Finite Group Databasebeta

← #11864555

Subgroups — #11864555

Order 1152.

2292 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64366C211yesmaximal, characteristic, complemented
2C32×#537C211yesmaximal, characteristic, complemented
3C2×#64159C211yesmaximal, characteristic, complemented
4C2×#64158C211yesmaximal, characteristic, complemented
5#10592933C211yesmaximal, characteristic, complemented
6#10592746C212yesmaximal, complemented
7#10592745C212yesmaximal, complemented
8#10592736C212yesmaximal, complemented
9#10592735C212yesmaximal, complemented
10#10592665C211yesmaximal, characteristic, complemented
11C2×#63636C211yesmaximal, characteristic, complemented
12#83424—34nomaximal
13C2×C32×#41C2211yescharacteristic
14C2×C32×#41C2211yescharacteristic, complemented
15#64366—24no—
16#64366—24no—
17C2×#3832—21no—
18C2×D4×C3⋊D3—21no—
19C2×D4×C3⋊D3—22no—
20C2×#3828C2212yes—
21C2×#3828C2211yescharacteristic
22C22×C4⋊(C3⋊D3)C2212yes—
23C22×C4⋊(C3⋊D3)C2211yescharacteristic
24C32×C22⋊Q8—21no—
25C32×C22⋊Q8C2212yescomplemented
26C32×#121—21no—
27C32×#121—22no—
28C32×#121C2212yescomplemented
29C2×C32×C4⋊C4C2211yescharacteristic, complemented
30#64159—24no—
31#64158—24no—
32C2×#3756C2211yescharacteristic, complemented
33C2×#3755C2212yescomplemented
34C2×#3755C2211yescharacteristic, complemented
35C2×#3754C2211yescharacteristic, complemented
36C2×#3753C2212yescomplemented
37C2×#3753C2211yescharacteristic, complemented
38C2×#3748C2211yescharacteristic, complemented
39C22×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
40#63657C2212yescomplemented
41#63657C2212yescomplemented
42#63637C2212yescomplemented
43#63636C2212yescomplemented
44#63636C2212yescomplemented
45#63636C2212yescomplemented
46C2×#1009—34no—
47C3×#537D314yescomplemented
48C2×#950—34no—
49C2×#949—34no—
50#5234—34no—
51#5047—38no—
52#5046—38no—
53#5037—38no—
54#5036—38no—
55#4966—34no—
56C2×#809—34no—
57C23×C3⋊D3—22no—
58C23×C3⋊D3—21no—
59C32×#41—22no—
60C32×#41—22no—
61C32×#41—22no—
62C32×#41—22no—
63C2×C32×Q8D411yescharacteristic
64C2×C32×Q8C2311yescharacteristic
65C2×C32×D4—21no—
66C2×C32×D4—22no—
67C2×C32×D4D412yes—
68C2×C32×D4D411yescharacteristic
69C2×C32×D4C2311yescharacteristic
70C22×C32×C4—21no—
71C22×C32×C4—21no—
72C22×C32×C4C2312yes—
73C22×C32×C4C2311yescharacteristic
74C2×#638—22no—
75C2×#638—21no—
76#3832—28no—
77D4×C3⋊D3—216no—
78D4×C3⋊D3—28no—
79#3828—22no—
80#3828—24no—
81#3828—24no—
82#3828D414yescomplemented
83C2×C4⋊(C3⋊D3)—22no—
84C2×C4⋊(C3⋊D3)—21no—
85C2×C4⋊(C3⋊D3)—22no—
86C2×C4⋊(C3⋊D3)—24no—
87C2×C4⋊(C3⋊D3)—24no—
88C2×C4⋊(C3⋊D3)D414yescomplemented
89C2×C4⋊(C3⋊D3)C2311yescharacteristic
90C2×C4⋊(C3⋊D3)C2312yes—
91C2×C4×C3⋊D3—22no—
92C2×C4×C3⋊D3—21no—
93C2×#634C2311yescharacteristic
94C32×C4⋊C4—21no—
95C32×C4⋊C4—22no—
96C32×C4⋊C4C2312yes—
97C32×C22⋊C4—22no—
98C32×C22⋊C4—22no—
99#3756—22no—
100#3755—24no—
101#3755—22no—
102#3754—22no—
103#3753—24no—
104#3753—22no—
105#3748—22no—
106C2×C3⋊(C3⋊C8)—22no—
107C2×C3⋊(C3⋊C8)C2311yescharacteristic
108C2×C3⋊(C3⋊C8)C2311yescharacteristic
109C2×C3⋊(C3⋊C8)C2312yes—
110#3012—91noSylow
111C2×C3×#41C2×D314yes—
112C2×C3×#41C2×D314yescomplemented
113#1009—616no—
114#1009—616no—
115C2×#239—64no—
116C2×D3×D4—68no—
117C2×D3×D4—64no—
118C2×#235—34no—
119C2×#235—38no—
120C22×D12—34no—
121C22×D12—38no—
122C3×C22⋊Q8—24no—
123C3×C22⋊Q8C2×D318yescomplemented
124C3×#121—24no—
125C3×#121—28no—
126C3×#121C2×D318yescomplemented
127C2×C3×C4⋊C4C2×D314yescomplemented
128#950—616no—
129#949—616no—
130C2×#216—34no—
131C2×#215—34no—
132C2×#215—38no—
133C2×#214—34no—
134C2×#213—34no—
135C2×#213—38no—
136C2×#208—34no—
137C22×C3⋊C8—34no—
138#830—38no—
139#830—38no—
140#810—38no—
141#809—38no—
142#809—38no—
143#809—38no—
144C23×C32—21no—
145C23×C32C2×D412yes—
146C23×C32C2×D411yescharacteristic
147C22×C3⋊D3—22no—
148C22×C3⋊D3—21no—
149C22×C3⋊D3—28no—
150C22×C3⋊D3—24no—
151C22×C3⋊D3—81no—
152C22×C3⋊D3—82no—
153C22×C3⋊D3—81no—
154C22×C3⋊D3—82no—
155C32×Q8—22no—
156C32×Q8—21no—
157C32×Q8—21no—
158C32×D4—24no—
159C32×D4—22no—
160C32×D4—21no—
161C32×D4—21no—
162C32×D4—42no—
163C32×D4—82no—
164C2×C32×C4—22no—
165C2×C32×C4—22no—
166C2×C32×C4—22no—
167C2×C32×C4—21no—
168C2×C32×C4—42no—
169C2×C32×C4—81no—
170C2×C32×C4C2×D412yes—
171C2×C32×C4C2×D412yes—
172C2×C32×C4C2×D412yes—
173C2×C32×C4C2×D411yescharacteristic
174C2×C32×C4C2411yescharacteristic
175#638—28no—
176#638—24no—
177C2×C3⋊Dic3—21no—
178C4⋊(C3⋊D3)—28no—
179C4⋊(C3⋊D3)—24no—
180C4⋊(C3⋊D3)—21no—
181C4⋊(C3⋊D3)—22no—
182C4⋊(C3⋊D3)C2×D412yes—
183C4⋊(C3⋊D3)C2×D414yes—
184C4×C3⋊D3—28no—
185C4×C3⋊D3—24no—
186#634—21no—
187#634C2×D412yes—
188C3⋊(C3⋊C8)—21no—
189C3⋊(C3⋊C8)—21no—
190C3⋊(C3⋊C8)—22no—
191C2×H(5,𝔽2)—91no—
192C2×AGL(1,ℤ/8)—91no—
193C2×#135—91no—
194#537C3⋊D311yescharacteristic, complemented
195#452—92no—
196#450—92no—
197#449—92no—
198#447—92no—
199#417—91no—
200C2×#102—91no—
201#408—91no—
202C23×D3—68no—
203C23×D3—64no—
204C3×#41—28no—
205C3×#41—28no—
206C3×#41—28no—
207C3×#41—28no—
208C2×C3×Q8#6714yes—
209C2×C3×Q8C22×D314yes—
210C2×C3×D4—24no—
211C2×C3×D4—28no—
212C2×C3×D4#6718yes—
213C2×C3×D4#6714yes—
214C2×C3×D4C22×D314yes—
215C22×C3×C4—24no—
216C22×C3×C4—24no—
217C22×C3×C4C22×D318yes—
218C22×C3×C4C22×D314yes—
219C2×#67—68no—
220C2×#67—64no—
221#239—632no—
222D3×D4—664no—
223D3×D4—632no—
224#235—316no—
225#235—616no—
226#235—616no—
227#235—68no—
228C2×D12—316no—
229C2×D12—38no—
230C2×D12—34no—
231C2×D12—616no—
232C2×D12—616no—
233C2×D12—68no—
234C2×D12—68no—
235C2×D12—64no—
236C2×C4×D3—68no—
237C2×C4×D3—64no—
238C2×Dic6—34no—
239C3×C4⋊C4—24no—
240C3×C4⋊C4—28no—
241C3×C4⋊C4C22×D318yes—
242C3×C22⋊C4—28no—
243C3×C22⋊C4—28no—
244#216—68no—
245#215—616no—
246#215—68no—
247#214—68no—
248#213—616no—
249#213—68no—
250#208—68no—
251C2×C3⋊C8—34no—
252C2×C3⋊C8—34no—
253C2×C3⋊C8—38no—
254C2×C3⋊C8—68no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32—41no—
261C22×C32—41no—
262C22×C32—41no—
263C22×C32#12012yes—
264C22×C32C22×D411yescharacteristic
265C2×C3⋊D3—24no—
266C2×C3⋊D3—22no—
267C2×C3⋊D3—42no—
268C2×C3⋊D3—41no—
269C2×C3⋊D3—42no—
270C2×C3⋊D3—41no—
271C2×C3⋊D3—84no—
272C2×C3⋊D3—88no—
273C32×C4—22no—
274C32×C4—41no—
275C32×C4—42no—
276C32×C4#12012yes—
277C32×C4C22×D411yescharacteristic
278C32×C4C22×D411yescharacteristic
279C3⋊Dic3—22no—
280H(5,𝔽2)—184no—
281H(5,𝔽2)—184no—
282C2×#41—91no—
283C2×#41—92no—
284C2×#41—181no—
285C2×#41C2×C3⋊D311yescharacteristic
286C2×#41C2×C3⋊D311yescharacteristic, complemented
287C22×D4—91no—
288C22×D4—92no—
289C22×D4—182no—
290C22×D4—181no—
291AGL(1,ℤ/8)—184no—
292#135—184no—
293C2×Q16—91no—
294C2×SD16—91no—
295C2×SD16—92no—
296C2×SD16—91no—
297C2×D8—91no—
298C2×D8—92no—
299C2×M16—91no—
300C22×C8—91no—
301C22⋊Q8—21no—
302C22⋊Q8C2×C3⋊D312yescomplemented
303#121—21no—
304#121—22no—
305#121C2×C3⋊D312yescomplemented
306C2×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
307#103—92no—
308#102—92no—
309#102—92no—
310#102—92no—
311C22⋊C8—92no—
312C22⋊C8—92no—
313C23×C3—24no—
314C23×C3C2×#6718yes—
315C23×C3C2×#6714yes—
316C22×D3—68no—
317C22×D3—64no—
318C22×D3—632no—
319C22×D3—616no—
320C22×D3—244no—
321C22×D3—248no—
322C22×D3—244no—
323C22×D3—248no—
324C3×Q8—28no—
325C3×Q8—24no—
326C3×Q8—24no—
327C3×D4—216no—
328C3×D4—28no—
329C3×D4—24no—
330C3×D4—24no—
331C3×D4—48no—
332C3×D4—88no—
333C2×C3×C4—28no—
334C2×C3×C4—28no—
335C2×C3×C4—28no—
336C2×C3×C4—24no—
337C2×C3×C4—48no—
338C2×C3×C4—84no—
339C2×C3×C4D3×D418yes—
340C2×C3×C4D3×D418yes—
341C2×C3×C4C2×#6718yes—
342C2×C3×C4C2×#6714yes—
343C2×C3×C4C23×D314yes—
344#67—632no—
345#67—616no—
346C2×Dic3—64no—
347D12—316no—
348D12—38no—
349D12—632no—
350D12—616no—
351D12—68no—
352D12—64no—
353C4×D3—632no—
354C4×D3—616no—
355Dic6—38no—
356Dic6—64no—
357C3⋊C8—64no—
358C3⋊C8—64no—
359C3⋊C8—68no—
360C2×C32—22no—
361C2×C32—42no—
362C2×C32—81no—
363C2×C32C2×#12011yescharacteristic
364C2×C32#57611yescharacteristic
365C2×C32#57711yescharacteristic
366C3⋊D3—44no—
367C3⋊D3—42no—
368C24—182no—
369C24—181no—
370#41—22no—
371#41—22no—
372#41—22no—
373#41—22no—
374#41—94no—
375#41—184no—
376#41—184no—
377#41—188no—
378#41—182no—
379C2×Q8—91no—
380C2×Q8#63811yescharacteristic
381C2×Q8C22×C3⋊D311yescharacteristic
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—92no—
385C2×D4—91no—
386C2×D4—94no—
387C2×D4—184no—
388C2×D4—184no—
389C2×D4—1816no—
390C2×D4—188no—
391C2×D4—182no—
392C2×D4—181no—
393C2×D4—182no—
394C2×D4—182no—
395C2×D4—181no—
396C2×D4#63812yes—
397C2×D4#63811yescharacteristic
398C2×D4C22×C3⋊D311yescharacteristic
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—182no—
402C22×C4—181no—
403C22×C4C22×C3⋊D312yes—
404C22×C4C22×C3⋊D311yescharacteristic
405Q16—182no—
406SD16—182no—
407SD16—182no—
408SD16—184no—
409D8—182no—
410D8—184no—
411M16—182no—
412C2×C8—91no—
413C2×C8—91no—
414C2×C8—92no—
415C2×C8—182no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4C22×C3⋊D312yes—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22×C3—24no—
422C22×C3—24no—
423C22×C3—28no—
424C22×C3—28no—
425C22×C3—28no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—44no—
429C22×C3#93718yes—
430C22×C3C22×#6714yes—
431C2×D3—616no—
432C2×D3—68no—
433C2×D3—128no—
434C2×D3—124no—
435C2×D3—128no—
436C2×D3—124no—
437C2×D3—2416no—
438C2×D3—2432no—
439C3×C4—28no—
440C3×C4—44no—
441C3×C4—48no—
442C3×C4#93718yes—
443C3×C4C2×D3×D414yes—
444C3×C4C2×D3×D414yes—
445Dic3—68no—
446C32#301211yescharacteristic, complemented, Sylow
447C23—21no—
448C23—188no—
449C23—182no—
450C23—181no—
451C23—184no—
452C23—721no—
453C23—721no—
454C23—722no—
455C23—722no—
456C23C2×#63812yes—
457C23C2×#63811yescharacteristic
458Q8—22no—
459Q8—21no—
460Q8—21no—
461Q8—92no—
462Q8—181no—
463D4—24no—
464D4—21no—
465D4—21no—
466D4—22no—
467D4—42no—
468D4—82no—
469D4—94no—
470D4—92no—
471D4—188no—
472D4—188no—
473D4—184no—
474D4—181no—
475D4—182no—
476D4—184no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—42no—
482C2×C4—81no—
483C2×C4—188no—
484C2×C4—181no—
485C2×C4—184no—
486C2×C4D4×C3⋊D312yes—
487C2×C4D4×C3⋊D312yes—
488C2×C4C2×#63812yes—
489C2×C4C2×#63811yescharacteristic
490C2×C4C23×C3⋊D311yescharacteristic
491C8—181no—
492C8—181no—
493C8—182no—
494C2×C3—28no—
495C2×C3—48no—
496C2×C3—84no—
497C2×C3C2×#93714yes—
498C2×C3#583514yes—
499C2×C3#583714yes—
500D3—1216no—
501D3—128no—
502C22—21no—
503C22—21no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—184no—
511C22—182no—
512C22—362no—
513C22—361no—
514C22—362no—
515C22—361no—
516C22—724no—
517C22—728no—
518C22#6414612yes—
519C22C22×#63811yescharacteristic
520C4—22no—
521C4—41no—
522C4—42no—
523C4—182no—
524C4#6414612yes—
525C4C2×D4×C3⋊D311yescharacteristic
526C4C2×D4×C3⋊D311yescharacteristic
527C3#8342414yescomplemented
528C2—22no—
529C2—42no—
530C2—81no—
531C2—364no—
532C2—362no—
533C2C2×#6414611yescharacteristic
534C2#1059630611yescharacteristic
535C2#1059630811yescharacteristic
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#537C2
socleC22×C32C22×D4
hypercenter#537C3⋊D3
hypocenterC32#3012
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#64366C2semidirect
2. C32×#537C2semidirect
3. C2×#64159C2semidirect
4. C2×#64158C2semidirect
5. #10592933C2semidirect
6. #10592746C2semidirect
7. #10592745C2semidirect
8. #10592736C2semidirect
9. #10592735C2semidirect
10. #10592665C2semidirect
11. C2×#63636C2semidirect
14. C2×C32×#41C22semidirect
25. C32×C22⋊Q8C22semidirect
28. C32×#121C22semidirect
29. C2×C32×C4⋊C4C22semidirect
32. C2×#3756C22semidirect
33. C2×#3755C22semidirect
34. C2×#3755C22semidirect
35. C2×#3754C22semidirect
36. C2×#3753C22semidirect
37. C2×#3753C22semidirect
38. C2×#3748C22semidirect
39. C22×C3⋊(C3⋊C8)C22semidirect
40. #63657C22semidirect
41. #63657C22semidirect
42. #63637C22semidirect
43. #63636C22semidirect
44. #63636C22semidirect
45. #63636C22semidirect
47. C3×#537D3semidirect
82. #3828D4semidirect
88. C2×C4⋊(C3⋊D3)D4semidirect
112. C2×C3×#41C2×D3semidirect
123. C3×C22⋊Q8C2×D3semidirect
126. C3×#121C2×D3semidirect
127. C2×C3×C4⋊C4C2×D3semidirect
194. #537C3⋊D3semidirect
286. C2×#41C2×C3⋊D3semidirect
302. C22⋊Q8C2×C3⋊D3semidirect
305. #121C2×C3⋊D3semidirect
306. C2×C4⋊C4C2×C3⋊D3semidirect
446. C32#3012semidirect
527. C3#83424semidirect

Maximal subgroups

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

subgroupindex
C2×#643662
C32×#5372
C2×#641592
C2×#641582
#105929332
#105927462
#105927452
#105927362
#105927352
#105926652
C2×#636362
#834243