Finite Group Databasebeta

← #80350

Subgroups — #80350

Order 384.

620 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 133 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×D3×D8C212yesmaximal, complemented
2#5717C211yesmaximal, characteristic, complemented
3D4×D12C211yesmaximal, characteristic, complemented
4C3×#459C211yesmaximal, characteristic, complemented
5#5156C212yesmaximal, complemented
6#5073C211yesmaximal, characteristic, complemented
7C4×#213C211yesmaximal, characteristic, complemented
8#4843C211yesmaximal, characteristic, complemented
9#4834C211yesmaximal, characteristic, complemented
10#4775C212yesmaximal, complemented
11#4773C212yesmaximal, complemented
12D4×D8—31nomaximal, Sylow
13C2×D3×D4—21no—
14C2×D3×D4C2212yescomplemented
15C2×D3×D4C2212yescomplemented
16C22×D12—21no—
17C2×C3×D8C2212yescomplemented
18C3×#127C2211yescharacteristic, complemented
19C3×C4×D4C2211yescharacteristic, complemented
20#938—21no—
21#938C2212yescomplemented
22#937—22no—
23C2×#213C2211yescharacteristic
24C2×#213C2212yescomplemented
25C2×#213C2212yescomplemented
26#930—21no—
27D3×D8—28no—
28C2×D24C2212yescomplemented
29C2×D3×C8C2212yescomplemented
30#895C2212yescomplemented
31#882—22no—
32#874C2211yescharacteristic, complemented
33C4×D12C2211yescharacteristic
34C4×D12C2211yescharacteristic, complemented
35C3×C4⋊C8C2211yescharacteristic, complemented
36C3×#102C2212yescomplemented
37#832C2211yescharacteristic, complemented
38#821C2212yescomplemented
39(C2×D3)⋊C8C2212yescomplemented
40#809C2211yescharacteristic, complemented
41#807C2211yescharacteristic, complemented
42C4×C3⋊C8C2211yescharacteristic, complemented
43C22×D8—32no—
44D42—31no—
45D42—31no—
46#493—31no—
47#466—32no—
48#459—31no—
49#459D311yescharacteristic, complemented
50#447—32no—
51#447—32no—
52C4×D8—31no—
53C8×D4—31no—
54C23×D3—22no—
55C23×D3—22no—
56C2×C3×D4—21no—
57C2×C3×D4—21no—
58C2×C3×D4C2312yes—
59C2×C3×D4C2311yescharacteristic
60C22×C3×C4—21no—
61C2×#67—22no—
62C2×#67—22no—
63C2×#67—22no—
64D3×D4—24no—
65D3×D4—24no—
66D3×D4—42no—
67D3×D4—42no—
68D3×D4—44no—
69C2×D12—22no—
70C2×D12—21no—
71C2×D12—21no—
72C2×D12—44no—
73C2×D12D411yescharacteristic
74C2×D12C2312yes—
75C2×D12C2311yescharacteristic
76C2×C4×D3—21no—
77C2×C4×D3C2312yes—
78C3×D8—24no—
79C2×C3×C8C2312yes—
80C3×C4⋊C4C2311yescharacteristic
81C3×C22⋊C4—21no—
82C3×C42C2311yescharacteristic
83#217—22no—
84#213—24no—
85#213—42no—
86#213D414yescomplemented
87(C2×D3)⋊C4—21no—
88(C2×D3)⋊C4—22no—
89(C2×D3)⋊C4D412yes—
90C4⋊Dic3D411yescharacteristic
91C4⋊Dic3C2311yescharacteristic
92C2×C3⋊C8C2311yescharacteristic
93C2×C3⋊C8C2311yescharacteristic
94D24—24no—
95D3×C8—24no—
96C22×D4—32no—
97C22×D4—32no—
98C22×D4—61no—
99C22×D4—61no—
100C2×D8—31no—
101C2×D8—32no—
102C2×D8—32no—
103C2×D8—32no—
104C2×D8—68no—
105C2×D8C2×D312yescomplemented
106C22×C8—32no—
107#127—31no—
108#127C2×D311yescharacteristic, complemented
109#121—32no—
110#121—32no—
111#121—61no—
112#121—61no—
113#120—62no—
114#120—62no—
115C4×D4—31no—
116C4×D4—31no—
117C4×D4C2×D311yescharacteristic, complemented
118#107—31no—
119C4⋊C8C2×D311yescharacteristic, complemented
120#102—31no—
121#102—32no—
122#102—31no—
123#102C2×D312yescomplemented
124C22⋊C8—32no—
125C4×C8—31no—
126C23×C3—21no—
127C23×C3—22no—
128C22×D3—21no—
129C22×D3—22no—
130C22×D3—44no—
131C22×D3—44no—
132C22×D3—44no—
133C22×D3—44no—
134C22×D3—82no—
135C22×D3—82no—
136C22×D3C2×D412yes—
137C3×D4—22no—
138C3×D4—21no—
139C3×D4—22no—
140C3×D4—42no—
141C3×D4—81no—
142C3×D4C2×D412yescomplemented
143C2×C3×C4—21no—
144C2×C3×C4—42no—
145C2×C3×C4C2×D411yescharacteristic
146C2×C3×C4C2×D411yescharacteristic
147C2×C3×C4C2411yescharacteristic
148#67—44no—
149#67—44no—
150#67—44no—
151C2×Dic3—21no—
152C2×Dic3C2×D412yes—
153D12—22no—
154D12—21no—
155D12—22no—
156D12—44no—
157D12—42no—
158D12—81no—
159D12D814yescomplemented
160D12C2×D412yescomplemented
161C4×D3—22no—
162C4×D3—22no—
163C4×D3—42no—
164C3×C8—22no—
165C3⋊C8—21no—
166C3⋊C8C2×D412yescomplemented
167C24—62no—
168C24—62no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—31no—
172C2×D4—31no—
173C2×D4—32no—
174C2×D4—62no—
175C2×D4—62no—
176C2×D4—62no—
177C2×D4—64no—
178C2×D4—64no—
179C2×D4—62no—
180C2×D4—61no—
181C2×D4—61no—
182C2×D4—124no—
183C2×D4—122no—
184C2×D4—122no—
185C2×D4—124no—
186C2×D4C22×D312yes—
187C2×D4C22×D311yescharacteristic
188C22×C4—21no—
189C22×C4—32no—
190C22×C4—61no—
191D8—24no—
192D8—34no—
193D8—64no—
194D8—64no—
195D8—122no—
196C2×C8—31no—
197C2×C8—31no—
198C2×C8—64no—
199C2×C8C22×D312yes—
200C4⋊C4—31no—
201C4⋊C4—31no—
202C4⋊C4C22×D311yescharacteristic
203C22⋊C4—21no—
204C22⋊C4—32no—
205C22⋊C4—62no—
206C22⋊C4—62no—
207C22⋊C4—61no—
208C42C22×D311yescharacteristic
209C22×C3—22no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3—42no—
215C22×C3C22×D411yescharacteristic
216C2×D3—22no—
217C2×D3—22no—
218C2×D3—22no—
219C2×D3—41no—
220C2×D3—41no—
221C2×D3—42no—
222C2×D3—42no—
223C2×D3—42no—
224C2×D3—88no—
225C2×D3—88no—
226C2×D3C2×D814yescomplemented
227C3×C4—21no—
228C3×C4—41no—
229C3×C4C2×D812yescomplemented
230C3×C4C22×D411yescharacteristic
231C3×C4H(5,𝔽2)11yescharacteristic
232Dic3—22no—
233Dic3—41no—
234C23—21no—
235C23—22no—
236C23—32no—
237C23—61no—
238C23—62no—
239C23—124no—
240C23—124no—
241C23—124no—
242C23—124no—
243C23—242no—
244C23—242no—
245D4—22no—
246D4—21no—
247D4—22no—
248D4—34no—
249D4—32no—
250D4—42no—
251D4—62no—
252D4—61no—
253D4—62no—
254D4—81no—
255D4—124no—
256D4—124no—
257D4—124no—
258D4—124no—
259D4—122no—
260D4—241no—
261D4D3×D412yescomplemented
262C2×C4—21no—
263C2×C4—32no—
264C2×C4—42no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—61no—
268C2×C4—122no—
269C2×C4D3×D411yescharacteristic
270C2×C4D3×D411yescharacteristic
271C2×C4C23×D311yescharacteristic
272C8—22no—
273C8—32no—
274C8—61no—
275C2×C3—42no—
276C2×C3—82no—
277C2×C3D4211yescharacteristic
278C2×C3C22×D811yescharacteristic
279C2×C3#57611yescharacteristic
280D3—24no—
281D3—42no—
282D3—82no—
283C22—22no—
284C22—34no—
285C22—41no—
286C22—41no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—62no—
291C22—62no—
292C22—62no—
293C22—121no—
294C22—121no—
295C22—122no—
296C22—122no—
297C22—122no—
298C22—248no—
299C22—248no—
300C22C2×D3×D411yescharacteristic
301C4—21no—
302C4—41no—
303C4—62no—
304C4—121no—
305C4D3×D812yescomplemented
306C4C2×D3×D411yescharacteristic
307C4#100911yescharacteristic
308C3D4×D811yescharacteristic, complemented, Sylow
309C2—42no—
310C2—64no—
311C2—82no—
312C2—122no—
313C2—242no—
314C2#560911yescharacteristic
315C2C2×D3×D811yescharacteristic
316C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3D4×D8
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×D3×D8C2semidirect
2. #5717C2semidirect
3. D4×D12C2semidirect
4. C3×#459C2semidirect
5. #5156C2semidirect
6. #5073C2semidirect
7. C4×#213C2semidirect
8. #4843C2semidirect
9. #4834C2semidirect
10. #4775C2semidirect
11. #4773C2semidirect
14. C2×D3×D4C22semidirect
15. C2×D3×D4C22semidirect
17. C2×C3×D8C22semidirect
18. C3×#127C22semidirect
19. C3×C4×D4C22semidirect
21. #938C22semidirect
24. C2×#213C22semidirect
25. C2×#213C22semidirect
28. C2×D24C22semidirect
29. C2×D3×C8C22semidirect
30. #895C22semidirect
32. #874C22semidirect
34. C4×D12C22semidirect
35. C3×C4⋊C8C22semidirect
36. C3×#102C22semidirect
37. #832C22semidirect
38. #821C22semidirect
39. (C2×D3)⋊C8C22semidirect
40. #809C22semidirect
41. #807C22semidirect
42. C4×C3⋊C8C22semidirect
49. #459D3semidirect
86. #213D4semidirect
105. C2×D8C2×D3semidirect
108. #127C2×D3semidirect
117. C4×D4C2×D3semidirect
119. C4⋊C8C2×D3semidirect
123. #102C2×D3semidirect
142. C3×D4C2×D4semidirect
159. D12D8semidirect
160. D12C2×D4semidirect
166. C3⋊C8C2×D4semidirect
226. C2×D3C2×D8semidirect
229. C3×C4C2×D8semidirect
261. D4D3×D4semidirect
305. C4D3×D8semidirect
308. C3D4×D8semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D82
#57172
D4×D122
C3×#4592
#51562
#50732
C4×#2132
#48432
#48342
#47752
#47732
D4×D83