Finite Group Databasebeta

← #87204

Subgroups — #87204

Order 384.

1464 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5966C211yesmaximal, characteristic, complemented
2D3×H(5,𝔽2)C211yesmaximal, characteristic, complemented
3C3×#576C211yesmaximal, characteristic, complemented
4#5836C211yesmaximal, characteristic, complemented
5#5835C211yesmaximal, characteristic, complemented
6#5775C213yesmaximal, complemented
7#5774C213yesmaximal, complemented
8#5773C213yesmaximal, complemented
9D3×AGL(1,ℤ/8)C213yesmaximal, complemented
10#5770C213yesmaximal, complemented
11#5768C213yesmaximal, complemented
12#5757C213yesmaximal, complemented
13C2×#911C213yesmaximal, complemented
14#5751C211yesmaximal, characteristic, complemented
15#5750C211yesmaximal, characteristic, complemented
16#3585—31nomaximal, Sylow
17C3×H(5,𝔽2)C2211yescharacteristic, complemented
18C3×H(5,𝔽2)C2211yescharacteristic, complemented
19#1010—21no—
20#1010C2213yescomplemented
21#1010C2211yescharacteristic, complemented
22#1009—21no—
23#1009C2213yescomplemented
24#1009C2211yescharacteristic, complemented
25D3×#41—21no—
26D3×#41—21no—
27D3×#41C2211yescharacteristic
28D3×#41C2213yescomplemented
29D3×#41C2213yescomplemented
30#1004—23no—
31#1004—23no—
32#1004C2213yescomplemented
33#1004C2213yescomplemented
34C2×#237—23no—
35C2×#237C2213yescomplemented
36C2×D3×D4—23no—
37C2×D3×D4C2213yescomplemented
38C3×AGL(1,ℤ/8)C2213yescomplemented
39C3×AGL(1,ℤ/8)C2213yescomplemented
40C3×#135C2213yescomplemented
41C2×C3×D8C2213yescomplemented
42C3×#131C2211yescharacteristic, complemented
43#951C2213yescomplemented
44#950C2213yescomplemented
45#950C2213yescomplemented
46#949C2213yescomplemented
47#948C2211yescharacteristic, complemented
48C2×#214C2213yescomplemented
49#932C2213yescomplemented
50#932C2213yescomplemented
51C2×#213C2213yescomplemented
52#918C2211yescharacteristic, complemented
53#918C2213yescomplemented
54#916C2216yescomplemented
55#915C2213yescomplemented
56#915C2213yescomplemented
57#914C2213yescomplemented
58#914C2213yescomplemented
59D3×SD16C2216yescomplemented
60#912C2216yescomplemented
61#911C2213yescomplemented
62#911C2216yescomplemented
63#911C2216yescomplemented
64#911C2216yescomplemented
65#911C2213yescomplemented
66D3×D8C2216yescomplemented
67#909C2213yescomplemented
68#908C2213yescomplemented
69#907C2213yescomplemented
70D3×M16C2213yescomplemented
71#904C2213yescomplemented
72C2×#204C2213yescomplemented
73#901C2213yescomplemented
74C2×ΓL(1,𝔽25)C2213yescomplemented
75#585—31no—
76C2×H(5,𝔽2)—31no—
77#577—31no—
78#577—33no—
79#577—33no—
80#577—31no—
81#576—31no—
82#576—33no—
83#576—33no—
84#576D311yescharacteristic, complemented
85#575—33no—
86#575—33no—
87C2×AGL(1,ℤ/8)—33no—
88C2×AGL(1,ℤ/8)—33no—
89#568—31no—
90C23×D3—23no—
91C3×#41—21no—
92C3×#41—21no—
93C3×#41C2313yes—
94C3×#41C2311yescharacteristic
95C3×#41C2313yescomplemented
96C2×C3×D4—23no—
97C2×C3×D4—23no—
98C2×C3×D4C2313yes—
99C2×C3×D4C2313yescomplemented
100C2×#67—23no—
101C2×#67—26no—
102C2×#67—26no—
103C2×#67—23no—
104C22×Dic3—23no—
105#239—21no—
106#239D411yescharacteristic
107#239C2311yescharacteristic
108#239C2313yes—
109D3×Q8—21no—
110D3×Q8D411yescharacteristic
111D3×Q8C2311yescharacteristic
112D3×Q8C2313yescomplemented
113#237—23no—
114#237—26no—
115#237—23no—
116#237—23no—
117#237—23no—
118#237—23no—
119#237—23no—
120#237—23no—
121#237D413yes—
122#237C2313yes—
123#237C2313yes—
124#237C2316yescomplemented
125D3×D4—23no—
126D3×D4—23no—
127D3×D4—23no—
128D3×D4—26no—
129D3×D4—23no—
130D3×D4—23no—
131D3×D4—23no—
132D3×D4—23no—
133D3×D4D413yes—
134D3×D4C2316yes—
135D3×D4C2313yes—
136D3×D4C2313yescomplemented
137#235—23no—
138#235—23no—
139#235—23no—
140#235—23no—
141#235C2313yes—
142#235C2313yes—
143#235C2313yescomplemented
144C2×D12—23no—
145C2×D12C2313yes—
146C2×C4×D3—23no—
147C2×C4×D3—23no—
148C2×C4×D3C2313yes—
149C2×Dic6—23no—
150C2×Dic6C2313yescomplemented
151C3×Q16C2311yescharacteristic, complemented
152C3×SD16C2313yescomplemented
153C3×SD16C2313yescomplemented
154C3×D8C2316yescomplemented
155C3×D8C2313yescomplemented
156C3×M16C2313yescomplemented
157C2×C3×C8C2313yescomplemented
158#216C2311yescharacteristic, complemented
159#216C2313yescomplemented
160#215C2313yescomplemented
161#215C2311yescharacteristic, complemented
162#214C2313yescomplemented
163#214C2316yescomplemented
164#214C2313yescomplemented
165#213C2313yescomplemented
166#213C2316yescomplemented
167#213C2313yescomplemented
168#208C2313yescomplemented
169C2×C3⋊C8C2313yescomplemented
170Dic12C2313yescomplemented
171D24C2313yescomplemented
172#204C2313yescomplemented
173#204C2316yescomplemented
174#204C2311yescharacteristic, complemented
175ΓL(1,𝔽25)C2311yescharacteristic
176ΓL(1,𝔽25)C2313yes—
177ΓL(1,𝔽25)C2313yescomplemented
178ΓL(1,𝔽25)C2313yescomplemented
179D3×C8C2316yescomplemented
1802−1+4—31no—
1812−1+4—33no—
1822−1+4—61no—
183H(5,𝔽2)—33no—
184H(5,𝔽2)—31no—
185H(5,𝔽2)—33no—
186H(5,𝔽2)—33no—
187H(5,𝔽2)—63no—
188H(5,𝔽2)—63no—
189H(5,𝔽2)—61no—
190H(5,𝔽2)C2×D311yescharacteristic, complemented
191H(5,𝔽2)C2×D311yescharacteristic, complemented
192C2×#41—31no—
193C2×#41—33no—
194C2×#41—33no—
195C2×#41—33no—
196C2×#41—63no—
197C2×#41—61no—
198C2×#41—61no—
199C22×D4—33no—
200C22×D4—63no—
201#137—33no—
202#137—31no—
203#137—33no—
204#137—33no—
205#137—33no—
206#137—33no—
207AGL(1,ℤ/8)—33no—
208AGL(1,ℤ/8)—33no—
209AGL(1,ℤ/8)—33no—
210AGL(1,ℤ/8)—33no—
211AGL(1,ℤ/8)—33no—
212AGL(1,ℤ/8)—36no—
213AGL(1,ℤ/8)—33no—
214AGL(1,ℤ/8)—36no—
215AGL(1,ℤ/8)—36no—
216AGL(1,ℤ/8)—33no—
217AGL(1,ℤ/8)—33no—
218AGL(1,ℤ/8)C2×D313yescomplemented
219AGL(1,ℤ/8)C2×D313yescomplemented
220#135—33no—
221#135—33no—
222#135—36no—
223#135—36no—
224#135—33no—
225#135C2×D313yescomplemented
226C2×SD16—33no—
227C2×SD16—36no—
228C2×SD16—33no—
229C2×D8—33no—
230C2×D8—36no—
231C2×D8C2×D313yescomplemented
232#131—31no—
233#131—33no—
234#131—33no—
235#131C2×D311yescharacteristic, complemented
236C2×M16—33no—
237C2×M16—33no—
238C23×C3—23no—
239C23×C3—23no—
240C22×D3—23no—
241C22×D3—23no—
242C22×D3—23no—
243C22×D3—26no—
244C22×D3—86no—
245C22×D3C2×D413yes—
246C3×Q8C2×D411yescharacteristic
247C3×Q8C2411yescharacteristic
248C3×Q8C2411yescharacteristic
249C3×D4—23no—
250C3×D4—23no—
251C3×D4—23no—
252C3×D4—23no—
253C3×D4C2×D413yes—
254C3×D4C2413yes—
255C3×D4C2413yes—
256C2×C3×C4—23no—
257C2×C3×C4—23no—
258C2×C3×C4C2413yes—
259#67—23no—
260#67—23no—
261#67—23no—
262#67—23no—
263#67—26no—
264#67—26no—
265#67—23no—
266#67—23no—
267#67C2×D416yes—
268C2×Dic3—23no—
269C2×Dic3—26no—
270C2×Dic3—23no—
271C2×Dic3—23no—
272C2×Dic3C2×D413yes—
273D12—23no—
274D12—21no—
275D12—23no—
276D12C2×D413yes—
277D12C2413yes—
278D12C2411yescharacteristic
279C4×D3—21no—
280C4×D3—23no—
281C4×D3—23no—
282C4×D3—23no—
283C4×D3—21no—
284C4×D3—23no—
285C4×D3C2×D413yes—
286C4×D3C2×D413yes—
287C4×D3C2413yes—
288C4×D3C2411yescharacteristic
289Dic6—23no—
290Dic6—23no—
291Dic6—21no—
292Dic6C2×D413yes—
293Dic6C2413yes—
294Dic6C2411yescharacteristic
295C3×C8C2413yes—
296C3×C8C2411yescharacteristic
297C3⋊C8C2413yes—
298C3⋊C8C2411yescharacteristic
299C24—63no—
300#41—21no—
301#41—21no—
302#41—31no—
303#41—33no—
304#41—33no—
305#41—33no—
306#41—31no—
307#41—36no—
308#41—33no—
309#41—33no—
310#41—33no—
311#41—33no—
312#41—63no—
313#41—63no—
314#41—63no—
315#41—63no—
316#41—63no—
317#41—63no—
318#41—63no—
319#41—63no—
320#41—66no—
321#41—63no—
322#41—63no—
323#41—61no—
324#41—63no—
325#41C22×D313yes—
326#41C22×D311yescharacteristic
327#41C22×D313yescomplemented
328C2×Q8—31no—
329C2×Q8—33no—
330C2×Q8—31no—
331C2×Q8—33no—
332C2×Q8—63no—
333C2×Q8—61no—
334C2×D4—23no—
335C2×D4—23no—
336C2×D4—33no—
337C2×D4—33no—
338C2×D4—36no—
339C2×D4—33no—
340C2×D4—33no—
341C2×D4—63no—
342C2×D4—63no—
343C2×D4—63no—
344C2×D4—63no—
345C2×D4—66no—
346C2×D4—63no—
347C2×D4—63no—
348C2×D4—63no—
349C2×D4—63no—
350C2×D4—66no—
351C2×D4—66no—
352C2×D4—63no—
353C2×D4—63no—
354C2×D4C22×D313yes—
355C2×D4C22×D313yescomplemented
356C22×C4—33no—
357C22×C4—63no—
358C22×C4—63no—
359C22×C4—63no—
360Q16—33no—
361Q16—31no—
362Q16—33no—
363Q16C22×D311yescharacteristic, complemented
364SD16—33no—
365SD16—31no—
366SD16—33no—
367SD16—36no—
368SD16—33no—
369SD16—36no—
370SD16—33no—
371SD16—31no—
372SD16C22×D313yescomplemented
373SD16C22×D313yescomplemented
374D8—33no—
375D8—33no—
376D8—36no—
377D8—33no—
378D8C22×D316yescomplemented
379D8C22×D313yescomplemented
380M16—33no—
381M16—33no—
382M16—31no—
383M16—33no—
384M16—33no—
385M16C22×D313yescomplemented
386C2×C8—33no—
387C2×C8—36no—
388C2×C8C22×D313yescomplemented
389C22×C3—23no—
390C22×C3—23no—
391C22×C3—83no—
392C22×C3—83no—
393C22×C3C22×D413yes—
394C2×D3—21no—
395C2×D3—23no—
396C2×D3—43no—
397C2×D3—46no—
398C2×D3—83no—
399C2×D3—86no—
400C2×D3—83no—
401C2×D3C22×D411yescharacteristic
402C2×D3C22×D413yes—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4C22×D413yes—
406C3×C4C2511yescharacteristic
407Dic3—23no—
408Dic3—21no—
409Dic3C22×D413yes—
410Dic3C22×D411yescharacteristic
411C23—23no—
412C23—23no—
413C23—33no—
414C23—63no—
415C23—63no—
416C23—66no—
417C23—63no—
418C23—246no—
419Q8—33no—
420Q8—33no—
421Q8—31no—
422Q8—63no—
423Q8—61no—
424Q8—63no—
425Q8D3×D411yescharacteristic
426Q8C23×D311yescharacteristic
427Q8C23×D311yescharacteristic
428D4—23no—
429D4—23no—
430D4—23no—
431D4—23no—
432D4—36no—
433D4—33no—
434D4—31no—
435D4—33no—
436D4—63no—
437D4—61no—
438D4—63no—
439D4—66no—
440D4—63no—
441D4—63no—
442D4—63no—
443D4—66no—
444D4—63no—
445D4—63no—
446D4—63no—
447D4D3×D413yes—
448D4C23×D313yes—
449D4C23×D313yes—
450C2×C4—23no—
451C2×C4—23no—
452C2×C4—33no—
453C2×C4—33no—
454C2×C4—33no—
455C2×C4—33no—
456C2×C4—31no—
457C2×C4—61no—
458C2×C4—63no—
459C2×C4—61no—
460C2×C4—63no—
461C2×C4—63no—
462C2×C4—63no—
463C2×C4—66no—
464C2×C4—63no—
465C2×C4—63no—
466C2×C4—63no—
467C2×C4C23×D313yes—
468C8—31no—
469C8—33no—
470C8C23×D313yes—
471C8C23×D311yescharacteristic
472C2×C3—23no—
473C2×C3—43no—
474C2×C3—43no—
475C2×C3C23×D411yescharacteristic
476D3—21no—
477D3—23no—
478D3—43no—
479D3—41no—
480C22—23no—
481C22—23no—
482C22—31no—
483C22—33no—
484C22—61no—
485C22—63no—
486C22—83no—
487C22—83no—
488C22—123no—
489C22—126no—
490C22—243no—
491C22—246no—
492C22—243no—
493C22C2×D3×D413yes—
494C4—21no—
495C4—21no—
496C4—31no—
497C4—33no—
498C4—61no—
499C4—63no—
500C4C2×D3×D413yes—
501C4C24×D311yescharacteristic
502C3#358511yescharacteristic, complemented, Sylow
503C2—23no—
504C2—43no—
505C2—43no—
506C2—61no—
507C2—63no—
508C2—123no—
509C2—121no—
510C2C22×D3×D411yescharacteristic
subgroupG/H
centerC2C22×D3×D4
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC3×#576C2
socleC2×C3C23×D4
hypercenter#576D3
hypocenterC3#3585
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ #41 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #5966C2semidirect
2. D3×H(5,𝔽2)C2semidirect
3. C3×#576C2semidirect
4. #5836C2semidirect
5. #5835C2semidirect
6. #5775C2semidirect
7. #5774C2semidirect
8. #5773C2semidirect
9. D3×AGL(1,ℤ/8)C2semidirect
10. #5770C2semidirect
11. #5768C2semidirect
12. #5757C2semidirect
13. C2×#911C2semidirect
14. #5751C2semidirect
15. #5750C2semidirect
17. C3×H(5,𝔽2)C22semidirect
18. C3×H(5,𝔽2)C22semidirect
20. #1010C22semidirect
21. #1010C22semidirect
23. #1009C22semidirect
24. #1009C22semidirect
28. D3×#41C22semidirect
29. D3×#41C22semidirect
32. #1004C22semidirect
33. #1004C22semidirect
35. C2×#237C22semidirect
37. C2×D3×D4C22semidirect
38. C3×AGL(1,ℤ/8)C22semidirect
39. C3×AGL(1,ℤ/8)C22semidirect
40. C3×#135C22semidirect
41. C2×C3×D8C22semidirect
42. C3×#131C22semidirect
43. #951C22semidirect
44. #950C22semidirect
45. #950C22semidirect
46. #949C22semidirect
47. #948C22semidirect
48. C2×#214C22semidirect
49. #932C22semidirect
50. #932C22semidirect
51. C2×#213C22semidirect
52. #918C22semidirect
53. #918C22semidirect
54. #916C22semidirect
55. #915C22semidirect
56. #915C22semidirect
57. #914C22semidirect
58. #914C22semidirect
59. D3×SD16C22semidirect
60. #912C22semidirect
61. #911C22semidirect
62. #911C22semidirect
63. #911C22semidirect
64. #911C22semidirect
65. #911C22semidirect
66. D3×D8C22semidirect
67. #909C22semidirect
68. #908C22semidirect
69. #907C22semidirect
70. D3×M16C22semidirect
71. #904C22semidirect
72. C2×#204C22semidirect
73. #901C22semidirect
74. C2×ΓL(1,𝔽25)C22semidirect
84. #576D3semidirect
95. C3×#41C23semidirect
99. C2×C3×D4C23semidirect
112. D3×Q8C23semidirect
124. #237C23semidirect
136. D3×D4C23semidirect
143. #235C23semidirect
150. C2×Dic6C23semidirect
151. C3×Q16C23semidirect
152. C3×SD16C23semidirect
153. C3×SD16C23semidirect
154. C3×D8C23semidirect
155. C3×D8C23semidirect
156. C3×M16C23semidirect
157. C2×C3×C8C23semidirect
158. #216C23semidirect
159. #216C23semidirect
160. #215C23semidirect
161. #215C23semidirect
162. #214C23semidirect
163. #214C23semidirect
164. #214C23semidirect
165. #213C23semidirect
166. #213C23semidirect
167. #213C23semidirect
168. #208C23semidirect
169. C2×C3⋊C8C23semidirect
170. Dic12C23semidirect
171. D24C23semidirect
172. #204C23semidirect
173. #204C23semidirect
174. #204C23semidirect
177. ΓL(1,𝔽25)C23semidirect
178. ΓL(1,𝔽25)C23semidirect
179. D3×C8C23semidirect
190. H(5,𝔽2)C2×D3semidirect
191. H(5,𝔽2)C2×D3semidirect
218. AGL(1,ℤ/8)C2×D3semidirect
219. AGL(1,ℤ/8)C2×D3semidirect
225. #135C2×D3semidirect
231. C2×D8C2×D3semidirect
235. #131C2×D3semidirect
327. #41C22×D3semidirect
355. C2×D4C22×D3semidirect
363. Q16C22×D3semidirect
372. SD16C22×D3semidirect
373. SD16C22×D3semidirect
378. D8C22×D3semidirect
379. D8C22×D3semidirect
385. M16C22×D3semidirect
388. C2×C8C22×D3semidirect
502. C3#3585semidirect

Maximal subgroups

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

subgroupindex
#59662
D3×H(5,𝔽2)2
C3×#5762
#58362
#58352
#57752
#57742
#57732
D3×AGL(1,ℤ/8)2
#57702
#57682
#57572
C2×#9112
#57512
#57502
#35853