Finite Group Databasebeta

← #16857212

Subgroups — #16857212

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×#11758488C211yesmaximal, characteristic, complemented
2C2×SD16×D31C211yesmaximal, characteristic, complemented
3#14521707C211yesmaximal, characteristic, complemented
4D4×D124C211yesmaximal, characteristic, complemented
5#14521439C211yesmaximal, characteristic, complemented
6#14521298C211yesmaximal, characteristic, complemented
7#14521284C211yesmaximal, characteristic, complemented
8#14521250C211yesmaximal, characteristic, complemented
9#14521198C211yesmaximal, characteristic, complemented
10#14521195C211yesmaximal, characteristic, complemented
11#14521189C211yesmaximal, characteristic, complemented
12#14521181C211yesmaximal, characteristic, complemented
13#14521149C211yesmaximal, characteristic, complemented
14#14521119C211yesmaximal, characteristic, complemented
15C31×#463C211yesmaximal, characteristic, complemented
16C2×Q8×D31C2211yescharacteristic, complemented
17C2×#92497C2211yescharacteristic, complemented
18C2×D4×D31C2211yescharacteristic
19C2×D4×D31C2211yescharacteristic
20C22×D124—21no—
21C2×SD16×C31C2211yescharacteristic
22C2×D8×C31C2211yescharacteristic
23C31×#124C2211yescharacteristic, complemented
24C4×D4×C31C2211yescharacteristic
25#11758523C2211yescharacteristic, complemented
26C2×#92481C2211yescharacteristic, complemented
27C2×#92481C2211yescharacteristic, complemented
28#11758515C2211yescharacteristic, complemented
29C2×#92480C2211yescharacteristic
30C2×#92479C2211yescharacteristic, complemented
31C2×#92479C2211yescharacteristic, complemented
32#11758507—21no—
33SD16×D31—24no—
34#11758488—24no—
35C2×#92470C2211yescharacteristic, complemented
36C2×#92470C2211yescharacteristic, complemented
37C2×#92469C2211yescharacteristic, complemented
38C2×C8×D31C2211yescharacteristic, complemented
39#11758472C2211yescharacteristic
40#11758472C2211yescharacteristic
41#11758462—21no—
42#11758461—21no—
43#11758459—21no—
44#11758459—21no—
45(C22⋊D31)⋊C4—21no—
46#11758455—21no—
47#11758451C2211yescharacteristic
48C4×D124C2211yescharacteristic
49C4×Dic62C2211yescharacteristic, complemented
50C31×C4⋊C8C2211yescharacteristic, complemented
51C31×#103C2211yescharacteristic, complemented
52C31×#102C2211yescharacteristic, complemented
53#11758420C2211yescharacteristic
54#11758409C2211yescharacteristic, complemented
55#11758409C2211yescharacteristic, complemented
56(C2×D31)⋊C8C2211yescharacteristic, complemented
57(C2×D31)⋊C8C2211yescharacteristic, complemented
58#11758398C2211yescharacteristic, complemented
59#11758395C2211yescharacteristic, complemented
60#11758391C2211yescharacteristic, complemented
61C23×D31—21no—
62C23×D31—21no—
63C2×Q8×C31C2311yescharacteristic
64C2×D4×C31C2311yescharacteristic
65C2×D4×C31C2311yescharacteristic
66C22×C4×C31—21no—
67C2×C22⋊D31—21no—
68C2×C22⋊D31—21no—
69C2×C22⋊D31—21no—
70C2×C22⋊D31—21no—
71C22×Dic31—21no—
72Q8×D31—22no—
73Q8×D31—41no—
74#92497—22no—
75#92497—41no—
76D4×D31—22no—
77D4×D31—22no—
78D4×D31—41no—
79D4×D31—41no—
80C2×D124—21no—
81C2×D124—21no—
82C2×D124—21no—
83C2×D124—21no—
84C2×D124—44no—
85C2×D124D411yescharacteristic
86C2×D124C2311yescharacteristic
87C2×D124C2311yescharacteristic
88C2×C4×D31—21no—
89C2×C4×D31C2311yescharacteristic
90C2×C4×D31C2311yescharacteristic
91C2×Dic62—21no—
92C2×Dic62C2311yescharacteristic
93SD16×C31—22no—
94D8×C31—22no—
95C2×C8×C31C2311yescharacteristic
96C2×C8×C31C2311yescharacteristic
97C4⋊C4×C31C2311yescharacteristic
98C22⋊C4×C31—21no—
99C22⋊C4×C31—21no—
100C22⋊C4×C31—21no—
101C42×C31C2311yescharacteristic
102C22⋊Dic31—21no—
103#92481—22no—
104#92481—41no—
105#92480D414yescomplemented
106#92479—22no—
107#92479—41no—
108(C2×D31)⋊C4—21no—
109(C2×D31)⋊C4—21no—
110(C2×D31)⋊C4—21no—
111(C2×D31)⋊C4D411yescharacteristic
112(C2×D31)⋊C4D411yescharacteristic
113C4⋊Dic31D411yescharacteristic
114C4⋊Dic31C2311yescharacteristic
115Dic31⋊C4—21no—
116Dic31⋊C4—21no—
117C4×Dic31—21no—
118C2×C31⋊C8C2311yescharacteristic
119C2×C31⋊C8C2311yescharacteristic
120#92470—22no—
121#92470—22no—
122#92469—22no—
123C8×D31—22no—
124C23×C31—21no—
125C23×C31—21no—
126C22×D31—21no—
127C22×D31—21no—
128C22×D31—42no—
129C22×D31—42no—
130C22×D31—42no—
131C22×D31—42no—
132C22×D31—81no—
133C22×D31—81no—
134C22×D31C2×D411yescharacteristic
135C22×D31C2×D411yescharacteristic
136Q8×C31—21no—
137Q8×C31—21no—
138D4×C31—21no—
139D4×C31—21no—
140D4×C31—21no—
141D4×C31C2×D412yes—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—42no—
145C2×C4×C31C2×D411yescharacteristic
146C2×C4×C31C2×D411yescharacteristic
147C2×C4×C31C2411yescharacteristic
148C22⋊D31—42no—
149C22⋊D31—42no—
150C22⋊D31—42no—
151C22⋊D31—42no—
152C2×Dic31—21no—
153C2×Dic31—21no—
154C2×Dic31—42no—
155C2×Dic31C2×D411yescharacteristic
156C2×Dic31C2×D411yescharacteristic
157D124—22no—
158D124—21no—
159D124—21no—
160D124—21no—
161D124—21no—
162D124—42no—
163D124—42no—
164D124—42no—
165D124—81no—
166C4×D31—21no—
167C4×D31—21no—
168C4×D31—21no—
169C4×D31—21no—
170C4×D31—42no—
171Dic62—21no—
172Dic62—42no—
173Dic62C2×D412yescomplemented
174C8×C31—21no—
175C8×C31—21no—
176C31⋊C8—21no—
177C31⋊C8C2×D412yescomplemented
178#3265—311nomaximal, Sylow
179C22×C31—22no—
180C22×C31—41no—
181C22×C31—41no—
182C22×C31—41no—
183C22×C31—41no—
184C22×C31—41no—
185C22×C31C22×D411yescharacteristic
186C2×D31—21no—
187C2×D31—21no—
188C2×D31—21no—
189C2×D31—21no—
190C2×D31—21no—
191C2×D31—41no—
192C2×D31—41no—
193C2×D31—41no—
194C2×D31—41no—
195C2×D31—41no—
196C2×D31—41no—
197C2×D31—84no—
198C2×D31—84no—
199C2×D31AGL(1,ℤ/8)12yescomplemented
200C4×C31—21no—
201C4×C31—21no—
202C4×C31—41no—
203C4×C31—41no—
204C4×C31C22×D411yescharacteristic
205C4×C31H(5,𝔽2)11yescharacteristic
206Dic31—22no—
207Dic31—21no—
208Dic31—21no—
209Dic31—41no—
210C2×AGL(1,ℤ/8)—311no—
211C22×SD16—311no—
212#548—311no—
213D42—311no—
214#496—311no—
215#468—311no—
216#466—311no—
217#463D3111yescharacteristic, complemented
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—81no—
227C2×C31D4211yescharacteristic
228C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
229C2×C31#57711yescharacteristic
230D31—22no—
231D31—41no—
232D31—81no—
233D31—81no—
234C2×#41—311no—
235C22×Q8—311no—
236C22×D4—311no—
237C22×D4—311no—
238C22×D4—621no—
239AGL(1,ℤ/8)—624no—
240C2×SD16—311no—
241C2×SD16—311no—
242C2×SD16—311no—
243C2×SD16—311no—
244C2×SD16—311no—
245C2×SD16—624no—
246C2×SD16C2×D3111yescharacteristic
247C2×D8—311no—
248C2×D8—311no—
249C2×D8C2×D3111yescharacteristic
250C2×M16—311no—
251C22×C8—311no—
252#127—311no—
253#124—621no—
254#124C2×D3111yescharacteristic, complemented
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—621no—
267C4×D4C2×D3111yescharacteristic
268#107—311no—
269C4⋊C8C2×D3111yescharacteristic, complemented
270#103—311no—
271#103C2×D3111yescharacteristic, complemented
272#102—311no—
273#102—311no—
274#102—311no—
275#102C2×D3111yescharacteristic, complemented
276C22⋊C8—311no—
277C22⋊C8—311no—
278#97—311no—
279C31#326511yescharacteristic, complemented, Sylow
280C24—621no—
281C24—621no—
282#41—622no—
283#41—1241no—
284C2×Q8—311no—
285C2×Q8—621no—
286C2×Q8—622no—
287C2×Q8—1241no—
288C2×Q8C22×D3111yescharacteristic
289C2×D4—311no—
290C2×D4—311no—
291C2×D4—311no—
292C2×D4—621no—
293C2×D4—621no—
294C2×D4—621no—
295C2×D4—621no—
296C2×D4—622no—
297C2×D4—622no—
298C2×D4—621no—
299C2×D4—621no—
300C2×D4—621no—
301C2×D4—621no—
302C2×D4—1244no—
303C2×D4—1241no—
304C2×D4—1241no—
305C2×D4C22×D3111yescharacteristic
306C2×D4C22×D3111yescharacteristic
307C22×C4—21no—
308C22×C4—311no—
309C22×C4—311no—
310C22×C4—621no—
311C22×C4—621no—
312SD16—22no—
313SD16—314no—
314SD16—622no—
315SD16—622no—
316SD16—622no—
317SD16—1241no—
318D8—22no—
319D8—622no—
320D8—1241no—
321M16—622no—
322C2×C8—311no—
323C2×C8—311no—
324C2×C8—622no—
325C2×C8C22×D3111yescharacteristic
326C2×C8C22×D3111yescharacteristic
327C4⋊C4—311no—
328C4⋊C4—311no—
329C4⋊C4—621no—
330C4⋊C4—621no—
331C4⋊C4C22×D3111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—311no—
336C22⋊C4—311no—
337C22⋊C4—621no—
338C22⋊C4—621no—
339C22⋊C4—621no—
340C22⋊C4—621no—
341C42—621no—
342C42C22×D3111yescharacteristic
343C23—21no—
344C23—21no—
345C23—311no—
346C23—311no—
347C23—621no—
348C23—621no—
349C23—1242no—
350C23—1242no—
351C23—1242no—
352C23—1242no—
353C23—2481no—
354C23—2481no—
355Q8—21no—
356Q8—21no—
357Q8—312no—
358Q8—621no—
359Q8—1242no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—622no—
364D4—621no—
365D4—621no—
366D4—621no—
367D4—621no—
368D4—1242no—
369D4—1242no—
370D4—1242no—
371D4—1242no—
372D4—1242no—
373D4—1242no—
374D4—1242no—
375D4—2481no—
376D4D4×D3112yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—311no—
381C2×C4—311no—
382C2×C4—621no—
383C2×C4—621no—
384C2×C4—621no—
385C2×C4—621no—
386C2×C4—621no—
387C2×C4—621no—
388C2×C4—1242no—
389C2×C4—1242no—
390C2×C4D4×D3111yescharacteristic
391C2×C4D4×D3111yescharacteristic
392C2×C4C23×D3111yescharacteristic
393C8—21no—
394C8—21no—
395C8—312no—
396C8—621no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—312no—
404C22—621no—
405C22—621no—
406C22—621no—
407C22—621no—
408C22—621no—
409C22—1241no—
410C22—1241no—
411C22—1241no—
412C22—1241no—
413C22—1241no—
414C22—1241no—
415C22—2484no—
416C22—2484no—
417C22C2×D4×D3111yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—41no—
422C4—621no—
423C4—621no—
424C4—622no—
425C4—1241no—
426C4C2×D4×D3111yescharacteristic
427C4#1175856611yescharacteristic
428C2—42no—
429C2—81no—
430C2—622no—
431C2—1241no—
432C2—2481no—
433C2—2481no—
434C2#1452168311yescharacteristic
435C2AGL(1,ℤ/8)×D3111yescharacteristic
436C2#1452185611yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#463C2
socleC22×C31C22×D4
hypercenter#463D31
hypocenterC31#3265
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×#11758488C2semidirect
2. C2×SD16×D31C2semidirect
3. #14521707C2semidirect
4. D4×D124C2semidirect
5. #14521439C2semidirect
6. #14521298C2semidirect
7. #14521284C2semidirect
8. #14521250C2semidirect
9. #14521198C2semidirect
10. #14521195C2semidirect
11. #14521189C2semidirect
12. #14521181C2semidirect
13. #14521149C2semidirect
14. #14521119C2semidirect
15. C31×#463C2semidirect
16. C2×Q8×D31C22semidirect
17. C2×#92497C22semidirect
23. C31×#124C22semidirect
25. #11758523C22semidirect
26. C2×#92481C22semidirect
27. C2×#92481C22semidirect
28. #11758515C22semidirect
30. C2×#92479C22semidirect
31. C2×#92479C22semidirect
35. C2×#92470C22semidirect
36. C2×#92470C22semidirect
37. C2×#92469C22semidirect
38. C2×C8×D31C22semidirect
49. C4×Dic62C22semidirect
50. C31×C4⋊C8C22semidirect
51. C31×#103C22semidirect
52. C31×#102C22semidirect
54. #11758409C22semidirect
55. #11758409C22semidirect
56. (C2×D31)⋊C8C22semidirect
57. (C2×D31)⋊C8C22semidirect
58. #11758398C22semidirect
59. #11758395C22semidirect
60. #11758391C22semidirect
105. #92480D4semidirect
173. Dic62C2×D4semidirect
177. C31⋊C8C2×D4semidirect
199. C2×D31AGL(1,ℤ/8)semidirect
217. #463D31semidirect
254. #124C2×D31semidirect
269. C4⋊C8C2×D31semidirect
271. #103C2×D31semidirect
275. #102C2×D31semidirect
279. C31#3265semidirect

Maximal subgroups

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

subgroupindex
C2×#117584882
C2×SD16×D312
#145217072
D4×D1242
#145214392
#145212982
#145212842
#145212502
#145211982
#145211952
#145211892
#145211812
#145211492
#145211192
C31×#4632
#326531