Finite Group Databasebeta

← #10704587

Subgroups — #10704587

Order 768.

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

normal subgroups only

This group has 578 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×#384C211yesmaximal, characteristic
2#71552C212yesmaximal
3#67635C214yesmaximal, complemented
4#11512—31nomaximal, Sylow
5C22×C3×C4⋊C4—22no—
6C22×C3×C4⋊C4C411yescharacteristic
7C22×C3×C4⋊C4C2211yescharacteristic
8C22×C3×C42C411yescharacteristic
9C3×#384—22no—
10C3×#384C412yes—
11C3×#384C2212yes—
12#5207—22no—
13#4528—22no—
14#4528C2214yes—
15C2×#384D311yescharacteristic
16#1463—32no—
17#1325—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—24no—
24C2×C3×C4⋊C4—24no—
25C2×C3×C4⋊C4—41no—
26C2×C3×C4⋊C4—41no—
27C2×C3×C4⋊C4—41no—
28C2×C3×C4⋊C4—41no—
29C2×C3×C4⋊C4—48no—
30C2×C3×C4⋊C4C2×C412yes—
31C2×C3×C4⋊C4C2×C411yescharacteristic
32C2×C3×C4⋊C4D412yes—
33C2×C3×C4⋊C4C2311yescharacteristic
34C2×C3×C42—21no—
35C2×C3×C42—24no—
36C2×C3×C42C2×C412yes—
37C2×#208—42no—
38#830—24no—
39#830—42no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4Dic311yescharacteristic
42C22×C4⋊C4C2×D311yescharacteristic
43C22×C42Dic311yescharacteristic
44#407—62no—
45#384—22no—
46#384Dic312yes—
47#384C2×D312yes—
48#324—34no—
49#324—62no—
50C24×C3C22⋊C411yescharacteristic
51C22×C3×C4—24no—
52C22×C3×C4—24no—
53C22×C3×C4—24no—
54C22×C3×C4—21no—
55C22×C3×C4—21no—
56C22×C3×C4—22no—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4—22no—
60C22×C3×C4—22no—
61C22×C3×C4—22no—
62C22×C3×C4—44no—
63C22×C3×C4—44no—
64C22×C3×C4—41no—
65C22×C3×C4—41no—
66C22×C3×C4C22⋊C411yescharacteristic
67C22×C3×C4C22⋊C412yes—
68C22×C3×C4C22×C411yescharacteristic
69C22×C3×C4C2×D412yes—
70C3×C4⋊C4—41no—
71C3×C4⋊C4—42no—
72C3×C4⋊C4—41no—
73C3×C4⋊C4—42no—
74C3×C4⋊C4—48no—
75C3×C4⋊C4—48no—
76C3×C4⋊C4—42no—
77C3×C42—24no—
78C3×C42—22no—
79C3×C42M1614yescomplemented
80#208—84no—
81C2×C3⋊C8—44no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4C2×Dic311yescharacteristic
85C23×C4#6712yes—
86C2×M16—122no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—48no—
95C2×C4⋊C4C2×Dic312yes—
96C2×C4⋊C4C2×Dic311yescharacteristic
97C2×C4⋊C4#6712yes—
98C2×C4⋊C4C22×D311yescharacteristic
99C2×C42—21no—
100C2×C42—24no—
101C2×C42C2×Dic312yes—
102C22⋊C8—64no—
103C22⋊C8—122no—
104C23×C3—22no—
105C23×C3—22no—
106C23×C3—21no—
107C23×C3—21no—
108C23×C3#9911yescharacteristic
109C23×C3#9911yescharacteristic
110C23×C3C2×C22⋊C411yescharacteristic
111C2×C3×C4—24no—
112C2×C3×C4—216no—
113C2×C3×C4—24no—
114C2×C3×C4—24no—
115C2×C3×C4—24no—
116C2×C3×C4—21no—
117C2×C3×C4—24no—
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4—21no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—44no—
124C2×C3×C4—42no—
125C2×C3×C4—42no—
126C2×C3×C4—44no—
127C2×C3×C4—42no—
128C2×C3×C4—82no—
129C2×C3×C4—82no—
130C2×C3×C4—82no—
131C2×C3×C4#10112yes—
132C2×C3×C4C2×M1612yes—
133C3⋊C8—84no—
134C24#21711yescharacteristic
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4#21711yescharacteristic
151C22×C4#21712yes—
152C22×C4C22×Dic311yescharacteristic
153C22×C4C2×#6712yes—
154M16—244no—
155C2×C8—124no—
156C4⋊C4—41no—
157C4⋊C4—42no—
158C4⋊C4—41no—
159C4⋊C4—42no—
160C4⋊C4—48no—
161C4⋊C4—48no—
162C4⋊C4—42no—
163C42—24no—
164C42—22no—
165C42#20814yescomplemented
166C22×C3—24no—
167C22×C3—24no—
168C22×C3—21no—
169C22×C3—21no—
170C22×C3—21no—
171C22×C3—21no—
172C22×C3—21no—
173C22×C3—21no—
174C22×C3#35312yes—
175C22×C3#35612yes—
176C22×C3#40711yescharacteristic
177C22×C3C2×#9911yescharacteristic
178C22×C3C2×#10111yescharacteristic
179C3×C4—28no—
180C3×C4—22no—
181C3×C4—22no—
182C3×C4—42no—
183C3×C4—42no—
184C3×C4—82no—
185C23—21no—
186C23—21no—
187C23—22no—
188C23—22no—
189C23#83411yescharacteristic
190C23#83411yescharacteristic
191C23C2×#21711yescharacteristic
192C2×C4—24no—
193C2×C4—216no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—44no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4#83612yes—
213C2×C4C2×#20812yes—
214C8—244no—
215C2×C3—21no—
216C2×C3—22no—
217C2×C3—22no—
218C2×C3—21no—
219C2×C3#146311yescharacteristic
220C2×C3C2×#35311yescharacteristic
221C2×C3C2×#35611yescharacteristic
222C22—24no—
223C22—24no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22#454512yes—
231C22#454812yes—
232C22#520711yescharacteristic
233C22C2×#83411yescharacteristic
234C22C2×#83611yescharacteristic
235C4—28no—
236C4—22no—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—82no—
241C3#1151211yescharacteristic, complemented, Sylow
242C2—21no—
243C2—21no—
244C2—22no—
245C2—22no—
246C2#7155211yescharacteristic
247C2C2×#454511yescharacteristic
248C2C2×#454811yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC2×C3×#384C2
socleC22×C3C2×#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenterC2×#384D3
hypocenterC3#11512
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
3. #67635C2semidirect
79. C3×C42M16semidirect
165. C42#208semidirect
241. C3#11512semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3842
#715522
#676352
#115123