Finite Group Databasebeta

← #86317

Subgroups — #86317

Order 384.

1540 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1010C211yesmaximal, characteristic, complemented
2C2×#1004C211yesmaximal, characteristic, complemented
3C3×#535C211yesmaximal, characteristic, complemented
4#5832C211yesmaximal, characteristic, complemented
5C2×#936C211yesmaximal, characteristic, complemented
6#5716C212yesmaximal, complemented
7#5715C212yesmaximal, complemented
8#5675C212yesmaximal, complemented
9D3×#124C212yesmaximal, complemented
10#5612C214yesmaximal, complemented
11#5598C214yesmaximal, complemented
12#5591C218yesmaximal, complemented
13#5526C211yesmaximal, characteristic, complemented
14#5522C211yesmaximal, characteristic, complemented
15#3500—31nomaximal, Sylow
16C2×C3×#41C2211yescharacteristic, complemented
17C22×C3×D4C2211yescharacteristic
18C22×#67C2214yes—
19#1010—28no—
20C2×D3×Q8C2212yescomplemented
21#1004—28no—
22C2×#237C2212yes—
23C2×#237C2212yes—
24C2×#237C2212yescomplemented
25C2×#237C2214yescomplemented
26C2×D3×D4C2212yes—
27C2×D3×D4C2212yes—
28C2×#235C2211yescharacteristic
29C2×#235C2211yescharacteristic
30C2×#235C2212yescomplemented
31C22×Dic6C2211yescharacteristic, complemented
32C22×Dic6C2212yescomplemented
33C3×#127C2212yescomplemented
34C3×#124C2212yescomplemented
35C3×#121C2214yescomplemented
36C3×#120C2214yescomplemented
37C3×#117C2211yescharacteristic, complemented
38C2×#217C2214yescomplemented
39#947C2211yescharacteristic, complemented
40#944C2211yescharacteristic, complemented
41#940C2211yescharacteristic, complemented
42#939C2218yescomplemented
43#939C2214yescomplemented
44#938C2218yescomplemented
45#937C2218yescomplemented
46#936C2212yescomplemented
47#936C2212yescomplemented
48#936C2211yescharacteristic, complemented
49#936C2214yescomplemented
50D4×Dic3C2218yescomplemented
51C4×#67C2214yescomplemented
52C2×C4×Dic3C2211yescharacteristic, complemented
53#897C2214yescomplemented
54#893C2212yescomplemented
55#887C2212yescomplemented
56#886C2214yescomplemented
57#885C2218yescomplemented
58#883C2218yescomplemented
59#67⋊C4C2214yescomplemented
60D3×C22⋊C4C2218yescomplemented
61#878C2218yescomplemented
62#875C2212yescomplemented
63#872C2212yescomplemented
64C42×D3C2212yescomplemented
65#869C2212yescomplemented
66C2×2−1+4—31no—
67C2×H(5,𝔽2)—31no—
68#548—34no—
69#546—34no—
70#546—38no—
71#536—32no—
72#536—31no—
73#535—32no—
74#535D311yescharacteristic, complemented
75#532—31no—
76#532—32no—
77C2×#124—31no—
78C2×#124—32no—
79C4×#41—31no—
80C24×C3C2312yes—
81C23×D3C2314yes—
82C3×#41—24no—
83C2×C3×Q8C2311yescharacteristic
84C2×C3×D4—24no—
85C2×C3×D4C2312yes—
86C2×C3×D4C2312yes—
87C2×C3×D4C2311yescharacteristic
88C2×C3×D4C2312yes—
89C2×C3×D4C2318yes—
90C22×C3×C4C2312yes—
91C22×C3×C4C2311yescharacteristic
92C2×#67—216no—
93C2×#67C2314yes—
94C2×#67C2314yes—
95C2×#67C2312yes—
96C2×#67C2314yes—
97C2×#67C2314yes—
98C2×#67C2312yes—
99C22×Dic3C2314yes—
100C22×Dic3C2312yes—
101C22×Dic3C2314yes—
102C22×Dic3C2312yes—
103D3×Q8—28no—
104#237—216no—
105#237—28no—
106#237—28no—
107#237—28no—
108D3×D4—28no—
109D3×D4—28no—
110#235—24no—
111#235—28no—
112#235D418yescomplemented
113C2×D12C2311yescharacteristic
114C2×D12C2311yescharacteristic
115C2×C4×D3C2314yes—
116C2×C4×D3C2312yes—
117C2×C4×D3C2314yes—
118C2×C4×D3C2312yes—
119C2×Dic6—24no—
120C2×Dic6—28no—
121C2×Dic6C2311yescharacteristic
122C2×Dic6C2312yes—
123C2×Dic6C2314yes—
124C2×Dic6C2312yes—
125C2×Dic6C2311yescharacteristic
126C3×C4⋊C4C2312yes—
127C3×C22⋊C4C2318yes—
128C3×C22⋊C4C2312yes—
129C3×C42C2312yes—
130#217C2318yes—
131#217C2318yes—
132#217C2314yes—
133(C2×D3)⋊C4C2314yes—
134(C2×D3)⋊C4C2318yes—
135(C2×D3)⋊C4C2314yes—
136C4⋊Dic3C2318yes—
137Dic3⋊C4C2314yes—
138Dic3⋊C4C2314yes—
139C4×Dic3C2311yescharacteristic
140C4×Dic3C2314yes—
141C4×Dic3C2311yescharacteristic
142C4×Dic3C2312yes—
1432−1+4—68no—
144H(5,𝔽2)—68no—
145C2×#41—32no—
146C2×#41—32no—
147C2×#41—34no—
148C2×#41—32no—
149C2×#41—31no—
150C2×#41—31no—
151C2×#41—32no—
152C2×#41C2×D311yescharacteristic, complemented
153C22×Q8—32no—
154C22×Q8—31no—
155C22×Q8—32no—
156C22×D4—34no—
157C22×D4—32no—
158C22×D4—32no—
159C22×D4C2×D311yescharacteristic
160C4⋊Q8—31no—
161C4⋊Q8—32no—
162#127—31no—
163#127C2×D312yescomplemented
164#124—31no—
165#124—32no—
166#124—32no—
167#124—34no—
168#124—31no—
169#124—38no—
170#124—32no—
171#124C2×D312yescomplemented
172#123—38no—
173#123—34no—
174C22⋊Q8—34no—
175C22⋊Q8—38no—
176#121—38no—
177#121—34no—
178#121—38no—
179#121C2×D314yescomplemented
180#120—38no—
181#120C2×D314yescomplemented
182C4×Q8—32no—
183C4×D4—38no—
184C4×D4—34no—
185C4×D4—32no—
186C4×D4—34no—
187#117—32no—
188#117C2×D311yescharacteristic, complemented
189C2×C22⋊C4—34no—
190C2×C22⋊C4—38no—
191C2×C42—31no—
192C2×C42—32no—
193C23×C3—24no—
194C23×C3—44no—
195C23×C3C2414yes—
196C23×C3C2412yes—
197C23×C3C2411yescharacteristic
198C22×D3—28no—
199C22×D3—48no—
200C22×D3C2412yes—
201C22×D3C2412yes—
202C3×Q8—22no—
203C3×D4—24no—
204C3×D4—24no—
205C3×D4—24no—
206C3×D4—22no—
207C3×D4—48no—
208C2×C3×C4—24no—
209C2×C3×C4C2×D414yes—
210C2×C3×C4C2414yes—
211C2×C3×C4C2412yes—
212C2×C3×C4C2411yescharacteristic
213C2×C3×C4C2411yescharacteristic
214#67—24no—
215#67—28no—
216#67—28no—
217#67—28no—
218#67—44no—
219#67C2×D418yes—
220C2×Dic3—24no—
221C2×Dic3—28no—
222C2×Dic3—28no—
223C2×Dic3—42no—
224C2×Dic3C2412yes—
225C2×Dic3C2412yes—
226C2×Dic3C2412yes—
227C2×Dic3C2414yes—
228C2×Dic3C2412yes—
229D12—22no—
230D12C2×D414yes—
231C4×D3—28no—
232C4×D3—24no—
233C4×D3—44no—
234C4×D3C2×D418yes—
235Dic6—24no—
236Dic6—24no—
237Dic6—22no—
238Dic6—28no—
239Dic6C2×D414yes—
240C24—34no—
241C24C22×D312yes—
242#41—24no—
243#41—38no—
244#41—68no—
245#41—68no—
246#41—68no—
247#41—616no—
248#41—68no—
249#41—64no—
250C2×Q8—31no—
251C2×Q8—34no—
252C2×Q8—32no—
253C2×Q8—32no—
254C2×Q8—31no—
255C2×Q8—68no—
256C2×Q8—68no—
257C2×Q8—64no—
258C2×Q8C22×D311yescharacteristic
259C2×D4—24no—
260C2×D4—32no—
261C2×D4—34no—
262C2×D4—34no—
263C2×D4—32no—
264C2×D4—34no—
265C2×D4—34no—
266C2×D4—31no—
267C2×D4—31no—
268C2×D4—68no—
269C2×D4—68no—
270C2×D4—616no—
271C2×D4C22×D312yes—
272C2×D4C22×D312yes—
273C2×D4C22×D312yes—
274C2×D4C22×D311yescharacteristic
275C2×D4C22×D318yes—
276C22×C4—32no—
277C22×C4—32no—
278C22×C4—34no—
279C22×C4—34no—
280C22×C4—32no—
281C22×C4—34no—
282C22×C4—32no—
283C22×C4—34no—
284C22×C4C22×D312yes—
285C22×C4C22×D311yescharacteristic
286C4⋊C4—34no—
287C4⋊C4—34no—
288C4⋊C4—38no—
289C4⋊C4C22×D312yes—
290C22⋊C4—38no—
291C22⋊C4—38no—
292C22⋊C4—34no—
293C22⋊C4—34no—
294C22⋊C4—38no—
295C22⋊C4—34no—
296C22⋊C4C22×D318yes—
297C22⋊C4C22×D312yes—
298C42—31no—
299C42—31no—
300C42—34no—
301C42—32no—
302C42C22×D312yes—
303C22×C3—22no—
304C22×C3—22no—
305C22×C3—24no—
306C22×C3—24no—
307C22×C3—28no—
308C22×C3—84no—
309C22×C3C22×D412yes—
310C22×C3C2511yescharacteristic
311C2×D3—24no—
312C2×D3—22no—
313C2×D3—24no—
314C2×D3—88no—
315C2×D3C22×D414yes—
316C3×C4—22no—
317C3×C4—24no—
318C3×C4C22×D412yes—
319C3×C4C22×D412yes—
320Dic3—24no—
321Dic3—22no—
322Dic3—22no—
323Dic3—22no—
324Dic3C22×D414yes—
325C23—24no—
326C23—32no—
327C23—32no—
328C23—44no—
329C23—68no—
330C23—128no—
331C23C23×D314yes—
332C23C23×D312yes—
333C23C23×D311yescharacteristic
334Q8—22no—
335Q8—34no—
336Q8—68no—
337Q8—64no—
338Q8—64no—
339Q8—62no—
340D4—24no—
341D4—24no—
342D4—24no—
343D4—22no—
344D4—38no—
345D4—34no—
346D4—48no—
347D4—68no—
348D4—68no—
349D4—68no—
350D4—64no—
351D4—62no—
352D4—124no—
353C2×C4—24no—
354C2×C4—32no—
355C2×C4—38no—
356C2×C4—32no—
357C2×C4—32no—
358C2×C4—34no—
359C2×C4—32no—
360C2×C4—68no—
361C2×C4—68no—
362C2×C4—68no—
363C2×C4—64no—
364C2×C4—64no—
365C2×C4—124no—
366C2×C4—122no—
367C2×C4D3×D414yes—
368C2×C4C23×D314yes—
369C2×C4C23×D312yes—
370C2×C4C23×D311yescharacteristic
371C2×C4C23×D311yescharacteristic
372C2×C3—22no—
373C2×C3—44no—
374C2×C3—42no—
375C2×C3C23×D411yescharacteristic
376C2×C3#58512yes—
377D3—24no—
378D3—42no—
379C22—22no—
380C22—22no—
381C22—24no—
382C22—24no—
383C22—28no—
384C22—34no—
385C22—64no—
386C22—62no—
387C22—64no—
388C22—84no—
389C22—248no—
390C22C2×D3×D412yes—
391C22C24×D311yescharacteristic
392C4—22no—
393C4—24no—
394C4—34no—
395C4—62no—
396C4—64no—
397C4—62no—
398C4—62no—
399C4C2×D3×D412yes—
400C4C2×D3×D412yes—
401C3#350011yescharacteristic, complemented, Sylow
402C2—22no—
403C2—44no—
404C2—42no—
405C2—64no—
406C2—122no—
407C2C22×D3×D411yescharacteristic
408C2#596612yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#535C2
hypercenter#535D3
hypocenterC3#3500
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#1010C2semidirect
2. C2×#1004C2semidirect
3. C3×#535C2semidirect
4. #5832C2semidirect
5. C2×#936C2semidirect
6. #5716C2semidirect
7. #5715C2semidirect
8. #5675C2semidirect
9. D3×#124C2semidirect
10. #5612C2semidirect
11. #5598C2semidirect
12. #5591C2semidirect
13. #5526C2semidirect
14. #5522C2semidirect
16. C2×C3×#41C22semidirect
20. C2×D3×Q8C22semidirect
24. C2×#237C22semidirect
25. C2×#237C22semidirect
30. C2×#235C22semidirect
31. C22×Dic6C22semidirect
32. C22×Dic6C22semidirect
33. C3×#127C22semidirect
34. C3×#124C22semidirect
35. C3×#121C22semidirect
36. C3×#120C22semidirect
37. C3×#117C22semidirect
38. C2×#217C22semidirect
39. #947C22semidirect
40. #944C22semidirect
41. #940C22semidirect
42. #939C22semidirect
43. #939C22semidirect
44. #938C22semidirect
45. #937C22semidirect
46. #936C22semidirect
47. #936C22semidirect
48. #936C22semidirect
49. #936C22semidirect
50. D4×Dic3C22semidirect
51. C4×#67C22semidirect
52. C2×C4×Dic3C22semidirect
53. #897C22semidirect
54. #893C22semidirect
55. #887C22semidirect
56. #886C22semidirect
57. #885C22semidirect
58. #883C22semidirect
59. #67⋊C4C22semidirect
60. D3×C22⋊C4C22semidirect
61. #878C22semidirect
62. #875C22semidirect
63. #872C22semidirect
64. C42×D3C22semidirect
65. #869C22semidirect
74. #535D3semidirect
112. #235D4semidirect
152. C2×#41C2×D3semidirect
163. #127C2×D3semidirect
171. #124C2×D3semidirect
179. #121C2×D3semidirect
181. #120C2×D3semidirect
188. #117C2×D3semidirect
401. C3#3500semidirect

Maximal subgroups

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

subgroupindex
C2×#10102
C2×#10042
C3×#5352
#58322
C2×#9362
#57162
#57152
#56752
D3×#1242
#56122
#55982
#55912
#55262
#55222
#35003