Finite Group Databasebeta

← #16857503

Subgroups — #16857503

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#14521861C211yesmaximal, characteristic, complemented
2#14521859C211yesmaximal, characteristic, complemented
3#14521858C211yesmaximal, characteristic, complemented
4#14521855C211yesmaximal, characteristic, complemented
5#14521420C211yesmaximal, characteristic, complemented
6#14521315C211yesmaximal, characteristic, complemented
7#14521313C211yesmaximal, characteristic, complemented
8#14521311C211yesmaximal, characteristic, complemented
9#14521234C211yesmaximal, characteristic, complemented
10#14521230C211yesmaximal, characteristic, complemented
11#14521219C211yesmaximal, characteristic, complemented
12#14521215C211yesmaximal, characteristic, complemented
13#14521165C211yesmaximal, characteristic, complemented
14#14521164C211yesmaximal, characteristic, complemented
15C31×#471C211yesmaximal, characteristic, complemented
16#11758567C2211yescharacteristic
17#11758566C2211yescharacteristic, complemented
18#11758564C2211yescharacteristic
19#11758561C2211yescharacteristic, complemented
20C31×#137C2211yescharacteristic, complemented
21C31×AGL(1,ℤ/8)C2211yescharacteristic
22C2×SD16×C31C2211yescharacteristic, complemented
23C31×#131C2211yescharacteristic, complemented
24#11758527C2211yescharacteristic
25#11758521C2211yescharacteristic, complemented
26#11758519C2211yescharacteristic, complemented
27C2×#92481C2211yescharacteristic, complemented
28#11758517C2211yescharacteristic
29C2×#92480C2211yescharacteristic, complemented
30#11758509C2211yescharacteristic
31#11758496—21no—
32#11758495—21no—
33#11758493—21no—
34#11758492—21no—
35#11758492—21no—
36#11758491—21no—
37#11758491—21no—
38SD16×D31—21no—
39SD16×D31—21no—
40SD16×D31—21no—
41#11758489—21no—
42#11758488—21no—
43#11758486—21no—
44#11758486C2211yescharacteristic, complemented
45#11758486C2211yescharacteristic, complemented
46#11758485—21no—
47#11758485C2211yescharacteristic
48#11758485C2211yescharacteristic
49#11758484C2211yescharacteristic, complemented
50#11758484C2211yescharacteristic, complemented
51#11758481—21no—
52#11758481C2211yescharacteristic
53C2×#92470—21no—
54C2×#92470C2211yescharacteristic, complemented
55#11758478C2211yescharacteristic, complemented
56C31×#108C2211yescharacteristic, complemented
57C31×#101C2211yescharacteristic, complemented
58C31×#100C2211yescharacteristic, complemented
59#11758425C2211yescharacteristic, complemented
60#11758413C2211yescharacteristic, complemented
61#11758413C2211yescharacteristic, complemented
62#11758413C2211yescharacteristic, complemented
63#11758412C2211yescharacteristic, complemented
64#11758411C2211yescharacteristic, complemented
65#11758410C2211yescharacteristic, complemented
66C8×Dic31C2211yescharacteristic, complemented
67#41×C31C2311yescharacteristic
68C2×Q8×C31C2311yescharacteristic
69C2×D4×C31C2311yescharacteristic
70C2×C22⋊D31—41no—
71C2×C22⋊D31—41no—
72#92499—21no—
73#92499—21no—
74#92499—21no—
75Q8×D31—21no—
76Q8×D31—21no—
77Q8×D31—21no—
78#92497—21no—
79#92497—21no—
80#92497—21no—
81#92497—41no—
82D4×D31—21no—
83D4×D31—21no—
84D4×D31—21no—
85D4×D31—41no—
86#92495—21no—
87#92495—21no—
88#92495—41no—
89#92495C2311yescharacteristic
90#92495C2311yescharacteristic
91#92495C2311yescharacteristic
92C2×D124—21no—
93C2×D124C2311yescharacteristic
94C2×Dic62—21no—
95C2×Dic62C2311yescharacteristic
96Q16×C31—21no—
97SD16×C31—21no—
98SD16×C31—21no—
99SD16×C31—21no—
100SD16×C31—21no—
101D8×C31—21no—
102M16×C31—21no—
103M16×C31C2311yescharacteristic
104M16×C31C2311yescharacteristic
105M16×C31C2311yescharacteristic
106C2×C8×C31—21no—
107C2×C8×C31C2311yescharacteristic
108#92482—21no—
109#92482D411yescharacteristic, complemented
110#92481—21no—
111#92481—21no—
112#92481—21no—
113#92481D411yescharacteristic, complemented
114#92480—21no—
115#92480—21no—
116#92480—21no—
117#92480D411yescharacteristic, complemented
118#92479—21no—
119#92479D411yescharacteristic, complemented
120Dic31⋊C4—41no—
121C4×Dic31C2311yescharacteristic
122#92474C2311yescharacteristic
123C2×C31⋊C8C2311yescharacteristic
124Dic124—21no—
125Dic124—21no—
126Dic124D411yescharacteristic, complemented
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#92470D411yescharacteristic, complemented
137#92470D411yescharacteristic, complemented
138#92469—21no—
139#92469—21no—
140#92469—21no—
141C8×D31—21no—
142C8×D31—21no—
143C8×D31—21no—
144C23×C31—21no—
145C22×D31—21no—
146C22×D31—41no—
147C22×D31—41no—
148Q8×C31—21no—
149Q8×C31—21no—
150Q8×C31C2×D411yescharacteristic
151D4×C31—21no—
152D4×C31—21no—
153D4×C31—21no—
154D4×C31C2×D411yescharacteristic
155C2×C4×C31—21no—
156C2×C4×C31—21no—
157C2×C4×C31C2411yescharacteristic
158C22⋊D31—21no—
159C22⋊D31—21no—
160C22⋊D31—21no—
161C22⋊D31—41no—
162C22⋊D31—41no—
163C22⋊D31—41no—
164C22⋊D31—41no—
165C22⋊D31—82no—
166C2×Dic31—21no—
167C2×Dic31—21no—
168C2×Dic31—41no—
169C2×Dic31—41no—
170D124—21no—
171D124—21no—
172D124—41no—
173D124—41no—
174D124C2×D411yescharacteristic
175D124C2×D411yescharacteristic
176D124C2×D411yescharacteristic
177C4×D31—21no—
178C4×D31—21no—
179C4×D31—21no—
180C4×D31—41no—
181C4×D31—41no—
182C4×D31—41no—
183C4×D31—41no—
184Dic62—21no—
185Dic62—21no—
186Dic62—41no—
187Dic62—41no—
188Dic62C2×D411yescharacteristic
189Dic62C2×D411yescharacteristic
190Dic62C2×D411yescharacteristic
191C8×C31—21no—
192C8×C31—21no—
193C8×C31—21no—
194C8×C31C2×D411yescharacteristic
195C8×C31C2×D411yescharacteristic
196C31⋊C8—21no—
197C31⋊C8C2×D411yescharacteristic
198C31⋊C8C2×D411yescharacteristic
199#3291—311nomaximal, Sylow
200C22×C31—21no—
201C22×C31—41no—
202C22×C31—81no—
203C22×C31H(5,𝔽2)11yescharacteristic
204C2×D31—21no—
205C2×D31—21no—
206C2×D31—21no—
207C2×D31—41no—
208C2×D31—81no—
209C2×D31—161no—
210C2×D31—161no—
211C4×C31—21no—
212C4×C31—41no—
213C4×C31C22×D411yescharacteristic
214C4×C31C22×D411yescharacteristic
215Dic31—22no—
216Dic31—21no—
217Dic31—21no—
218Dic31—21no—
219Dic31—41no—
220#577—311no—
221#577—311no—
222#577—311no—
223#577—311no—
224#492—311no—
225#471—311no—
226#471—311no—
227#471—311no—
228#471D3111yescharacteristic, complemented
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—81no—
238C2×C31D4211yescharacteristic
239D31—41no—
240D31—41no—
241D31—41no—
242D31—81no—
2432−1+4—311no—
2442−1+4—311no—
245H(5,𝔽2)—311no—
246H(5,𝔽2)—311no—
247#137—311no—
248#137—311no—
249#137—311no—
250#137—621no—
251#137—621no—
252#137—621no—
253#137—621no—
254#137C2×D3111yescharacteristic, complemented
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)C2×D3111yescharacteristic
263#135—311no—
264#135—311no—
265#135—621no—
266#135—621no—
267#135—621no—
268#135—621no—
269C2×SD16—311no—
270C2×SD16—311no—
271C2×SD16—311no—
272C2×SD16—621no—
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#108C2×D3111yescharacteristic, complemented
285#104—311no—
286#104—311no—
287#104—311no—
288#104—311no—
289#101—311no—
290#101C2×D3111yescharacteristic, complemented
291#100—311no—
292#100C2×D3111yescharacteristic, complemented
293C4×C8—311no—
294C31#329111yescharacteristic, complemented, Sylow
295#41—311no—
296#41—311no—
297#41—311no—
298#41—621no—
299#41—621no—
300#41—621no—
301#41—621no—
302#41—621no—
303#41—621no—
304#41—621no—
305#41—621no—
306#41—1241no—
307#41—1241no—
308#41C22×D3111yescharacteristic
309C2×Q8—311no—
310C2×Q8—621no—
311C2×Q8—621no—
312C2×Q8—621no—
313C2×Q8—621no—
314C2×Q8C22×D3111yescharacteristic
315C2×D4—311no—
316C2×D4—621no—
317C2×D4—621no—
318C2×D4—621no—
319C2×D4—621no—
320C2×D4—1241no—
321C2×D4—1241no—
322C2×D4—1241no—
323C2×D4C22×D3111yescharacteristic
324Q16—21no—
325Q16—311no—
326Q16—311no—
327Q16—621no—
328Q16—621no—
329Q16—621no—
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—311no—
335SD16—311no—
336SD16—311no—
337SD16—311no—
338SD16—621no—
339SD16—621no—
340SD16—621no—
341SD16—621no—
342SD16—621no—
343SD16—621no—
344SD16—621no—
345SD16—621no—
346SD16—621no—
347SD16—621no—
348SD16—621no—
349SD16—621no—
350D8—21no—
351D8—311no—
352D8—311no—
353D8—621no—
354D8—621no—
355D8—621no—
356M16—21no—
357M16—311no—
358M16—621no—
359M16—621no—
360M16—621no—
361M16C22×D3111yescharacteristic
362M16C22×D3111yescharacteristic
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—621no—
374C23—1241no—
375C23—1241no—
376Q8—21no—
377Q8—21no—
378Q8—311no—
379Q8—311no—
380Q8—311no—
381Q8—621no—
382Q8—621no—
383Q8—1241no—
384Q8—1241no—
385Q8D4×D3111yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—311no—
390D4—311no—
391D4—311no—
392D4—621no—
393D4—621no—
394D4—621no—
395D4—621no—
396D4—621no—
397D4—1241no—
398D4—1241no—
399D4—1241no—
400D4—1241no—
401D4—1241no—
402D4—1241no—
403D4—2482no—
404D4D4×D3111yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—621no—
408C2×C4—621no—
409C2×C4—621no—
410C2×C4—621no—
411C2×C4—621no—
412C2×C4—1241no—
413C2×C4—1241no—
414C2×C4—1241no—
415C2×C4—1241no—
416C2×C4—1241no—
417C2×C4—1241no—
418C2×C4C23×D3111yescharacteristic
419C8—21no—
420C8—21no—
421C8—21no—
422C8—311no—
423C8—311no—
424C8—621no—
425C8D4×D3111yescharacteristic
426C8D4×D3111yescharacteristic
427C22—21no—
428C22—41no—
429C22—81no—
430C22—621no—
431C22—621no—
432C22—621no—
433C22—1241no—
434C22—4961no—
435C22—4961no—
436C22—2481no—
437C22#1175856611yescharacteristic
438C4—21no—
439C4—41no—
440C4—622no—
441C4—621no—
442C4—621no—
443C4—621no—
444C4—1241no—
445C4C2×D4×D3111yescharacteristic
446C4C2×D4×D3111yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—1241no—
451C2—1241no—
452C2—1241no—
453C2—2481no—
454C2#1452168311yescharacteristic
subgroupG/H
centerC2#14521683
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#471C2
socleC2×C31D42
hypercenter#471D31
hypocenterC31#3291
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #14521861C2semidirect
2. #14521859C2semidirect
3. #14521858C2semidirect
4. #14521855C2semidirect
5. #14521420C2semidirect
6. #14521315C2semidirect
7. #14521313C2semidirect
8. #14521311C2semidirect
9. #14521234C2semidirect
10. #14521230C2semidirect
11. #14521219C2semidirect
12. #14521215C2semidirect
13. #14521165C2semidirect
14. #14521164C2semidirect
15. C31×#471C2semidirect
17. #11758566C22semidirect
19. #11758561C22semidirect
20. C31×#137C22semidirect
22. C2×SD16×C31C22semidirect
23. C31×#131C22semidirect
25. #11758521C22semidirect
26. #11758519C22semidirect
27. C2×#92481C22semidirect
29. C2×#92480C22semidirect
44. #11758486C22semidirect
45. #11758486C22semidirect
49. #11758484C22semidirect
50. #11758484C22semidirect
54. C2×#92470C22semidirect
55. #11758478C22semidirect
56. C31×#108C22semidirect
57. C31×#101C22semidirect
58. C31×#100C22semidirect
59. #11758425C22semidirect
60. #11758413C22semidirect
61. #11758413C22semidirect
62. #11758413C22semidirect
63. #11758412C22semidirect
64. #11758411C22semidirect
65. #11758410C22semidirect
66. C8×Dic31C22semidirect
109. #92482D4semidirect
113. #92481D4semidirect
117. #92480D4semidirect
119. #92479D4semidirect
126. Dic124D4semidirect
129. D248D4semidirect
136. #92470D4semidirect
137. #92470D4semidirect
228. #471D31semidirect
254. #137C2×D31semidirect
276. C2×SD16C2×D31semidirect
280. #131C2×D31semidirect
284. #108C2×D31semidirect
290. #101C2×D31semidirect
292. #100C2×D31semidirect
294. C31#3291semidirect

Maximal subgroups

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

subgroupindex
#145218612
#145218592
#145218582
#145218552
#145214202
#145213152
#145213132
#145213112
#145212342
#145212302
#145212192
#145212152
#145211652
#145211642
C31×#4712
#329131