Finite Group Databasebeta

← #77677

Subgroups — #77677

Order 384.

700 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 141 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×D3×C22⋊C4C212yesmaximal
2C3×#397C211yesmaximal, characteristic, complemented
3#4989C212yesmaximal, complemented
4#4955C211yesmaximal, characteristic
5#4953C211yesmaximal, characteristic, complemented
6#4952C212yesmaximal, complemented
7#4668C212yesmaximal, complemented
8#4664C212yesmaximal, complemented
9#4664C212yesmaximal, complemented
10#2483—31nomaximal, Sylow
11C24×D3C2211yescharacteristic
12C22×C4×D3C2212yes—
13C22×C4×D3C2212yes—
14C2×C3×C4⋊C4C2211yescharacteristic, complemented
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×#217C2212yes—
18C2×#217C2211yescharacteristic
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×C4⋊Dic3C2211yescharacteristic, complemented
24C2×Dic3⋊C4C2212yescomplemented
25D3×C22⋊C4—28no—
26C3×#95C2211yescharacteristic, complemented
27C3×#95C2212yescomplemented
28C3⋊#95C2212yescomplemented
29C3⋊#95C2211yescharacteristic, complemented
30C3⋊#95C2212yescomplemented
31C3⋊#95C2212yescomplemented
32C3⋊#95C2212yescomplemented
33C22×C22⋊C4—32no—
34#397—32no—
35#397—31no—
36#397D311yescharacteristic, complemented
37#385—32no—
38#385—32no—
39#381—32no—
40#381—32no—
41#379—31no—
42C24×C3C2311yescharacteristic
43C23×D3—24no—
44C23×D3—24no—
45C23×D3—41no—
46C23×D3—41no—
47C23×D3D414yes—
48C23×D3C2312yes—
49C22×C3×C4C2312yes—
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2312yes—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C2×C4×D3—24no—
57C2×C4×D3—24no—
58C2×C4×D3—24no—
59C2×C4×D3—24no—
60C2×C4×D3—42no—
61C2×C4×D3—42no—
62C3×C4⋊C4—41no—
63C3×C22⋊C4—24no—
64C3×C22⋊C4—41no—
65#217—24no—
66#217—41no—
67(C2×D3)⋊C4—24no—
68(C2×D3)⋊C4—24no—
69(C2×D3)⋊C4—24no—
70(C2×D3)⋊C4—24no—
71C4⋊Dic3—41no—
72Dic3⋊C4—42no—
73C25—31no—
74C23×C4—32no—
75C23×C4—32no—
76C2×C4⋊C4—32no—
77C2×C4⋊C4—31no—
78C2×C4⋊C4C2×D311yescharacteristic, complemented
79C2×C22⋊C4—32no—
80C2×C22⋊C4—32no—
81C2×C22⋊C4—32no—
82C2×C22⋊C4—32no—
83C2×C22⋊C4—32no—
84C2×C22⋊C4—31no—
85C2×C22⋊C4—68no—
86C2×C22⋊C4C2×D312yes—
87C2×C22⋊C4C2×D311yescharacteristic
88#95—31no—
89#95—32no—
90#95—32no—
91#95—32no—
92#95—32no—
93#95C2×D311yescharacteristic, complemented
94#95C2×D312yescomplemented
95C23×C3—22no—
96C23×C3—22no—
97C23×C3—21no—
98C23×C3—21no—
99C23×C3C2×D412yes—
100C23×C3C2411yescharacteristic
101C22×D3—22no—
102C22×D3—22no—
103C22×D3—22no—
104C22×D3—22no—
105C22×D3—216no—
106C22×D3—44no—
107C22×D3—44no—
108C22×D3—44no—
109C22×D3—48no—
110C22×D3C2×D414yes—
111C22×D3#4114yes—
112C22×D3#4114yes—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4—22no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C4×D3—48no—
133C4×D3—48no—
134C24—32no—
135C24—34no—
136C24—64no—
137C24—64no—
138C24—121no—
139C24—121no—
140C24C22×D311yescharacteristic
141C22×C4—32no—
142C22×C4—32no—
143C22×C4—32no—
144C22×C4—64no—
145C22×C4—64no—
146C22×C4—64no—
147C22×C4—64no—
148C22×C4—122no—
149C22×C4—122no—
150C22×C4C22×D312yes—
151C22×C4C22×D311yescharacteristic
152C22×C4C22×D311yescharacteristic
153C22×C4C22×D312yes—
154C4⋊C4—41no—
155C4⋊C4—122no—
156C4⋊C4—121no—
157C22⋊C4—24no—
158C22⋊C4—41no—
159C22⋊C4—64no—
160C22⋊C4—64no—
161C22⋊C4—64no—
162C22⋊C4—64no—
163C22⋊C4—64no—
164C22⋊C4—121no—
165C22×C3—24no—
166C22×C3—22no—
167C22×C3—41no—
168C22×C3—41no—
169C22×C3—42no—
170C22×C3C22×D411yescharacteristic
171C22×C3C2×#4112yes—
172C22×C3C2×#4112yes—
173C22×C3H(5,𝔽2)11yescharacteristic
174C22×C3H(5,𝔽2)11yescharacteristic
175C2×D3—24no—
176C2×D3—24no—
177C2×D3—24no—
178C2×D3—24no—
179C2×D3—24no—
180C2×D3—24no—
181C2×D3—416no—
182C2×D3#12318yes—
183C3×C4—42no—
184C3×C4—42no—
185C3×C4—41no—
186C3×C4—41no—
187Dic3—42no—
188Dic3—42no—
189Dic3—42no—
190C23—21no—
191C23—22no—
192C23—22no—
193C23—21no—
194C23—34no—
195C23—34no—
196C23—34no—
197C23—616no—
198C23—62no—
199C23—62no—
200C23—62no—
201C23—62no—
202C23—124no—
203C23—124no—
204C23—124no—
205C23—128no—
206C23D3×D412yes—
207C23C23×D311yescharacteristic
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4—62no—
226C2×C4—62no—
227C2×C4—128no—
228C2×C4—128no—
229C2×C3—42no—
230C2×C3C2×#12312yes—
231C2×C3#53411yescharacteristic
232C2×C3#55111yescharacteristic
233C2×C3#55111yescharacteristic
234C2×C3#55112yes—
235D3—28no—
236C22—24no—
237C22—22no—
238C22—38no—
239C22—41no—
240C22—41no—
241C22—42no—
242C22—64no—
243C22—64no—
244C22—64no—
245C22—64no—
246C22—64no—
247C22—64no—
248C22—1216no—
249C22C2×D3×D411yescharacteristic
250C22#100411yescharacteristic
251C22D3×#4112yes—
252C22D3×#4112yes—
253C22#100911yescharacteristic
254C4—42no—
255C4—41no—
256C4—41no—
257C4—42no—
258C4—122no—
259C4—122no—
260C4—122no—
261C3#248311yescharacteristic, complemented, Sylow
262C2—42no—
263C2—68no—
264C2#560511yescharacteristic
265C2D3×#12312yes—
266C2#565911yescharacteristic
267C2#567911yescharacteristic
268C2#570512yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#397C2
prime radicalC24×D3C22
hypercenter#397D3
hypocenterC3#2483
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C3×#397C2semidirect
3. #4989C2semidirect
5. #4953C2semidirect
6. #4952C2semidirect
7. #4668C2semidirect
8. #4664C2semidirect
9. #4664C2semidirect
14. C2×C3×C4⋊C4C22semidirect
23. C2×C4⋊Dic3C22semidirect
24. C2×Dic3⋊C4C22semidirect
26. C3×#95C22semidirect
27. C3×#95C22semidirect
28. C3⋊#95C22semidirect
29. C3⋊#95C22semidirect
30. C3⋊#95C22semidirect
31. C3⋊#95C22semidirect
32. C3⋊#95C22semidirect
36. #397D3semidirect
78. C2×C4⋊C4C2×D3semidirect
93. #95C2×D3semidirect
94. #95C2×D3semidirect
261. C3#2483semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C22⋊C42
C3×#3972
#49892
#49552
#49532
#49522
#46682
#46642
#24833