Finite Group Databasebeta

← C3×#15762 #10634369

Subgroups — C3×#15762 #10634369

Order 768.

2018 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C212yesmaximal
2C2×C3×#395C214yesmaximal
3C3×#1446C211yesmaximal, characteristic
4C3×#1444C214yesmaximal
5C3×#1436C214yesmaximal
6#15762C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×Q8C2212yes—
8C23×C3×Q8C2211yescharacteristic
9C2×C3×C4⋊Q8—28no—
10C2×C3×C4⋊Q8—216no—
11C22×C3×C4⋊C4C2214yes—
12C22×C3×C4⋊C4C2214yes—
13C22×C3×C4⋊C4C2214yes—
14C22×C3×C42C2212yes—
15C3×#395—216no—
16C2×C3×#95C2218yes—
17C2×C3×#95C2214yes—
18C2×C3×#95C2212yes—
19C2×C3×#95C2214yes—
20C22×C4⋊Q8C2×C312yes—
21C2×#395C2×C314yes—
22#1446C2×C311yescharacteristic
23#1444C2×C314yes—
24#1436C2×C314yes—
25C22×C3×Q8—28no—
26C22×C3×Q8—28no—
27C22×C3×Q8—24no—
28C22×C3×Q8—28no—
29C22×C3×Q8—42no—
30C22×C3×Q8—44no—
31C22×C3×Q8D414yes—
32C23×C3×C4C2314yes—
33C23×C3×C4C2311yescharacteristic
34C23×C3×C4C2314yes—
35C23×C3×C4C2312yes—
36C23×C3×C4C2314yes—
37C3×C4⋊Q8—264no—
38C2×C3×C4⋊C4—28no—
39C2×C3×C4⋊C4—28no—
40C2×C3×C4⋊C4—28no—
41C2×C3×C4⋊C4—28no—
42C2×C3×C4⋊C4—28no—
43C2×C3×C4⋊C4—216no—
44C2×C3×C4⋊C4—44no—
45C2×C3×C4⋊C4D4116yes—
46C2×C3×C42—24no—
47C2×C3×C42—28no—
48C3×#95—28no—
49C3×#95—216no—
50C3×#95—42no—
51C3×#95Q8116yescomplemented
52C23×Q8C22×C312yes—
53C23×Q8C22×C311yescharacteristic
54C2×C4⋊Q8—28no—
55C2×C4⋊Q8—216no—
56C22×C4⋊C4C22×C314yes—
57C22×C4⋊C4C22×C314yes—
58C22×C4⋊C4C22×C314yes—
59C22×C42C22×C312yes—
60#395—216no—
61C2×#95C22×C318yes—
62C2×#95C22×C314yes—
63C2×#95C22×C312yes—
64C2×#95C22×C314yes—
65C24×C3C2411yescharacteristic
66C2×C3×Q8—216no—
67C2×C3×Q8—216no—
68C2×C3×Q8—44no—
69C2×C3×Q8—416no—
70C2×C3×Q8—432no—
71C2×C3×Q8—48no—
72C2×C3×Q8—48no—
73C22×C3×C4—24no—
74C22×C3×C4—24no—
75C22×C3×C4—24no—
76C22×C3×C4—24no—
77C22×C3×C4—28no—
78C22×C3×C4—28no—
79C22×C3×C4—28no—
80C22×C3×C4—24no—
81C22×C3×C4—24no—
82C22×C3×C4—24no—
83C22×C3×C4—24no—
84C22×C3×C4—22no—
85C22×C3×C4—28no—
86C22×C3×C4—24no—
87C22×C3×C4—28no—
88C22×C3×C4C2×D412yes—
89C22×C3×C4C2×D418yes—
90C22×C3×C4C2×D418yes—
91C22×C3×C4C2×D414yes—
92C22×C3×C4C2×D418yes—
93C22×C3×C4C2×Q818yes—
94C22×C3×C4C2×Q8116yes—
95C3×C4⋊C4—232no—
96C3×C4⋊C4—232no—
97C3×C4⋊C4—416no—
98C3×C42—216no—
99C22×Q8—28no—
100C22×Q8—28no—
101C22×Q8—24no—
102C22×Q8—28no—
103C22×Q8—42no—
104C22×Q8—44no—
105C22×Q8C3×D414yes—
106C23×C4C23×C314yes—
107C23×C4C23×C311yescharacteristic
108C23×C4C23×C314yes—
109C23×C4C23×C312yes—
110C23×C4C23×C314yes—
111C4⋊Q8—264no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—216no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4C3×D4116yes—
120C2×C42—24no—
121C2×C42—28no—
122#95—28no—
123#95—216no—
124#95—42no—
125#95C3×Q8116yescomplemented
126C23×C3C22×D411yescharacteristic
127C23×C3C22×D414yes—
128C23×C3C22×Q814yes—
129C23×C3H(5,𝔽2)12yes—
130C23×C32−1+414yes—
131C3×Q8—416no—
132C3×Q8—432no—
133C2×C3×C4—28no—
134C2×C3×C4—24no—
135C2×C3×C4—24no—
136C2×C3×C4—28no—
137C2×C3×C4—216no—
138C2×C3×C4—216no—
139C2×C3×C4—216no—
140C2×C3×C4—24no—
141C2×C3×C4—28no—
142C2×C3×C4—22no—
143C2×C3×C4—28no—
144C2×C3×C4—216no—
145C2×C3×C4—216no—
146C2×C3×C4—44no—
147C2×C3×C4—44no—
148C2×C3×C4—48no—
149C2×C3×C4—44no—
150C2×C3×C4—48no—
151C2×C3×C4—44no—
152C2×C3×C4—44no—
153C2×C3×C4#12018yes—
154C2×C3×C4C4⋊Q8116yescomplemented
155C24C24×C311yescharacteristic
156C2×Q8—216no—
157C2×Q8—216no—
158C2×Q8—44no—
159C2×Q8—416no—
160C2×Q8—432no—
161C2×Q8—48no—
162C2×Q8—48no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—28no—
168C22×C4—28no—
169C22×C4—28no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—28no—
176C22×C4—24no—
177C22×C4—28no—
178C22×C4C2×C3×D412yes—
179C22×C4C2×C3×D418yes—
180C22×C4C2×C3×D418yes—
181C22×C4C2×C3×D414yes—
182C22×C4C2×C3×D418yes—
183C22×C4C2×C3×Q818yes—
184C22×C4C2×C3×Q8116yes—
185C4⋊C4—232no—
186C4⋊C4—232no—
187C4⋊C4—416no—
188C42—216no—
189C22×C3C2×#12012yes—
190C22×C3C2×C4⋊Q814yes—
191C22×C3#53411yescharacteristic
192C22×C3#53614yes—
193C22×C3#54414yes—
194C22×C3D4214yes—
195C22×C3D4×Q818yes—
196C22×C3D4×Q814yes—
197C22×C3Q8212yes—
198C22×C3#56312yes—
199C3×C4—28no—
200C3×C4—216no—
201C3×C4—42no—
202C3×C4—48no—
203C3×C4—48no—
204C23C22×C3×D411yescharacteristic
205C23C22×C3×D414yes—
206C23C22×C3×Q814yes—
207C23C3×H(5,𝔽2)12yes—
208C23C3×2−1+414yes—
209Q8—416no—
210Q8—432no—
211C2×C4—28no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—28no—
215C2×C4—216no—
216C2×C4—216no—
217C2×C4—216no—
218C2×C4—24no—
219C2×C4—28no—
220C2×C4—22no—
221C2×C4—28no—
222C2×C4—216no—
223C2×C4—216no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—48no—
228C2×C4—44no—
229C2×C4—48no—
230C2×C4—44no—
231C2×C4C3×#12018yes—
232C2×C4C3×C4⋊Q8116yescomplemented
233C2×C3#240311yescharacteristic
234C2×C3#243414yes—
235C2×C3#255114yes—
236C2×C3#267714yes—
237C2×C3#273412yes—
238C22C2×C3×#12012yes—
239C22C2×C3×C4⋊Q814yes—
240C22C3×#53411yescharacteristic
241C22C3×#53614yes—
242C22C3×#54414yes—
243C22C3×D4214yes—
244C22C3×D4×Q818yes—
245C22C3×D4×Q814yes—
246C22C3×Q8212yes—
247C22C3×#56312yes—
248C4—28no—
249C4—216no—
250C4—42no—
251C4—48no—
252C4—48no—
253C3#1576211yescharacteristic, direct product factor, Sylow
254C2C3×#240311yescharacteristic
255C2C3×#243414yes—
256C2C3×#255114yes—
257C2C3×#267714yes—
258C2C3×#273412yes—
subgroupG/H
center, socle, prime radicalC24×C3C24
commutator, Frattini, cosocleC24C24×C3
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
6. #15762C3direct (N × H)
51. C3×#95Q8semidirect
125. #95C3×Q8semidirect
154. C2×C3×C4C4⋊Q8semidirect
232. C2×C4C3×C4⋊Q8semidirect
253. C3#15762direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C2×C3×#3952
C3×#14462
C3×#14442
C3×#14362
#157623