Finite Group Databasebeta

← #87206

Subgroups — #87206

Order 384.

1424 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5968C211yesmaximal, characteristic, complemented
2#5966C211yesmaximal, characteristic, complemented
3C3×#577C211yesmaximal, characteristic, complemented
4#5838C211yesmaximal, characteristic, complemented
5#5835C211yesmaximal, characteristic, complemented
6#5779C213yesmaximal, complemented
7#5777C213yesmaximal, complemented
8#5774C213yesmaximal, complemented
9#5773C213yesmaximal, complemented
10#5770C213yesmaximal, complemented
11D3×#135C213yesmaximal, complemented
12#5762C213yesmaximal, complemented
13C2×#916C213yesmaximal, complemented
14#5751C211yesmaximal, characteristic, complemented
15D3×#131C211yesmaximal, characteristic, complemented
16#3584—31nomaximal, Sylow
17C3×2−1+4C2211yescharacteristic, complemented
18C3×H(5,𝔽2)C2211yescharacteristic, complemented
19#1010—21no—
20#1010C2213yescomplemented
21#1010C2211yescharacteristic, complemented
22#1009—23no—
23#1009C2213yescomplemented
24#1009C2211yescharacteristic, complemented
25D3×#41—21no—
26D3×#41—23no—
27D3×#41C2211yescharacteristic
28D3×#41C2213yescomplemented
29D3×#41C2213yescomplemented
30#1007—21no—
31#1007C2213yescomplemented
32C2×#239—21no—
33C2×#239C2213yescomplemented
34#1004—23no—
35#1004C2213yescomplemented
36C2×#237—23no—
37C2×#237C2213yescomplemented
38C3×#137C2213yescomplemented
39C3×AGL(1,ℤ/8)C2213yescomplemented
40C3×#135C2213yescomplemented
41C2×C3×SD16C2213yescomplemented
42C3×#131C2211yescharacteristic, complemented
43#951C2213yescomplemented
44#950C2213yescomplemented
45#950C2213yescomplemented
46#949C2213yescomplemented
47#948C2211yescharacteristic, complemented
48C2×#216C2213yescomplemented
49#942C2213yescomplemented
50#932C2213yescomplemented
51C2×#213C2213yescomplemented
52#919C2213yescomplemented
53#918C2216yescomplemented
54D3×Q16C2213yescomplemented
55#916C2213yescomplemented
56#916C2216yescomplemented
57#916C2216yescomplemented
58#916C2216yescomplemented
59#916C2211yescharacteristic, complemented
60#915C2216yescomplemented
61#914C2216yescomplemented
62D3×SD16C2213yescomplemented
63D3×SD16C2213yescomplemented
64#912C2213yescomplemented
65#911C2216yescomplemented
66D3×D8C2213yescomplemented
67#909C2213yescomplemented
68#908C2213yescomplemented
69#907C2213yescomplemented
70D3×M16C2213yescomplemented
71#904C2213yescomplemented
72C2×#204C2213yescomplemented
73#901C2213yescomplemented
74C2×D3×C8C2213yescomplemented
75#585—31no—
76#585—31no—
77#578—31no—
78#578—33no—
79#577—33no—
80#577—33no—
81#577—31no—
82#577D311yescharacteristic, complemented
83#576—31no—
84#576—33no—
85#575—33no—
86#575—33no—
87C2×#135—33no—
88C2×#135—33no—
89C2×#131—31no—
90C3×#41—21no—
91C3×#41—23no—
92C3×#41C2313yes—
93C3×#41C2313yes—
94C3×#41C2311yescharacteristic
95C2×C3×Q8—21no—
96C2×C3×Q8C2313yes—
97C2×C3×D4—23no—
98C2×C3×D4C2313yes—
99C2×#67—23no—
100C2×#67—26no—
101C22×Dic3—23no—
102#239—21no—
103#239—23no—
104#239—23no—
105#239—23no—
106#239D411yescharacteristic
107#239C2316yes—
108#239C2313yes—
109#239C2311yescharacteristic
110D3×Q8—21no—
111D3×Q8—23no—
112D3×Q8D411yescharacteristic
113D3×Q8C2313yes—
114D3×Q8C2313yes—
115#237—23no—
116#237—26no—
117#237—23no—
118#237—23no—
119#237—23no—
120#237—23no—
121#237—23no—
122#237D413yes—
123#237C2316yes—
124#237C2311yescharacteristic
125#237C2313yes—
126D3×D4—23no—
127D3×D4—23no—
128D3×D4—23no—
129D3×D4—26no—
130D3×D4—23no—
131D3×D4D413yes—
132D3×D4C2313yes—
133D3×D4C2313yes—
134#235—26no—
135#235—23no—
136#235—23no—
137#235—23no—
138#235—23no—
139#235C2313yes—
140#235C2313yes—
141#235C2313yes—
142C2×D12—23no—
143C2×D12—23no—
144C2×D12C2313yes—
145C2×C4×D3—23no—
146C2×C4×D3—23no—
147C2×C4×D3—23no—
148C2×C4×D3C2313yes—
149C2×Dic6—23no—
150C2×Dic6C2313yes—
151C3×Q16C2313yes—
152C3×SD16C2311yescharacteristic
153C3×SD16C2313yes—
154C3×SD16C2316yes—
155C3×D8C2313yes—
156C3×M16C2313yes—
157C2×C3×C8C2313yes—
158#216C2316yes—
159#216C2313yes—
160#216C2311yescharacteristic
161#215C2313yes—
162#215C2313yes—
163#214C2313yes—
164#214C2313yes—
165#213C2311yescharacteristic
166#213C2313yes—
167#213C2316yes—
168#208C2313yes—
169C2×C3⋊C8C2313yes—
170Dic12C2313yes—
171D24C2313yes—
172#204C2313yes—
173#204C2316yes—
174#204C2311yescharacteristic
175ΓL(1,𝔽25)C2316yes—
176D3×C8C2311yescharacteristic
177D3×C8C2313yes—
178D3×C8C2313yes—
179D3×C8C2313yes—
1802−1+4—33no—
1812−1+4—31no—
1822−1+4—33no—
1832−1+4—61no—
1842−1+4—61no—
1852−1+4C2×D311yescharacteristic, complemented
186H(5,𝔽2)—33no—
187H(5,𝔽2)—31no—
188H(5,𝔽2)—33no—
189H(5,𝔽2)—63no—
190H(5,𝔽2)—63no—
191H(5,𝔽2)C2×D311yescharacteristic, complemented
192C2×#41—31no—
193C2×#41—33no—
194C2×#41—33no—
195C2×#41—33no—
196C2×#41—33no—
197C2×#41—63no—
198C2×#41—61no—
199C2×#41—61no—
200C2×#41—63no—
201#137—33no—
202#137—33no—
203#137—36no—
204#137—36no—
205#137—33no—
206#137C2×D313yescomplemented
207AGL(1,ℤ/8)—33no—
208AGL(1,ℤ/8)—33no—
209AGL(1,ℤ/8)—36no—
210AGL(1,ℤ/8)—36no—
211AGL(1,ℤ/8)—33no—
212AGL(1,ℤ/8)C2×D313yescomplemented
213#135—33no—
214#135—33no—
215#135—33no—
216#135—31no—
217#135—33no—
218#135—36no—
219#135—36no—
220#135—36no—
221#135—33no—
222#135—33no—
223#135C2×D313yescomplemented
224C2×Q16—33no—
225C2×Q16—33no—
226C2×SD16—33no—
227C2×SD16—33no—
228C2×SD16—33no—
229C2×SD16C2×D313yescomplemented
230C2×D8—33no—
231C2×D8—33no—
232#131—31no—
233#131—33no—
234#131—33no—
235#131C2×D311yescharacteristic, complemented
236C2×M16—33no—
237C22×C8—33no—
238C23×C3—23no—
239C22×D3—23no—
240C22×D3—23no—
241C22×D3—23no—
242C22×D3C2×D413yes—
243C3×Q8—23no—
244C3×Q8C2×D411yescharacteristic
245C3×Q8C2411yescharacteristic
246C3×Q8C2413yes—
247C3×D4—23no—
248C3×D4—23no—
249C3×D4—23no—
250C3×D4C2×D413yes—
251C3×D4C2413yes—
252C3×D4C2411yescharacteristic
253C2×C3×C4—23no—
254C2×C3×C4—23no—
255C2×C3×C4—23no—
256C2×C3×C4C2413yes—
257#67—23no—
258#67—23no—
259#67—21no—
260#67—26no—
261#67—23no—
262#67—23no—
263#67—23no—
264#67C2×D416yes—
265C2×Dic3—23no—
266C2×Dic3—26no—
267C2×Dic3—21no—
268C2×Dic3—23no—
269C2×Dic3C2×D413yes—
270D12—26no—
271D12—23no—
272D12—21no—
273D12—23no—
274D12C2×D413yes—
275D12C2413yes—
276D12C2411yescharacteristic
277C4×D3—23no—
278C4×D3—23no—
279C4×D3—23no—
280C4×D3—23no—
281C4×D3—21no—
282C4×D3—23no—
283C4×D3—26no—
284C4×D3C2×D413yes—
285C4×D3C2×D413yes—
286C4×D3C2411yescharacteristic
287C4×D3C2413yes—
288Dic6—23no—
289Dic6—23no—
290Dic6—23no—
291Dic6C2×D413yes—
292Dic6C2413yes—
293Dic6C2411yescharacteristic
294C3×C8C2411yescharacteristic
295C3×C8C2413yes—
296C3⋊C8C2411yescharacteristic
297C3⋊C8C2413yes—
298#41—21no—
299#41—23no—
300#41—31no—
301#41—33no—
302#41—33no—
303#41—31no—
304#41—33no—
305#41—36no—
306#41—33no—
307#41—31no—
308#41—36no—
309#41—33no—
310#41—33no—
311#41—63no—
312#41—63no—
313#41—66no—
314#41—63no—
315#41—63no—
316#41—63no—
317#41—63no—
318#41—63no—
319#41—63no—
320#41—63no—
321#41—63no—
322#41—66no—
323#41—63no—
324#41—63no—
325#41—61no—
326#41—63no—
327#41C22×D313yes—
328#41C22×D313yes—
329#41C22×D311yescharacteristic
330C2×Q8—21no—
331C2×Q8—31no—
332C2×Q8—33no—
333C2×Q8—33no—
334C2×Q8—33no—
335C2×Q8—63no—
336C2×Q8—63no—
337C2×Q8—61no—
338C2×Q8C22×D313yes—
339C2×D4—23no—
340C2×D4—33no—
341C2×D4—33no—
342C2×D4—33no—
343C2×D4—33no—
344C2×D4—63no—
345C2×D4—63no—
346C2×D4—66no—
347C2×D4—63no—
348C2×D4—63no—
349C2×D4—63no—
350C2×D4—66no—
351C2×D4—63no—
352C2×D4—63no—
353C2×D4C22×D313yes—
354C22×C4—33no—
355C22×C4—63no—
356C22×C4—63no—
357C22×C4—63no—
358C22×C4—63no—
359Q16—31no—
360Q16—33no—
361Q16—36no—
362Q16—33no—
363Q16C22×D313yes—
364SD16—33no—
365SD16—33no—
366SD16—33no—
367SD16—33no—
368SD16—31no—
369SD16—33no—
370SD16—36no—
371SD16C22×D311yescharacteristic
372SD16C22×D313yes—
373SD16C22×D316yes—
374D8—33no—
375D8—31no—
376D8—36no—
377D8—33no—
378D8C22×D313yes—
379M16—33no—
380M16—36no—
381M16C22×D313yes—
382C2×C8—33no—
383C2×C8—31no—
384C2×C8—33no—
385C2×C8—33no—
386C2×C8—33no—
387C2×C8C22×D313yes—
388C22×C3—23no—
389C22×C3—21no—
390C22×C3—83no—
391C22×C3C22×D413yes—
392C2×D3—23no—
393C2×D3—21no—
394C2×D3—43no—
395C2×D3—83no—
396C2×D3—83no—
397C2×D3—83no—
398C2×D3C22×D413yes—
399C2×D3C22×D411yescharacteristic
400C3×C4—21no—
401C3×C4—23no—
402C3×C4C22×D413yes—
403C3×C4C2511yescharacteristic
404Dic3—21no—
405Dic3—23no—
406Dic3C22×D413yes—
407Dic3C22×D411yescharacteristic
408C23—23no—
409C23—33no—
410C23—63no—
411C23—63no—
412C23—63no—
413Q8—23no—
414Q8—33no—
415Q8—31no—
416Q8—33no—
417Q8—63no—
418Q8—63no—
419Q8—63no—
420Q8D3×D411yescharacteristic
421Q8C23×D311yescharacteristic
422Q8C23×D313yes—
423D4—23no—
424D4—23no—
425D4—23no—
426D4—36no—
427D4—33no—
428D4—33no—
429D4—31no—
430D4—61no—
431D4—63no—
432D4—66no—
433D4—63no—
434D4—63no—
435D4—61no—
436D4—63no—
437D4—66no—
438D4—63no—
439D4—63no—
440D4—63no—
441D4D3×D413yes—
442D4C23×D313yes—
443D4C23×D311yescharacteristic
444C2×C4—23no—
445C2×C4—23no—
446C2×C4—23no—
447C2×C4—33no—
448C2×C4—33no—
449C2×C4—33no—
450C2×C4—31no—
451C2×C4—33no—
452C2×C4—61no—
453C2×C4—63no—
454C2×C4—66no—
455C2×C4—63no—
456C2×C4—63no—
457C2×C4—61no—
458C2×C4—63no—
459C2×C4—66no—
460C2×C4—63no—
461C2×C4—63no—
462C2×C4—63no—
463C2×C4C23×D313yes—
464C8—31no—
465C8—33no—
466C8C23×D311yescharacteristic
467C8C23×D313yes—
468C2×C3—23no—
469C2×C3—41no—
470C2×C3—43no—
471C2×C3C23×D411yescharacteristic
472D3—21no—
473D3—23no—
474D3—43no—
475D3—41no—
476C22—21no—
477C22—23no—
478C22—31no—
479C22—33no—
480C22—63no—
481C22—61no—
482C22—83no—
483C22—123no—
484C22—243no—
485C22—243no—
486C22—243no—
487C22C2×D3×D413yes—
488C4—21no—
489C4—23no—
490C4—31no—
491C4—33no—
492C4—61no—
493C4—63no—
494C4C2×D3×D413yes—
495C4C24×D311yescharacteristic
496C3#358411yescharacteristic, complemented, Sylow
497C2—23no—
498C2—43no—
499C2—41no—
500C2—61no—
501C2—63no—
502C2—123no—
503C2—121no—
504C2C22×D3×D411yescharacteristic
subgroupG/H
centerC2C22×D3×D4
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC3×#577C2
socleC2×C3C23×D4
hypercenter#577D3
hypocenterC3#3584
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ #41 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #5968C2semidirect
2. #5966C2semidirect
3. C3×#577C2semidirect
4. #5838C2semidirect
5. #5835C2semidirect
6. #5779C2semidirect
7. #5777C2semidirect
8. #5774C2semidirect
9. #5773C2semidirect
10. #5770C2semidirect
11. D3×#135C2semidirect
12. #5762C2semidirect
13. C2×#916C2semidirect
14. #5751C2semidirect
15. D3×#131C2semidirect
17. C3×2−1+4C22semidirect
18. C3×H(5,𝔽2)C22semidirect
20. #1010C22semidirect
21. #1010C22semidirect
23. #1009C22semidirect
24. #1009C22semidirect
28. D3×#41C22semidirect
29. D3×#41C22semidirect
31. #1007C22semidirect
33. C2×#239C22semidirect
35. #1004C22semidirect
37. C2×#237C22semidirect
38. C3×#137C22semidirect
39. C3×AGL(1,ℤ/8)C22semidirect
40. C3×#135C22semidirect
41. C2×C3×SD16C22semidirect
42. C3×#131C22semidirect
43. #951C22semidirect
44. #950C22semidirect
45. #950C22semidirect
46. #949C22semidirect
47. #948C22semidirect
48. C2×#216C22semidirect
49. #942C22semidirect
50. #932C22semidirect
51. C2×#213C22semidirect
52. #919C22semidirect
53. #918C22semidirect
54. D3×Q16C22semidirect
55. #916C22semidirect
56. #916C22semidirect
57. #916C22semidirect
58. #916C22semidirect
59. #916C22semidirect
60. #915C22semidirect
61. #914C22semidirect
62. D3×SD16C22semidirect
63. D3×SD16C22semidirect
64. #912C22semidirect
65. #911C22semidirect
66. D3×D8C22semidirect
67. #909C22semidirect
68. #908C22semidirect
69. #907C22semidirect
70. D3×M16C22semidirect
71. #904C22semidirect
72. C2×#204C22semidirect
73. #901C22semidirect
74. C2×D3×C8C22semidirect
82. #577D3semidirect
185. 2−1+4C2×D3semidirect
191. H(5,𝔽2)C2×D3semidirect
206. #137C2×D3semidirect
212. AGL(1,ℤ/8)C2×D3semidirect
223. #135C2×D3semidirect
229. C2×SD16C2×D3semidirect
235. #131C2×D3semidirect
496. C3#3584semidirect

Maximal subgroups

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

subgroupindex
#59682
#59662
C3×#5772
#58382
#58352
#57792
#57772
#57742
#57732
#57702
D3×#1352
#57622
C2×#9162
#57512
D3×#1312
#35843