Finite Group Databasebeta

← #16857195

Subgroups — #16857195

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
2C22×#92479C211yesmaximal, characteristic, complemented
3#14521690C211yesmaximal, characteristic, complemented
4D4×C22⋊D31C211yesmaximal, characteristic, complemented
5#14521435C211yesmaximal, characteristic, complemented
6#14521301C211yesmaximal, characteristic, complemented
7#14521280C211yesmaximal, characteristic, complemented
8#14521266C211yesmaximal, characteristic, complemented
9#14521243C211yesmaximal, characteristic, complemented
10#14521193C211yesmaximal, characteristic, complemented
11#14521185C211yesmaximal, characteristic, complemented
12#14521184C211yesmaximal, characteristic, complemented
13#14521158C211yesmaximal, characteristic, complemented
14#14521117C211yesmaximal, characteristic, complemented
15C31×#450C211yesmaximal, characteristic, complemented
16C22×D4×C31C2211yescharacteristic
17C22×C22⋊D31—21no—
18C2×#92499C2211yescharacteristic, complemented
19C2×D4×D31C2211yescharacteristic, complemented
20C22×D124C2211yescharacteristic, complemented
21C2×SD16×C31C2211yescharacteristic, complemented
22C2×SD16×C31C2211yescharacteristic, complemented
23C31×C22⋊Q8C2211yescharacteristic, complemented
24#11758530—21no—
25#11758524C2211yescharacteristic, complemented
26C2×#92481C2211yescharacteristic, complemented
27#11758517C2211yescharacteristic, complemented
28#11758516—21no—
29#11758515C2211yescharacteristic, complemented
30#11758514—21no—
31D4×Dic31C2211yescharacteristic, complemented
32C2×#92479—24no—
33C2×#92479C2211yescharacteristic
34C2×#92479C2211yescharacteristic
35C2×#92479C2211yescharacteristic, complemented
36C2×#92479C2211yescharacteristic, complemented
37#11758507C2211yescharacteristic
38C4×C22⋊D31C2211yescharacteristic
39C22×C31⋊C8C2211yescharacteristic, complemented
40#11758491—24no—
41C2×D248C2211yescharacteristic, complemented
42C2×D248C2211yescharacteristic, complemented
43C2×#92469C2211yescharacteristic, complemented
44#11758472—21no—
45#11758472C2211yescharacteristic, complemented
46#11758471—21no—
47#11758470C2211yescharacteristic, complemented
48#11758459—21no—
49C22⋊C4×D31—21no—
50C31×#102C2211yescharacteristic, complemented
51C31×#102C2211yescharacteristic, complemented
52C31×C22⋊C8C2211yescharacteristic, complemented
53#11758423C2211yescharacteristic, complemented
54#11758409C2211yescharacteristic, complemented
55#11758409C2211yescharacteristic, complemented
56(C2×D31)⋊C8C2211yescharacteristic, complemented
57#11758403C2211yescharacteristic, complemented
58Dic31⋊C8C2211yescharacteristic, complemented
59#11758397C2211yescharacteristic, complemented
60#11758396C2211yescharacteristic, complemented
61C24×C31—21no—
62C23×D31—21no—
63C23×D31—21no—
64C2×Q8×C31C2311yescharacteristic
65C2×D4×C31—22no—
66C2×D4×C31—21no—
67C2×D4×C31—41no—
68C2×D4×C31C2311yescharacteristic
69C2×D4×C31C2311yescharacteristic
70C22×C4×C31C2311yescharacteristic
71C2×C22⋊D31—21no—
72C2×C22⋊D31—21no—
73C2×C22⋊D31—21no—
74C2×C22⋊D31—21no—
75C2×C22⋊D31—21no—
76C2×C22⋊D31—44no—
77C2×C22⋊D31D411yescharacteristic
78C22×Dic31—21no—
79#92499—22no—
80#92499—41no—
81D4×D31—22no—
82D4×D31—41no—
83C2×D124—22no—
84C2×D124—21no—
85C2×D124—21no—
86C2×D124—21no—
87C2×D124—41no—
88C2×D124C2311yescharacteristic
89C2×D124C2311yescharacteristic
90C2×D124C2311yescharacteristic
91C2×C4×D31—21no—
92C2×C4×D31—21no—
93C2×C4×D31C2311yescharacteristic
94SD16×C31—22no—
95SD16×C31—41no—
96C2×C8×C31C2311yescharacteristic
97C2×C8×C31C2311yescharacteristic
98C4⋊C4×C31—21no—
99C4⋊C4×C31C2311yescharacteristic
100C22⋊C4×C31—21no—
101C22⋊Dic31—21no—
102C22⋊Dic31—21no—
103C22⋊Dic31D411yescharacteristic
104#92481—22no—
105#92479—22no—
106#92479—22no—
107#92479—22no—
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)⋊C4—21no—
114(C2×D31)⋊C4D411yescharacteristic
115C4⋊Dic31C2311yescharacteristic
116Dic31⋊C4D411yescharacteristic
117C4×Dic31C2311yescharacteristic
118C2×C31⋊C8—22no—
119C2×C31⋊C8C2311yescharacteristic
120C2×C31⋊C8C2311yescharacteristic
121D248—22no—
122D248—41no—
123#92469—22no—
124C23×C31—21no—
125C23×C31—21no—
126C23×C31—42no—
127C23×C31—42no—
128C23×C31—81no—
129C23×C31C2×D411yescharacteristic
130C22×D31—21no—
131C22×D31—21no—
132C22×D31—21no—
133C22×D31—42no—
134C22×D31—42no—
135C22×D31—42no—
136C22×D31—42no—
137C22×D31—81no—
138C22×D31—81no—
139C22×D31C2×D411yescharacteristic
140Q8×C31—21no—
141Q8×C31—21no—
142D4×C31—21no—
143D4×C31—21no—
144D4×C31—21no—
145D4×C31—42no—
146D4×C31C2×D412yes—
147C2×C4×C31—21no—
148C2×C4×C31—21no—
149C2×C4×C31—21no—
150C2×C4×C31—21no—
151C2×C4×C31C2×D411yescharacteristic
152C2×C4×C31C2411yescharacteristic
153C22⋊D31—22no—
154C22⋊D31—42no—
155C22⋊D31—42no—
156C22⋊D31—42no—
157C22⋊D31—42no—
158C22⋊D31—81no—
159C2×Dic31—21no—
160C2×Dic31—42no—
161C2×Dic31C2×D411yescharacteristic
162C2×Dic31C2×D411yescharacteristic
163C2×Dic31C2×D411yescharacteristic
164D124—21no—
165D124—21no—
166D124—21no—
167D124—21no—
168D124—21no—
169D124—42no—
170D124—42no—
171D124—81no—
172D124C2×D412yescomplemented
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—21no—
183C22×C31—21no—
184C22×C31—22no—
185C22×C31—41no—
186C22×C31—41no—
187C22×C31—41no—
188C22×C31—41no—
189C22×C31—41no—
190C22×C31—84no—
191C22×C31AGL(1,ℤ/8)12yescomplemented
192C22×C31C22×D411yescharacteristic
193C2×D31—21no—
194C2×D31—22no—
195C2×D31—21no—
196C2×D31—21no—
197C2×D31—41no—
198C2×D31—41no—
199C2×D31—41no—
200C2×D31—41no—
201C2×D31—41no—
202C2×D31—41no—
203C2×D31—41no—
204C2×D31—41no—
205C2×D31—84no—
206C2×D31—84no—
207C4×C31—21no—
208C4×C31—41no—
209C4×C31—41no—
210C4×C31C22×D411yescharacteristic
211C4×C31H(5,𝔽2)11yescharacteristic
212Dic31—21no—
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#463—311no—
224#459—311no—
225#450D3111yescharacteristic, complemented
226#449—311no—
227#447—311no—
228#447—311no—
229#442—311no—
230#435—311no—
231C2×C31—22no—
232C2×C31—42no—
233C2×C31—81no—
234C2×C31D4211yescharacteristic
235C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
236C2×C31#57611yescharacteristic
237D31—41no—
238D31—42no—
239D31—81no—
240D31—81no—
241C2×#41—311no—
242C22×D4—311no—
243C22×D4—311no—
244C22×D4—621no—
245C22×D4C2×D3111yescharacteristic
246AGL(1,ℤ/8)—624no—
247C2×SD16—311no—
248C2×SD16C2×D3111yescharacteristic, complemented
249C2×SD16C2×D3111yescharacteristic, complemented
250C2×D8—311no—
251C2×D8—311no—
252C2×D8—311no—
253C2×D8—311no—
254C2×D8—311no—
255C2×D8—311no—
256C2×D8—624no—
257C2×M16—311no—
258C22×C8—311no—
259#127—311no—
260#124—311no—
261#123—621no—
262C22⋊Q8C2×D3111yescharacteristic, complemented
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×D4—311no—
274C2×C22⋊C4—621no—
275#106—311no—
276C4⋊C8—311no—
277#103—311no—
278#102—311no—
279#102—311no—
280#102—311no—
281#102C2×D3111yescharacteristic, complemented
282#102C2×D3111yescharacteristic, complemented
283C22⋊C8—311no—
284C22⋊C8C2×D3111yescharacteristic, complemented
285#97—311no—
286C31#326411yescharacteristic, complemented, Sylow
287C24—21no—
288C24—621no—
289C24—621no—
290#41—622no—
291#41—1241no—
292C2×Q8C22×D3111yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—311no—
297C2×D4—311no—
298C2×D4—311no—
299C2×D4—311no—
300C2×D4—622no—
301C2×D4—621no—
302C2×D4—621no—
303C2×D4—621no—
304C2×D4—621no—
305C2×D4—621no—
306C2×D4—622no—
307C2×D4—621no—
308C2×D4—621no—
309C2×D4—621no—
310C2×D4—1241no—
311C2×D4—1241no—
312C2×D4—1244no—
313C2×D4C22×D3111yescharacteristic
314C2×D4C22×D3111yescharacteristic
315C22×C4—311no—
316C22×C4—621no—
317C22×C4—621no—
318C22×C4—621no—
319C22×C4C22×D3111yescharacteristic
320SD16—22no—
321SD16—41no—
322SD16—622no—
323D8—314no—
324D8—622no—
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—21no—
336C4⋊C4—311no—
337C4⋊C4—311no—
338C4⋊C4C22×D3111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—311no—
341C22⋊C4—311no—
342C22⋊C4—621no—
343C22⋊C4—621no—
344C22⋊C4—621no—
345C22⋊C4—621no—
346C22⋊C4—621no—
347C22⋊C4—621no—
348C22⋊C4—621no—
349C42—311no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—311no—
356C23—621no—
357C23—621no—
358C23—621no—
359C23—1242no—
360C23—1242no—
361C23—1242no—
362C23—1242no—
363C23—2481no—
364C23—2481no—
365C23D4×D3111yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—42no—
372D4—312no—
373D4—622no—
374D4—621no—
375D4—621no—
376D4—621no—
377D4—621no—
378D4—621no—
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—21no—
391C2×C4—21no—
392C2×C4—311no—
393C2×C4—311no—
394C2×C4—311no—
395C2×C4—621no—
396C2×C4—621no—
397C2×C4—621no—
398C2×C4—1242no—
399C2×C4—1242no—
400C2×C4—2481no—
401C2×C4D4×D3111yescharacteristic
402C2×C4C23×D3111yescharacteristic
403C8—21no—
404C8—21no—
405C8—312no—
406C8—621no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—84no—
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#1175849112yescomplemented
431C22C2×D4×D3111yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—621no—
436C4—621no—
437C4—621no—
438C4—1241no—
439C4C2×D4×D3111yescharacteristic
440C4#1175856111yescharacteristic
441C2—22no—
442C2—42no—
443C2—81no—
444C2—1241no—
445C2—1242no—
446C2—2481no—
447C2—2481no—
448C2#1452168111yescharacteristic
449C2C2×#1175849111yescharacteristic
450C2#1452185311yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#450C2
socleC22×C31C22×D4
hypercenter#450D31
hypocenterC31#3264
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#11758491C2semidirect
2. C22×#92479C2semidirect
3. #14521690C2semidirect
4. D4×C22⋊D31C2semidirect
5. #14521435C2semidirect
6. #14521301C2semidirect
7. #14521280C2semidirect
8. #14521266C2semidirect
9. #14521243C2semidirect
10. #14521193C2semidirect
11. #14521185C2semidirect
12. #14521184C2semidirect
13. #14521158C2semidirect
14. #14521117C2semidirect
15. C31×#450C2semidirect
18. C2×#92499C22semidirect
19. C2×D4×D31C22semidirect
20. C22×D124C22semidirect
21. C2×SD16×C31C22semidirect
22. C2×SD16×C31C22semidirect
23. C31×C22⋊Q8C22semidirect
25. #11758524C22semidirect
26. C2×#92481C22semidirect
27. #11758517C22semidirect
29. #11758515C22semidirect
31. D4×Dic31C22semidirect
35. C2×#92479C22semidirect
36. C2×#92479C22semidirect
39. C22×C31⋊C8C22semidirect
41. C2×D248C22semidirect
42. C2×D248C22semidirect
43. C2×#92469C22semidirect
45. #11758472C22semidirect
47. #11758470C22semidirect
50. C31×#102C22semidirect
51. C31×#102C22semidirect
52. C31×C22⋊C8C22semidirect
53. #11758423C22semidirect
54. #11758409C22semidirect
55. #11758409C22semidirect
56. (C2×D31)⋊C8C22semidirect
57. #11758403C22semidirect
58. Dic31⋊C8C22semidirect
59. #11758397C22semidirect
60. #11758396C22semidirect
108. #92479D4semidirect
172. D124C2×D4semidirect
180. C31⋊C8C2×D4semidirect
191. C22×C31AGL(1,ℤ/8)semidirect
225. #450D31semidirect
248. C2×SD16C2×D31semidirect
249. C2×SD16C2×D31semidirect
262. C22⋊Q8C2×D31semidirect
281. #102C2×D31semidirect
282. #102C2×D31semidirect
284. C22⋊C8C2×D31semidirect
286. C31#3264semidirect
430. C22#11758491semidirect

Maximal subgroups

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

subgroupindex
C2×#117584912
C22×#924792
#145216902
D4×C22⋊D312
#145214352
#145213012
#145212802
#145212662
#145212432
#145211932
#145211852
#145211842
#145211582
#145211172
C31×#4502
#326431