Finite Group Databasebeta

← #86715

Subgroups — #86715

Order 384.

1540 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1009C211yesmaximal, characteristic, complemented
2C2×#1007C211yesmaximal, characteristic, complemented
3C3×#548C211yesmaximal, characteristic, complemented
4#5831C211yesmaximal, characteristic, complemented
5#5817C211yesmaximal, characteristic, complemented
6#5676C213yesmaximal, complemented
7D3×#124C213yesmaximal, complemented
8#5630C213yesmaximal, complemented
9#5628C213yesmaximal, complemented
10#5612C213yesmaximal, complemented
11#5607C213yesmaximal, complemented
12#5576C211yesmaximal, characteristic, complemented
13C4×#239C211yesmaximal, characteristic, complemented
14#5550C213yesmaximal, complemented
15#5548C213yesmaximal, complemented
16#3500—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic, complemented
19#1009—28no—
20#1007—28no—
21C2×#239C2211yescharacteristic
22C2×#239C2211yescharacteristic
23C2×#239C2213yescomplemented
24C2×D3×Q8C2213yescomplemented
25C2×D3×Q8C2211yescharacteristic, complemented
26C2×D3×D4C2213yes—
27C2×D3×D4C2213yes—
28C2×#235C2213yes—
29C2×#235C2213yescomplemented
30C2×#235C2213yescomplemented
31C22×D12C2213yes—
32C3×#124C2213yescomplemented
33C3×C22⋊Q8C2213yescomplemented
34C3×#121C2213yescomplemented
35C3×C4×Q8C2211yescharacteristic, complemented
36C3×C4×D4C2213yescomplemented
37#947C2213yescomplemented
38#946C2213yescomplemented
39Q8×Dic3C2211yescharacteristic, complemented
40#940C2213yescomplemented
41#938C2213yescomplemented
42#938C2213yescomplemented
43#937C2216yescomplemented
44#936C2213yescomplemented
45#930C2213yescomplemented
46C4×#67C2213yescomplemented
47C4×#67C2213yescomplemented
48#924C2213yescomplemented
49#897C2213yescomplemented
50#895C2213yescomplemented
51#895C2213yescomplemented
52#894C2216yescomplemented
53#893C2213yescomplemented
54#892C2213yescomplemented
55#888C2213yescomplemented
56#885C2213yescomplemented
57#885C2216yescomplemented
58#885C2211yescharacteristic, complemented
59#884C2213yescomplemented
60#884C2213yescomplemented
61#883C2216yescomplemented
62#882C2216yescomplemented
63D3×C22⋊C4C2216yescomplemented
64D3×C22⋊C4C2216yescomplemented
65#875C2213yescomplemented
66C4×D12C2213yescomplemented
67C4×D12C2213yescomplemented
68C42×D3C2213yescomplemented
69C2×2−1+4—31no—
70C2×H(5,𝔽2)—31no—
71#548—31no—
72#548—31no—
73#548—31no—
74#548D311yescharacteristic, complemented
75#546—33no—
76#546—33no—
77#546—33no—
78#546—33no—
79#536—33no—
80#535—33no—
81#532—33no—
82C2×#124—33no—
83C4×#41—31no—
84C23×D3C2316yes—
85C3×#41—24no—
86C2×C3×Q8—24no—
87C2×C3×Q8C2313yes—
88C2×C3×Q8C2311yescharacteristic
89C2×C3×D4C2313yes—
90C2×C3×D4C2313yes—
91C22×C3×C4C2313yes—
92C22×C3×C4C2313yes—
93C2×#67C2313yes—
94C2×#67C2313yes—
95C2×#67C2313yes—
96C2×#67C2311yescharacteristic
97C2×#67C2313yes—
98C2×#67C2313yes—
99#239—24no—
100#239—212no—
101#239D418yescomplemented
102D3×Q8—24no—
103D3×Q8—212no—
104D3×D4—212no—
105D3×D4—212no—
106#235—212no—
107#235—212no—
108#235—212no—
109C2×D12—212no—
110C2×D12C2313yes—
111C2×D12C2313yes—
112C2×D12C2313yes—
113C2×D12C2313yes—
114C2×D12C2313yes—
115C2×C4×D3C2313yes—
116C2×C4×D3C2313yes—
117C2×C4×D3C2316yes—
118C2×C4×D3C2313yes—
119C2×C4×D3C2313yes—
120C2×C4×D3C2313yes—
121C2×Dic6C2313yes—
122C2×Dic6C2313yes—
123C2×Dic6C2311yescharacteristic
124C3×C4⋊C4C2313yes—
125C3×C4⋊C4C2313yes—
126C3×C22⋊C4C2313yes—
127C3×C22⋊C4C2316yes—
128C3×C22⋊C4C2311yescharacteristic
129C3×C42C2313yes—
130#217C2313yes—
131#217C2316yes—
132#217C2311yescharacteristic
133(C2×D3)⋊C4C2311yescharacteristic
134(C2×D3)⋊C4C2313yes—
135(C2×D3)⋊C4C2316yes—
136(C2×D3)⋊C4C2316yes—
137(C2×D3)⋊C4C2313yes—
138(C2×D3)⋊C4C2316yes—
139(C2×D3)⋊C4C2311yescharacteristic
140C4⋊Dic3C2313yes—
141C4⋊Dic3C2313yes—
142Dic3⋊C4C2313yes—
143Dic3⋊C4C2313yes—
144C4×Dic3C2311yescharacteristic
145C4×Dic3C2313yes—
146C4×Dic3C2313yes—
1472−1+4—68no—
148H(5,𝔽2)—68no—
149C2×#41—31no—
150C2×#41—33no—
151C2×#41—31no—
152C2×#41—33no—
153C2×#41—33no—
154C2×#41—33no—
155C2×#41C2×D311yescharacteristic
156C22×Q8—33no—
157C22×Q8—31no—
158C22×Q8C2×D311yescharacteristic, complemented
159C22×D4—33no—
160C22×D4—33no—
161C22×D4—33no—
162C4⋊Q8—33no—
163#127—33no—
164#124—33no—
165#124—33no—
166#124—33no—
167#124—36no—
168#124—31no—
169#124—33no—
170#124C2×D313yescomplemented
171#123—36no—
172#123—36no—
173C22⋊Q8—33no—
174C22⋊Q8—33no—
175C22⋊Q8—33no—
176C22⋊Q8C2×D313yescomplemented
177#121—33no—
178#121—33no—
179#121—33no—
180#121—33no—
181#121—33no—
182#121—33no—
183#121—33no—
184#121C2×D313yescomplemented
185#120—36no—
186#120—36no—
187C4×Q8—31no—
188C4×Q8C2×D311yescharacteristic, complemented
189C4×D4—33no—
190C4×D4—33no—
191C4×D4—33no—
192C4×D4—33no—
193C4×D4—33no—
194C4×D4C2×D313yescomplemented
195#117—33no—
196C2×C22⋊C4—36no—
197C2×C22⋊C4—36no—
198C2×C42—33no—
199C23×C3C2413yes—
200C23×C3C2411yescharacteristic
201C22×D3—212no—
202C22×D3—46no—
203C22×D3—46no—
204C22×D3C2413yes—
205C22×D3C2411yescharacteristic
206C22×D3C2413yes—
207C3×Q8—26no—
208C3×Q8C2×D414yes—
209C3×D4—26no—
210C3×D4—43no—
211C2×C3×C4—26no—
212C2×C3×C4—26no—
213C2×C3×C4C2413yes—
214C2×C3×C4C2411yescharacteristic
215C2×C3×C4C2411yescharacteristic
216C2×C3×C4C2413yes—
217C2×C3×C4C2413yes—
218#67—26no—
219#67—26no—
220#67—22no—
221#67—26no—
222#67—43no—
223#67—43no—
224C2×Dic3C2411yescharacteristic
225C2×Dic3C2413yes—
226C2×Dic3C2411yescharacteristic
227C2×Dic3C2413yes—
228C2×Dic3C2411yescharacteristic
229D12—26no—
230D12—26no—
231D12—26no—
232D12—43no—
233D12C2×D4112yes—
234C4×D3—212no—
235C4×D3—26no—
236C4×D3—26no—
237C4×D3—43no—
238C4×D3—43no—
239C4×D3C2×D4112yes—
240Dic6—26no—
241Dic6—22no—
242Dic6—26no—
243C24—36no—
244#41—24no—
245#41—38no—
246#41—612no—
247#41—612no—
248#41—612no—
249#41—612no—
250#41—64no—
251C2×Q8—24no—
252C2×Q8—33no—
253C2×Q8—33no—
254C2×Q8—31no—
255C2×Q8—612no—
256C2×Q8—64no—
257C2×Q8C22×D313yes—
258C2×Q8C22×D311yescharacteristic
259C2×D4—33no—
260C2×D4—33no—
261C2×D4—33no—
262C2×D4—31no—
263C2×D4—33no—
264C2×D4—33no—
265C2×D4—33no—
266C2×D4—33no—
267C2×D4—33no—
268C2×D4—33no—
269C2×D4—33no—
270C2×D4—612no—
271C2×D4—612no—
272C2×D4—612no—
273C2×D4C22×D313yes—
274C2×D4C22×D313yes—
275C22×C4—33no—
276C22×C4—36no—
277C22×C4—33no—
278C22×C4—33no—
279C22×C4—33no—
280C22×C4—33no—
281C22×C4C22×D313yes—
282C22×C4C22×D313yes—
283C4⋊C4—33no—
284C4⋊C4—33no—
285C4⋊C4—33no—
286C4⋊C4—33no—
287C4⋊C4C22×D313yes—
288C4⋊C4C22×D313yes—
289C22⋊C4—33no—
290C22⋊C4—36no—
291C22⋊C4—31no—
292C22⋊C4—33no—
293C22⋊C4—33no—
294C22⋊C4—36no—
295C22⋊C4—36no—
296C22⋊C4—36no—
297C22⋊C4—31no—
298C22⋊C4—31no—
299C22⋊C4C22×D313yes—
300C22⋊C4C22×D316yes—
301C22⋊C4C22×D311yescharacteristic
302C42—31no—
303C42—33no—
304C42—33no—
305C42C22×D313yes—
306C22×C3—21no—
307C22×C3—23no—
308C22×C3—21no—
309C22×C3—23no—
310C22×C3—21no—
311C22×C3—23no—
312C22×C3C2511yescharacteristic
313C2×D3—23no—
314C2×D3—23no—
315C2×D3—21no—
316C2×D3—23no—
317C2×D3—21no—
318C2×D3—23no—
319C2×D3—21no—
320C2×D3—23no—
321C2×D3—812no—
322C2×D3C22×D416yes—
323C3×C4—23no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—23no—
327C3×C4C22×D416yes—
328Dic3—23no—
329Dic3—21no—
330Dic3—21no—
331Dic3—23no—
332Dic3C22×D412yes—
333C23—33no—
334C23—33no—
335C23—31no—
336C23—612no—
337C23—126no—
338C23—126no—
339C23C23×D313yes—
340C23C23×D311yescharacteristic
341Q8—26no—
342Q8—66no—
343Q8—62no—
344Q8—66no—
345Q8D3×D414yes—
346D4—26no—
347D4—312no—
348D4—43no—
349D4—66no—
350D4—66no—
351D4—66no—
352D4—62no—
353D4—66no—
354D4—66no—
355D4—66no—
356D4—123no—
357D4—123no—
358D4—123no—
359C2×C4—26no—
360C2×C4—26no—
361C2×C4—31no—
362C2×C4—312no—
363C2×C4—31no—
364C2×C4—33no—
365C2×C4—33no—
366C2×C4—31no—
367C2×C4—612no—
368C2×C4—66no—
369C2×C4—66no—
370C2×C4—123no—
371C2×C4—123no—
372C2×C4C23×D311yescharacteristic
373C2×C4C23×D313yes—
374C2×C4C23×D313yes—
375C2×C4C23×D311yescharacteristic
376C2×C4C23×D313yes—
377C2×C3—43no—
378C2×C3—41no—
379C2×C3C23×D411yescharacteristic
380C2×C3#58511yescharacteristic
381C2×C3#58511yescharacteristic
382D3—26no—
383D3—41no—
384D3—43no—
385C22—21no—
386C22—23no—
387C22—21no—
388C22—23no—
389C22—21no—
390C22—23no—
391C22—36no—
392C22—63no—
393C22—63no—
394C22—61no—
395C22—63no—
396C22—61no—
397C22—63no—
398C22—61no—
399C22—63no—
400C22—2412no—
401C22C24×D311yescharacteristic
402C4—21no—
403C4—23no—
404C4—23no—
405C4—21no—
406C4—32no—
407C4—61no—
408C4—63no—
409C4—61no—
410C4—63no—
411C4C2×D3×D416yes—
412C3#350011yescharacteristic, complemented, Sylow
413C2—43no—
414C2—41no—
415C2—66no—
416C2—123no—
417C2—121no—
418C2C22×D3×D411yescharacteristic
419C2#596411yescharacteristic
420C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#548C2
hypercenter#548D3
hypocenterC3#3500
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#1009C2semidirect
2. C2×#1007C2semidirect
3. C3×#548C2semidirect
4. #5831C2semidirect
5. #5817C2semidirect
6. #5676C2semidirect
7. D3×#124C2semidirect
8. #5630C2semidirect
9. #5628C2semidirect
10. #5612C2semidirect
11. #5607C2semidirect
12. #5576C2semidirect
13. C4×#239C2semidirect
14. #5550C2semidirect
15. #5548C2semidirect
18. C22×C3×Q8C22semidirect
23. C2×#239C22semidirect
24. C2×D3×Q8C22semidirect
25. C2×D3×Q8C22semidirect
29. C2×#235C22semidirect
30. C2×#235C22semidirect
32. C3×#124C22semidirect
33. C3×C22⋊Q8C22semidirect
34. C3×#121C22semidirect
35. C3×C4×Q8C22semidirect
36. C3×C4×D4C22semidirect
37. #947C22semidirect
38. #946C22semidirect
39. Q8×Dic3C22semidirect
40. #940C22semidirect
41. #938C22semidirect
42. #938C22semidirect
43. #937C22semidirect
44. #936C22semidirect
45. #930C22semidirect
46. C4×#67C22semidirect
47. C4×#67C22semidirect
48. #924C22semidirect
49. #897C22semidirect
50. #895C22semidirect
51. #895C22semidirect
52. #894C22semidirect
53. #893C22semidirect
54. #892C22semidirect
55. #888C22semidirect
56. #885C22semidirect
57. #885C22semidirect
58. #885C22semidirect
59. #884C22semidirect
60. #884C22semidirect
61. #883C22semidirect
62. #882C22semidirect
63. D3×C22⋊C4C22semidirect
64. D3×C22⋊C4C22semidirect
65. #875C22semidirect
66. C4×D12C22semidirect
67. C4×D12C22semidirect
68. C42×D3C22semidirect
74. #548D3semidirect
101. #239D4semidirect
158. C22×Q8C2×D3semidirect
170. #124C2×D3semidirect
176. C22⋊Q8C2×D3semidirect
184. #121C2×D3semidirect
188. C4×Q8C2×D3semidirect
194. C4×D4C2×D3semidirect
412. C3#3500semidirect

Maximal subgroups

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

subgroupindex
C2×#10092
C2×#10072
C3×#5482
#58312
#58172
#56762
D3×#1242
#56302
#56282
#56122
#56072
#55762
C4×#2392
#55502
#55482
#35003