Finite Group Databasebeta

← #74146

Subgroups — #74146

Order 384.

1102 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C23×C4×D3C211yesmaximal, characteristic, complemented
2C2×#929C211yesmaximal, characteristic, complemented
3C2×#886C211yesmaximal, characteristic, complemented
4C2×#882C211yesmaximal, characteristic
5C3×#381C211yesmaximal, characteristic, complemented
6#4955C211yesmaximal, characteristic, complemented
7#4943C211yesmaximal, characteristic, complemented
8#4668C214yesmaximal, complemented
9#4665C214yesmaximal, complemented
10#2849—31nomaximal, Sylow
11C24×D3C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C22×#67C2211yescharacteristic
14C23×Dic3C2211yescharacteristic
15C22×D12C2212yes—
16C22×C4×D3—24no—
17C22×C4×D3—24no—
18C22×C4×D3—24no—
19C22×C4×D3C2214yes—
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×#217C2211yescharacteristic
23#929—24no—
24C2×(C2×D3)⋊C4C2214yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yescomplemented
27C2×(C2×D3)⋊C4C2212yescomplemented
28C2×C4⋊Dic3C2212yescomplemented
29C2×Dic3⋊C4C2212yescomplemented
30#886—24no—
31#882—24no—
32C3×#95C2214yescomplemented
33C3⋊#95C2214yescomplemented
34C24×C4—31no—
35C2×#123—31no—
36C2×#123—31no—
37C2×#120—31no—
38#386—34no—
39#385—34no—
40#381—31no—
41#381D311yescharacteristic, complemented
42#379—31no—
43C24×C3C2311yescharacteristic
44C23×D3—22no—
45C23×D3—22no—
46C23×D3—28no—
47C23×D3D414yes—
48C23×D3C2311yescharacteristic
49C23×D3C2312yes—
50C22×C3×C4—22no—
51C22×C3×C4—22no—
52C22×C3×C4D414yescomplemented
53C22×C3×C4C2312yes—
54C22×C3×C4C2312yes—
55C22×C3×C4C2312yes—
56C2×#67—22no—
57C2×#67—22no—
58C2×#67—22no—
59C22×Dic3—22no—
60C22×Dic3—22no—
61C22×Dic3D414yescomplemented
62C22×Dic3C2312yes—
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2312yes—
65C2×D12—24no—
66C2×D12—28no—
67C2×C4×D3—232no—
68C2×C4×D3—28no—
69C2×C4×D3—28no—
70C2×C4×D3—28no—
71C3×C22⋊C4—22no—
72C3×C22⋊C4—22no—
73#217—22no—
74(C2×D3)⋊C4—24no—
75(C2×D3)⋊C4—28no—
76(C2×D3)⋊C4—24no—
77(C2×D3)⋊C4—24no—
78C4⋊Dic3—24no—
79Dic3⋊C4—24no—
80C25—31no—
81C22×D4—31no—
82C22×D4—32no—
83C23×C4—31no—
84C23×C4—34no—
85C23×C4—64no—
86C23×C4—64no—
87C23×C4—64no—
88C23×C4C2×D311yescharacteristic
89#123—64no—
90#123—64no—
91#120—64no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—32no—
94C2×C22⋊C4—31no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—34no—
99C2×C22⋊C4C2×D311yescharacteristic
100C2×C22⋊C4C2×D311yescharacteristic
101#95—34no—
102#95C2×D314yescomplemented
103C23×C3—24no—
104C23×C3C2×D412yes—
105C23×C3C2×D412yes—
106C23×C3C2×D412yes—
107C23×C3C2411yescharacteristic
108C22×D3—216no—
109C22×D3—216no—
110C22×D3—28no—
111C22×D3—22no—
112C22×D3—22no—
113C22×D3—24no—
114C22×D3—216no—
115C22×D3—21no—
116C22×D3—21no—
117C22×D3—21no—
118C22×D3—24no—
119C22×D3C2×D414yes—
120C22×D3#4118yes—
121C2×C3×C4—28no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—24no—
127C2×C3×C4—24no—
128C2×C3×C4C2×D414yes—
129#67—44no—
130C2×Dic3—28no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—21no—
134C2×Dic3—22no—
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—24no—
138C2×Dic3C2×D414yes—
139D12—48no—
140C4×D3—232no—
141C24—32no—
142C24—34no—
143C24—31no—
144C24—68no—
145C24—62no—
146C24—62no—
147C24C22×D311yescharacteristic
148C2×D4—64no—
149C2×D4—68no—
150C2×D4—62no—
151C2×D4—62no—
152C2×D4—62no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—34no—
156C22×C4—32no—
157C22×C4—32no—
158C22×C4—31no—
159C22×C4—68no—
160C22×C4—68no—
161C22×C4—632no—
162C22×C4—62no—
163C22×C4—62no—
164C22×C4—68no—
165C22×C4D1214yescomplemented
166C22×C4C22×D312yes—
167C22×C4C22×D312yes—
168C22×C4C22×D312yes—
169C4⋊C4—64no—
170C4⋊C4—64no—
171C22⋊C4—22no—
172C22⋊C4—22no—
173C22⋊C4—64no—
174C22⋊C4—64no—
175C22⋊C4—68no—
176C22⋊C4—62no—
177C22⋊C4—64no—
178C22×C3—24no—
179C22×C3—24no—
180C22×C3—24no—
181C22×C3#12014yes—
182C22×C3C22×D411yescharacteristic
183C22×C3C22×D411yescharacteristic
184C22×C3C22×D411yescharacteristic
185C22×C3C2×#4114yes—
186C2×D3—232no—
187C2×D3—24no—
188C2×D3—24no—
189C2×D3—28no—
190C2×D3—28no—
191C2×D3—41no—
192C2×D3—42no—
193C2×D3—42no—
194C2×D3—42no—
195C2×D3#12318yescomplemented
196C3×C4—24no—
197C3×C4—42no—
198C3×C4—42no—
199Dic3—24no—
200Dic3—41no—
201Dic3—42no—
202C23—24no—
203C23—34no—
204C23—38no—
205C23—616no—
206C23—616no—
207C23—616no—
208C23—68no—
209C23—62no—
210C23—62no—
211C23—64no—
212C23—61no—
213C23—61no—
214C23—61no—
215C23—64no—
216C23C2×D1212yes—
217C23D3×D412yes—
218C23D3×D412yes—
219C23C23×D311yescharacteristic
220D4—128no—
221D4—124no—
222C2×C4—28no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—34no—
230C2×C4—632no—
231C2×C4—68no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—62no—
235C2×C4—61no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—64no—
239C2×C4C2×D1214yes—
240C2×C3—24no—
241C2×C3C2×#12011yescharacteristic
242C2×C3C2×#12312yes—
243C2×C3#52512yes—
244C2×C3#52512yes—
245D3—28no—
246D3—81no—
247C22—24no—
248C22—24no—
249C22—24no—
250C22—38no—
251C22—632no—
252C22—64no—
253C22—68no—
254C22—64no—
255C22—68no—
256C22—122no—
257C22—122no—
258C22—121no—
259C22—122no—
260C22#88214yes—
261C22C22×D1211yescharacteristic
262C22C2×D3×D411yescharacteristic
263C22C2×D3×D411yescharacteristic
264C22D3×#4114yes—
265C4—24no—
266C4—42no—
267C4—42no—
268C4—64no—
269C4—121no—
270C4—122no—
271C3#284911yescharacteristic, complemented, Sylow
272C2—24no—
273C2—68no—
274C2—241no—
275C2C2×#88211yescharacteristic
276C2#552412yes—
277C2D3×#12312yes—
278C2#565612yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#381C2
prime radicalC2×#882C2
hypercenter#381D3
hypocenterC3#2849
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C23×C4×D3C2semidirect
2. C2×#929C2semidirect
3. C2×#886C2semidirect
5. C3×#381C2semidirect
6. #4955C2semidirect
7. #4943C2semidirect
8. #4668C2semidirect
9. #4665C2semidirect
26. C2×(C2×D3)⋊C4C22semidirect
27. C2×(C2×D3)⋊C4C22semidirect
28. C2×C4⋊Dic3C22semidirect
29. C2×Dic3⋊C4C22semidirect
32. C3×#95C22semidirect
33. C3⋊#95C22semidirect
41. #381D3semidirect
52. C22×C3×C4D4semidirect
61. C22×Dic3D4semidirect
102. #95C2×D3semidirect
165. C22×C4D12semidirect
195. C2×D3#123semidirect
271. C3#2849semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D32
C2×#9292
C2×#8862
C2×#8822
C3×#3812
#49552
#49432
#46682
#46652
#28493