Finite Group Databasebeta

← #10691216

Subgroups — #10691216

Order 768.

478 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2621C211yesmaximal, characteristic
2#82209C211yesmaximal, characteristic
3#71773C212yesmaximal, complemented
4#71571C211yesmaximal, characteristic
5#67631C212yesmaximal, complemented
6#9196—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×C22⋊Q8—21no—
9C2×C3×#117C2211yescharacteristic
10C3×#398—21no—
11C3×#398C412yes—
12C3×#397—21no—
13C3×#397C412yes—
14C3×#391—21no—
15C3×#382C2211yescharacteristic
16#5207—21no—
17#5000—24no—
18#4925C2212yes—
19#4924C2212yes—
20C4×#208—21no—
21#4547—24no—
22#4528—21no—
23#4525—21no—
24#2921—31no—
25#2621D311yescharacteristic
26#1546—32no—
27#1458—31no—
28#1286—32no—
29C22×C3×Q8—21no—
30C23×C3×C4D411yescharacteristic
31C3×C4⋊Q8—22no—
32C3×C4⋊Q8—22no—
33C3×C22⋊Q8—44no—
34C3×#117—22no—
35C3×#117—22no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4C2×C412yes—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4C2×C411yescharacteristic
44C2×C3×C22⋊C4D411yescharacteristic
45C2×C3×C42D411yescharacteristic
46C2×C3×C42C2311yescharacteristic
47C2×#208—22no—
48C3×#95—22no—
49C3×#95—22no—
50C3×#95—21no—
51C3×#95C2×C412yes—
52#830—24no—
53#830—22no—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)—22no—
56C4⋊(C3⋊C8)D414yes—
57#803—22no—
58C4×C3⋊C8—22no—
59C4×C3⋊C8—22no—
60C2×C4⋊Q8C2×D311yescharacteristic
61C2×C22⋊Q8—21no—
62C2×#117C2×D311yescharacteristic
63#433—64no—
64#431—32no—
65#423—32no—
66#407—61no—
67C4×M16—61no—
68#398—21no—
69#398Dic312yes—
70#397—21no—
71#397Dic312yes—
72#391—21no—
73#382C2×D311yescharacteristic
74#332—64no—
75#324—61no—
76#323—61no—
77C24×C3C22⋊C411yescharacteristic
78C2×C3×Q8—41no—
79C2×C3×Q8—41no—
80C2×C3×Q8—42no—
81C2×C3×Q8—42no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22×C411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C3×C4⋊C4—22no—
95C3×C4⋊C4—22no—
96C3×C4⋊C4—24no—
97C3×C4⋊C4—44no—
98C3×C4⋊C4—42no—
99C3×C4⋊C4—42no—
100C3×C22⋊C4—22no—
101C3×C22⋊C4—22no—
102C3×C22⋊C4—44no—
103C3×C42—21no—
104C3×C42—21no—
105C3×C42C2×D412yes—
106C3×C42#4112yes—
107#208—42no—
108#208—42no—
109C2×C3⋊C8—24no—
110C2×C3⋊C8—42no—
111C22×Q8—21no—
112C23×C4#6711yescharacteristic
113C2×M16—62no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C22⋊Q8—44no—
117#117—22no—
118#117—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4C2×Dic312yes—
123C2×C4⋊C4C2×Dic311yescharacteristic
124C2×C4⋊C4#6711yescharacteristic
125C2×C22⋊C4—22no—
126C2×C22⋊C4C2×Dic311yescharacteristic
127C2×C22⋊C4#6711yescharacteristic
128C2×C42#6711yescharacteristic
129C2×C42C22×D311yescharacteristic
130C4⋊C8—34no—
131C4⋊C8—62no—
132C4⋊C8—62no—
133C22⋊C8—64no—
134C22⋊C8—62no—
135#97—62no—
136C4×C8—62no—
137C4×C8—62no—
138#95—22no—
139#95—22no—
140#95—21no—
141#95C2×Dic312yes—
142C23×C3—21no—
143C23×C3—21no—
144C23×C3—21no—
145C23×C3—41no—
146C23×C3—41no—
147C23×C3C2×C22⋊C411yescharacteristic
148C3×Q8—82no—
149C3×Q8—82no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—41no—
160C2×C3×C4—41no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4C4×D412yes—
168C2×C3×C4#12011yescharacteristic
169C2×C3×C4#12112yes—
170C2×C3×C4#12311yescharacteristic
171C2×C3×C4#13512yes—
172C2×C3×C4AGL(1,ℤ/8)12yes—
173C3⋊C8—44no—
174C3⋊C8—82no—
175C24#21711yescharacteristic
176C2×Q8—42no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—42no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4#21711yescharacteristic
187C22×C4#21711yescharacteristic
188C22×C4#21711yescharacteristic
189C22×C4C22×Dic311yescharacteristic
190C22×C4C2×#6711yescharacteristic
191C22×C4C2×#6711yescharacteristic
192M16—122no—
193M16—122no—
194C2×C8—64no—
195C2×C8—122no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—24no—
199C4⋊C4—44no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C22⋊C4—22no—
203C22⋊C4—22no—
204C22⋊C4—44no—
205C42—21no—
206C42—21no—
207C42D3×D412yes—
208C42#23712yes—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—42no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3—81no—
216C22×C3#38611yescharacteristic
217C22×C3#41611yescharacteristic
218C22×C3#41811yescharacteristic
219C3×C4—22no—
220C3×C4—22no—
221C3×C4—41no—
222C3×C4—41no—
223C3×C4—41no—
224C3×C4—82no—
225C3×C4—82no—
226C3×C4#44912yes—
227C3×C4#48212yes—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—41no—
232C23—41no—
233C23C2×#21711yescharacteristic
234Q8—82no—
235Q8—82no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4#93212yes—
254C2×C4D4×Dic312yes—
255C2×C4#93511yescharacteristic
256C2×C4#93711yescharacteristic
257C2×C4#93912yes—
258C2×C4#95012yes—
259C8—124no—
260C8—242no—
261C2×C3—21no—
262C2×C3—21no—
263C2×C3—81no—
264C2×C3#187511yescharacteristic
265C2×C3#190911yescharacteristic
266C2×C3#191511yescharacteristic
267C22—21no—
268C22—21no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22#521511yescharacteristic
275C22#522011yescharacteristic
276C22#523311yescharacteristic
277C4—22no—
278C4—22no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—82no—
283C4—82no—
284C4#503512yes—
285C4#503712yes—
286C3#919611yescharacteristic, complemented, Sylow
287C2—21no—
288C2—21no—
289C2—81no—
290C2#7167811yescharacteristic
291C2#7186511yescharacteristic
292C2#7186611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2621C2
socleC22×C3#386
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2621D3
hypocenterC3#9196
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2621

Complements

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

NH ≅ G/Ntype
3. #71773C2semidirect
5. #67631C2semidirect
286. C3#9196semidirect

Maximal subgroups

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

subgroupindex
C3×#26212
#822092
#717732
#715712
#676312
#91963