Finite Group Databasebeta

← #15685162

Subgroups — #15685162

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#13093226C211yesmaximal, characteristic, complemented
2#13093223C211yesmaximal, characteristic, complemented
3#13093222C211yesmaximal, characteristic, complemented
4#13093221C211yesmaximal, characteristic, complemented
5#13092803C211yesmaximal, characteristic, complemented
6#13092688C211yesmaximal, characteristic, complemented
7#13092686C211yesmaximal, characteristic, complemented
8#13092683C211yesmaximal, characteristic, complemented
9#13092593C211yesmaximal, characteristic, complemented
10#13092592C211yesmaximal, characteristic, complemented
11#13092587C211yesmaximal, characteristic, complemented
12#13092583C211yesmaximal, characteristic, complemented
13#13092538C211yesmaximal, characteristic, complemented
14#13092537C211yesmaximal, characteristic, complemented
15C23×#471C211yesmaximal, characteristic, complemented
16#10625160C2211yescharacteristic
17#10625160C2211yescharacteristic, complemented
18#10625158C2211yescharacteristic, complemented
19#10625155C2211yescharacteristic, complemented
20C23×#137C2211yescharacteristic, complemented
21C23×AGL(1,ℤ/8)C2211yescharacteristic
22C2×SD16×C23C2211yescharacteristic, complemented
23C23×#131C2211yescharacteristic, complemented
24#10625121C2211yescharacteristic
25#10625118C2211yescharacteristic, complemented
26#10625113C2211yescharacteristic, complemented
27C2×#67274C2211yescharacteristic, complemented
28#10625111C2211yescharacteristic, complemented
29#10625103C2211yescharacteristic, complemented
30C2×#67272C2211yescharacteristic, complemented
31#10625090—21no—
32#10625089—21no—
33#10625087—21no—
34#10625087—21no—
35#10625086—21no—
36#10625085—21no—
37#10625085—21no—
38#10625085—21no—
39SD16×D23—21no—
40SD16×D23—21no—
41#10625082—21no—
42D8×D23—21no—
43#10625079—21no—
44#10625079—21no—
45#10625079C2211yescharacteristic
46#10625079C2211yescharacteristic, complemented
47#10625079C2211yescharacteristic, complemented
48#10625079C2211yescharacteristic, complemented
49#10625078C2211yescharacteristic, complemented
50#10625078C2211yescharacteristic, complemented
51#10625075—21no—
52#10625075C2211yescharacteristic
53C2×D184—21no—
54C2×D184C2211yescharacteristic, complemented
55#10625072C2211yescharacteristic, complemented
56C23×#108C2211yescharacteristic, complemented
57C23×#101C2211yescharacteristic, complemented
58C23×#100C2211yescharacteristic, complemented
59#10625019C2211yescharacteristic, complemented
60#10625007C2211yescharacteristic, complemented
61#10625007C2211yescharacteristic, complemented
62#10625007C2211yescharacteristic, complemented
63#10625005C2211yescharacteristic, complemented
64#10625005C2211yescharacteristic, complemented
65#10625004C2211yescharacteristic, complemented
66#10624997C2211yescharacteristic, complemented
67#41×C23C2311yescharacteristic
68C2×Q8×C23C2311yescharacteristic
69C2×D4×C23C2311yescharacteristic
70C2×C22⋊D23—41no—
71C2×C22⋊D23—41no—
72#67292—21no—
73#67292—21no—
74#67292—21no—
75#67292—21no—
76Q8×D23—21no—
77Q8×D23—21no—
78#67290—21no—
79#67290—21no—
80D4×D23—21no—
81D4×D23—21no—
82D4×D23—21no—
83D4×D23—21no—
84D4×D23—41no—
85D4×D23—41no—
86#67288—21no—
87#67288—21no—
88#67288—41no—
89#67288C2311yescharacteristic
90#67288C2311yescharacteristic
91#67288C2311yescharacteristic
92C2×D92—21no—
93C2×D92—21no—
94C2×D92C2311yescharacteristic
95C2×D92C2311yescharacteristic
96Q16×C23—21no—
97SD16×C23—21no—
98SD16×C23—21no—
99SD16×C23—21no—
100SD16×C23—21no—
101D8×C23—21no—
102M16×C23—21no—
103M16×C23C2311yescharacteristic
104M16×C23C2311yescharacteristic
105M16×C23C2311yescharacteristic
106C2×C8×C23—21no—
107C2×C8×C23C2311yescharacteristic
108#67275—21no—
109#67275D411yescharacteristic, complemented
110#67274—21no—
111#67274—21no—
112#67274—21no—
113#67274D411yescharacteristic, complemented
114#67273—21no—
115#67273D411yescharacteristic, complemented
116#67272—21no—
117#67272—21no—
118#67272—21no—
119#67272D411yescharacteristic, complemented
120(C2×D23)⋊C4—41no—
121C4×Dic23C2311yescharacteristic
122#67267C2311yescharacteristic
123C2×C23⋊C8C2311yescharacteristic
124Dic92D411yescharacteristic, complemented
125D184—21no—
126D184—21no—
127D184—21no—
128D184—21no—
129D184—21no—
130D184—21no—
131D184D411yescharacteristic, complemented
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—
145C22×D23—21no—
146C22×D23—21no—
147C22×D23—41no—
148C22×D23—41no—
149C22×D23—41no—
150Q8×C23—21no—
151Q8×C23—21no—
152Q8×C23C2×D411yescharacteristic
153D4×C23—21no—
154D4×C23—21no—
155D4×C23—21no—
156D4×C23C2×D411yescharacteristic
157C2×C4×C23—21no—
158C2×C4×C23—21no—
159C2×C4×C23C2411yescharacteristic
160C22⋊D23—21no—
161C22⋊D23—21no—
162C22⋊D23—21no—
163C22⋊D23—41no—
164C22⋊D23—41no—
165C22⋊D23—41no—
166C22⋊D23—41no—
167C22⋊D23—82no—
168C2×Dic23—21no—
169C2×Dic23—41no—
170D92—21no—
171D92—21no—
172D92—21no—
173D92—21no—
174D92—41no—
175D92—41no—
176D92—41no—
177D92C2×D411yescharacteristic
178D92C2×D411yescharacteristic
179D92C2×D411yescharacteristic
180C4×D23—21no—
181C4×D23—21no—
182C4×D23—21no—
183C4×D23—41no—
184C4×D23—41no—
185C4×D23—41no—
186C4×D23—41no—
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—81no—
203C22×C23H(5,𝔽2)11yescharacteristic
204C2×D23—21no—
205C2×D23—21no—
206C2×D23—21no—
207C2×D23—41no—
208C2×D23—41no—
209C2×D23—81no—
210C2×D23—81no—
211C2×D23—161no—
212C2×D23—161no—
213C2×D23—161no—
214C4×C23—21no—
215C4×C23—41no—
216C4×C23C22×D411yescharacteristic
217C4×C23C22×D411yescharacteristic
218Dic23—21no—
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#471D2311yescharacteristic, complemented
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×C23D4211yescharacteristic
241D23—41no—
242D23—41no—
243D23—41no—
244D23—81no—
245D23—81no—
2462−1+4—231no—
247H(5,𝔽2)—231no—
248H(5,𝔽2)—231no—
249H(5,𝔽2)—231no—
250#137—231no—
251#137—461no—
252#137—461no—
253#137C2×D2311yescharacteristic, complemented
254AGL(1,ℤ/8)—231no—
255AGL(1,ℤ/8)—231no—
256AGL(1,ℤ/8)—231no—
257AGL(1,ℤ/8)—231no—
258AGL(1,ℤ/8)—231no—
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)—461no—
265AGL(1,ℤ/8)C2×D2311yescharacteristic
266#135—231no—
267#135—231no—
268#135—461no—
269#135—461no—
270#135—461no—
271#135—461no—
272C2×SD16—231no—
273C2×SD16—461no—
274C2×SD16—461no—
275C2×SD16C2×D2311yescharacteristic, complemented
276C2×D8—231no—
277C2×D8—231no—
278C2×D8—461no—
279C2×D8—461no—
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#101C2×D2311yescharacteristic, complemented
293#100—231no—
294#100—231no—
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—461no—
312C2×Q8—461no—
313C2×Q8C22×D2311yescharacteristic
314C2×D4—231no—
315C2×D4—231no—
316C2×D4—461no—
317C2×D4—461no—
318C2×D4—461no—
319C2×D4—461no—
320C2×D4—461no—
321C2×D4—461no—
322C2×D4—921no—
323C2×D4—921no—
324C2×D4—921no—
325C2×D4—921no—
326C2×D4C22×D2311yescharacteristic
327Q16—21no—
328Q16—231no—
329Q16—231no—
330Q16—461no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—21no—
335SD16—231no—
336SD16—231no—
337SD16—231no—
338SD16—231no—
339SD16—461no—
340SD16—461no—
341SD16—461no—
342SD16—461no—
343SD16—461no—
344SD16—461no—
345SD16—461no—
346SD16—461no—
347D8—21no—
348D8—231no—
349D8—231no—
350D8—461no—
351D8—461no—
352D8—461no—
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—461no—
377C23—461no—
378C23—921no—
379C23—921no—
380C23—921no—
381Q8—21no—
382Q8—21no—
383Q8—231no—
384Q8—231no—
385Q8—231no—
386Q8—921no—
387Q8D4×D2311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—231no—
392D4—231no—
393D4—231no—
394D4—461no—
395D4—461no—
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—21no—
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—81no—
433C22—461no—
434C22—461no—
435C22—461no—
436C22—921no—
437C22—921no—
438C22—3681no—
439C22—3681no—
440C22—3681no—
441C22—1841no—
442C22—1841no—
443C22#1062516011yescharacteristic
444C4—21no—
445C4—41no—
446C4—461no—
447C4—461no—
448C4—461no—
449C4—921no—
450C4C2×D4×D2311yescharacteristic
451C4C2×D4×D2311yescharacteristic
452C2—21no—
453C2—41no—
454C2—81no—
455C2—921no—
456C2—921no—
457C2—921no—
458C2—1841no—
459C2—1841no—
460C2#1309305111yescharacteristic
subgroupG/H
centerC2#13093051
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#471C2
socleC2×C23D42
hypercenter#471D23
hypocenterC23#3288
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #13093226C2semidirect
2. #13093223C2semidirect
3. #13093222C2semidirect
4. #13093221C2semidirect
5. #13092803C2semidirect
6. #13092688C2semidirect
7. #13092686C2semidirect
8. #13092683C2semidirect
9. #13092593C2semidirect
10. #13092592C2semidirect
11. #13092587C2semidirect
12. #13092583C2semidirect
13. #13092538C2semidirect
14. #13092537C2semidirect
15. C23×#471C2semidirect
17. #10625160C22semidirect
18. #10625158C22semidirect
19. #10625155C22semidirect
20. C23×#137C22semidirect
22. C2×SD16×C23C22semidirect
23. C23×#131C22semidirect
25. #10625118C22semidirect
26. #10625113C22semidirect
27. C2×#67274C22semidirect
28. #10625111C22semidirect
29. #10625103C22semidirect
30. C2×#67272C22semidirect
46. #10625079C22semidirect
47. #10625079C22semidirect
48. #10625079C22semidirect
49. #10625078C22semidirect
50. #10625078C22semidirect
54. C2×D184C22semidirect
55. #10625072C22semidirect
56. C23×#108C22semidirect
57. C23×#101C22semidirect
58. C23×#100C22semidirect
59. #10625019C22semidirect
60. #10625007C22semidirect
61. #10625007C22semidirect
62. #10625007C22semidirect
63. #10625005C22semidirect
64. #10625005C22semidirect
65. #10625004C22semidirect
66. #10624997C22semidirect
109. #67275D4semidirect
113. #67274D4semidirect
115. #67273D4semidirect
119. #67272D4semidirect
124. Dic92D4semidirect
131. D184D4semidirect
136. #67263D4semidirect
137. #67263D4semidirect
230. #471D23semidirect
253. #137C2×D23semidirect
275. C2×SD16C2×D23semidirect
283. #131C2×D23semidirect
287. #108C2×D23semidirect
292. #101C2×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
#130932262
#130932232
#130932222
#130932212
#130928032
#130926882
#130926862
#130926832
#130925932
#130925922
#130925872
#130925832
#130925382
#130925372
C23×#4712
#328823