Finite Group Databasebeta

← #15685160

Subgroups — #15685160

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#13093229C211yesmaximal, characteristic, complemented
2#13093227C211yesmaximal, characteristic, complemented
3#13093221C211yesmaximal, characteristic, complemented
4#13093218C211yesmaximal, characteristic, complemented
5#13092805C211yesmaximal, characteristic, complemented
6#13092686C211yesmaximal, characteristic, complemented
7#13092681C211yesmaximal, characteristic, complemented
8#13092679C211yesmaximal, characteristic, complemented
9#13092597C211yesmaximal, characteristic, complemented
10#13092588C211yesmaximal, characteristic, complemented
11#13092587C211yesmaximal, characteristic, complemented
12#13092583C211yesmaximal, characteristic, complemented
13#13092538C211yesmaximal, characteristic, complemented
14#13092537C211yesmaximal, characteristic, complemented
15C23×#472C211yesmaximal, characteristic, complemented
16#10625161C2211yescharacteristic, complemented
17#10625160C2211yescharacteristic, complemented
18#10625155C2211yescharacteristic
19#10625155C2211yescharacteristic, complemented
20C23×AGL(1,ℤ/8)C2211yescharacteristic, complemented
21C23×AGL(1,ℤ/8)C2211yescharacteristic, complemented
22C2×D8×C23C2211yescharacteristic, complemented
23C23×#131C2211yescharacteristic, complemented
24#10625121C2211yescharacteristic
25#10625111C2211yescharacteristic, complemented
26#10625107C2211yescharacteristic, complemented
27C2×#67273C2211yescharacteristic, complemented
28#10625103C2211yescharacteristic
29#10625103C2211yescharacteristic, complemented
30C2×#67272C2211yescharacteristic, complemented
31#10625087—21no—
32#10625086—21no—
33#10625085—21no—
34SD16×D23—21no—
35#10625083—21no—
36#10625083—21no—
37#10625082—21no—
38#10625082—21no—
39#10625082—21no—
40#10625082—21no—
41D8×D23—21no—
42D8×D23—21no—
43#10625080—21no—
44#10625080C2211yescharacteristic, complemented
45#10625080C2211yescharacteristic, complemented
46#10625079—21no—
47#10625079C2211yescharacteristic
48#10625079C2211yescharacteristic, complemented
49#10625078C2211yescharacteristic, complemented
50#10625078C2211yescharacteristic, complemented
51#10625075—21no—
52#10625075C2211yescharacteristic
53C2×#67263—21no—
54C2×#67263C2211yescharacteristic, complemented
55#10625072C2211yescharacteristic, complemented
56C23×#108C2211yescharacteristic, complemented
57C23×#100C2211yescharacteristic, complemented
58C23×#100C2211yescharacteristic, complemented
59#10625019C2211yescharacteristic, complemented
60#10625007C2211yescharacteristic, complemented
61#10625007C2211yescharacteristic, complemented
62#10625007C2211yescharacteristic, complemented
63#10625006C2211yescharacteristic, complemented
64#10625005C2211yescharacteristic, complemented
65#10625004C2211yescharacteristic, complemented
66#10624997C2211yescharacteristic, complemented
67#41×C23C2311yescharacteristic
68C2×D4×C23C2311yescharacteristic
69C2×D4×C23C2311yescharacteristic
70C2×C22⋊D23—41no—
71C2×C22⋊D23—41no—
72C2×C22⋊D23—41no—
73#67292—21no—
74Q8×D23—21no—
75#67290—21no—
76#67290—21no—
77#67290—21no—
78#67290—21no—
79#67290—21no—
80#67290—41no—
81D4×D23—21no—
82D4×D23—21no—
83D4×D23—21no—
84D4×D23—21no—
85D4×D23—21no—
86D4×D23—41no—
87#67288—21no—
88#67288—21no—
89#67288C2311yescharacteristic
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—
100D8×C23—21no—
101D8×C23—21no—
102M16×C23—21no—
103M16×C23C2311yescharacteristic
104M16×C23C2311yescharacteristic
105M16×C23C2311yescharacteristic
106C2×C8×C23—21no—
107C2×C8×C23C2311yescharacteristic
108C22⋊Dic23—41no—
109#67275D411yescharacteristic, complemented
110#67274D411yescharacteristic, complemented
111#67273—21no—
112#67273—21no—
113#67273—21no—
114#67273—21no—
115#67273D411yescharacteristic, complemented
116#67272—21no—
117#67272—21no—
118#67272—21no—
119#67272—21no—
120#67272D411yescharacteristic, complemented
121C4×Dic23C2311yescharacteristic
122#67267C2311yescharacteristic
123C2×C23⋊C8C2311yescharacteristic
124Dic92—21no—
125Dic92—21no—
126Dic92D411yescharacteristic, complemented
127D184—21no—
128D184—21no—
129D184D411yescharacteristic, complemented
130#67263—21no—
131#67263—21no—
132#67263—21no—
133#67263—21no—
134#67263—21no—
135#67263—21no—
136#67263D411yescharacteristic, complemented
137#67263D411yescharacteristic, complemented
138#67262—21no—
139#67262—21no—
140#67262—21no—
141C8×D23—21no—
142C8×D23—21no—
143C8×D23—21no—
144C23×C23—21no—
145C23×C23—21no—
146C22×D23—21no—
147C22×D23—41no—
148C22×D23—41no—
149C22×D23—41no—
150Q8×C23C2×D411yescharacteristic
151D4×C23—21no—
152D4×C23—21no—
153D4×C23—21no—
154D4×C23—21no—
155D4×C23—21no—
156D4×C23C2×D411yescharacteristic
157C2×C4×C23—21no—
158C2×C4×C23C2411yescharacteristic
159C22⋊D23—21no—
160C22⋊D23—21no—
161C22⋊D23—21no—
162C22⋊D23—41no—
163C22⋊D23—41no—
164C22⋊D23—41no—
165C22⋊D23—41no—
166C22⋊D23—41no—
167C22⋊D23—41no—
168C22⋊D23—82no—
169C2×Dic23—21no—
170C2×Dic23—21no—
171C2×Dic23—41no—
172C2×Dic23—41no—
173C2×Dic23—41no—
174D92—21no—
175D92—21no—
176D92—41no—
177D92C2×D411yescharacteristic
178D92C2×D411yescharacteristic
179D92C2×D411yescharacteristic
180C4×D23—21no—
181C4×D23—21no—
182C4×D23—21no—
183C4×D23—41no—
184C4×D23—41no—
185Dic46—21no—
186Dic46—21no—
187Dic46—41no—
188Dic46C2×D411yescharacteristic
189Dic46C2×D411yescharacteristic
190Dic46C2×D411yescharacteristic
191C8×C23—21no—
192C8×C23—21no—
193C8×C23—21no—
194C8×C23C2×D411yescharacteristic
195C8×C23C2×D411yescharacteristic
196C23⋊C8—21no—
197C23⋊C8C2×D411yescharacteristic
198C23⋊C8C2×D411yescharacteristic
199#3288—231nomaximal, Sylow
200C22×C23—21no—
201C22×C23—41no—
202C22×C23—41no—
203C22×C23—81no—
204C22×C23—81no—
205C22×C23H(5,𝔽2)11yescharacteristic
206C2×D23—21no—
207C2×D23—21no—
208C2×D23—21no—
209C2×D23—41no—
210C2×D23—81no—
211C2×D23—161no—
212C2×D23—161no—
213C2×D23—161no—
214C4×C23—21no—
215C4×C23C22×D411yescharacteristic
216C4×C23C22×D411yescharacteristic
217Dic23—21no—
218Dic23—21no—
219Dic23—21no—
220Dic23—41no—
221Dic23—41no—
222#577—231no—
223#577—231no—
224#576—231no—
225#576—231no—
226#496—231no—
227#472—231no—
228#472D2311yescharacteristic, complemented
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—81no—
240C2×C23—81no—
241C2×C23D4211yescharacteristic
242D23—41no—
243D23—41no—
244D23—41no—
245D23—81no—
2462−1+4—231no—
247H(5,𝔽2)—231no—
248H(5,𝔽2)—231no—
249H(5,𝔽2)—231no—
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, complemented
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—461no—
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#108C2×D2311yescharacteristic, complemented
288#104—231no—
289#104—231no—
290#104—231no—
291#104—231no—
292#101—231no—
293#100—231no—
294#100C2×D2311yescharacteristic, complemented
295#100C2×D2311yescharacteristic, complemented
296#97—231no—
297C23#328811yescharacteristic, complemented, Sylow
298#41—231no—
299#41—231no—
300#41—231no—
301#41—461no—
302#41—461no—
303#41—461no—
304#41—461no—
305#41—461no—
306#41—461no—
307#41—461no—
308#41—461no—
309#41—921no—
310#41C22×D2311yescharacteristic
311C2×Q8—231no—
312C2×Q8—461no—
313C2×Q8—461no—
314C2×D4—231no—
315C2×D4—461no—
316C2×D4—461no—
317C2×D4—461no—
318C2×D4—461no—
319C2×D4—461no—
320C2×D4—461no—
321C2×D4—921no—
322C2×D4—921no—
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—21no—
350D8—21no—
351D8—231no—
352D8—231no—
353D8—461no—
354D8—461no—
355D8—461no—
356D8—461no—
357D8—461no—
358D8—461no—
359M16—21no—
360M16—231no—
361M16—461no—
362M16—461no—
363M16—461no—
364M16C22×D2311yescharacteristic
365M16C22×D2311yescharacteristic
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—461no—
378C23—921no—
379C23—921no—
380C23—921no—
381Q8—231no—
382Q8—231no—
383Q8—231no—
384Q8—461no—
385Q8—461no—
386Q8—921no—
387Q8D4×D2311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—231no—
394D4—231no—
395D4—231no—
396D4—461no—
397D4—461no—
398D4—461no—
399D4—461no—
400D4—461no—
401D4—921no—
402D4—921no—
403D4—921no—
404D4—921no—
405D4—921no—
406D4—921no—
407D4—921no—
408D4—1842no—
409D4D4×D2311yescharacteristic
410C2×C4—21no—
411C2×C4—461no—
412C2×C4—461no—
413C2×C4—461no—
414C2×C4—461no—
415C2×C4—461no—
416C2×C4—921no—
417C2×C4—921no—
418C2×C4—921no—
419C2×C4—921no—
420C2×C4—921no—
421C2×C4C23×D2311yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—231no—
426C8—231no—
427C8—461no—
428C8D4×D2311yescharacteristic
429C8D4×D2311yescharacteristic
430C22—21no—
431C22—41no—
432C22—41no—
433C22—81no—
434C22—81no—
435C22—461no—
436C22—461no—
437C22—461no—
438C22—921no—
439C22—3681no—
440C22—3681no—
441C22—3681no—
442C22—1841no—
443C22#1062516011yescharacteristic
444C4—21no—
445C4—461no—
446C4—461no—
447C4—461no—
448C4—921no—
449C4—921no—
450C4C2×D4×D2311yescharacteristic
451C4C2×D4×D2311yescharacteristic
452C2—21no—
453C2—41no—
454C2—81no—
455C2—81no—
456C2—921no—
457C2—921no—
458C2—921no—
459C2—1841no—
460C2#1309305111yescharacteristic
subgroupG/H
centerC2#13093051
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#472C2
socleC2×C23D42
hypercenter#472D23
hypocenterC23#3288
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #13093229C2semidirect
2. #13093227C2semidirect
3. #13093221C2semidirect
4. #13093218C2semidirect
5. #13092805C2semidirect
6. #13092686C2semidirect
7. #13092681C2semidirect
8. #13092679C2semidirect
9. #13092597C2semidirect
10. #13092588C2semidirect
11. #13092587C2semidirect
12. #13092583C2semidirect
13. #13092538C2semidirect
14. #13092537C2semidirect
15. C23×#472C2semidirect
16. #10625161C22semidirect
17. #10625160C22semidirect
19. #10625155C22semidirect
20. C23×AGL(1,ℤ/8)C22semidirect
21. C23×AGL(1,ℤ/8)C22semidirect
22. C2×D8×C23C22semidirect
23. C23×#131C22semidirect
25. #10625111C22semidirect
26. #10625107C22semidirect
27. C2×#67273C22semidirect
29. #10625103C22semidirect
30. C2×#67272C22semidirect
44. #10625080C22semidirect
45. #10625080C22semidirect
48. #10625079C22semidirect
49. #10625078C22semidirect
50. #10625078C22semidirect
54. C2×#67263C22semidirect
55. #10625072C22semidirect
56. C23×#108C22semidirect
57. C23×#100C22semidirect
58. C23×#100C22semidirect
59. #10625019C22semidirect
60. #10625007C22semidirect
61. #10625007C22semidirect
62. #10625007C22semidirect
63. #10625006C22semidirect
64. #10625005C22semidirect
65. #10625004C22semidirect
66. #10624997C22semidirect
109. #67275D4semidirect
110. #67274D4semidirect
115. #67273D4semidirect
120. #67272D4semidirect
126. Dic92D4semidirect
129. D184D4semidirect
136. #67263D4semidirect
137. #67263D4semidirect
228. #472D23semidirect
264. AGL(1,ℤ/8)C2×D23semidirect
265. AGL(1,ℤ/8)C2×D23semidirect
279. C2×D8C2×D23semidirect
283. #131C2×D23semidirect
287. #108C2×D23semidirect
294. #100C2×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
#130932292
#130932272
#130932212
#130932182
#130928052
#130926862
#130926812
#130926792
#130925972
#130925882
#130925872
#130925832
#130925382
#130925372
C23×#4722
#328823