Finite Group Databasebeta

← #16857472

Subgroups — #16857472

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#14521860C211yesmaximal, characteristic, complemented
2#14521859C211yesmaximal, characteristic, complemented
3#14521853C211yesmaximal, characteristic, complemented
4#14521852C211yesmaximal, characteristic, complemented
5#14521436C211yesmaximal, characteristic, complemented
6#14521318C211yesmaximal, characteristic, complemented
7#14521317C211yesmaximal, characteristic, complemented
8#14521311C211yesmaximal, characteristic, complemented
9#14521310C211yesmaximal, characteristic, complemented
10#14521228C211yesmaximal, characteristic, complemented
11#14521223C211yesmaximal, characteristic, complemented
12#14521216C211yesmaximal, characteristic, complemented
13#14521168C211yesmaximal, characteristic, complemented
14#14521168C211yesmaximal, characteristic, complemented
15C31×#453C211yesmaximal, characteristic, complemented
16C31×H(5,𝔽2)C2211yescharacteristic, complemented
17#11758567C2211yescharacteristic, complemented
18#11758566C2211yescharacteristic
19#11758561C2211yescharacteristic, complemented
20C31×AGL(1,ℤ/8)C2211yescharacteristic
21C31×AGL(1,ℤ/8)C2211yescharacteristic, complemented
22C31×#127C2211yescharacteristic, complemented
23#11758528—21no—
24#11758528C2211yescharacteristic, complemented
25#11758527—21no—
26#11758527—21no—
27#11758527C2211yescharacteristic
28#11758527C2211yescharacteristic
29#11758526—21no—
30#11758526C2211yescharacteristic, complemented
31#11758525C2211yescharacteristic, complemented
32#11758525C2211yescharacteristic, complemented
33C2×#92480—21no—
34C2×#92480C2211yescharacteristic, complemented
35C2×#92480C2211yescharacteristic, complemented
36#11758509—21no—
37#11758509—21no—
38#11758509C2211yescharacteristic
39#11758509C2211yescharacteristic, complemented
40C2×#92479—21no—
41C2×#92479C2211yescharacteristic, complemented
42C2×#92479C2211yescharacteristic, complemented
43#11758493—21no—
44#11758492—21no—
45#11758491—21no—
46SD16×D31—21no—
47#11758489—21no—
48#11758488—21no—
49#11758488—21no—
50D8×D31—21no—
51#11758486C2211yescharacteristic, complemented
52#11758485C2211yescharacteristic, complemented
53#11758484C2211yescharacteristic, complemented
54#11758484C2211yescharacteristic, complemented
55#11758452C2211yescharacteristic, complemented
56C31×#104C2211yescharacteristic, complemented
57C31×#104C2211yescharacteristic, complemented
58C31×#100C2211yescharacteristic, complemented
59#11758421C2211yescharacteristic, complemented
60#11758412C2211yescharacteristic, complemented
61#11758411C2211yescharacteristic, complemented
62#11758410C2211yescharacteristic, complemented
63#11758410C2211yescharacteristic, complemented
64#11758393C2211yescharacteristic, complemented
65#11758393C2211yescharacteristic, complemented
66#11758391C2211yescharacteristic, complemented
67#41×C31—21no—
68#41×C31—21no—
69#41×C31C2311yescharacteristic
70#41×C31C2311yescharacteristic
71C2×D4×C31—21no—
72C2×D4×C31—21no—
73C2×D4×C31—41no—
74C2×D4×C31—41no—
75C2×D4×C31C2311yescharacteristic
76C2×D4×C31C2311yescharacteristic
77C2×C22⋊D31—41no—
78#92499—21no—
79Q8×D31—21no—
80#92497—21no—
81#92497—21no—
82#92497—21no—
83#92497—41no—
84D4×D31—21no—
85D4×D31—21no—
86D4×D31—21no—
87D4×D31—41no—
88#92495—21no—
89#92495—21no—
90#92495C2311yescharacteristic
91#92495C2311yescharacteristic
92C2×D124—21no—
93C2×D124C2311yescharacteristic
94C2×Dic62—21no—
95C2×Dic62C2311yescharacteristic
96SD16×C31—21no—
97SD16×C31—21no—
98D8×C31—21no—
99D8×C31—21no—
100M16×C31C2311yescharacteristic
101M16×C31C2311yescharacteristic
102C42×C31C2311yescharacteristic
103#92482—21no—
104#92482D411yescharacteristic, complemented
105#92482D411yescharacteristic, complemented
106#92481—21no—
107#92481D411yescharacteristic, complemented
108#92481D411yescharacteristic, complemented
109#92480—21no—
110#92480—21no—
111#92480—21no—
112#92480—21no—
113#92480—21no—
114#92480—21no—
115#92480—21no—
116#92480D411yescharacteristic, complemented
117#92480D411yescharacteristic, complemented
118#92479—21no—
119#92479—21no—
120#92479—21no—
121#92479—21no—
122#92479—21no—
123#92479—21no—
124#92479—21no—
125#92479D411yescharacteristic, complemented
126#92479D411yescharacteristic, complemented
127(C2×D31)⋊C4—41no—
128#92474—21no—
129#92474—21no—
130#92474C2311yescharacteristic
131#92474C2311yescharacteristic
132C2×C31⋊C8—21no—
133C2×C31⋊C8—21no—
134C2×C31⋊C8C2311yescharacteristic
135C2×C31⋊C8C2311yescharacteristic
136Dic124—21no—
137D248—21no—
138#92470—21no—
139#92470—21no—
140#92469—21no—
141#92469—21no—
142C8×D31—21no—
143C8×D31—21no—
144C23×C31—21no—
145C23×C31—21no—
146C23×C31—41no—
147C22×D31—21no—
148C22×D31—41no—
149C22×D31—41no—
150Q8×C31C2×D411yescharacteristic
151Q8×C31C2×D411yescharacteristic
152D4×C31—21no—
153D4×C31—21no—
154D4×C31—21no—
155D4×C31—21no—
156D4×C31—21no—
157D4×C31—21no—
158D4×C31—41no—
159D4×C31—41no—
160D4×C31—82no—
161D4×C31C2×D411yescharacteristic
162D4×C31C2×D411yescharacteristic
163C2×C4×C31—21no—
164C2×C4×C31—21no—
165C2×C4×C31—21no—
166C2×C4×C31—41no—
167C2×C4×C31C2411yescharacteristic
168C22⋊D31—21no—
169C22⋊D31—21no—
170C22⋊D31—41no—
171C22⋊D31—41no—
172C22⋊D31—41no—
173C22⋊D31—41no—
174C2×Dic31—21no—
175C2×Dic31—41no—
176C2×Dic31—41no—
177D124—21no—
178D124—21no—
179D124—41no—
180D124C2×D411yescharacteristic
181D124C2×D411yescharacteristic
182C4×D31—21no—
183C4×D31—21no—
184C4×D31—41no—
185C4×D31—41no—
186Dic62—21no—
187Dic62—21no—
188Dic62—41no—
189Dic62C2×D411yescharacteristic
190Dic62C2×D411yescharacteristic
191C8×C31—21no—
192C8×C31—21no—
193C31⋊C8—21no—
194C31⋊C8—21no—
195C31⋊C8C2×D411yescharacteristic
196C31⋊C8C2×D411yescharacteristic
197C31⋊C8C2×D411yescharacteristic
198C31⋊C8C2×D411yescharacteristic
199#3288—311nomaximal, Sylow
200C22×C31—21no—
201C22×C31—21no—
202C22×C31—41no—
203C22×C31—41no—
204C22×C31—81no—
205C22×C31—81no—
206C22×C31—161no—
207C22×C31H(5,𝔽2)11yescharacteristic
208C2×D31—21no—
209C2×D31—21no—
210C2×D31—41no—
211C2×D31—81no—
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#453D3111yescharacteristic, complemented
235#444—311no—
236#444—311no—
237C2×C31—21no—
238C2×C31—41no—
239C2×C31—41no—
240C2×C31—81no—
241C2×C31—81no—
242C2×C31D4211yescharacteristic
243D31—41no—
244D31—41no—
245D31—81no—
2462−1+4—311no—
247H(5,𝔽2)—311no—
248H(5,𝔽2)—311no—
249H(5,𝔽2)C2×D3111yescharacteristic, complemented
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)—621no—
259AGL(1,ℤ/8)—621no—
260AGL(1,ℤ/8)—621no—
261AGL(1,ℤ/8)—621no—
262AGL(1,ℤ/8)—621no—
263AGL(1,ℤ/8)—621no—
264AGL(1,ℤ/8)C2×D3111yescharacteristic
265AGL(1,ℤ/8)C2×D3111yescharacteristic, complemented
266#135—311no—
267#135—311no—
268#135—621no—
269#135—621no—
270#135—621no—
271#135—621no—
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#131—311no—
283#131—311no—
284#127C2×D3111yescharacteristic, complemented
285#124—311no—
286#108—311no—
287#108—311no—
288#104—311no—
289#104—311no—
290#104C2×D3111yescharacteristic, complemented
291#104C2×D3111yescharacteristic, complemented
292#101—311no—
293#100—311no—
294#100—311no—
295#100C2×D3111yescharacteristic, complemented
296#97—311no—
297C31#328811yescharacteristic, complemented, Sylow
298#41—21no—
299#41—21no—
300#41—311no—
301#41—311no—
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—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—311no—
319C2×D4—621no—
320C2×D4—621no—
321C2×D4—621no—
322C2×D4—621no—
323C2×D4—1241no—
324C2×D4—1241no—
325C2×D4C22×D3111yescharacteristic
326C2×D4C22×D3111yescharacteristic
327Q16—311no—
328Q16—311no—
329Q16—621no—
330Q16—621no—
331SD16—21no—
332SD16—21no—
333SD16—311no—
334SD16—311no—
335SD16—311no—
336SD16—311no—
337SD16—621no—
338SD16—621no—
339SD16—621no—
340SD16—621no—
341SD16—621no—
342SD16—621no—
343SD16—621no—
344SD16—621no—
345SD16—621no—
346SD16—621no—
347D8—21no—
348D8—21no—
349D8—311no—
350D8—311no—
351D8—621no—
352D8—621no—
353D8—621no—
354D8—621no—
355D8—621no—
356D8—621no—
357D8—621no—
358D8—621no—
359M16—311no—
360M16—311no—
361M16—621no—
362M16—621no—
363M16—621no—
364M16—621no—
365M16C22×D3111yescharacteristic
366M16C22×D3111yescharacteristic
367C2×C8—311no—
368C2×C8—311no—
369C2×C8—621no—
370C2×C8—621no—
371C2×C8—621no—
372C2×C8—621no—
373C22⋊C4—1241no—
374C42C22×D3111yescharacteristic
375C23—21no—
376C23—21no—
377C23—41no—
378C23—621no—
379C23—1241no—
380C23—1241no—
381Q8—311no—
382Q8—311no—
383Q8—621no—
384Q8—621no—
385Q8—1241no—
386Q8D4×D3111yescharacteristic
387Q8D4×D3111yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—41no—
395D4—41no—
396D4—82no—
397D4—311no—
398D4—311no—
399D4—621no—
400D4—621no—
401D4—621no—
402D4—621no—
403D4—1241no—
404D4—1241no—
405D4—1241no—
406D4—1241no—
407D4—1241no—
408D4D4×D3111yescharacteristic
409D4D4×D3111yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—621no—
415C2×C4—621no—
416C2×C4—621no—
417C2×C4—1241no—
418C2×C4—1241no—
419C2×C4—1241no—
420C2×C4—1241no—
421C2×C4C23×D3111yescharacteristic
422C8—21no—
423C8—21no—
424C8—311no—
425C8—311no—
426C8—311no—
427C8—311no—
428C8—621no—
429C8—621no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—81no—
435C22—81no—
436C22—161no—
437C22—621no—
438C22—621no—
439C22—1241no—
440C22—4961no—
441C22—4961no—
442C22—2481no—
443C22#1175856111yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—621no—
448C4—621no—
449C4—1241no—
450C4C2×D4×D3111yescharacteristic
451C4C2×D4×D3111yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2—81no—
457C2—1241no—
458C2—1241no—
459C2—2481no—
460C2#1452168111yescharacteristic
subgroupG/H
centerC2#14521681
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#453C2
socleC2×C31D42
hypercenter#453D31
hypocenterC31#3288
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #14521860C2semidirect
2. #14521859C2semidirect
3. #14521853C2semidirect
4. #14521852C2semidirect
5. #14521436C2semidirect
6. #14521318C2semidirect
7. #14521317C2semidirect
8. #14521311C2semidirect
9. #14521310C2semidirect
10. #14521228C2semidirect
11. #14521223C2semidirect
12. #14521216C2semidirect
13. #14521168C2semidirect
14. #14521168C2semidirect
15. C31×#453C2semidirect
16. C31×H(5,𝔽2)C22semidirect
17. #11758567C22semidirect
19. #11758561C22semidirect
21. C31×AGL(1,ℤ/8)C22semidirect
22. C31×#127C22semidirect
24. #11758528C22semidirect
30. #11758526C22semidirect
31. #11758525C22semidirect
32. #11758525C22semidirect
34. C2×#92480C22semidirect
35. C2×#92480C22semidirect
39. #11758509C22semidirect
41. C2×#92479C22semidirect
42. C2×#92479C22semidirect
51. #11758486C22semidirect
52. #11758485C22semidirect
53. #11758484C22semidirect
54. #11758484C22semidirect
55. #11758452C22semidirect
56. C31×#104C22semidirect
57. C31×#104C22semidirect
58. C31×#100C22semidirect
59. #11758421C22semidirect
60. #11758412C22semidirect
61. #11758411C22semidirect
62. #11758410C22semidirect
63. #11758410C22semidirect
64. #11758393C22semidirect
65. #11758393C22semidirect
66. #11758391C22semidirect
104. #92482D4semidirect
105. #92482D4semidirect
107. #92481D4semidirect
108. #92481D4semidirect
116. #92480D4semidirect
117. #92480D4semidirect
125. #92479D4semidirect
126. #92479D4semidirect
234. #453D31semidirect
249. H(5,𝔽2)C2×D31semidirect
265. AGL(1,ℤ/8)C2×D31semidirect
284. #127C2×D31semidirect
290. #104C2×D31semidirect
291. #104C2×D31semidirect
295. #100C2×D31semidirect
297. C31#3288semidirect

Maximal subgroups

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

subgroupindex
#145218602
#145218592
#145218532
#145218522
#145214362
#145213182
#145213172
#145213112
#145213102
#145212282
#145212232
#145212162
#145211682
C31×#4532
#328831