Finite Group Databasebeta

← #10688708

Subgroups — #10688708

Order 768.

566 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#74053C211yesmaximal, characteristic
2#69990C212yesmaximal
3C3×#1306C211yesmaximal, characteristic
4#67686C212yesmaximal
5#67629C211yesmaximal, characteristic
6#8753—31nomaximal, Sylow
7C23×Dic6C2211yescharacteristic
8C2×#877C412yes—
9C3×#407C2212yes—
10C3×#381C2211yescharacteristic
11#5211C2211yescharacteristic
12#5207C2212yes—
13C2×#824—24no—
14#4943C412yes—
15#4646—24no—
16#2310—31no—
17#1784—32no—
18#1306—32no—
19#1306—31no—
20#1306D311yescharacteristic
21C23×C3×C4C2311yescharacteristic
22C23×Dic3C2×C411yescharacteristic
23C23×Dic3C2×C411yescharacteristic
24C22×Dic6—22no—
25C22×Dic6—21no—
26C22×Dic6—21no—
27C22×Dic6—44no—
28C22×Dic6D414yes—
29C2×C3×M16—22no—
30C2×C3×C22⋊C4C2×C412yes—
31C2×#217C2×C412yes—
32C2×Dic3⋊C4—22no—
33C2×C4×Dic3—22no—
34C2×#208—22no—
35#877—44no—
36C3×C22⋊C8—22no—
37C3×C22⋊C8—22no—
38C3×#95—22no—
39C3⋊#95—22no—
40C3⋊#95—24no—
41#830—22no—
42#830—22no—
43#824—48no—
44C23×Q8—31no—
45C2×#117—32no—
46C2×#101—64no—
47#407—32no—
48#407C2×D312yes—
49#391—64no—
50#381—31no—
51#381—32no—
52#381C2×D311yescharacteristic
53C24×C3C22×C411yescharacteristic
54C22×C3×C4—22no—
55C22×C3×C4—42no—
56C22×C3×C4C2×D412yes—
57C22×C3×C4#4112yes—
58C22×C3×C4#4111yescharacteristic
59C22×C3×C4#4111yescharacteristic
60C22×Dic3—21no—
61C22×Dic3—21no—
62C22×Dic3—24no—
63C22×Dic3—21no—
64C22×Dic3—21no—
65C22×Dic3—22no—
66C22×Dic3—42no—
67C22×Dic3C22⋊C412yes—
68C22×Dic3C22⋊C412yes—
69C2×Dic6—24no—
70C2×Dic6—24no—
71C2×Dic6—28no—
72C2×Dic6—44no—
73C2×Dic6—48no—
74C2×Dic6—44no—
75C2×Dic6—44no—
76C3×M16—44no—
77C2×C3×C8—42no—
78C3×C22⋊C4—42no—
79#217—42no—
80Dic3⋊C4—44no—
81C4×Dic3—44no—
82#208—44no—
83C2×C3⋊C8—42no—
84C22×Q8—34no—
85C22×Q8—62no—
86C22×Q8—61no—
87C22×Q8—61no—
88C22×Q8—124no—
89C23×C4—31no—
90C23×C4—31no—
91C23×C4C22×D311yescharacteristic
92C2×M16—22no—
93C2×M16—62no—
94#117—124no—
95C2×C4⋊C4—62no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4C4×D312yes—
98C2×C42—62no—
99#101—128no—
100C22⋊C8—22no—
101C22⋊C8—22no—
102C22⋊C8—62no—
103C22⋊C8—62no—
104#95—22no—
105#95—64no—
106#95—62no—
107C23×C3—22no—
108C23×C3—42no—
109C23×C3C2×C22⋊C411yescharacteristic
110C23×C3#11711yescharacteristic
111C23×C3#11711yescharacteristic
112C2×C3×C4—24no—
113C2×C3×C4—42no—
114C2×C3×C4—41no—
115C2×C3×C4—41no—
116C2×C3×C4—44no—
117C2×C3×C4—42no—
118C2×C3×C4#12312yes—
119C2×C3×C4#12312yes—
120C2×Dic3—24no—
121C2×Dic3—24no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—24no—
126C2×Dic3—42no—
127C2×Dic3—42no—
128C2×Dic3—44no—
129C2×Dic3—42no—
130C2×Dic3#9914yes—
131Dic6—48no—
132Dic6—48no—
133C3×C8—82no—
134C3⋊C8—82no—
135C24C2×C4×D311yescharacteristic
136C2×Q8—68no—
137C2×Q8—64no—
138C2×Q8—64no—
139C2×Q8—124no—
140C2×Q8—128no—
141C2×Q8—124no—
142C2×Q8—124no—
143C22×C4—22no—
144C22×C4—32no—
145C22×C4—32no—
146C22×C4—42no—
147C22×C4—64no—
148C22×C4—61no—
149C22×C4—61no—
150C22×C4—61no—
151C22×C4—61no—
152C22×C4—62no—
153C22×C4—122no—
154C22×C4#23512yes—
155C22×C4D3×D412yes—
156C22×C4#23711yescharacteristic
157C22×C4#23911yescharacteristic
158M16—44no—
159M16—124no—
160C2×C8—42no—
161C2×C8—122no—
162C4⋊C4—124no—
163C22⋊C4—42no—
164C22⋊C4—122no—
165C42—124no—
166C22×C3—22no—
167C22×C3—21no—
168C22×C3—21no—
169C22×C3—22no—
170C22×C3—22no—
171C22×C3—44no—
172C22×C3#38111yescharacteristic
173C22×C3C2×#9912yes—
174C3×C4—22no—
175C3×C4—22no—
176C3×C4—82no—
177Dic3—24no—
178Dic3—42no—
179Dic3—82no—
180C23—22no—
181C23—42no—
182C23#87211yescharacteristic
183C23D3×C22⋊C411yescharacteristic
184C23#89211yescharacteristic
185Q8—128no—
186Q8—128no—
187C2×C4—24no—
188C2×C4—34no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—44no—
193C2×C4—42no—
194C2×C4—64no—
195C2×C4—64no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—64no—
200C2×C4—122no—
201C2×C4—122no—
202C2×C4—124no—
203C2×C4—122no—
204C2×C4#88312yes—
205C2×C4#89412yes—
206C8—82no—
207C8—242no—
208C2×C3—22no—
209C2×C3—41no—
210C2×C3—41no—
211C2×C3#181911yescharacteristic
212C2×C3#183012yes—
213C22—22no—
214C22—21no—
215C22—21no—
216C22—22no—
217C22—22no—
218C22—44no—
219C22#466411yescharacteristic
220C22D3×#9912yes—
221C4—22no—
222C4—22no—
223C4—64no—
224C4—82no—
225C4—122no—
226C4—242no—
227C3#875311yescharacteristic, complemented, Sylow
228C2—22no—
229C2—41no—
230C2—41no—
231C2#6896911yescharacteristic
232C2#6902912yes—
subgroupG/H
centerC22#4664
commutator, prime radicalC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1306C2
socleC22×C3#381
cosocleC23×C3×C4C23
hypercenter#1306D3
hypocenterC3#8753
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
227. C3#8753semidirect

Maximal subgroups

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

subgroupindex
#740532
#699902
C3×#13062
#676862
#676292
#87533