Finite Group Databasebeta

← #78112

Subgroups — #78112

Order 384.

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

normal subgroups only

This group has 816 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×#895C212yesmaximal, complemented
2C2×#882C212yesmaximal, complemented
3C3×#399C211yesmaximal, characteristic, complemented
4#4992C212yesmaximal, complemented
5#4991C212yesmaximal, complemented
6#4960C211yesmaximal, characteristic, complemented
7#4956C212yesmaximal, complemented
8#4955C211yesmaximal, characteristic, complemented
9#4665C212yesmaximal, complemented
10#2668—31nomaximal, Sylow
11C24×D3C2211yescharacteristic
12C22×#67C2212yes—
13C22×D12C2212yes—
14C22×D12C2212yes—
15C22×D12C2211yescharacteristic, complemented
16C22×C4×D3C2212yescomplemented
17C2×C3×C4⋊C4C2212yescomplemented
18C2×C3×C4⋊C4C2211yescharacteristic, complemented
19C2×C3×C22⋊C4C2212yes—
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×#217C2211yescharacteristic
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yescomplemented
27C2×(C2×D3)⋊C4C2212yescomplemented
28#895—28no—
29#882—28no—
30C3×#95C2211yescharacteristic, complemented
31C3⋊#95C2212yescomplemented
32C3⋊#95C2212yescomplemented
33C3⋊#95C2211yescharacteristic, complemented
34C2×#121—32no—
35C2×#120—32no—
36#399D311yescharacteristic, complemented
37#397—32no—
38#394—32no—
39#392—31no—
40#386—32no—
41#386—32no—
42#379—31no—
43C24×C3C2311yescharacteristic
44C23×D3—22no—
45C23×D3—24no—
46C23×D3—42no—
47C23×D3—41no—
48C23×D3—41no—
49C23×D3C2312yes—
50C23×D3C2312yes—
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2312yes—
55C2×#67—24no—
56C2×#67—24no—
57C2×#67—42no—
58C22×Dic3C2312yes—
59C22×Dic3C2312yes—
60C2×D12—24no—
61C2×D12—24no—
62C2×D12—24no—
63C2×D12—42no—
64C2×D12—42no—
65C2×D12—42no—
66C2×D12—41no—
67C2×D12D418yescomplemented
68C2×C4×D3—24no—
69C2×C4×D3—42no—
70C2×C4×D3—42no—
71C3×C4⋊C4—24no—
72C3×C4⋊C4—41no—
73C3×C22⋊C4—24no—
74C3×C22⋊C4—41no—
75#217—41no—
76(C2×D3)⋊C4—24no—
77(C2×D3)⋊C4—24no—
78(C2×D3)⋊C4—24no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—42no—
81(C2×D3)⋊C4—42no—
82C25—31no—
83C22×D4—32no—
84C22×D4—32no—
85C22×D4—32no—
86C22×D4—31no—
87C23×C4—32no—
88#121—68no—
89#120—68no—
90C2×C4⋊C4C2×D312yescomplemented
91C2×C4⋊C4C2×D311yescharacteristic, complemented
92C2×C22⋊C4—32no—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4C2×D312yes—
100C2×C22⋊C4C2×D311yescharacteristic
101#95—32no—
102#95—32no—
103#95—31no—
104#95C2×D311yescharacteristic, complemented
105C23×C3—22no—
106C23×C3—21no—
107C23×C3—21no—
108C23×C3—22no—
109C23×C3#4112yes—
110C23×C3C2411yescharacteristic
111C22×D3—22no—
112C22×D3—22no—
113C22×D3—22no—
114C22×D3—22no—
115C22×D3—22no—
116C22×D3—22no—
117C22×D3—22no—
118C22×D3—22no—
119C22×D3—22no—
120C22×D3—22no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—44no—
124C22×D3—48no—
125C22×D3—48no—
126C22×D3—44no—
127C22×D3—44no—
128C22×D3C2×D414yes—
129C22×D3C2×D414yes—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—22no—
139C2×C3×C4C2×D414yes—
140#67—48no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147D12—48no—
148D12—44no—
149D12—48no—
150C4×D3—48no—
151C24—32no—
152C24—32no—
153C24—64no—
154C24—62no—
155C24—121no—
156C24—121no—
157C24—122no—
158C24C22×D311yescharacteristic
159C2×D4—38no—
160C2×D4—64no—
161C2×D4—64no—
162C2×D4—64no—
163C2×D4—64no—
164C2×D4—64no—
165C2×D4—122no—
166C2×D4—122no—
167C2×D4—121no—
168C2×D4—122no—
169C2×D4—122no—
170C22×C4—32no—
171C22×C4—32no—
172C22×C4—64no—
173C22×C4—122no—
174C22×C4—122no—
175C22×C4C22×D312yes—
176C22×C4C22×D311yescharacteristic
177C22×C4C22×D311yescharacteristic
178C22×C4C22×D312yes—
179C4⋊C4—24no—
180C4⋊C4—41no—
181C22⋊C4—24no—
182C22⋊C4—41no—
183C22⋊C4—64no—
184C22⋊C4—64no—
185C22⋊C4—64no—
186C22⋊C4—64no—
187C22⋊C4—122no—
188C22⋊C4—122no—
189C22⋊C4—121no—
190C22×C3—22no—
191C22×C3—24no—
192C22×C3—42no—
193C22×C3—41no—
194C22×C3—41no—
195C22×C3C22×D412yes—
196C22×C3C2×#4111yescharacteristic
197C22×C3H(5,𝔽2)11yescharacteristic
198C22×C3H(5,𝔽2)11yescharacteristic
199C22×C3H(5,𝔽2)12yes—
200C2×D3—24no—
201C2×D3—24no—
202C2×D3—24no—
203C2×D3—24no—
204C2×D3—24no—
205C2×D3—24no—
206C2×D3—416no—
207C2×D3—42no—
208C2×D3—42no—
209C2×D3—42no—
210C2×D3—42no—
211C2×D3—42no—
212C2×D3—42no—
213C2×D3—42no—
214C2×D3—42no—
215C3×C4—24no—
216C3×C4—42no—
217C3×C4—41no—
218C3×C4—41no—
219Dic3—42no—
220Dic3—42no—
221C23—22no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—34no—
226C23—34no—
227C23—62no—
228C23—62no—
229C23—62no—
230C23—62no—
231C23—62no—
232C23—62no—
233C23—62no—
234C23—62no—
235C23—62no—
236C23—62no—
237C23—62no—
238C23—62no—
239C23—128no—
240C23—124no—
241C23—124no—
242C23—128no—
243C23—124no—
244C23#23912yes—
245C23C23×D311yescharacteristic
246D4—128no—
247D4—124no—
248D4—128no—
249D4—128no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4—128no—
266C2×C4D3×D414yes—
267C2×C3—42no—
268C2×C3#53412yes—
269C2×C3#53811yescharacteristic
270C2×C3D4211yescharacteristic
271C2×C3#54612yes—
272C2×C3#56011yescharacteristic
273D3—44no—
274D3—44no—
275C22—22no—
276C22—24no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—64no—
281C22—64no—
282C22—64no—
283C22—64no—
284C22—64no—
285C22—64no—
286C22—1216no—
287C22—122no—
288C22—122no—
289C22—122no—
290C22—122no—
291C22—122no—
292C22—122no—
293C22—122no—
294C22—122no—
295C22C2×D3×D412yes—
296C22#100411yescharacteristic
297C22C2×#23911yescharacteristic
298C22#100911yescharacteristic
299C22#100912yes—
300C4—24no—
301C4—41no—
302C4—42no—
303C4—41no—
304C4—122no—
305C4—122no—
306C3#266811yescharacteristic, complemented, Sylow
307C2—42no—
308C2—124no—
309C2—124no—
310C2#563012yes—
311C2#564411yescharacteristic
312C2#565312yes—
313C2#566611yescharacteristic
314C2#571711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
hypercenter#399D3
hypocenterC3#2668
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#895C2semidirect
2. C2×#882C2semidirect
3. C3×#399C2semidirect
4. #4992C2semidirect
5. #4991C2semidirect
6. #4960C2semidirect
7. #4956C2semidirect
8. #4955C2semidirect
9. #4665C2semidirect
15. C22×D12C22semidirect
16. C22×C4×D3C22semidirect
17. C2×C3×C4⋊C4C22semidirect
18. C2×C3×C4⋊C4C22semidirect
26. C2×(C2×D3)⋊C4C22semidirect
27. C2×(C2×D3)⋊C4C22semidirect
30. C3×#95C22semidirect
31. C3⋊#95C22semidirect
32. C3⋊#95C22semidirect
33. C3⋊#95C22semidirect
36. #399D3semidirect
67. C2×D12D4semidirect
90. C2×C4⋊C4C2×D3semidirect
91. C2×C4⋊C4C2×D3semidirect
104. #95C2×D3semidirect
306. C3#2668semidirect

Maximal subgroups

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

subgroupindex
C2×#8952
C2×#8822
C3×#3992
#49922
#49912
#49602
#49562
#49552
#46652
#26683