Finite Group Databasebeta

← #16857502

Subgroups — #16857502

Order 3968.

460 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521860C211yesmaximal, characteristic, complemented
2#14521857C211yesmaximal, characteristic, complemented
3#14521856C211yesmaximal, characteristic, complemented
4#14521420C211yesmaximal, characteristic, complemented
5#14521314C211yesmaximal, characteristic, complemented
6#14521312C211yesmaximal, characteristic, complemented
7#14521310C211yesmaximal, characteristic, complemented
8#14521309C211yesmaximal, characteristic, complemented
9#14521232C211yesmaximal, characteristic, complemented
10#14521230C211yesmaximal, characteristic, complemented
11#14521217C211yesmaximal, characteristic, complemented
12#14521215C211yesmaximal, characteristic, complemented
13#14521166C211yesmaximal, characteristic, complemented
14#14521164C211yesmaximal, characteristic, complemented
15C31×#577C211yesmaximal, characteristic, complemented
16C31×2−1+4C2211yescharacteristic
17C31×H(5,𝔽2)C2211yescharacteristic, complemented
18#11758567C2211yescharacteristic
19#11758566C2211yescharacteristic, complemented
20C31×#137—21no—
21C31×#137C2211yescharacteristic, complemented
22C31×AGL(1,ℤ/8)—21no—
23C31×AGL(1,ℤ/8)C2211yescharacteristic
24C31×#135—21no—
25C31×#135C2211yescharacteristic
26C2×SD16×C31—21no—
27C2×SD16×C31C2211yescharacteristic, complemented
28C31×#131C2211yescharacteristic, complemented
29#11758528—21no—
30#11758528C2211yescharacteristic, complemented
31#11758528C2211yescharacteristic, complemented
32#11758527—21no—
33#11758527—21no—
34#11758527C2211yescharacteristic
35#11758526—21no—
36#11758526C2211yescharacteristic
37#11758526C2211yescharacteristic
38#11758525C2211yescharacteristic, complemented
39#11758525C2211yescharacteristic, complemented
40#11758521C2211yescharacteristic, complemented
41#11758519—21no—
42#11758519C2211yescharacteristic, complemented
43C2×#92481—21no—
44C2×#92481C2211yescharacteristic, complemented
45#11758517C2211yescharacteristic
46C2×#92480—21no—
47C2×#92480C2211yescharacteristic, complemented
48#11758509—21no—
49#11758509C2211yescharacteristic
50#11758495—21no—
51#11758493—21no—
52SD16×D31—21no—
53#11758488—21no—
54C2×#92470C2211yescharacteristic, complemented
55#11758478C2211yescharacteristic, complemented
56#11758425C2211yescharacteristic, complemented
57#11758425C2211yescharacteristic, complemented
58#11758425C2211yescharacteristic, complemented
59#11758424C2211yescharacteristic, complemented
60#11758421C2211yescharacteristic, complemented
61#11758413C2211yescharacteristic, complemented
62#11758412C2211yescharacteristic, complemented
63#11758411C2211yescharacteristic, complemented
64#11758410C2211yescharacteristic, complemented
65#11758407C2211yescharacteristic, complemented
66C8×Dic31C2211yescharacteristic, complemented
67#41×C31—21no—
68#41×C31—21no—
69#41×C31—21no—
70#41×C31—21no—
71#41×C31—41no—
72#41×C31C2311yescharacteristic
73#41×C31C2311yescharacteristic
74#41×C31C2311yescharacteristic
75C2×Q8×C31—21no—
76C2×Q8×C31—21no—
77C2×Q8×C31C2311yescharacteristic
78C2×D4×C31—21no—
79C2×D4×C31—21no—
80C2×D4×C31—41no—
81C2×D4×C31C2311yescharacteristic
82C2×C22⋊D31—41no—
83#92499—21no—
84Q8×D31—21no—
85#92497—21no—
86#92497—41no—
87D4×D31—21no—
88D4×D31—41no—
89#92495—21no—
90#92495—21no—
91#92495C2311yescharacteristic
92C2×D124—21no—
93C2×D124C2311yescharacteristic
94C2×Dic62—21no—
95C2×Dic62C2311yescharacteristic
96Q16×C31—21no—
97Q16×C31D411yescharacteristic, complemented
98SD16×C31—21no—
99SD16×C31—21no—
100SD16×C31—21no—
101SD16×C31—21no—
102SD16×C31D411yescharacteristic, complemented
103SD16×C31D411yescharacteristic, complemented
104D8×C31—21no—
105D8×C31D411yescharacteristic, complemented
106M16×C31—21no—
107M16×C31C2311yescharacteristic
108C2×C8×C31—21no—
109C2×C8×C31C2311yescharacteristic
110#92482—21no—
111#92482—21no—
112#92482—21no—
113#92482D411yescharacteristic, complemented
114#92481—21no—
115#92481—21no—
116#92481—21no—
117#92481—21no—
118#92481—21no—
119#92481D411yescharacteristic, complemented
120#92480—21no—
121#92480—21no—
122#92480—21no—
123#92480—21no—
124#92480—21no—
125#92480D411yescharacteristic, complemented
126#92479—21no—
127#92479—21no—
128#92479—21no—
129#92479D411yescharacteristic, complemented
130Dic31⋊C4—41no—
131C4×Dic31C2311yescharacteristic
132#92474—21no—
133#92474—21no—
134#92474C2311yescharacteristic
135#92474C2311yescharacteristic
136#92474C2311yescharacteristic
137C2×C31⋊C8—21no—
138C2×C31⋊C8—21no—
139C2×C31⋊C8C2311yescharacteristic
140#92470—21no—
141#92470—21no—
142#92469—21no—
143C8×D31—21no—
144C23×C31—21no—
145C23×C31—41no—
146C22×D31—21no—
147C22×D31—41no—
148Q8×C31—21no—
149Q8×C31—21no—
150Q8×C31—41no—
151Q8×C31C2×D411yescharacteristic
152Q8×C31C2×D411yescharacteristic
153Q8×C31C2×D411yescharacteristic
154D4×C31—21no—
155D4×C31—21no—
156D4×C31—21no—
157D4×C31—21no—
158D4×C31—21no—
159D4×C31—41no—
160D4×C31—41no—
161D4×C31—41no—
162D4×C31C2×D411yescharacteristic
163D4×C31C2×D411yescharacteristic
164D4×C31C2×D411yescharacteristic
165C2×C4×C31—21no—
166C2×C4×C31—21no—
167C2×C4×C31—21no—
168C2×C4×C31—21no—
169C2×C4×C31—41no—
170C2×C4×C31—41no—
171C2×C4×C31—41no—
172C2×C4×C31C2411yescharacteristic
173C22⋊D31—21no—
174C22⋊D31—41no—
175C22⋊D31—41no—
176C22⋊D31—82no—
177C2×Dic31—21no—
178C2×Dic31—21no—
179C2×Dic31—41no—
180D124—21no—
181D124—21no—
182D124—41no—
183D124C2×D411yescharacteristic
184C4×D31—21no—
185C4×D31—41no—
186C4×D31—41no—
187Dic62—21no—
188Dic62—21no—
189Dic62—41no—
190Dic62C2×D411yescharacteristic
191C8×C31—21no—
192C8×C31C2×D411yescharacteristic
193C8×C31C2×D411yescharacteristic
194C31⋊C8—21no—
195C31⋊C8—21no—
196C31⋊C8—21no—
197C31⋊C8C2×D411yescharacteristic
198C31⋊C8C2×D411yescharacteristic
199#3291—311nomaximal, Sylow
200C22×C31—21no—
201C22×C31—21no—
202C22×C31—21no—
203C22×C31—41no—
204C22×C31—81no—
205C22×C31—161no—
206C22×C31H(5,𝔽2)11yescharacteristic
207C2×D31—21no—
208C2×D31—41no—
209C2×D31—81no—
210C2×D31—161no—
211C4×C31—21no—
212C4×C31—21no—
213C4×C31—21no—
214C4×C31—41no—
215C4×C31C22×D411yescharacteristic
216C4×C31C22×D411yescharacteristic
217Dic31—22no—
218Dic31—21no—
219Dic31—41no—
220#577—311no—
221#577—311no—
222#577—311no—
223#577D3111yescharacteristic, complemented
224#492—311no—
225#471—311no—
226#471—311no—
227#471—311no—
228#471—311no—
229#456—311no—
230#456—311no—
231#453—311no—
232#453—311no—
233#443—311no—
234#443—311no—
235C2×C31—21no—
236C2×C31—41no—
237C2×C31—41no—
238C2×C31—41no—
239C2×C31—81no—
240C2×C31D4211yescharacteristic
241D31—41no—
242D31—81no—
2432−1+4—311no—
2442−1+4C2×D3111yescharacteristic
245H(5,𝔽2)—311no—
246H(5,𝔽2)C2×D3111yescharacteristic, complemented
247#137—21no—
248#137—311no—
249#137—311no—
250#137—311no—
251#137—621no—
252#137—621no—
253#137—621no—
254#137C2×D3111yescharacteristic, complemented
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—311no—
257AGL(1,ℤ/8)—311no—
258AGL(1,ℤ/8)—311no—
259AGL(1,ℤ/8)—621no—
260AGL(1,ℤ/8)—621no—
261AGL(1,ℤ/8)—621no—
262AGL(1,ℤ/8)C2×D3111yescharacteristic
263#135—21no—
264#135—311no—
265#135—621no—
266#135—621no—
267#135—621no—
268#135C2×D3111yescharacteristic
269C2×SD16—21no—
270C2×SD16—311no—
271C2×SD16—311no—
272C2×SD16—311no—
273C2×SD16—621no—
274C2×SD16—621no—
275C2×SD16—621no—
276C2×SD16C2×D3111yescharacteristic, complemented
277#131—311no—
278#131—311no—
279#131—311no—
280#131C2×D3111yescharacteristic, complemented
281C4⋊Q8—311no—
282#127—311no—
283#108—311no—
284#108—311no—
285#104—311no—
286#104—311no—
287#104—311no—
288#104—311no—
289#101—311no—
290#101—311no—
291#100—311no—
292#100—311no—
293C4×C8—311no—
294C31#329111yescharacteristic, complemented, Sylow
295#41—21no—
296#41—21no—
297#41—21no—
298#41—21no—
299#41—41no—
300#41—311no—
301#41—621no—
302#41—621no—
303#41—621no—
304#41—621no—
305#41—1241no—
306#41C22×D3111yescharacteristic
307#41C22×D3111yescharacteristic
308#41C22×D3111yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—311no—
312C2×Q8—621no—
313C2×Q8—621no—
314C2×Q8C22×D3111yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—311no—
319C2×D4—621no—
320C2×D4—621no—
321C2×D4—1241no—
322C2×D4—1241no—
323C2×D4C22×D3111yescharacteristic
324Q16—21no—
325Q16—311no—
326Q16—621no—
327Q16—621no—
328Q16—621no—
329Q16C22⋊D3111yescharacteristic, complemented
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—311no—
335SD16—311no—
336SD16—621no—
337SD16—621no—
338SD16—621no—
339SD16—621no—
340SD16—621no—
341SD16—621no—
342SD16—621no—
343SD16—621no—
344SD16—621no—
345SD16—621no—
346SD16—621no—
347SD16—621no—
348SD16C22⋊D3111yescharacteristic, complemented
349SD16C22⋊D3111yescharacteristic, complemented
350D8—21no—
351D8—311no—
352D8—621no—
353D8—621no—
354D8—621no—
355D8C22⋊D3111yescharacteristic, complemented
356M16—21no—
357M16—311no—
358M16—311no—
359M16—311no—
360M16—621no—
361M16—621no—
362M16—621no—
363M16C22×D3111yescharacteristic
364C2×C8—21no—
365C2×C8—311no—
366C2×C8—621no—
367C2×C8—621no—
368C2×C8—621no—
369C2×C8C22×D3111yescharacteristic
370C4⋊C4—1241no—
371C42—311no—
372C23—21no—
373C23—41no—
374C23—621no—
375C23—1241no—
376Q8—21no—
377Q8—21no—
378Q8—41no—
379Q8—311no—
380Q8—621no—
381Q8—621no—
382Q8—1241no—
383Q8D4×D3111yescharacteristic
384Q8C2×C22⋊D3111yescharacteristic
385Q8C2×C22⋊D3111yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—311no—
395D4—621no—
396D4—621no—
397D4—621no—
398D4—1241no—
399D4—1241no—
400D4—1241no—
401D4—2482no—
402D4D4×D3111yescharacteristic
403D4C2×C22⋊D3111yescharacteristic
404D4C2×C22⋊D3111yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—621no—
413C2×C4—621no—
414C2×C4—621no—
415C2×C4—1241no—
416C2×C4—1241no—
417C2×C4—1241no—
418C2×C4C23×D3111yescharacteristic
419C8—21no—
420C8—311no—
421C8—311no—
422C8—621no—
423C8—621no—
424C8—621no—
425C8C2×C22⋊D3111yescharacteristic
426C8C2×C22⋊D3111yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—41no—
431C22—81no—
432C22—161no—
433C22—621no—
434C22—1241no—
435C22—4961no—
436C22—2481no—
437C22#1175856111yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—622no—
443C4—621no—
444C4—1241no—
445C4C2×D4×D3111yescharacteristic
446C4C22×C22⋊D3111yescharacteristic
447C2—21no—
448C2—41no—
449C2—41no—
450C2—41no—
451C2—81no—
452C2—1241no—
453C2—2481no—
454C2D4×C22⋊D3111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#577C2
socleC2×C31D42
hypercenter#577D31
hypocenterC31#3291
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #14521860C2semidirect
2. #14521857C2semidirect
3. #14521856C2semidirect
4. #14521420C2semidirect
5. #14521314C2semidirect
6. #14521312C2semidirect
7. #14521310C2semidirect
8. #14521309C2semidirect
9. #14521232C2semidirect
10. #14521230C2semidirect
11. #14521217C2semidirect
12. #14521215C2semidirect
13. #14521166C2semidirect
14. #14521164C2semidirect
15. C31×#577C2semidirect
17. C31×H(5,𝔽2)C22semidirect
19. #11758566C22semidirect
21. C31×#137C22semidirect
27. C2×SD16×C31C22semidirect
28. C31×#131C22semidirect
30. #11758528C22semidirect
31. #11758528C22semidirect
38. #11758525C22semidirect
39. #11758525C22semidirect
40. #11758521C22semidirect
42. #11758519C22semidirect
44. C2×#92481C22semidirect
47. C2×#92480C22semidirect
54. C2×#92470C22semidirect
55. #11758478C22semidirect
56. #11758425C22semidirect
57. #11758425C22semidirect
58. #11758425C22semidirect
59. #11758424C22semidirect
60. #11758421C22semidirect
61. #11758413C22semidirect
62. #11758412C22semidirect
63. #11758411C22semidirect
64. #11758410C22semidirect
65. #11758407C22semidirect
66. C8×Dic31C22semidirect
97. Q16×C31D4semidirect
102. SD16×C31D4semidirect
103. SD16×C31D4semidirect
105. D8×C31D4semidirect
113. #92482D4semidirect
119. #92481D4semidirect
125. #92480D4semidirect
129. #92479D4semidirect
223. #577D31semidirect
246. H(5,𝔽2)C2×D31semidirect
254. #137C2×D31semidirect
276. C2×SD16C2×D31semidirect
280. #131C2×D31semidirect
294. C31#3291semidirect
329. Q16C22⋊D31semidirect
348. SD16C22⋊D31semidirect
349. SD16C22⋊D31semidirect
355. D8C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
#145218602
#145218572
#145218562
#145214202
#145213142
#145213122
#145213102
#145213092
#145212322
#145212302
#145212172
#145212152
#145211662
#145211642
C31×#5772
#329131