Finite Group Databasebeta

← #86244

Subgroups — #86244

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×#1009C211yesmaximal, characteristic, complemented
3C3×#532C211yesmaximal, characteristic, complemented
4#5830C211yesmaximal, characteristic, complemented
5#5825C211yesmaximal, characteristic, complemented
6#5724C212yesmaximal, complemented
7#5715C212yesmaximal, complemented
8D3×#124C212yesmaximal, complemented
9#5668C212yesmaximal, complemented
10#5610C218yesmaximal, complemented
11#5552C214yesmaximal, complemented
12#5550C214yesmaximal, complemented
13C2×#875C211yesmaximal, characteristic, complemented
14C4×#235C211yesmaximal, characteristic, complemented
15#3500—31nomaximal, Sylow
16C2×C3×#41C2211yescharacteristic
17C2×C3×#41C2211yescharacteristic, complemented
18#1010—28no—
19#1009—28no—
20C2×#239C2212yes—
21C2×D3×Q8C2212yescomplemented
22C2×#237C2212yescomplemented
23C2×#237C2214yescomplemented
24C2×D3×D4C2212yes—
25C2×D3×D4C2214yes—
26C2×#235C2212yes—
27C2×#235C2211yescharacteristic
28C2×#235C2212yescomplemented
29C22×D12C2211yescharacteristic
30C22×D12C2212yes—
31C22×Dic6C2211yescharacteristic, complemented
32C22×Dic6C2212yescomplemented
33C3×C4⋊Q8C2211yescharacteristic, complemented
34C3×#127C2211yescharacteristic, complemented
35C3×#124C2211yescharacteristic, complemented
36C3×#124C2211yescharacteristic, complemented
37C3×#121C2214yescomplemented
38C3×C4×D4C2214yescomplemented
39C2×C3×C42C2211yescharacteristic, complemented
40#947C2212yescomplemented
41#944C2212yescomplemented
42#940C2212yescomplemented
43#939C2218yescomplemented
44#937C2218yescomplemented
45#937C2214yescomplemented
46#936C2212yescomplemented
47#935C2214yescomplemented
48#930C2214yescomplemented
49C4×#67C2214yescomplemented
50C2×(C2×D3)⋊C4C2214yescomplemented
51#926C2211yescharacteristic, complemented
52#924C2214yescomplemented
53#896C2218yescomplemented
54#894C2218yescomplemented
55#885C2218yescomplemented
56#884C2218yescomplemented
57#67⋊C4C2218yescomplemented
58D3×C22⋊C4C2218yescomplemented
59#875C2212yescomplemented
60#875C2212yescomplemented
61#875C2214yescomplemented
62C4×D12C2212yescomplemented
63#872C2212yescomplemented
64C42×D3C2212yescomplemented
65C4×Dic6C2212yescomplemented
66C2×2−1+4—31no—
67C2×H(5,𝔽2)—31no—
68#548—34no—
69#546—38no—
70#546—34no—
71#536—31no—
72#536—32no—
73#535—31no—
74#535—32no—
75#532—32no—
76#532D311yescharacteristic, complemented
77C2×#124—32no—
78C2×#124—31no—
79C4×#41—31no—
80C23×D3C2314yes—
81C23×D3C2312yes—
82C3×#41—24no—
83C3×#41—24no—
84C2×C3×Q8C2311yescharacteristic
85C2×C3×Q8C2311yescharacteristic
86C2×C3×D4C2312yes—
87C2×C3×D4C2311yescharacteristic
88C2×C3×D4C2312yes—
89C2×C3×D4C2311yescharacteristic
90C2×C3×D4C2314yes—
91C22×C3×C4C2312yes—
92C22×C3×C4C2312yes—
93C22×C3×C4C2312yes—
94C22×C3×C4C2311yescharacteristic
95C2×#67C2314yes—
96C2×#67C2314yes—
97C2×#67C2318yes—
98C2×#67C2312yes—
99C22×Dic3C2314yes—
100C22×Dic3C2312yes—
101#239—28no—
102D3×Q8—28no—
103#237—216no—
104#237—28no—
105D3×D4—216no—
106D3×D4—28no—
107#235—28no—
108#235—28no—
109#235D418yescomplemented
110C2×D12—24no—
111C2×D12—28no—
112C2×D12C2312yes—
113C2×D12C2312yes—
114C2×D12C2314yes—
115C2×D12C2311yescharacteristic
116C2×C4×D3C2314yes—
117C2×C4×D3C2314yes—
118C2×C4×D3C2314yes—
119C2×C4×D3C2312yes—
120C2×Dic6—24no—
121C2×Dic6—28no—
122C2×Dic6C2312yes—
123C2×Dic6C2312yes—
124C2×Dic6C2314yes—
125C2×Dic6C2311yescharacteristic
126C3×C4⋊C4C2314yes—
127C3×C22⋊C4C2314yes—
128C3×C22⋊C4C2314yes—
129C3×C42C2312yes—
130C3×C42C2311yescharacteristic
131C3×C42C2311yescharacteristic
132#217C2318yes—
133#217C2312yes—
134(C2×D3)⋊C4C2314yes—
135(C2×D3)⋊C4C2318yes—
136(C2×D3)⋊C4C2318yes—
137(C2×D3)⋊C4C2318yes—
138C4⋊Dic3C2312yes—
139Dic3⋊C4C2314yes—
140Dic3⋊C4C2318yes—
141C4×Dic3C2314yes—
142C4×Dic3C2312yes—
1432−1+4—68no—
144H(5,𝔽2)—68no—
145C2×#41—32no—
146C2×#41—32no—
147C2×#41—34no—
148C2×#41—31no—
149C2×#41—32no—
150C2×#41—32no—
151C2×#41C2×D311yescharacteristic
152C2×#41C2×D311yescharacteristic, complemented
153C22×Q8—32no—
154C22×Q8—31no—
155C22×Q8—32no—
156C22×D4—32no—
157C22×D4—34no—
158C22×D4—31no—
159C22×D4—32no—
160C4⋊Q8—32no—
161C4⋊Q8C2×D311yescharacteristic, complemented
162#127—32no—
163#127C2×D311yescharacteristic, complemented
164#124—32no—
165#124—32no—
166#124—38no—
167#124—32no—
168#124—32no—
169#124—34no—
170#124C2×D311yescharacteristic, complemented
171#124C2×D311yescharacteristic, complemented
172#123—34no—
173#123—38no—
174C22⋊Q8—34no—
175C22⋊Q8—38no—
176#121—38no—
177#121—34no—
178#121—38no—
179#121C2×D314yescomplemented
180#120—38no—
181#120—34no—
182C4×Q8—32no—
183C4×D4—34no—
184C4×D4—38no—
185C4×D4—32no—
186C4×D4C2×D314yescomplemented
187#117—31no—
188#117—32no—
189C2×C22⋊C4—34no—
190C2×C22⋊C4—38no—
191C2×C42—32no—
192C2×C42C2×D311yescharacteristic, complemented
193C23×C3C2412yes—
194C23×C3C2412yes—
195C23×C3C2411yescharacteristic
196C22×D3—24no—
197C22×D3—28no—
198C22×D3—48no—
199C22×D3—44no—
200C22×D3C2414yes—
201C22×D3C2412yes—
202C3×Q8—22no—
203C3×Q8—22no—
204C3×D4—24no—
205C3×D4—24no—
206C3×D4—22no—
207C3×D4—22no—
208C3×D4—44no—
209C2×C3×C4—24no—
210C2×C3×C4—24no—
211C2×C3×C4—42no—
212C2×C3×C4C2×D414yes—
213C2×C3×C4C2412yes—
214C2×C3×C4C2412yes—
215C2×C3×C4C2412yes—
216C2×C3×C4C2412yes—
217C2×C3×C4C2411yescharacteristic
218C2×C3×C4C2411yescharacteristic
219#67—28no—
220#67—28no—
221#67—48no—
222#67C2×D418yes—
223C2×Dic3—24no—
224C2×Dic3—28no—
225C2×Dic3C2414yes—
226C2×Dic3C2414yes—
227C2×Dic3C2412yes—
228D12—28no—
229D12—24no—
230D12—24no—
231D12C2×D414yes—
232C4×D3—28no—
233C4×D3—28no—
234C4×D3—44no—
235C4×D3C2×D418yes—
236Dic6—24no—
237Dic6—24no—
238Dic6—28no—
239Dic6C2×D414yes—
240C24—34no—
241C24—32no—
242#41—24no—
243#41—24no—
244#41—38no—
245#41—68no—
246#41—68no—
247#41—68no—
248#41—68no—
249#41—616no—
250C2×Q8—31no—
251C2×Q8—34no—
252C2×Q8—32no—
253C2×Q8—32no—
254C2×Q8—68no—
255C2×Q8—68no—
256C2×Q8—64no—
257C2×Q8C22×D311yescharacteristic
258C2×Q8C22×D311yescharacteristic
259C2×D4—32no—
260C2×D4—34no—
261C2×D4—34no—
262C2×D4—38no—
263C2×D4—31no—
264C2×D4—34no—
265C2×D4—32no—
266C2×D4—32no—
267C2×D4—68no—
268C2×D4—68no—
269C2×D4—616no—
270C2×D4—64no—
271C2×D4C22×D312yes—
272C2×D4C22×D312yes—
273C2×D4C22×D311yescharacteristic
274C2×D4C22×D311yescharacteristic
275C2×D4C22×D314yes—
276C22×C4—32no—
277C22×C4—32no—
278C22×C4—34no—
279C22×C4—34no—
280C22×C4—34no—
281C22×C4—34no—
282C22×C4C22×D312yes—
283C22×C4C22×D312yes—
284C22×C4C22×D312yes—
285C22×C4C22×D311yescharacteristic
286C4⋊C4—34no—
287C4⋊C4—32no—
288C4⋊C4—38no—
289C4⋊C4C22×D314yes—
290C22⋊C4—38no—
291C22⋊C4—32no—
292C22⋊C4—38no—
293C22⋊C4—38no—
294C22⋊C4—38no—
295C22⋊C4—34no—
296C22⋊C4C22×D314yes—
297C22⋊C4C22×D314yes—
298C42—32no—
299C42—34no—
300C42C22×D311yescharacteristic
301C42C22×D311yescharacteristic
302C42C22×D312yes—
303C22×C3—22no—
304C22×C3—22no—
305C22×C3—22no—
306C22×C3—24no—
307C22×C3—24no—
308C22×C3C22×D412yes—
309C22×C3C2511yescharacteristic
310C2×D3—24no—
311C2×D3—24no—
312C2×D3—28no—
313C2×D3—88no—
314C2×D3—84no—
315C2×D3C22×D414yes—
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—22no—
319C3×C4—22no—
320C3×C4C22×D412yes—
321C3×C4C22×D412yes—
322Dic3—24no—
323Dic3—24no—
324Dic3C22×D414yes—
325C23—32no—
326C23—34no—
327C23—68no—
328C23—64no—
329C23—124no—
330C23—128no—
331C23C23×D312yes—
332C23C23×D312yes—
333C23C23×D311yescharacteristic
334Q8—22no—
335Q8—22no—
336Q8—34no—
337Q8—68no—
338Q8—64no—
339Q8—64no—
340D4—24no—
341D4—24no—
342D4—22no—
343D4—22no—
344D4—38no—
345D4—34no—
346D4—44no—
347D4—68no—
348D4—68no—
349D4—68no—
350D4—64no—
351D4—64no—
352D4—128no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—32no—
356C2×C4—38no—
357C2×C4—34no—
358C2×C4—34no—
359C2×C4—42no—
360C2×C4—68no—
361C2×C4—68no—
362C2×C4—68no—
363C2×C4—64no—
364C2×C4—124no—
365C2×C4D3×D414yes—
366C2×C4C23×D312yes—
367C2×C4C23×D312yes—
368C2×C4C23×D312yes—
369C2×C4C23×D312yes—
370C2×C4C23×D311yescharacteristic
371C2×C4C23×D311yescharacteristic
372C2×C3—22no—
373C2×C3—42no—
374C2×C3—42no—
375C2×C3C23×D411yescharacteristic
376C2×C3#58512yes—
377D3—24no—
378D3—44no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—24no—
383C22—24no—
384C22—34no—
385C22—64no—
386C22—64no—
387C22—68no—
388C22—244no—
389C22—248no—
390C22C2×D3×D412yes—
391C22C24×D311yescharacteristic
392C4—22no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—34no—
397C4—64no—
398C4—64no—
399C4C2×D3×D412yes—
400C4C2×D3×D412yes—
401C3#350011yescharacteristic, complemented, Sylow
402C2—22no—
403C2—42no—
404C2—42no—
405C2—64no—
406C2—124no—
407C2C22×D3×D411yescharacteristic
408C2#596412yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#532C2
hypercenter#532D3
hypocenterC3#3500
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#1010C2semidirect
2. C2×#1009C2semidirect
3. C3×#532C2semidirect
4. #5830C2semidirect
5. #5825C2semidirect
6. #5724C2semidirect
7. #5715C2semidirect
8. D3×#124C2semidirect
9. #5668C2semidirect
10. #5610C2semidirect
11. #5552C2semidirect
12. #5550C2semidirect
13. C2×#875C2semidirect
14. C4×#235C2semidirect
17. C2×C3×#41C22semidirect
21. C2×D3×Q8C22semidirect
22. C2×#237C22semidirect
23. C2×#237C22semidirect
28. C2×#235C22semidirect
31. C22×Dic6C22semidirect
32. C22×Dic6C22semidirect
33. C3×C4⋊Q8C22semidirect
34. C3×#127C22semidirect
35. C3×#124C22semidirect
36. C3×#124C22semidirect
37. C3×#121C22semidirect
38. C3×C4×D4C22semidirect
39. C2×C3×C42C22semidirect
40. #947C22semidirect
41. #944C22semidirect
42. #940C22semidirect
43. #939C22semidirect
44. #937C22semidirect
45. #937C22semidirect
46. #936C22semidirect
47. #935C22semidirect
48. #930C22semidirect
49. C4×#67C22semidirect
50. C2×(C2×D3)⋊C4C22semidirect
51. #926C22semidirect
52. #924C22semidirect
53. #896C22semidirect
54. #894C22semidirect
55. #885C22semidirect
56. #884C22semidirect
57. #67⋊C4C22semidirect
58. D3×C22⋊C4C22semidirect
59. #875C22semidirect
60. #875C22semidirect
61. #875C22semidirect
62. C4×D12C22semidirect
63. #872C22semidirect
64. C42×D3C22semidirect
65. C4×Dic6C22semidirect
76. #532D3semidirect
109. #235D4semidirect
152. C2×#41C2×D3semidirect
161. C4⋊Q8C2×D3semidirect
163. #127C2×D3semidirect
170. #124C2×D3semidirect
171. #124C2×D3semidirect
179. #121C2×D3semidirect
186. C4×D4C2×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×#10102
C2×#10092
C3×#5322
#58302
#58252
#57242
#57152
D3×#1242
#56682
#56102
#55522
#55502
C2×#8752
C4×#2352
#35003