Finite Group Databasebeta

← #15682662

Subgroups — #15682662

Order 2944.

784 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10625160C211yesmaximal, characteristic, complemented
2C2×#10625085C211yesmaximal, characteristic, complemented
3C2×D8×D23C211yesmaximal, characteristic, complemented
4#13092983C211yesmaximal, characteristic, complemented
5#13092565C211yesmaximal, characteristic, complemented
6#13092562C211yesmaximal, characteristic, complemented
7#13092561C211yesmaximal, characteristic, complemented
8#13092553C211yesmaximal, characteristic, complemented
9#13092552C211yesmaximal, characteristic, complemented
10#13092550C211yesmaximal, characteristic, complemented
11#13092549C211yesmaximal, characteristic, complemented
12#13092420C211yesmaximal, characteristic, complemented
13#13092412C211yesmaximal, characteristic, complemented
14#13092382C211yesmaximal, characteristic, complemented
15C23×#449C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic, complemented
17#10625160—24no—
18#10625160—24no—
19C2×#67292C2211yescharacteristic
20C2×#67292C2211yescharacteristic, complemented
21C2×D4×D23—21no—
22C2×D4×D23—21no—
23C2×D4×D23C2211yescharacteristic
24C2×D4×D23C2211yescharacteristic, complemented
25C2×D4×D23C2211yescharacteristic, complemented
26C2×#67288—21no—
27C2×#67288C2211yescharacteristic
28C22×D92—21no—
29C22×D92C2211yescharacteristic, complemented
30C2×SD16×C23C2211yescharacteristic, complemented
31C2×D8×C23C2211yescharacteristic, complemented
32C23×#121C2211yescharacteristic, complemented
33C2×#67274C2211yescharacteristic, complemented
34#10625111—21no—
35#10625109C2211yescharacteristic, complemented
36#10625108—21no—
37C2×#67272C2211yescharacteristic, complemented
38C2×#67272C2211yescharacteristic, complemented
39C2×#67272C2211yescharacteristic, complemented
40#10625101C2211yescharacteristic, complemented
41#10625085—24no—
42D8×D23—24no—
43C2×D184C2211yescharacteristic, complemented
44C2×D184C2211yescharacteristic, complemented
45C2×#67262C2211yescharacteristic, complemented
46C2×C8×D23C2211yescharacteristic, complemented
47#10625066C2211yescharacteristic, complemented
48#10625063C2211yescharacteristic, complemented
49#10625056—21no—
50#10625053—21no—
51C23×#103C2211yescharacteristic, complemented
52C23×#102C2211yescharacteristic, complemented
53C23×C22⋊C8C2211yescharacteristic, complemented
54#10625017C2211yescharacteristic, complemented
55#10625014C2211yescharacteristic, complemented
56C22⋊(C23⋊C8)C2211yescharacteristic, complemented
57#10625003C2211yescharacteristic, complemented
58#10625003C2211yescharacteristic, complemented
59(C2×D23)⋊C8C2211yescharacteristic, complemented
60(C2×D23)⋊C8C2211yescharacteristic, complemented
61#10624991C2211yescharacteristic, complemented
62#10624991C2211yescharacteristic, complemented
63C23×D23—21no—
64C23×D23—21no—
65C23×D23—21no—
66C23×D23—21no—
67#41×C23—22no—
68#41×C23—22no—
69C2×Q8×C23C2311yescharacteristic
70C2×D4×C23—21no—
71C2×D4×C23—21no—
72C2×D4×C23C2311yescharacteristic
73C2×D4×C23C2311yescharacteristic
74C22×C4×C23—21no—
75C22×C4×C23C2311yescharacteristic
76C2×C22⋊D23—21no—
77C2×C22⋊D23—21no—
78C2×C22⋊D23—21no—
79C2×C22⋊D23—21no—
80C2×C22⋊D23—21no—
81C2×C22⋊D23D411yescharacteristic
82C2×C22⋊D23D411yescharacteristic
83#67292—22no—
84#67292—22no—
85#67292—22no—
86#67292D414yescomplemented
87D4×D23—28no—
88D4×D23—28no—
89D4×D23—22no—
90D4×D23—22no—
91D4×D23—22no—
92D4×D23—22no—
93D4×D23—22no—
94D4×D23D414yescomplemented
95#67288—28no—
96#67288—22no—
97#67288—22no—
98C2×D92—28no—
99C2×D92—21no—
100C2×D92—21no—
101C2×D92—21no—
102C2×D92—22no—
103C2×D92—22no—
104C2×D92—21no—
105C2×D92D411yescharacteristic
106C2×D92C2311yescharacteristic
107C2×D92C2311yescharacteristic
108C2×D92C2311yescharacteristic
109C2×C4×D23—21no—
110C2×C4×D23—21no—
111C2×C4×D23C2311yescharacteristic
112C2×C4×D23C2311yescharacteristic
113C2×Dic46D411yescharacteristic
114SD16×C23—22no—
115D8×C23—22no—
116C2×C8×C23C2311yescharacteristic
117C2×C8×C23C2311yescharacteristic
118C4⋊C4×C23C2311yescharacteristic
119C22⋊C4×C23—21no—
120C22⋊Dic23—21no—
121#67274—22no—
122#67272—22no—
123#67272—22no—
124#67272—22no—
125(C2×D23)⋊C4—21no—
126(C2×D23)⋊C4—21no—
127(C2×D23)⋊C4—21no—
128C4⋊Dic23C2311yescharacteristic
129C4×Dic23—21no—
130C2×C23⋊C8C2311yescharacteristic
131C2×C23⋊C8C2311yescharacteristic
132D184—22no—
133D184—22no—
134#67262—22no—
135C8×D23—22no—
136C23×C23—21no—
137C23×C23—21no—
138C23×C23C2×D411yescharacteristic
139C22×D23—21no—
140C22×D23—21no—
141C22×D23—24no—
142C22×D23—24no—
143C22×D23—24no—
144C22×D23—21no—
145C22×D23—42no—
146C22×D23—81no—
147C22×D23—81no—
148C22×D23—81no—
149C22×D23—81no—
150C22×D23—81no—
151C22×D23—81no—
152C22×D23—81no—
153C22×D23—81no—
154C22×D23C2×D411yescharacteristic
155C22×D23C2×D411yescharacteristic
156Q8×C23—21no—
157Q8×C23C2×D412yes—
158D4×C23—24no—
159D4×C23—21no—
160D4×C23—21no—
161D4×C23—21no—
162D4×C23—81no—
163D4×C23C2×D412yes—
164C2×C4×C23—24no—
165C2×C4×C23—21no—
166C2×C4×C23—21no—
167C2×C4×C23—21no—
168C2×C4×C23—21no—
169C2×C4×C23C2×D411yescharacteristic
170C2×C4×C23C2411yescharacteristic
171C22⋊D23—24no—
172C22⋊D23—22no—
173C22⋊D23—22no—
174C22⋊D23—24no—
175C22⋊D23—42no—
176C22⋊D23—81no—
177C22⋊D23—81no—
178C2×Dic23—21no—
179C2×Dic23C2×D411yescharacteristic
180C2×Dic23C2×D411yescharacteristic
181D92—24no—
182D92—24no—
183D92—22no—
184D92—24no—
185D92—21no—
186D92—21no—
187D92—21no—
188D92—21no—
189D92—81no—
190D92C2×D412yes—
191D92C2×D412yes—
192C4×D23—24no—
193C4×D23—21no—
194C4×D23—21no—
195C4×D23—24no—
196C4×D23C2×D412yes—
197C4×D23C2×D412yes—
198Dic46—22no—
199C8×C23—21no—
200C8×C23—21no—
201C23⋊C8—21no—
202C23⋊C8—21no—
203#3008—231nomaximal, Sylow
204C22×C23—21no—
205C22×C23—22no—
206C22×C23—21no—
207C22×C23—21no—
208C22×C23—41no—
209C22×C23—41no—
210C22×C23—41no—
211C22×C23—41no—
212C22×C23—41no—
213C22×C23C22×D411yescharacteristic
214C2×D23—21no—
215C2×D23—21no—
216C2×D23—21no—
217C2×D23—22no—
218C2×D23—22no—
219C2×D23—21no—
220C2×D23—21no—
221C2×D23—41no—
222C2×D23—41no—
223C2×D23—41no—
224C2×D23—41no—
225C2×D23—41no—
226C2×D23—41no—
227C2×D23—41no—
228C2×D23—84no—
229C2×D23—84no—
230C2×D23—84no—
231C2×D23—84no—
232C2×D23#12012yes—
233C4×C23—21no—
234C4×C23—22no—
235C4×C23—41no—
236C4×C23C22×D411yescharacteristic
237C4×C23C22×D411yescharacteristic
238Dic23—21no—
239Dic23—41no—
240Dic23#12012yes—
241C2×H(5,𝔽2)—231no—
242C2×AGL(1,ℤ/8)—231no—
243C22×D8—231no—
244#535—231no—
245#449—231no—
246#449—231no—
247#449—231no—
248#449D2311yescharacteristic, complemented
249#447—231no—
250#447—231no—
251#447—231no—
252#447—231no—
253#418—231no—
254#416—231no—
255#408—231no—
256C2×C23—41no—
257C2×C23—42no—
258C2×C23—81no—
259C2×C23C2×#12011yescharacteristic
260C2×C23#57611yescharacteristic
261C2×C23#57611yescharacteristic
262D23—22no—
263D23—41no—
264D23—42no—
265D23—42no—
266D23—81no—
267H(5,𝔽2)—464no—
268H(5,𝔽2)—464no—
269C2×#41—231no—
270C2×#41—231no—
271C2×#41—231no—
272C2×#41—461no—
273C2×#41C2×D2311yescharacteristic, complemented
274C22×D4—231no—
275C22×D4—231no—
276C22×D4—231no—
277C22×D4—231no—
278C22×D4—461no—
279C22×D4—461no—
280C22×D4—461no—
281AGL(1,ℤ/8)—464no—
282C2×SD16—231no—
283C2×SD16C2×D2311yescharacteristic, complemented
284C2×D8—231no—
285C2×D8—231no—
286C2×D8—231no—
287C2×D8—231no—
288C2×D8—231no—
289C2×D8—464no—
290C2×D8C2×D2311yescharacteristic, complemented
291C2×M16—231no—
292C22×C8—231no—
293#127—461no—
294#124—461no—
295#121—231no—
296#121—231no—
297#121—231no—
298#121C2×D2311yescharacteristic, complemented
299#120—461no—
300#120—461no—
301#117—231no—
302#103—231no—
303#103C2×D2311yescharacteristic, complemented
304#102—231no—
305#102—231no—
306#102—231no—
307#102—231no—
308#102—231no—
309#102C2×D2311yescharacteristic, complemented
310C22⋊C8—231no—
311C22⋊C8—231no—
312C22⋊C8—231no—
313C22⋊C8C2×D2311yescharacteristic, complemented
314C23#300811yescharacteristic, complemented, Sylow
315C24—461no—
316C24—461no—
317C24—461no—
318C24—461no—
319#41—22no—
320#41—22no—
321#41—234no—
322#41—462no—
323#41—462no—
324#41—468no—
325#41—462no—
326#41—462no—
327#41—462no—
328C2×Q8—231no—
329C2×Q8C22×D2311yescharacteristic
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—234no—
333C2×D4—231no—
334C2×D4—231no—
335C2×D4—231no—
336C2×D4—231no—
337C2×D4—231no—
338C2×D4—231no—
339C2×D4—462no—
340C2×D4—462no—
341C2×D4—461no—
342C2×D4—461no—
343C2×D4—468no—
344C2×D4—461no—
345C2×D4—462no—
346C2×D4—462no—
347C2×D4—462no—
348C2×D4—461no—
349C2×D4—461no—
350C2×D4—461no—
351C2×D4—462no—
352C2×D4—468no—
353C2×D4—468no—
354C2×D4—462no—
355C2×D4—461no—
356C2×D4—461no—
357C2×D4—461no—
358C2×D4C22×D2311yescharacteristic
359C2×D4C22×D2311yescharacteristic
360C22×C4—21no—
361C22×C4—231no—
362C22×C4—231no—
363C22×C4—461no—
364C22×C4—461no—
365C22×C4C22×D2311yescharacteristic
366SD16—22no—
367SD16—462no—
368D8—22no—
369D8—462no—
370D8—462no—
371D8—462no—
372D8—462no—
373D8—462no—
374M16—462no—
375C2×C8—231no—
376C2×C8—231no—
377C2×C8—462no—
378C2×C8C22×D2311yescharacteristic
379C2×C8C22×D2311yescharacteristic
380C4⋊C4—231no—
381C4⋊C4C22×D2311yescharacteristic
382C22⋊C4—21no—
383C22⋊C4—461no—
384C22⋊C4—461no—
385C22⋊C4—461no—
386C22⋊C4—461no—
387C42—461no—
388C23—21no—
389C23—21no—
390C23—231no—
391C23—231no—
392C23—464no—
393C23—464no—
394C23—464no—
395C23—461no—
396C23—461no—
397C23—461no—
398C23—922no—
399C23—1841no—
400C23—1841no—
401C23—1841no—
402C23—1841no—
403C23—1841no—
404C23—1841no—
405C23—1841no—
406C23—1841no—
407C23D4×D2311yescharacteristic
408Q8—21no—
409Q8—462no—
410Q8D4×D2312yes—
411D4—24no—
412D4—21no—
413D4—21no—
414D4—21no—
415D4—81no—
416D4—232no—
417D4—232no—
418D4—462no—
419D4—464no—
420D4—464no—
421D4—464no—
422D4—462no—
423D4—464no—
424D4—462no—
425D4—464no—
426D4—461no—
427D4—461no—
428D4—461no—
429D4—461no—
430D4—922no—
431D4—1841no—
432D4—1841no—
433D4—1841no—
434D4D4×D2312yes—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—231no—
441C2×C4—232no—
442C2×C4—232no—
443C2×C4—231no—
444C2×C4—464no—
445C2×C4—464no—
446C2×C4—461no—
447C2×C4—461no—
448C2×C4—461no—
449C2×C4D4×D2311yescharacteristic
450C2×C4C23×D2311yescharacteristic
451C8—21no—
452C8—21no—
453C8—461no—
454C8—461no—
455C22—21no—
456C22—22no—
457C22—21no—
458C22—21no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—232no—
465C22—461no—
466C22—461no—
467C22—462no—
468C22—462no—
469C22—461no—
470C22—461no—
471C22—461no—
472C22—921no—
473C22—921no—
474C22—921no—
475C22—921no—
476C22—921no—
477C22—921no—
478C22—921no—
479C22—1844no—
480C22—1844no—
481C22—1844no—
482C22—1844no—
483C22C2×D4×D2311yescharacteristic
484C4—21no—
485C4—22no—
486C4—41no—
487C4—232no—
488C4—461no—
489C4—921no—
490C4C2×D4×D2311yescharacteristic
491C4C2×D4×D2311yescharacteristic
492C2—41no—
493C2—42no—
494C2—81no—
495C2—462no—
496C2—921no—
497C2—922no—
498C2—922no—
499C2—1841no—
500C2#120×D2311yescharacteristic
501C2#1309321911yescharacteristic
502C2#1309322111yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#449C2
socleC22×C23C22×D4
hypercenter#449D23
hypocenterC23#3008
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#10625160C2semidirect
2. C2×#10625085C2semidirect
3. C2×D8×D23C2semidirect
4. #13092983C2semidirect
5. #13092565C2semidirect
6. #13092562C2semidirect
7. #13092561C2semidirect
8. #13092553C2semidirect
9. #13092552C2semidirect
10. #13092550C2semidirect
11. #13092549C2semidirect
12. #13092420C2semidirect
13. #13092412C2semidirect
14. #13092382C2semidirect
15. C23×#449C2semidirect
16. C2×#41×C23C22semidirect
20. C2×#67292C22semidirect
24. C2×D4×D23C22semidirect
25. C2×D4×D23C22semidirect
29. C22×D92C22semidirect
30. C2×SD16×C23C22semidirect
31. C2×D8×C23C22semidirect
32. C23×#121C22semidirect
33. C2×#67274C22semidirect
35. #10625109C22semidirect
37. C2×#67272C22semidirect
38. C2×#67272C22semidirect
39. C2×#67272C22semidirect
40. #10625101C22semidirect
43. C2×D184C22semidirect
44. C2×D184C22semidirect
45. C2×#67262C22semidirect
46. C2×C8×D23C22semidirect
47. #10625066C22semidirect
48. #10625063C22semidirect
51. C23×#103C22semidirect
52. C23×#102C22semidirect
53. C23×C22⋊C8C22semidirect
54. #10625017C22semidirect
55. #10625014C22semidirect
56. C22⋊(C23⋊C8)C22semidirect
57. #10625003C22semidirect
58. #10625003C22semidirect
59. (C2×D23)⋊C8C22semidirect
60. (C2×D23)⋊C8C22semidirect
61. #10624991C22semidirect
62. #10624991C22semidirect
86. #67292D4semidirect
94. D4×D23D4semidirect
248. #449D23semidirect
273. C2×#41C2×D23semidirect
283. C2×SD16C2×D23semidirect
290. C2×D8C2×D23semidirect
298. #121C2×D23semidirect
303. #103C2×D23semidirect
309. #102C2×D23semidirect
313. C22⋊C8C2×D23semidirect
314. C23#3008semidirect

Maximal subgroups

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

subgroupindex
C2×#106251602
C2×#106250852
C2×D8×D232
#130929832
#130925652
#130925622
#130925612
#130925532
#130925522
#130925502
#130925492
#130924202
#130924122
#130923822
C23×#4492
#300823