Finite Group Databasebeta

← #16857187

Subgroups — #16857187

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×#11758509C211yesmaximal, characteristic, complemented
2C22×#92479C211yesmaximal, characteristic, complemented
3#14521695C211yesmaximal, characteristic, complemented
4#14521436C211yesmaximal, characteristic, complemented
5#14521300C211yesmaximal, characteristic, complemented
6#14521276C211yesmaximal, characteristic, complemented
7#14521264C211yesmaximal, characteristic, complemented
8#14521241C211yesmaximal, characteristic, complemented
9#14521200C211yesmaximal, characteristic, complemented
10#14521192C211yesmaximal, characteristic, complemented
11#14521183C211yesmaximal, characteristic, complemented
12#14521182C211yesmaximal, characteristic, complemented
13#14521159C211yesmaximal, characteristic, complemented
14#14521121C211yesmaximal, characteristic, complemented
15D42×C31C211yesmaximal, characteristic, complemented
16C22×D4×C31—21no—
17C22×D4×C31C2211yescharacteristic
18C22×D4×C31C2211yescharacteristic, complemented
19C2×#92495C2211yescharacteristic, complemented
20C22×D124C2211yescharacteristic, complemented
21C31×#127C2211yescharacteristic, complemented
22C31×#121—21no—
23C31×#121C2211yescharacteristic
24C31×#121C2211yescharacteristic, complemented
25C31×#120—21no—
26C31×#120—21no—
27C4×D4×C31C2211yescharacteristic
28C4×D4×C31C2211yescharacteristic, complemented
29#11758514—21no—
30C2×#92480C2211yescharacteristic, complemented
31C2×#92480C2211yescharacteristic, complemented
32C2×#92480C2211yescharacteristic, complemented
33#11758509—24no—
34C2×#92479—24no—
35C2×#92479C2211yescharacteristic
36C2×#92479C2211yescharacteristic
37C2×#92479C2211yescharacteristic, complemented
38C2×#92479C2211yescharacteristic, complemented
39C2×#92479C2211yescharacteristic, complemented
40C2×#92479C2211yescharacteristic, complemented
41#11758507C2211yescharacteristic, complemented
42#11758501C2211yescharacteristic, complemented
43C2×#92474C2211yescharacteristic, complemented
44C22×C31⋊C8C2211yescharacteristic, complemented
45#11758471—21no—
46#11758461—21no—
47C22⋊C4×D31—21no—
48#11758452C2211yescharacteristic, complemented
49C4×D124C2211yescharacteristic, complemented
50#11758420C2211yescharacteristic, complemented
51#11758420C2211yescharacteristic, complemented
52#11758420C2211yescharacteristic, complemented
53C22⋊(C31⋊C8)C2211yescharacteristic, complemented
54C22⋊(C31⋊C8)C2211yescharacteristic, complemented
55#11758398C2211yescharacteristic, complemented
56#11758397C2211yescharacteristic, complemented
57#11758397C2211yescharacteristic, complemented
58#11758396C2211yescharacteristic, complemented
59C4⋊(C31⋊C8)C2211yescharacteristic, complemented
60#11758391C2211yescharacteristic, complemented
61C24×C31—21no—
62C24×C31—21no—
63C23×D31—21no—
64C2×D4×C31—22no—
65C2×D4×C31—22no—
66C2×D4×C31—21no—
67C2×D4×C31—21no—
68C2×D4×C31—21no—
69C2×D4×C31—21no—
70C2×D4×C31—21no—
71C2×D4×C31—21no—
72C2×D4×C31—41no—
73C2×D4×C31—41no—
74C2×D4×C31—44no—
75C2×D4×C31D411yescharacteristic
76C2×D4×C31C2311yescharacteristic
77C2×D4×C31C2311yescharacteristic
78C2×D4×C31C2311yescharacteristic
79C22×C4×C31—21no—
80C22×C4×C31C2311yescharacteristic
81C22×C4×C31C2311yescharacteristic
82C2×C22⋊D31—21no—
83C2×C22⋊D31—21no—
84#92495—22no—
85#92495—41no—
86C2×D124—22no—
87C2×D124—21no—
88C2×D124—41no—
89C2×D124C2311yescharacteristic
90C2×D124C2311yescharacteristic
91C2×C4×D31—21no—
92C2×C4×D31—21no—
93C2×Dic62C2311yescharacteristic
94C4⋊C4×C31D411yescharacteristic
95C4⋊C4×C31C2311yescharacteristic
96C22⋊C4×C31—21no—
97C22⋊C4×C31—21no—
98C22⋊C4×C31—21no—
99C22⋊C4×C31D411yescharacteristic
100C22⋊C4×C31D411yescharacteristic
101C42×C31C2311yescharacteristic
102C22⋊Dic31—21no—
103#92480—22no—
104#92480—22no—
105#92480—41no—
106#92479—22no—
107#92479—22no—
108#92479—22no—
109#92479—22no—
110#92479—41no—
111#92479D414yescomplemented
112(C2×D31)⋊C4—21no—
113(C2×D31)⋊C4—21no—
114(C2×D31)⋊C4—21no—
115(C2×D31)⋊C4—21no—
116C4⋊Dic31C2311yescharacteristic
117Dic31⋊C4—21no—
118#92474—22no—
119C2×C31⋊C8—22no—
120C2×C31⋊C8C2311yescharacteristic
121C2×C31⋊C8C2311yescharacteristic
122C2×C31⋊C8C2311yescharacteristic
123C2×C31⋊C8C2311yescharacteristic
124C23×C31—21no—
125C23×C31—21no—
126C23×C31—21no—
127C23×C31—42no—
128C23×C31—42no—
129C23×C31—42no—
130C23×C31—42no—
131C23×C31—81no—
132C23×C31—81no—
133C23×C31C2×D411yescharacteristic
134C23×C31C2×D411yescharacteristic
135C22×D31—21no—
136C22×D31—21no—
137C22×D31—42no—
138C22×D31—42no—
139C22×D31—81no—
140D4×C31—22no—
141D4×C31—21no—
142D4×C31—21no—
143D4×C31—21no—
144D4×C31—21no—
145D4×C31—21no—
146D4×C31—42no—
147D4×C31—42no—
148D4×C31—42no—
149D4×C31—42no—
150D4×C31—42no—
151D4×C31—81no—
152D4×C31C2×D412yes—
153C2×C4×C31—21no—
154C2×C4×C31—21no—
155C2×C4×C31—21no—
156C2×C4×C31—21no—
157C2×C4×C31—21no—
158C2×C4×C31—42no—
159C2×C4×C31C2×D411yescharacteristic
160C2×C4×C31C2×D411yescharacteristic
161C2×C4×C31C2×D411yescharacteristic
162C2×C4×C31C2×D411yescharacteristic
163C2×C4×C31C2411yescharacteristic
164C22⋊D31—42no—
165C22⋊D31—81no—
166C2×Dic31—21no—
167C2×Dic31—21no—
168D124—21no—
169D124—21no—
170D124—21no—
171D124—42no—
172D124C2×D412yescomplemented
173C4×D31—42no—
174C4×D31—81no—
175Dic62—21no—
176Dic62—21no—
177C31⋊C8—21no—
178C31⋊C8—21no—
179C31⋊C8—21no—
180C31⋊C8C2×D412yescomplemented
181#3264—311nomaximal, Sylow
182C22×C31—21no—
183C22×C31—21no—
184C22×C31—21no—
185C22×C31—22no—
186C22×C31—21no—
187C22×C31—21no—
188C22×C31—41no—
189C22×C31—41no—
190C22×C31—41no—
191C22×C31—41no—
192C22×C31—41no—
193C22×C31—41no—
194C22×C31—41no—
195C22×C31—41no—
196C22×C31—84no—
197C22×C31—84no—
198C22×C31AGL(1,ℤ/8)12yescomplemented
199C22×C31C22×D411yescharacteristic
200C2×D31—22no—
201C2×D31—41no—
202C2×D31—41no—
203C2×D31—41no—
204C2×D31—41no—
205C2×D31—41no—
206C2×D31—84no—
207C4×C31—21no—
208C4×C31—21no—
209C4×C31—21no—
210C4×C31—21no—
211C4×C31—41no—
212C4×C31C22×D411yescharacteristic
213C4×C31H(5,𝔽2)11yescharacteristic
214Dic31—41no—
215Dic31—41no—
216C2×AGL(1,ℤ/8)—311no—
217C22×D8—311no—
218#546—311no—
219D42D3111yescharacteristic, complemented
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#435—311no—
231C2×C31—22no—
232C2×C31—41no—
233C2×C31—42no—
234C2×C31—81no—
235C2×C31—81no—
236C2×C31D4211yescharacteristic
237C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
238C2×C31#57611yescharacteristic
239D31—42no—
240D31—81no—
241C2×#41—311no—
242C22×D4—21no—
243C22×D4—311no—
244C22×D4C2×D3111yescharacteristic
245C22×D4C2×D3111yescharacteristic, complemented
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×M16—311no—
258C22×C8—311no—
259#127C2×D3111yescharacteristic, complemented
260#124—311no—
261#123—621no—
262C22⋊Q8—311no—
263#121—21no—
264#121—311no—
265#121—621no—
266#121C2×D3111yescharacteristic
267#121C2×D3111yescharacteristic, complemented
268#120—21no—
269#120—21no—
270#120—621no—
271C4×D4—311no—
272C4×D4C2×D3111yescharacteristic
273C4×D4C2×D3111yescharacteristic, complemented
274C2×C22⋊C4—621no—
275#106—311no—
276C4⋊C8—311no—
277#103—311no—
278#102—311no—
279#102—311no—
280#102—311no—
281#102—311no—
282#102—311no—
283C22⋊C8—311no—
284C22⋊C8—311no—
285#97—311no—
286C31#326411yescharacteristic, complemented, Sylow
287C24—21no—
288C24—21no—
289C24—621no—
290#41—622no—
291#41—1241no—
292C2×Q8—311no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—44no—
304C2×D4—311no—
305C2×D4—311no—
306C2×D4—622no—
307C2×D4—621no—
308C2×D4—621no—
309C2×D4—621no—
310C2×D4—1241no—
311C2×D4C22⋊D3111yescharacteristic
312C2×D4C22×D3111yescharacteristic
313C2×D4C22×D3111yescharacteristic
314C2×D4C22×D3111yescharacteristic
315C22×C4—21no—
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—622no—
330C2×C8—311no—
331C2×C8—311no—
332C2×C8—311no—
333C2×C8—311no—
334C2×C8—622no—
335C4⋊C4—311no—
336C4⋊C4—621no—
337C4⋊C4C22⋊D3111yescharacteristic
338C4⋊C4C22×D3111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—621no—
343C22⋊C4—621no—
344C22⋊C4—621no—
345C22⋊C4—621no—
346C22⋊C4—621no—
347C22⋊C4C22⋊D3111yescharacteristic
348C22⋊C4C22⋊D3111yescharacteristic
349C42C22×D3111yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—81no—
358C23—81no—
359C23—621no—
360C23—621no—
361C23—1242no—
362C23—1242no—
363C23—2481no—
364C23C2×C22⋊D3111yescharacteristic
365C23C2×C22⋊D3111yescharacteristic
366Q8—621no—
367Q8—621no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—42no—
378D4—42no—
379D4—81no—
380D4—312no—
381D4—621no—
382D4—621no—
383D4—621no—
384D4—1242no—
385D4—1242no—
386D4—2481no—
387D4D4×D3112yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—621no—
395C2×C4—621no—
396C2×C4—1242no—
397C2×C4—2481no—
398C2×C4C2×C22⋊D3111yescharacteristic
399C2×C4C2×C22⋊D3111yescharacteristic
400C2×C4C2×C22⋊D3111yescharacteristic
401C2×C4C2×C22⋊D3111yescharacteristic
402C2×C4C23×D3111yescharacteristic
403C8—312no—
404C8—621no—
405C8—621no—
406C8—621no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—622no—
424C22—1241no—
425C22—1241no—
426C22—1241no—
427C22—1241no—
428C22—1241no—
429C22—2484no—
430C22#1175850912yescomplemented
431C22C22×C22⋊D3111yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—1241no—
438C4—1241no—
439C4C2×D4×D3111yescharacteristic
440C4#1175856111yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—1242no—
447C2—2481no—
448C2D4×C22⋊D3111yescharacteristic
449C2C2×#1175850911yescharacteristic
450C2#1452185211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingD42×C31C2
socleC22×C31C22×D4
hypercenterD42D31
hypocenterC31#3264
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#11758509C2semidirect
2. C22×#92479C2semidirect
3. #14521695C2semidirect
4. #14521436C2semidirect
5. #14521300C2semidirect
6. #14521276C2semidirect
7. #14521264C2semidirect
8. #14521241C2semidirect
9. #14521200C2semidirect
10. #14521192C2semidirect
11. #14521183C2semidirect
12. #14521182C2semidirect
13. #14521159C2semidirect
14. #14521121C2semidirect
15. D42×C31C2semidirect
18. C22×D4×C31C22semidirect
19. C2×#92495C22semidirect
20. C22×D124C22semidirect
21. C31×#127C22semidirect
24. C31×#121C22semidirect
28. C4×D4×C31C22semidirect
30. C2×#92480C22semidirect
31. C2×#92480C22semidirect
32. C2×#92480C22semidirect
37. C2×#92479C22semidirect
38. C2×#92479C22semidirect
39. C2×#92479C22semidirect
40. C2×#92479C22semidirect
41. #11758507C22semidirect
42. #11758501C22semidirect
43. C2×#92474C22semidirect
44. C22×C31⋊C8C22semidirect
48. #11758452C22semidirect
49. C4×D124C22semidirect
50. #11758420C22semidirect
51. #11758420C22semidirect
52. #11758420C22semidirect
53. C22⋊(C31⋊C8)C22semidirect
54. C22⋊(C31⋊C8)C22semidirect
55. #11758398C22semidirect
56. #11758397C22semidirect
57. #11758397C22semidirect
58. #11758396C22semidirect
59. C4⋊(C31⋊C8)C22semidirect
60. #11758391C22semidirect
111. #92479D4semidirect
172. D124C2×D4semidirect
180. C31⋊C8C2×D4semidirect
198. C22×C31AGL(1,ℤ/8)semidirect
219. D42D31semidirect
245. C22×D4C2×D31semidirect
259. #127C2×D31semidirect
267. #121C2×D31semidirect
273. C4×D4C2×D31semidirect
286. C31#3264semidirect
430. C22#11758509semidirect

Maximal subgroups

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

subgroupindex
C2×#117585092
C22×#924792
#145216952
#145214362
#145213002
#145212762
#145212642
#145212412
#145212002
#145211922
#145211832
#145211822
#145211592
#145211212
D42×C312
#326431