Finite Group Databasebeta

← #15685161

Subgroups — #15685161

Order 2944.

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#13093228C211yesmaximal, characteristic, complemented
2#13093224C211yesmaximal, characteristic, complemented
3#13093220C211yesmaximal, characteristic, complemented
4#13092805C211yesmaximal, characteristic, complemented
5#13092685C211yesmaximal, characteristic, complemented
6#13092684C211yesmaximal, characteristic, complemented
7#13092680C211yesmaximal, characteristic, complemented
8#13092678C211yesmaximal, characteristic, complemented
9#13092595C211yesmaximal, characteristic, complemented
10#13092588C211yesmaximal, characteristic, complemented
11#13092585C211yesmaximal, characteristic, complemented
12#13092583C211yesmaximal, characteristic, complemented
13#13092539C211yesmaximal, characteristic, complemented
14#13092537C211yesmaximal, characteristic, complemented
15C23×#576C211yesmaximal, characteristic, complemented
16C23×H(5,𝔽2)C2211yescharacteristic
17C23×H(5,𝔽2)C2211yescharacteristic, complemented
18#10625161C2211yescharacteristic, complemented
19#10625160C2211yescharacteristic, complemented
20C23×AGL(1,ℤ/8)—21no—
21C23×AGL(1,ℤ/8)—21no—
22C23×AGL(1,ℤ/8)C2211yescharacteristic
23C23×AGL(1,ℤ/8)C2211yescharacteristic, complemented
24C23×#135—21no—
25C23×#135C2211yescharacteristic
26C2×D8×C23—21no—
27C2×D8×C23C2211yescharacteristic, complemented
28C23×#131C2211yescharacteristic, complemented
29#10625122—21no—
30#10625122C2211yescharacteristic, complemented
31#10625122C2211yescharacteristic, complemented
32#10625121—21no—
33#10625121—21no—
34#10625121C2211yescharacteristic
35#10625120—21no—
36#10625120C2211yescharacteristic
37#10625120C2211yescharacteristic, complemented
38#10625119C2211yescharacteristic, complemented
39#10625119C2211yescharacteristic, complemented
40#10625111C2211yescharacteristic, complemented
41#10625107C2211yescharacteristic, complemented
42C2×#67273—21no—
43C2×#67273C2211yescharacteristic, complemented
44#10625103—21no—
45#10625103—21no—
46#10625103C2211yescharacteristic, complemented
47#10625103C2211yescharacteristic, complemented
48C2×#67272—21no—
49C2×#67272C2211yescharacteristic, complemented
50#10625089—21no—
51#10625087—21no—
52SD16×D23—21no—
53#10625082—21no—
54C2×#67263C2211yescharacteristic, complemented
55#10625072C2211yescharacteristic, complemented
56#10625019C2211yescharacteristic, complemented
57#10625019C2211yescharacteristic, complemented
58#10625019C2211yescharacteristic, complemented
59#10625015C2211yescharacteristic, complemented
60#10625015C2211yescharacteristic, complemented
61#10625007C2211yescharacteristic, complemented
62#10625006C2211yescharacteristic, complemented
63#10625005C2211yescharacteristic, complemented
64#10625004C2211yescharacteristic, complemented
65#10625001C2211yescharacteristic, complemented
66#10624997C2211yescharacteristic, complemented
67#41×C23—21no—
68#41×C23—21no—
69#41×C23—21no—
70#41×C23—21no—
71#41×C23C2311yescharacteristic
72#41×C23C2311yescharacteristic
73#41×C23C2311yescharacteristic
74C2×D4×C23—21no—
75C2×D4×C23—21no—
76C2×D4×C23—21no—
77C2×D4×C23—21no—
78C2×D4×C23—41no—
79C2×D4×C23—41no—
80C2×D4×C23C2311yescharacteristic
81C2×D4×C23C2311yescharacteristic
82C2×C22⋊D23—41no—
83#67292—21no—
84Q8×D23—21no—
85#67290—21no—
86#67290—41no—
87D4×D23—21no—
88D4×D23—41no—
89#67288—21no—
90#67288—21no—
91#67288C2311yescharacteristic
92C2×D92—21no—
93C2×D92C2311yescharacteristic
94C2×Dic46—21no—
95C2×Dic46C2311yescharacteristic
96Q16×C23D411yescharacteristic, complemented
97SD16×C23—21no—
98SD16×C23—21no—
99SD16×C23D411yescharacteristic, complemented
100SD16×C23D411yescharacteristic, complemented
101D8×C23—21no—
102D8×C23—21no—
103D8×C23—21no—
104D8×C23—21no—
105D8×C23D411yescharacteristic, complemented
106M16×C23—21no—
107M16×C23C2311yescharacteristic
108C2×C8×C23—21no—
109C2×C8×C23C2311yescharacteristic
110C22⋊Dic23—41no—
111#67275—21no—
112#67275—21no—
113#67275D411yescharacteristic, complemented
114#67274—21no—
115#67274—21no—
116#67274D411yescharacteristic, complemented
117#67273—21no—
118#67273—21no—
119#67273—21no—
120#67273—21no—
121#67273—21no—
122#67273—21no—
123#67273D411yescharacteristic, complemented
124#67272—21no—
125#67272—21no—
126#67272—21no—
127#67272—21no—
128#67272—21no—
129#67272—21no—
130#67272D411yescharacteristic, complemented
131C4×Dic23C2311yescharacteristic
132#67267—21no—
133#67267—21no—
134#67267C2311yescharacteristic
135#67267C2311yescharacteristic
136#67267C2311yescharacteristic
137C2×C23⋊C8—21no—
138C2×C23⋊C8—21no—
139C2×C23⋊C8C2311yescharacteristic
140#67263—21no—
141#67263—21no—
142#67262—21no—
143C8×D23—21no—
144C23×C23—21no—
145C23×C23—21no—
146C23×C23—41no—
147C23×C23—41no—
148C22×D23—21no—
149C22×D23—41no—
150Q8×C23C2×D411yescharacteristic
151Q8×C23C2×D411yescharacteristic
152Q8×C23C2×D411yescharacteristic
153D4×C23—21no—
154D4×C23—21no—
155D4×C23—21no—
156D4×C23—21no—
157D4×C23—21no—
158D4×C23—21no—
159D4×C23—21no—
160D4×C23—41no—
161D4×C23—41no—
162D4×C23—41no—
163D4×C23—41no—
164D4×C23C2×D411yescharacteristic
165D4×C23C2×D411yescharacteristic
166D4×C23C2×D411yescharacteristic
167C2×C4×C23—21no—
168C2×C4×C23—21no—
169C2×C4×C23—21no—
170C2×C4×C23—41no—
171C2×C4×C23—41no—
172C2×C4×C23C2411yescharacteristic
173C22⋊D23—21no—
174C22⋊D23—41no—
175C22⋊D23—41no—
176C22⋊D23—82no—
177C2×Dic23—21no—
178C2×Dic23—21no—
179C2×Dic23—41no—
180D92—21no—
181D92—21no—
182D92—41no—
183D92C2×D411yescharacteristic
184C4×D23—21no—
185C4×D23—41no—
186C4×D23—41no—
187Dic46—21no—
188Dic46—21no—
189Dic46—41no—
190Dic46C2×D411yescharacteristic
191C8×C23—21no—
192C8×C23C2×D411yescharacteristic
193C8×C23C2×D411yescharacteristic
194C23⋊C8—21no—
195C23⋊C8—21no—
196C23⋊C8—21no—
197C23⋊C8C2×D411yescharacteristic
198C23⋊C8C2×D411yescharacteristic
199#3288—231nomaximal, Sylow
200C22×C23—21no—
201C22×C23—21no—
202C22×C23—21no—
203C22×C23—41no—
204C22×C23—41no—
205C22×C23—81no—
206C22×C23—81no—
207C22×C23—161no—
208C22×C23—161no—
209C22×C23H(5,𝔽2)11yescharacteristic
210C2×D23—21no—
211C2×D23—41no—
212C2×D23—81no—
213C2×D23—161no—
214C4×C23—21no—
215C4×C23—21no—
216C4×C23—21no—
217C4×C23C22×D411yescharacteristic
218C4×C23C22×D411yescharacteristic
219Dic23—21no—
220Dic23—41no—
221Dic23—41no—
222#577—231no—
223#577—231no—
224#576—231no—
225#576D2311yescharacteristic, complemented
226#496—231no—
227#472—231no—
228#472—231no—
229#471—231no—
230#471—231no—
231#455—231no—
232#454—231no—
233#453—231no—
234#453—231no—
235#444—231no—
236#444—231no—
237C2×C23—21no—
238C2×C23—41no—
239C2×C23—41no—
240C2×C23—41no—
241C2×C23—81no—
242C2×C23—81no—
243C2×C23D4211yescharacteristic
244D23—41no—
245D23—81no—
2462−1+4—231no—
247H(5,𝔽2)—231no—
248H(5,𝔽2)C2×D2311yescharacteristic
249H(5,𝔽2)C2×D2311yescharacteristic, complemented
250#137—231no—
251#137—231no—
252#137—461no—
253#137—461no—
254AGL(1,ℤ/8)—21no—
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—231no—
257AGL(1,ℤ/8)—231no—
258AGL(1,ℤ/8)—231no—
259AGL(1,ℤ/8)—231no—
260AGL(1,ℤ/8)—461no—
261AGL(1,ℤ/8)—461no—
262AGL(1,ℤ/8)—461no—
263AGL(1,ℤ/8)—461no—
264AGL(1,ℤ/8)C2×D2311yescharacteristic
265AGL(1,ℤ/8)C2×D2311yescharacteristic, complemented
266#135—21no—
267#135—231no—
268#135—461no—
269#135—461no—
270#135—461no—
271#135C2×D2311yescharacteristic
272C2×SD16—231no—
273C2×SD16—231no—
274C2×SD16—461no—
275C2×SD16—461no—
276C2×D8—21no—
277C2×D8—231no—
278C2×D8—461no—
279C2×D8C2×D2311yescharacteristic, complemented
280#131—231no—
281#131—231no—
282#131—231no—
283#131C2×D2311yescharacteristic, complemented
284#127—231no—
285#124—231no—
286#108—231no—
287#108—231no—
288#104—231no—
289#104—231no—
290#104—231no—
291#104—231no—
292#101—231no—
293#100—231no—
294#100—231no—
295#100—231no—
296#97—231no—
297C23#328811yescharacteristic, complemented, Sylow
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—231no—
303#41—461no—
304#41—461no—
305#41—461no—
306#41—461no—
307#41—921no—
308#41C22×D2311yescharacteristic
309#41C22×D2311yescharacteristic
310#41C22×D2311yescharacteristic
311C2×Q8—231no—
312C2×Q8—461no—
313C2×Q8—461no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—231no—
321C2×D4—461no—
322C2×D4—461no—
323C2×D4—921no—
324C2×D4—921no—
325C2×D4C22×D2311yescharacteristic
326C2×D4C22×D2311yescharacteristic
327Q16—231no—
328Q16—461no—
329Q16—461no—
330Q16C22⋊D2311yescharacteristic, complemented
331SD16—21no—
332SD16—21no—
333SD16—231no—
334SD16—231no—
335SD16—461no—
336SD16—461no—
337SD16—461no—
338SD16—461no—
339SD16—461no—
340SD16—461no—
341SD16—461no—
342SD16—461no—
343SD16—461no—
344SD16—461no—
345SD16C22⋊D2311yescharacteristic, complemented
346SD16C22⋊D2311yescharacteristic, complemented
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—231no—
352D8—461no—
353D8—461no—
354D8—461no—
355D8—461no—
356D8—461no—
357D8—461no—
358D8C22⋊D2311yescharacteristic, complemented
359M16—21no—
360M16—231no—
361M16—231no—
362M16—231no—
363M16—461no—
364M16—461no—
365M16—461no—
366M16C22×D2311yescharacteristic
367C2×C8—21no—
368C2×C8—231no—
369C2×C8—461no—
370C2×C8—461no—
371C2×C8—461no—
372C2×C8C22×D2311yescharacteristic
373C22⋊C4—921no—
374C42—231no—
375C23—21no—
376C23—21no—
377C23—41no—
378C23—41no—
379C23—461no—
380C23—921no—
381Q8—231no—
382Q8—461no—
383Q8—461no—
384Q8—921no—
385Q8D4×D2311yescharacteristic
386Q8C2×C22⋊D2311yescharacteristic
387Q8C2×C22⋊D2311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—21no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—231no—
400D4—461no—
401D4—461no—
402D4—461no—
403D4—921no—
404D4—921no—
405D4—921no—
406D4—1842no—
407D4D4×D2311yescharacteristic
408D4C2×C22⋊D2311yescharacteristic
409D4C2×C22⋊D2311yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—461no—
416C2×C4—461no—
417C2×C4—461no—
418C2×C4—921no—
419C2×C4—921no—
420C2×C4—921no—
421C2×C4C23×D2311yescharacteristic
422C8—21no—
423C8—231no—
424C8—231no—
425C8—461no—
426C8—461no—
427C8—461no—
428C8C2×C22⋊D2311yescharacteristic
429C8C2×C22⋊D2311yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22—161no—
438C22—161no—
439C22—461no—
440C22—921no—
441C22—3681no—
442C22—1841no—
443C22#1062515511yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—461no—
448C4—921no—
449C4—921no—
450C4C2×D4×D2311yescharacteristic
451C4C22×C22⋊D2311yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—81no—
458C2—921no—
459C2—1841no—
460C2D4×C22⋊D2311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#576C2
socleC2×C23D42
hypercenter#576D23
hypocenterC23#3288
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #13093228C2semidirect
2. #13093224C2semidirect
3. #13093220C2semidirect
4. #13092805C2semidirect
5. #13092685C2semidirect
6. #13092684C2semidirect
7. #13092680C2semidirect
8. #13092678C2semidirect
9. #13092595C2semidirect
10. #13092588C2semidirect
11. #13092585C2semidirect
12. #13092583C2semidirect
13. #13092539C2semidirect
14. #13092537C2semidirect
15. C23×#576C2semidirect
17. C23×H(5,𝔽2)C22semidirect
18. #10625161C22semidirect
19. #10625160C22semidirect
23. C23×AGL(1,ℤ/8)C22semidirect
27. C2×D8×C23C22semidirect
28. C23×#131C22semidirect
30. #10625122C22semidirect
31. #10625122C22semidirect
37. #10625120C22semidirect
38. #10625119C22semidirect
39. #10625119C22semidirect
40. #10625111C22semidirect
41. #10625107C22semidirect
43. C2×#67273C22semidirect
46. #10625103C22semidirect
47. #10625103C22semidirect
49. C2×#67272C22semidirect
54. C2×#67263C22semidirect
55. #10625072C22semidirect
56. #10625019C22semidirect
57. #10625019C22semidirect
58. #10625019C22semidirect
59. #10625015C22semidirect
60. #10625015C22semidirect
61. #10625007C22semidirect
62. #10625006C22semidirect
63. #10625005C22semidirect
64. #10625004C22semidirect
65. #10625001C22semidirect
66. #10624997C22semidirect
96. Q16×C23D4semidirect
99. SD16×C23D4semidirect
100. SD16×C23D4semidirect
105. D8×C23D4semidirect
113. #67275D4semidirect
116. #67274D4semidirect
123. #67273D4semidirect
130. #67272D4semidirect
225. #576D23semidirect
249. H(5,𝔽2)C2×D23semidirect
265. AGL(1,ℤ/8)C2×D23semidirect
279. C2×D8C2×D23semidirect
283. #131C2×D23semidirect
297. C23#3288semidirect
330. Q16C22⋊D23semidirect
345. SD16C22⋊D23semidirect
346. SD16C22⋊D23semidirect
358. D8C22⋊D23semidirect

Maximal subgroups

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

subgroupindex
#130932282
#130932242
#130932202
#130928052
#130926852
#130926842
#130926802
#130926782
#130925952
#130925882
#130925852
#130925832
#130925392
#130925372
C23×#5762
#328823