Finite Group Databasebeta

← #15683123

Subgroups — #15683123

Order 2944.

466 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13093213C211yesmaximal, characteristic, complemented
2C2×#10625082C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×D23C211yesmaximal, characteristic, complemented
4#13093182C211yesmaximal, characteristic, complemented
5#13092686C211yesmaximal, characteristic, complemented
6#13092685C211yesmaximal, characteristic, complemented
7#13092684C211yesmaximal, characteristic, complemented
8#13092681C211yesmaximal, characteristic, complemented
9#13092680C211yesmaximal, characteristic, complemented
10#13092679C211yesmaximal, characteristic, complemented
11#13092678C211yesmaximal, characteristic, complemented
12#13092464C211yesmaximal, characteristic, complemented
13#13092398C211yesmaximal, characteristic, complemented
14#100×D23C211yesmaximal, characteristic, complemented
15C23×#472C211yesmaximal, characteristic, complemented
16#41×D23C2211yescharacteristic
17C2×#67290C2211yescharacteristic
18C2×D4×D23C2211yescharacteristic
19C23×AGL(1,ℤ/8)C2211yescharacteristic
20C23×AGL(1,ℤ/8)C2211yescharacteristic, complemented
21C2×D8×C23C2211yescharacteristic, complemented
22C23×#131C2211yescharacteristic, complemented
23#10625122C2211yescharacteristic, complemented
24#10625120C2211yescharacteristic
25#10625119C2211yescharacteristic, complemented
26C2×#67273C2211yescharacteristic, complemented
27#10625103C2211yescharacteristic
28#10625103C2211yescharacteristic, complemented
29C2×#67272C2211yescharacteristic, complemented
30#10625087—21no—
31#10625086—21no—
32#10625085—21no—
33SD16×D23—21no—
34#10625083—21no—
35#10625082—21no—
36#10625082—21no—
37#10625082—21no—
38#10625082—21no—
39#10625082—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×#67262—21no—
54C2×#67262C2211yescharacteristic
55C23×#108C2211yescharacteristic, complemented
56C23×#100C2211yescharacteristic, complemented
57C23×#100C2211yescharacteristic, complemented
58#10625015C2211yescharacteristic, 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×D4×C23C2311yescharacteristic
70C2×D4×C23C2311yescharacteristic
71C2×C22⋊D23—21no—
72C2×C22⋊D23—21no—
73C22×Dic23—21no—
74#67292D411yescharacteristic
75Q8×D23D411yescharacteristic
76#67290—21no—
77#67290—21no—
78#67290—21no—
79#67290—21no—
80#67290—21no—
81#67290D411yescharacteristic
82D4×D23—21no—
83D4×D23—21no—
84D4×D23—21no—
85D4×D23—21no—
86D4×D23—21no—
87D4×D23D411yescharacteristic
88#67288—21no—
89#67288C2311yescharacteristic
90C2×D92C2311yescharacteristic
91C2×C4×D23—21no—
92C2×C4×D23C2311yescharacteristic
93C2×Dic46C2311yescharacteristic
94SD16×C23—21no—
95SD16×C23—21no—
96D8×C23—21no—
97D8×C23—21no—
98D8×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#67274—21no—
108#67273—21no—
109#67273—21no—
110#67273—21no—
111#67273—21no—
112#67273—21no—
113#67272—21no—
114#67272—21no—
115#67272—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—
135#67262D411yescharacteristic
136#67262D411yescharacteristic
137#67262D411yescharacteristic
138#67262D411yescharacteristic
139C8×D23—21no—
140C8×D23—21no—
141C8×D23—21no—
142C8×D23—21no—
143C8×D23—21no—
144C23×C23—21no—
145C23×C23—21no—
146C22×D23—21no—
147C22×D23—21no—
148C22×D23—41no—
149C22×D23—41no—
150C22×D23—82no—
151C22×D23#4111yescharacteristic
152Q8×C23C2×D411yescharacteristic
153D4×C23—21no—
154D4×C23—21no—
155D4×C23—21no—
156D4×C23—21no—
157D4×C23—21no—
158D4×C23C2×D411yescharacteristic
159C2×C4×C23—21no—
160C2×C4×C23C2411yescharacteristic
161C22⋊D23—21no—
162C22⋊D23—21no—
163C22⋊D23—21no—
164C22⋊D23—41no—
165C22⋊D23—41no—
166C22⋊D23—41no—
167C22⋊D23—41no—
168C2×Dic23—21no—
169C2×Dic23—21no—
170C2×Dic23—41no—
171C2×Dic23—41no—
172C2×Dic23#4111yescharacteristic
173D92—21no—
174D92—21no—
175D92—21no—
176D92C2×D411yescharacteristic
177C4×D23—21no—
178C4×D23—21no—
179C4×D23—21no—
180C4×D23C2×D411yescharacteristic
181C4×D23C2×D411yescharacteristic
182C4×D23C2×D411yescharacteristic
183C4×D23C2×D411yescharacteristic
184Dic46—21no—
185Dic46—21no—
186Dic46—21no—
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
198#3068—231nomaximal, Sylow
199C22×C23—21no—
200C22×C23—41no—
201C22×C23—41no—
202C22×C23—81no—
203C22×C23—81no—
204C22×C23C2×#4111yescharacteristic
205C2×D23—21no—
206C2×D23—41no—
207C2×D23—41no—
208C2×D23—81no—
209C2×D23—82no—
210C2×D23—82no—
211C2×D23—81no—
212C2×D23#12111yescharacteristic
213C2×D23#12111yescharacteristic
214C4×C23—21no—
215C4×C23C22×D411yescharacteristic
216C4×C23C22×D411yescharacteristic
217Dic23—21no—
218Dic23—41no—
219Dic23#12111yescharacteristic
220Dic23#12111yescharacteristic
221#575—231no—
222C2×AGL(1,ℤ/8)—231no—
223C2×AGL(1,ℤ/8)—231no—
224#568—231no—
225#472—231no—
226#472—231no—
227#472—231no—
228#472D2311yescharacteristic, complemented
229#471—231no—
230#471—231no—
231#471—231no—
232#471—231no—
233#430—231no—
234#413—231no—
235C2×#100—231no—
236C2×C23—21no—
237C2×C23—41no—
238C2×C23—81no—
239C2×C23—81no—
240C2×C23C2×#12111yescharacteristic
241D23—21no—
242D23—21no—
243D23—41no—
244D23—81no—
245C2×#41—231no—
246C2×#41—231no—
247C22×D4—231no—
248#137—231no—
249#137—231no—
250#137—461no—
251AGL(1,ℤ/8)—231no—
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)—461no—
258AGL(1,ℤ/8)—461no—
259AGL(1,ℤ/8)—461no—
260AGL(1,ℤ/8)—461no—
261AGL(1,ℤ/8)—461no—
262AGL(1,ℤ/8)C2×D2311yescharacteristic
263AGL(1,ℤ/8)C2×D2311yescharacteristic, complemented
264#135—461no—
265#135—461no—
266C2×SD16—231no—
267C2×SD16—231no—
268C2×SD16—461no—
269C2×D8—231no—
270C2×D8—461no—
271C2×D8C2×D2311yescharacteristic, complemented
272#131—231no—
273#131—231no—
274#131—231no—
275#131—461no—
276#131—461no—
277#131C2×D2311yescharacteristic, complemented
278C2×M16—231no—
279C2×M16—231no—
280C2×M16—231no—
281C2×M16—231no—
282C2×M16—461no—
283C2×M16—461no—
284#108—231no—
285#108—231no—
286#108—231no—
287#108C2×D2311yescharacteristic, complemented
288#101—231no—
289#101—231no—
290#100—231no—
291#100—231no—
292#100—231no—
293#100—231no—
294#100C2×D2311yescharacteristic, complemented
295#100C2×D2311yescharacteristic, complemented
296C23#306811yescharacteristic, complemented, Sylow
297C24—461no—
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#41C22×D2311yescharacteristic
308C2×Q8—231no—
309C2×Q8—231no—
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×D4—461no—
319C2×D4C22×D2311yescharacteristic
320C2×D4C22×D2311yescharacteristic
321C22×C4—231no—
322C22×C4—461no—
323C22×C4—461no—
324Q16—461no—
325Q16—461no—
326SD16—21no—
327SD16—21no—
328SD16—461no—
329SD16—461no—
330SD16—461no—
331SD16—461no—
332SD16—461no—
333SD16—461no—
334SD16—461no—
335SD16—461no—
336SD16—461no—
337SD16—461no—
338D8—21no—
339D8—21no—
340D8—21no—
341D8—21no—
342D8—461no—
343D8—461no—
344D8—461no—
345D8—461no—
346D8—461no—
347D8—461no—
348M16—21no—
349M16—231no—
350M16—231no—
351M16—231no—
352M16—231no—
353M16—231no—
354M16—231no—
355M16—231no—
356M16—461no—
357M16—461no—
358M16—461no—
359M16—461no—
360M16—461no—
361M16—461no—
362M16C22×D2311yescharacteristic
363M16C22×D2311yescharacteristic
364M16C22×D2311yescharacteristic
365C2×C8—21no—
366C2×C8—231no—
367C2×C8—461no—
368C2×C8—461no—
369C2×C8—461no—
370C2×C8—461no—
371C2×C8—461no—
372C2×C8—461no—
373C2×C8C22×D2311yescharacteristic
374C23—21no—
375C23—21no—
376C23—231no—
377C23—461no—
378C23—461no—
379C23—921no—
380C23—921no—
381C23—1842no—
382Q8—231no—
383Q8—461no—
384Q8—461no—
385Q8—461no—
386Q8D4×D2311yescharacteristic
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—231no—
393D4—461no—
394D4—461no—
395D4—461no—
396D4—461no—
397D4—461no—
398D4—461no—
399D4—921no—
400D4—921no—
401D4—921no—
402D4—921no—
403D4D4×D2311yescharacteristic
404C2×C4—21no—
405C2×C4—231no—
406C2×C4—231no—
407C2×C4—231no—
408C2×C4—231no—
409C2×C4—231no—
410C2×C4—461no—
411C2×C4—461no—
412C2×C4—461no—
413C2×C4—461no—
414C2×C4—461no—
415C2×C4—921no—
416C2×C4—921no—
417C2×C4C23×D2311yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—231no—
422C8—231no—
423C8—461no—
424C8—461no—
425C8—461no—
426C8D4×D2311yescharacteristic
427C8D4×D2311yescharacteristic
428C22—21no—
429C22—41no—
430C22—41no—
431C22—81no—
432C22—81no—
433C22—231no—
434C22—231no—
435C22—461no—
436C22—921no—
437C22—921no—
438C22—1841no—
439C22—1841no—
440C22—1842no—
441C22—1842no—
442C22#41×D2311yescharacteristic
443C4—21no—
444C4—231no—
445C4—231no—
446C4—461no—
447C4—921no—
448C4C2×D4×D2311yescharacteristic
449C4C2×D4×D2311yescharacteristic
450C2—21no—
451C2—41no—
452C2—81no—
453C2—81no—
454C2—461no—
455C2—461no—
456C2—921no—
457C2—1841no—
458C2#121×D2311yescharacteristic
subgroupG/H
centerC2#121×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#472C2
socleC2×C23C2×#121
hypercenter#472D23
hypocenterC23#3068
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. #13093213C2semidirect
2. C2×#10625082C2semidirect
3. AGL(1,ℤ/8)×D23C2semidirect
4. #13093182C2semidirect
5. #13092686C2semidirect
6. #13092685C2semidirect
7. #13092684C2semidirect
8. #13092681C2semidirect
9. #13092680C2semidirect
10. #13092679C2semidirect
11. #13092678C2semidirect
12. #13092464C2semidirect
13. #13092398C2semidirect
14. #100×D23C2semidirect
15. C23×#472C2semidirect
20. C23×AGL(1,ℤ/8)C22semidirect
21. C2×D8×C23C22semidirect
22. C23×#131C22semidirect
23. #10625122C22semidirect
25. #10625119C22semidirect
26. C2×#67273C22semidirect
28. #10625103C22semidirect
29. C2×#67272C22semidirect
42. #10625080C22semidirect
45. #10625078C22semidirect
50. C2×#67263C22semidirect
52. #10625072C22semidirect
55. C23×#108C22semidirect
56. C23×#100C22semidirect
57. C23×#100C22semidirect
58. #10625015C22semidirect
59. #10625015C22semidirect
60. #10625006C22semidirect
61. #10625006C22semidirect
62. #10625005C22semidirect
63. #10625005C22semidirect
64. #10625004C22semidirect
65. #10625004C22semidirect
66. #10625001C22semidirect
228. #472D23semidirect
263. AGL(1,ℤ/8)C2×D23semidirect
271. C2×D8C2×D23semidirect
277. #131C2×D23semidirect
287. #108C2×D23semidirect
294. #100C2×D23semidirect
295. #100C2×D23semidirect
296. C23#3068semidirect

Maximal subgroups

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

subgroupindex
#130932132
C2×#106250822
AGL(1,ℤ/8)×D232
#130931822
#130926862
#130926852
#130926842
#130926812
#130926802
#130926792
#130926782
#130924642
#130923982
#100×D232
C23×#4722
#306823