Finite Group Databasebeta

← #76229

Subgroups — #76229

Order 384.

540 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#892C211yesmaximal, characteristic, complemented
2C2×C4×D12C211yesmaximal, characteristic
3C3×#389C211yesmaximal, characteristic, complemented
4#4991C212yesmaximal
5#4987C211yesmaximal, characteristic, complemented
6#4987C211yesmaximal, characteristic, complemented
7#4987C211yesmaximal, characteristic, complemented
8#4980C212yesmaximal, complemented
9#4928C211yesmaximal, characteristic, complemented
10#4675C212yesmaximal, complemented
11#4669C212yesmaximal, complemented
12#2485—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×C4×D3C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C2×C3×C4⋊C4C2212yes—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C42C2211yescharacteristic
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×C4⋊Dic3C2211yescharacteristic
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×Dic3⋊C4C2212yes—
30C2×C4×Dic3C2212yes—
31#892—24no—
32C4×D12—24no—
33C3×#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4×D4—31no—
39C2×#117—31no—
40#397—32no—
41#394—32no—
42#389—31no—
43#389D311yescharacteristic, complemented
44#388—32no—
45#387—32no—
46#380—31no—
47#380—31no—
48#380—31no—
49C23×D3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2312yes—
59C22×Dic3C2311yescharacteristic
60C2×D12—22no—
61C2×D12—22no—
62C2×D12—22no—
63C2×C4×D3—22no—
64C2×C4×D3—22no—
65C2×C4×D3—22no—
66C2×C4×D3—22no—
67C2×C4×D3—22no—
68C2×C4×D3D414yes—
69C3×C4⋊C4—22no—
70C3×C4⋊C4—22no—
71C3×C4⋊C4—24no—
72C3×C42—22no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4—24no—
75(C2×D3)⋊C4—24no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4—42no—
78(C2×D3)⋊C4—42no—
79C4⋊Dic3—22no—
80C4⋊Dic3—22no—
81C4⋊Dic3—22no—
82Dic3⋊C4—42no—
83C4×Dic3—24no—
84C22×D4—31no—
85C23×C4—31no—
86C23×C4—31no—
87C4×D4—64no—
88#117—64no—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4C2×D311yescharacteristic
94C2×C4⋊C4C2×D311yescharacteristic
95C2×C4⋊C4C2×D312yes—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—31no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C42—32no—
103C2×C42C2×D311yescharacteristic
104#95—32no—
105#95—32no—
106#95—32no—
107#95—32no—
108#95C2×D312yes—
109C23×C3C2411yescharacteristic
110C22×D3—21no—
111C22×D3—21no—
112C22×D3—21no—
113C22×D3—21no—
114C22×D3—21no—
115C22×D3—21no—
116C22×D3—21no—
117C22×D3—21no—
118C22×D3—21no—
119C22×D3—21no—
120C22×D3—21no—
121C22×D3—21no—
122C22×D3C2×D412yes—
123C22×D3#4112yes—
124C2×C3×C4—21no—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—21no—
133C2×C3×C4—22no—
134C2×C3×C4—21no—
135C2×C3×C4C2×D412yes—
136C2×C3×C4#4112yes—
137C2×C3×C4#4112yes—
138C2×C3×C4#4112yes—
139C2×Dic3—21no—
140C2×Dic3—21no—
141C2×Dic3—22no—
142C2×Dic3—21no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3C2×D412yes—
151D12—28no—
152C4×D3—28no—
153C4×D3—44no—
154C24—31no—
155C24—31no—
156C2×D4—62no—
157C2×D4—62no—
158C2×D4—62no—
159C22×C4—31no—
160C22×C4—34no—
161C22×C4—31no—
162C22×C4—32no—
163C22×C4—32no—
164C22×C4—62no—
165C22×C4—62no—
166C22×C4—62no—
167C22×C4—62no—
168C22×C4—62no—
169C22×C4C22×D312yes—
170C22×C4C22×D312yes—
171C22×C4C22×D311yescharacteristic
172C22×C4C22×D311yescharacteristic
173C22×C4C22×D311yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—24no—
177C4⋊C4—62no—
178C4⋊C4—62no—
179C4⋊C4—62no—
180C4⋊C4—122no—
181C22⋊C4—62no—
182C22⋊C4—64no—
183C22⋊C4—62no—
184C22⋊C4—64no—
185C22⋊C4—122no—
186C22⋊C4—122no—
187C42—22no—
188C42—64no—
189C22×C3C22×D411yescharacteristic
190C22×C3C2×#4111yescharacteristic
191C22×C3C2×#4111yescharacteristic
192C22×C3C2×#4111yescharacteristic
193C22×C3C2×#4111yescharacteristic
194C22×C3H(5,𝔽2)11yescharacteristic
195C22×C32−1+411yescharacteristic
196C2×D3—22no—
197C2×D3—22no—
198C2×D3—22no—
199C2×D3—22no—
200C2×D3—22no—
201C2×D3—22no—
202C2×D3—41no—
203C2×D3—41no—
204C2×D3—41no—
205C2×D3—41no—
206C2×D3—41no—
207C2×D3—41no—
208C2×D3—41no—
209C2×D3—41no—
210C3×C4—22no—
211C3×C4—42no—
212C3×C4—41no—
213C3×C4—42no—
214C3×C4#12314yes—
215Dic3—22no—
216Dic3—41no—
217Dic3—42no—
218Dic3—42no—
219C23—32no—
220C23—32no—
221C23—61no—
222C23—61no—
223C23—61no—
224C23—61no—
225C23—61no—
226C23—61no—
227C23—61no—
228C23—61no—
229C23—61no—
230C23—61no—
231C23—61no—
232C23—61no—
233C23C23×D311yescharacteristic
234D4—68no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—32no—
247C2×C4—61no—
248C2×C4—61no—
249C2×C4—68no—
250C2×C4—61no—
251C2×C4—62no—
252C2×C4—61no—
253C2×C4—62no—
254C2×C4—61no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4—124no—
260C2×C4#23512yes—
261C2×C4D3×D412yes—
262C2×C4#23712yes—
263C2×C4#23912yes—
264C2×C3C2×#12311yescharacteristic
265C2×C3#52511yescharacteristic
266C2×C3#53711yescharacteristic
267C2×C3#55211yescharacteristic
268C2×C3#55311yescharacteristic
269C2×C3#55511yescharacteristic
270C2×C3#55611yescharacteristic
271D3—42no—
272D3—42no—
273C22—62no—
274C22—62no—
275C22—62no—
276C22—62no—
277C22—62no—
278C22—62no—
279C22—121no—
280C22—121no—
281C22—121no—
282C22—121no—
283C22—121no—
284C22—121no—
285C22—121no—
286C22—121no—
287C22C2×#23511yescharacteristic
288C22C2×D3×D411yescharacteristic
289C22C2×#23711yescharacteristic
290C22C2×#23911yescharacteristic
291C22#100711yescharacteristic
292C22D3×#4111yescharacteristic
293C22#100911yescharacteristic
294C4—22no—
295C4—42no—
296C4—42no—
297C4—41no—
298C4—62no—
299C4—122no—
300C4—121no—
301C4—122no—
302C4#89414yes—
303C3#248511yescharacteristic, complemented, Sylow
304C2—122no—
305C2—122no—
306C2C2×#89411yescharacteristic
307C2#556311yescharacteristic
308C2#558111yescharacteristic
309C2#560611yescharacteristic
310C2#562911yescharacteristic
311C2#569511yescharacteristic
312C2#573211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#389C2
prime radicalC22×D12C22
hypercenter#389D3
hypocenterC3#2485
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#892C2semidirect
3. C3×#389C2semidirect
5. #4987C2semidirect
6. #4987C2semidirect
7. #4987C2semidirect
8. #4980C2semidirect
9. #4928C2semidirect
10. #4675C2semidirect
11. #4669C2semidirect
43. #389D3semidirect
303. C3#2485semidirect

Maximal subgroups

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

subgroupindex
C2×#8922
C2×C4×D122
C3×#3892
#49912
#49872
#49802
#49282
#46752
#46692
#24853