Finite Group Databasebeta

← #16854976

Subgroups — #16854976

Order 3968.

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×#11758566C211yesmaximal, characteristic, complemented
2C2×#11758491C211yesmaximal, characteristic, complemented
3C2×D8×D31C211yesmaximal, characteristic, complemented
4#14521615C211yesmaximal, characteristic, complemented
5#14521197C211yesmaximal, characteristic, complemented
6#14521194C211yesmaximal, characteristic, complemented
7#14521193C211yesmaximal, characteristic, complemented
8#14521185C211yesmaximal, characteristic, complemented
9#14521184C211yesmaximal, characteristic, complemented
10#14521182C211yesmaximal, characteristic, complemented
11#14521181C211yesmaximal, characteristic, complemented
12#14521052C211yesmaximal, characteristic, complemented
13#14521044C211yesmaximal, characteristic, complemented
14#14521014C211yesmaximal, characteristic, complemented
15C31×#449C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic, complemented
17#11758566—24no—
18#11758566—24no—
19C2×#92499C2211yescharacteristic
20C2×#92499C2211yescharacteristic, complemented
21C2×D4×D31—21no—
22C2×D4×D31—21no—
23C2×D4×D31C2211yescharacteristic
24C2×D4×D31C2211yescharacteristic, complemented
25C2×D4×D31C2211yescharacteristic, complemented
26C2×#92495—21no—
27C2×#92495C2211yescharacteristic
28C22×D124—21no—
29C22×D124C2211yescharacteristic, complemented
30C2×SD16×C31C2211yescharacteristic, complemented
31C2×D8×C31C2211yescharacteristic, complemented
32C31×#121C2211yescharacteristic, complemented
33C2×#92481C2211yescharacteristic, complemented
34#11758517—21no—
35#11758515C2211yescharacteristic, complemented
36#11758514—21no—
37C2×#92479C2211yescharacteristic, complemented
38C2×#92479C2211yescharacteristic, complemented
39C2×#92479C2211yescharacteristic, complemented
40#11758507C2211yescharacteristic, complemented
41#11758491—24no—
42D8×D31—24no—
43C2×D248C2211yescharacteristic, complemented
44C2×D248C2211yescharacteristic, complemented
45C2×#92469C2211yescharacteristic, complemented
46C2×C8×D31C2211yescharacteristic, complemented
47#11758472C2211yescharacteristic, complemented
48#11758469C2211yescharacteristic, complemented
49#11758462—21no—
50#11758459—21no—
51C31×#103C2211yescharacteristic, complemented
52C31×#102C2211yescharacteristic, complemented
53C31×C22⋊C8C2211yescharacteristic, complemented
54#11758423C2211yescharacteristic, complemented
55#11758420C2211yescharacteristic, complemented
56C22⋊(C31⋊C8)C2211yescharacteristic, complemented
57#11758409C2211yescharacteristic, complemented
58#11758409C2211yescharacteristic, complemented
59(C2×D31)⋊C8C2211yescharacteristic, complemented
60(C2×D31)⋊C8C2211yescharacteristic, complemented
61#11758397C2211yescharacteristic, complemented
62#11758397C2211yescharacteristic, complemented
63C23×D31—21no—
64C23×D31—21no—
65C23×D31—21no—
66C23×D31—21no—
67#41×C31—22no—
68#41×C31—22no—
69C2×Q8×C31C2311yescharacteristic
70C2×D4×C31—21no—
71C2×D4×C31—21no—
72C2×D4×C31C2311yescharacteristic
73C2×D4×C31C2311yescharacteristic
74C22×C4×C31—21no—
75C22×C4×C31C2311yescharacteristic
76C2×C22⋊D31—21no—
77C2×C22⋊D31—21no—
78C2×C22⋊D31—21no—
79C2×C22⋊D31—21no—
80C2×C22⋊D31—21no—
81C2×C22⋊D31D411yescharacteristic
82C2×C22⋊D31D411yescharacteristic
83#92499—22no—
84#92499—22no—
85#92499—22no—
86#92499D414yescomplemented
87D4×D31—28no—
88D4×D31—28no—
89D4×D31—22no—
90D4×D31—22no—
91D4×D31—22no—
92D4×D31—22no—
93D4×D31—22no—
94D4×D31D414yescomplemented
95#92495—28no—
96#92495—22no—
97#92495—22no—
98C2×D124—28no—
99C2×D124—21no—
100C2×D124—21no—
101C2×D124—21no—
102C2×D124—22no—
103C2×D124—22no—
104C2×D124—21no—
105C2×D124D411yescharacteristic
106C2×D124C2311yescharacteristic
107C2×D124C2311yescharacteristic
108C2×D124C2311yescharacteristic
109C2×C4×D31—21no—
110C2×C4×D31—21no—
111C2×C4×D31C2311yescharacteristic
112C2×C4×D31C2311yescharacteristic
113C2×Dic62D411yescharacteristic
114SD16×C31—22no—
115D8×C31—22no—
116C2×C8×C31C2311yescharacteristic
117C2×C8×C31C2311yescharacteristic
118C4⋊C4×C31C2311yescharacteristic
119C22⋊C4×C31—21no—
120C22⋊Dic31—21no—
121#92481—22no—
122#92479—22no—
123#92479—22no—
124#92479—22no—
125(C2×D31)⋊C4—21no—
126(C2×D31)⋊C4—21no—
127(C2×D31)⋊C4—21no—
128C4⋊Dic31C2311yescharacteristic
129C4×Dic31—21no—
130C2×C31⋊C8C2311yescharacteristic
131C2×C31⋊C8C2311yescharacteristic
132D248—22no—
133D248—22no—
134#92469—22no—
135C8×D31—22no—
136C23×C31—21no—
137C23×C31—21no—
138C23×C31C2×D411yescharacteristic
139C22×D31—21no—
140C22×D31—21no—
141C22×D31—24no—
142C22×D31—24no—
143C22×D31—24no—
144C22×D31—21no—
145C22×D31—42no—
146C22×D31—81no—
147C22×D31—81no—
148C22×D31—81no—
149C22×D31—81no—
150C22×D31—81no—
151C22×D31—81no—
152C22×D31—81no—
153C22×D31—81no—
154C22×D31C2×D411yescharacteristic
155C22×D31C2×D411yescharacteristic
156Q8×C31—21no—
157Q8×C31C2×D412yes—
158D4×C31—24no—
159D4×C31—21no—
160D4×C31—21no—
161D4×C31—21no—
162D4×C31—81no—
163D4×C31C2×D412yes—
164C2×C4×C31—24no—
165C2×C4×C31—21no—
166C2×C4×C31—21no—
167C2×C4×C31—21no—
168C2×C4×C31—21no—
169C2×C4×C31C2×D411yescharacteristic
170C2×C4×C31C2411yescharacteristic
171C22⋊D31—24no—
172C22⋊D31—22no—
173C22⋊D31—22no—
174C22⋊D31—24no—
175C22⋊D31—42no—
176C22⋊D31—81no—
177C22⋊D31—81no—
178C2×Dic31—21no—
179C2×Dic31C2×D411yescharacteristic
180C2×Dic31C2×D411yescharacteristic
181D124—24no—
182D124—24no—
183D124—22no—
184D124—24no—
185D124—21no—
186D124—21no—
187D124—21no—
188D124—21no—
189D124—81no—
190D124C2×D412yes—
191D124C2×D412yes—
192C4×D31—24no—
193C4×D31—21no—
194C4×D31—21no—
195C4×D31—24no—
196C4×D31C2×D412yes—
197C4×D31C2×D412yes—
198Dic62—22no—
199C8×C31—21no—
200C8×C31—21no—
201C31⋊C8—21no—
202C31⋊C8—21no—
203#3008—311nomaximal, Sylow
204C22×C31—21no—
205C22×C31—22no—
206C22×C31—21no—
207C22×C31—21no—
208C22×C31—41no—
209C22×C31—41no—
210C22×C31—41no—
211C22×C31—41no—
212C22×C31—41no—
213C22×C31C22×D411yescharacteristic
214C2×D31—21no—
215C2×D31—21no—
216C2×D31—21no—
217C2×D31—22no—
218C2×D31—22no—
219C2×D31—21no—
220C2×D31—21no—
221C2×D31—41no—
222C2×D31—41no—
223C2×D31—41no—
224C2×D31—41no—
225C2×D31—41no—
226C2×D31—41no—
227C2×D31—41no—
228C2×D31—84no—
229C2×D31—84no—
230C2×D31—84no—
231C2×D31—84no—
232C2×D31#12012yes—
233C4×C31—21no—
234C4×C31—22no—
235C4×C31—41no—
236C4×C31C22×D411yescharacteristic
237C4×C31C22×D411yescharacteristic
238Dic31—21no—
239Dic31—41no—
240Dic31#12012yes—
241C2×H(5,𝔽2)—311no—
242C2×AGL(1,ℤ/8)—311no—
243C22×D8—311no—
244#535—311no—
245#449—311no—
246#449—311no—
247#449—311no—
248#449D3111yescharacteristic, complemented
249#447—311no—
250#447—311no—
251#447—311no—
252#447—311no—
253#418—311no—
254#416—311no—
255#408—311no—
256C2×C31—41no—
257C2×C31—42no—
258C2×C31—81no—
259C2×C31C2×#12011yescharacteristic
260C2×C31#57611yescharacteristic
261C2×C31#57611yescharacteristic
262D31—22no—
263D31—41no—
264D31—42no—
265D31—42no—
266D31—81no—
267H(5,𝔽2)—624no—
268H(5,𝔽2)—624no—
269C2×#41—311no—
270C2×#41—311no—
271C2×#41—311no—
272C2×#41—621no—
273C2×#41C2×D3111yescharacteristic, complemented
274C22×D4—311no—
275C22×D4—311no—
276C22×D4—311no—
277C22×D4—311no—
278C22×D4—621no—
279C22×D4—621no—
280C22×D4—621no—
281AGL(1,ℤ/8)—624no—
282C2×SD16—311no—
283C2×SD16C2×D3111yescharacteristic, complemented
284C2×D8—311no—
285C2×D8—311no—
286C2×D8—311no—
287C2×D8—311no—
288C2×D8—311no—
289C2×D8—624no—
290C2×D8C2×D3111yescharacteristic, complemented
291C2×M16—311no—
292C22×C8—311no—
293#127—621no—
294#124—621no—
295#121—311no—
296#121—311no—
297#121—311no—
298#121C2×D3111yescharacteristic, complemented
299#120—621no—
300#120—621no—
301#117—311no—
302#103—311no—
303#103C2×D3111yescharacteristic, complemented
304#102—311no—
305#102—311no—
306#102—311no—
307#102—311no—
308#102—311no—
309#102C2×D3111yescharacteristic, complemented
310C22⋊C8—311no—
311C22⋊C8—311no—
312C22⋊C8—311no—
313C22⋊C8C2×D3111yescharacteristic, complemented
314C31#300811yescharacteristic, complemented, Sylow
315C24—621no—
316C24—621no—
317C24—621no—
318C24—621no—
319#41—22no—
320#41—22no—
321#41—314no—
322#41—622no—
323#41—622no—
324#41—628no—
325#41—622no—
326#41—622no—
327#41—622no—
328C2×Q8—311no—
329C2×Q8C22×D3111yescharacteristic
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—314no—
333C2×D4—311no—
334C2×D4—311no—
335C2×D4—311no—
336C2×D4—311no—
337C2×D4—311no—
338C2×D4—311no—
339C2×D4—622no—
340C2×D4—622no—
341C2×D4—621no—
342C2×D4—621no—
343C2×D4—628no—
344C2×D4—621no—
345C2×D4—622no—
346C2×D4—622no—
347C2×D4—622no—
348C2×D4—621no—
349C2×D4—621no—
350C2×D4—621no—
351C2×D4—622no—
352C2×D4—628no—
353C2×D4—628no—
354C2×D4—622no—
355C2×D4—621no—
356C2×D4—621no—
357C2×D4—621no—
358C2×D4C22×D3111yescharacteristic
359C2×D4C22×D3111yescharacteristic
360C22×C4—21no—
361C22×C4—311no—
362C22×C4—311no—
363C22×C4—621no—
364C22×C4—621no—
365C22×C4C22×D3111yescharacteristic
366SD16—22no—
367SD16—622no—
368D8—22no—
369D8—622no—
370D8—622no—
371D8—622no—
372D8—622no—
373D8—622no—
374M16—622no—
375C2×C8—311no—
376C2×C8—311no—
377C2×C8—622no—
378C2×C8C22×D3111yescharacteristic
379C2×C8C22×D3111yescharacteristic
380C4⋊C4—311no—
381C4⋊C4C22×D3111yescharacteristic
382C22⋊C4—21no—
383C22⋊C4—621no—
384C22⋊C4—621no—
385C22⋊C4—621no—
386C22⋊C4—621no—
387C42—621no—
388C23—21no—
389C23—21no—
390C23—311no—
391C23—311no—
392C23—624no—
393C23—624no—
394C23—624no—
395C23—621no—
396C23—621no—
397C23—621no—
398C23—1242no—
399C23—2481no—
400C23—2481no—
401C23—2481no—
402C23—2481no—
403C23—2481no—
404C23—2481no—
405C23—2481no—
406C23—2481no—
407C23D4×D3111yescharacteristic
408Q8—21no—
409Q8—622no—
410Q8D4×D3112yes—
411D4—24no—
412D4—21no—
413D4—21no—
414D4—21no—
415D4—81no—
416D4—312no—
417D4—312no—
418D4—622no—
419D4—624no—
420D4—624no—
421D4—624no—
422D4—622no—
423D4—624no—
424D4—622no—
425D4—624no—
426D4—621no—
427D4—621no—
428D4—621no—
429D4—621no—
430D4—1242no—
431D4—2481no—
432D4—2481no—
433D4—2481no—
434D4D4×D3112yes—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—311no—
441C2×C4—312no—
442C2×C4—312no—
443C2×C4—311no—
444C2×C4—624no—
445C2×C4—624no—
446C2×C4—621no—
447C2×C4—621no—
448C2×C4—621no—
449C2×C4D4×D3111yescharacteristic
450C2×C4C23×D3111yescharacteristic
451C8—21no—
452C8—21no—
453C8—621no—
454C8—621no—
455C22—21no—
456C22—22no—
457C22—21no—
458C22—21no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—312no—
465C22—621no—
466C22—621no—
467C22—622no—
468C22—622no—
469C22—621no—
470C22—621no—
471C22—621no—
472C22—1241no—
473C22—1241no—
474C22—1241no—
475C22—1241no—
476C22—1241no—
477C22—1241no—
478C22—1241no—
479C22—2484no—
480C22—2484no—
481C22—2484no—
482C22—2484no—
483C22C2×D4×D3111yescharacteristic
484C4—21no—
485C4—22no—
486C4—41no—
487C4—312no—
488C4—621no—
489C4—1241no—
490C4C2×D4×D3111yescharacteristic
491C4C2×D4×D3111yescharacteristic
492C2—41no—
493C2—42no—
494C2—81no—
495C2—622no—
496C2—1241no—
497C2—1242no—
498C2—1242no—
499C2—2481no—
500C2#120×D3111yescharacteristic
501C2#1452185111yescharacteristic
502C2#1452185311yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#449C2
socleC22×C31C22×D4
hypercenter#449D31
hypocenterC31#3008
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#11758566C2semidirect
2. C2×#11758491C2semidirect
3. C2×D8×D31C2semidirect
4. #14521615C2semidirect
5. #14521197C2semidirect
6. #14521194C2semidirect
7. #14521193C2semidirect
8. #14521185C2semidirect
9. #14521184C2semidirect
10. #14521182C2semidirect
11. #14521181C2semidirect
12. #14521052C2semidirect
13. #14521044C2semidirect
14. #14521014C2semidirect
15. C31×#449C2semidirect
16. C2×#41×C31C22semidirect
20. C2×#92499C22semidirect
24. C2×D4×D31C22semidirect
25. C2×D4×D31C22semidirect
29. C22×D124C22semidirect
30. C2×SD16×C31C22semidirect
31. C2×D8×C31C22semidirect
32. C31×#121C22semidirect
33. C2×#92481C22semidirect
35. #11758515C22semidirect
37. C2×#92479C22semidirect
38. C2×#92479C22semidirect
39. C2×#92479C22semidirect
40. #11758507C22semidirect
43. C2×D248C22semidirect
44. C2×D248C22semidirect
45. C2×#92469C22semidirect
46. C2×C8×D31C22semidirect
47. #11758472C22semidirect
48. #11758469C22semidirect
51. C31×#103C22semidirect
52. C31×#102C22semidirect
53. C31×C22⋊C8C22semidirect
54. #11758423C22semidirect
55. #11758420C22semidirect
56. C22⋊(C31⋊C8)C22semidirect
57. #11758409C22semidirect
58. #11758409C22semidirect
59. (C2×D31)⋊C8C22semidirect
60. (C2×D31)⋊C8C22semidirect
61. #11758397C22semidirect
62. #11758397C22semidirect
86. #92499D4semidirect
94. D4×D31D4semidirect
248. #449D31semidirect
273. C2×#41C2×D31semidirect
283. C2×SD16C2×D31semidirect
290. C2×D8C2×D31semidirect
298. #121C2×D31semidirect
303. #103C2×D31semidirect
309. #102C2×D31semidirect
313. C22⋊C8C2×D31semidirect
314. C31#3008semidirect

Maximal subgroups

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

subgroupindex
C2×#117585662
C2×#117584912
C2×D8×D312
#145216152
#145211972
#145211942
#145211932
#145211852
#145211842
#145211822
#145211812
#145210522
#145210442
#145210142
C31×#4492
#300831