Finite Group Databasebeta

← #16857200

Subgroups — #16857200

Order 3968.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D31C211yesmaximal, characteristic, complemented
2#14521706C211yesmaximal, characteristic, complemented
3D4×C22⋊D31C211yesmaximal, characteristic, complemented
4#14521437C211yesmaximal, characteristic, complemented
5#14521295C211yesmaximal, characteristic, complemented
6#14521282C211yesmaximal, characteristic, complemented
7#14521262C211yesmaximal, characteristic, complemented
8#14521247C211yesmaximal, characteristic, complemented
9#14521200C211yesmaximal, characteristic, complemented
10#14521194C211yesmaximal, characteristic, complemented
11#14521186C211yesmaximal, characteristic, complemented
12#14521181C211yesmaximal, characteristic, complemented
13#14521150C211yesmaximal, characteristic, complemented
14#14521118C211yesmaximal, characteristic, complemented
15C2×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic, complemented
17C22×D4×C31C2211yescharacteristic
18C22×C22⋊D31—21no—
19C2×Q8×D31C2211yescharacteristic, complemented
20C2×D4×D31C2211yescharacteristic
21C31×AGL(1,ℤ/8)—24no—
22C2×SD16×C31C2211yescharacteristic
23C2×SD16×C31C2211yescharacteristic, complemented
24C2×D8×C31C2211yescharacteristic
25C2×D8×C31C2211yescharacteristic, complemented
26C2×M16×C31C2211yescharacteristic, complemented
27#11758530—21no—
28Q8×Dic31C2211yescharacteristic, complemented
29C2×#92481C2211yescharacteristic, complemented
30#11758517C2211yescharacteristic
31#11758516—21no—
32#11758515—21no—
33#11758515C2211yescharacteristic
34#11758515C2211yescharacteristic, complemented
35#11758514—21no—
36#11758513—21no—
37#11758513C2211yescharacteristic, complemented
38D4×Dic31C2211yescharacteristic
39C2×#92480C2211yescharacteristic
40C2×#92480C2211yescharacteristic, complemented
41C2×#92479C2211yescharacteristic, complemented
42#11758507C2211yescharacteristic
43C4×C22⋊D31—21no—
44C4×C22⋊D31C2211yescharacteristic
45#11758501—21no—
46#11758501C2211yescharacteristic, complemented
47SD16×D31—24no—
48C2×#92470C2211yescharacteristic, complemented
49C2×C8×D31C2211yescharacteristic, complemented
50#11758423C2211yescharacteristic, complemented
51#11758420C2211yescharacteristic
52#11758420C2211yescharacteristic, complemented
53#11758420C2211yescharacteristic, complemented
54C22⋊(C31⋊C8)C2211yescharacteristic, complemented
55#11758409C2211yescharacteristic, complemented
56(C2×D31)⋊C8C2211yescharacteristic, complemented
57#11758406C2211yescharacteristic, complemented
58#11758404C2211yescharacteristic, complemented
59#11758403C2211yescharacteristic, complemented
60Dic31⋊C8C2211yescharacteristic, complemented
61C24×C31—21no—
62C23×D31—21no—
63#41×C31—22no—
64#41×C31—41no—
65C2×Q8×C31C2311yescharacteristic
66C2×D4×C31—22no—
67C2×D4×C31—21no—
68C2×D4×C31—21no—
69C2×D4×C31—41no—
70C2×D4×C31C2311yescharacteristic
71C2×D4×C31C2311yescharacteristic
72C2×D4×C31C2311yescharacteristic
73C22×C4×C31—21no—
74C22×C4×C31C2311yescharacteristic
75C2×C22⋊D31—21no—
76C2×C22⋊D31—21no—
77C2×C22⋊D31—21no—
78C2×C22⋊D31—21no—
79C2×C22⋊D31—21no—
80C2×C22⋊D31—21no—
81C2×C22⋊D31—44no—
82C2×C22⋊D31D411yescharacteristic
83C22×Dic31—21no—
84Q8×D31—22no—
85Q8×D31—41no—
86D4×D31—22no—
87D4×D31—41no—
88C2×D124C2311yescharacteristic
89C2×C4×D31—21no—
90C2×C4×D31C2311yescharacteristic
91C2×Dic62—21no—
92C2×Dic62C2311yescharacteristic
93SD16×C31—22no—
94SD16×C31D414yescomplemented
95D8×C31—22no—
96D8×C31—22no—
97M16×C31—22no—
98C2×C8×C31C2311yescharacteristic
99C2×C8×C31C2311yescharacteristic
100C22⋊Dic31—21no—
101C22⋊Dic31—21no—
102C22⋊Dic31—21no—
103C22⋊Dic31—21no—
104C22⋊Dic31—21no—
105C22⋊Dic31—21no—
106C22⋊Dic31D411yescharacteristic
107#92481—22no—
108#92480—22no—
109#92480—41no—
110#92479—41no—
111(C2×D31)⋊C4—21no—
112(C2×D31)⋊C4D411yescharacteristic
113C4⋊Dic31—21no—
114C4⋊Dic31C2311yescharacteristic
115C4⋊Dic31C2311yescharacteristic
116Dic31⋊C4—21no—
117Dic31⋊C4D411yescharacteristic
118C4×Dic31—21no—
119C4×Dic31C2311yescharacteristic
120C2×C31⋊C8C2311yescharacteristic
121C2×C31⋊C8C2311yescharacteristic
122#92470—22no—
123C8×D31—22no—
124C23×C31—21no—
125C23×C31—21no—
126C23×C31—21no—
127C23×C31—42no—
128C23×C31—42no—
129C23×C31—81no—
130C23×C31C2×D411yescharacteristic
131C22×D31—21no—
132C22×D31—42no—
133C22×D31—42no—
134C22×D31—81no—
135C22×D31C2×D411yescharacteristic
136Q8×C31—21no—
137Q8×C31C2×D412yescomplemented
138D4×C31—21no—
139D4×C31—21no—
140D4×C31—21no—
141D4×C31—21no—
142D4×C31—21no—
143D4×C31—42no—
144D4×C31—42no—
145D4×C31C2×D412yes—
146C2×C4×C31—21no—
147C2×C4×C31—21no—
148C2×C4×C31—21no—
149C2×C4×C31—42no—
150C2×C4×C31C2×D411yescharacteristic
151C2×C4×C31C2411yescharacteristic
152C22⋊D31—22no—
153C22⋊D31—42no—
154C22⋊D31—42no—
155C22⋊D31—42no—
156C22⋊D31—42no—
157C22⋊D31—42no—
158C22⋊D31—81no—
159C2×Dic31—21no—
160C2×Dic31—21no—
161C2×Dic31—21no—
162C2×Dic31—42no—
163C2×Dic31C2×D411yescharacteristic
164C2×Dic31C2×D411yescharacteristic
165C2×Dic31C2×D411yescharacteristic
166D124—21no—
167D124—21no—
168C4×D31—21no—
169C4×D31—21no—
170C4×D31—42no—
171Dic62—21no—
172Dic62—21no—
173Dic62—42no—
174C8×C31—21no—
175C8×C31C2×D412yescomplemented
176C31⋊C8—21no—
177C31⋊C8—21no—
178#3265—311nomaximal, Sylow
179C22×C31—22no—
180C22×C31—21no—
181C22×C31—21no—
182C22×C31—21no—
183C22×C31—41no—
184C22×C31—41no—
185C22×C31—41no—
186C22×C31—41no—
187C22×C31—41no—
188C22×C31—41no—
189C22×C31—41no—
190C22×C31—41no—
191C22×C31—84no—
192C22×C31C22×D411yescharacteristic
193C2×D31—21no—
194C2×D31—21no—
195C2×D31—41no—
196C2×D31—41no—
197C2×D31—41no—
198C2×D31—84no—
199C2×D31AGL(1,ℤ/8)12yescomplemented
200C4×C31—22no—
201C4×C31—21no—
202C4×C31C22×D411yescharacteristic
203C4×C31H(5,𝔽2)11yescharacteristic
204Dic31—21no—
205Dic31—21no—
206Dic31—21no—
207Dic31—41no—
208Dic31—41no—
209Dic31—41no—
210C2×AGL(1,ℤ/8)D3111yescharacteristic, complemented
211C22×SD16—311no—
212#548—311no—
213D42—311no—
214#496—311no—
215#468—311no—
216#466—311no—
217#463—311no—
218#460—311no—
219#450—311no—
220#449—311no—
221#448—311no—
222#447—311no—
223#440—311no—
224#435—311no—
225C2×C31—42no—
226C2×C31—41no—
227C2×C31—81no—
228C2×C31—81no—
229C2×C31D4211yescharacteristic
230C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
231C2×C31#57711yescharacteristic
232D31—22no—
233D31—81no—
234C2×#41C2×D3111yescharacteristic, complemented
235C22×Q8—311no—
236C22×D4—311no—
237C22×D4—621no—
238C22×D4C2×D3111yescharacteristic
239AGL(1,ℤ/8)—24no—
240C2×SD16—311no—
241C2×SD16—311no—
242C2×SD16—311no—
243C2×SD16—311no—
244C2×SD16—624no—
245C2×SD16C2×D3111yescharacteristic
246C2×SD16C2×D3111yescharacteristic, complemented
247C2×D8—311no—
248C2×D8C2×D3111yescharacteristic
249C2×D8C2×D3111yescharacteristic, complemented
250C2×M16C2×D3111yescharacteristic, complemented
251C22×C8—311no—
252#127—311no—
253#124—311no—
254#124—621no—
255C22⋊Q8—311no—
256C22⋊Q8—621no—
257#121—311no—
258#121—311no—
259#121—311no—
260#121—621no—
261#121—621no—
262#120—621no—
263#120—621no—
264C4×Q8—311no—
265C4×D4—311no—
266C4×D4—311no—
267C4×D4—621no—
268#107—311no—
269C4⋊C8—311no—
270#103—311no—
271#103—311no—
272#102—311no—
273#102—311no—
274#102—311no—
275#102—311no—
276C22⋊C8—311no—
277C22⋊C8—311no—
278#97—311no—
279C31#326511yescharacteristic, complemented, Sylow
280C24—21no—
281C24—621no—
282#41—22no—
283#41—41no—
284C2×Q8—311no—
285C2×Q8—621no—
286C2×Q8—622no—
287C2×Q8—1241no—
288C2×Q8C22×D3111yescharacteristic
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—311no—
294C2×D4—311no—
295C2×D4—621no—
296C2×D4—621no—
297C2×D4—621no—
298C2×D4—622no—
299C2×D4—621no—
300C2×D4—621no—
301C2×D4—621no—
302C2×D4—1241no—
303C2×D4—1244no—
304C2×D4C22×D3111yescharacteristic
305C2×D4C22×D3111yescharacteristic
306C2×D4C22×D3111yescharacteristic
307C22×C4—21no—
308C22×C4—311no—
309C22×C4—621no—
310C22×C4—621no—
311C22×C4C22×D3111yescharacteristic
312SD16—22no—
313SD16—622no—
314SD16—622no—
315SD16—622no—
316SD16—1241no—
317SD16C22⋊D3114yescomplemented
318D8—22no—
319D8—22no—
320D8—1241no—
321M16—22no—
322C2×C8—311no—
323C2×C8—311no—
324C2×C8—622no—
325C2×C8C22×D3111yescharacteristic
326C2×C8C22×D3111yescharacteristic
327C4⋊C4—311no—
328C4⋊C4—311no—
329C4⋊C4—311no—
330C4⋊C4—621no—
331C4⋊C4—621no—
332C22⋊C4—311no—
333C22⋊C4—311no—
334C22⋊C4—621no—
335C22⋊C4—621no—
336C22⋊C4—621no—
337C22⋊C4—621no—
338C22⋊C4—621no—
339C22⋊C4—621no—
340C22⋊C4—621no—
341C42—311no—
342C42—621no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—311no—
350C23—621no—
351C23—1242no—
352C23—1242no—
353C23—2481no—
354C23D4×D3111yescharacteristic
355Q8—21no—
356Q8—621no—
357Q8—621no—
358Q8—1242no—
359Q8C2×C22⋊D3112yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—42no—
366D4—42no—
367D4—622no—
368D4—621no—
369D4—621no—
370D4—1242no—
371D4—1242no—
372D4—1242no—
373D4—1242no—
374D4—1242no—
375D4—2481no—
376D4C2×C22⋊D3112yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—311no—
382C2×C4—311no—
383C2×C4—311no—
384C2×C4—621no—
385C2×C4—621no—
386C2×C4—621no—
387C2×C4—621no—
388C2×C4—621no—
389C2×C4—1242no—
390C2×C4—1242no—
391C2×C4D4×D3111yescharacteristic
392C2×C4C23×D3111yescharacteristic
393C8—21no—
394C8—621no—
395C8—621no—
396C8C2×C22⋊D3112yescomplemented
397C22—21no—
398C22—21no—
399C22—22no—
400C22—21no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—312no—
411C22—621no—
412C22—621no—
413C22—1241no—
414C22—1241no—
415C22—1241no—
416C22—2484no—
417C22C2×D4×D3111yescharacteristic
418C4—21no—
419C4—22no—
420C4—621no—
421C4—621no—
422C4—621no—
423C4—1241no—
424C4—1241no—
425C4—1241no—
426C4#1175856111yescharacteristic
427C4C22×C22⋊D3111yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—81no—
432C2—622no—
433C2—2481no—
434C2D4×C22⋊D3111yescharacteristic
435C2AGL(1,ℤ/8)×D3111yescharacteristic
436C2#1452185911yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×AGL(1,ℤ/8)C2
socleC22×C31C22×D4
hypercenterC2×AGL(1,ℤ/8)D31
hypocenterC31#3265
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D31C2semidirect
2. #14521706C2semidirect
3. D4×C22⋊D31C2semidirect
4. #14521437C2semidirect
5. #14521295C2semidirect
6. #14521282C2semidirect
7. #14521262C2semidirect
8. #14521247C2semidirect
9. #14521200C2semidirect
10. #14521194C2semidirect
11. #14521186C2semidirect
12. #14521181C2semidirect
13. #14521150C2semidirect
14. #14521118C2semidirect
15. C2×C31×AGL(1,ℤ/8)C2semidirect
16. C2×#41×C31C22semidirect
19. C2×Q8×D31C22semidirect
23. C2×SD16×C31C22semidirect
25. C2×D8×C31C22semidirect
26. C2×M16×C31C22semidirect
28. Q8×Dic31C22semidirect
29. C2×#92481C22semidirect
34. #11758515C22semidirect
37. #11758513C22semidirect
40. C2×#92480C22semidirect
41. C2×#92479C22semidirect
46. #11758501C22semidirect
48. C2×#92470C22semidirect
49. C2×C8×D31C22semidirect
50. #11758423C22semidirect
52. #11758420C22semidirect
53. #11758420C22semidirect
54. C22⋊(C31⋊C8)C22semidirect
55. #11758409C22semidirect
56. (C2×D31)⋊C8C22semidirect
57. #11758406C22semidirect
58. #11758404C22semidirect
59. #11758403C22semidirect
60. Dic31⋊C8C22semidirect
94. SD16×C31D4semidirect
137. Q8×C31C2×D4semidirect
175. C8×C31C2×D4semidirect
199. C2×D31AGL(1,ℤ/8)semidirect
210. C2×AGL(1,ℤ/8)D31semidirect
234. C2×#41C2×D31semidirect
246. C2×SD16C2×D31semidirect
249. C2×D8C2×D31semidirect
250. C2×M16C2×D31semidirect
279. C31#3265semidirect
317. SD16C22⋊D31semidirect
359. Q8C2×C22⋊D31semidirect
396. C8C2×C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D312
#145217062
D4×C22⋊D312
#145214372
#145212952
#145212822
#145212622
#145212472
#145212002
#145211942
#145211862
#145211812
#145211502
#145211182
C2×C31×AGL(1,ℤ/8)2
#326531