Finite Group Databasebeta

← #76379

Subgroups — #76379

Order 384.

668 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 149 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×#939C212yesmaximal, complemented
2C2×#886C212yesmaximal, complemented
3C3×#390C211yesmaximal, characteristic, complemented
4#4994C212yesmaximal, complemented
5#4961C212yesmaximal, complemented
6#4960C212yesmaximal, complemented
7#4950C212yesmaximal, complemented
8#4939C211yesmaximal, characteristic, complemented
9#4928C211yesmaximal, characteristic, complemented
10#2682—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C22×#67C2212yes—
13C22×#67C2212yes—
14C23×Dic3C2212yes—
15C22×D12C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2212yes—
18C2×C3×C22⋊C4C2212yes—
19C2×C3×C42C2211yescharacteristic
20C2×#217C2212yes—
21#939—28no—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×C4⋊Dic3C2212yes—
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31#886—28no—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C2×#123—32no—
35C2×#121—32no—
36#396—32no—
37#394—32no—
38#392—32no—
39#390—31no—
40#390D311yescharacteristic, complemented
41#389—31no—
42#385—32no—
43C24×C3C2312yes—
44C23×D3C2312yes—
45C2×C3×D4—41no—
46C2×C3×D4—41no—
47C2×C3×D4D414yescomplemented
48C22×C3×C4C2312yes—
49C22×C3×C4C2311yescharacteristic
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C2×#67—24no—
53C2×#67—24no—
54C2×#67—24no—
55C2×#67—42no—
56C2×#67—42no—
57C2×#67—42no—
58C22×Dic3—24no—
59C22×Dic3—24no—
60C22×Dic3—42no—
61C22×Dic3C2312yes—
62C22×Dic3C2312yes—
63C22×Dic3C2312yes—
64C2×D12—22no—
65C2×D12—41no—
66C2×D12—41no—
67C3×C4⋊C4—41no—
68C3×C22⋊C4—24no—
69C3×C22⋊C4—24no—
70C3×C42—41no—
71#217—24no—
72(C2×D3)⋊C4—24no—
73(C2×D3)⋊C4—24no—
74(C2×D3)⋊C4—24no—
75(C2×D3)⋊C4—42no—
76C4⋊Dic3—24no—
77C4⋊Dic3—41no—
78C4⋊Dic3—41no—
79C4⋊Dic3D414yescomplemented
80Dic3⋊C4—24no—
81C22×D4—32no—
82C22×D4—32no—
83C22×D4—31no—
84C22×D4C2×D311yescharacteristic
85C23×C4—32no—
86#123—68no—
87#121—68no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4C2×D311yescharacteristic
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4C2×D312yes—
100C2×C22⋊C4C2×D312yes—
101C2×C42C2×D311yescharacteristic
102#95—32no—
103#95—32no—
104C23×C3—22no—
105C23×C3—22no—
106C23×C3—22no—
107C23×C3—22no—
108C23×C3—22no—
109C23×C3—22no—
110C23×C3C2×D414yes—
111C23×C3C2411yescharacteristic
112C22×D3—22no—
113C22×D3—22no—
114C22×D3—22no—
115C22×D3—22no—
116C22×D3—22no—
117C22×D3—22no—
118C22×D3—22no—
119C3×D4—44no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4—21no—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4C2×D412yes—
130C2×C3×C4C2×D412yes—
131C2×C3×C4#4112yes—
132#67—48no—
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—48no—
143C2×Dic3C2×D414yes—
144D12—44no—
145C24—32no—
146C24C22×D312yes—
147C2×D4—41no—
148C2×D4—41no—
149C2×D4—64no—
150C2×D4—64no—
151C2×D4—64no—
152C2×D4—62no—
153C2×D4—121no—
154C2×D4—121no—
155C2×D4—122no—
156C2×D4—122no—
157C2×D4—122no—
158C2×D4D1214yescomplemented
159C22×C4—32no—
160C22×C4—32no—
161C22×C4—32no—
162C22×C4—64no—
163C22×C4—64no—
164C22×C4—122no—
165C22×C4C22×D312yes—
166C22×C4C22×D311yescharacteristic
167C22×C4C22×D311yescharacteristic
168C22×C4C22×D311yescharacteristic
169C4⋊C4—34no—
170C4⋊C4—41no—
171C4⋊C4—64no—
172C4⋊C4—64no—
173C4⋊C4—121no—
174C4⋊C4—121no—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—64no—
178C22⋊C4—64no—
179C22⋊C4—64no—
180C22⋊C4—64no—
181C22⋊C4—122no—
182C42—41no—
183C22×C3—24no—
184C22×C3—24no—
185C22×C3—24no—
186C22×C3—42no—
187C22×C3—42no—
188C22×C3—42no—
189C22×C3—42no—
190C22×C3C22×D411yescharacteristic
191C22×C3C22×D411yescharacteristic
192C22×C3C2×#4111yescharacteristic
193C22×C3H(5,𝔽2)11yescharacteristic
194C22×C3H(5,𝔽2)11yescharacteristic
195C22×C3H(5,𝔽2)11yescharacteristic
196C22×C32−1+411yescharacteristic
197C2×D3—42no—
198C2×D3—42no—
199C2×D3—42no—
200C2×D3—42no—
201C2×D3—42no—
202C2×D3—42no—
203C2×D3—42no—
204C3×C4—22no—
205C3×C4—22no—
206C3×C4—22no—
207C3×C4—42no—
208Dic3—24no—
209Dic3—42no—
210Dic3—42no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—62no—
218C23—62no—
219C23—62no—
220C23—62no—
221C23—62no—
222C23—62no—
223C23—62no—
224C23C2×D1214yes—
225C23C23×D311yescharacteristic
226D4—44no—
227D4—124no—
228D4—128no—
229D4—128no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—34no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—62no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4—128no—
249C2×C4C2×D1212yes—
250C2×C4D3×D412yes—
251C2×C4#23712yes—
252C2×C3—44no—
253C2×C3#53411yescharacteristic
254C2×C3#53711yescharacteristic
255C2×C3#53811yescharacteristic
256C2×C3D4211yescharacteristic
257C2×C3#54711yescharacteristic
258C2×C3#55011yescharacteristic
259C2×C3#56211yescharacteristic
260D3—82no—
261C22—24no—
262C22—24no—
263C22—24no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—122no—
269C22—122no—
270C22—122no—
271C22—122no—
272C22—122no—
273C22—122no—
274C22—122no—
275C22C22×D1211yescharacteristic
276C22C2×D3×D411yescharacteristic
277C22C2×#23711yescharacteristic
278C22#100411yescharacteristic
279C22#100411yescharacteristic
280C22#100911yescharacteristic
281C22#101011yescharacteristic
282C4—22no—
283C4—22no—
284C4—22no—
285C4—42no—
286C4—64no—
287C4—122no—
288C4—122no—
289C3#268211yescharacteristic, complemented, Sylow
290C2—44no—
291C2—242no—
292C2#549311yescharacteristic
293C2D4×D1211yescharacteristic
294C2#555511yescharacteristic
295C2#559711yescharacteristic
296C2#561711yescharacteristic
297C2#568411yescharacteristic
298C2#571811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#390C2
hypercenter#390D3
hypocenterC3#2682
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#939C2semidirect
2. C2×#886C2semidirect
3. C3×#390C2semidirect
4. #4994C2semidirect
5. #4961C2semidirect
6. #4960C2semidirect
7. #4950C2semidirect
8. #4939C2semidirect
9. #4928C2semidirect
40. #390D3semidirect
47. C2×C3×D4D4semidirect
79. C4⋊Dic3D4semidirect
158. C2×D4D12semidirect
289. C3#2682semidirect

Maximal subgroups

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

subgroupindex
C2×#9392
C2×#8862
C3×#3902
#49942
#49612
#49602
#49502
#49392
#49282
#26823