
242C22#572213yes—
243C22#573016yes—
244C22#573413yes—
245C22#579813yes—
246C22C2×Q8×Dic313yes—
247C22#582711yescharacteristic
248C4—212no—
249C4—46no—
250C4—42no—
251C4—1212no—
252C4—242no—
253C3#1460111yescharacteristic, complemented, Sylow
254C2#7723311yescharacteristic
255C2#7723513yes—
256C2#7724011yescharacteristic
257C2#7727213yes—
258C2#8258011yescharacteristic
259C2#8274513yes—
260C2#8323013yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#395C2
socle, prime radical, cosocleC24×C3C24
hypercenterC2×#395D3
hypocenterC3#14601
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
12. C3×#395C4semidirect
37. C2×C4⋊Dic3Q8semidirect
51. #395Dic3semidirect
253. C3#14601semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3952
#721342
#715652
#715632
#694072
#146013