Finite Group Databasebeta

← #15685158

Subgroups — #15685158

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#13093227C211yesmaximal, characteristic, complemented
3#13093221C211yesmaximal, characteristic, complemented
4#13093220C211yesmaximal, characteristic, complemented
5#13092804C211yesmaximal, characteristic, complemented
6#13092686C211yesmaximal, characteristic, complemented
7#13092685C211yesmaximal, characteristic, complemented
8#13092679C211yesmaximal, characteristic, complemented
9#13092678C211yesmaximal, characteristic, complemented
10#13092596C211yesmaximal, characteristic, complemented
11#13092591C211yesmaximal, characteristic, complemented
12#13092584C211yesmaximal, characteristic, complemented
13#13092536C211yesmaximal, characteristic, complemented
14#13092536C211yesmaximal, characteristic, complemented
15C23×#453C211yesmaximal, characteristic, complemented
16C23×H(5,𝔽2)C2211yescharacteristic, complemented
17#10625161C2211yescharacteristic, complemented
18#10625160C2211yescharacteristic
19#10625155C2211yescharacteristic, complemented
20C23×AGL(1,ℤ/8)C2211yescharacteristic
21C23×AGL(1,ℤ/8)C2211yescharacteristic, complemented
22C23×#127C2211yescharacteristic, complemented
23#10625122—21no—
24#10625122C2211yescharacteristic, complemented
25#10625121—21no—
26#10625121—21no—
27#10625121C2211yescharacteristic
28#10625121C2211yescharacteristic
29#10625120—21no—
30#10625120C2211yescharacteristic, complemented
31#10625119C2211yescharacteristic, complemented
32#10625119C2211yescharacteristic, complemented
33C2×#67273—21no—
34C2×#67273C2211yescharacteristic, complemented
35C2×#67273C2211yescharacteristic, complemented
36#10625103—21no—
37#10625103—21no—
38#10625103C2211yescharacteristic
39#10625103C2211yescharacteristic, complemented
40C2×#67272—21no—
41C2×#67272C2211yescharacteristic, complemented
42C2×#67272C2211yescharacteristic, complemented
43#10625087—21no—
44#10625086—21no—
45#10625085—21no—
46SD16×D23—21no—
47#10625083—21no—
48#10625082—21no—
49#10625082—21no—
50D8×D23—21no—
51#10625080C2211yescharacteristic, complemented
52#10625079C2211yescharacteristic, complemented
53#10625078C2211yescharacteristic, complemented
54#10625078C2211yescharacteristic, complemented
55#10625046C2211yescharacteristic, complemented
56C23×#104C2211yescharacteristic, complemented
57C23×#104C2211yescharacteristic, complemented
58C23×#100C2211yescharacteristic, complemented
59#10625015C2211yescharacteristic, complemented
60#10625006C2211yescharacteristic, complemented
61#10625005C2211yescharacteristic, complemented
62#10625004C2211yescharacteristic, complemented
63#10625004C2211yescharacteristic, complemented
64#10624987C2211yescharacteristic, complemented
65#10624987C2211yescharacteristic, complemented
66#10624985C2211yescharacteristic, complemented
67#41×C23—21no—
68#41×C23—21no—
69#41×C23C2311yescharacteristic
70#41×C23C2311yescharacteristic
71C2×D4×C23—21no—
72C2×D4×C23—21no—
73C2×D4×C23—41no—
74C2×D4×C23—41no—
75C2×D4×C23C2311yescharacteristic
76C2×D4×C23C2311yescharacteristic
77C2×C22⋊D23—41no—
78#67292—21no—
79Q8×D23—21no—
80#67290—21no—
81#67290—21no—
82#67290—21no—
83#67290—41no—
84D4×D23—21no—
85D4×D23—21no—
86D4×D23—21no—
87D4×D23—41no—
88#67288—21no—
89#67288—21no—
90#67288C2311yescharacteristic
91#67288C2311yescharacteristic
92C2×D92—21no—
93C2×D92C2311yescharacteristic
94C2×Dic46—21no—
95C2×Dic46C2311yescharacteristic
96SD16×C23—21no—
97SD16×C23—21no—
98D8×C23—21no—
99D8×C23—21no—
100M16×C23C2311yescharacteristic
101M16×C23C2311yescharacteristic
102C42×C23C2311yescharacteristic
103#67275—21no—
104#67275D411yescharacteristic, complemented
105#67275D411yescharacteristic, complemented
106#67274—21no—
107#67274D411yescharacteristic, complemented
108#67274D411yescharacteristic, complemented
109#67273—21no—
110#67273—21no—
111#67273—21no—
112#67273—21no—
113#67273—21no—
114#67273—21no—
115#67273—21no—
116#67273D411yescharacteristic, complemented
117#67273D411yescharacteristic, complemented
118#67272—21no—
119#67272—21no—
120#67272—21no—
121#67272—21no—
122#67272—21no—
123#67272—21no—
124#67272—21no—
125#67272D411yescharacteristic, complemented
126#67272D411yescharacteristic, complemented
127(C2×D23)⋊C4—41no—
128#67267—21no—
129#67267—21no—
130#67267C2311yescharacteristic
131#67267C2311yescharacteristic
132C2×C23⋊C8—21no—
133C2×C23⋊C8—21no—
134C2×C23⋊C8C2311yescharacteristic
135C2×C23⋊C8C2311yescharacteristic
136Dic92—21no—
137D184—21no—
138#67263—21no—
139#67263—21no—
140#67262—21no—
141#67262—21no—
142C8×D23—21no—
143C8×D23—21no—
144C23×C23—21no—
145C23×C23—21no—
146C23×C23—41no—
147C22×D23—21no—
148C22×D23—41no—
149C22×D23—41no—
150Q8×C23C2×D411yescharacteristic
151Q8×C23C2×D411yescharacteristic
152D4×C23—21no—
153D4×C23—21no—
154D4×C23—21no—
155D4×C23—21no—
156D4×C23—21no—
157D4×C23—21no—
158D4×C23—41no—
159D4×C23—41no—
160D4×C23—82no—
161D4×C23C2×D411yescharacteristic
162D4×C23C2×D411yescharacteristic
163C2×C4×C23—21no—
164C2×C4×C23—21no—
165C2×C4×C23—21no—
166C2×C4×C23—41no—
167C2×C4×C23C2411yescharacteristic
168C22⋊D23—21no—
169C22⋊D23—21no—
170C22⋊D23—41no—
171C22⋊D23—41no—
172C22⋊D23—41no—
173C22⋊D23—41no—
174C2×Dic23—21no—
175C2×Dic23—41no—
176C2×Dic23—41no—
177D92—21no—
178D92—21no—
179D92—41no—
180D92C2×D411yescharacteristic
181D92C2×D411yescharacteristic
182C4×D23—21no—
183C4×D23—21no—
184C4×D23—41no—
185C4×D23—41no—
186Dic46—21no—
187Dic46—21no—
188Dic46—41no—
189Dic46C2×D411yescharacteristic
190Dic46C2×D411yescharacteristic
191C8×C23—21no—
192C8×C23—21no—
193C23⋊C8—21no—
194C23⋊C8—21no—
195C23⋊C8C2×D411yescharacteristic
196C23⋊C8C2×D411yescharacteristic
197C23⋊C8C2×D411yescharacteristic
198C23⋊C8C2×D411yescharacteristic
199#3288—231nomaximal, Sylow
200C22×C23—21no—
201C22×C23—21no—
202C22×C23—41no—
203C22×C23—41no—
204C22×C23—81no—
205C22×C23—81no—
206C22×C23—161no—
207C22×C23H(5,𝔽2)11yescharacteristic
208C2×D23—21no—
209C2×D23—21no—
210C2×D23—41no—
211C2×D23—81no—
212C2×D23—161no—
213C2×D23—161no—
214C4×C23—21no—
215C4×C23—21no—
216C4×C23—41no—
217C4×C23C22×D411yescharacteristic
218C4×C23C22×D411yescharacteristic
219Dic23—21no—
220Dic23—21no—
221Dic23—41no—
222#577—231no—
223#577—231no—
224#576—231no—
225#576—231no—
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#453D2311yescharacteristic, complemented
235#444—231no—
236#444—231no—
237C2×C23—21no—
238C2×C23—41no—
239C2×C23—41no—
240C2×C23—81no—
241C2×C23—81no—
242C2×C23D4211yescharacteristic
243D23—41no—
244D23—41no—
245D23—81no—
2462−1+4—231no—
247H(5,𝔽2)—231no—
248H(5,𝔽2)—231no—
249H(5,𝔽2)C2×D2311yescharacteristic, complemented
250#137—231no—
251#137—231no—
252#137—461no—
253#137—461no—
254AGL(1,ℤ/8)—231no—
255AGL(1,ℤ/8)—231no—
256AGL(1,ℤ/8)—231no—
257AGL(1,ℤ/8)—231no—
258AGL(1,ℤ/8)—461no—
259AGL(1,ℤ/8)—461no—
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—231no—
267#135—231no—
268#135—461no—
269#135—461no—
270#135—461no—
271#135—461no—
272C2×SD16—231no—
273C2×SD16—231no—
274C2×SD16—461no—
275C2×SD16—461no—
276C2×D8—231no—
277C2×D8—231no—
278C2×D8—461no—
279C2×D8—461no—
280#131—231no—
281#131—231no—
282#131—231no—
283#131—231no—
284#127C2×D2311yescharacteristic, complemented
285#124—231no—
286#108—231no—
287#108—231no—
288#104—231no—
289#104—231no—
290#104C2×D2311yescharacteristic, complemented
291#104C2×D2311yescharacteristic, complemented
292#101—231no—
293#100—231no—
294#100—231no—
295#100C2×D2311yescharacteristic, complemented
296#97—231no—
297C23#328811yescharacteristic, complemented, Sylow
298#41—21no—
299#41—21no—
300#41—231no—
301#41—231no—
302#41—461no—
303#41—461no—
304#41—461no—
305#41—461no—
306#41—461no—
307#41—461no—
308#41—921no—
309#41C22×D2311yescharacteristic
310#41C22×D2311yescharacteristic
311C2×Q8—231no—
312C2×Q8—461no—
313C2×Q8—461no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—231no—
319C2×D4—461no—
320C2×D4—461no—
321C2×D4—461no—
322C2×D4—461no—
323C2×D4—921no—
324C2×D4—921no—
325C2×D4C22×D2311yescharacteristic
326C2×D4C22×D2311yescharacteristic
327Q16—231no—
328Q16—231no—
329Q16—461no—
330Q16—461no—
331SD16—21no—
332SD16—21no—
333SD16—231no—
334SD16—231no—
335SD16—231no—
336SD16—231no—
337SD16—461no—
338SD16—461no—
339SD16—461no—
340SD16—461no—
341SD16—461no—
342SD16—461no—
343SD16—461no—
344SD16—461no—
345SD16—461no—
346SD16—461no—
347D8—21no—
348D8—21no—
349D8—231no—
350D8—231no—
351D8—461no—
352D8—461no—
353D8—461no—
354D8—461no—
355D8—461no—
356D8—461no—
357D8—461no—
358D8—461no—
359M16—231no—
360M16—231no—
361M16—461no—
362M16—461no—
363M16—461no—
364M16—461no—
365M16C22×D2311yescharacteristic
366M16C22×D2311yescharacteristic
367C2×C8—231no—
368C2×C8—231no—
369C2×C8—461no—
370C2×C8—461no—
371C2×C8—461no—
372C2×C8—461no—
373C22⋊C4—921no—
374C42C22×D2311yescharacteristic
375C23—21no—
376C23—21no—
377C23—41no—
378C23—461no—
379C23—921no—
380C23—921no—
381Q8—231no—
382Q8—231no—
383Q8—461no—
384Q8—461no—
385Q8—921no—
386Q8D4×D2311yescharacteristic
387Q8D4×D2311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—41no—
395D4—41no—
396D4—82no—
397D4—231no—
398D4—231no—
399D4—461no—
400D4—461no—
401D4—461no—
402D4—461no—
403D4—921no—
404D4—921no—
405D4—921no—
406D4—921no—
407D4—921no—
408D4D4×D2311yescharacteristic
409D4D4×D2311yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—461no—
415C2×C4—461no—
416C2×C4—461no—
417C2×C4—921no—
418C2×C4—921no—
419C2×C4—921no—
420C2×C4—921no—
421C2×C4C23×D2311yescharacteristic
422C8—21no—
423C8—21no—
424C8—231no—
425C8—231no—
426C8—231no—
427C8—231no—
428C8—461no—
429C8—461no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—81no—
435C22—81no—
436C22—161no—
437C22—461no—
438C22—461no—
439C22—921no—
440C22—3681no—
441C22—3681no—
442C22—1841no—
443C22#1062515511yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—461no—
448C4—461no—
449C4—921no—
450C4C2×D4×D2311yescharacteristic
451C4C2×D4×D2311yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2—81no—
457C2—921no—
458C2—921no—
459C2—1841no—
460C2#1309304911yescharacteristic
subgroupG/H
centerC2#13093049
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#453C2
socleC2×C23D42
hypercenter#453D23
hypocenterC23#3288
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #13093228C2semidirect
2. #13093227C2semidirect
3. #13093221C2semidirect
4. #13093220C2semidirect
5. #13092804C2semidirect
6. #13092686C2semidirect
7. #13092685C2semidirect
8. #13092679C2semidirect
9. #13092678C2semidirect
10. #13092596C2semidirect
11. #13092591C2semidirect
12. #13092584C2semidirect
13. #13092536C2semidirect
14. #13092536C2semidirect
15. C23×#453C2semidirect
16. C23×H(5,𝔽2)C22semidirect
17. #10625161C22semidirect
19. #10625155C22semidirect
21. C23×AGL(1,ℤ/8)C22semidirect
22. C23×#127C22semidirect
24. #10625122C22semidirect
30. #10625120C22semidirect
31. #10625119C22semidirect
32. #10625119C22semidirect
34. C2×#67273C22semidirect
35. C2×#67273C22semidirect
39. #10625103C22semidirect
41. C2×#67272C22semidirect
42. C2×#67272C22semidirect
51. #10625080C22semidirect
52. #10625079C22semidirect
53. #10625078C22semidirect
54. #10625078C22semidirect
55. #10625046C22semidirect
56. C23×#104C22semidirect
57. C23×#104C22semidirect
58. C23×#100C22semidirect
59. #10625015C22semidirect
60. #10625006C22semidirect
61. #10625005C22semidirect
62. #10625004C22semidirect
63. #10625004C22semidirect
64. #10624987C22semidirect
65. #10624987C22semidirect
66. #10624985C22semidirect
104. #67275D4semidirect
105. #67275D4semidirect
107. #67274D4semidirect
108. #67274D4semidirect
116. #67273D4semidirect
117. #67273D4semidirect
125. #67272D4semidirect
126. #67272D4semidirect
234. #453D23semidirect
249. H(5,𝔽2)C2×D23semidirect
265. AGL(1,ℤ/8)C2×D23semidirect
284. #127C2×D23semidirect
290. #104C2×D23semidirect
291. #104C2×D23semidirect
295. #100C2×D23semidirect
297. C23#3288semidirect

Maximal subgroups

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

subgroupindex
#130932282
#130932272
#130932212
#130932202
#130928042
#130926862
#130926852
#130926792
#130926782
#130925962
#130925912
#130925842
#130925362
C23×#4532
#328823