Finite Group Databasebeta

← #10692872

Subgroups — #10692872

Order 768.

468 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 111 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#82436C211yesmaximal, characteristic
3#71777C212yesmaximal
4#71625C211yesmaximal, characteristic
5#67621C212yesmaximal
6#9452—31nomaximal, Sylow
7C2×C3×C4⋊Q8C412yes—
8C2×C3×C4⋊Q8C2211yescharacteristic
9C2×C3×C4×Q8C2211yescharacteristic
10C3×#398—21no—
11C3×#398—21no—
12C3×#391C412yes—
13C3×#389—21no—
14C3×#389—21no—
15C3×C4×C4⋊C4C2211yescharacteristic
16#5024—24no—
17#4926—21no—
18#4926C2212yes—
19#4926C2212yes—
20#4925—21no—
21#4546—24no—
22#4528—21no—
23#4528—21no—
24#2983—31no—
25#2546D311yescharacteristic
26#1558—32no—
27#1467—31no—
28#1285—32no—
29C22×C3×Q8C2×C412yes—
30C22×C3×Q8C2×C411yescharacteristic
31C3×C4⋊Q8—22no—
32C3×C4⋊Q8—22no—
33C3×C4⋊Q8—44no—
34C3×C4×Q8—22no—
35C3×C4×Q8—22no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C42C2×C412yes—
45C2×C3×C42D411yescharacteristic
46C2×C3×C42D411yescharacteristic
47C2×C3×C42C2311yescharacteristic
48C3×#95—22no—
49C3×#95—22no—
50C3×#95—21no—
51#830—22no—
52#830—22no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)—22no—
56C4⋊(C3⋊C8)D414yes—
57#803—22no—
58#803—22no—
59#803—22no—
60C4×C3⋊C8—22no—
61C2×C4⋊Q8Dic312yes—
62C2×C4⋊Q8C2×D311yescharacteristic
63C2×C4×Q8C2×D311yescharacteristic
64#446—64no—
65#432—32no—
66#432—32no—
67#432—61no—
68#431—61no—
69#398—21no—
70#398—21no—
71#391Dic312yes—
72#389—21no—
73#389—21no—
74C4×C4⋊C4C2×D311yescharacteristic
75#333—64no—
76#324—61no—
77#324—61no—
78C2×C3×Q8—22no—
79C2×C3×Q8—22no—
80C2×C3×Q8—24no—
81C2×C3×Q8—41no—
82C2×C3×Q8—42no—
83C2×C3×Q8—42no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4C22⋊C411yescharacteristic
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C3×C4⋊C4—24no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—22no—
96C3×C4⋊C4—22no—
97C3×C4⋊C4—44no—
98C3×C4⋊C4—44no—
99C3×C4⋊C4—41no—
100C3×C4⋊C4—42no—
101C3×C4⋊C4—41no—
102C3×C42—21no—
103C3×C42—21no—
104C3×C42—21no—
105C3×C42—21no—
106C3×C42—42no—
107C3×C42C2×D412yes—
108C3×C42#4112yes—
109C2×C3⋊C8—24no—
110C2×C3⋊C8—42no—
111C22×Q8C2×Dic312yes—
112C22×Q8C2×Dic311yescharacteristic
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—44no—
116C4×Q8—22no—
117C4×Q8—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4C2×Dic311yescharacteristic
124C2×C4⋊C4#6711yescharacteristic
125C2×C4⋊C4#6711yescharacteristic
126C2×C42C2×Dic312yes—
127C2×C42#6711yescharacteristic
128C2×C42#6711yescharacteristic
129C2×C42C22×D311yescharacteristic
130C4⋊C8—34no—
131C4⋊C8—62no—
132C4⋊C8—62no—
133C4⋊C8—62no—
134C22⋊C8—62no—
135C22⋊C8—62no—
136#97—62no—
137#97—62no—
138#97—62no—
139C4×C8—62no—
140#95—22no—
141#95—22no—
142#95—21no—
143C23×C3C2×C22⋊C411yescharacteristic
144C3×Q8—44no—
145C3×Q8—84no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—22no—
151C2×C3×C4—24no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—41no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4—41no—
161C2×C3×C4—41no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4#10012yes—
166C2×C3×C4C4×D412yes—
167C2×C3×C4#12011yescharacteristic
168C2×C3×C4#12112yes—
169C2×C3×C4#12311yescharacteristic
170C3⋊C8—44no—
171C3⋊C8—82no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—41no—
176C2×Q8—42no—
177C2×Q8—42no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4#21711yescharacteristic
181C22×C4#21711yescharacteristic
182C22×C4#21711yescharacteristic
183C22×C4#21711yescharacteristic
184C22×C4C22×Dic311yescharacteristic
185C22×C4C2×#6711yescharacteristic
186C22×C4C2×#6711yescharacteristic
187C2×C8—64no—
188C2×C8—122no—
189C4⋊C4—24no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—42no—
203C42D3×D412yes—
204C42#23712yes—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3#38611yescharacteristic
208C22×C3C2×#10011yescharacteristic
209C22×C3#41311yescharacteristic
210C3×C4—21no—
211C3×C4—22no—
212C3×C4—22no—
213C3×C4—22no—
214C3×C4—44no—
215C3×C4—41no—
216C3×C4—41no—
217C3×C4—41no—
218C3×C4—82no—
219C3×C4#45612yes—
220C23C2×#21711yescharacteristic
221Q8—44no—
222Q8—84no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4#83312yes—
243C2×C4D4×Dic312yes—
244C2×C4#93511yescharacteristic
245C2×C4#93711yescharacteristic
246C2×C4#93912yes—
247C8—124no—
248C8—242no—
249C2×C3—21no—
250C2×C3—21no—
251C2×C3#188611yescharacteristic
252C2×C3#190011yescharacteristic
253C2×C3#190911yescharacteristic
254C22—21no—
255C22—21no—
256C22C2×#83311yescharacteristic
257C22#522011yescharacteristic
258C22#523711yescharacteristic
259C4—22no—
260C4—22no—
261C4—21no—
262C4—22no—
263C4—44no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—82no—
268C4#509512yes—
269C3#945211yescharacteristic, complemented, Sylow
270C2—21no—
271C2—21no—
272C2#7168311yescharacteristic
273C2#7178411yescharacteristic
274C2#7186511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2546C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2546D3
hypocenterC3#9452
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2546

Complements

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

NH ≅ G/Ntype
269. C3#9452semidirect

Maximal subgroups

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

subgroupindex
C3×#25462
#824362
#717772
#716252
#676212
#94523