Finite Group Databasebeta

← #16857478

Subgroups — #16857478

Order 3968.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521861C211yesmaximal, characteristic, complemented
2#14521856C211yesmaximal, characteristic, complemented
3#14521854C211yesmaximal, characteristic, complemented
4#14521851C211yesmaximal, characteristic, complemented
5#14521438C211yesmaximal, characteristic, complemented
6#14521320C211yesmaximal, characteristic, complemented
7#14521319C211yesmaximal, characteristic, complemented
8#14521313C211yesmaximal, characteristic, complemented
9#14521312C211yesmaximal, characteristic, complemented
10#14521228C211yesmaximal, characteristic, complemented
11#14521223C211yesmaximal, characteristic, complemented
12#14521218C211yesmaximal, characteristic, complemented
13#14521218C211yesmaximal, characteristic, complemented
14#14521172C211yesmaximal, characteristic, complemented
15C31×#444C211yesmaximal, characteristic, complemented
16#11758567C2211yescharacteristic, complemented
17#11758566C2211yescharacteristic
18#11758566C2211yescharacteristic, complemented
19#11758566C2211yescharacteristic, complemented
20C31×#135C2211yescharacteristic
21C31×#131C2211yescharacteristic, complemented
22C31×#131C2211yescharacteristic, complemented
23#11758528C2211yescharacteristic, complemented
24#11758526C2211yescharacteristic
25#11758526C2211yescharacteristic, complemented
26#11758526C2211yescharacteristic, complemented
27#11758496—21no—
28#11758495—21no—
29#11758493—21no—
30#11758491—21no—
31#11758491—21no—
32SD16×D31—21no—
33#11758488—21no—
34D8×D31—21no—
35#11758486—21no—
36#11758486C2211yescharacteristic, complemented
37#11758485—21no—
38#11758485—21no—
39#11758485—21no—
40#11758485C2211yescharacteristic
41#11758485C2211yescharacteristic, complemented
42#11758485C2211yescharacteristic, complemented
43#11758481—21no—
44#11758481—21no—
45#11758481C2211yescharacteristic
46C2×D248—21no—
47C2×D248C2211yescharacteristic, complemented
48C2×D248C2211yescharacteristic, complemented
49C2×#92470—21no—
50C2×#92470C2211yescharacteristic, complemented
51C2×#92470C2211yescharacteristic, complemented
52#11758478C2211yescharacteristic, complemented
53#11758478C2211yescharacteristic, complemented
54#11758452C2211yescharacteristic, complemented
55#11758451C2211yescharacteristic, complemented
56C31×#108C2211yescharacteristic, complemented
57C31×#104C2211yescharacteristic, complemented
58C31×#104C2211yescharacteristic, complemented
59C31×#97C2211yescharacteristic, complemented
60#11758412C2211yescharacteristic, complemented
61#11758411C2211yescharacteristic, complemented
62#11758411C2211yescharacteristic, complemented
63#11758411C2211yescharacteristic, complemented
64#11758407C2211yescharacteristic, complemented
65#11758393C2211yescharacteristic, complemented
66#11758393C2211yescharacteristic, complemented
67#41×C31C2311yescharacteristic
68#41×C31C2311yescharacteristic
69#92499—21no—
70#92499—21no—
71#92499—21no—
72Q8×D31—21no—
73#92497—21no—
74#92497—41no—
75D4×D31—21no—
76D4×D31—21no—
77D4×D31—21no—
78D4×D31—41no—
79D4×D31—41no—
80D4×D31—41no—
81#92495—21no—
82#92495—21no—
83#92495—21no—
84#92495—21no—
85#92495C2311yescharacteristic
86#92495C2311yescharacteristic
87C2×D124—21no—
88C2×D124—21no—
89C2×D124—21no—
90C2×D124—41no—
91C2×D124C2311yescharacteristic
92C2×D124C2311yescharacteristic
93C2×D124C2311yescharacteristic
94C2×Dic62—21no—
95C2×Dic62C2311yescharacteristic
96Q16×C31D411yescharacteristic, complemented
97SD16×C31D411yescharacteristic, complemented
98SD16×C31D411yescharacteristic, complemented
99D8×C31D411yescharacteristic, complemented
100M16×C31—21no—
101M16×C31—21no—
102M16×C31C2311yescharacteristic
103M16×C31C2311yescharacteristic
104C2×C8×C31—21no—
105C2×C8×C31—21no—
106C2×C8×C31C2311yescharacteristic
107C2×C8×C31C2311yescharacteristic
108C42×C31C2311yescharacteristic
109#92482—21no—
110#92481—21no—
111#92481—21no—
112#92481—21no—
113#92480—21no—
114#92479—21no—
115#92479—21no—
116#92479—21no—
117(C2×D31)⋊C4—41no—
118#92474C2311yescharacteristic
119#92474C2311yescharacteristic
120Dic124—21no—
121Dic124D411yescharacteristic, complemented
122D248—21no—
123D248—21no—
124D248—21no—
125D248—21no—
126D248—21no—
127D248—21no—
128D248—21no—
129D248D411yescharacteristic, complemented
130#92470—21no—
131#92470—21no—
132#92470—21no—
133#92470—21no—
134#92470—21no—
135#92470—21no—
136#92470—21no—
137#92470—21no—
138#92470D411yescharacteristic, complemented
139#92470D411yescharacteristic, complemented
140#92469—21no—
141#92469—21no—
142C8×D31—21no—
143C8×D31—21no—
144C22×D31—21no—
145C22×D31—21no—
146C22×D31—21no—
147C22×D31—41no—
148C22×D31—41no—
149C22×D31—41no—
150Q8×C31C2×D411yescharacteristic
151Q8×C31C2×D411yescharacteristic
152D4×C31—21no—
153D4×C31—21no—
154D4×C31C2×D411yescharacteristic
155D4×C31C2×D411yescharacteristic
156C2×C4×C31—21no—
157C2×C4×C31—21no—
158C2×C4×C31—21no—
159C2×C4×C31C2411yescharacteristic
160C22⋊D31—21no—
161C22⋊D31—21no—
162C22⋊D31—41no—
163C22⋊D31—41no—
164C22⋊D31—41no—
165C22⋊D31—41no—
166C2×Dic31—21no—
167C2×Dic31—41no—
168D124—21no—
169D124—21no—
170D124—21no—
171D124—21no—
172D124—21no—
173D124—21no—
174D124—41no—
175D124—41no—
176D124—41no—
177D124—82no—
178D124C2×D411yescharacteristic
179D124C2×D411yescharacteristic
180C4×D31—21no—
181C4×D31—21no—
182C4×D31—41no—
183C4×D31—41no—
184C4×D31—41no—
185C4×D31—41no—
186Dic62—21no—
187Dic62—21no—
188Dic62—41no—
189Dic62C2×D411yescharacteristic
190Dic62C2×D411yescharacteristic
191C8×C31—21no—
192C8×C31—21no—
193C8×C31C2×D411yescharacteristic
194C8×C31C2×D411yescharacteristic
195C8×C31C2×D411yescharacteristic
196C8×C31C2×D411yescharacteristic
197C31⋊C8—21no—
198C31⋊C8—21no—
199#3288—311nomaximal, Sylow
200C22×C31—21no—
201C22×C31—21no—
202C22×C31H(5,𝔽2)11yescharacteristic
203C2×D31—21no—
204C2×D31—21no—
205C2×D31—41no—
206C2×D31—41no—
207C2×D31—41no—
208C2×D31—81no—
209C2×D31—81no—
210C2×D31—81no—
211C2×D31—161no—
212C2×D31—161no—
213C2×D31—161no—
214C4×C31—21no—
215C4×C31—21no—
216C4×C31—41no—
217C4×C31C22×D411yescharacteristic
218C4×C31C22×D411yescharacteristic
219Dic31—21no—
220Dic31—21no—
221Dic31—41no—
222#577—311no—
223#577—311no—
224#576—311no—
225#576—311no—
226#496—311no—
227#472—311no—
228#472—311no—
229#471—311no—
230#471—311no—
231#455—311no—
232#454—311no—
233#453—311no—
234#453—311no—
235#444—311no—
236#444D3111yescharacteristic, complemented
237C2×C31—21no—
238C2×C31—41no—
239C2×C31—41no—
240C2×C31D4211yescharacteristic
241D31—41no—
242D31—41no—
243D31—81no—
244D31—81no—
245D31—81no—
2462−1+4—311no—
247H(5,𝔽2)—311no—
248H(5,𝔽2)—311no—
249H(5,𝔽2)—311no—
250#137—311no—
251#137—311no—
252#137—621no—
253#137—621no—
254AGL(1,ℤ/8)—311no—
255AGL(1,ℤ/8)—311no—
256AGL(1,ℤ/8)—311no—
257AGL(1,ℤ/8)—311no—
258AGL(1,ℤ/8)—311no—
259AGL(1,ℤ/8)—311no—
260AGL(1,ℤ/8)—621no—
261AGL(1,ℤ/8)—621no—
262AGL(1,ℤ/8)—621no—
263AGL(1,ℤ/8)—621no—
264AGL(1,ℤ/8)—621no—
265AGL(1,ℤ/8)—621no—
266#135—311no—
267#135—621no—
268#135—621no—
269#135—621no—
270#135—621no—
271#135C2×D3111yescharacteristic
272C2×SD16—311no—
273C2×SD16—311no—
274C2×SD16—621no—
275C2×SD16—621no—
276C2×D8—311no—
277C2×D8—311no—
278C2×D8—621no—
279C2×D8—621no—
280#131—311no—
281#131—311no—
282#131C2×D3111yescharacteristic, complemented
283#131C2×D3111yescharacteristic, complemented
284#127—311no—
285#124—311no—
286#108—311no—
287#108C2×D3111yescharacteristic, complemented
288#104—311no—
289#104—311no—
290#104C2×D3111yescharacteristic, complemented
291#104C2×D3111yescharacteristic, complemented
292#101—311no—
293#100—311no—
294#100—311no—
295#100—311no—
296#97C2×D3111yescharacteristic, complemented
297C31#328811yescharacteristic, complemented, Sylow
298#41—311no—
299#41—311no—
300#41—621no—
301#41—621no—
302#41—621no—
303#41—621no—
304#41—621no—
305#41—621no—
306#41—621no—
307#41—621no—
308#41—1241no—
309#41C22×D3111yescharacteristic
310#41C22×D3111yescharacteristic
311C2×Q8—311no—
312C2×Q8—621no—
313C2×Q8—621no—
314C2×D4—311no—
315C2×D4—311no—
316C2×D4—311no—
317C2×D4—621no—
318C2×D4—621no—
319C2×D4—621no—
320C2×D4—621no—
321C2×D4—621no—
322C2×D4—621no—
323C2×D4—1241no—
324C2×D4—1241no—
325C2×D4—1241no—
326C2×D4—1241no—
327Q16—311no—
328Q16—621no—
329Q16—621no—
330Q16D12411yescharacteristic, complemented
331SD16—311no—
332SD16—311no—
333SD16—621no—
334SD16—621no—
335SD16—621no—
336SD16—621no—
337SD16—621no—
338SD16—621no—
339SD16—621no—
340SD16—621no—
341SD16—621no—
342SD16—621no—
343SD16—621no—
344SD16—621no—
345SD16D12411yescharacteristic, complemented
346SD16D12411yescharacteristic, complemented
347D8—311no—
348D8—621no—
349D8—621no—
350D8—621no—
351D8—621no—
352D8—621no—
353D8—621no—
354D8—621no—
355D8—621no—
356D8—621no—
357D8—621no—
358D8D12411yescharacteristic, complemented
359M16—21no—
360M16—21no—
361M16—311no—
362M16—311no—
363M16—621no—
364M16—621no—
365M16C22×D3111yescharacteristic
366M16C22×D3111yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—621no—
370C2×C8—621no—
371C2×C8C22×D3111yescharacteristic
372C2×C8C22×D3111yescharacteristic
373C22⋊C4—1241no—
374C42C22×D3111yescharacteristic
375C23—621no—
376C23—621no—
377C23—621no—
378C23—1241no—
379C23—1241no—
380C23—1241no—
381Q8—311no—
382Q8—311no—
383Q8—621no—
384Q8—621no—
385Q8—1241no—
386Q8C2×D12411yescharacteristic
387Q8C2×D12411yescharacteristic
388D4—21no—
389D4—21no—
390D4—311no—
391D4—311no—
392D4—621no—
393D4—621no—
394D4—621no—
395D4—621no—
396D4—621no—
397D4—621no—
398D4—621no—
399D4—621no—
400D4—1241no—
401D4—1241no—
402D4—1241no—
403D4—1241no—
404D4—1241no—
405D4—1241no—
406D4—1241no—
407D4—2482no—
408D4C2×D12411yescharacteristic
409D4C2×D12411yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—621no—
414C2×C4—621no—
415C2×C4—621no—
416C2×C4—1241no—
417C2×C4—1241no—
418C2×C4—1241no—
419C2×C4—1241no—
420C2×C4—1241no—
421C2×C4C23×D3111yescharacteristic
422C8—21no—
423C8—21no—
424C8—621no—
425C8—621no—
426C8C2×D12411yescharacteristic
427C8C2×D12411yescharacteristic
428C8D4×D3111yescharacteristic
429C8D4×D3111yescharacteristic
430C22—21no—
431C22—21no—
432C22—621no—
433C22—621no—
434C22—1241no—
435C22—1241no—
436C22—1241no—
437C22—4961no—
438C22—4961no—
439C22—4961no—
440C22—2481no—
441C22—2481no—
442C22—2481no—
443C22#1175856611yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—621no—
448C4—621no—
449C4—1241no—
450C4C22×D12411yescharacteristic
451C4C2×D4×D3111yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—1241no—
456C2—1241no—
457C2—2481no—
458C2—2481no—
459C2—2481no—
460C2D4×D12411yescharacteristic
subgroupG/H
centerC2D4×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#444C2
socleC2×C31D42
hypercenter#444D31
hypocenterC31#3288
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #14521861C2semidirect
2. #14521856C2semidirect
3. #14521854C2semidirect
4. #14521851C2semidirect
5. #14521438C2semidirect
6. #14521320C2semidirect
7. #14521319C2semidirect
8. #14521313C2semidirect
9. #14521312C2semidirect
10. #14521228C2semidirect
11. #14521223C2semidirect
12. #14521218C2semidirect
13. #14521218C2semidirect
14. #14521172C2semidirect
15. C31×#444C2semidirect
16. #11758567C22semidirect
18. #11758566C22semidirect
19. #11758566C22semidirect
21. C31×#131C22semidirect
22. C31×#131C22semidirect
23. #11758528C22semidirect
25. #11758526C22semidirect
26. #11758526C22semidirect
36. #11758486C22semidirect
41. #11758485C22semidirect
42. #11758485C22semidirect
47. C2×D248C22semidirect
48. C2×D248C22semidirect
50. C2×#92470C22semidirect
51. C2×#92470C22semidirect
52. #11758478C22semidirect
53. #11758478C22semidirect
54. #11758452C22semidirect
55. #11758451C22semidirect
56. C31×#108C22semidirect
57. C31×#104C22semidirect
58. C31×#104C22semidirect
59. C31×#97C22semidirect
60. #11758412C22semidirect
61. #11758411C22semidirect
62. #11758411C22semidirect
63. #11758411C22semidirect
64. #11758407C22semidirect
65. #11758393C22semidirect
66. #11758393C22semidirect
96. Q16×C31D4semidirect
97. SD16×C31D4semidirect
98. SD16×C31D4semidirect
99. D8×C31D4semidirect
121. Dic124D4semidirect
129. D248D4semidirect
138. #92470D4semidirect
139. #92470D4semidirect
236. #444D31semidirect
282. #131C2×D31semidirect
283. #131C2×D31semidirect
287. #108C2×D31semidirect
290. #104C2×D31semidirect
291. #104C2×D31semidirect
296. #97C2×D31semidirect
297. C31#3288semidirect
330. Q16D124semidirect
345. SD16D124semidirect
346. SD16D124semidirect
358. D8D124semidirect

Maximal subgroups

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

subgroupindex
#145218612
#145218562
#145218542
#145218512
#145214382
#145213202
#145213192
#145213132
#145213122
#145212282
#145212232
#145212182
#145211722
C31×#4442
#328831