Finite Group Databasebeta

← #76767

Subgroups — #76767

Order 384.

796 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#937C212yesmaximal, complemented
2C2×#885C212yesmaximal, complemented
3C3×#392C211yesmaximal, characteristic, complemented
4#5223C212yesmaximal, complemented
5#5221C212yesmaximal, complemented
6#4961C212yesmaximal, complemented
7#4955C211yesmaximal, characteristic, complemented
8#4947C211yesmaximal, characteristic, complemented
9#4670C212yesmaximal, complemented
10#2670—31nomaximal, Sylow
11C24×D3C2211yescharacteristic
12C22×C3×D4C2212yes—
13C22×C3×D4C2211yescharacteristic, complemented
14C22×#67C2212yes—
15C22×#67C2212yes—
16C22×Dic6C2211yescharacteristic, complemented
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2212yescomplemented
19C2×#217C2212yes—
20C2×#217C2211yescharacteristic
21C2×#217C2212yescomplemented
22C2×#217C2212yescomplemented
23#937—28no—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×C4⋊Dic3C2212yescomplemented
29C2×C4×Dic3C2212yescomplemented
30#885—28no—
31C3×#95C2211yescharacteristic, complemented
32C3⋊#95C2212yescomplemented
33C3⋊#95C2211yescharacteristic, complemented
34C2×#124—32no—
35C2×#120—32no—
36#394—32no—
37#393—31no—
38#392—32no—
39#392D311yescharacteristic, complemented
40#390—32no—
41#388—32no—
42#379—31no—
43C24×C3C2312yes—
44C24×C3C2311yescharacteristic
45C23×D3—22no—
46C23×D3—24no—
47C23×D3—42no—
48C23×D3—41no—
49C23×D3—41no—
50C23×D3C2312yes—
51C2×C3×D4—24no—
52C2×C3×D4—24no—
53C2×C3×D4—42no—
54C2×C3×D4—42no—
55C2×C3×D4—41no—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2312yes—
59C2×#67—24no—
60C2×#67—24no—
61C2×#67—24no—
62C2×#67—42no—
63C2×#67—42no—
64C2×#67—42no—
65C22×Dic3C2312yes—
66C22×Dic3C2312yes—
67C22×Dic3C2312yes—
68C2×Dic6—24no—
69C2×Dic6—41no—
70C3×C22⋊C4—24no—
71C3×C22⋊C4—41no—
72#217—24no—
73#217—24no—
74#217—41no—
75#217—42no—
76(C2×D3)⋊C4—24no—
77(C2×D3)⋊C4—24no—
78(C2×D3)⋊C4—42no—
79(C2×D3)⋊C4D418yescomplemented
80C4⋊Dic3—42no—
81C4×Dic3—24no—
82C25—31no—
83C22×Q8—31no—
84C22×D4—32no—
85C22×D4—32no—
86C22×D4C2×D312yes—
87C22×D4C2×D311yescharacteristic, complemented
88#124—68no—
89#120—68no—
90C2×C4⋊C4—32no—
91C2×C22⋊C4—32no—
92C2×C22⋊C4—32no—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4C2×D311yescharacteristic
100C2×C22⋊C4C2×D312yescomplemented
101C2×C42—32no—
102#95—32no—
103#95—31no—
104#95C2×D311yescharacteristic, complemented
105C23×C3—22no—
106C23×C3—21no—
107C23×C3—21no—
108C23×C3—22no—
109C23×C3—22no—
110C23×C3—22no—
111C23×C3—22no—
112C23×C3—22no—
113C23×C3—22no—
114C23×C3—22no—
115C23×C3—22no—
116C23×C3#4112yes—
117C23×C3C2411yescharacteristic
118C22×D3—22no—
119C22×D3—22no—
120C22×D3—22no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—44no—
125C22×D3—48no—
126C22×D3—48no—
127C22×D3—44no—
128C22×D3—44no—
129C22×D3C2×D414yes—
130C3×D4—48no—
131C3×D4—44no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4C2×D414yes—
139#67—48no—
140#67—48no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3C2×D414yes—
150Dic6—44no—
151C24—32no—
152C24—64no—
153C24—62no—
154C24—121no—
155C24—122no—
156C24—121no—
157C24C22×D312yes—
158C24C22×D311yescharacteristic
159C2×Q8—64no—
160C2×Q8—121no—
161C2×D4—24no—
162C2×D4—24no—
163C2×D4—42no—
164C2×D4—41no—
165C2×D4—42no—
166C2×D4—64no—
167C2×D4—64no—
168C2×D4—64no—
169C2×D4—122no—
170C2×D4—122no—
171C2×D4—122no—
172C22×C4—32no—
173C22×C4—32no—
174C22×C4—32no—
175C22×C4C22×D311yescharacteristic
176C22×C4C22×D311yescharacteristic
177C22×C4C22×D312yes—
178C4⋊C4—122no—
179C22⋊C4—24no—
180C22⋊C4—38no—
181C22⋊C4—41no—
182C22⋊C4—64no—
183C22⋊C4—64no—
184C22⋊C4—64no—
185C22⋊C4—64no—
186C22⋊C4—122no—
187C22⋊C4—121no—
188C22⋊C4—122no—
189C42—64no—
190C22×C3—22no—
191C22×C3—24no—
192C22×C3—42no—
193C22×C3—41no—
194C22×C3—41no—
195C22×C3—42no—
196C22×C3—42no—
197C22×C3—42no—
198C22×C3—42no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3—42no—
202C22×C3C22×D412yes—
203C22×C3C2×#4111yescharacteristic
204C22×C3H(5,𝔽2)12yes—
205C22×C3H(5,𝔽2)11yescharacteristic
206C22×C3H(5,𝔽2)11yescharacteristic
207C2×D3—24no—
208C2×D3—24no—
209C2×D3—24no—
210C2×D3—416no—
211C2×D3—42no—
212C2×D3—42no—
213C2×D3—42no—
214C2×D3—42no—
215C3×C4—24no—
216C3×C4—41no—
217C3×C4—41no—
218Dic3—24no—
219Dic3—42no—
220Dic3—42no—
221C23—22no—
222C23—21no—
223C23—22no—
224C23—21no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—34no—
233C23—62no—
234C23—62no—
235C23—62no—
236C23—62no—
237C23—62no—
238C23—62no—
239C23—128no—
240C23—124no—
241C23—124no—
242C23—128no—
243C23—124no—
244C23#23712yes—
245C23C23×D311yescharacteristic
246Q8—124no—
247D4—48no—
248D4—44no—
249D4—128no—
250D4—128no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—34no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4D3×D414yes—
267C2×C3—42no—
268C2×C3—82no—
269C2×C3#53512yes—
270C2×C3#53811yescharacteristic
271C2×C3D4211yescharacteristic
272C2×C3#54612yes—
273C2×C3H(3,𝔽4)11yescharacteristic
274D3—44no—
275C22—24no—
276C22—22no—
277C22—41no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—64no—
288C22—64no—
289C22—64no—
290C22—1216no—
291C22—122no—
292C22—122no—
293C22—122no—
294C22—122no—
295C22C2×D3×D412yes—
296C22C2×#23711yescharacteristic
297C22#100411yescharacteristic
298C22#100412yes—
299C22#100911yescharacteristic
300C4—24no—
301C4—41no—
302C4—41no—
303C4—64no—
304C4—122no—
305C4—122no—
306C3#267011yescharacteristic, complemented, Sylow
307C2—42no—
308C2—82no—
309C2—124no—
310C2#559112yes—
311C2#559411yescharacteristic
312C2#560712yes—
313C2#561711yescharacteristic
314C2#571711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2670
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#937C2semidirect
2. C2×#885C2semidirect
3. C3×#392C2semidirect
4. #5223C2semidirect
5. #5221C2semidirect
6. #4961C2semidirect
7. #4955C2semidirect
8. #4947C2semidirect
9. #4670C2semidirect
13. C22×C3×D4C22semidirect
16. C22×Dic6C22semidirect
18. C2×C3×C22⋊C4C22semidirect
21. C2×#217C22semidirect
22. C2×#217C22semidirect
28. C2×C4⋊Dic3C22semidirect
29. C2×C4×Dic3C22semidirect
31. C3×#95C22semidirect
32. C3⋊#95C22semidirect
33. C3⋊#95C22semidirect
39. #392D3semidirect
79. (C2×D3)⋊C4D4semidirect
87. C22×D4C2×D3semidirect
100. C2×C22⋊C4C2×D3semidirect
104. #95C2×D3semidirect
306. C3#2670semidirect

Maximal subgroups

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

subgroupindex
C2×#9372
C2×#8852
C3×#3922
#52232
#52212
#49612
#49552
#49472
#46702
#26703