Finite Group Databasebeta

← #74621

Subgroups — #74621

Order 384.

668 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 149 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×(C2×D3)⋊C4C211yesmaximal, characteristic
2C2×#878C211yesmaximal, characteristic, complemented
3C3×#385C211yesmaximal, characteristic, complemented
4#5210C211yesmaximal, characteristic, complemented
5#4994C212yesmaximal, complemented
6#4993C212yesmaximal, complemented
7#4955C211yesmaximal, characteristic
8#4955C211yesmaximal, characteristic
9#4942C211yesmaximal, characteristic, complemented
10#4676C212yesmaximal, complemented
11#4673C212yesmaximal, complemented
12#2626—31nomaximal, Sylow
13C24×D3C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×Dic3C2211yescharacteristic
16C22×Dic6C2211yescharacteristic, complemented
17C2×C3×C4⋊C4C2212yescomplemented
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×#217C2211yescharacteristic
21C2×#217C2211yescharacteristic
22C2×(C2×D3)⋊C4—24no—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×C4⋊Dic3C2211yescharacteristic, complemented
30C2×Dic3⋊C4C2211yescharacteristic, complemented
31C2×Dic3⋊C4C2211yescharacteristic, complemented
32C2×Dic3⋊C4C2212yescomplemented
33C2×Dic3⋊C4C2212yescomplemented
34#878—24no—
35C3×#95C2212yescomplemented
36C3⋊#95C2212yescomplemented
37C3⋊#95C2212yescomplemented
38C2×C22⋊Q8—31no—
39C22×C22⋊C4—31no—
40#399—32no—
41#396—32no—
42#396—32no—
43#393—32no—
44#385—31no—
45#385D311yescharacteristic, complemented
46#380—31no—
47#379—31no—
48#379—31no—
49C24×C3C2311yescharacteristic
50C23×D3—22no—
51C23×D3—22no—
52C23×D3—42no—
53C23×D3—42no—
54C23×D3Q814yescomplemented
55C23×D3C2312yes—
56C22×C3×C4—22no—
57C22×C3×C4—42no—
58C22×C3×C4C2312yes—
59C22×C3×C4C2312yes—
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2311yescharacteristic
62C22×Dic3—42no—
63C22×Dic3D414yes—
64C22×Dic3C2312yes—
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2311yescharacteristic
69C2×Dic6—22no—
70C2×Dic6—42no—
71C3×C4⋊C4—42no—
72C3×C22⋊C4—22no—
73C3×C22⋊C4—22no—
74#217—22no—
75#217—22no—
76(C2×D3)⋊C4—24no—
77(C2×D3)⋊C4—24no—
78(C2×D3)⋊C4—42no—
79(C2×D3)⋊C4—42no—
80(C2×D3)⋊C4—42no—
81(C2×D3)⋊C4—42no—
82C4⋊Dic3—22no—
83Dic3⋊C4—22no—
84Dic3⋊C4—22no—
85Dic3⋊C4—42no—
86Dic3⋊C4—42no—
87C25—31no—
88C22×Q8—31no—
89C23×C4—31no—
90C23×C4C2×D311yescharacteristic
91C22⋊Q8—64no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4—31no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4—32no—
97C2×C4⋊C4C2×D312yescomplemented
98C2×C22⋊C4—31no—
99C2×C22⋊C4—31no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—64no—
107C2×C22⋊C4C2×D311yescharacteristic
108C2×C22⋊C4C2×D311yescharacteristic
109#95—32no—
110#95—32no—
111#95C2×D312yescomplemented
112C23×C3—22no—
113C23×C3—22no—
114C23×C3C2×D412yes—
115C23×C3C2×Q812yes—
116C23×C3#4112yes—
117C23×C3C2411yescharacteristic
118C22×D3—28no—
119C22×D3—22no—
120C22×D3—22no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—22no—
125C22×D3—48no—
126C22×D3—44no—
127C22×D3—44no—
128C22×D3—48no—
129C22×D3C2×Q814yes—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—44no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—44no—
151C2×Dic3C2×D412yes—
152C2×Dic3C2×D412yes—
153Dic6—44no—
154C24—32no—
155C24—34no—
156C24—62no—
157C24—62no—
158C24—122no—
159C24—122no—
160C24C22×D311yescharacteristic
161C2×Q8—62no—
162C2×Q8—122no—
163C22×C4—22no—
164C22×C4—31no—
165C22×C4—31no—
166C22×C4—34no—
167C22×C4—32no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—42no—
171C22×C4—122no—
172C22×C4C22×D312yes—
173C22×C4C22×D312yes—
174C22×C4C22×D311yescharacteristic
175C22×C4C22×D311yescharacteristic
176C4⋊C4—42no—
177C4⋊C4—62no—
178C4⋊C4—62no—
179C4⋊C4—62no—
180C4⋊C4—122no—
181C4⋊C4—122no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—64no—
185C22⋊C4—64no—
186C22⋊C4—62no—
187C22⋊C4—62no—
188C22⋊C4—122no—
189C22⋊C4—122no—
190C22⋊C4—122no—
191C22⋊C4—122no—
192C22×C3—22no—
193C22×C3—22no—
194C22×C3—24no—
195C22×C3—24no—
196C22×C3C22⋊Q814yes—
197C22×C3C22×D411yescharacteristic
198C22×C3C22×Q811yescharacteristic
199C22×C3C2×#4111yescharacteristic
200C22×C3H(5,𝔽2)12yes—
201C22×C3H(5,𝔽2)12yes—
202C2×D3—24no—
203C2×D3—24no—
204C2×D3—24no—
205C2×D3—416no—
206C2×D3—42no—
207C2×D3—42no—
208C2×D3—42no—
209C2×D3—42no—
210C3×C4—41no—
211C3×C4—41no—
212C3×C4—42no—
213C3×C4—42no—
214Dic3—22no—
215Dic3—22no—
216Dic3—41no—
217Dic3—41no—
218Dic3—42no—
219C23—22no—
220C23—22no—
221C23—34no—
222C23—68no—
223C23—62no—
224C23—62no—
225C23—62no—
226C23—62no—
227C23—62no—
228C23—62no—
229C23—128no—
230C23—124no—
231C23—128no—
232C23—124no—
233C23#23512yes—
234C23D3×D412yes—
235C23D3×Q812yes—
236C23C23×D311yescharacteristic
237Q8—124no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—32no—
249C2×C4—32no—
250C2×C4—44no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—61no—
255C2×C4—61no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—124no—
261C2×C3—24no—
262C2×C3C2×C22⋊Q811yescharacteristic
263C2×C3#53411yescharacteristic
264C2×C3#53511yescharacteristic
265C2×C3#53812yes—
266C2×C3#54312yes—
267D3—44no—
268C22—24no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—64no—
273C22—64no—
274C22—64no—
275C22—1216no—
276C22—122no—
277C22—122no—
278C22—122no—
279C22—122no—
280C22#89614yes—
281C22C2×#23511yescharacteristic
282C22C2×D3×D411yescharacteristic
283C22#100412yes—
284C22C2×D3×Q811yescharacteristic
285C22#100912yes—
286C4—41no—
287C4—42no—
288C4—42no—
289C4—41no—
290C4—62no—
291C4—62no—
292C4—121no—
293C4—122no—
294C4—121no—
295C3#262611yescharacteristic, complemented, Sylow
296C2—24no—
297C2—124no—
298C2C2×#89611yescharacteristic
299C2#556012yes—
300C2#559211yescharacteristic
301C2#563412yes—
302C2#565311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
prime radicalC24×D3C22
hypercenter#385D3
hypocenterC3#2626
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#878C2semidirect
3. C3×#385C2semidirect
4. #5210C2semidirect
5. #4994C2semidirect
6. #4993C2semidirect
9. #4942C2semidirect
10. #4676C2semidirect
11. #4673C2semidirect
16. C22×Dic6C22semidirect
17. C2×C3×C4⋊C4C22semidirect
29. C2×C4⋊Dic3C22semidirect
30. C2×Dic3⋊C4C22semidirect
31. C2×Dic3⋊C4C22semidirect
32. C2×Dic3⋊C4C22semidirect
33. C2×Dic3⋊C4C22semidirect
35. C3×#95C22semidirect
36. C3⋊#95C22semidirect
37. C3⋊#95C22semidirect
45. #385D3semidirect
54. C23×D3Q8semidirect
97. C2×C4⋊C4C2×D3semidirect
111. #95C2×D3semidirect
295. C3#2626semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×D3)⋊C42
C2×#8782
C3×#3852
#52102
#49942
#49932
#49552
#49422
#46762
#46732
#26263