Finite Group Databasebeta

← #76428

Subgroups — #76428

Order 384.

1104 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C22×D3×D4C211yesmaximal, characteristic
2C2×#892C211yesmaximal, characteristic, complemented
3C2×#872C211yesmaximal, characteristic, complemented
4C3×#390C211yesmaximal, characteristic, complemented
5#5221C211yesmaximal, characteristic, complemented
6#4988C211yesmaximal, characteristic, complemented
7#4957C214yesmaximal, complemented
8#4955C214yesmaximal, complemented
9#4939C211yesmaximal, characteristic, complemented
10#2309—31nomaximal, Sylow
11C24×D3C2212yes—
12C22×C3×D4C2211yescharacteristic
13C22×#67C2212yes—
14C2×D3×D4—24no—
15C2×D3×D4—24no—
16C2×D3×D4C418yescomplemented
17C22×D12C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic, complemented
20C2×C3×C22⋊C4C2214yescomplemented
21C2×C3×C42C2211yescharacteristic, complemented
22C2×#217C2214yescomplemented
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2214yescomplemented
26C2×(C2×D3)⋊C4C2214yescomplemented
27C2×C4⋊Dic3C2211yescharacteristic, complemented
28C2×Dic3⋊C4C2212yescomplemented
29C2×C4×Dic3C2212yescomplemented
30C2×C4×Dic3C2211yescharacteristic, complemented
31#892—24no—
32#872—24no—
33C23×D4—31no—
34C2×#117—31no—
35C2×#117—31no—
36#390—31no—
37#390—31no—
38#390—34no—
39#390—31no—
40#390D311yescharacteristic, complemented
41#379—34no—
42C24×C3C2312yes—
43C23×D3—24no—
44C23×D3—24no—
45C23×D3—44no—
46C23×D3—44no—
47C23×D3C2×C418yes—
48C23×D3C2312yes—
49C23×D3C2311yescharacteristic
50C2×C3×D4—22no—
51C2×C3×D4—22no—
52C2×C3×D4C2×C414yes—
53C22×C3×C4C2312yes—
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C2×#67—24no—
57C2×#67—24no—
58C2×#67C2×C418yes—
59C22×Dic3C2312yes—
60C22×Dic3C2312yes—
61C22×Dic3C2311yescharacteristic
62D3×D4—232no—
63C2×D12—22no—
64C2×D12—22no—
65C2×D12C2×C414yes—
66C2×C4×D3C2×C414yes—
67C2×C4×D3D414yes—
68C2×C4×D3D414yes—
69C3×C4⋊C4—22no—
70C3×C22⋊C4—44no—
71C3×C42—22no—
72#217—44no—
73(C2×D3)⋊C4—24no—
74(C2×D3)⋊C4—24no—
75(C2×D3)⋊C4—44no—
76(C2×D3)⋊C4—44no—
77C4⋊Dic3—22no—
78Dic3⋊C4—24no—
79C4×Dic3—22no—
80C4×Dic3—24no—
81C25—32no—
82C22×D4—32no—
83C22×D4—38no—
84C22×D4—31no—
85C22×D4—64no—
86C22×D4—64no—
87C22×D4C2×D311yescharacteristic
88C23×C4—31no—
89#117—64no—
90#117—64no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4C2×D311yescharacteristic, complemented
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—34no—
97C2×C22⋊C4—34no—
98C2×C22⋊C4—34no—
99C2×C22⋊C4C2×D314yescomplemented
100C2×C42—32no—
101C2×C42—31no—
102C2×C42C2×D311yescharacteristic, complemented
103C23×C3—22no—
104C23×C3—22no—
105C23×C3—24no—
106C23×C3—24no—
107C23×C3C22×C414yes—
108C23×C3C2411yescharacteristic
109C22×D3—22no—
110C22×D3—22no—
111C22×D3—22no—
112C22×D3—22no—
113C22×D3—24no—
114C22×D3—24no—
115C22×D3—216no—
116C22×D3—416no—
117C22×D3—48no—
118C22×D3—48no—
119C22×D3—416no—
120C22×D3C22×C414yes—
121C22×D3C22×C412yes—
122C22×D3C2×D412yes—
123C22×D3C2×D412yes—
124C3×D4—28no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—24no—
129C2×C3×C4—22no—
130C2×C3×C4C22×C412yes—
131C2×C3×C4C2×D412yes—
132C2×C3×C4C2×D412yes—
133#67—216no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—24no—
138C2×Dic3—22no—
139C2×Dic3C22×C412yes—
140C2×Dic3C2×D412yes—
141C2×Dic3C2×D412yes—
142D12—28no—
143C4×D3C22⋊C4116yescomplemented
144C24—31no—
145C24—32no—
146C24—38no—
147C24—64no—
148C24—64no—
149C24—124no—
150C24—124no—
151C24C22×D312yes—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—38no—
155C2×D4—34no—
156C2×D4—632no—
157C2×D4—64no—
158C2×D4—64no—
159C2×D4—62no—
160C2×D4—62no—
161C2×D4C4×D314yes—
162C22×C4—31no—
163C22×C4—34no—
164C22×C4—34no—
165C22×C4—34no—
166C22×C4—32no—
167C22×C4—32no—
168C22×C4C22×D312yes—
169C22×C4C22×D312yes—
170C22×C4C22×D311yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—64no—
173C4⋊C4—62no—
174C22⋊C4—44no—
175C22⋊C4—64no—
176C22⋊C4—64no—
177C22⋊C4—124no—
178C22⋊C4—124no—
179C22⋊C4—124no—
180C42—22no—
181C42—64no—
182C42—62no—
183C22×C3—24no—
184C22×C3—28no—
185C22×C3—42no—
186C22×C3—42no—
187C22×C3—44no—
188C22×C3C23×C411yescharacteristic
189C22×C3C22×D411yescharacteristic
190C22×C3C22×D411yescharacteristic
191C22×C3H(5,𝔽2)12yes—
192C22×C3H(5,𝔽2)12yes—
193C2×D3—24no—
194C2×D3—22no—
195C2×D3—22no—
196C2×D3—24no—
197C2×D3—24no—
198C2×D3—28no—
199C2×D3—432no—
200C2×D3—42no—
201C2×D3—42no—
202C2×D3—44no—
203C2×D3C2×C22⋊C414yes—
204C3×C4—42no—
205C3×C4—42no—
206C3×C4C2×C22⋊C414yes—
207Dic3—42no—
208Dic3—42no—
209Dic3C2×C22⋊C414yes—
210C23—22no—
211C23—22no—
212C23—24no—
213C23—24no—
214C23—32no—
215C23—32no—
216C23—32no—
217C23—34no—
218C23—616no—
219C23—62no—
220C23—62no—
221C23—62no—
222C23—62no—
223C23—64no—
224C23—64no—
225C23—1216no—
226C23—128no—
227C23—128no—
228C23—1216no—
229C23C2×C4×D314yes—
230C23C23×D311yescharacteristic
231D4—28no—
232D4—616no—
233D4—68no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—32no—
240C2×C4—32no—
241C2×C4—32no—
242C2×C4—316no—
243C2×C4—62no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—64no—
247C2×C4—62no—
248C2×C4C2×C4×D312yes—
249C2×C4D3×D412yes—
250C2×C4D3×D412yes—
251C2×C3—44no—
252C2×C3C22×C22⋊C411yescharacteristic
253C2×C3#51812yes—
254C2×C3#53511yescharacteristic
255C2×C3#53511yescharacteristic
256C2×C3#53512yes—
257D3—24no—
258D3—44no—
259C22—24no—
260C22—28no—
261C22—34no—
262C22—42no—
263C22—42no—
264C22—44no—
265C22—62no—
266C22—62no—
267C22—64no—
268C22—64no—
269C22—64no—
270C22—68no—
271C22—1232no—
272C22—122no—
273C22—122no—
274C22—124no—
275C22C22×C4×D311yescharacteristic
276C22C2×D3×D411yescharacteristic
277C22C2×D3×D411yescharacteristic
278C22#100412yes—
279C22#100912yes—
280C4—34no—
281C4—42no—
282C4—42no—
283C4—122no—
284C4—122no—
285C4D3×C22⋊C414yes—
286C3#230911yescharacteristic, complemented, Sylow
287C2—44no—
288C2—64no—
289C2—124no—
290C2C2×D3×C22⋊C411yescharacteristic
291C2#554512yes—
292C2#560712yes—
293C2#567411yescharacteristic
294C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#390C2
socle, cosocleC23×C3C24
prime radicalC22×D3×D4C2
hypercenter#390D3
hypocenterC3#2309
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#892C2semidirect
3. C2×#872C2semidirect
4. C3×#390C2semidirect
5. #5221C2semidirect
6. #4988C2semidirect
7. #4957C2semidirect
8. #4955C2semidirect
9. #4939C2semidirect
16. C2×D3×D4C4semidirect
19. C2×C3×C4⋊C4C22semidirect
20. C2×C3×C22⋊C4C22semidirect
21. C2×C3×C42C22semidirect
22. C2×#217C22semidirect
25. C2×(C2×D3)⋊C4C22semidirect
26. C2×(C2×D3)⋊C4C22semidirect
27. C2×C4⋊Dic3C22semidirect
28. C2×Dic3⋊C4C22semidirect
29. C2×C4×Dic3C22semidirect
30. C2×C4×Dic3C22semidirect
40. #390D3semidirect
93. C2×C4⋊C4C2×D3semidirect
99. C2×C22⋊C4C2×D3semidirect
102. C2×C42C2×D3semidirect
143. C4×D3C22⋊C4semidirect
286. C3#2309semidirect

Maximal subgroups

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

subgroupindex
C22×D3×D42
C2×#8922
C2×#8722
C3×#3902
#52212
#49882
#49572
#49552
#49392
#23093