Finite Group Databasebeta

← #10704584

Subgroups — #10704584

Order 768.

490 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 107 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×C3×#382C211yesmaximal, characteristic
2#71552C212yesmaximal
3#67632C214yesmaximal, complemented
4#11511—31nomaximal, Sylow
5C22×C3×C4⋊C4C411yescharacteristic
6C22×C3×C4⋊C4C2211yescharacteristic
7C22×C3×C42C411yescharacteristic
8C3×#382—22no—
9C3×#382C412yes—
10C3×#382C2212yes—
11C2×C3×#95—22no—
12#5207—22no—
13#4528—22no—
14#4528C2214yes—
15C2×#382D311yescharacteristic
16#1463—32no—
17#1324—34no—
18C23×C3×C4—21no—
19C23×C3×C4—21no—
20C23×C3×C4C2×C411yescharacteristic
21C23×C3×C4D412yes—
22C2×C3×C4⋊C4—21no—
23C2×C3×C4⋊C4—41no—
24C2×C3×C4⋊C4—41no—
25C2×C3×C4⋊C4—41no—
26C2×C3×C4⋊C4—41no—
27C2×C3×C4⋊C4C2×C412yes—
28C2×C3×C4⋊C4C2×C411yescharacteristic
29C2×C3×C4⋊C4D412yes—
30C2×C3×C4⋊C4C2311yescharacteristic
31C2×C3×C42—21no—
32C2×C3×C42—24no—
33C2×C3×C42C2×C412yes—
34C2×#208—42no—
35C3×#95—24no—
36C3×#95—24no—
37#830—24no—
38#830—42no—
39C22×C4⋊C4Dic311yescharacteristic
40C22×C4⋊C4C2×D311yescharacteristic
41C22×C42Dic311yescharacteristic
42#407—62no—
43#382—22no—
44#382Dic312yes—
45#382C2×D312yes—
46C2×#95—22no—
47#324—34no—
48#324—62no—
49C24×C3C22⋊C411yescharacteristic
50C22×C3×C4—21no—
51C22×C3×C4—21no—
52C22×C3×C4—22no—
53C22×C3×C4—22no—
54C22×C3×C4—24no—
55C22×C3×C4—22no—
56C22×C3×C4—22no—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4—42no—
60C22×C3×C4—42no—
61C22×C3×C4—44no—
62C22×C3×C4—44no—
63C22×C3×C4—41no—
64C22×C3×C4—41no—
65C22×C3×C4C22⋊C411yescharacteristic
66C22×C3×C4C22⋊C412yes—
67C22×C3×C4C22×C411yescharacteristic
68C22×C3×C4C2×D412yes—
69C3×C4⋊C4—41no—
70C3×C4⋊C4—42no—
71C3×C4⋊C4—41no—
72C3×C4⋊C4—42no—
73C3×C4⋊C4—42no—
74C3×C42—24no—
75C3×C42—22no—
76C3×C42M1614yescomplemented
77#208—84no—
78C2×C3⋊C8—44no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4C2×Dic311yescharacteristic
82C23×C4#6712yes—
83C2×M16—122no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4C2×Dic312yes—
90C2×C4⋊C4C2×Dic311yescharacteristic
91C2×C4⋊C4#6712yes—
92C2×C4⋊C4C22×D311yescharacteristic
93C2×C42—21no—
94C2×C42—24no—
95C2×C42C2×Dic312yes—
96C22⋊C8—64no—
97C22⋊C8—122no—
98#95—24no—
99#95—24no—
100C23×C3—22no—
101C23×C3—22no—
102C23×C3—21no—
103C23×C3—21no—
104C23×C3#9911yescharacteristic
105C23×C3#9911yescharacteristic
106C23×C3C2×C22⋊C411yescharacteristic
107C2×C3×C4—24no—
108C2×C3×C4—24no—
109C2×C3×C4—21no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—21no—
113C2×C3×C4—24no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—48no—
117C2×C3×C4—42no—
118C2×C3×C4—42no—
119C2×C3×C4—44no—
120C2×C3×C4—42no—
121C2×C3×C4—42no—
122C2×C3×C4—44no—
123C2×C3×C4—42no—
124C2×C3×C4—82no—
125C2×C3×C4—82no—
126C2×C3×C4—82no—
127C2×C3×C4#10112yes—
128C2×C3×C4C2×M1612yes—
129C3⋊C8—84no—
130C24#21711yescharacteristic
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—24no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4#21711yescharacteristic
147C22×C4#21712yes—
148C22×C4C22×Dic311yescharacteristic
149C22×C4C2×#6712yes—
150M16—244no—
151C2×C8—124no—
152C4⋊C4—41no—
153C4⋊C4—42no—
154C4⋊C4—41no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C42—24no—
158C42—22no—
159C42#20814yescomplemented
160C22×C3—24no—
161C22×C3—24no—
162C22×C3—21no—
163C22×C3—21no—
164C22×C3—21no—
165C22×C3—21no—
166C22×C3—21no—
167C22×C3—21no—
168C22×C3#35412yes—
169C22×C3#35512yes—
170C22×C3#40711yescharacteristic
171C22×C3C2×#9911yescharacteristic
172C22×C3C2×#10111yescharacteristic
173C3×C4—22no—
174C3×C4—22no—
175C3×C4—44no—
176C3×C4—42no—
177C3×C4—42no—
178C3×C4—82no—
179C23—21no—
180C23—21no—
181C23—22no—
182C23—22no—
183C23#83411yescharacteristic
184C23#83411yescharacteristic
185C23C2×#21711yescharacteristic
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—48no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4#83612yes—
207C2×C4C2×#20812yes—
208C8—244no—
209C2×C3—21no—
210C2×C3—22no—
211C2×C3—22no—
212C2×C3—21no—
213C2×C3#146311yescharacteristic
214C2×C3C2×#35411yescharacteristic
215C2×C3C2×#35511yescharacteristic
216C22—24no—
217C22—24no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22#454112yes—
225C22#454212yes—
226C22#520711yescharacteristic
227C22C2×#83411yescharacteristic
228C22C2×#83611yescharacteristic
229C4—22no—
230C4—22no—
231C4—44no—
232C4—42no—
233C4—42no—
234C4—82no—
235C3#1151111yescharacteristic, complemented, Sylow
236C2—21no—
237C2—21no—
238C2—22no—
239C2—22no—
240C2#7155211yescharacteristic
241C2C2×#454111yescharacteristic
242C2C2×#454211yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC2×C3×#382C2
socleC22×C3C2×#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenterC2×#382D3
hypocenterC3#11511
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
3. #67632C2semidirect
76. C3×C42M16semidirect
159. C42#208semidirect
235. C3#11511semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3822
#715522
#676322
#115113