Finite Group Databasebeta

← #15685176

Subgroups — #15685176

Order 2944.

454 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13093232C211yesmaximal, characteristic, complemented
2#13093227C211yesmaximal, characteristic, complemented
3#13093225C211yesmaximal, characteristic, complemented
4#13093221C211yesmaximal, characteristic, complemented
5#13092799C211yesmaximal, characteristic, complemented
6#13092691C211yesmaximal, characteristic, complemented
7#13092685C211yesmaximal, characteristic, complemented
8#13092683C211yesmaximal, characteristic, complemented
9#13092678C211yesmaximal, characteristic, complemented
10#13092596C211yesmaximal, characteristic, complemented
11#13092594C211yesmaximal, characteristic, complemented
12#13092591C211yesmaximal, characteristic, complemented
13#13092531C211yesmaximal, characteristic, complemented
14#13092531C211yesmaximal, characteristic, complemented
15C23×#454C211yesmaximal, characteristic, complemented
16C23×2−1+4C2211yescharacteristic, complemented
17#10625161C2211yescharacteristic, complemented
18#10625160C2211yescharacteristic
19#10625155C2211yescharacteristic
20C23×AGL(1,ℤ/8)C2211yescharacteristic
21C23×AGL(1,ℤ/8)C2211yescharacteristic, complemented
22C23×#124C2211yescharacteristic, 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×#67275—21no—
34C2×#67275C2211yescharacteristic, complemented
35#10625113—21no—
36#10625113—21no—
37#10625113C2211yescharacteristic
38#10625113C2211yescharacteristic, complemented
39C2×#67274—21no—
40C2×#67274C2211yescharacteristic, complemented
41C2×#67273C2211yescharacteristic, complemented
42C2×#67272C2211yescharacteristic
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
52#10625079C2211yescharacteristic
53#10625078C2211yescharacteristic
54#10625078C2211yescharacteristic, complemented
55#10625046C2211yescharacteristic, complemented
56C23×#104C2211yescharacteristic, complemented
57C23×#104C2211yescharacteristic, complemented
58C23×#101C2211yescharacteristic, complemented
59#10625015C2211yescharacteristic, complemented
60#10625006C2211yescharacteristic, complemented
61#10625005C2211yescharacteristic, complemented
62#10625004C2211yescharacteristic, complemented
63#10625004C2211yescharacteristic, complemented
64#10624987C2211yescharacteristic
65#10624987C2211yescharacteristic, complemented
66C4×C23⋊C8C2211yescharacteristic, complemented
67#41×C23—21no—
68#41×C23—21no—
69#41×C23—41no—
70#41×C23C2311yescharacteristic
71#41×C23C2311yescharacteristic
72C2×Q8×C23—21no—
73C2×Q8×C23—21no—
74C2×Q8×C23C2311yescharacteristic
75C2×D4×C23C2311yescharacteristic
76C2×C22⋊D23—41no—
77#67292—21no—
78Q8×D23—21no—
79#67290—21no—
80#67290—21no—
81#67290—21no—
82#67290—41no—
83D4×D23—21no—
84D4×D23—21no—
85D4×D23—21no—
86D4×D23—41no—
87#67288—21no—
88#67288—21no—
89#67288C2311yescharacteristic
90#67288C2311yescharacteristic
91C2×D92—21no—
92C2×D92C2311yescharacteristic
93C2×Dic46—21no—
94C2×Dic46C2311yescharacteristic
95SD16×C23—21no—
96SD16×C23—21no—
97D8×C23—21no—
98D8×C23—21no—
99M16×C23C2311yescharacteristic
100M16×C23C2311yescharacteristic
101C22⋊C4×C23—41no—
102C42×C23C2311yescharacteristic
103#67275—21no—
104#67275—21no—
105#67275—21no—
106#67275—21no—
107#67275—21no—
108#67275D411yescharacteristic
109#67275D411yescharacteristic
110#67274—21no—
111#67274—21no—
112#67274—21no—
113#67274—21no—
114#67274—21no—
115#67274D411yescharacteristic
116#67274D411yescharacteristic
117#67273—21no—
118#67273—21no—
119#67273—21no—
120#67273D411yescharacteristic
121#67273D411yescharacteristic
122#67272—21no—
123#67272—21no—
124#67272—21no—
125#67272D411yescharacteristic
126#67272D411yescharacteristic
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—
145C22×D23—21no—
146C22×D23—41no—
147C22×D23—41no—
148Q8×C23—21no—
149Q8×C23—21no—
150Q8×C23—41no—
151Q8×C23C2×D411yescharacteristic
152Q8×C23C2×D411yescharacteristic
153D4×C23—21no—
154D4×C23—21no—
155D4×C23—21no—
156D4×C23—21no—
157D4×C23—41no—
158D4×C23C2×D411yescharacteristic
159D4×C23C2×D411yescharacteristic
160C2×C4×C23—21no—
161C2×C4×C23—21no—
162C2×C4×C23—21no—
163C2×C4×C23—21no—
164C2×C4×C23—41no—
165C2×C4×C23—41no—
166C2×C4×C23C2411yescharacteristic
167C22⋊D23—21no—
168C22⋊D23—21no—
169C22⋊D23—41no—
170C22⋊D23—41no—
171C22⋊D23—41no—
172C22⋊D23—41no—
173C2×Dic23—21no—
174C2×Dic23—41no—
175C2×Dic23—41no—
176D92—21no—
177D92—21no—
178D92—41no—
179D92C2×D411yescharacteristic
180D92C2×D411yescharacteristic
181C4×D23—21no—
182C4×D23—21no—
183C4×D23—41no—
184C4×D23—41no—
185Dic46—21no—
186Dic46—21no—
187Dic46—41no—
188Dic46C2×D411yescharacteristic
189Dic46C2×D411yescharacteristic
190C8×C23—21no—
191C8×C23—21no—
192C23⋊C8—21no—
193C23⋊C8—21no—
194C23⋊C8C2×D411yescharacteristic
195C23⋊C8C2×D411yescharacteristic
196C23⋊C8C2×D411yescharacteristic
197C23⋊C8C2×D411yescharacteristic
198#3290—231nomaximal, Sylow
199C22×C23—21no—
200C22×C23—21no—
201C22×C23—41no—
202C22×C23—81no—
203C22×C23H(5,𝔽2)11yescharacteristic
204C2×D23—21no—
205C2×D23—21no—
206C2×D23—41no—
207C2×D23—81no—
208C2×D23—161no—
209C2×D23—161no—
210C4×C23—21no—
211C4×C23—21no—
212C4×C23—41no—
213C4×C23—41no—
214C4×C23C22×D411yescharacteristic
215C4×C23C22×D411yescharacteristic
216Dic23—21no—
217Dic23—21no—
218Dic23—41no—
219#578—231no—
220#577—231no—
221#577—231no—
222#576—231no—
223#495—231no—
224#473—231no—
225#472—231no—
226#471—231no—
227#471—231no—
228#455—231no—
229#455—231no—
230#454—231no—
231#454D2311yescharacteristic, complemented
232#443—231no—
233#443—231no—
234C2×C23—21no—
235C2×C23—41no—
236C2×C23—41no—
237C2×C23—81no—
238C2×C23D4211yescharacteristic
239D23—41no—
240D23—41no—
241D23—81no—
2422−1+4—231no—
2432−1+4C2×D2311yescharacteristic, complemented
244H(5,𝔽2)—231no—
245H(5,𝔽2)—231no—
246#137—231no—
247#137—231no—
248#137—231no—
249#137—231no—
250#137—461no—
251#137—461no—
252#137—461no—
253#137—461no—
254AGL(1,ℤ/8)—231no—
255AGL(1,ℤ/8)—231no—
256AGL(1,ℤ/8)—461no—
257AGL(1,ℤ/8)—461no—
258AGL(1,ℤ/8)—461no—
259AGL(1,ℤ/8)—461no—
260AGL(1,ℤ/8)C2×D2311yescharacteristic
261AGL(1,ℤ/8)C2×D2311yescharacteristic, complemented
262#135—231no—
263#135—231no—
264#135—461no—
265#135—461no—
266#135—461no—
267#135—461no—
268C2×Q16—231no—
269C2×Q16—461no—
270C2×SD16—231no—
271C2×SD16—231no—
272C2×SD16—461no—
273C2×SD16—461no—
274C2×D8—231no—
275C2×D8—461no—
276#131—231no—
277#131—231no—
278#131—231no—
279#131—231no—
280#124—231no—
281#124C2×D2311yescharacteristic, complemented
282#108—231no—
283#108—231no—
284#104—231no—
285#104—231no—
286#104C2×D2311yescharacteristic, complemented
287#104C2×D2311yescharacteristic, complemented
288#101—231no—
289#101C2×D2311yescharacteristic, complemented
290#100—231no—
291#100—231no—
292C4×C8—231no—
293C23#329011yescharacteristic, complemented, Sylow
294#41—21no—
295#41—21no—
296#41—41no—
297#41—231no—
298#41—231no—
299#41—461no—
300#41—461no—
301#41—461no—
302#41—461no—
303#41—461no—
304#41—461no—
305#41—921no—
306#41C22×D2311yescharacteristic
307#41C22×D2311yescharacteristic
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—231no—
311C2×Q8—461no—
312C2×Q8—461no—
313C2×Q8C22×D2311yescharacteristic
314C2×D4—231no—
315C2×D4—461no—
316C2×D4—461no—
317C2×D4—461no—
318C2×D4—461no—
319C2×D4—921no—
320C2×D4—921no—
321C2×D4C22×D2311yescharacteristic
322Q16—231no—
323Q16—231no—
324Q16—461no—
325Q16—461no—
326Q16—461no—
327Q16—461no—
328Q16—461no—
329Q16—461no—
330SD16—21no—
331SD16—21no—
332SD16—231no—
333SD16—231no—
334SD16—231no—
335SD16—231no—
336SD16—461no—
337SD16—461no—
338SD16—461no—
339SD16—461no—
340SD16—461no—
341SD16—461no—
342SD16—461no—
343SD16—461no—
344SD16—461no—
345SD16—461no—
346D8—21no—
347D8—21no—
348D8—231no—
349D8—231no—
350D8—461no—
351D8—461no—
352D8—461no—
353D8—461no—
354M16—231no—
355M16—231no—
356M16—461no—
357M16—461no—
358M16—461no—
359M16—461no—
360M16C22×D2311yescharacteristic
361M16C22×D2311yescharacteristic
362C2×C8—231no—
363C2×C8—231no—
364C2×C8—461no—
365C2×C8—461no—
366C2×C8—461no—
367C2×C8—461no—
368C22⋊C4—41no—
369C22⋊C4—921no—
370C42C22×D2311yescharacteristic
371C23—21no—
372C23—461no—
373C23—921no—
374C23—921no—
375Q8—21no—
376Q8—21no—
377Q8—41no—
378Q8—231no—
379Q8—231no—
380Q8—461no—
381Q8—461no—
382Q8—921no—
383Q8D4×D2311yescharacteristic
384Q8D4×D2311yescharacteristic
385D4—21no—
386D4—21no—
387D4—21no—
388D4—21no—
389D4—41no—
390D4—231no—
391D4—231no—
392D4—461no—
393D4—461no—
394D4—461no—
395D4—461no—
396D4—921no—
397D4—921no—
398D4—921no—
399D4—921no—
400D4—921no—
401D4D4×D2311yescharacteristic
402D4D4×D2311yescharacteristic
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—461no—
410C2×C4—461no—
411C2×C4—461no—
412C2×C4—921no—
413C2×C4—921no—
414C2×C4—921no—
415C2×C4—921no—
416C2×C4C23×D2311yescharacteristic
417C8—21no—
418C8—21no—
419C8—231no—
420C8—231no—
421C8—231no—
422C8—231no—
423C8—461no—
424C8—461no—
425C22—21no—
426C22—21no—
427C22—41no—
428C22—81no—
429C22—461no—
430C22—461no—
431C22—921no—
432C22—3681no—
433C22—3681no—
434C22—1841no—
435C22#1062515511yescharacteristic
436C4—21no—
437C4—21no—
438C4—41no—
439C4—41no—
440C4—461no—
441C4—461no—
442C4—921no—
443C4C2×D4×D2311yescharacteristic
444C4C2×D4×D2311yescharacteristic
445C2—21no—
446C2—41no—
447C2—41no—
448C2—81no—
449C2—921no—
450C2—921no—
451C2—1841no—
452C2#1309304911yescharacteristic
subgroupG/H
centerC2#13093049
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#454C2
socleC2×C23D42
hypercenter#454D23
hypocenterC23#3290
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #13093232C2semidirect
2. #13093227C2semidirect
3. #13093225C2semidirect
4. #13093221C2semidirect
5. #13092799C2semidirect
6. #13092691C2semidirect
7. #13092685C2semidirect
8. #13092683C2semidirect
9. #13092678C2semidirect
10. #13092596C2semidirect
11. #13092594C2semidirect
12. #13092591C2semidirect
13. #13092531C2semidirect
14. #13092531C2semidirect
15. C23×#454C2semidirect
16. C23×2−1+4C22semidirect
17. #10625161C22semidirect
21. C23×AGL(1,ℤ/8)C22semidirect
22. C23×#124C22semidirect
24. #10625122C22semidirect
30. #10625120C22semidirect
31. #10625119C22semidirect
32. #10625119C22semidirect
34. C2×#67275C22semidirect
38. #10625113C22semidirect
40. C2×#67274C22semidirect
41. C2×#67273C22semidirect
54. #10625078C22semidirect
55. #10625046C22semidirect
56. C23×#104C22semidirect
57. C23×#104C22semidirect
58. C23×#101C22semidirect
59. #10625015C22semidirect
60. #10625006C22semidirect
61. #10625005C22semidirect
62. #10625004C22semidirect
63. #10625004C22semidirect
65. #10624987C22semidirect
66. C4×C23⋊C8C22semidirect
231. #454D23semidirect
243. 2−1+4C2×D23semidirect
261. AGL(1,ℤ/8)C2×D23semidirect
281. #124C2×D23semidirect
286. #104C2×D23semidirect
287. #104C2×D23semidirect
289. #101C2×D23semidirect
293. C23#3290semidirect

Maximal subgroups

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

subgroupindex
#130932322
#130932272
#130932252
#130932212
#130927992
#130926912
#130926852
#130926832
#130926782
#130925962
#130925942
#130925912
#130925312
C23×#4542
#329023