Finite Group Databasebeta

← #86076

Subgroups — #86076

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×#1007C211yesmaximal, characteristic, complemented
2C2×#1004C211yesmaximal, characteristic, complemented
3C2×C3×#124C211yesmaximal, characteristic, complemented
4#5816C211yesmaximal, characteristic, complemented
5#5805C211yesmaximal, characteristic, complemented
6#5677C214yesmaximal, complemented
7#5676C214yesmaximal, complemented
8#5675C212yesmaximal, complemented
9#5674C212yesmaximal, complemented
10D3×#124C212yesmaximal, complemented
11#5668C212yesmaximal, complemented
12#5490C218yesmaximal, complemented
13#5477C211yesmaximal, characteristic, complemented
14C4×#235C211yesmaximal, characteristic, complemented
15#3500—31nomaximal, Sylow
16C22×C3×Q8C2211yescharacteristic, complemented
17C22×C3×D4C2211yescharacteristic
18C22×#67C2214yes—
19#1007—28no—
20C2×#239C2212yescomplemented
21C2×#239C2212yescomplemented
22C2×D3×Q8C2212yescomplemented
23C2×D3×Q8C2212yescomplemented
24#1004—28no—
25C2×#237C2212yes—
26C2×#237C2212yes—
27C2×D3×D4C2212yes—
28C2×D3×D4C2212yes—
29C2×#235C2211yescharacteristic
30C2×#235C2211yescharacteristic
31C2×#235C2211yescharacteristic, complemented
32C2×#235C2214yescomplemented
33C3×#124C2212yescomplemented
34C3×#124C2212yescomplemented
35C3×#124C2214yescomplemented
36C2×C3×C22⋊C4C2214yescomplemented
37C2×C3×C42C2211yescharacteristic, complemented
38#953C2214yescomplemented
39#947C2212yescomplemented
40#946C2214yescomplemented
41#944C2212yescomplemented
42#940C2212yescomplemented
43#938C2214yescomplemented
44#936C2212yescomplemented
45#930C2214yescomplemented
46#929C2214yescomplemented
47C4×#67C2214yescomplemented
48#926C2211yescharacteristic, complemented
49#885C2218yescomplemented
50#884C2218yescomplemented
51#884C2218yescomplemented
52#883C2218yescomplemented
53#882C2218yescomplemented
54#67⋊C4C2218yescomplemented
55D3×C22⋊C4C2218yescomplemented
56#878C2218yescomplemented
57#875C2211yescharacteristic, complemented
58#875C2211yescharacteristic, complemented
59#874C2211yescharacteristic, complemented
60C4×D12C2212yescomplemented
61C4×D12C2214yescomplemented
62#872C2212yescomplemented
63C42×D3C2212yescomplemented
64#869C2211yescharacteristic, complemented
65C4×Dic6C2212yescomplemented
66C2×2−1+4—31no—
67C2×H(5,𝔽2)—31no—
68#548—34no—
69#546—34no—
70#546—38no—
71#536—31no—
72#536—32no—
73#535—31no—
74#535—32no—
75#532—32no—
76#532—31no—
77C2×#124—32no—
78C2×#124D311yescharacteristic, complemented
79C4×#41—31no—
80C24×C3C2312yes—
81C23×D3C2314yes—
82C2×C3×Q8—24no—
83C2×C3×Q8C2312yes—
84C2×C3×Q8C2312yes—
85C2×C3×D4—24no—
86C2×C3×D4C2312yes—
87C2×C3×D4C2312yes—
88C22×C3×C4C2312yes—
89C22×C3×C4C2312yes—
90C22×C3×C4C2311yescharacteristic
91C2×#67—216no—
92C2×#67C2314yes—
93C2×#67C2314yes—
94C2×#67C2312yes—
95C2×#67C2312yes—
96C2×#67C2318yes—
97C2×#67C2312yes—
98C22×Dic3C2314yes—
99#239—28no—
100#239—28no—
101D3×Q8—28no—
102D3×Q8—28no—
103#237—28no—
104#237—28no—
105D3×D4—28no—
106D3×D4—28no—
107#235—24no—
108#235—24no—
109#235—216no—
110#235D418yescomplemented
111C2×D12C2311yescharacteristic
112C2×D12C2311yescharacteristic
113C2×D12C2314yes—
114C2×D12C2314yes—
115C2×D12C2311yescharacteristic
116C2×C4×D3C2312yes—
117C2×C4×D3C2312yes—
118C2×C4×D3C2314yes—
119C2×C4×D3C2314yes—
120C2×C4×D3C2314yes—
121C2×C4×D3C2312yes—
122C2×Dic6C2311yescharacteristic
123C2×Dic6C2311yescharacteristic
124C2×Dic6C2314yes—
125C2×Dic6C2311yescharacteristic
126C3×C22⋊C4C2318yes—
127C3×C22⋊C4C2318yes—
128C3×C42C2312yes—
129C3×C42C2311yescharacteristic
130C3×C42C2311yescharacteristic
131#217C2318yes—
132#217C2312yes—
133(C2×D3)⋊C4C2314yes—
134(C2×D3)⋊C4C2314yes—
135(C2×D3)⋊C4C2314yes—
136(C2×D3)⋊C4C2318yes—
137C4⋊Dic3C2312yes—
138C4⋊Dic3C2314yes—
139Dic3⋊C4C2314yes—
140Dic3⋊C4C2318yes—
141C4×Dic3C2314yes—
142C4×Dic3C2312yes—
1432−1+4—68no—
144H(5,𝔽2)—68no—
145C2×#41—32no—
146C2×#41—32no—
147C2×#41—32no—
148C2×#41—32no—
149C2×#41—31no—
150C2×#41—31no—
151C2×#41—31no—
152C2×#41—34no—
153C22×Q8—32no—
154C22×Q8—32no—
155C22×Q8C2×D311yescharacteristic, complemented
156C22×D4—34no—
157C22×D4—32no—
158C22×D4—32no—
159C22×D4C2×D311yescharacteristic
160C4⋊Q8—32no—
161C4⋊Q8—31no—
162#127—32no—
163#127—31no—
164#124—32no—
165#124—32no—
166#124—38no—
167#124—31no—
168#124—31no—
169#124C2×D312yescomplemented
170#124C2×D312yescomplemented
171#124C2×D314yescomplemented
172#123—34no—
173#123—38no—
174C22⋊Q8—34no—
175C22⋊Q8—38no—
176#121—34no—
177#121—34no—
178#121—38no—
179#121—38no—
180#120—34no—
181#120—38no—
182C4×Q8—32no—
183C4×D4—34no—
184C4×D4—38no—
185C4×D4—32no—
186C4×D4—34no—
187#117—31no—
188#117—32no—
189C2×C22⋊C4—38no—
190C2×C22⋊C4C2×D314yescomplemented
191C2×C42—32no—
192C2×C42C2×D311yescharacteristic, complemented
193C23×C3—24no—
194C23×C3—44no—
195C23×C3C2414yes—
196C23×C3C2411yescharacteristic
197C22×D3—28no—
198C22×D3—48no—
199C22×D3C2412yes—
200C22×D3C2412yes—
201C22×D3C2412yes—
202C3×Q8—24no—
203C3×Q8—24no—
204C3×D4—24no—
205C3×D4—24no—
206C2×C3×C4—24no—
207C2×C3×C4—42no—
208C2×C3×C4C2×D414yes—
209C2×C3×C4C2412yes—
210C2×C3×C4C2412yes—
211C2×C3×C4C2414yes—
212C2×C3×C4C2411yescharacteristic
213C2×C3×C4C2411yescharacteristic
214#67—24no—
215#67—24no—
216#67—28no—
217#67—28no—
218#67—48no—
219#67C2×D418yes—
220C2×Dic3—28no—
221C2×Dic3C2414yes—
222C2×Dic3C2412yes—
223C2×Dic3C2412yes—
224C2×Dic3C2412yes—
225D12—28no—
226D12—22no—
227D12—22no—
228D12—44no—
229D12C2×D414yes—
230C4×D3—28no—
231C4×D3—24no—
232C4×D3—24no—
233C4×D3—28no—
234C4×D3—44no—
235C4×D3C2×D418yes—
236Dic6—22no—
237Dic6—22no—
238Dic6—28no—
239Dic6C2×D414yes—
240C24—34no—
241C24C22×D312yes—
242#41—38no—
243#41—616no—
244#41—68no—
245#41—68no—
246#41—68no—
247#41—68no—
248#41—64no—
249#41—64no—
250C2×Q8—24no—
251C2×Q8—31no—
252C2×Q8—34no—
253C2×Q8—31no—
254C2×Q8—31no—
255C2×Q8—68no—
256C2×Q8—68no—
257C2×Q8C22×D312yes—
258C2×Q8C22×D312yes—
259C2×D4—24no—
260C2×D4—32no—
261C2×D4—32no—
262C2×D4—34no—
263C2×D4—32no—
264C2×D4—34no—
265C2×D4—38no—
266C2×D4—31no—
267C2×D4—34no—
268C2×D4—31no—
269C2×D4—31no—
270C2×D4—34no—
271C2×D4—68no—
272C2×D4—68no—
273C2×D4—616no—
274C2×D4C22×D312yes—
275C2×D4C22×D312yes—
276C22×C4—32no—
277C22×C4—34no—
278C22×C4—34no—
279C22×C4—34no—
280C22×C4—32no—
281C22×C4—32no—
282C22×C4—34no—
283C22×C4C22×D312yes—
284C22×C4C22×D312yes—
285C22×C4C22×D311yescharacteristic
286C4⋊C4—34no—
287C4⋊C4—32no—
288C4⋊C4—38no—
289C4⋊C4—34no—
290C22⋊C4—38no—
291C22⋊C4—32no—
292C22⋊C4—34no—
293C22⋊C4—38no—
294C22⋊C4—34no—
295C22⋊C4—34no—
296C22⋊C4C22×D318yes—
297C22⋊C4C22×D318yes—
298C42—32no—
299C42—34no—
300C42C22×D311yescharacteristic
301C42C22×D311yescharacteristic
302C42C22×D312yes—
303C22×C3—22no—
304C22×C3—24no—
305C22×C3—28no—
306C22×C3—84no—
307C22×C3C22×D412yes—
308C22×C3C2511yescharacteristic
309C2×D3—24no—
310C2×D3—22no—
311C2×D3—22no—
312C2×D3—24no—
313C2×D3—24no—
314C2×D3—88no—
315C2×D3C22×D414yes—
316C3×C4—24no—
317C3×C4—22no—
318C3×C4—22no—
319C3×C4C22×D412yes—
320C3×C4C22×D412yes—
321Dic3—22no—
322Dic3—22no—
323Dic3—24no—
324Dic3C22×D414yes—
325C23—24no—
326C23—32no—
327C23—32no—
328C23—32no—
329C23—44no—
330C23—68no—
331C23—128no—
332C23C23×D314yes—
333C23C23×D311yescharacteristic
334Q8—24no—
335Q8—24no—
336Q8—34no—
337Q8—68no—
338Q8—62no—
339Q8—62no—
340D4—24no—
341D4—24no—
342D4—38no—
343D4—34no—
344D4—68no—
345D4—68no—
346D4—68no—
347D4—64no—
348D4—64no—
349D4—62no—
350D4—62no—
351D4—124no—
352D4—128no—
353C2×C4—24no—
354C2×C4—32no—
355C2×C4—38no—
356C2×C4—34no—
357C2×C4—32no—
358C2×C4—32no—
359C2×C4—42no—
360C2×C4—68no—
361C2×C4—68no—
362C2×C4—68no—
363C2×C4—64no—
364C2×C4—64no—
365C2×C4—124no—
366C2×C4D3×D414yes—
367C2×C4C23×D312yes—
368C2×C4C23×D312yes—
369C2×C4C23×D314yes—
370C2×C4C23×D311yescharacteristic
371C2×C4C23×D311yescharacteristic
372C2×C3—22no—
373C2×C3—44no—
374C2×C3C23×D411yescharacteristic
375C2×C3#58512yes—
376D3—24no—
377D3—42no—
378D3—42no—
379C22—22no—
380C22—24no—
381C22—28no—
382C22—34no—
383C22—64no—
384C22—62no—
385C22—62no—
386C22—64no—
387C22—64no—
388C22—84no—
389C22—248no—
390C22C2×D3×D412yes—
391C22C24×D311yescharacteristic
392C4—24no—
393C4—22no—
394C4—22no—
395C4—34no—
396C4—62no—
397C4—62no—
398C4—64no—
399C4C2×D3×D412yes—
400C4C2×D3×D412yes—
401C3#350011yescharacteristic, complemented, Sylow
402C2—22no—
403C2—44no—
404C2—64no—
405C2—122no—
406C2—122no—
407C2C22×D3×D411yescharacteristic
408C2#596412yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC2×C3×#124C2
hypercenterC2×#124D3
hypocenterC3#3500
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#1007C2semidirect
2. C2×#1004C2semidirect
3. C2×C3×#124C2semidirect
4. #5816C2semidirect
5. #5805C2semidirect
6. #5677C2semidirect
7. #5676C2semidirect
8. #5675C2semidirect
9. #5674C2semidirect
10. D3×#124C2semidirect
11. #5668C2semidirect
12. #5490C2semidirect
13. #5477C2semidirect
14. C4×#235C2semidirect
16. C22×C3×Q8C22semidirect
20. C2×#239C22semidirect
21. C2×#239C22semidirect
22. C2×D3×Q8C22semidirect
23. C2×D3×Q8C22semidirect
31. C2×#235C22semidirect
32. C2×#235C22semidirect
33. C3×#124C22semidirect
34. C3×#124C22semidirect
35. C3×#124C22semidirect
36. C2×C3×C22⋊C4C22semidirect
37. C2×C3×C42C22semidirect
38. #953C22semidirect
39. #947C22semidirect
40. #946C22semidirect
41. #944C22semidirect
42. #940C22semidirect
43. #938C22semidirect
44. #936C22semidirect
45. #930C22semidirect
46. #929C22semidirect
47. C4×#67C22semidirect
48. #926C22semidirect
49. #885C22semidirect
50. #884C22semidirect
51. #884C22semidirect
52. #883C22semidirect
53. #882C22semidirect
54. #67⋊C4C22semidirect
55. D3×C22⋊C4C22semidirect
56. #878C22semidirect
57. #875C22semidirect
58. #875C22semidirect
59. #874C22semidirect
60. C4×D12C22semidirect
61. C4×D12C22semidirect
62. #872C22semidirect
63. C42×D3C22semidirect
64. #869C22semidirect
65. C4×Dic6C22semidirect
78. C2×#124D3semidirect
110. #235D4semidirect
155. C22×Q8C2×D3semidirect
169. #124C2×D3semidirect
170. #124C2×D3semidirect
171. #124C2×D3semidirect
190. C2×C22⋊C4C2×D3semidirect
192. C2×C42C2×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×#10072
C2×#10042
C2×C3×#1242
#58162
#58052
#56772
#56762
#56752
#56742
D3×#1242
#56682
#54902
#54772
C4×#2352
#35003