Finite Group Databasebeta

← #87208

Subgroups — #87208

Order 384.

1412 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×2−1+4C211yesmaximal, characteristic, complemented
2#5966C211yesmaximal, characteristic, complemented
3C3×#577C211yesmaximal, characteristic, complemented
4#5838C211yesmaximal, characteristic, complemented
5#5836C211yesmaximal, characteristic, complemented
6#5779C213yesmaximal, complemented
7D3×#137C213yesmaximal, complemented
8#5775C213yesmaximal, complemented
9#5773C213yesmaximal, complemented
10#5771C213yesmaximal, complemented
11#5768C213yesmaximal, complemented
12#5762C213yesmaximal, complemented
13C2×#915C213yesmaximal, complemented
14#5753C211yesmaximal, characteristic, complemented
15#5750C211yesmaximal, characteristic, complemented
16#3586—31nomaximal, Sylow
17C3×2−1+4C2211yescharacteristic, complemented
18C3×H(5,𝔽2)C2211yescharacteristic, complemented
19#1010—21no—
20#1010—23no—
21#1010C2213yescomplemented
22#1010C2213yescomplemented
23#1010C2211yescharacteristic, complemented
24#1010C2211yescharacteristic, complemented
25D3×#41—21no—
26D3×#41—23no—
27D3×#41C2213yes—
28D3×#41C2211yescharacteristic
29D3×#41C2213yescomplemented
30#1007—21no—
31#1007C2213yescomplemented
32C2×D3×Q8—21no—
33C2×D3×Q8C2213yescomplemented
34#1004—23no—
35#1004C2213yescomplemented
36C2×#237—23no—
37C2×#237C2213yes—
38C3×#137C2213yescomplemented
39C3×AGL(1,ℤ/8)C2213yescomplemented
40C3×#135C2213yescomplemented
41C2×C3×SD16C2213yescomplemented
42C3×#131C2211yescharacteristic, complemented
43#951C2213yescomplemented
44#951C2213yescomplemented
45#950C2213yescomplemented
46#950C2213yescomplemented
47#948C2211yescharacteristic, complemented
48C2×#216C2213yescomplemented
49#942C2213yescomplemented
50C2×#214C2213yescomplemented
51#932C2213yescomplemented
52#918C2213yescomplemented
53#918C2213yescomplemented
54D3×Q16C2216yescomplemented
55#916C2216yescomplemented
56#915C2213yescomplemented
57#915C2216yescomplemented
58#915C2216yescomplemented
59#915C2216yescomplemented
60#915C2213yescomplemented
61#914C2213yescomplemented
62#914C2211yescharacteristic, complemented
63D3×SD16C2216yescomplemented
64#912C2216yescomplemented
65#911C2213yescomplemented
66#911C2213yescomplemented
67#909C2213yescomplemented
68#909C2213yescomplemented
69#907C2213yescomplemented
70D3×M16C2213yescomplemented
71C2×Dic12C2213yescomplemented
72#904C2213yescomplemented
73#901C2213yescomplemented
74C2×ΓL(1,𝔽25)C2213yescomplemented
75#585—31no—
76C2×2−1+4—31no—
77#578—31no—
78#578—33no—
79#578—33no—
80#578—31no—
81#577—31no—
82#577—33no—
83#577—33no—
84#577D311yescharacteristic, complemented
85#575—33no—
86#575—33no—
87C2×#137—33no—
88C2×#137—33no—
89#568—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—23no—
103#239D411yescharacteristic
104#239C2313yes—
105#239C2311yescharacteristic
106D3×Q8—21no—
107D3×Q8—21no—
108D3×Q8—23no—
109D3×Q8—23no—
110D3×Q8—23no—
111D3×Q8D411yescharacteristic
112D3×Q8C2316yes—
113D3×Q8C2313yes—
114D3×Q8C2313yes—
115#237—23no—
116#237—23no—
117#237—26no—
118#237—26no—
119#237—23no—
120#237—23no—
121#237—23no—
122#237—23no—
123#237—23no—
124#237D413yes—
125#237C2313yes—
126#237C2313yes—
127#237C2316yes—
128D3×D4—23no—
129D3×D4—23no—
130D3×D4—23no—
131D3×D4D413yes—
132D3×D4C2311yescharacteristic
133D3×D4C2313yes—
134#235—26no—
135#235—23no—
136#235—23no—
137#235—23no—
138#235—23no—
139#235C2313yes—
140#235C2313yes—
141#235C2313yes—
142C2×C4×D3—23no—
143C2×C4×D3—23no—
144C2×C4×D3—23no—
145C2×C4×D3C2313yes—
146C2×Dic6—23no—
147C2×Dic6—23no—
148C2×Dic6—23no—
149C2×Dic6C2313yes—
150C2×Dic6C2313yes—
151C3×Q16C2313yes—
152C3×SD16C2316yes—
153C3×SD16C2313yes—
154C3×SD16C2311yescharacteristic
155C3×D8C2313yes—
156C3×M16C2313yes—
157C2×C3×C8C2313yes—
158#216C2313yes—
159#216C2316yes—
160#216C2313yes—
161#215C2313yes—
162#215C2311yescharacteristic
163#214C2313yes—
164#214C2313yes—
165#214C2316yes—
166#213C2311yescharacteristic
167#213C2313yes—
168#208C2313yes—
169C2×C3⋊C8C2313yes—
170Dic12C2313yes—
171Dic12C2316yes—
172D24C2311yescharacteristic
173#204C2313yes—
174#204C2313yes—
175ΓL(1,𝔽25)C2311yescharacteristic
176ΓL(1,𝔽25)C2313yes—
177ΓL(1,𝔽25)C2313yes—
178ΓL(1,𝔽25)C2313yes—
179D3×C8C2316yes—
1802−1+4—33no—
1812−1+4—33no—
1822−1+4—31no—
1832−1+4—31no—
1842−1+4—33no—
1852−1+4—61no—
1862−1+4—61no—
1872−1+4—63no—
1882−1+4C2×D311yescharacteristic, complemented
189H(5,𝔽2)—33no—
190H(5,𝔽2)—63no—
191H(5,𝔽2)C2×D311yescharacteristic, complemented
192C2×#41—31no—
193C2×#41—33no—
194C2×#41—33no—
195C2×#41—33no—
196C2×#41—63no—
197C2×#41—63no—
198C2×#41—61no—
199C22×Q8—33no—
200C22×Q8—61no—
201#137—33no—
202#137—33no—
203#137—33no—
204#137—33no—
205#137—33no—
206#137—36no—
207#137—33no—
208#137—36no—
209#137—36no—
210#137—33no—
211#137—33no—
212#137—33no—
213#137C2×D313yescomplemented
214AGL(1,ℤ/8)—33no—
215AGL(1,ℤ/8)—31no—
216AGL(1,ℤ/8)—33no—
217AGL(1,ℤ/8)—33no—
218AGL(1,ℤ/8)—33no—
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×Q16—33no—
227C2×Q16—36no—
228C2×Q16—33no—
229C2×SD16—33no—
230C2×SD16—36no—
231C2×SD16C2×D313yescomplemented
232#131—31no—
233#131—33no—
234#131—33no—
235#131C2×D311yescharacteristic, complemented
236C2×M16—33no—
237C2×M16—33no—
238C23×C3—23no—
239C22×D3—21no—
240C22×D3—23no—
241C22×D3C2×D413yes—
242C3×Q8—23no—
243C3×Q8C2×D411yescharacteristic
244C3×Q8C2413yes—
245C3×Q8C2411yescharacteristic
246C3×D4—23no—
247C3×D4—23no—
248C3×D4—23no—
249C3×D4C2×D413yes—
250C3×D4C2411yescharacteristic
251C3×D4C2413yes—
252C2×C3×C4—23no—
253C2×C3×C4—23no—
254C2×C3×C4—23no—
255C2×C3×C4C2413yes—
256#67—23no—
257#67—23no—
258#67—23no—
259#67—23no—
260#67—26no—
261#67—21no—
262#67—23no—
263#67C2×D416yes—
264C2×Dic3—23no—
265C2×Dic3—23no—
266C2×Dic3—23no—
267C2×Dic3—26no—
268C2×Dic3—23no—
269C2×Dic3C2×D413yes—
270D12—21no—
271D12—23no—
272D12C2×D413yes—
273D12C2411yescharacteristic
274D12C2411yescharacteristic
275C4×D3—23no—
276C4×D3—23no—
277C4×D3—26no—
278C4×D3—23no—
279C4×D3—23no—
280C4×D3—21no—
281C4×D3—23no—
282C4×D3C2×D413yes—
283C4×D3C2×D413yes—
284C4×D3C2413yes—
285C4×D3C2411yescharacteristic
286Dic6—23no—
287Dic6—23no—
288Dic6—23no—
289Dic6—26no—
290Dic6—23no—
291Dic6C2×D413yes—
292Dic6C2413yes—
293Dic6C2413yes—
294C3×C8C2413yes—
295C3×C8C2411yescharacteristic
296C3⋊C8C2413yes—
297C3⋊C8C2411yescharacteristic
298#41—21no—
299#41—23no—
300#41—31no—
301#41—33no—
302#41—33no—
303#41—33no—
304#41—31no—
305#41—33no—
306#41—36no—
307#41—33no—
308#41—33no—
309#41—33no—
310#41—63no—
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—66no—
320#41—66no—
321#41—63no—
322#41—63no—
323#41—63no—
324#41—63no—
325#41C22×D313yes—
326#41C22×D313yes—
327#41C22×D311yescharacteristic
328C2×Q8—21no—
329C2×Q8—31no—
330C2×Q8—36no—
331C2×Q8—33no—
332C2×Q8—33no—
333C2×Q8—33no—
334C2×Q8—33no—
335C2×Q8—63no—
336C2×Q8—63no—
337C2×Q8—63no—
338C2×Q8—63no—
339C2×Q8—63no—
340C2×Q8—63no—
341C2×Q8—61no—
342C2×Q8—61no—
343C2×Q8C22×D313yes—
344C2×D4—23no—
345C2×D4—33no—
346C2×D4—33no—
347C2×D4—31no—
348C2×D4—63no—
349C2×D4—63no—
350C2×D4—63no—
351C2×D4—66no—
352C2×D4—63no—
353C2×D4C22×D313yes—
354C22×C4—33no—
355C22×C4—63no—
356C22×C4—63no—
357C22×C4—63no—
358C22×C4—63no—
359Q16—33no—
360Q16—33no—
361Q16—36no—
362Q16—36no—
363Q16—33no—
364Q16C22×D313yes—
365SD16—33no—
366SD16—31no—
367SD16—33no—
368SD16—36no—
369SD16—33no—
370SD16—33no—
371SD16—33no—
372SD16C22×D316yes—
373SD16C22×D313yes—
374SD16C22×D311yescharacteristic
375D8—33no—
376D8—31no—
377D8—31no—
378D8C22×D313yes—
379M16—33no—
380M16—33no—
381M16—31no—
382M16—33no—
383M16—33no—
384M16C22×D313yes—
385C2×C8—33no—
386C2×C8—36no—
387C2×C8C22×D313yes—
388C22×C3—21no—
389C22×C3—23no—
390C22×C3—83no—
391C22×C3C22×D413yes—
392C2×D3—21no—
393C2×D3—21no—
394C2×D3—43no—
395C2×D3—42no—
396C2×D3—83no—
397C2×D3C22×D411yescharacteristic
398C2×D3C22×D413yes—
399C3×C4—21no—
400C3×C4—23no—
401C3×C4C22×D413yes—
402C3×C4C2511yescharacteristic
403Dic3—23no—
404Dic3—23no—
405Dic3C22×D413yes—
406Dic3C22×D411yescharacteristic
407C23—23no—
408C23—33no—
409C23—61no—
410C23—63no—
411Q8—23no—
412Q8—33no—
413Q8—33no—
414Q8—33no—
415Q8—63no—
416Q8—66no—
417Q8—63no—
418Q8—63no—
419Q8—63no—
420Q8D3×D411yescharacteristic
421Q8C23×D313yes—
422Q8C23×D311yescharacteristic
423D4—23no—
424D4—23no—
425D4—23no—
426D4—36no—
427D4—33no—
428D4—31no—
429D4—31no—
430D4—61no—
431D4—63no—
432D4—63no—
433D4—61no—
434D4—63no—
435D4—66no—
436D4—63no—
437D4—63no—
438D4—63no—
439D4D3×D413yes—
440D4C23×D311yescharacteristic
441D4C23×D313yes—
442C2×C4—23no—
443C2×C4—23no—
444C2×C4—23no—
445C2×C4—33no—
446C2×C4—33no—
447C2×C4—33no—
448C2×C4—33no—
449C2×C4—31no—
450C2×C4—61no—
451C2×C4—63no—
452C2×C4—63no—
453C2×C4—63no—
454C2×C4—66no—
455C2×C4—63no—
456C2×C4—66no—
457C2×C4—63no—
458C2×C4—63no—
459C2×C4—63no—
460C2×C4—63no—
461C2×C4—63no—
462C2×C4C23×D313yes—
463C8—31no—
464C8—33no—
465C8C23×D313yes—
466C8C23×D311yescharacteristic
467C2×C3—23no—
468C2×C3—41no—
469C2×C3—43no—
470C2×C3C23×D411yescharacteristic
471D3—21no—
472D3—23no—
473D3—41no—
474D3—41no—
475C22—21no—
476C22—23no—
477C22—31no—
478C22—33no—
479C22—61no—
480C22—61no—
481C22—83no—
482C22—123no—
483C22—122no—
484C22—243no—
485C22C2×D3×D413yes—
486C4—21no—
487C4—23no—
488C4—31no—
489C4—33no—
490C4—63no—
491C4—63no—
492C4C2×D3×D413yes—
493C4C24×D311yescharacteristic
494C3#358611yescharacteristic, complemented, Sylow
495C2—23no—
496C2—43no—
497C2—41no—
498C2—61no—
499C2—63no—
500C2—121no—
501C2—121no—
502C2C22×D3×D411yescharacteristic
subgroupG/H
centerC2C22×D3×D4
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC3×#577C2
socleC2×C3C23×D4
hypercenter#577D3
hypocenterC3#3586
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. D3×2−1+4C2semidirect
2. #5966C2semidirect
3. C3×#577C2semidirect
4. #5838C2semidirect
5. #5836C2semidirect
6. #5779C2semidirect
7. D3×#137C2semidirect
8. #5775C2semidirect
9. #5773C2semidirect
10. #5771C2semidirect
11. #5768C2semidirect
12. #5762C2semidirect
13. C2×#915C2semidirect
14. #5753C2semidirect
15. #5750C2semidirect
17. C3×2−1+4C22semidirect
18. C3×H(5,𝔽2)C22semidirect
21. #1010C22semidirect
22. #1010C22semidirect
23. #1010C22semidirect
24. #1010C22semidirect
29. D3×#41C22semidirect
31. #1007C22semidirect
33. C2×D3×Q8C22semidirect
35. #1004C22semidirect
38. C3×#137C22semidirect
39. C3×AGL(1,ℤ/8)C22semidirect
40. C3×#135C22semidirect
41. C2×C3×SD16C22semidirect
42. C3×#131C22semidirect
43. #951C22semidirect
44. #951C22semidirect
45. #950C22semidirect
46. #950C22semidirect
47. #948C22semidirect
48. C2×#216C22semidirect
49. #942C22semidirect
50. C2×#214C22semidirect
51. #932C22semidirect
52. #918C22semidirect
53. #918C22semidirect
54. D3×Q16C22semidirect
55. #916C22semidirect
56. #915C22semidirect
57. #915C22semidirect
58. #915C22semidirect
59. #915C22semidirect
60. #915C22semidirect
61. #914C22semidirect
62. #914C22semidirect
63. D3×SD16C22semidirect
64. #912C22semidirect
65. #911C22semidirect
66. #911C22semidirect
67. #909C22semidirect
68. #909C22semidirect
69. #907C22semidirect
70. D3×M16C22semidirect
71. C2×Dic12C22semidirect
72. #904C22semidirect
73. #901C22semidirect
74. C2×ΓL(1,𝔽25)C22semidirect
84. #577D3semidirect
188. 2−1+4C2×D3semidirect
191. H(5,𝔽2)C2×D3semidirect
213. #137C2×D3semidirect
219. AGL(1,ℤ/8)C2×D3semidirect
225. #135C2×D3semidirect
231. C2×SD16C2×D3semidirect
235. #131C2×D3semidirect
494. C3#3586semidirect

Maximal subgroups

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

subgroupindex
D3×2−1+42
#59662
C3×#5772
#58382
#58362
#57792
D3×#1372
#57752
#57732
#57712
#57682
#57622
C2×#9152
#57532
#57502
#35863