Finite Group Databasebeta

← #76434

Subgroups — #76434

Order 384.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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×#937C211yesmaximal, characteristic
2C3×#390C211yesmaximal, characteristic, complemented
3#5220C211yesmaximal, characteristic, complemented
4#4990C212yesmaximal, complemented
5#4958C212yesmaximal, complemented
6#4957C212yesmaximal, complemented
7#4955C212yesmaximal, complemented
8#4944C212yesmaximal, complemented
9#4940C212yesmaximal, complemented
10#2375—31nomaximal, Sylow
11C24×D3C2211yescharacteristic
12C22×C3×D4C2211yescharacteristic
13C22×#67C2212yes—
14C23×Dic3C2211yescharacteristic, complemented
15C2×C3×C4⋊C4C2211yescharacteristic, complemented
16C2×C3×C22⋊C4C2212yes—
17C2×C3×C22⋊C4C2212yescomplemented
18C2×C3×C42C2211yescharacteristic, complemented
19C2×#217C2212yes—
20C2×#217C2211yescharacteristic
21C2×#217C2211yescharacteristic, complemented
22#937C418yescomplemented
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×Dic3⋊C4C2212yescomplemented
29C2×C4×Dic3C2212yescomplemented
30C3⋊#95C2212yescomplemented
31C3⋊#95C2212yescomplemented
32C3⋊#95C2212yescomplemented
33C2×#120—31no—
34#390—32no—
35#390D311yescharacteristic, complemented
36#388—32no—
37#388—32no—
38#388—32no—
39#386—31no—
40#386—32no—
41#379—32no—
42C24×C3C2311yescharacteristic
43C24×C3C2311yescharacteristic
44C23×D3—41no—
45C23×D3—41no—
46C23×D3—42no—
47C23×D3—42no—
48C23×D3C2×C414yescomplemented
49C23×D3C2312yes—
50C2×C3×D4—41no—
51C2×C3×D4—41no—
52C2×C3×D4C2×C414yes—
53C22×C3×C4C2312yes—
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C2×#67—42no—
57C2×#67—42no—
58C2×#67C2×C418yes—
59C22×Dic3—41no—
60C22×Dic3—42no—
61C22×Dic3C2312yes—
62C22×Dic3C2312yes—
63C22×Dic3C2312yes—
64C3×C4⋊C4—41no—
65C3×C22⋊C4—42no—
66C3×C22⋊C4—42no—
67C3×C42—41no—
68#217—42no—
69#217—41no—
70#217C2×C414yescomplemented
71(C2×D3)⋊C4—42no—
72(C2×D3)⋊C4—42no—
73(C2×D3)⋊C4—42no—
74(C2×D3)⋊C4—42no—
75(C2×D3)⋊C4C2×C418yescomplemented
76Dic3⋊C4—42no—
77C4×Dic3—42no—
78C25—31no—
79C22×D4—32no—
80C22×D4C2×D311yescharacteristic
81C23×C4—31no—
82#120—38no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4C2×D311yescharacteristic, complemented
85C2×C22⋊C4—31no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4—32no—
88C2×C22⋊C4—31no—
89C2×C22⋊C4—32no—
90C2×C22⋊C4—32no—
91C2×C22⋊C4—32no—
92C2×C22⋊C4—32no—
93C2×C22⋊C4C2×D312yes—
94C2×C22⋊C4C2×D312yescomplemented
95C2×C42—32no—
96C2×C42C2×D311yescharacteristic, complemented
97#95—32no—
98#95—32no—
99#95—32no—
100C23×C3—21no—
101C23×C3—22no—
102C23×C3—22no—
103C23×C3—22no—
104C23×C3—22no—
105C23×C3—21no—
106C23×C3—21no—
107C23×C3—21no—
108C23×C3C22×C412yes—
109C23×C3C22×C412yes—
110C23×C3C2411yescharacteristic
111C22×D3—22no—
112C22×D3—22no—
113C22×D3—22no—
114C22×D3—22no—
115C22×D3—22no—
116C22×D3—22no—
117C22×D3—48no—
118C22×D3—44no—
119C22×D3—44no—
120C22×D3—48no—
121C22×D3—44no—
122C22×D3C22×C414yes—
123C3×D4—44no—
124C2×C3×C4—21no—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4C22×C412yes—
133#67—48no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—44no—
143C2×Dic3C22×C414yes—
144C24—32no—
145C24—34no—
146C24—122no—
147C24—121no—
148C24—121no—
149C24—122no—
150C24C22×D311yescharacteristic
151C24C22×D311yescharacteristic
152C2×D4—38no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—122no—
156C2×D4—122no—
157C2×D4C4×D314yes—
158C22×C4—32no—
159C22×C4—32no—
160C22×C4—32no—
161C22×C4—121no—
162C22×C4—122no—
163C22×C4C22×D312yes—
164C22×C4C22×D312yes—
165C22×C4C22×D311yescharacteristic
166C4⋊C4—41no—
167C4⋊C4—122no—
168C22⋊C4—34no—
169C22⋊C4—38no—
170C22⋊C4—42no—
171C22⋊C4—42no—
172C22⋊C4—122no—
173C22⋊C4—122no—
174C22⋊C4—122no—
175C22⋊C4—122no—
176C22⋊C4—121no—
177C22⋊C4—122no—
178C42—41no—
179C42—122no—
180C22×C3—24no—
181C22×C3—22no—
182C22×C3—22no—
183C22×C3—24no—
184C22×C3—41no—
185C22×C3—42no—
186C22×C3—41no—
187C22×C3—42no—
188C22×C3—41no—
189C22×C3—41no—
190C22×C3C23×C411yescharacteristic
191C22×C3H(5,𝔽2)12yes—
192C22×C3H(5,𝔽2)11yescharacteristic
193C22×C3H(5,𝔽2)11yescharacteristic
194C22×C3H(5,𝔽2)12yes—
195C2×D3—24no—
196C2×D3—24no—
197C2×D3—24no—
198C2×D3—416no—
199C2×D3—42no—
200C2×D3—42no—
201C2×D3—42no—
202C2×D3—42no—
203C3×C4—22no—
204C3×C4—42no—
205C3×C4—42no—
206Dic3—24no—
207Dic3—42no—
208Dic3—42no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—22no—
216C23—22no—
217C23—34no—
218C23—62no—
219C23—62no—
220C23—62no—
221C23—62no—
222C23—62no—
223C23—62no—
224C23—124no—
225C23—128no—
226C23—124no—
227C23—124no—
228C23—128no—
229C23C2×C4×D312yes—
230C23C2×C4×D312yes—
231C23C23×D311yescharacteristic
232D4—44no—
233D4—128no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—34no—
243C2×C4—62no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4—62no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—124no—
252C2×C4C2×C4×D312yes—
253C2×C3—42no—
254C2×C3—42no—
255C2×C3#51811yescharacteristic
256C2×C3#51812yes—
257C2×C3#56011yescharacteristic
258C2×C3#56012yes—
259C2×C3H(3,𝔽4)11yescharacteristic
260D3—44no—
261C22—22no—
262C22—22no—
263C22—24no—
264C22—24no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—42no—
270C22—42no—
271C22—64no—
272C22—64no—
273C22—64no—
274C22—1216no—
275C22—122no—
276C22—122no—
277C22—122no—
278C22—122no—
279C22C22×C4×D311yescharacteristic
280C22#100411yescharacteristic
281C22#100412yes—
282C22#100411yescharacteristic
283C22#100912yes—
284C4—22no—
285C4—42no—
286C4—42no—
287C4—64no—
288C4—122no—
289C4—122no—
290C3#237511yescharacteristic, complemented, Sylow
291C2—42no—
292C2—42no—
293C2—124no—
294C2#548611yescharacteristic
295C2#554512yes—
296C2#562012yes—
297C2#568311yescharacteristic
298C2#572011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#390C2
socle, cosocleC23×C3C24
prime radicalC2×#937C2
hypercenter#390D3
hypocenterC3#2375
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C3×#390C2semidirect
3. #5220C2semidirect
4. #4990C2semidirect
5. #4958C2semidirect
6. #4957C2semidirect
7. #4955C2semidirect
8. #4944C2semidirect
9. #4940C2semidirect
14. C23×Dic3C22semidirect
15. C2×C3×C4⋊C4C22semidirect
17. C2×C3×C22⋊C4C22semidirect
18. C2×C3×C42C22semidirect
21. C2×#217C22semidirect
22. #937C4semidirect
28. C2×Dic3⋊C4C22semidirect
29. C2×C4×Dic3C22semidirect
30. C3⋊#95C22semidirect
31. C3⋊#95C22semidirect
32. C3⋊#95C22semidirect
35. #390D3semidirect
48. C23×D3C2×C4semidirect
70. #217C2×C4semidirect
75. (C2×D3)⋊C4C2×C4semidirect
84. C2×C4⋊C4C2×D3semidirect
94. C2×C22⋊C4C2×D3semidirect
96. C2×C42C2×D3semidirect
290. C3#2375semidirect

Maximal subgroups

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

subgroupindex
C2×#9372
C3×#3902
#52202
#49902
#49582
#49572
#49552
#49442
#49402
#23753