Finite Group Databasebeta

← #16857196

Subgroups — #16857196

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×#11758485C211yesmaximal, characteristic, complemented
2C22×D248C211yesmaximal, characteristic, complemented
3#14521695C211yesmaximal, characteristic, complemented
4D4×D124C211yesmaximal, characteristic, complemented
5#14521438C211yesmaximal, characteristic, complemented
6#14521302C211yesmaximal, characteristic, complemented
7#14521278C211yesmaximal, characteristic, complemented
8#14521265C211yesmaximal, characteristic, complemented
9#14521245C211yesmaximal, characteristic, complemented
10#14521202C211yesmaximal, characteristic, complemented
11#14521196C211yesmaximal, characteristic, complemented
12#14521185C211yesmaximal, characteristic, complemented
13#14521185C211yesmaximal, characteristic, complemented
14#14521162C211yesmaximal, characteristic, complemented
15C31×#435C211yesmaximal, characteristic, complemented
16C2×D4×D31—21no—
17C2×#92495C2211yescharacteristic, complemented
18C22×D124C2211yescharacteristic
19C22×D124C2211yescharacteristic, complemented
20C22×D124C2211yescharacteristic, complemented
21C2×M16×C31C2211yescharacteristic, complemented
22C22×C8×C31C2211yescharacteristic, complemented
23C4×D4×C31C2211yescharacteristic
24#11758514—21no—
25#11758507C2211yescharacteristic
26#11758507C2211yescharacteristic, complemented
27#11758507C2211yescharacteristic, complemented
28#11758501C2211yescharacteristic, complemented
29#11758485—24no—
30C2×D248—24no—
31C2×D248C2211yescharacteristic
32C2×D248C2211yescharacteristic
33C2×D248C2211yescharacteristic, complemented
34C2×D248C2211yescharacteristic, complemented
35C2×D248C2211yescharacteristic, complemented
36C2×D248C2211yescharacteristic, complemented
37C2×#92470C2211yescharacteristic, complemented
38C2×#92470C2211yescharacteristic, complemented
39C2×#92470C2211yescharacteristic, complemented
40#11758472—21no—
41#11758471—21no—
42#11758461—21no—
43#11758459—21no—
44#11758459—21no—
45C22⋊C4×D31—21no—
46#11758452C2211yescharacteristic, complemented
47#11758451C2211yescharacteristic, complemented
48C4×D124C2211yescharacteristic, complemented
49C4×D124C2211yescharacteristic, complemented
50C31×C4⋊C8C2211yescharacteristic, complemented
51C31×C22⋊C8C2211yescharacteristic, complemented
52C31×C22⋊C8C2211yescharacteristic, complemented
53C31×#97C2211yescharacteristic, complemented
54#11758409C2211yescharacteristic, complemented
55#11758409C2211yescharacteristic, complemented
56#11758409C2211yescharacteristic, complemented
57#11758409C2211yescharacteristic, complemented
58#11758409C2211yescharacteristic, complemented
59#11758405C2211yescharacteristic, complemented
60#11758404C2211yescharacteristic, complemented
61C23×D31—21no—
62C23×D31—21no—
63C23×D31—21no—
64C2×D4×C31D411yescharacteristic
65C22×C4×C31C2311yescharacteristic
66C22×C4×C31C2311yescharacteristic
67C2×C22⋊D31—21no—
68C2×C22⋊D31—21no—
69C2×C22⋊D31—21no—
70C2×C22⋊D31—21no—
71D4×D31—44no—
72#92495—22no—
73#92495—41no—
74C2×D124—22no—
75C2×D124—22no—
76C2×D124—22no—
77C2×D124—21no—
78C2×D124—21no—
79C2×D124—21no—
80C2×D124—21no—
81C2×D124—21no—
82C2×D124—41no—
83C2×D124—41no—
84C2×D124—41no—
85C2×D124C2311yescharacteristic
86C2×D124C2311yescharacteristic
87C2×D124C2311yescharacteristic
88C2×D124C2311yescharacteristic
89C2×D124C2311yescharacteristic
90C2×C4×D31—21no—
91C2×C4×D31—21no—
92C2×C4×D31—21no—
93C2×Dic62C2311yescharacteristic
94M16×C31—22no—
95C2×C8×C31—22no—
96C2×C8×C31C2311yescharacteristic
97C2×C8×C31C2311yescharacteristic
98C2×C8×C31C2311yescharacteristic
99C2×C8×C31C2311yescharacteristic
100C4⋊C4×C31D411yescharacteristic
101C22⋊C4×C31D411yescharacteristic
102C22⋊C4×C31D411yescharacteristic
103C42×C31C2311yescharacteristic
104C22⋊Dic31—21no—
105(C2×D31)⋊C4—21no—
106(C2×D31)⋊C4—21no—
107(C2×D31)⋊C4—21no—
108(C2×D31)⋊C4—21no—
109(C2×D31)⋊C4—21no—
110(C2×D31)⋊C4—21no—
111(C2×D31)⋊C4—21no—
112C4⋊Dic31C2311yescharacteristic
113C4⋊Dic31C2311yescharacteristic
114Dic31⋊C4—21no—
115D248—22no—
116D248—22no—
117D248—22no—
118D248—22no—
119D248—41no—
120D248D414yescomplemented
121#92470—22no—
122#92470—22no—
123#92470—41no—
124C23×C31C2×D411yescharacteristic
125C23×C31C2×D411yescharacteristic
126C22×D31—21no—
127C22×D31—21no—
128C22×D31—21no—
129C22×D31—21no—
130C22×D31—21no—
131C22×D31—42no—
132C22×D31—42no—
133C22×D31—42no—
134C22×D31—42no—
135C22×D31—42no—
136C22×D31—42no—
137C22×D31—81no—
138C22×D31—81no—
139C22×D31—81no—
140D4×C31—22no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—21no—
145C2×C4×C31C2×D411yescharacteristic
146C2×C4×C31C2×D411yescharacteristic
147C2×C4×C31C2×D411yescharacteristic
148C2×C4×C31C2×D411yescharacteristic
149C2×C4×C31C2411yescharacteristic
150C22⋊D31—42no—
151C22⋊D31—42no—
152C22⋊D31—42no—
153C22⋊D31—81no—
154C2×Dic31—21no—
155C2×Dic31—21no—
156C2×Dic31—21no—
157D124—21no—
158D124—21no—
159D124—21no—
160D124—21no—
161D124—21no—
162D124—21no—
163D124—21no—
164D124—21no—
165D124—42no—
166D124—42no—
167D124—42no—
168D124—42no—
169D124—81no—
170D124C2×D412yes—
171D124C2×D412yescomplemented
172C4×D31—42no—
173C4×D31—42no—
174C4×D31—81no—
175Dic62—21no—
176Dic62—21no—
177C8×C31—21no—
178C8×C31—21no—
179C8×C31—21no—
180C8×C31C2×D412yescomplemented
181#3264—311nomaximal, Sylow
182C22×C31—21no—
183C22×C31—21no—
184C22×C31—21no—
185C22×C31—21no—
186C22×C31—21no—
187C22×C31AGL(1,ℤ/8)12yescomplemented
188C22×C31C22×D411yescharacteristic
189C2×D31—22no—
190C2×D31—22no—
191C2×D31—41no—
192C2×D31—41no—
193C2×D31—41no—
194C2×D31—41no—
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—41no—
204C2×D31—84no—
205C2×D31—84no—
206C2×D31—84no—
207C4×C31—21no—
208C4×C31—21no—
209C4×C31—21no—
210C4×C31—21no—
211C4×C31C22×D411yescharacteristic
212C4×C31H(5,𝔽2)11yescharacteristic
213Dic31—41no—
214Dic31—41no—
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#463—311no—
224#459—311no—
225#450—311no—
226#449—311no—
227#447—311no—
228#447—311no—
229#442—311no—
230#435D3111yescharacteristic, complemented
231C2×C31—22no—
232C2×C31—41no—
233C2×C31D4211yescharacteristic
234C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
235C2×C31#57611yescharacteristic
236D31—42no—
237D31—42no—
238D31—81no—
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×SD16—311no—
250C2×D8—311no—
251C2×D8—311no—
252C2×D8—311no—
253C2×D8—311no—
254C2×D8—311no—
255C2×D8—311no—
256C2×D8—624no—
257C2×M16C2×D3111yescharacteristic, complemented
258C22×C8C2×D3111yescharacteristic, complemented
259#127—311no—
260#124—311no—
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
274C2×C22⋊C4—621no—
275#106—311no—
276C4⋊C8C2×D3111yescharacteristic, complemented
277#103—311no—
278#102—311no—
279#102—311no—
280#102—311no—
281#102—311no—
282#102—311no—
283C22⋊C8C2×D3111yescharacteristic, complemented
284C22⋊C8C2×D3111yescharacteristic, complemented
285#97C2×D3111yescharacteristic, complemented
286C31#326411yescharacteristic, complemented, Sylow
287C24—621no—
288C24—621no—
289C24—621no—
290#41—622no—
291#41—1241no—
292C2×Q8—311no—
293C2×D4—311no—
294C2×D4—311no—
295C2×D4—311no—
296C2×D4—311no—
297C2×D4—311no—
298C2×D4—622no—
299C2×D4—622no—
300C2×D4—622no—
301C2×D4—621no—
302C2×D4—621no—
303C2×D4—621no—
304C2×D4—621no—
305C2×D4—621no—
306C2×D4—621no—
307C2×D4—621no—
308C2×D4—621no—
309C2×D4—621no—
310C2×D4—1241no—
311C2×D4—1241no—
312C2×D4—1241no—
313C2×D4—1244no—
314C2×D4D12411yescharacteristic
315C22×C4—621no—
316C22×C4—621no—
317C22×C4—621no—
318C22×C4C22×D3111yescharacteristic
319C22×C4C22×D3111yescharacteristic
320SD16—622no—
321SD16—622no—
322SD16—1241no—
323D8—314no—
324D8—622no—
325D8—622no—
326D8—622no—
327D8—622no—
328D8—1241no—
329M16—22no—
330C2×C8—22no—
331C2×C8C22×D3111yescharacteristic
332C2×C8C22×D3111yescharacteristic
333C2×C8C22×D3111yescharacteristic
334C2×C8C22×D3111yescharacteristic
335C4⋊C4—311no—
336C4⋊C4—311no—
337C4⋊C4—621no—
338C4⋊C4D12411yescharacteristic
339C22⋊C4—621no—
340C22⋊C4—621no—
341C22⋊C4—621no—
342C22⋊C4—621no—
343C22⋊C4—621no—
344C22⋊C4—621no—
345C22⋊C4—621no—
346C22⋊C4—621no—
347C22⋊C4D12411yescharacteristic
348C22⋊C4D12411yescharacteristic
349C42C22×D3111yescharacteristic
350C23—621no—
351C23—621no—
352C23—621no—
353C23—621no—
354C23—621no—
355C23—1242no—
356C23—1242no—
357C23—1242no—
358C23—1242no—
359C23—1242no—
360C23—1242no—
361C23—2481no—
362C23—2481no—
363C23—2481no—
364C23C2×D12411yescharacteristic
365C23C2×D12411yescharacteristic
366Q8—621no—
367Q8—621no—
368D4—22no—
369D4—312no—
370D4—312no—
371D4—621no—
372D4—621no—
373D4—621no—
374D4—621no—
375D4—621no—
376D4—621no—
377D4—621no—
378D4—621no—
379D4—1242no—
380D4—1242no—
381D4—1242no—
382D4—1242no—
383D4—1242no—
384D4—1242no—
385D4—1242no—
386D4—2481no—
387D4—2481no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—621no—
393C2×C4—621no—
394C2×C4—621no—
395C2×C4—1242no—
396C2×C4—1242no—
397C2×C4—2481no—
398C2×C4C2×D12411yescharacteristic
399C2×C4C2×D12411yescharacteristic
400C2×C4C2×D12411yescharacteristic
401C2×C4C2×D12411yescharacteristic
402C2×C4C23×D3111yescharacteristic
403C8—21no—
404C8—21no—
405C8—21no—
406C8D4×D3112yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—622no—
413C22—622no—
414C22—1241no—
415C22—1241no—
416C22—1241no—
417C22—1241no—
418C22—1241no—
419C22—1241no—
420C22—1241no—
421C22—1241no—
422C22—1241no—
423C22—1241no—
424C22—1241no—
425C22—1241no—
426C22—1241no—
427C22—2484no—
428C22—2484no—
429C22—2484no—
430C22#1175848512yescomplemented
431C22C22×D12411yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—1241no—
437C4—1241no—
438C4—1241no—
439C4C2×D4×D3111yescharacteristic
440C4#1175856611yescharacteristic
441C2—22no—
442C2—41no—
443C2—1242no—
444C2—1242no—
445C2—2481no—
446C2—2481no—
447C2—2481no—
448C2D4×D12411yescharacteristic
449C2C2×#1175848511yescharacteristic
450C2#1452185411yescharacteristic
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#435C2
socleC22×C31C22×D4
hypercenter#435D31
hypocenterC31#3264
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#11758485C2semidirect
2. C22×D248C2semidirect
3. #14521695C2semidirect
4. D4×D124C2semidirect
5. #14521438C2semidirect
6. #14521302C2semidirect
7. #14521278C2semidirect
8. #14521265C2semidirect
9. #14521245C2semidirect
10. #14521202C2semidirect
11. #14521196C2semidirect
12. #14521185C2semidirect
13. #14521185C2semidirect
14. #14521162C2semidirect
15. C31×#435C2semidirect
17. C2×#92495C22semidirect
19. C22×D124C22semidirect
20. C22×D124C22semidirect
21. C2×M16×C31C22semidirect
22. C22×C8×C31C22semidirect
26. #11758507C22semidirect
27. #11758507C22semidirect
28. #11758501C22semidirect
33. C2×D248C22semidirect
34. C2×D248C22semidirect
35. C2×D248C22semidirect
36. C2×D248C22semidirect
37. C2×#92470C22semidirect
38. C2×#92470C22semidirect
39. C2×#92470C22semidirect
46. #11758452C22semidirect
47. #11758451C22semidirect
48. C4×D124C22semidirect
49. C4×D124C22semidirect
50. C31×C4⋊C8C22semidirect
51. C31×C22⋊C8C22semidirect
52. C31×C22⋊C8C22semidirect
53. C31×#97C22semidirect
54. #11758409C22semidirect
55. #11758409C22semidirect
56. #11758409C22semidirect
57. #11758409C22semidirect
58. #11758409C22semidirect
59. #11758405C22semidirect
60. #11758404C22semidirect
120. D248D4semidirect
171. D124C2×D4semidirect
180. C8×C31C2×D4semidirect
187. C22×C31AGL(1,ℤ/8)semidirect
230. #435D31semidirect
257. C2×M16C2×D31semidirect
258. C22×C8C2×D31semidirect
276. C4⋊C8C2×D31semidirect
283. C22⋊C8C2×D31semidirect
284. C22⋊C8C2×D31semidirect
285. #97C2×D31semidirect
286. C31#3264semidirect
406. C8D4×D31semidirect
430. C22#11758485semidirect

Maximal subgroups

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

subgroupindex
C2×#117584852
C22×D2482
#145216952
D4×D1242
#145214382
#145213022
#145212782
#145212652
#145212452
#145212022
#145211962
#145211852
#145211622
C31×#4352
#326431