Finite Group Databasebeta

← #76774

Subgroups — #76774

Order 384.

712 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 125 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×#937C212yesmaximal, complemented
2C3×#392C211yesmaximal, characteristic, complemented
3#5223C212yesmaximal, complemented
4#5222C212yesmaximal, complemented
5#4961C212yesmaximal, complemented
6#4959C212yesmaximal, complemented
7#4955C211yesmaximal, characteristic, complemented
8#4953C211yesmaximal, characteristic, complemented
9#4675C212yesmaximal, complemented
10#2812—31nomaximal, Sylow
11C24×D3C2211yescharacteristic
12C22×C3×D4C2212yes—
13C22×C3×D4C2211yescharacteristic, complemented
14C22×#67C2212yes—
15C22×#67C2212yes—
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2212yescomplemented
18C2×#217C2212yes—
19C2×#217C2211yescharacteristic
20C2×#217C2212yescomplemented
21C2×#217C2212yescomplemented
22#937—28no—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×C4⋊Dic3C2211yescharacteristic, complemented
28C2×Dic3⋊C4C2212yescomplemented
29C3×#95C2211yescharacteristic, complemented
30C3⋊#95C2212yescomplemented
31C3⋊#95C2212yescomplemented
32C3⋊#95C2211yescharacteristic, complemented
33C2×#120—32no—
34#397—31no—
35#397—32no—
36#394—32no—
37#394—32no—
38#394—32no—
39#392—32no—
40#392D311yescharacteristic, complemented
41#379—31no—
42C24×C3C2312yes—
43C24×C3C2311yescharacteristic
44C23×D3—42no—
45C23×D3—42no—
46C23×D3—41no—
47C23×D3—41no—
48C23×D3D414yescomplemented
49C23×D3C2312yes—
50C2×C3×D4—24no—
51C2×C3×D4—42no—
52C2×C3×D4—42no—
53C2×C3×D4—42no—
54C2×C3×D4—41no—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2312yes—
57C22×C3×C4C2311yescharacteristic
58C2×#67—24no—
59C2×#67—24no—
60C2×#67—42no—
61C2×#67—42no—
62C2×#67—42no—
63C2×#67—42no—
64C22×Dic3C2312yes—
65C22×Dic3C2312yes—
66C22×Dic3C2312yes—
67C3×C22⋊C4—41no—
68C3×C22⋊C4—42no—
69#217—24no—
70#217—41no—
71#217—42no—
72#217—42no—
73(C2×D3)⋊C4—24no—
74(C2×D3)⋊C4—24no—
75(C2×D3)⋊C4—42no—
76(C2×D3)⋊C4—42no—
77C4⋊Dic3—41no—
78Dic3⋊C4—42no—
79C25—31no—
80C22×D4—32no—
81C22×D4—32no—
82C22×D4C2×D312yes—
83C22×D4C2×D311yescharacteristic, complemented
84#120—68no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4—31no—
87C2×C22⋊C4—32no—
88C2×C22⋊C4—32no—
89C2×C22⋊C4—32no—
90C2×C22⋊C4—31no—
91C2×C22⋊C4—32no—
92C2×C22⋊C4—32no—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4C2×D311yescharacteristic
96C2×C22⋊C4C2×D312yescomplemented
97#95—32no—
98#95—31no—
99#95—32no—
100#95C2×D311yescharacteristic, complemented
101C23×C3—21no—
102C23×C3—21no—
103C23×C3—22no—
104C23×C3—22no—
105C23×C3—22no—
106C23×C3—22no—
107C23×C3—22no—
108C23×C3—22no—
109C23×C3—22no—
110C23×C3—22no—
111C23×C3—22no—
112C23×C3C2×D412yes—
113C23×C3C2411yescharacteristic
114C22×D3—22no—
115C22×D3—22no—
116C22×D3—22no—
117C22×D3—22no—
118C22×D3—22no—
119C22×D3—22no—
120C22×D3—44no—
121C22×D3—48no—
122C22×D3—44no—
123C22×D3—48no—
124C22×D3—44no—
125C22×D3C2×D414yes—
126C3×D4—48no—
127C3×D4—44no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—21no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—21no—
135#67—48no—
136#67—48no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C24—32no—
147C24—34no—
148C24—121no—
149C24—122no—
150C24—122no—
151C24—121no—
152C24C22×D312yes—
153C24C22×D311yescharacteristic
154C2×D4—24no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—41no—
158C2×D4—42no—
159C2×D4—64no—
160C2×D4—64no—
161C2×D4—122no—
162C2×D4—122no—
163C2×D4—122no—
164C2×D4—122no—
165C22×C4—32no—
166C22×C4—32no—
167C22×C4—32no—
168C22×C4C22×D311yescharacteristic
169C22×C4C22×D312yes—
170C22×C4C22×D311yescharacteristic
171C4⋊C4—122no—
172C4⋊C4—121no—
173C22⋊C4—41no—
174C22⋊C4—42no—
175C22⋊C4—64no—
176C22⋊C4—64no—
177C22⋊C4—64no—
178C22⋊C4—122no—
179C22⋊C4—122no—
180C22⋊C4—122no—
181C22⋊C4—121no—
182C22⋊C4—122no—
183C22×C3—24no—
184C22×C3—22no—
185C22×C3—41no—
186C22×C3—42no—
187C22×C3—41no—
188C22×C3—42no—
189C22×C3—42no—
190C22×C3—42no—
191C22×C3—42no—
192C22×C3—42no—
193C22×C3—42no—
194C22×C3—42no—
195C22×C3C22×D411yescharacteristic
196C22×C3H(5,𝔽2)12yes—
197C22×C3H(5,𝔽2)11yescharacteristic
198C22×C3H(5,𝔽2)12yes—
199C22×C3H(5,𝔽2)11yescharacteristic
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—42no—
209C3×C4—41no—
210C3×C4—41no—
211Dic3—42no—
212Dic3—42no—
213Dic3—42no—
214C23—21no—
215C23—22no—
216C23—22no—
217C23—21no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—34no—
226C23—62no—
227C23—62no—
228C23—62no—
229C23—62no—
230C23—62no—
231C23—62no—
232C23—128no—
233C23—124no—
234C23—124no—
235C23—124no—
236C23—128no—
237C23D3×D412yes—
238C23C23×D311yescharacteristic
239D4—48no—
240D4—44no—
241D4—128no—
242D4—128no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C3—42no—
260C2×C3—82no—
261C2×C3#53412yes—
262C2×C3#53411yescharacteristic
263C2×C3#56011yescharacteristic
264C2×C3#56012yes—
265C2×C3H(3,𝔽4)11yescharacteristic
266D3—44no—
267C22—24no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—64no—
280C22—64no—
281C22—64no—
282C22—1216no—
283C22—122no—
284C22—122no—
285C22—122no—
286C22—122no—
287C22C2×D3×D411yescharacteristic
288C22#100412yes—
289C22#100411yescharacteristic
290C22#100412yes—
291C22#100911yescharacteristic
292C4—41no—
293C4—42no—
294C4—41no—
295C4—122no—
296C4—122no—
297C4—122no—
298C3#281211yescharacteristic, complemented, Sylow
299C2—42no—
300C2—82no—
301C2—124no—
302C2#558912yes—
303C2#559411yescharacteristic
304C2#560511yescharacteristic
305C2#562012yes—
306C2#572011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2812
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#937C2semidirect
2. C3×#392C2semidirect
3. #5223C2semidirect
4. #5222C2semidirect
5. #4961C2semidirect
6. #4959C2semidirect
7. #4955C2semidirect
8. #4953C2semidirect
9. #4675C2semidirect
13. C22×C3×D4C22semidirect
17. C2×C3×C22⋊C4C22semidirect
20. C2×#217C22semidirect
21. C2×#217C22semidirect
27. C2×C4⋊Dic3C22semidirect
28. C2×Dic3⋊C4C22semidirect
29. C3×#95C22semidirect
30. C3⋊#95C22semidirect
31. C3⋊#95C22semidirect
32. C3⋊#95C22semidirect
40. #392D3semidirect
48. C23×D3D4semidirect
83. C22×D4C2×D3semidirect
96. C2×C22⋊C4C2×D3semidirect
100. #95C2×D3semidirect
298. C3#2812semidirect

Maximal subgroups

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

subgroupindex
C2×#9372
C3×#3922
#52232
#52222
#49612
#49592
#49552
#49532
#46752
#28123