Finite Group Databasebeta

← #16857205

Subgroups — #16857205

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
4#14521684C211yesmaximal, characteristic, complemented
5#14521292C211yesmaximal, characteristic, complemented
6#14521282C211yesmaximal, characteristic, complemented
7#14521261C211yesmaximal, characteristic, complemented
8#14521246C211yesmaximal, characteristic, complemented
9#14521198C211yesmaximal, characteristic, complemented
10#14521191C211yesmaximal, characteristic, complemented
11#14521186C211yesmaximal, characteristic, complemented
12#14521181C211yesmaximal, characteristic, complemented
13#14521151C211yesmaximal, characteristic, complemented
14#14521122C211yesmaximal, characteristic, complemented
15C31×#496C211yesmaximal, characteristic, complemented
16C2×Q8×D31C2211yescharacteristic, complemented
17C2×#92497C2211yescharacteristic, complemented
18C2×D4×D31—21no—
19C2×D4×D31C2211yescharacteristic
20C2×D4×D31C2211yescharacteristic
21C2×SD16×C31C2211yescharacteristic, complemented
22C2×SD16×C31C2211yescharacteristic, complemented
23C2×D8×C31C2211yescharacteristic, complemented
24C2×D8×C31C2211yescharacteristic, complemented
25C31×#127C2211yescharacteristic
26C31×#124C2211yescharacteristic, complemented
27#11758523C2211yescharacteristic, complemented
28C2×#92481C2211yescharacteristic
29#11758515—21no—
30#11758515C2211yescharacteristic
31#11758515C2211yescharacteristic
32#11758515C2211yescharacteristic, complemented
33#11758514—21no—
34#11758514—21no—
35C2×#92480C2211yescharacteristic, complemented
36C2×#92480C2211yescharacteristic, complemented
37C2×#92479C2211yescharacteristic
38SD16×D31—24no—
39#11758488—24no—
40C2×#92470C2211yescharacteristic
41C2×#92469C2211yescharacteristic, complemented
42C2×C8×D31C2211yescharacteristic, complemented
43#11758462—21no—
44#11758461—21no—
45(C22⋊D31)⋊C4—21no—
46#11758455—21no—
47C4×D124C2211yescharacteristic
48C4×D124C2211yescharacteristic
49C4×Dic62C2211yescharacteristic, complemented
50C31×#97C2211yescharacteristic, complemented
51#11758423C2211yescharacteristic, complemented
52#11758420C2211yescharacteristic, complemented
53#11758420C2211yescharacteristic, complemented
54#11758420C2211yescharacteristic, complemented
55#11758409C2211yescharacteristic
56(C2×D31)⋊C8C2211yescharacteristic, complemented
57(C2×D31)⋊C8C2211yescharacteristic, complemented
58#11758406C2211yescharacteristic, complemented
59#11758404C2211yescharacteristic, complemented
60C4⋊(C31⋊C8)C2211yescharacteristic, complemented
61C23×D31—21no—
62C23×D31—21no—
63C2×Q8×C31C2311yescharacteristic
64C2×D4×C31—21no—
65C2×D4×C31—21no—
66C2×D4×C31C2311yescharacteristic
67C2×D4×C31C2311yescharacteristic
68C2×D4×C31C2311yescharacteristic
69C2×C22⋊D31—21no—
70C2×C22⋊D31—21no—
71C2×C22⋊D31—21no—
72C2×C22⋊D31—21no—
73C2×C22⋊D31—21no—
74C2×C22⋊D31—21no—
75C22×Dic31—21no—
76Q8×D31—22no—
77Q8×D31—41no—
78#92497—22no—
79#92497—41no—
80D4×D31—22no—
81D4×D31—22no—
82D4×D31—41no—
83D4×D31—41no—
84D4×D31—44no—
85C2×D124D411yescharacteristic
86C2×D124C2311yescharacteristic
87C2×C4×D31—21no—
88C2×C4×D31—21no—
89C2×C4×D31C2311yescharacteristic
90C2×C4×D31C2311yescharacteristic
91C2×Dic62—21no—
92C2×Dic62C2311yescharacteristic
93SD16×C31—22no—
94SD16×C31—41no—
95D8×C31—22no—
96D8×C31—41no—
97C2×C8×C31C2311yescharacteristic
98C2×C8×C31C2311yescharacteristic
99C22⋊C4×C31—21no—
100C22⋊C4×C31—21no—
101C42×C31C2311yescharacteristic
102C22⋊Dic31—21no—
103C22⋊Dic31—21no—
104C22⋊Dic31—21no—
105#92481—22no—
106#92480—22no—
107#92480—22no—
108#92479—22no—
109(C2×D31)⋊C4—21no—
110(C2×D31)⋊C4—21no—
111(C2×D31)⋊C4D411yescharacteristic
112(C2×D31)⋊C4D411yescharacteristic
113C4⋊Dic31D411yescharacteristic
114C4⋊Dic31C2311yescharacteristic
115C4⋊Dic31C2311yescharacteristic
116Dic31⋊C4—21no—
117Dic31⋊C4—21no—
118C4×Dic31—21no—
119C2×C31⋊C8C2311yescharacteristic
120C2×C31⋊C8C2311yescharacteristic
121#92470D414yescomplemented
122#92469—22no—
123C8×D31—22no—
124C23×C31—21no—
125C23×C31—21no—
126C23×C31—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×C31—21no—
142D4×C31—21no—
143D4×C31—21no—
144D4×C31—42no—
145D4×C31—81no—
146C2×C4×C31—21no—
147C2×C4×C31C2×D411yescharacteristic
148C2×C4×C31C2×D411yescharacteristic
149C2×C4×C31C2411yescharacteristic
150C22⋊D31—42no—
151C22⋊D31—42no—
152C22⋊D31—42no—
153C22⋊D31—42no—
154C22⋊D31—42no—
155C22⋊D31—42no—
156C2×Dic31—21no—
157C2×Dic31—21no—
158C2×Dic31—21no—
159C2×Dic31—42no—
160C2×Dic31C2×D411yescharacteristic
161C2×Dic31C2×D411yescharacteristic
162D124—22no—
163D124—21no—
164D124C2×D412yes—
165C4×D31—21no—
166C4×D31—21no—
167C4×D31—21no—
168C4×D31—21no—
169C4×D31—42no—
170C4×D31—42no—
171Dic62—21no—
172Dic62—42no—
173Dic62C2×D412yescomplemented
174C8×C31—21no—
175C8×C31C2×D412yescomplemented
176C31⋊C8—21no—
177C31⋊C8—21no—
178#3265—311nomaximal, Sylow
179C22×C31—41no—
180C22×C31—41no—
181C22×C31—41no—
182C22×C31—41no—
183C22×C31—41no—
184C22×C31—41no—
185C22×C31—41no—
186C22×C31—41no—
187C22×C31—41no—
188C22×C31C22×D411yescharacteristic
189C2×D31—21no—
190C2×D31—21no—
191C2×D31—21no—
192C2×D31—22no—
193C2×D31—21no—
194C2×D31—21no—
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×C31C22×D411yescharacteristic
204C4×C31H(5,𝔽2)11yescharacteristic
205Dic31—22no—
206Dic31—21no—
207Dic31—21no—
208Dic31—41no—
209Dic31—41no—
210C2×AGL(1,ℤ/8)—311no—
211C22×SD16—311no—
212#548—311no—
213D42—311no—
214#496D3111yescharacteristic, complemented
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—81no—
226C2×C31—81no—
227C2×C31—81no—
228C2×C31D4211yescharacteristic
229C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
230C2×C31#57711yescharacteristic
231D31—22no—
232D31—41no—
233D31—42no—
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—624no—
245C2×SD16C2×D3111yescharacteristic, complemented
246C2×SD16C2×D3111yescharacteristic, complemented
247C2×D8—311no—
248C2×D8C2×D3111yescharacteristic, complemented
249C2×D8C2×D3111yescharacteristic, complemented
250C2×M16—311no—
251C22×C8—311no—
252#127C2×D3111yescharacteristic
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—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#97C2×D3111yescharacteristic, complemented
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—21no—
290C2×D4—21no—
291C2×D4—311no—
292C2×D4—311no—
293C2×D4—621no—
294C2×D4—621no—
295C2×D4—622no—
296C2×D4—622no—
297C2×D4—621no—
298C2×D4—621no—
299C2×D4—621no—
300C2×D4—621no—
301C2×D4—1241no—
302C2×D4—1241no—
303C2×D4—1244no—
304C2×D4C22×D3111yescharacteristic
305C2×D4C22×D3111yescharacteristic
306C2×D4C22×D3111yescharacteristic
307C22×C4—311no—
308C22×C4—311no—
309C22×C4—621no—
310C22×C4—621no—
311C22×C4—621no—
312SD16—22no—
313SD16—41no—
314SD16—314no—
315SD16—622no—
316SD16—622no—
317SD16—622no—
318D8—22no—
319D8—41no—
320D8—622no—
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—311no—
330C4⋊C4—621no—
331C4⋊C4—621no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—311no—
335C22⋊C4—311no—
336C22⋊C4—621no—
337C22⋊C4—621no—
338C22⋊C4—621no—
339C22⋊C4—621no—
340C22⋊C4—621no—
341C42—621no—
342C42C22×D3111yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—311no—
347C23—311no—
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—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—81no—
368D4—312no—
369D4—622no—
370D4—621no—
371D4—1242no—
372D4—1242no—
373D4—1242no—
374D4—1242no—
375D4—1242no—
376D4—1242no—
377C2×C4—21no—
378C2×C4—311no—
379C2×C4—311no—
380C2×C4—621no—
381C2×C4—621no—
382C2×C4—621no—
383C2×C4—621no—
384C2×C4—621no—
385C2×C4—621no—
386C2×C4—621no—
387C2×C4—1242no—
388C2×C4—1242no—
389C2×C4—1242no—
390C2×C4D4×D3111yescharacteristic
391C2×C4D4×D3111yescharacteristic
392C2×C4C23×D3111yescharacteristic
393C8—21no—
394C8—621no—
395C8—621no—
396C8D4×D3112yescomplemented
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—312no—
407C22—621no—
408C22—621no—
409C22—621no—
410C22—621no—
411C22—622no—
412C22—621no—
413C22—1241no—
414C22—1241no—
415C22—2484no—
416C22—2484no—
417C22C2×D4×D3111yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—621no—
422C4—621no—
423C4—622no—
424C4—1241no—
425C4—1241no—
426C4C2×D4×D3111yescharacteristic
427C4#1175856111yescharacteristic
428C2—81no—
429C2—81no—
430C2—81no—
431C2—622no—
432C2—1241no—
433C2—1242no—
434C2#1452168411yescharacteristic
435C2AGL(1,ℤ/8)×D3111yescharacteristic
436C2#1452185911yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#496C2
socleC22×C31C22×D4
hypercenter#496D31
hypocenterC31#3265
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#11758488C2semidirect
2. C2×SD16×D31C2semidirect
3. #14521707C2semidirect
4. #14521684C2semidirect
5. #14521292C2semidirect
6. #14521282C2semidirect
7. #14521261C2semidirect
8. #14521246C2semidirect
9. #14521198C2semidirect
10. #14521191C2semidirect
11. #14521186C2semidirect
12. #14521181C2semidirect
13. #14521151C2semidirect
14. #14521122C2semidirect
15. C31×#496C2semidirect
16. C2×Q8×D31C22semidirect
17. C2×#92497C22semidirect
21. C2×SD16×C31C22semidirect
22. C2×SD16×C31C22semidirect
23. C2×D8×C31C22semidirect
24. C2×D8×C31C22semidirect
26. C31×#124C22semidirect
27. #11758523C22semidirect
32. #11758515C22semidirect
35. C2×#92480C22semidirect
36. C2×#92480C22semidirect
41. C2×#92469C22semidirect
42. C2×C8×D31C22semidirect
49. C4×Dic62C22semidirect
50. C31×#97C22semidirect
51. #11758423C22semidirect
52. #11758420C22semidirect
53. #11758420C22semidirect
54. #11758420C22semidirect
56. (C2×D31)⋊C8C22semidirect
57. (C2×D31)⋊C8C22semidirect
58. #11758406C22semidirect
59. #11758404C22semidirect
60. C4⋊(C31⋊C8)C22semidirect
121. #92470D4semidirect
173. Dic62C2×D4semidirect
175. C8×C31C2×D4semidirect
199. C2×D31AGL(1,ℤ/8)semidirect
214. #496D31semidirect
245. C2×SD16C2×D31semidirect
246. C2×SD16C2×D31semidirect
248. C2×D8C2×D31semidirect
249. C2×D8C2×D31semidirect
254. #124C2×D31semidirect
278. #97C2×D31semidirect
279. C31#3265semidirect
396. C8D4×D31semidirect

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
#145216842
#145212922
#145212822
#145212612
#145212462
#145211982
#145211912
#145211862
#145211812
#145211512
#145211222
C31×#4962
#326531