Finite Group Databasebeta

← #74132

Subgroups — #74132

Order 384.

672 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 145 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×#381C211yesmaximal, characteristic, complemented
4#5211C211yesmaximal, characteristic, complemented
5#4955C211yesmaximal, characteristic
6#4955C211yesmaximal, characteristic
7Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
8#4675C212yesmaximal, complemented
9#4673C212yesmaximal, complemented
10#4670C212yesmaximal, complemented
11#4669C212yesmaximal, complemented
12#2561—31nomaximal, Sylow
13C24×D3C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×Dic3C2211yescharacteristic
16C22×Dic6C2211yescharacteristic, complemented
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×#217C2211yescharacteristic
20C2×#217C2211yescharacteristic
21C2×(C2×D3)⋊C4—24no—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×C4⋊Dic3C2211yescharacteristic, complemented
29C2×Dic3⋊C4C2211yescharacteristic, complemented
30C2×Dic3⋊C4C2211yescharacteristic, complemented
31C2×C4×Dic3C2212yescomplemented
32#878—24no—
33C3×#95C2212yescomplemented
34C3×#95C2212yescomplemented
35C3⋊#95C2212yescomplemented
36C3⋊#95C2212yescomplemented
37C3⋊#95C2212yescomplemented
38C2×C22⋊Q8—31no—
39C22×C22⋊C4—31no—
40#397—32no—
41#393—32no—
42#388—32no—
43#388—32no—
44#381—31no—
45#381D311yescharacteristic, complemented
46#379—31no—
47#379—31no—
48C4×C22⋊C4—31no—
49C24×C3C2311yescharacteristic
50C23×D3—22no—
51C23×D3—24no—
52C23×D3—22no—
53C23×D3—22no—
54C23×D3—42no—
55C23×D3C2312yes—
56C22×C3×C4—22no—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4C2312yes—
60C22×C3×C4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C22×Dic3—42no—
64C22×Dic3D414yes—
65C22×Dic3C2312yes—
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2311yescharacteristic
70C2×Dic6—22no—
71C2×Dic6—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—24no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—42no—
81(C2×D3)⋊C4—42no—
82C4⋊Dic3—22no—
83Dic3⋊C4—22no—
84Dic3⋊C4—22no—
85C4×Dic3—42no—
86C25—31no—
87C22×Q8—31no—
88C23×C4—31no—
89C23×C4C2×D311yescharacteristic
90C22⋊Q8—64no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4—31no—
94C2×C22⋊C4—31no—
95C2×C22⋊C4—31no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—64no—
103C2×C22⋊C4C2×D311yescharacteristic
104C2×C22⋊C4C2×D311yescharacteristic
105C2×C42—32no—
106#95—32no—
107#95—32no—
108#95—32no—
109#95C2×D312yescomplemented
110#95C2×D312yescomplemented
111C23×C3—22no—
112C23×C3—22no—
113C23×C3C2×D412yes—
114C23×C3#4112yes—
115C23×C3#4112yes—
116C23×C3C2411yescharacteristic
117C22×D3—28no—
118C22×D3—22no—
119C22×D3—22no—
120C22×D3—22no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—48no—
125C22×D3—48no—
126C22×D3—44no—
127C22×D3—44no—
128C22×D3#4114yes—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—22no—
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—44no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—21no—
143C2×Dic3—21no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—44no—
150C2×Dic3C2×D412yes—
151C2×Dic3C2×D412yes—
152Dic6—44no—
153C24—32no—
154C24—64no—
155C24—62no—
156C24—62no—
157C24—62no—
158C24—122no—
159C24C22×D311yescharacteristic
160C2×Q8—62no—
161C2×Q8—122no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—34no—
168C22×C4—32no—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—122no—
172C22×C4C22×D312yes—
173C22×C4C22×D312yes—
174C22×C4C22×D311yescharacteristic
175C22×C4C22×D311yescharacteristic
176C4⋊C4—62no—
177C4⋊C4—62no—
178C4⋊C4—62no—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—64no—
182C22⋊C4—64no—
183C22⋊C4—64no—
184C22⋊C4—64no—
185C22⋊C4—62no—
186C22⋊C4—62no—
187C22⋊C4—122no—
188C22⋊C4—122no—
189C42—122no—
190C22×C3—22no—
191C22×C3—22no—
192C22×C3—24no—
193C22×C3—24no—
194C22×C3#12414yes—
195C22×C3C22×D411yescharacteristic
196C22×C3C2×#4111yescharacteristic
197C22×C3C2×#4111yescharacteristic
198C22×C3C2×#4112yes—
199C22×C3H(5,𝔽2)12yes—
200C2×D3—24no—
201C2×D3—24no—
202C2×D3—24no—
203C2×D3—416no—
204C2×D3—42no—
205C2×D3—42no—
206C2×D3—42no—
207C2×D3—42no—
208C3×C4—41no—
209C3×C4—42no—
210C3×C4—41no—
211C3×C4—42no—
212Dic3—22no—
213Dic3—22no—
214Dic3—41no—
215Dic3—41no—
216Dic3—42no—
217C23—22no—
218C23—22no—
219C23—34no—
220C23—68no—
221C23—62no—
222C23—62no—
223C23—62no—
224C23—62no—
225C23—62no—
226C23—62no—
227C23—128no—
228C23—128no—
229C23—124no—
230C23—124no—
231C23#23512yes—
232C23D3×D412yes—
233C23#23712yes—
234C23C23×D311yescharacteristic
235Q8—124no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—32no—
247C2×C4—32no—
248C2×C4—44no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—61no—
253C2×C4—61no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—124no—
259C2×C3—24no—
260C2×C3#52511yescharacteristic
261C2×C3C2×#12411yescharacteristic
262C2×C3#53411yescharacteristic
263C2×C3#55112yes—
264C2×C3#55112yes—
265D3—44no—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—24no—
270C22—64no—
271C22—64no—
272C22—64no—
273C22—1216no—
274C22—122no—
275C22—122no—
276C22—122no—
277C22—122no—
278C22#88514yes—
279C22C2×#23511yescharacteristic
280C22C2×D3×D411yescharacteristic
281C22C2×#23711yescharacteristic
282C22D3×#4112yes—
283C22#100912yes—
284C4—41no—
285C4—41no—
286C4—42no—
287C4—42no—
288C4—62no—
289C4—62no—
290C4—121no—
291C4—121no—
292C4—122no—
293C3#256111yescharacteristic, complemented, Sylow
294C2—24no—
295C2—124no—
296C2C2×#88511yescharacteristic
297C2#552712yes—
298C2#565311yescharacteristic
299C2#565611yescharacteristic
300C2#565812yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#381C2
prime radicalC24×D3C22
hypercenter#381D3
hypocenterC3#2561
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#878C2semidirect
3. C3×#381C2semidirect
4. #5211C2semidirect
7. Dic3×C22⋊C4C2semidirect
8. #4675C2semidirect
9. #4673C2semidirect
10. #4670C2semidirect
11. #4669C2semidirect
16. C22×Dic6C22semidirect
28. C2×C4⋊Dic3C22semidirect
29. C2×Dic3⋊C4C22semidirect
30. C2×Dic3⋊C4C22semidirect
31. C2×C4×Dic3C22semidirect
33. C3×#95C22semidirect
34. C3×#95C22semidirect
35. C3⋊#95C22semidirect
36. C3⋊#95C22semidirect
37. C3⋊#95C22semidirect
45. #381D3semidirect
109. #95C2×D3semidirect
110. #95C2×D3semidirect
293. C3#2561semidirect

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×#3812
#52112
#49552
Dic3×C22⋊C42
#46752
#46732
#46702
#46692
#25613