Finite Group Databasebeta

← #11867107

Subgroups — #11867107

Order 1152.

1644 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2#10596299C211yesmaximal, characteristic, complemented
3C2×#64147C211yesmaximal, characteristic, complemented
4C2×M16×C3⋊D3C211yesmaximal, characteristic, complemented
5#10592931C211yesmaximal, characteristic, complemented
6#10592914C211yesmaximal, characteristic, complemented
7#10592873C212yesmaximal, complemented
8#10592870C212yesmaximal, complemented
9#10592858C212yesmaximal, complemented
10#10592856C212yesmaximal, complemented
11#10592820C211yesmaximal, characteristic, complemented
12#83950—34nomaximal
13C2×C32×#41C2211yescharacteristic
14C22×C32×D4C2211yescharacteristic
15C22×#638—21no—
16C22×C4×C3⋊D3C2211yescharacteristic
17C32×AGL(1,ℤ/8)—24no—
18C2×C32×SD16C2212yescomplemented
19C2×C32×D8C2212yescomplemented
20C2×C32×M16C2211yescharacteristic
21C2×#3757—21no—
22#64155C2211yescharacteristic, complemented
23#64155C2211yescharacteristic, complemented
24#64147—21no—
25#64147—21no—
26#64147C2212yes—
27#64147C2212yes—
28#64147C2211yescharacteristic, complemented
29#64147C2211yescharacteristic, complemented
30#64146—21no—
31#64144—21no—
32C4×#638—21no—
33C4×#638—21no—
34#64135C2211yescharacteristic
35C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
36C2×#3748C2211yescharacteristic
37M16×C3⋊D3—24no—
38C2×#3743C2212yescomplemented
39C2×C8×C3⋊D3C2212yescomplemented
40#63662C2212yescomplemented
41#63659C2212yescomplemented
42#63659C2212yescomplemented
43#63659C2212yescomplemented
44#63645C2212yescomplemented
45#63644C2212yescomplemented
46C2×C3×AGL(1,ℤ/8)D314yescomplemented
47#5828—34no—
48C2×#938—34no—
49C2×D3×M16—34no—
50#5232—34no—
51#5215—34no—
52#5174—38no—
53#5171—38no—
54#5159—38no—
55#5157—38no—
56#5121—34no—
57C24×C32—21no—
58C23×C3⋊D3D411yescharacteristic
59C32×#41—22no—
60C32×#41—22no—
61C2×C32×Q8C2311yescharacteristic
62C2×C32×D4—21no—
63C2×C32×D4—22no—
64C2×C32×D4—21no—
65C2×C32×D4—22no—
66C2×C32×D4C2312yes—
67C2×C32×D4C2311yescharacteristic
68C22×C32×C4—21no—
69C22×C32×C4C2311yescharacteristic
70C2×#638—22no—
71C2×#638—22no—
72C2×#638—21no—
73C2×#638—21no—
74C2×#638—44no—
75C22×C3⋊Dic3—21no—
76C22×C3⋊Dic3D411yescharacteristic
77C2×C4×C3⋊D3—22no—
78C2×C4×C3⋊D3—22no—
79C2×C4×C3⋊D3D411yescharacteristic
80C2×C4×C3⋊D3D411yescharacteristic
81C2×C4×C3⋊D3C2311yescharacteristic
82C2×C4×C3⋊D3C2311yescharacteristic
83C32×SD16—24no—
84C32×D8—24no—
85C32×M16D414yescomplemented
86C2×C32×C8C2312yes—
87#3757—21no—
88#3757—21no—
89#3757—21no—
90#3757—22no—
91#3757—22no—
92#3752—21no—
93#3752—21no—
94C4⋊(C3⋊Dic3)—21no—
95C4⋊(C3⋊Dic3)C2312yes—
96C4⋊(C3⋊Dic3)C2311yescharacteristic
97C4⋊(C3⋊Dic3)C2311yescharacteristic
98#3750—21no—
99#3750—21no—
100C4×C3⋊Dic3—21no—
101#3748—22no—
102C2×C3⋊(C3⋊C8)C2312yes—
103#3743—24no—
104C8×C3⋊D3—24no—
105#3055—91noSylow
106C2×C3×#41C2×D314yes—
107C22×C3×D4C2×D314yes—
108C22×#67—64no—
109C22×C4×D3—34no—
110C3×AGL(1,ℤ/8)—216no—
111C2×C3×SD16C2×D318yescomplemented
112C2×C3×D8C2×D318yescomplemented
113C2×C3×M16C2×D314yes—
114C2×#217—64no—
115#946—34no—
116#946—34no—
117#938—34no—
118#938—38no—
119#938—34no—
120#938—38no—
121#938—64no—
122#938—64no—
123#937—64no—
124#935—64no—
125C4×#67—64no—
126C4×#67—64no—
127#926—34no—
128C2×C4⋊Dic3—34no—
129C2×#208—34no—
130D3×M16—616no—
131C2×ΓL(1,𝔽25)—38no—
132C2×D3×C8—38no—
133#835—38no—
134#832—38no—
135#832—38no—
136#832—38no—
137#818—38no—
138#817—38no—
139C23×C32—22no—
140C23×C32—21no—
141C23×C32—42no—
142C23×C32—42no—
143C23×C32—42no—
144C23×C32C2×D411yescharacteristic
145C22×C3⋊D3—22no—
146C22×C3⋊D3—22no—
147C22×C3⋊D3—41no—
148C22×C3⋊D3C2×D411yescharacteristic
149C22×C3⋊D3C2×D411yescharacteristic
150C32×Q8—21no—
151C32×Q8—21no—
152C32×D4—21no—
153C32×D4—22no—
154C32×D4—22no—
155C32×D4—21no—
156C32×D4—42no—
157C32×D4—42no—
158C2×C32×C4—21no—
159C2×C32×C4—42no—
160C2×C32×C4C2×D411yescharacteristic
161C2×C32×C4C2×D412yes—
162C2×C32×C4#4112yes—
163C2×C32×C4C2411yescharacteristic
164#638—42no—
165#638—44no—
166#638—42no—
167#638—44no—
168C2×C3⋊Dic3—22no—
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3—22no—
172C2×C3⋊Dic3—42no—
173C2×C3⋊Dic3C2×D411yescharacteristic
174C2×C3⋊Dic3C2×D411yescharacteristic
175C4×C3⋊D3—22no—
176C4×C3⋊D3—22no—
177C4×C3⋊D3—21no—
178C4×C3⋊D3—21no—
179C4×C3⋊D3—21no—
180C4×C3⋊D3—21no—
181C32×C8C2×D414yescomplemented
182C3⋊(C3⋊C8)—22no—
183C2×AGL(1,ℤ/8)C3⋊D311yescharacteristic, complemented
184C22×M16—91no—
185#525—91no—
186C2×#121—91no—
187#469—92no—
188#468—92no—
189#466—92no—
190#465—92no—
191#428—91no—
192#418—91no—
193#417—91no—
194C24×C3—24no—
195C23×D3—34no—
196C3×#41—28no—
197C3×#41—28no—
198C2×C3×Q8C22×D314yes—
199C2×C3×D4—24no—
200C2×C3×D4—24no—
201C2×C3×D4—28no—
202C2×C3×D4—28no—
203C2×C3×D4C22×D314yes—
204C2×C3×D4C22×D318yes—
205C22×C3×C4—24no—
206C22×C3×C4C22×D314yes—
207C2×#67—64no—
208C2×#67—64no—
209C2×#67—68no—
210C2×#67—68no—
211C2×#67—1216no—
212C22×Dic3—34no—
213C22×Dic3—64no—
214C2×C4×D3—34no—
215C2×C4×D3—34no—
216C2×C4×D3—34no—
217C2×C4×D3—34no—
218C2×C4×D3—68no—
219C2×C4×D3—68no—
220C3×SD16—216no—
221C3×D8—216no—
222C3×M16#67116yescomplemented
223C2×C3×C8C22×D318yes—
224#217—64no—
225#217—64no—
226#217—68no—
227#217—68no—
228#217—64no—
229(C2×D3)⋊C4—64no—
230(C2×D3)⋊C4—64no—
231C4⋊Dic3—34no—
232C4⋊Dic3—38no—
233C4⋊Dic3—34no—
234C4⋊Dic3—64no—
235Dic3⋊C4—64no—
236Dic3⋊C4—64no—
237C4×Dic3—64no—
238#208—68no—
239C2×C3⋊C8—38no—
240ΓL(1,𝔽25)—616no—
241D3×C8—616no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—41no—
245C22×C32—42no—
246C22×C32—42no—
247C22×C32—41no—
248C22×C32—41no—
249C22×C32—42no—
250C22×C32—84no—
251C22×C32#12112yes—
252C22×C32C22×D411yescharacteristic
253C2×C3⋊D3—21no—
254C2×C3⋊D3—21no—
255C2×C3⋊D3—21no—
256C2×C3⋊D3—21no—
257C2×C3⋊D3—21no—
258C2×C3⋊D3—44no—
259C2×C3⋊D3AGL(1,ℤ/8)12yescomplemented
260C32×C4—41no—
261C32×C4#12112yes—
262C32×C4C22×D411yescharacteristic
263C32×C4C2×#4111yescharacteristic
264C3⋊Dic3—21no—
265C3⋊Dic3—21no—
266C3⋊Dic3—42no—
267C3⋊Dic3—41no—
268C3⋊Dic3—41no—
269C2×#41C2×C3⋊D311yescharacteristic
270C22×D4—181no—
271C22×D4C2×C3⋊D311yescharacteristic
272C23×C4—91no—
273AGL(1,ℤ/8)—24no—
274C2×SD16C2×C3⋊D312yescomplemented
275C2×D8C2×C3⋊D312yescomplemented
276C2×M16—91no—
277C2×M16—92no—
278C2×M16—184no—
279C2×M16C2×C3⋊D311yescharacteristic
280C22×C8—92no—
281#123—181no—
282C22⋊Q8—91no—
283C22⋊Q8—91no—
284#121—92no—
285#121—92no—
286#121—91no—
287#121—91no—
288#121—181no—
289#121—181no—
290#120—181no—
291C4×D4—181no—
292C4×D4—181no—
293#117—91no—
294C2×C4⋊C4—91no—
295C2×C22⋊C4—181no—
296#107—92no—
297#106—92no—
298#103—92no—
299#102—92no—
300#102—92no—
301#102—92no—
302C23×C3—24no—
303C23×C3—28no—
304C23×C3—48no—
305C23×C3—48no—
306C23×C3—48no—
307C23×C3D3×D414yes—
308C22×D3—34no—
309C22×D3—34no—
310C22×D3—68no—
311C22×D3—68no—
312C22×D3—124no—
313C3×Q8—24no—
314C3×Q8—24no—
315C3×D4—28no—
316C3×D4—24no—
317C3×D4—24no—
318C3×D4—28no—
319C3×D4—48no—
320C3×D4—48no—
321C2×C3×C4—24no—
322C2×C3×C4—48no—
323C2×C3×C4D3×D414yes—
324C2×C3×C4#23718yes—
325C2×C3×C4C2×#6718yes—
326C2×C3×C4C23×D314yes—
327#67—128no—
328#67—1216no—
329#67—128no—
330#67—1216no—
331C2×Dic3—34no—
332C2×Dic3—34no—
333C2×Dic3—68no—
334C2×Dic3—64no—
335C2×Dic3—64no—
336C2×Dic3—68no—
337C2×Dic3—128no—
338C4×D3—68no—
339C4×D3—68no—
340C4×D3—64no—
341C4×D3—64no—
342C4×D3—64no—
343C4×D3—64no—
344C3×C8C2×#67116yescomplemented
345C3⋊C8—68no—
346C2×C32—22no—
347C2×C32—81no—
348C2×C32—82no—
349C2×C32C2×#12111yescharacteristic
350C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
351C2×C32#57511yescharacteristic
352C3⋊D3—22no—
353C3⋊D3—41no—
354C24—21no—
355C24—91no—
356#41—22no—
357#41—22no—
358C2×Q8C22×C3⋊D311yescharacteristic
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—181no—
364C2×D4—181no—
365C2×D4—182no—
366C2×D4—182no—
367C2×D4—364no—
368C2×D4C22×C3⋊D311yescharacteristic
369C2×D4C22×C3⋊D312yes—
370C22×C4—21no—
371C22×C4—91no—
372C22×C4—91no—
373C22×C4—91no—
374C22×C4—91no—
375C22×C4—91no—
376C22×C4—182no—
377C22×C4—182no—
378C22×C4—181no—
379C22×C4C22×C3⋊D311yescharacteristic
380SD16—24no—
381D8—24no—
382M16—182no—
383M16—184no—
384M16#63814yescomplemented
385C2×C8—92no—
386C2×C8—184no—
387C2×C8C22×C3⋊D312yes—
388C4⋊C4—91no—
389C4⋊C4—91no—
390C4⋊C4—92no—
391C4⋊C4—181no—
392C4⋊C4—181no—
393C4⋊C4—181no—
394C22⋊C4—181no—
395C22⋊C4—181no—
396C22⋊C4—182no—
397C22⋊C4—182no—
398C22⋊C4—181no—
399C22⋊C4—181no—
400C22⋊C4—181no—
401C42—181no—
402C22×C3—24no—
403C22×C3—24no—
404C22×C3—48no—
405C22×C3—44no—
406C22×C3—44no—
407C22×C3—48no—
408C22×C3—44no—
409C22×C3—48no—
410C22×C3—816no—
411C22×C3#93818yes—
412C22×C3C2×D3×D414yes—
413C2×D3—38no—
414C2×D3—64no—
415C2×D3—64no—
416C2×D3—64no—
417C2×D3—64no—
418C2×D3—64no—
419C2×D3—1216no—
420C3×C4—44no—
421C3×C4#93818yes—
422C3×C4C2×#23714yes—
423C3×C4C22×#6714yes—
424Dic3—64no—
425Dic3—64no—
426Dic3—128no—
427Dic3—124no—
428Dic3—124no—
429C32#305511yescharacteristic, complemented, Sylow
430C23—21no—
431C23—22no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—91no—
436C23—91no—
437C23—182no—
438C23—182no—
439C23—361no—
440C23D4×C3⋊D311yescharacteristic
441Q8—21no—
442Q8—21no—
443D4—22no—
444D4—21no—
445D4—21no—
446D4—22no—
447D4—42no—
448D4—42no—
449D4—364no—
450D4—362no—
451D4—364no—
452D4—362no—
453C2×C4—21no—
454C2×C4—42no—
455C2×C4—91no—
456C2×C4—91no—
457C2×C4—182no—
458C2×C4—181no—
459C2×C4—181no—
460C2×C4—181no—
461C2×C4—181no—
462C2×C4—182no—
463C2×C4—182no—
464C2×C4—181no—
465C2×C4—181no—
466C2×C4—182no—
467C2×C4—362no—
468C2×C4D4×C3⋊D311yescharacteristic
469C2×C4#383012yes—
470C2×C4C2×#63812yes—
471C2×C4C23×C3⋊D311yescharacteristic
472C8—182no—
473C8C2×#63814yescomplemented
474C2×C3—28no—
475C2×C3—88no—
476C2×C3—84no—
477C2×C3D3×AGL(1,ℤ/8)14yes—
478C2×C3#577314yes—
479C2×C3C2×#93814yes—
480D3—68no—
481D3—124no—
482C22—21no—
483C22—21no—
484C22—42no—
485C22—41no—
486C22—41no—
487C22—42no—
488C22—41no—
489C22—42no—
490C22—84no—
491C22—92no—
492C22—181no—
493C22—181no—
494C22—181no—
495C22—181no—
496C22—181no—
497C22—364no—
498C22#6414712yes—
499C22C2×D4×C3⋊D311yescharacteristic
500C4—41no—
501C4—181no—
502C4—181no—
503C4—362no—
504C4—361no—
505C4—361no—
506C4#6414712yes—
507C4C2×#383011yescharacteristic
508C4C22×#63811yescharacteristic
509C3#8395014yescomplemented
510C2—22no—
511C2—82no—
512C2—81no—
513C2—182no—
514C2—361no—
515C2C3⋊D3×AGL(1,ℤ/8)11yescharacteristic
516C2#1059624411yescharacteristic
517C2C2×#6414711yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC2×C32×AGL(1,ℤ/8)C2
socleC22×C32C22×D4
hypercenterC2×AGL(1,ℤ/8)C3⋊D3
hypocenterC32#3055
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×C32×AGL(1,ℤ/8)C2semidirect
2. #10596299C2semidirect
3. C2×#64147C2semidirect
4. C2×M16×C3⋊D3C2semidirect
5. #10592931C2semidirect
6. #10592914C2semidirect
7. #10592873C2semidirect
8. #10592870C2semidirect
9. #10592858C2semidirect
10. #10592856C2semidirect
11. #10592820C2semidirect
18. C2×C32×SD16C22semidirect
19. C2×C32×D8C22semidirect
22. #64155C22semidirect
23. #64155C22semidirect
28. #64147C22semidirect
29. #64147C22semidirect
38. C2×#3743C22semidirect
39. C2×C8×C3⋊D3C22semidirect
40. #63662C22semidirect
41. #63659C22semidirect
42. #63659C22semidirect
43. #63659C22semidirect
44. #63645C22semidirect
45. #63644C22semidirect
46. C2×C3×AGL(1,ℤ/8)D3semidirect
85. C32×M16D4semidirect
111. C2×C3×SD16C2×D3semidirect
112. C2×C3×D8C2×D3semidirect
181. C32×C8C2×D4semidirect
183. C2×AGL(1,ℤ/8)C3⋊D3semidirect
222. C3×M16#67semidirect
259. C2×C3⋊D3AGL(1,ℤ/8)semidirect
274. C2×SD16C2×C3⋊D3semidirect
275. C2×D8C2×C3⋊D3semidirect
344. C3×C8C2×#67semidirect
384. M16#638semidirect
429. C32#3055semidirect
473. C8C2×#638semidirect
509. C3#83950semidirect

Maximal subgroups

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

subgroupindex
C2×C32×AGL(1,ℤ/8)2
#105962992
C2×#641472
C2×M16×C3⋊D32
#105929312
#105929142
#105928732
#105928702
#105928582
#105928562
#105928202
#839503