Finite Group Databasebeta

← #78102

Subgroups — #78102

Order 384.

780 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 161 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×#896C212yesmaximal, complemented
2C2×D3×C22⋊C4C212yesmaximal
3C3×#399C211yesmaximal, characteristic, complemented
4#4989C212yesmaximal, complemented
5#4987C212yesmaximal, complemented
6#4955C211yesmaximal, characteristic
7#4948C212yesmaximal, complemented
8#4947C211yesmaximal, characteristic, complemented
9#4668C212yesmaximal, complemented
10#2437—31nomaximal, Sylow
11C24×D3C2211yescharacteristic
12C22×C4×D3C2212yes—
13C22×C4×D3C2212yes—
14C22×Dic6C2211yescharacteristic, complemented
15C2×C3×C4⋊C4C2212yescomplemented
16C2×C3×C4⋊C4C2211yescharacteristic, complemented
17C2×C3×C22⋊C4C2212yes—
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×#217C2212yes—
20C2×#217C2211yescharacteristic
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×C4⋊Dic3C2212yescomplemented
26C2×Dic3⋊C4C2212yescomplemented
27C2×Dic3⋊C4C2212yescomplemented
28#896—28no—
29D3×C22⋊C4—28no—
30C3×#95C2211yescharacteristic, complemented
31C3⋊#95C2212yescomplemented
32C3⋊#95C2211yescharacteristic, complemented
33C3⋊#95C2212yescomplemented
34C2×C22⋊Q8—32no—
35C22×C22⋊C4—32no—
36#399D311yescharacteristic, complemented
37#396—32no—
38#393—31no—
39#385—32no—
40#385—32no—
41#380—32no—
42#379—31no—
43C24×C3C2311yescharacteristic
44C23×D3—24no—
45C23×D3—24no—
46C23×D3—41no—
47C23×D3—41no—
48C23×D3Q814yescomplemented
49C23×D3C2312yes—
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2312yes—
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C22×Dic3C2312yes—
57C2×C4×D3—24no—
58C2×C4×D3—24no—
59C2×C4×D3—24no—
60C2×C4×D3—24no—
61C2×C4×D3—42no—
62C2×C4×D3—42no—
63C2×Dic6—24no—
64C2×Dic6—41no—
65C3×C4⋊C4—24no—
66C3×C4⋊C4—41no—
67C3×C22⋊C4—24no—
68C3×C22⋊C4—41no—
69#217—24no—
70#217—41no—
71(C2×D3)⋊C4—24no—
72(C2×D3)⋊C4—24no—
73(C2×D3)⋊C4—24no—
74(C2×D3)⋊C4D418yes—
75C4⋊Dic3—42no—
76Dic3⋊C4—24no—
77Dic3⋊C4—24no—
78C25—31no—
79C22×Q8—31no—
80C23×C4—32no—
81C23×C4—32no—
82C22⋊Q8—68no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4C2×D312yescomplemented
87C2×C4⋊C4C2×D311yescharacteristic, complemented
88C2×C22⋊C4—32no—
89C2×C22⋊C4—32no—
90C2×C22⋊C4—32no—
91C2×C22⋊C4—32no—
92C2×C22⋊C4—32no—
93C2×C22⋊C4—31no—
94C2×C22⋊C4—68no—
95C2×C22⋊C4C2×D312yes—
96C2×C22⋊C4C2×D311yescharacteristic
97#95—31no—
98#95—32no—
99#95—32no—
100#95C2×D311yescharacteristic, complemented
101C23×C3—22no—
102C23×C3—22no—
103C23×C3—21no—
104C23×C3—21no—
105C23×C3C2×Q812yes—
106C23×C3C2411yescharacteristic
107C22×D3—22no—
108C22×D3—22no—
109C22×D3—22no—
110C22×D3—22no—
111C22×D3—216no—
112C22×D3—44no—
113C22×D3—44no—
114C22×D3—48no—
115C22×D3—44no—
116C22×D3C2×D414yes—
117C22×D3C2×Q814yes—
118C22×D3#4114yes—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—22no—
128C2×C3×C4C2×D414yes—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3C2×D414yes—
138C4×D3—48no—
139C4×D3—48no—
140Dic6—44no—
141C24—32no—
142C24—34no—
143C24—64no—
144C24—64no—
145C24—121no—
146C24—121no—
147C24C22×D311yescharacteristic
148C2×Q8—64no—
149C2×Q8—121no—
150C22×C4—32no—
151C22×C4—32no—
152C22×C4—32no—
153C22×C4—64no—
154C22×C4—64no—
155C22×C4—64no—
156C22×C4—64no—
157C22×C4—122no—
158C22×C4—122no—
159C22×C4C22×D311yescharacteristic
160C22×C4C22×D311yescharacteristic
161C22×C4C22×D312yes—
162C22×C4C22×D312yes—
163C4⋊C4—24no—
164C4⋊C4—41no—
165C4⋊C4—64no—
166C4⋊C4—64no—
167C4⋊C4—122no—
168C22⋊C4—24no—
169C22⋊C4—38no—
170C22⋊C4—41no—
171C22⋊C4—64no—
172C22⋊C4—64no—
173C22⋊C4—64no—
174C22⋊C4—64no—
175C22⋊C4—121no—
176C22×C3—24no—
177C22×C3—22no—
178C22×C3—41no—
179C22×C3—42no—
180C22×C3—41no—
181C22×C3C22×D412yes—
182C22×C3C22×Q811yescharacteristic
183C22×C3C2×#4112yes—
184C22×C3H(5,𝔽2)11yescharacteristic
185C22×C3H(5,𝔽2)11yescharacteristic
186C2×D3—24no—
187C2×D3—24no—
188C2×D3—24no—
189C2×D3—24no—
190C2×D3—24no—
191C2×D3—24no—
192C2×D3—416no—
193C2×D3C22⋊Q818yescomplemented
194C3×C4—24no—
195C3×C4—41no—
196C3×C4—42no—
197C3×C4—41no—
198Dic3—24no—
199Dic3—42no—
200Dic3—42no—
201C23—22no—
202C23—21no—
203C23—22no—
204C23—21no—
205C23—34no—
206C23—34no—
207C23—34no—
208C23—616no—
209C23—62no—
210C23—62no—
211C23—62no—
212C23—62no—
213C23—124no—
214C23—128no—
215C23—124no—
216C23—124no—
217C23D3×Q812yes—
218C23C23×D311yescharacteristic
219Q8—124no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—34no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—128no—
239C2×C4—128no—
240C2×C4D3×D414yes—
241C2×C3—42no—
242C2×C3C2×C22⋊Q812yes—
243C2×C3#54311yescharacteristic
244C2×C3D4211yescharacteristic
245C2×C3#54612yes—
246C2×C3#55111yescharacteristic
247D3—28no—
248C22—22no—
249C22—24no—
250C22—38no—
251C22—42no—
252C22—41no—
253C22—41no—
254C22—64no—
255C22—64no—
256C22—64no—
257C22—64no—
258C22—64no—
259C22—64no—
260C22—1216no—
261C22C2×D3×D412yes—
262C22#100411yescharacteristic
263C22C2×D3×Q811yescharacteristic
264C22D3×#4112yes—
265C22#100911yescharacteristic
266C4—24no—
267C4—41no—
268C4—42no—
269C4—41no—
270C4—64no—
271C4—122no—
272C4—122no—
273C3#243711yescharacteristic, complemented, Sylow
274C2—42no—
275C2—68no—
276C2D3×C22⋊Q812yes—
277C2#563411yescharacteristic
278C2#565412yes—
279C2#565911yescharacteristic
280C2#571711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radicalC24×D3C22
hypercenter#399D3
hypocenterC3#2437
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#896C2semidirect
3. C3×#399C2semidirect
4. #4989C2semidirect
5. #4987C2semidirect
7. #4948C2semidirect
8. #4947C2semidirect
9. #4668C2semidirect
14. C22×Dic6C22semidirect
15. C2×C3×C4⋊C4C22semidirect
16. C2×C3×C4⋊C4C22semidirect
25. C2×C4⋊Dic3C22semidirect
26. C2×Dic3⋊C4C22semidirect
27. C2×Dic3⋊C4C22semidirect
30. C3×#95C22semidirect
31. C3⋊#95C22semidirect
32. C3⋊#95C22semidirect
33. C3⋊#95C22semidirect
36. #399D3semidirect
48. C23×D3Q8semidirect
86. C2×C4⋊C4C2×D3semidirect
87. C2×C4⋊C4C2×D3semidirect
100. #95C2×D3semidirect
193. C2×D3C22⋊Q8semidirect
273. C3#2437semidirect

Maximal subgroups

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

subgroupindex
C2×#8962
C2×D3×C22⋊C42
C3×#3992
#49892
#49872
#49552
#49482
#49472
#46682
#24373