Finite Group Databasebeta

← #10704830

Subgroups — #10704830

Order 768.

394 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C3×#2546C211yesmaximal, characteristic
2#71625C211yesmaximal, characteristic
3#71624C211yesmaximal, characteristic
4#67643C214yesmaximal
5#11568—31nomaximal, Sylow
6C2×C3×C4⋊Q8C411yescharacteristic
7C2×C3×C4⋊Q8C412yes—
8C2×C3×C4×Q8C2211yescharacteristic
9C3×#398—21no—
10C3×#398—21no—
11C3×#391C2212yes—
12C3×#389—21no—
13C3×#389—21no—
14C3×C4×C4⋊C4C411yescharacteristic
15#4926—21no—
16#4925—21no—
17#4528—21no—
18#4528—21no—
19#4528C2212yes—
20#4528C2212yes—
21#2546D311yescharacteristic
22#1467—31no—
23#1461—31no—
24#1319—34no—
25C22×C3×Q8C2×C412yes—
26C22×C3×Q8C2×C411yescharacteristic
27C3×C4⋊Q8—41no—
28C3×C4⋊Q8—41no—
29C3×C4⋊Q8—44no—
30C3×C4×Q8—41no—
31C3×C4×Q8—41no—
32C2×C3×C4⋊C4—21no—
33C2×C3×C4⋊C4—21no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C4⋊C4C2311yescharacteristic
40C2×C3×C42C2×C411yescharacteristic
41C2×C3×C42C2×C412yes—
42C2×C3×C42D411yescharacteristic
43C2×C3×C42D411yescharacteristic
44C3×#95—21no—
45C3×#95—22no—
46C3×#95—22no—
47#830—22no—
48#830—22no—
49C4⋊(C3⋊C8)—41no—
50C4⋊(C3⋊C8)—41no—
51#803—41no—
52C4×C3⋊C8—41no—
53C2×C4⋊Q8Dic311yescharacteristic
54C2×C4⋊Q8Dic312yes—
55C2×C4×Q8C2×D311yescharacteristic
56#432—61no—
57#431—61no—
58#398—21no—
59#398—21no—
60#391C2×D312yes—
61#389—21no—
62#389—21no—
63C4×C4⋊C4Dic311yescharacteristic
64#324—32no—
65#324—32no—
66#324—61no—
67#324—61no—
68C2×C3×Q8—22no—
69C2×C3×Q8—22no—
70C2×C3×Q8—24no—
71C2×C3×Q8—41no—
72C2×C3×Q8—42no—
73C2×C3×Q8—42no—
74C22×C3×C4—22no—
75C22×C3×C4—22no—
76C22×C3×C4C22⋊C411yescharacteristic
77C22×C3×C4C22⋊C411yescharacteristic
78C22×C3×C4C22⋊C411yescharacteristic
79C22×C3×C4C22⋊C411yescharacteristic
80C22×C3×C4C22×C411yescharacteristic
81C22×C3×C4C2×D411yescharacteristic
82C22×C3×C4C2×D411yescharacteristic
83C3×C4⋊C4—24no—
84C3×C4⋊C4—41no—
85C3×C4⋊C4—41no—
86C3×C4⋊C4—41no—
87C3×C4⋊C4—42no—
88C3×C4⋊C4—44no—
89C3×C4⋊C4—41no—
90C3×C4⋊C4—41no—
91C3×C4⋊C4—44no—
92C3×C42—21no—
93C3×C42—21no—
94C3×C42—21no—
95C3×C42—21no—
96C3×C42—21no—
97C3×C42—21no—
98C3×C42—42no—
99C2×C3⋊C8—42no—
100C2×C3⋊C8—42no—
101C22×Q8C2×Dic312yes—
102C22×Q8C2×Dic311yescharacteristic
103C4⋊Q8—41no—
104C4⋊Q8—41no—
105C4⋊Q8—44no—
106C4×Q8—41no—
107C4×Q8—41no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4#6711yescharacteristic
114C2×C4⋊C4#6711yescharacteristic
115C2×C4⋊C4C22×D311yescharacteristic
116C2×C42C2×Dic311yescharacteristic
117C2×C42C2×Dic312yes—
118C2×C42#6711yescharacteristic
119C2×C42#6711yescharacteristic
120C4⋊C8—121no—
121C4⋊C8—121no—
122C22⋊C8—62no—
123C22⋊C8—62no—
124#97—121no—
125C4×C8—121no—
126#95—21no—
127#95—22no—
128#95—22no—
129C23×C3C2×C22⋊C411yescharacteristic
130C3×Q8—44no—
131C3×Q8—84no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—22no—
137C2×C3×C4—24no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—42no—
145C2×C3×C4—42no—
146C2×C3×C4—41no—
147C2×C3×C4—41no—
148C2×C3×C4—41no—
149C2×C3×C4—42no—
150C2×C3×C4—42no—
151C2×C3×C4—42no—
152C2×C3×C4#9911yescharacteristic
153C2×C3×C4#9911yescharacteristic
154C2×C3×C4#10012yes—
155C2×C3×C4#13112yes—
156C3⋊C8—82no—
157C3⋊C8—82no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8—24no—
161C2×Q8—41no—
162C2×Q8—42no—
163C2×Q8—42no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4#21711yescharacteristic
167C22×C4#21711yescharacteristic
168C22×C4#21711yescharacteristic
169C22×C4#21711yescharacteristic
170C22×C4C22×Dic311yescharacteristic
171C22×C4C2×#6711yescharacteristic
172C22×C4C2×#6711yescharacteristic
173C2×C8—122no—
174C2×C8—122no—
175C4⋊C4—24no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—44no—
180C4⋊C4—42no—
181C4⋊C4—44no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—42no—
191C22×C3—21no—
192C22×C3—21no—
193C22×C3#40811yescharacteristic
194C22×C3C2×#9911yescharacteristic
195C22×C3C2×#10011yescharacteristic
196C3×C4—21no—
197C3×C4—22no—
198C3×C4—22no—
199C3×C4—44no—
200C3×C4—41no—
201C3×C4—41no—
202C3×C4—41no—
203C3×C4—41no—
204C3×C4—82no—
205C3×C4#35612yes—
206C23C2×#21711yescharacteristic
207Q8—44no—
208Q8—84no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4#83312yes—
230C2×C4#83411yescharacteristic
231C2×C4#83411yescharacteristic
232C2×C4#94812yes—
233C8—242no—
234C8—242no—
235C2×C3—21no—
236C2×C3—21no—
237C2×C3#147111yescharacteristic
238C2×C3#212911yescharacteristic
239C2×C3C2×#35611yescharacteristic
240C22—21no—
241C22—21no—
242C22C2×#83311yescharacteristic
243C22C2×#83411yescharacteristic
244C22#523411yescharacteristic
245C4—21no—
246C4—22no—
247C4—22no—
248C4—44no—
249C4—41no—
250C4—41no—
251C4—41no—
252C4—41no—
253C4—82no—
254C4#454812yes—
255C3#1156811yescharacteristic, complemented, Sylow
256C2—21no—
257C2—21no—
258C2#7167611yescharacteristic
259C2C2×#454811yescharacteristic
260C2#7187111yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2546C2
socleC22×C3C2×#99
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#2546D3
hypocenterC3#11568
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
255. C3#11568semidirect

Maximal subgroups

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

subgroupindex
C3×#25462
#716252
#716242
#676432
#115683