Finite Group Databasebeta

← D23×#471 #15683092

Subgroups — D23×#471 #15683092

Order 2944.

468 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#137×D23C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D23C211yesmaximal, characteristic
3C2×SD16×D23C211yesmaximal, characteristic, complemented
4#131×D23C211yesmaximal, characteristic, complemented
5#13092683C211yesmaximal, characteristic, complemented
6#13092682C211yesmaximal, characteristic, complemented
7#13092681C211yesmaximal, characteristic, complemented
8#13092680C211yesmaximal, characteristic, complemented
9#13092679C211yesmaximal, characteristic, complemented
10#13092678C211yesmaximal, characteristic, complemented
11#13092677C211yesmaximal, characteristic, complemented
12#108×D23C211yesmaximal, characteristic, complemented
13#101×D23C211yesmaximal, characteristic, complemented
14#100×D23C211yesmaximal, characteristic, complemented
15C23×#471C211yesmaximal, characteristic, complemented
16#41×D23C2211yescharacteristic
17C2×Q8×D23C2211yescharacteristic
18C2×D4×D23C2211yescharacteristic
19C23×#137C2211yescharacteristic, complemented
20C23×AGL(1,ℤ/8)C2211yescharacteristic
21C2×SD16×C23C2211yescharacteristic, complemented
22C23×#131C2211yescharacteristic, complemented
23#10625122C2211yescharacteristic, complemented
24#10625120C2211yescharacteristic
25#10625119C2211yescharacteristic, complemented
26#10625113C2211yescharacteristic, complemented
27C2×#67274C2211yescharacteristic, complemented
28C2×#67273C2211yescharacteristic, complemented
29#10625103C2211yescharacteristic
30#10625089—21no—
31Q16×D23—21no—
32#10625086—21no—
33#10625085—21no—
34SD16×D23—21no—
35SD16×D23—21no—
36SD16×D23—21no—
37SD16×D23—21no—
38SD16×D23—21no—
39SD16×D23—21no—
40#10625082—21no—
41D8×D23—21no—
42#10625080C2211yescharacteristic, complemented
43#10625079C2211yescharacteristic
44#10625078—21no—
45#10625078C2211yescharacteristic, complemented
46M16×D23—21no—
47M16×D23C2211yescharacteristic
48M16×D23C2211yescharacteristic
49M16×D23C2211yescharacteristic
50C2×#67263C2211yescharacteristic, complemented
51#10625072—21no—
52#10625072C2211yescharacteristic, complemented
53C2×C8×D23—21no—
54C2×C8×D23C2211yescharacteristic
55C23×#108C2211yescharacteristic, complemented
56C23×#101C2211yescharacteristic, complemented
57C23×#100C2211yescharacteristic, complemented
58#10625018C2211yescharacteristic, complemented
59#10625015C2211yescharacteristic, complemented
60#10625006C2211yescharacteristic, complemented
61#10625006C2211yescharacteristic, complemented
62#10625005C2211yescharacteristic, complemented
63#10625005C2211yescharacteristic, complemented
64#10625004C2211yescharacteristic, complemented
65#10625004C2211yescharacteristic, complemented
66#10625001C2211yescharacteristic, complemented
67C23×D23—21no—
68#41×C23C2311yescharacteristic
69C2×Q8×C23C2311yescharacteristic
70C2×D4×C23C2311yescharacteristic
71C2×C22⋊D23—21no—
72#67292D411yescharacteristic
73Q8×D23—21no—
74Q8×D23—21no—
75Q8×D23—21no—
76Q8×D23—21no—
77Q8×D23D411yescharacteristic
78#67290—21no—
79#67290D411yescharacteristic
80D4×D23—21no—
81D4×D23—21no—
82D4×D23—21no—
83D4×D23—21no—
84D4×D23—21no—
85D4×D23D411yescharacteristic
86#67288—21no—
87#67288C2311yescharacteristic
88C2×D92C2311yescharacteristic
89C2×C4×D23—21no—
90C2×C4×D23—21no—
91C2×C4×D23C2311yescharacteristic
92C2×Dic46—21no—
93C2×Dic46C2311yescharacteristic
94Q16×C23—21no—
95SD16×C23—21no—
96SD16×C23—21no—
97SD16×C23—21no—
98SD16×C23—21no—
99D8×C23—21no—
100M16×C23—21no—
101M16×C23C2311yescharacteristic
102M16×C23C2311yescharacteristic
103M16×C23C2311yescharacteristic
104C2×C8×C23—21no—
105C2×C8×C23C2311yescharacteristic
106#67275—21no—
107#67275—21no—
108#67274—21no—
109#67274—21no—
110#67274—21no—
111#67274—21no—
112#67273—21no—
113#67273—21no—
114#67273—21no—
115#67273—21no—
116#67272—21no—
117#67272—21no—
118#67267—21no—
119#67267C2311yescharacteristic
120#67267C2311yescharacteristic
121#67267C2311yescharacteristic
122C2×C23⋊C8—21no—
123C2×C23⋊C8C2311yescharacteristic
124Dic92—21no—
125D184—21no—
126#67263—21no—
127#67263—21no—
128#67263—21no—
129#67263—21no—
130#67262—21no—
131#67262—21no—
132#67262—21no—
133#67262—21no—
134#67262—21no—
135C8×D23—21no—
136C8×D23—21no—
137C8×D23—21no—
138C8×D23—21no—
139C8×D23—21no—
140C8×D23D411yescharacteristic
141C8×D23D411yescharacteristic
142C8×D23D411yescharacteristic
143C8×D23D411yescharacteristic
144C23×C23—21no—
145C22×D23—21no—
146C22×D23—21no—
147C22×D23—41no—
148C22×D23—41no—
149C22×D23—82no—
150C22×D23#4111yescharacteristic
151Q8×C23—21no—
152Q8×C23—21no—
153Q8×C23C2×D411yescharacteristic
154D4×C23—21no—
155D4×C23—21no—
156D4×C23—21no—
157D4×C23C2×D411yescharacteristic
158C2×C4×C23—21no—
159C2×C4×C23—21no—
160C2×C4×C23C2411yescharacteristic
161C22⋊D23—21no—
162C22⋊D23—21no—
163C22⋊D23—21no—
164C22⋊D23—41no—
165C22⋊D23—41no—
166C2×Dic23—21no—
167C2×Dic23—21no—
168C2×Dic23#4111yescharacteristic
169D92—21no—
170D92—21no—
171D92—21no—
172D92C2×D411yescharacteristic
173C4×D23—21no—
174C4×D23—21no—
175C4×D23—21no—
176C4×D23—41no—
177C4×D23—41no—
178C4×D23C2×D411yescharacteristic
179C4×D23C2×D411yescharacteristic
180C4×D23C2×D411yescharacteristic
181C4×D23C2×D411yescharacteristic
182Dic46—21no—
183Dic46—21no—
184Dic46—21no—
185Dic46—41no—
186Dic46—41no—
187Dic46C2×D411yescharacteristic
188C8×C23—21no—
189C8×C23—21no—
190C8×C23—21no—
191C8×C23C2×D411yescharacteristic
192C8×C23C2×D411yescharacteristic
193C23⋊C8—21no—
194C23⋊C8—21no—
195C23⋊C8—21no—
196C23⋊C8C2×D411yescharacteristic
197C23⋊C8C2×D411yescharacteristic
198C2×#471—231nomaximal, Sylow
199C22×C23—21no—
200C22×C23—41no—
201C22×C23—81no—
202C22×C23C2×#4111yescharacteristic
203C2×D23—22no—
204C2×D23—21no—
205C2×D23—42no—
206C2×D23—41no—
207C2×D23—82no—
208C2×D23—82no—
209C2×D23—81no—
210C2×D23#12111yescharacteristic
211C2×D23#12111yescharacteristic
212C4×C23—21no—
213C4×C23—41no—
214C4×C23C22×D411yescharacteristic
215C4×C23C22×D411yescharacteristic
216Dic23—21no—
217Dic23—41no—
218Dic23#12111yescharacteristic
219Dic23#12111yescharacteristic
220C2×#137—231no—
221C2×AGL(1,ℤ/8)—231no—
222C22×SD16—231no—
223C2×#131—231no—
224#471—231no—
225#471—231no—
226#471—231no—
227#471—231no—
228#471—231no—
229#471—231no—
230#471—231no—
231#471D2311yescharacteristic, direct product factor
232C2×#108—231no—
233C2×#101—231no—
234C2×#100—231no—
235C2×C23—21no—
236C2×C23—41no—
237C2×C23—81no—
238C2×C23C2×#12111yescharacteristic
239D23—21no—
240D23—41no—
241D23—81no—
242D23#47112yesdirect product factor
243C2×#41—231no—
244C22×Q8—231no—
245C22×D4—231no—
246#137—231no—
247#137—231no—
248#137—231no—
249#137—461no—
250#137—461no—
251#137C2×D2311yescharacteristic, complemented
252AGL(1,ℤ/8)—231no—
253AGL(1,ℤ/8)—231no—
254AGL(1,ℤ/8)—231no—
255AGL(1,ℤ/8)—461no—
256AGL(1,ℤ/8)—461no—
257AGL(1,ℤ/8)C2×D2311yescharacteristic
258C2×Q16—461no—
259C2×SD16—231no—
260C2×SD16—231no—
261C2×SD16—231no—
262C2×SD16—461no—
263C2×SD16—461no—
264C2×SD16—461no—
265C2×SD16—461no—
266C2×SD16—461no—
267C2×SD16—461no—
268C2×SD16C2×D2311yescharacteristic, complemented
269C2×D8—461no—
270#131—231no—
271#131—231no—
272#131—231no—
273#131—461no—
274#131—461no—
275#131C2×D2311yescharacteristic, complemented
276C2×M16—231no—
277C2×M16—231no—
278C2×M16—231no—
279C2×M16—461no—
280C22×C8—231no—
281C22×C8—461no—
282#108—231no—
283#108—231no—
284#108—231no—
285#108C2×D2311yescharacteristic, complemented
286#101—231no—
287#101—231no—
288#101—231no—
289#101C2×D2311yescharacteristic, complemented
290#100—231no—
291#100—231no—
292#100—231no—
293#100C2×D2311yescharacteristic, complemented
294C23C2×#47111yescharacteristic, complemented, Sylow
295C24—461no—
296#41—231no—
297#41—231no—
298#41—231no—
299#41—461no—
300#41—461no—
301#41C22×D2311yescharacteristic
302C2×Q8—231no—
303C2×Q8—231no—
304C2×Q8—461no—
305C2×Q8—461no—
306C2×Q8—461no—
307C2×Q8—461no—
308C2×Q8—461no—
309C2×Q8C22×D2311yescharacteristic
310C2×D4—231no—
311C2×D4—231no—
312C2×D4—461no—
313C2×D4—461no—
314C2×D4—461no—
315C2×D4—461no—
316C2×D4—461no—
317C2×D4—461no—
318C2×D4C22×D2311yescharacteristic
319C22×C4—231no—
320C22×C4—461no—
321C22×C4—461no—
322Q16—21no—
323Q16—461no—
324Q16—461no—
325Q16—461no—
326SD16—21no—
327SD16—21no—
328SD16—21no—
329SD16—21no—
330SD16—461no—
331SD16—461no—
332SD16—461no—
333SD16—461no—
334SD16—461no—
335SD16—461no—
336SD16—461no—
337SD16—461no—
338SD16—461no—
339SD16—461no—
340SD16—461no—
341SD16—461no—
342D8—21no—
343D8—461no—
344D8—461no—
345D8—461no—
346M16—21no—
347M16—231no—
348M16—231no—
349M16—231no—
350M16—461no—
351M16—461no—
352M16—461no—
353M16—461no—
354M16—461no—
355M16—461no—
356M16C22×D2311yescharacteristic
357M16C22×D2311yescharacteristic
358M16C22×D2311yescharacteristic
359C2×C8—21no—
360C2×C8—231no—
361C2×C8—231no—
362C2×C8—231no—
363C2×C8—231no—
364C2×C8—231no—
365C2×C8—461no—
366C2×C8—461no—
367C2×C8—461no—
368C2×C8—461no—
369C2×C8—461no—
370C2×C8—461no—
371C2×C8C22×D2311yescharacteristic
372C23—21no—
373C23—231no—
374C23—461no—
375C23—461no—
376C23—921no—
377C23—921no—
378C23—1842no—
379Q8—21no—
380Q8—21no—
381Q8—231no—
382Q8—461no—
383Q8—461no—
384Q8—461no—
385Q8—921no—
386Q8—921no—
387Q8D4×D2311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—231no—
392D4—461no—
393D4—461no—
394D4—461no—
395D4—461no—
396D4—461no—
397D4—461no—
398D4—921no—
399D4—921no—
400D4D4×D2311yescharacteristic
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—231no—
404C2×C4—231no—
405C2×C4—231no—
406C2×C4—231no—
407C2×C4—231no—
408C2×C4—461no—
409C2×C4—461no—
410C2×C4—461no—
411C2×C4—461no—
412C2×C4—461no—
413C2×C4—921no—
414C2×C4—921no—
415C2×C4C23×D2311yescharacteristic
416C8—21no—
417C8—21no—
418C8—21no—
419C8—231no—
420C8—231no—
421C8—461no—
422C8—461no—
423C8—461no—
424C8D4×D2311yescharacteristic
425C8D4×D2311yescharacteristic
426C22—21no—
427C22—41no—
428C22—81no—
429C22—231no—
430C22—231no—
431C22—462no—
432C22—461no—
433C22—922no—
434C22—921no—
435C22—1841no—
436C22—1842no—
437C22—1842no—
438C22#41×D2311yescharacteristic
439C4—21no—
440C4—41no—
441C4—231no—
442C4—231no—
443C4—461no—
444C4—921no—
445C4C2×D4×D2311yescharacteristic
446C4C2×D4×D2311yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—232no—
451C2—461no—
452C2—921no—
453C2—1841no—
454C2#121×D2311yescharacteristic
subgroupG/H
centerC2#121×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#471C2
socleC2×C23C2×#121
prime radicalAGL(1,ℤ/8)×D23C2
hypercenter#471D23
hypocenterC23C2×#471
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. #137×D23C2semidirect
3. C2×SD16×D23C2semidirect
4. #131×D23C2semidirect
5. #13092683C2semidirect
6. #13092682C2semidirect
7. #13092681C2semidirect
8. #13092680C2semidirect
9. #13092679C2semidirect
10. #13092678C2semidirect
11. #13092677C2semidirect
12. #108×D23C2semidirect
13. #101×D23C2semidirect
14. #100×D23C2semidirect
15. C23×#471C2semidirect
19. C23×#137C22semidirect
21. C2×SD16×C23C22semidirect
22. C23×#131C22semidirect
23. #10625122C22semidirect
25. #10625119C22semidirect
26. #10625113C22semidirect
27. C2×#67274C22semidirect
28. C2×#67273C22semidirect
42. #10625080C22semidirect
45. #10625078C22semidirect
50. C2×#67263C22semidirect
52. #10625072C22semidirect
55. C23×#108C22semidirect
56. C23×#101C22semidirect
57. C23×#100C22semidirect
58. #10625018C22semidirect
59. #10625015C22semidirect
60. #10625006C22semidirect
61. #10625006C22semidirect
62. #10625005C22semidirect
63. #10625005C22semidirect
64. #10625004C22semidirect
65. #10625004C22semidirect
66. #10625001C22semidirect
231. #471D23direct (N × H)
242. D23#471direct (N × H)
251. #137C2×D23semidirect
268. C2×SD16C2×D23semidirect
275. #131C2×D23semidirect
285. #108C2×D23semidirect
289. #101C2×D23semidirect
293. #100C2×D23semidirect
294. C23C2×#471semidirect

Maximal subgroups

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

subgroupindex
#137×D232
AGL(1,ℤ/8)×D232
C2×SD16×D232
#131×D232
#130926832
#130926822
#130926812
#130926802
#130926792
#130926782
#130926772
#108×D232
#101×D232
#100×D232
C23×#4712
C2×#47123