Finite Group Databasebeta

← #16857197

Subgroups — #16857197

Order 3968.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758491C211yesmaximal, characteristic, complemented
2C2×D8×D31C211yesmaximal, characteristic, complemented
3#14521692C211yesmaximal, characteristic, complemented
4D4×D124C211yesmaximal, characteristic, complemented
5#14521439C211yesmaximal, characteristic, complemented
6#14521301C211yesmaximal, characteristic, complemented
7#14521281C211yesmaximal, characteristic, complemented
8#14521245C211yesmaximal, characteristic, complemented
9#14521198C211yesmaximal, characteristic, complemented
10#14521197C211yesmaximal, characteristic, complemented
11#14521184C211yesmaximal, characteristic, complemented
12#14521181C211yesmaximal, characteristic, complemented
13#14521159C211yesmaximal, characteristic, complemented
14#14521119C211yesmaximal, characteristic, complemented
15C31×#463C211yesmaximal, characteristic, complemented
16C2×#92499C2211yescharacteristic, complemented
17C2×D4×D31C2211yescharacteristic
18C2×D4×D31C2211yescharacteristic, complemented
19C2×D4×D31C2211yescharacteristic, complemented
20C22×D124—21no—
21C2×SD16×C31C2211yescharacteristic, complemented
22C2×D8×C31C2211yescharacteristic
23C31×#124C2211yescharacteristic, complemented
24C4×D4×C31C2211yescharacteristic, complemented
25#11758523C2211yescharacteristic, complemented
26C2×#92481C2211yescharacteristic, complemented
27C2×#92481C2211yescharacteristic, complemented
28#11758515C2211yescharacteristic, complemented
29C2×#92479C2211yescharacteristic
30C2×#92479C2211yescharacteristic, complemented
31C2×#92479C2211yescharacteristic, complemented
32#11758507—21no—
33#11758491—24no—
34D8×D31—24no—
35C2×D248C2211yescharacteristic, complemented
36C2×D248C2211yescharacteristic, complemented
37C2×#92469C2211yescharacteristic, complemented
38C2×C8×D31C2211yescharacteristic, complemented
39#11758472C2211yescharacteristic
40#11758472C2211yescharacteristic, complemented
41#11758461—21no—
42#11758460—21no—
43#11758459—21no—
44#11758459—21no—
45#11758459—21no—
46C22⋊C4×D31—21no—
47#11758451C2211yescharacteristic, complemented
48C4×D124C2211yescharacteristic
49C4×D124C2211yescharacteristic, complemented
50C31×C4⋊C8C2211yescharacteristic, complemented
51C31×#103C2211yescharacteristic, complemented
52C31×#102C2211yescharacteristic, complemented
53#11758420C2211yescharacteristic, complemented
54#11758409C2211yescharacteristic, complemented
55#11758409C2211yescharacteristic, complemented
56(C2×D31)⋊C8C2211yescharacteristic, complemented
57(C2×D31)⋊C8C2211yescharacteristic, complemented
58#11758397C2211yescharacteristic, complemented
59#11758396C2211yescharacteristic, complemented
60#11758391C2211yescharacteristic, complemented
61C23×D31—21no—
62C23×D31—21no—
63C23×D31—21no—
64C2×Q8×C31C2311yescharacteristic
65C2×D4×C31C2311yescharacteristic
66C2×D4×C31C2311yescharacteristic
67C22×C4×C31—21no—
68C2×C22⋊D31—21no—
69C2×C22⋊D31—21no—
70C2×C22⋊D31—21no—
71C2×C22⋊D31—21no—
72#92499—22no—
73#92499—41no—
74D4×D31—22no—
75D4×D31—22no—
76D4×D31—22no—
77D4×D31—41no—
78D4×D31—41no—
79D4×D31—41no—
80C2×D124—21no—
81C2×D124—21no—
82C2×D124—21no—
83C2×D124—21no—
84C2×D124—21no—
85C2×D124—44no—
86C2×D124D411yescharacteristic
87C2×D124C2311yescharacteristic
88C2×D124C2311yescharacteristic
89C2×D124C2311yescharacteristic
90C2×C4×D31—21no—
91C2×C4×D31—21no—
92C2×C4×D31C2311yescharacteristic
93C2×C4×D31C2311yescharacteristic
94SD16×C31—22no—
95D8×C31—22no—
96C2×C8×C31C2311yescharacteristic
97C2×C8×C31C2311yescharacteristic
98C4⋊C4×C31C2311yescharacteristic
99C22⋊C4×C31—21no—
100C22⋊C4×C31—21no—
101C22⋊C4×C31—21no—
102C42×C31C2311yescharacteristic
103C22⋊Dic31—21no—
104#92481—22no—
105#92481—41no—
106#92479—22no—
107#92479—41no—
108#92479D414yescomplemented
109(C2×D31)⋊C4—21no—
110(C2×D31)⋊C4—21no—
111(C2×D31)⋊C4—21no—
112(C2×D31)⋊C4—21no—
113(C2×D31)⋊C4D411yescharacteristic
114(C2×D31)⋊C4D411yescharacteristic
115C4⋊Dic31D411yescharacteristic
116C4⋊Dic31C2311yescharacteristic
117Dic31⋊C4—21no—
118C2×C31⋊C8C2311yescharacteristic
119C2×C31⋊C8C2311yescharacteristic
120D248—22no—
121D248—22no—
122#92469—22no—
123C8×D31—22no—
124C23×C31—21no—
125C23×C31—21no—
126C22×D31—21no—
127C22×D31—21no—
128C22×D31—21no—
129C22×D31—42no—
130C22×D31—42no—
131C22×D31—42no—
132C22×D31—42no—
133C22×D31—42no—
134C22×D31—42no—
135C22×D31—81no—
136C22×D31—81no—
137C22×D31—81no—
138C22×D31C2×D411yescharacteristic
139C22×D31C2×D411yescharacteristic
140Q8×C31—21no—
141Q8×C31—21no—
142D4×C31—21no—
143D4×C31—21no—
144D4×C31—21no—
145D4×C31C2×D412yes—
146C2×C4×C31—21no—
147C2×C4×C31—21no—
148C2×C4×C31—42no—
149C2×C4×C31C2×D411yescharacteristic
150C2×C4×C31C2×D411yescharacteristic
151C2×C4×C31C2411yescharacteristic
152C22⋊D31—42no—
153C22⋊D31—42no—
154C22⋊D31—42no—
155C22⋊D31—81no—
156C2×Dic31—21no—
157C2×Dic31C2×D411yescharacteristic
158C2×Dic31C2×D411yescharacteristic
159D124—22no—
160D124—21no—
161D124—21no—
162D124—21no—
163D124—21no—
164D124—21no—
165D124—42no—
166D124—42no—
167D124—42no—
168D124—42no—
169D124—81no—
170D124C2×D412yescomplemented
171C4×D31—21no—
172C4×D31—21no—
173C4×D31—21no—
174C4×D31—21no—
175C4×D31—42no—
176C4×D31—81no—
177C8×C31—21no—
178C8×C31—21no—
179C31⋊C8—21no—
180C31⋊C8C2×D412yescomplemented
181#3264—311nomaximal, Sylow
182C22×C31—22no—
183C22×C31—41no—
184C22×C31—41no—
185C22×C31—41no—
186C22×C31—41no—
187C22×C31—41no—
188C22×C31C22×D411yescharacteristic
189C2×D31—21no—
190C2×D31—21no—
191C2×D31—21no—
192C2×D31—22no—
193C2×D31—21no—
194C2×D31—21no—
195C2×D31—41no—
196C2×D31—41no—
197C2×D31—41no—
198C2×D31—41no—
199C2×D31—41no—
200C2×D31—41no—
201C2×D31—41no—
202C2×D31—41no—
203C2×D31—84no—
204C2×D31—84no—
205C2×D31—84no—
206C2×D31AGL(1,ℤ/8)12yescomplemented
207C4×C31—21no—
208C4×C31—21no—
209C4×C31—41no—
210C4×C31—41no—
211C4×C31C22×D411yescharacteristic
212C4×C31H(5,𝔽2)11yescharacteristic
213Dic31—21no—
214Dic31—21no—
215Dic31—41no—
216C2×AGL(1,ℤ/8)—311no—
217C22×D8—311no—
218#546—311no—
219D42—311no—
220#496—311no—
221#469—311no—
222#465—311no—
223#463D3111yescharacteristic, complemented
224#459—311no—
225#450—311no—
226#449—311no—
227#447—311no—
228#447—311no—
229#442—311no—
230#435—311no—
231C2×C31—42no—
232C2×C31—81no—
233C2×C31D4211yescharacteristic
234C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
235C2×C31#57611yescharacteristic
236D31—22no—
237D31—41no—
238D31—42no—
239D31—81no—
240D31—81no—
241C2×#41—311no—
242C22×D4—311no—
243C22×D4—311no—
244C22×D4—311no—
245C22×D4—621no—
246AGL(1,ℤ/8)—624no—
247C2×SD16—311no—
248C2×SD16—311no—
249C2×SD16C2×D3111yescharacteristic, complemented
250C2×D8—311no—
251C2×D8—311no—
252C2×D8—311no—
253C2×D8—311no—
254C2×D8—311no—
255C2×D8—624no—
256C2×D8C2×D3111yescharacteristic
257C2×M16—311no—
258C22×C8—311no—
259#127—311no—
260#124C2×D3111yescharacteristic, complemented
261#123—621no—
262C22⋊Q8—311no—
263#121—311no—
264#121—311no—
265#121—311no—
266#121—621no—
267#121—621no—
268#120—621no—
269#120—621no—
270#120—621no—
271C4×D4—311no—
272C4×D4—311no—
273C4×D4C2×D3111yescharacteristic, complemented
274C2×C22⋊C4—621no—
275#106—311no—
276C4⋊C8C2×D3111yescharacteristic, complemented
277#103C2×D3111yescharacteristic, complemented
278#102—311no—
279#102—311no—
280#102—311no—
281#102—311no—
282#102C2×D3111yescharacteristic, complemented
283C22⋊C8—311no—
284C22⋊C8—311no—
285#97—311no—
286C31#326411yescharacteristic, complemented, Sylow
287C24—621no—
288C24—621no—
289C24—621no—
290#41—622no—
291#41—1241no—
292C2×Q8C22×D3111yescharacteristic
293C2×D4—311no—
294C2×D4—311no—
295C2×D4—311no—
296C2×D4—311no—
297C2×D4—621no—
298C2×D4—621no—
299C2×D4—621no—
300C2×D4—621no—
301C2×D4—621no—
302C2×D4—621no—
303C2×D4—622no—
304C2×D4—622no—
305C2×D4—622no—
306C2×D4—621no—
307C2×D4—621no—
308C2×D4—621no—
309C2×D4—1244no—
310C2×D4—1241no—
311C2×D4—1241no—
312C2×D4—1241no—
313C2×D4C22×D3111yescharacteristic
314C2×D4C22×D3111yescharacteristic
315C22×C4—21no—
316C22×C4—311no—
317C22×C4—311no—
318C22×C4—621no—
319C22×C4—621no—
320SD16—22no—
321SD16—622no—
322SD16—1241no—
323D8—22no—
324D8—314no—
325D8—622no—
326D8—622no—
327D8—622no—
328D8—1241no—
329M16—622no—
330C2×C8—311no—
331C2×C8—311no—
332C2×C8—622no—
333C2×C8C22×D3111yescharacteristic
334C2×C8C22×D3111yescharacteristic
335C4⋊C4—311no—
336C4⋊C4—311no—
337C4⋊C4—621no—
338C4⋊C4C22×D3111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—311no—
343C22⋊C4—311no—
344C22⋊C4—621no—
345C22⋊C4—621no—
346C22⋊C4—621no—
347C22⋊C4—621no—
348C22⋊C4—621no—
349C42C22×D3111yescharacteristic
350C23—21no—
351C23—21no—
352C23—311no—
353C23—311no—
354C23—621no—
355C23—621no—
356C23—621no—
357C23—1242no—
358C23—1242no—
359C23—1242no—
360C23—1242no—
361C23—1242no—
362C23—1242no—
363C23—2481no—
364C23—2481no—
365C23—2481no—
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—312no—
372D4—622no—
373D4—621no—
374D4—621no—
375D4—621no—
376D4—621no—
377D4—621no—
378D4—1242no—
379D4—1242no—
380D4—1242no—
381D4—1242no—
382D4—1242no—
383D4—1242no—
384D4—1242no—
385D4—2481no—
386D4—2481no—
387D4D4×D3112yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—311no—
392C2×C4—311no—
393C2×C4—621no—
394C2×C4—621no—
395C2×C4—621no—
396C2×C4—621no—
397C2×C4—621no—
398C2×C4—1242no—
399C2×C4—2481no—
400C2×C4D4×D3111yescharacteristic
401C2×C4D4×D3111yescharacteristic
402C2×C4C23×D3111yescharacteristic
403C8—21no—
404C8—21no—
405C8—312no—
406C8—621no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—312no—
414C22—621no—
415C22—621no—
416C22—621no—
417C22—622no—
418C22—621no—
419C22—621no—
420C22—1241no—
421C22—1241no—
422C22—1241no—
423C22—1241no—
424C22—1241no—
425C22—1241no—
426C22—1241no—
427C22—1241no—
428C22—2484no—
429C22—2484no—
430C22—2484no—
431C22C2×D4×D3111yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—621no—
437C4—621no—
438C4—1241no—
439C4C2×D4×D3111yescharacteristic
440C4#1175856611yescharacteristic
441C2—42no—
442C2—81no—
443C2—622no—
444C2—1241no—
445C2—1242no—
446C2—2481no—
447C2—2481no—
448C2#1452168311yescharacteristic
449C2AGL(1,ℤ/8)×D3111yescharacteristic
450C2#1452185111yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#463C2
socleC22×C31C22×D4
hypercenter#463D31
hypocenterC31#3264
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#11758491C2semidirect
2. C2×D8×D31C2semidirect
3. #14521692C2semidirect
4. D4×D124C2semidirect
5. #14521439C2semidirect
6. #14521301C2semidirect
7. #14521281C2semidirect
8. #14521245C2semidirect
9. #14521198C2semidirect
10. #14521197C2semidirect
11. #14521184C2semidirect
12. #14521181C2semidirect
13. #14521159C2semidirect
14. #14521119C2semidirect
15. C31×#463C2semidirect
16. C2×#92499C22semidirect
18. C2×D4×D31C22semidirect
19. C2×D4×D31C22semidirect
21. C2×SD16×C31C22semidirect
23. C31×#124C22semidirect
24. C4×D4×C31C22semidirect
25. #11758523C22semidirect
26. C2×#92481C22semidirect
27. C2×#92481C22semidirect
28. #11758515C22semidirect
30. C2×#92479C22semidirect
31. C2×#92479C22semidirect
35. C2×D248C22semidirect
36. C2×D248C22semidirect
37. C2×#92469C22semidirect
38. C2×C8×D31C22semidirect
40. #11758472C22semidirect
47. #11758451C22semidirect
49. C4×D124C22semidirect
50. C31×C4⋊C8C22semidirect
51. C31×#103C22semidirect
52. C31×#102C22semidirect
53. #11758420C22semidirect
54. #11758409C22semidirect
55. #11758409C22semidirect
56. (C2×D31)⋊C8C22semidirect
57. (C2×D31)⋊C8C22semidirect
58. #11758397C22semidirect
59. #11758396C22semidirect
60. #11758391C22semidirect
108. #92479D4semidirect
170. D124C2×D4semidirect
180. C31⋊C8C2×D4semidirect
206. C2×D31AGL(1,ℤ/8)semidirect
223. #463D31semidirect
249. C2×SD16C2×D31semidirect
260. #124C2×D31semidirect
273. C4×D4C2×D31semidirect
276. C4⋊C8C2×D31semidirect
277. #103C2×D31semidirect
282. #102C2×D31semidirect
286. C31#3264semidirect

Maximal subgroups

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

subgroupindex
C2×#117584912
C2×D8×D312
#145216922
D4×D1242
#145214392
#145213012
#145212812
#145212452
#145211982
#145211972
#145211842
#145211812
#145211592
#145211192
C31×#4632
#326431