Finite Group Databasebeta

← #74591

Subgroups — #74591

Order 384.

816 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×(C2×D3)⋊C4C211yesmaximal, characteristic
2C2×D3×C22⋊C4C211yesmaximal, characteristic
3C2×D3×C22⋊C4C211yesmaximal, characteristic
4C3×#385C211yesmaximal, characteristic, complemented
5#5210C211yesmaximal, characteristic, complemented
6#4989C212yesmaximal, complemented
7#4987C212yesmaximal, complemented
8#4950C211yesmaximal, characteristic, complemented
9#4946C211yesmaximal, characteristic, complemented
10#4668C212yesmaximal, complemented
11#4667C212yesmaximal, complemented
12#2315—31nomaximal, Sylow
13C24×D3C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×Dic3C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C22×C4×D3C2211yescharacteristic
18C22×C4×D3C2212yes—
19C2×C3×C4⋊C4C2212yescomplemented
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×#217C2211yescharacteristic
23C2×#217C2211yescharacteristic
24C2×(C2×D3)⋊C4—24no—
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2212yes—
30C2×C4⋊Dic3C2212yescomplemented
31C2×Dic3⋊C4C2212yescomplemented
32C2×Dic3⋊C4C2212yescomplemented
33D3×C22⋊C4—24no—
34D3×C22⋊C4C418yescomplemented
35C3×#95C2212yescomplemented
36C3⋊#95C2212yescomplemented
37C3⋊#95C2212yescomplemented
38C3⋊#95C2212yescomplemented
39C22×C22⋊C4—31no—
40C22×C22⋊C4—31no—
41C22×C22⋊C4—31no—
42#385—31no—
43#385—32no—
44#385—31no—
45#385—31no—
46#385—32no—
47#385D311yescharacteristic, complemented
48#380—32no—
49#380—32no—
50C24×C3C2311yescharacteristic
51C23×D3—42no—
52C23×D3—42no—
53C23×D3C2×C414yes—
54C23×D3D414yes—
55C23×D3Q814yes—
56C23×D3C2311yescharacteristic
57C23×D3C2311yescharacteristic
58C22×C3×C4—22no—
59C22×C3×C4—42no—
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2312yes—
62C22×C3×C4C2311yescharacteristic
63C22×C3×C4C2312yes—
64C22×Dic3—22no—
65C22×Dic3—42no—
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2312yes—
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2312yes—
70C2×C4×D3—22no—
71C2×C4×D3—22no—
72C2×C4×D3—42no—
73C2×C4×D3—42no—
74C2×C4×D3—42no—
75C2×C4×D3—42no—
76C2×C4×D3C2×C418yescomplemented
77C3×C4⋊C4—42no—
78C3×C22⋊C4—22no—
79C3×C22⋊C4C2×C414yescomplemented
80#217—22no—
81#217C2×C414yescomplemented
82(C2×D3)⋊C4—22no—
83(C2×D3)⋊C4—22no—
84(C2×D3)⋊C4—24no—
85(C2×D3)⋊C4—24no—
86(C2×D3)⋊C4C2×C418yescomplemented
87C4⋊Dic3—42no—
88Dic3⋊C4—42no—
89Dic3⋊C4—42no—
90C25—31no—
91C23×C4—32no—
92C23×C4—31no—
93C23×C4—31no—
94C23×C4—31no—
95C23×C4C2×D311yescharacteristic
96C2×C4⋊C4—32no—
97C2×C4⋊C4—32no—
98C2×C4⋊C4—32no—
99C2×C4⋊C4C2×D312yescomplemented
100C2×C22⋊C4—31no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—38no—
103C2×C22⋊C4—31no—
104C2×C22⋊C4—31no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—64no—
109C2×C22⋊C4—64no—
110C2×C22⋊C4C2×D311yescharacteristic
111C2×C22⋊C4C2×D311yescharacteristic
112#95—32no—
113#95—32no—
114#95—32no—
115#95C2×D312yescomplemented
116C23×C3—22no—
117C23×C3—22no—
118C23×C3C22×C412yes—
119C23×C3C2×D412yes—
120C23×C3C2×Q812yes—
121C23×C3C2411yescharacteristic
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—22no—
125C22×D3—22no—
126C22×D3—48no—
127C22×D3—44no—
128C22×D3—44no—
129C22×D3—48no—
130C22×D3C4⋊C4116yescomplemented
131C22×D3C22×C412yes—
132C22×D3C22×C412yes—
133C22×D3C2×D412yes—
134C22×D3C2×D412yes—
135C22×D3C2×Q812yes—
136C22×D3C2×Q812yes—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—44no—
147C2×C3×C4C22×C414yes—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—21no—
152C2×Dic3—21no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—44no—
158C2×Dic3C22×C414yes—
159C4×D3—48no—
160C4×D3—44no—
161C4×D3—44no—
162C24—31no—
163C24—31no—
164C24—34no—
165C24—34no—
166C24—34no—
167C24—122no—
168C24—122no—
169C24C22×D311yescharacteristic
170C22×C4—22no—
171C22×C4—32no—
172C22×C4—38no—
173C22×C4—31no—
174C22×C4—31no—
175C22×C4—32no—
176C22×C4—42no—
177C22×C4—62no—
178C22×C4—62no—
179C22×C4—62no—
180C22×C4—122no—
181C22×C4—122no—
182C22×C4—122no—
183C22×C4—122no—
184C22×C4—122no—
185C22×C4C22×D311yescharacteristic
186C22×C4C22×D312yes—
187C22×C4C22×D311yescharacteristic
188C22×C4C22×D312yes—
189C4⋊C4—42no—
190C4⋊C4—122no—
191C4⋊C4—122no—
192C4⋊C4—122no—
193C22⋊C4—22no—
194C22⋊C4—34no—
195C22⋊C4—38no—
196C22⋊C4—62no—
197C22⋊C4—62no—
198C22⋊C4—64no—
199C22⋊C4—64no—
200C22⋊C4—62no—
201C22⋊C4C4×D314yescomplemented
202C22×C3—24no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3—24no—
206C22×C3C2×C4⋊C414yes—
207C22×C3C23×C411yescharacteristic
208C22×C3C22×D411yescharacteristic
209C22×C3C22×Q811yescharacteristic
210C22×C3H(5,𝔽2)12yes—
211C22×C3H(5,𝔽2)12yes—
212C2×D3—24no—
213C2×D3—24no—
214C2×D3—22no—
215C2×D3—22no—
216C2×D3—22no—
217C2×D3—22no—
218C2×D3—24no—
219C2×D3—24no—
220C2×D3—416no—
221C2×D3C2×C4⋊C414yes—
222C2×D3C2×C4⋊C414yes—
223C3×C4—24no—
224C3×C4—41no—
225C3×C4—41no—
226C3×C4—42no—
227Dic3—24no—
228Dic3—41no—
229Dic3—41no—
230Dic3—42no—
231C23—22no—
232C23—22no—
233C23—32no—
234C23—32no—
235C23—32no—
236C23—32no—
237C23—32no—
238C23—32no—
239C23—316no—
240C23—62no—
241C23—62no—
242C23—62no—
243C23—62no—
244C23—128no—
245C23—124no—
246C23—128no—
247C23—124no—
248C23C2×C4×D312yes—
249C23D3×D412yes—
250C23D3×Q812yes—
251C23C23×D311yescharacteristic
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—34no—
262C2×C4—44no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4—128no—
273C2×C4—124no—
274C2×C4—124no—
275C2×C4—124no—
276C2×C4C2×C4×D314yes—
277C2×C3—24no—
278C2×C3C22×C4⋊C411yescharacteristic
279C2×C3#51812yes—
280C2×C3#53411yescharacteristic
281C2×C3#53411yescharacteristic
282C2×C3#54312yes—
283D3—24no—
284D3—24no—
285C22—24no—
286C22—22no—
287C22—24no—
288C22—22no—
289C22—34no—
290C22—34no—
291C22—64no—
292C22—64no—
293C22—62no—
294C22—62no—
295C22—64no—
296C22—64no—
297C22—62no—
298C22—62no—
299C22—1216no—
300C22D3×C4⋊C414yes—
301C22C22×C4×D311yescharacteristic
302C22C2×D3×D411yescharacteristic
303C22#100412yes—
304C22C2×D3×Q811yescharacteristic
305C22#100912yes—
306C4—24no—
307C4—41no—
308C4—42no—
309C4—41no—
310C4—64no—
311C4—121no—
312C4—122no—
313C4—121no—
314C3#231511yescharacteristic, complemented, Sylow
315C2—24no—
316C2—64no—
317C2—64no—
318C2C2×D3×C4⋊C411yescharacteristic
319C2#554512yes—
320C2#558911yescharacteristic
321C2#563412yes—
322C2#565311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#385C2
socle, cosocleC23×C3C24
prime radicalC24×D3C22
hypercenter#385D3
hypocenterC3#2315
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
4. C3×#385C2semidirect
5. #5210C2semidirect
6. #4989C2semidirect
7. #4987C2semidirect
8. #4950C2semidirect
9. #4946C2semidirect
10. #4668C2semidirect
11. #4667C2semidirect
19. C2×C3×C4⋊C4C22semidirect
30. C2×C4⋊Dic3C22semidirect
31. C2×Dic3⋊C4C22semidirect
32. C2×Dic3⋊C4C22semidirect
34. D3×C22⋊C4C4semidirect
35. C3×#95C22semidirect
36. C3⋊#95C22semidirect
37. C3⋊#95C22semidirect
38. C3⋊#95C22semidirect
47. #385D3semidirect
76. C2×C4×D3C2×C4semidirect
79. C3×C22⋊C4C2×C4semidirect
81. #217C2×C4semidirect
86. (C2×D3)⋊C4C2×C4semidirect
99. C2×C4⋊C4C2×D3semidirect
115. #95C2×D3semidirect
130. C22×D3C4⋊C4semidirect
201. C22⋊C4C4×D3semidirect
314. C3#2315semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×D3)⋊C42
C2×D3×C22⋊C42
C3×#3852
#52102
#49892
#49872
#49502
#49462
#46682
#46672
#23153