Finite Group Databasebeta

← #80085

Subgroups — #80085

Order 384.

784 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
1C2×#1009C211yesmaximal, characteristic, complemented
2C2×#914C211yesmaximal, characteristic, complemented
3C2×D3×D8C211yesmaximal, characteristic, complemented
4#5607C211yesmaximal, characteristic, complemented
5C3×#449C211yesmaximal, characteristic, complemented
6#5238C211yesmaximal, characteristic, complemented
7#5172C211yesmaximal, characteristic, complemented
8#5156C211yesmaximal, characteristic, complemented
9#5036C211yesmaximal, characteristic, complemented
10#4810C211yesmaximal, characteristic, complemented
11#4804C211yesmaximal, characteristic, complemented
12#4773C211yesmaximal, characteristic, complemented
13#4770C211yesmaximal, characteristic, complemented
14#4732C211yesmaximal, characteristic, complemented
15#4727C211yesmaximal, characteristic, complemented
16#3008—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic, complemented
18#1009—24no—
19#1009—24no—
20C2×#239C2211yescharacteristic
21C2×#239C2211yescharacteristic, complemented
22C2×D3×D4—21no—
23C2×D3×D4—21no—
24C2×D3×D4C2211yescharacteristic
25C2×D3×D4C2211yescharacteristic, complemented
26C2×D3×D4C2211yescharacteristic, complemented
27C2×#235—21no—
28C2×#235C2211yescharacteristic
29C22×D12—21no—
30C22×D12C2211yescharacteristic, complemented
31C2×C3×SD16C2211yescharacteristic, complemented
32C2×C3×D8C2211yescharacteristic, complemented
33C3×#121C2211yescharacteristic, complemented
34C2×#215C2211yescharacteristic, complemented
35#940—21no—
36#938C2211yescharacteristic, complemented
37#937—21no—
38C2×#213C2211yescharacteristic, complemented
39C2×#213C2211yescharacteristic, complemented
40C2×#213C2211yescharacteristic, complemented
41#930C2211yescharacteristic, complemented
42#914—24no—
43D3×D8—24no—
44C2×D24C2211yescharacteristic, complemented
45C2×D24C2211yescharacteristic, complemented
46C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
47C2×D3×C8C2211yescharacteristic, complemented
48#895C2211yescharacteristic, complemented
49#892C2211yescharacteristic, complemented
50#885—21no—
51#882—21no—
52C3×#103C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54C3×C22⋊C8C2211yescharacteristic, complemented
55#835C2211yescharacteristic, complemented
56#832C2211yescharacteristic, complemented
57#830C2211yescharacteristic, complemented
58#821C2211yescharacteristic, complemented
59#821C2211yescharacteristic, complemented
60(C2×D3)⋊C8C2211yescharacteristic, complemented
61(C2×D3)⋊C8C2211yescharacteristic, complemented
62#809C2211yescharacteristic, complemented
63#809C2211yescharacteristic, complemented
64C2×H(5,𝔽2)—31no—
65C2×AGL(1,ℤ/8)—31no—
66C22×D8—31no—
67#535—31no—
68#449—31no—
69#449—31no—
70#449—31no—
71#449D311yescharacteristic, complemented
72#447—31no—
73#447—31no—
74#447—31no—
75#447—31no—
76#418—31no—
77#416—31no—
78#408—31no—
79C23×D3—21no—
80C23×D3—21no—
81C23×D3—21no—
82C23×D3—21no—
83C3×#41—22no—
84C3×#41—22no—
85C2×C3×Q8C2311yescharacteristic
86C2×C3×D4—21no—
87C2×C3×D4—21no—
88C2×C3×D4C2311yescharacteristic
89C2×C3×D4C2311yescharacteristic
90C22×C3×C4—21no—
91C22×C3×C4C2311yescharacteristic
92C2×#67—21no—
93C2×#67—21no—
94C2×#67—21no—
95C2×#67—21no—
96C2×#67—21no—
97C2×#67D411yescharacteristic
98C2×#67D411yescharacteristic
99#239—22no—
100#239—22no—
101#239—22no—
102#239D414yescomplemented
103D3×D4—28no—
104D3×D4—28no—
105D3×D4—22no—
106D3×D4—22no—
107D3×D4—22no—
108D3×D4—22no—
109D3×D4—22no—
110D3×D4D414yescomplemented
111#235—28no—
112#235—22no—
113#235—22no—
114C2×D12—28no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—22no—
119C2×D12—22no—
120C2×D12—21no—
121C2×D12D411yescharacteristic
122C2×D12C2311yescharacteristic
123C2×D12C2311yescharacteristic
124C2×D12C2311yescharacteristic
125C2×C4×D3—21no—
126C2×C4×D3—21no—
127C2×C4×D3C2311yescharacteristic
128C2×C4×D3C2311yescharacteristic
129C2×Dic6D411yescharacteristic
130C3×SD16—22no—
131C3×D8—22no—
132C2×C3×C8C2311yescharacteristic
133C2×C3×C8C2311yescharacteristic
134C3×C4⋊C4C2311yescharacteristic
135C3×C22⋊C4—21no—
136#217—21no—
137#215—22no—
138#213—22no—
139#213—22no—
140#213—22no—
141(C2×D3)⋊C4—21no—
142(C2×D3)⋊C4—21no—
143(C2×D3)⋊C4—21no—
144C4⋊Dic3C2311yescharacteristic
145C4×Dic3—21no—
146C2×C3⋊C8C2311yescharacteristic
147C2×C3⋊C8C2311yescharacteristic
148D24—22no—
149D24—22no—
150ΓL(1,𝔽25)—22no—
151D3×C8—22no—
152H(5,𝔽2)—64no—
153H(5,𝔽2)—64no—
154C2×#41—31no—
155C2×#41—31no—
156C2×#41—31no—
157C2×#41—61no—
158C2×#41C2×D311yescharacteristic, complemented
159C22×D4—31no—
160C22×D4—31no—
161C22×D4—31no—
162C22×D4—31no—
163C22×D4—61no—
164C22×D4—61no—
165C22×D4—61no—
166AGL(1,ℤ/8)—64no—
167C2×SD16—31no—
168C2×SD16C2×D311yescharacteristic, complemented
169C2×D8—31no—
170C2×D8—31no—
171C2×D8—31no—
172C2×D8—31no—
173C2×D8—31no—
174C2×D8—64no—
175C2×D8C2×D311yescharacteristic, complemented
176C2×M16—31no—
177C22×C8—31no—
178#127—61no—
179#124—61no—
180#121—31no—
181#121—31no—
182#121—31no—
183#121C2×D311yescharacteristic, complemented
184#120—61no—
185#120—61no—
186#117—31no—
187#103—31no—
188#103C2×D311yescharacteristic, complemented
189#102—31no—
190#102—31no—
191#102—31no—
192#102—31no—
193#102—31no—
194#102C2×D311yescharacteristic, complemented
195C22⋊C8—31no—
196C22⋊C8—31no—
197C22⋊C8—31no—
198C22⋊C8C2×D311yescharacteristic, complemented
199C23×C3—21no—
200C23×C3—21no—
201C23×C3C2×D411yescharacteristic
202C22×D3—21no—
203C22×D3—21no—
204C22×D3—24no—
205C22×D3—24no—
206C22×D3—24no—
207C22×D3—21no—
208C22×D3—42no—
209C22×D3—81no—
210C22×D3—81no—
211C22×D3—81no—
212C22×D3—81no—
213C22×D3—81no—
214C22×D3—81no—
215C22×D3—81no—
216C22×D3—81no—
217C22×D3C2×D411yescharacteristic
218C22×D3C2×D411yescharacteristic
219C3×Q8—21no—
220C3×Q8C2×D412yes—
221C3×D4—24no—
222C3×D4—21no—
223C3×D4—21no—
224C3×D4—21no—
225C3×D4—81no—
226C3×D4C2×D412yes—
227C2×C3×C4—24no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4C2×D411yescharacteristic
233C2×C3×C4C2411yescharacteristic
234#67—24no—
235#67—22no—
236#67—22no—
237#67—24no—
238#67—42no—
239#67—81no—
240#67—81no—
241C2×Dic3—21no—
242C2×Dic3C2×D411yescharacteristic
243C2×Dic3C2×D411yescharacteristic
244D12—24no—
245D12—24no—
246D12—22no—
247D12—24no—
248D12—21no—
249D12—21no—
250D12—21no—
251D12—21no—
252D12—81no—
253D12C2×D412yes—
254D12C2×D412yes—
255C4×D3—24no—
256C4×D3—21no—
257C4×D3—21no—
258C4×D3—24no—
259C4×D3C2×D412yes—
260C4×D3C2×D412yes—
261Dic6—22no—
262C3×C8—21no—
263C3×C8—21no—
264C3⋊C8—21no—
265C3⋊C8—21no—
266C24—61no—
267C24—61no—
268C24—61no—
269C24—61no—
270#41—22no—
271#41—22no—
272#41—34no—
273#41—62no—
274#41—62no—
275#41—68no—
276#41—62no—
277#41—62no—
278#41—62no—
279C2×Q8—31no—
280C2×Q8C22×D311yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—34no—
284C2×D4—31no—
285C2×D4—31no—
286C2×D4—31no—
287C2×D4—31no—
288C2×D4—31no—
289C2×D4—31no—
290C2×D4—62no—
291C2×D4—62no—
292C2×D4—61no—
293C2×D4—61no—
294C2×D4—68no—
295C2×D4—61no—
296C2×D4—62no—
297C2×D4—62no—
298C2×D4—62no—
299C2×D4—61no—
300C2×D4—61no—
301C2×D4—61no—
302C2×D4—62no—
303C2×D4—68no—
304C2×D4—68no—
305C2×D4—62no—
306C2×D4—61no—
307C2×D4—61no—
308C2×D4—61no—
309C2×D4C22×D311yescharacteristic
310C2×D4C22×D311yescharacteristic
311C22×C4—21no—
312C22×C4—31no—
313C22×C4—31no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4C22×D311yescharacteristic
317SD16—22no—
318SD16—62no—
319D8—22no—
320D8—62no—
321D8—62no—
322D8—62no—
323D8—62no—
324D8—62no—
325M16—62no—
326C2×C8—31no—
327C2×C8—31no—
328C2×C8—62no—
329C2×C8C22×D311yescharacteristic
330C2×C8C22×D311yescharacteristic
331C4⋊C4—31no—
332C4⋊C4C22×D311yescharacteristic
333C22⋊C4—21no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22⋊C4—61no—
337C22⋊C4—61no—
338C42—61no—
339C22×C3—21no—
340C22×C3—22no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3C22×D411yescharacteristic
349C2×D3—21no—
350C2×D3—21no—
351C2×D3—21no—
352C2×D3—22no—
353C2×D3—22no—
354C2×D3—21no—
355C2×D3—21no—
356C2×D3—41no—
357C2×D3—41no—
358C2×D3—41no—
359C2×D3—41no—
360C2×D3—41no—
361C2×D3—41no—
362C2×D3—41no—
363C2×D3—84no—
364C2×D3—84no—
365C2×D3—84no—
366C2×D3—84no—
367C2×D3#12012yes—
368C3×C4—21no—
369C3×C4—22no—
370C3×C4—41no—
371C3×C4C22×D411yescharacteristic
372C3×C4C22×D411yescharacteristic
373Dic3—21no—
374Dic3—41no—
375Dic3#12012yes—
376C23—21no—
377C23—21no—
378C23—31no—
379C23—31no—
380C23—64no—
381C23—64no—
382C23—64no—
383C23—61no—
384C23—61no—
385C23—61no—
386C23—122no—
387C23—241no—
388C23—241no—
389C23—241no—
390C23—241no—
391C23—241no—
392C23—241no—
393C23—241no—
394C23—241no—
395C23D3×D411yescharacteristic
396Q8—21no—
397Q8—62no—
398Q8D3×D412yes—
399D4—24no—
400D4—21no—
401D4—21no—
402D4—21no—
403D4—32no—
404D4—32no—
405D4—62no—
406D4—64no—
407D4—64no—
408D4—64no—
409D4—62no—
410D4—64no—
411D4—62no—
412D4—64no—
413D4—61no—
414D4—61no—
415D4—61no—
416D4—61no—
417D4—81no—
418D4—122no—
419D4—241no—
420D4—241no—
421D4—241no—
422D4D3×D412yes—
423C2×C4—24no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—31no—
429C2×C4—32no—
430C2×C4—32no—
431C2×C4—31no—
432C2×C4—64no—
433C2×C4—64no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4D3×D411yescharacteristic
438C2×C4C23×D311yescharacteristic
439C8—21no—
440C8—21no—
441C8—61no—
442C8—61no—
443C2×C3—41no—
444C2×C3—42no—
445C2×C3—81no—
446C2×C3C2×#12011yescharacteristic
447C2×C3#57611yescharacteristic
448C2×C3#57611yescharacteristic
449D3—22no—
450D3—41no—
451D3—42no—
452D3—42no—
453D3—81no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—22no—
458C22—32no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—61no—
465C22—61no—
466C22—61no—
467C22—61no—
468C22—62no—
469C22—62no—
470C22—61no—
471C22—121no—
472C22—121no—
473C22—121no—
474C22—121no—
475C22—121no—
476C22—121no—
477C22—121no—
478C22—244no—
479C22—244no—
480C22—244no—
481C22—244no—
482C22C2×D3×D411yescharacteristic
483C4—21no—
484C4—22no—
485C4—32no—
486C4—41no—
487C4—61no—
488C4—121no—
489C4C2×D3×D411yescharacteristic
490C4C2×D3×D411yescharacteristic
491C3#300811yescharacteristic, complemented, Sylow
492C2—41no—
493C2—42no—
494C2—62no—
495C2—81no—
496C2—121no—
497C2—122no—
498C2—122no—
499C2—241no—
500C2D3×#12011yescharacteristic
501C2#576911yescharacteristic
502C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#449C2
socleC22×C3C22×D4
hypercenter#449D3
hypocenterC3#3008
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#1009C2semidirect
2. C2×#914C2semidirect
3. C2×D3×D8C2semidirect
4. #5607C2semidirect
5. C3×#449C2semidirect
6. #5238C2semidirect
7. #5172C2semidirect
8. #5156C2semidirect
9. #5036C2semidirect
10. #4810C2semidirect
11. #4804C2semidirect
12. #4773C2semidirect
13. #4770C2semidirect
14. #4732C2semidirect
15. #4727C2semidirect
17. C2×C3×#41C22semidirect
21. C2×#239C22semidirect
25. C2×D3×D4C22semidirect
26. C2×D3×D4C22semidirect
30. C22×D12C22semidirect
31. C2×C3×SD16C22semidirect
32. C2×C3×D8C22semidirect
33. C3×#121C22semidirect
34. C2×#215C22semidirect
36. #938C22semidirect
38. C2×#213C22semidirect
39. C2×#213C22semidirect
40. C2×#213C22semidirect
41. #930C22semidirect
44. C2×D24C22semidirect
45. C2×D24C22semidirect
46. C2×ΓL(1,𝔽25)C22semidirect
47. C2×D3×C8C22semidirect
48. #895C22semidirect
49. #892C22semidirect
52. C3×#103C22semidirect
53. C3×#102C22semidirect
54. C3×C22⋊C8C22semidirect
55. #835C22semidirect
56. #832C22semidirect
57. #830C22semidirect
58. #821C22semidirect
59. #821C22semidirect
60. (C2×D3)⋊C8C22semidirect
61. (C2×D3)⋊C8C22semidirect
62. #809C22semidirect
63. #809C22semidirect
71. #449D3semidirect
102. #239D4semidirect
110. D3×D4D4semidirect
158. C2×#41C2×D3semidirect
168. C2×SD16C2×D3semidirect
175. C2×D8C2×D3semidirect
183. #121C2×D3semidirect
188. #103C2×D3semidirect
194. #102C2×D3semidirect
198. C22⋊C8C2×D3semidirect
491. C3#3008semidirect

Maximal subgroups

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

subgroupindex
C2×#10092
C2×#9142
C2×D3×D82
#56072
C3×#4492
#52382
#51722
#51562
#50362
#48102
#48042
#47732
#47702
#47322
#47272
#30083