Finite Group Databasebeta

← #78110

Subgroups — #78110

Order 384.

668 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 143 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×#897C211yesmaximal, characteristic, complemented
2C2×D3×C22⋊C4C211yesmaximal, characteristic
3C3×#399C211yesmaximal, characteristic, complemented
4#4994C212yesmaximal, complemented
5#4993C211yesmaximal, characteristic, complemented
6#4989C212yesmaximal, complemented
7#4955C212yesmaximal
8#4948C212yesmaximal, complemented
9#4947C211yesmaximal, characteristic, complemented
10#4673C211yesmaximal, characteristic, complemented
11#4667C211yesmaximal, characteristic, complemented
12#2626—31nomaximal, Sylow
13C24×D3C2211yescharacteristic
14C22×C4×D3C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C22×Dic6C2211yescharacteristic, complemented
17C2×C3×C4⋊C4C2211yescharacteristic, complemented
18C2×C3×C4⋊C4C2212yescomplemented
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2212yes—
21C2×#217C2211yescharacteristic
22C2×#217C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×C4⋊Dic3C2212yescomplemented
30C2×Dic3⋊C4C2212yescomplemented
31C2×Dic3⋊C4C2212yescomplemented
32#897—24no—
33D3×C22⋊C4—24no—
34C3×#95C2211yescharacteristic, complemented
35C3⋊#95C2211yescharacteristic, complemented
36C3⋊#95C2212yescomplemented
37C3⋊#95C2212yescomplemented
38C2×C22⋊Q8—31no—
39C22×C22⋊C4—31no—
40#399—31no—
41#399D311yescharacteristic, complemented
42#396—32no—
43#396—32no—
44#393—31no—
45#393—31no—
46#385—32no—
47#380—31no—
48#379—32no—
49C24×C3C2311yescharacteristic
50C23×D3—22no—
51C23×D3—22no—
52C23×D3—42no—
53C23×D3—42no—
54C23×D3Q814yescomplemented
55C23×D3C2311yescharacteristic
56C23×D3C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2312yes—
59C22×C3×C4C2312yes—
60C22×C3×C4C2311yescharacteristic
61C22×Dic3C2312yes—
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2312yes—
64C22×Dic3C2311yescharacteristic
65C2×C4×D3—22no—
66C2×C4×D3—42no—
67C2×C4×D3—42no—
68C2×C4×D3D414yes—
69C2×Dic6—22no—
70C2×Dic6—42no—
71C3×C4⋊C4—22no—
72C3×C4⋊C4—42no—
73C3×C22⋊C4—22no—
74C3×C22⋊C4—42no—
75#217—22no—
76#217—42no—
77(C2×D3)⋊C4—24no—
78(C2×D3)⋊C4—24no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—42no—
82(C2×D3)⋊C4—42no—
83C4⋊Dic3—24no—
84Dic3⋊C4—42no—
85Dic3⋊C4—42no—
86C25—31no—
87C22×Q8—31no—
88C23×C4—31no—
89C23×C4—31no—
90C22⋊Q8—64no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4C2×D311yescharacteristic, complemented
95C2×C4⋊C4C2×D312yescomplemented
96C2×C22⋊C4—31no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—31no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—64no—
105C2×C22⋊C4C2×D311yescharacteristic
106C2×C22⋊C4C2×D312yes—
107#95—31no—
108#95—32no—
109#95—32no—
110#95C2×D311yescharacteristic, complemented
111C23×C3—21no—
112C23×C3—21no—
113C23×C3—22no—
114C23×C3—22no—
115C23×C3C2×Q812yes—
116C23×C3C2411yescharacteristic
117C22×D3—22no—
118C22×D3—22no—
119C22×D3—21no—
120C22×D3—21no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—28no—
124C22×D3—48no—
125C22×D3—44no—
126C22×D3—44no—
127C22×D3—48no—
128C22×D3C2×D412yes—
129C22×D3C2×Q812yes—
130C22×D3C2×Q812yes—
131C22×D3#4112yes—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—21no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4C2×D412yes—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3C2×D412yes—
152C4×D3—44no—
153C4×D3—44no—
154Dic6—44no—
155C24—31no—
156C24—31no—
157C24—34no—
158C24—62no—
159C24—62no—
160C24—122no—
161C24—122no—
162C24C22×D311yescharacteristic
163C2×Q8—62no—
164C2×Q8—122no—
165C22×C4—31no—
166C22×C4—34no—
167C22×C4—32no—
168C22×C4—32no—
169C22×C4—31no—
170C22×C4—62no—
171C22×C4—122no—
172C22×C4—122no—
173C22×C4C22×D311yescharacteristic
174C22×C4C22×D312yes—
175C22×C4C22×D312yes—
176C22×C4C22×D311yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—64no—
180C4⋊C4—122no—
181C4⋊C4—122no—
182C22⋊C4—22no—
183C22⋊C4—42no—
184C22⋊C4—62no—
185C22⋊C4—64no—
186C22⋊C4—64no—
187C22⋊C4—62no—
188C22⋊C4—62no—
189C22⋊C4—122no—
190C22⋊C4—122no—
191C22⋊C4—122no—
192C22×C3—22no—
193C22×C3—24no—
194C22×C3—41no—
195C22×C3—41no—
196C22×C3—42no—
197C22×C3C22×D411yescharacteristic
198C22×C3C22×Q811yescharacteristic
199C22×C3C2×#4111yescharacteristic
200C22×C3H(5,𝔽2)12yes—
201C22×C3H(5,𝔽2)12yes—
202C2×D3—22no—
203C2×D3—22no—
204C2×D3—24no—
205C2×D3—24no—
206C2×D3—22no—
207C2×D3—24no—
208C2×D3—416no—
209C2×D3—41no—
210C2×D3—41no—
211C2×D3—42no—
212C2×D3C22⋊Q814yes—
213C3×C4—22no—
214C3×C4—41no—
215C3×C4—42no—
216C3×C4—42no—
217Dic3—22no—
218Dic3—42no—
219Dic3—41no—
220Dic3—42no—
221C23—21no—
222C23—22no—
223C23—21no—
224C23—22no—
225C23—32no—
226C23—32no—
227C23—32no—
228C23—32no—
229C23—68no—
230C23—62no—
231C23—62no—
232C23—61no—
233C23—62no—
234C23—61no—
235C23—62no—
236C23—128no—
237C23—124no—
238C23—128no—
239C23—124no—
240C23D3×Q812yes—
241C23C23×D311yescharacteristic
242Q8—124no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—32no—
253C2×C4—62no—
254C2×C4—61no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—124no—
263C2×C4—124no—
264C2×C4D3×D412yes—
265C2×C3—42no—
266C2×C3C2×C22⋊Q811yescharacteristic
267C2×C3#53411yescharacteristic
268C2×C3#53511yescharacteristic
269C2×C3#53812yes—
270C2×C3#54312yes—
271D3—24no—
272D3—42no—
273C22—24no—
274C22—22no—
275C22—34no—
276C22—41no—
277C22—42no—
278C22—41no—
279C22—64no—
280C22—62no—
281C22—64no—
282C22—62no—
283C22—64no—
284C22—62no—
285C22—1216no—
286C22—121no—
287C22—122no—
288C22—121no—
289C22C2×D3×D411yescharacteristic
290C22#100412yes—
291C22C2×D3×Q811yescharacteristic
292C22D3×#4111yescharacteristic
293C22#100912yes—
294C4—22no—
295C4—42no—
296C4—41no—
297C4—42no—
298C4—62no—
299C4—122no—
300C4—122no—
301C4—121no—
302C3#262611yescharacteristic, complemented, Sylow
303C2—42no—
304C2—64no—
305C2—122no—
306C2D3×C22⋊Q811yescharacteristic
307C2#563412yes—
308C2#565311yescharacteristic
309C2#565712yes—
310C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radicalC24×D3C22
hypercenter#399D3
hypocenterC3#2626
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#897C2semidirect
3. C3×#399C2semidirect
4. #4994C2semidirect
5. #4993C2semidirect
6. #4989C2semidirect
8. #4948C2semidirect
9. #4947C2semidirect
10. #4673C2semidirect
11. #4667C2semidirect
16. C22×Dic6C22semidirect
17. C2×C3×C4⋊C4C22semidirect
18. C2×C3×C4⋊C4C22semidirect
29. C2×C4⋊Dic3C22semidirect
30. C2×Dic3⋊C4C22semidirect
31. C2×Dic3⋊C4C22semidirect
34. C3×#95C22semidirect
35. C3⋊#95C22semidirect
36. C3⋊#95C22semidirect
37. C3⋊#95C22semidirect
41. #399D3semidirect
54. C23×D3Q8semidirect
94. C2×C4⋊C4C2×D3semidirect
95. C2×C4⋊C4C2×D3semidirect
110. #95C2×D3semidirect
302. C3#2626semidirect

Maximal subgroups

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

subgroupindex
C2×#8972
C2×D3×C22⋊C42
C3×#3992
#49942
#49932
#49892
#49552
#49482
#49472
#46732
#46672
#26263