Finite Group Databasebeta

← #10692638

Subgroups — #10692638

Order 768.

644 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2591C211yesmaximal, characteristic
2#82144C211yesmaximal, characteristic, complemented
3#71673C212yesmaximal
4#71552C211yesmaximal, characteristic, complemented
5#67618C212yesmaximal, complemented
6#9410—31nomaximal, Sylow
7C2×C3×#121C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C2×#926—21no—
10C3×#400C412yescomplemented
11C3×#394C412yescomplemented
12C3×#388—22no—
13C3×#386—21no—
14C3×#386—21no—
15C3×C4×C22⋊C4C2211yescharacteristic
16C4×#217—21no—
17C4×#217C2212yes—
18#5207C2212yes—
19Dic3×C4⋊C4—24no—
20#4537—24no—
21#4528—22no—
22#2591D311yescharacteristic
23#2274—31no—
24#1507—32no—
25#1463—31no—
26#1297—32no—
27C22×C3×D4—21no—
28C22×C3×D4C2×C411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4C2311yescharacteristic
32C3×#121—22no—
33C3×#121—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C4⋊C4C2×C412yes—
38C2×C3×C4⋊C4D412yes—
39C2×C3×C22⋊C4—21no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C42—21no—
44C2×#217—22no—
45#926—28no—
46C2×C4⋊Dic3—22no—
47C2×C4×Dic3—22no—
48C2×C4×Dic3—22no—
49C2×C4×Dic3—22no—
50C2×#208—22no—
51C3×#95—22no—
52C3×#95—22no—
53C3×#95C2×C412yes—
54C3⋊#95—22no—
55C3⋊#95D414yes—
56#830—22no—
57#830—22no—
58C2×#121C2×D311yescharacteristic
59C2×#117—61no—
60C22×C4⋊C4C2×D311yescharacteristic
61#407—32no—
62#400Dic312yescomplemented
63#394Dic312yescomplemented
64#388—22no—
65#386—21no—
66#386—21no—
67C4×C4⋊C4—64no—
68C4×C22⋊C4—32no—
69C4×C22⋊C4—61no—
70C4×C22⋊C4C2×D311yescharacteristic
71#327—64no—
72#324—62no—
73C24×C3—21no—
74C24×C3C2×D411yescharacteristic
75C2×C3×D4—22no—
76C2×C3×D4—44no—
77C2×C3×D4—41no—
78C2×C3×D4—41no—
79C2×C3×D4—81no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—24no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—22no—
87C22×C3×C4—81no—
88C22×C3×C4—81no—
89C22×C3×C4C22⋊C412yes—
90C22×C3×C4C22⋊C412yes—
91C22×C3×C4C22×C411yescharacteristic
92C22×C3×C4C2×D412yes—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4#4112yes—
95C22×Dic3—24no—
96C3×C4⋊C4—24no—
97C3×C4⋊C4—24no—
98C3×C4⋊C4—22no—
99C3×C22⋊C4—42no—
100C3×C22⋊C4—44no—
101C3×C22⋊C4—42no—
102C3×C22⋊C4—82no—
103C3×C42—81no—
104#217—28no—
105C4⋊Dic3—28no—
106C4×Dic3—28no—
107C4×Dic3—82no—
108C4×Dic3—82no—
109#208—82no—
110C2×C3⋊C8—42no—
111C22×D4—21no—
112C22×D4C2×Dic311yescharacteristic
113C23×C4#6711yescharacteristic
114C23×C4#6711yescharacteristic
115C23×C4C22×D311yescharacteristic
116C2×M16—62no—
117#121—22no—
118#121—22no—
119#117—68no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—62no—
123C2×C4⋊C4C2×Dic311yescharacteristic
124C2×C4⋊C4C2×Dic312yes—
125C2×C4⋊C4#6712yes—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—62no—
131C2×C42—21no—
132C2×C42—62no—
133C2×C42—62no—
134C2×C42—62no—
135C22⋊C8—62no—
136C22⋊C8—62no—
137#95—22no—
138#95—22no—
139#95—34no—
140#95—62no—
141#95C2×Dic312yes—
142C23×C3—22no—
143C23×C3—22no—
144C23×C3—41no—
145C23×C3—42no—
146C23×C3—42no—
147C23×C3—41no—
148C23×C3—42no—
149C23×C3C2×C22⋊C411yescharacteristic
150C23×C3#12011yescharacteristic
151C23×C3#12311yescharacteristic
152C3×D4—82no—
153C3×D4—82no—
154C3×D4—82no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—24no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4—41no—
164C2×C3×C4—41no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—41no—
171C2×C3×C4—42no—
172C2×C3×C4—81no—
173C2×C3×C4#10414yes—
174C2×C3×C4C4×D412yes—
175C2×C3×C4#12112yes—
176C2×Dic3—28no—
177C2×Dic3—44no—
178C2×Dic3—44no—
179C3⋊C8—82no—
180C24—21no—
181C24C2×#6711yescharacteristic
182C2×D4—22no—
183C2×D4—44no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—81no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—24no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—64no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4#21712yes—
198C22×C4#21712yes—
199C22×C4D3×D412yes—
200C22×C4#23712yes—
201C22×C4C22×Dic311yescharacteristic
202C22×C4C2×#6711yescharacteristic
203M16—242no—
204C2×C8—122no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—22no—
208C4⋊C4—68no—
209C22⋊C4—42no—
210C22⋊C4—44no—
211C22⋊C4—42no—
212C22⋊C4—68no—
213C22⋊C4—82no—
214C42—68no—
215C42—81no—
216C42—242no—
217C42—242no—
218C22×C3—21no—
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—44no—
226C22×C3—84no—
227C22×C3—81no—
228C22×C3—81no—
229C22×C3—81no—
230C22×C3#38611yescharacteristic
231C22×C3C2×#10412yes—
232C22×C3#45412yes—
233C3×C4—24no—
234C3×C4—22no—
235C3×C4—22no—
236C3×C4—42no—
237C3×C4—82no—
238C3×C4—81no—
239Dic3—48no—
240C23—22no—
241C23—22no—
242C23—41no—
243C23—42no—
244C23—42no—
245C23—41no—
246C23—42no—
247C23#93511yescharacteristic
248C23#93711yescharacteristic
249C23C2×#21711yescharacteristic
250D4—82no—
251D4—82no—
252D4—82no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—68no—
271C2×C4—81no—
272C2×C4—124no—
273C2×C4—124no—
274C2×C4#83714yes—
275C2×C4D4×Dic312yes—
276C2×C4#93912yes—
277C8—242no—
278C2×C3—22no—
279C2×C3—22no—
280C2×C3—41no—
281C2×C3—161no—
282C2×C3#188811yescharacteristic
283C2×C3#189611yescharacteristic
284C2×C3#191311yescharacteristic
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—44no—
293C22—84no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22#506212yes—
298C22#522011yescharacteristic
299C22C2×#83712yes—
300C4—24no—
301C4—22no—
302C4—22no—
303C4—42no—
304C4—82no—
305C4—81no—
306C4—128no—
307C3#941011yescharacteristic, complemented, Sylow
308C2—22no—
309C2—22no—
310C2—41no—
311C2—161no—
312C2#7171111yescharacteristic
313C2#7172411yescharacteristic
314C2#7184811yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2591C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#2591D3
hypocenterC3#9410
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2591

Complements

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

NH ≅ G/Ntype
2. #82144C2semidirect
4. #71552C2semidirect
5. #67618C2semidirect
10. C3×#400C4semidirect
11. C3×#394C4semidirect
62. #400Dic3semidirect
63. #394Dic3semidirect
307. C3#9410semidirect

Maximal subgroups

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

subgroupindex
C3×#25912
#821442
#716732
#715522
#676182
#94103