Finite Group Databasebeta

← #87281

Subgroups — #87281

Order 384.

1660 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C2×#1004C211yesmaximal, characteristic, complemented
3#5833C212yesmaximal, complemented
4#5832C212yesmaximal, complemented
5#5829C214yesmaximal, complemented
6#5824C214yesmaximal, complemented
7#5806C214yesmaximal, complemented
8#5804C214yesmaximal, complemented
9C2×#939C214yesmaximal, complemented
10D4×#67C214yesmaximal, complemented
11C2×D4×Dic3C211yesmaximal, characteristic, complemented
12#3496—31nomaximal, Sylow
13C3×H(5,𝔽2)—28no—
14C2×C3×#41C2214yescomplemented
15C2×C3×#41C2212yescomplemented
16C22×C3×D4C2211yescharacteristic
17C22×C3×D4C2214yescomplemented
18C22×C3×D4C2214yescomplemented
19C22×#67C2214yescomplemented
20C22×#67C2214yescomplemented
21C23×Dic3C2212yescomplemented
22#1004—28no—
23C2×#237C2214yescomplemented
24C2×D3×D4C2214yescomplemented
25C2×#235C2212yescomplemented
26#953C2218yescomplemented
27C2×#217C2212yescomplemented
28C2×#217C2214yescomplemented
29#947C2212yescomplemented
30#946C2214yescomplemented
31#944C2212yescomplemented
32#940C2212yescomplemented
33#939C2218yescomplemented
34#939C2214yescomplemented
35#939C2218yescomplemented
36#939C2218yescomplemented
37#939C2214yescomplemented
38#938C2214yescomplemented
39#937C2218yescomplemented
40#936C2212yescomplemented
41#935C2218yescomplemented
42D4×Dic3C2214yescomplemented
43D4×Dic3C2214yescomplemented
44#930C2214yescomplemented
45#929C2218yescomplemented
46C4×#67C2214yescomplemented
47C4×#67C2214yescomplemented
48C2×(C2×D3)⋊C4C2214yescomplemented
49C2×C4⋊Dic3C2211yescharacteristic, complemented
50#924C2214yescomplemented
51C2×Dic3⋊C4C2214yescomplemented
52C2×C4×Dic3C2211yescharacteristic, complemented
53C2×H(5,𝔽2)—31no—
54C2×H(5,𝔽2)D311yescharacteristic, complemented
55#547—34no—
56#546—34no—
57#546—34no—
58D42—34no—
59#537—32no—
60#534—34no—
61#532—32no—
62C2×#121—34no—
63C2×C4×D4—31no—
64C24×C3C2314yes—
65C24×C3C2312yes—
66C23×D3C2314yes—
67C3×#41—216no—
68C3×#41—28no—
69C2×C3×Q8C2312yes—
70C2×C3×D4—216no—
71C2×C3×D4—216no—
72C2×C3×D4D418yescomplemented
73C2×C3×D4C2318yes—
74C2×C3×D4C2312yes—
75C2×C3×D4C2314yes—
76C2×C3×D4C2314yes—
77C22×C3×C4C2314yes—
78C22×C3×C4C2314yes—
79C22×C3×C4C2311yescharacteristic
80C2×#67—216no—
81C2×#67—48no—
82C2×#67C2314yes—
83C2×#67C2318yes—
84C2×#67C2314yes—
85C2×#67C2314yes—
86C2×#67C2318yes—
87C22×Dic3—44no—
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2314yes—
90C22×Dic3C2314yes—
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2312yes—
93C22×Dic3C2314yes—
94#237—216no—
95D3×D4—216no—
96#235—28no—
97C2×D12C2312yes—
98C2×C4×D3C2314yes—
99C2×Dic6C2312yes—
100#217C2318yes—
101#217C2314yes—
102#217C2314yes—
103#217C2318yes—
104(C2×D3)⋊C4C2314yes—
105(C2×D3)⋊C4C2318yes—
106(C2×D3)⋊C4C2314yes—
107C4⋊Dic3C2314yes—
108Dic3⋊C4C2314yes—
109Dic3⋊C4C2318yes—
110Dic3⋊C4C2314yes—
111C4×Dic3C2312yes—
112C4×Dic3C2312yes—
113H(5,𝔽2)—28no—
114H(5,𝔽2)—68no—
115C2×#41—34no—
116C2×#41—32no—
117C2×#41C2×D312yescomplemented
118C2×#41C2×D314yescomplemented
119C22×D4—34no—
120C22×D4—34no—
121C22×D4—34no—
122C22×D4C2×D311yescharacteristic
123C22×D4C2×D314yescomplemented
124C22×D4C2×D314yescomplemented
125C23×C4—32no—
126C4⋊Q8—32no—
127#127—32no—
128#124—32no—
129#124—32no—
130#123—38no—
131#123—38no—
132C22⋊Q8—34no—
133C22⋊Q8—34no—
134#121—34no—
135#121—38no—
136#121—38no—
137#121—38no—
138#121—34no—
139#121—34no—
140#121—34no—
141#120—38no—
142#120—38no—
143C4×D4—34no—
144C4×D4—34no—
145C4×D4—34no—
146C4×D4—34no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—34no—
149C2×C22⋊C4—32no—
150C2×C22⋊C4—34no—
151C2×C22⋊C4—34no—
152C2×C42—31no—
153C23×C3—28no—
154C23×C3—42no—
155C23×C3—42no—
156C23×C3—44no—
157C23×C3—44no—
158C23×C3C2×D418yescomplemented
159C23×C3C2414yes—
160C23×C3C2414yes—
161C23×C3C2411yescharacteristic
162C22×D3—28no—
163C22×D3—44no—
164C22×D3—44no—
165C22×D3C2414yes—
166C3×Q8—24no—
167C3×D4—216no—
168C3×D4—28no—
169C3×D4—24no—
170C3×D4C2×D4116yescomplemented
171C2×C3×C4—28no—
172C2×C3×C4—28no—
173C2×C3×C4C2×D414yescomplemented
174C2×C3×C4C2414yes—
175C2×C3×C4C2412yes—
176#67—28no—
177#67—216no—
178#67—28no—
179#67—48no—
180#67—44no—
181C2×Dic3—28no—
182C2×Dic3—22no—
183C2×Dic3—28no—
184C2×Dic3—44no—
185C2×Dic3—42no—
186C2×Dic3—41no—
187C2×Dic3C2412yes—
188C2×Dic3C2411yescharacteristic
189C2×Dic3C2414yes—
190C2×Dic3C2414yes—
191C2×Dic3C2411yescharacteristic
192D12—24no—
193C4×D3—28no—
194Dic6—24no—
195C24—34no—
196C24C22×D314yes—
197C24C22×D312yes—
198#41—216no—
199#41—28no—
200#41—616no—
201#41—68no—
202C2×Q8—32no—
203C2×Q8C22×D312yes—
204C2×D4—216no—
205C2×D4—216no—
206C2×D4—34no—
207C2×D4—38no—
208C2×D4—34no—
209C2×D4—34no—
210C2×D4—38no—
211C2×D4—32no—
212C2×D4—616no—
213C2×D4—616no—
214C2×D4—128no—
215C2×D4#6718yescomplemented
216C2×D4C22×D318yes—
217C2×D4C22×D314yes—
218C2×D4C22×D312yes—
219C2×D4C22×D314yes—
220C22×C4—31no—
221C22×C4—34no—
222C22×C4—34no—
223C22×C4—31no—
224C22×C4—34no—
225C22×C4—32no—
226C22×C4—34no—
227C22×C4—124no—
228C22×C4C22×D314yes—
229C22×C4C22×D314yes—
230C22×C4C22×D311yescharacteristic
231C4⋊C4—34no—
232C4⋊C4—34no—
233C4⋊C4—38no—
234C4⋊C4—34no—
235C22⋊C4—34no—
236C22⋊C4—38no—
237C22⋊C4—38no—
238C22⋊C4—34no—
239C22⋊C4—34no—
240C22⋊C4—38no—
241C22⋊C4—34no—
242C42—32no—
243C42—32no—
244C22×C3—21no—
245C22×C3—24no—
246C22×C3—21no—
247C22×C3—24no—
248C22×C3—24no—
249C22×C3—24no—
250C22×C3—24no—
251C22×C3—48no—
252C22×C3—88no—
253C22×C3C22×D418yes—
254C22×C3C22×D412yes—
255C22×C3C2511yescharacteristic
256C2×D3—24no—
257C2×D3—24no—
258C2×D3—24no—
259C2×D3—88no—
260C3×C4—24no—
261C3×C4C22×D414yes—
262Dic3—24no—
263Dic3—21no—
264Dic3—24no—
265Dic3—22no—
266Dic3H(5,𝔽2)12yescomplemented
267C23—28no—
268C23—34no—
269C23—42no—
270C23—42no—
271C23—44no—
272C23—44no—
273C23—68no—
274C23—124no—
275C23—124no—
276C23C2×#6718yescomplemented
277C23C23×D314yes—
278C23C23×D314yes—
279C23C23×D311yescharacteristic
280Q8—24no—
281Q8—64no—
282D4—216no—
283D4—28no—
284D4—24no—
285D4—68no—
286D4—616no—
287D4—68no—
288D4—64no—
289D4—128no—
290D4—124no—
291D4C2×#67116yescomplemented
292C2×C4—28no—
293C2×C4—28no—
294C2×C4—31no—
295C2×C4—32no—
296C2×C4—31no—
297C2×C4—34no—
298C2×C4—34no—
299C2×C4—62no—
300C2×C4—68no—
301C2×C4—68no—
302C2×C4—68no—
303C2×C4—124no—
304C2×C4—122no—
305C2×C4—121no—
306C2×C4C2×#6714yescomplemented
307C2×C4C23×D314yes—
308C2×C4C23×D312yes—
309C2×C3—28no—
310C2×C3—22no—
311C2×C3—44no—
312C2×C3C23×D411yescharacteristic
313C2×C3C2×H(5,𝔽2)11yescharacteristic
314C2×C3#58511yescharacteristic
315D3—44no—
316C22—21no—
317C22—24no—
318C22—21no—
319C22—24no—
320C22—24no—
321C22—24no—
322C22—24no—
323C22—48no—
324C22—64no—
325C22—64no—
326C22—64no—
327C22—88no—
328C22—248no—
329C22C22×#6718yes—
330C22C22×#6712yes—
331C22C24×D311yescharacteristic
332C4—24no—
333C4—32no—
334C4—61no—
335C4—64no—
336C4—62no—
337C4—64no—
338C4C22×#6714yes—
339C3#349611yescharacteristic, complemented, Sylow
340C2—22no—
341C2—28no—
342C2—44no—
343C2—124no—
344C2D3×H(5,𝔽2)11yescharacteristic
345C2#596611yescharacteristic
346C2C23×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC2×C3×H(5,𝔽2)C2
hypercenterC2×H(5,𝔽2)D3
hypocenterC3#3496
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×C3×H(5,𝔽2)C2semidirect
2. C2×#1004C2semidirect
3. #5833C2semidirect
4. #5832C2semidirect
5. #5829C2semidirect
6. #5824C2semidirect
7. #5806C2semidirect
8. #5804C2semidirect
9. C2×#939C2semidirect
10. D4×#67C2semidirect
11. C2×D4×Dic3C2semidirect
14. C2×C3×#41C22semidirect
15. C2×C3×#41C22semidirect
17. C22×C3×D4C22semidirect
18. C22×C3×D4C22semidirect
19. C22×#67C22semidirect
20. C22×#67C22semidirect
21. C23×Dic3C22semidirect
23. C2×#237C22semidirect
24. C2×D3×D4C22semidirect
25. C2×#235C22semidirect
26. #953C22semidirect
27. C2×#217C22semidirect
28. C2×#217C22semidirect
29. #947C22semidirect
30. #946C22semidirect
31. #944C22semidirect
32. #940C22semidirect
33. #939C22semidirect
34. #939C22semidirect
35. #939C22semidirect
36. #939C22semidirect
37. #939C22semidirect
38. #938C22semidirect
39. #937C22semidirect
40. #936C22semidirect
41. #935C22semidirect
42. D4×Dic3C22semidirect
43. D4×Dic3C22semidirect
44. #930C22semidirect
45. #929C22semidirect
46. C4×#67C22semidirect
47. C4×#67C22semidirect
48. C2×(C2×D3)⋊C4C22semidirect
49. C2×C4⋊Dic3C22semidirect
50. #924C22semidirect
51. C2×Dic3⋊C4C22semidirect
52. C2×C4×Dic3C22semidirect
54. C2×H(5,𝔽2)D3semidirect
72. C2×C3×D4D4semidirect
117. C2×#41C2×D3semidirect
118. C2×#41C2×D3semidirect
123. C22×D4C2×D3semidirect
124. C22×D4C2×D3semidirect
158. C23×C3C2×D4semidirect
170. C3×D4C2×D4semidirect
173. C2×C3×C4C2×D4semidirect
215. C2×D4#67semidirect
266. Dic3H(5,𝔽2)semidirect
276. C23C2×#67semidirect
291. D4C2×#67semidirect
306. C2×C4C2×#67semidirect
339. C3#3496semidirect

Maximal subgroups

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

subgroupindex
C2×C3×H(5,𝔽2)2
C2×#10042
#58332
#58322
#58292
#58242
#58062
#58042
C2×#9392
D4×#672
C2×D4×Dic32
#34963