Finite Group Databasebeta

← #11864561

Subgroups — #11864561

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×#64361C211yesmaximal, characteristic, complemented
2C2×C32×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3#10596304C211yesmaximal, characteristic, complemented
4C2×#64159C211yesmaximal, characteristic, complemented
5C2×#63659C211yesmaximal, characteristic, complemented
6#10592872C212yesmaximal, complemented
7#10592868C212yesmaximal, complemented
8#10592857C212yesmaximal, complemented
9#10592855C212yesmaximal, complemented
10#10592832C211yesmaximal, characteristic, complemented
11#10592828C211yesmaximal, characteristic, complemented
12#83949—34nomaximal
13C2×C32×#41C2211yescharacteristic, complemented
14C22×C32×D4C2211yescharacteristic
15C22×#638—22no—
16#64361—24no—
17#64361—24no—
18C2×#3830—21no—
19C2×#3830C2212yes—
20C2×D4×C3⋊D3—21no—
21C2×D4×C3⋊D3C2212yes—
22C2×#3828C2211yescharacteristic
23C2×#3828C2211yescharacteristic
24C32×AGL(1,ℤ/8)—24no—
25C2×C32×SD16C2212yescomplemented
26C2×C32×D8C2212yescomplemented
27C2×C32×M16C2211yescharacteristic, complemented
28#64159—24no—
29#64155C2212yescomplemented
30C2×#3756C2211yescharacteristic, complemented
31C2×#3755C2211yescharacteristic, complemented
32#64148—22no—
33#64147C2212yescomplemented
34C2×#3754C2211yescharacteristic, complemented
35C2×#3753C2211yescharacteristic, complemented
36#64139—21no—
37C2×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
38#64133—21no—
39C22×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
40#63659C2212yescomplemented
41#63659C2212yescomplemented
42#63648C2212yescomplemented
43#63647C2212yescomplemented
44#63647C2212yescomplemented
45#63643C2212yescomplemented
46C2×#1004—34no—
47C2×C3×AGL(1,ℤ/8)D314yescomplemented
48#5833—34no—
49C2×#950—34no—
50C2×#832—34no—
51#5173—38no—
52#5169—38no—
53#5158—38no—
54#5156—38no—
55#5133—34no—
56#5129—34no—
57C24×C32—21no—
58C23×C3⋊D3—22no—
59C32×#41—22no—
60C32×#41—22no—
61C2×C32×Q8C2311yescharacteristic
62C2×C32×D4—21no—
63C2×C32×D4—21no—
64C2×C32×D4—22no—
65C2×C32×D4D414yescomplemented
66C2×C32×D4C2311yescharacteristic
67C2×C32×D4C2312yes—
68C22×C32×C4—21no—
69C22×C32×C4C2311yescharacteristic
70C2×#638—216no—
71C2×#638—21no—
72C2×#638—22no—
73C2×#638—22no—
74C2×#638—22no—
75C2×#638D412yes—
76C22×C3⋊Dic3—22no—
77C22×C3⋊Dic3—21no—
78#3830—28no—
79#3830—24no—
80#3830—24no—
81D4×C3⋊D3—28no—
82D4×C3⋊D3—24no—
83D4×C3⋊D3—24no—
84#3828—22no—
85#3828—22no—
86#3828—22no—
87#3828D414yescomplemented
88C2×C4⋊(C3⋊D3)D411yescharacteristic
89C2×C4⋊(C3⋊D3)C2311yescharacteristic
90C2×C4×C3⋊D3—21no—
91C2×C4×C3⋊D3C2312yes—
92C2×#634D411yescharacteristic
93C2×#634C2311yescharacteristic
94C32×SD16—24no—
95C32×D8—24no—
96C32×M16—22no—
97C2×C32×C8C2312yes—
98#3757—22no—
99#3756—22no—
100#3755—22no—
101#3754—22no—
102#3753—22no—
103#3752—22no—
104C4⋊(C3⋊Dic3)—21no—
105C4⋊(C3⋊Dic3)C2312yes—
106#3750—22no—
107C2×C3⋊(C3⋊C8)—22no—
108C2×C3⋊(C3⋊C8)C2311yescharacteristic
109C2×C3⋊(C3⋊C8)C2311yescharacteristic
110#3012—91noSylow
111C2×C3×#41C2×D314yescomplemented
112C22×C3×D4C2×D314yes—
113C22×#67—68no—
114#1004—616no—
115#1004—616no—
116C2×#237—38no—
117C2×#237—64no—
118C2×D3×D4—38no—
119C2×D3×D4—64no—
120C2×#235—34no—
121C2×#235—34no—
122C3×AGL(1,ℤ/8)—216no—
123C2×C3×SD16C2×D318yescomplemented
124C2×C3×D8C2×D318yescomplemented
125C2×C3×M16C2×D314yescomplemented
126#950—616no—
127#946—38no—
128C2×#216—34no—
129C2×#215—34no—
130#939—68no—
131#938—38no—
132C2×#214—34no—
133C2×#213—34no—
134#930—64no—
135C2×C4⋊Dic3—34no—
136#924—64no—
137C22×C3⋊C8—34no—
138#832—38no—
139#832—38no—
140#821—38no—
141(C2×D3)⋊C8—38no—
142(C2×D3)⋊C8—38no—
143#816—38no—
144C23×C32—22no—
145C23×C32—24no—
146C23×C32—21no—
147C23×C32—81no—
148C23×C32—81no—
149C23×C32C2×D411yescharacteristic
150C22×C3⋊D3—21no—
151C22×C3⋊D3—28no—
152C22×C3⋊D3—82no—
153C22×C3⋊D3—82no—
154C22×C3⋊D3C2×D412yes—
155C32×Q8—21no—
156C32×Q8—21no—
157C32×D4—24no—
158C32×D4—22no—
159C32×D4—21no—
160C32×D4—21no—
161C32×D4—42no—
162C32×D4C2×D414yes—
163C2×C32×C4—21no—
164C2×C32×C4—42no—
165C2×C32×C4C2×D411yescharacteristic
166C2×C32×C4C2×D412yes—
167C2×C32×C4C2×D412yes—
168C2×C32×C4C2411yescharacteristic
169#638—28no—
170#638—28no—
171#638—24no—
172#638—24no—
173#638—82no—
174C2×C3⋊Dic3—28no—
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3—22no—
177C2×C3⋊Dic3—81no—
178C2×C3⋊Dic3C2×D412yes—
179C4⋊(C3⋊D3)—22no—
180C4⋊(C3⋊D3)—21no—
181C4⋊(C3⋊D3)C2×D412yes—
182C4×C3⋊D3—24no—
183C4×C3⋊D3—22no—
184C4×C3⋊D3—22no—
185#634—22no—
186#634—21no—
187#634C2×D412yes—
188C32×C8—22no—
189C3⋊(C3⋊C8)—21no—
190C3⋊(C3⋊C8)—21no—
191C2×H(5,𝔽2)—91no—
192C2×AGL(1,ℤ/8)C3⋊D311yescharacteristic, complemented
193C2×#135—91no—
194#537—91no—
195#452—92no—
196#450—92no—
197#449—92no—
198#447—92no—
199#417—91no—
200C2×#102—91no—
201#408—91no—
202C24×C3—24no—
203C23×D3—68no—
204C3×#41—28no—
205C3×#41—28no—
206C2×C3×Q8C22×D314yes—
207C2×C3×D4—24no—
208C2×C3×D4—24no—
209C2×C3×D4—28no—
210C2×C3×D4#67116yescomplemented
211C2×C3×D4C22×D314yes—
212C2×C3×D4C22×D318yes—
213C22×C3×C4—24no—
214C22×C3×C4C22×D314yes—
215C2×#67—38no—
216C2×#67—68no—
217C2×#67—68no—
218C2×#67—664no—
219C2×#67—64no—
220C2×#67—68no—
221C22×Dic3—68no—
222C22×Dic3—64no—
223#237—616no—
224#237—632no—
225#237—616no—
226D3×D4—616no—
227D3×D4—632no—
228D3×D4—616no—
229#235—316no—
230#235—68no—
231#235—68no—
232#235—68no—
233C2×D12—34no—
234C2×D12—34no—
235C2×C4×D3—38no—
236C2×C4×D3—64no—
237C2×Dic6—34no—
238C2×Dic6—34no—
239C3×SD16—216no—
240C3×D8—216no—
241C3×M16—28no—
242C2×C3×C8C22×D318yes—
243#217—68no—
244#216—68no—
245#215—68no—
246#214—68no—
247#213—68no—
248(C2×D3)⋊C4—68no—
249C4⋊Dic3—38no—
250C4⋊Dic3—64no—
251Dic3⋊C4—68no—
252C2×C3⋊C8—34no—
253C2×C3⋊C8—34no—
254C2×C3⋊C8—68no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—24no—
258C22×C32—42no—
259C22×C32—42no—
260C22×C32—41no—
261C22×C32—41no—
262C22×C32—41no—
263C22×C32—84no—
264C22×C32#12012yes—
265C22×C32C22×D411yescharacteristic
266C2×C3⋊D3—22no—
267C2×C3⋊D3—22no—
268C2×C3⋊D3—22no—
269C2×C3⋊D3—22no—
270C2×C3⋊D3—41no—
271C2×C3⋊D3—41no—
272C2×C3⋊D3—88no—
273C32×C4—41no—
274C32×C4#12012yes—
275C32×C4C22×D411yescharacteristic
276C32×C4C22×D411yescharacteristic
277C3⋊Dic3—22no—
278C3⋊Dic3—22no—
279C3⋊Dic3—42no—
280H(5,𝔽2)—184no—
281H(5,𝔽2)—184no—
282C2×#41—91no—
283C2×#41—92no—
284C2×#41—91no—
285C2×#41—181no—
286C2×#41C2×C3⋊D311yescharacteristic, complemented
287C22×D4—92no—
288C22×D4—181no—
289C22×D4—182no—
290C22×D4C2×C3⋊D311yescharacteristic
291AGL(1,ℤ/8)—24no—
292#135—184no—
293C2×Q16—91no—
294C2×SD16—91no—
295C2×SD16—91no—
296C2×SD16C2×C3⋊D312yescomplemented
297C2×D8—91no—
298C2×D8C2×C3⋊D312yescomplemented
299C2×M16C2×C3⋊D311yescharacteristic, complemented
300C22×C8—91no—
301C22⋊Q8—92no—
302C22⋊Q8—181no—
303#121—92no—
304#121—181no—
305#121—182no—
306C2×C4⋊C4—91no—
307#103—92no—
308#102—92no—
309#102—92no—
310#102—92no—
311C22⋊C8—92no—
312C22⋊C8—92no—
313C23×C3—28no—
314C23×C3—216no—
315C23×C3—24no—
316C23×C3—84no—
317C23×C3—84no—
318C23×C3D3×D414yes—
319C22×D3—38no—
320C22×D3—64no—
321C22×D3—632no—
322C22×D3—248no—
323C22×D3—248no—
324C3×Q8—24no—
325C3×Q8—24no—
326C3×D4—216no—
327C3×D4—28no—
328C3×D4—24no—
329C3×D4—24no—
330C3×D4—48no—
331C3×D4C2×#67116yes—
332C2×C3×C4—24no—
333C2×C3×C4—48no—
334C2×C3×C4D3×D414yes—
335C2×C3×C4D3×D418yes—
336C2×C3×C4C2×#6718yes—
337C2×C3×C4C23×D314yes—
338#67—632no—
339#67—632no—
340#67—616no—
341#67—616no—
342#67—248no—
343C2×Dic3—38no—
344C2×Dic3—632no—
345C2×Dic3—64no—
346C2×Dic3—68no—
347C2×Dic3—244no—
348D12—38no—
349D12—68no—
350D12—64no—
351C4×D3—616no—
352C4×D3—68no—
353C4×D3—68no—
354Dic6—38no—
355Dic6—68no—
356Dic6—64no—
357C3×C8—28no—
358C3⋊C8—64no—
359C3⋊C8—64no—
360C2×C32—22no—
361C2×C32—44no—
362C2×C32—81no—
363C2×C32C2×#12011yescharacteristic
364C2×C32#57611yescharacteristic
365C2×C32#57711yescharacteristic
366C3⋊D3—42no—
367C3⋊D3—42no—
368C24—21no—
369C24—182no—
370#41—22no—
371#41—22no—
372#41—94no—
373#41—184no—
374#41—188no—
375#41—184no—
376#41—182no—
377#41—182no—
378#41—182no—
379C2×Q8—91no—
380C2×Q8—91no—
381C2×Q8C22×C3⋊D311yescharacteristic
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4—92no—
386C2×D4—91no—
387C2×D4—91no—
388C2×D4—182no—
389C2×D4—184no—
390C2×D4—188no—
391C2×D4—182no—
392C2×D4—1816no—
393C2×D4—181no—
394C2×D4—184no—
395C2×D4—182no—
396C2×D4#63814yescomplemented
397C2×D4C22×C3⋊D311yescharacteristic
398C2×D4C22×C3⋊D312yes—
399C22×C4—21no—
400C22×C4—92no—
401C22×C4—182no—
402C22×C4—181no—
403C22×C4—181no—
404C22×C4C22×C3⋊D311yescharacteristic
405Q16—182no—
406SD16—24no—
407SD16—182no—
408SD16—182no—
409D8—24no—
410D8—182no—
411M16—22no—
412C2×C8—91no—
413C2×C8—91no—
414C2×C8—182no—
415C2×C8C22×C3⋊D312yes—
416C4⋊C4—92no—
417C4⋊C4—182no—
418C4⋊C4—181no—
419C22⋊C4—182no—
420C22⋊C4—182no—
421C22×C3—24no—
422C22×C3—24no—
423C22×C3—216no—
424C22×C3—48no—
425C22×C3—48no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—44no—
429C22×C3—816no—
430C22×C3#93718yes—
431C22×C3C2×D3×D414yes—
432C2×D3—68no—
433C2×D3—68no—
434C2×D3—68no—
435C2×D3—68no—
436C2×D3—124no—
437C2×D3—124no—
438C2×D3—2432no—
439C3×C4—44no—
440C3×C4#93718yes—
441C3×C4C2×D3×D414yes—
442C3×C4C22×#6714yes—
443Dic3—68no—
444Dic3—68no—
445Dic3—128no—
446C32#301211yescharacteristic, complemented, Sylow
447C23—22no—
448C23—24no—
449C23—21no—
450C23—81no—
451C23—81no—
452C23—92no—
453C23—188no—
454C23—181no—
455C23—722no—
456C23—722no—
457C23D4×C3⋊D311yescharacteristic
458Q8—21no—
459Q8—21no—
460Q8—92no—
461Q8—181no—
462Q8—182no—
463D4—22no—
464D4—21no—
465D4—21no—
466D4—24no—
467D4—42no—
468D4—92no—
469D4—188no—
470D4—188no—
471D4—184no—
472D4—184no—
473D4—181no—
474D4—182no—
475D4—722no—
476D4C2×#63814yes—
477C2×C4—21no—
478C2×C4—42no—
479C2×C4—92no—
480C2×C4—188no—
481C2×C4—181no—
482C2×C4—182no—
483C2×C4—182no—
484C2×C4—184no—
485C2×C4—182no—
486C2×C4—721no—
487C2×C4D4×C3⋊D312yes—
488C2×C4D4×C3⋊D311yescharacteristic
489C2×C4C2×#63812yes—
490C2×C4C23×C3⋊D311yescharacteristic
491C8—22no—
492C8—181no—
493C8—181no—
494C2×C3—28no—
495C2×C3—416no—
496C2×C3—84no—
497C2×C3#577414yes—
498C2×C3#577514yes—
499C2×C3C2×#93714yes—
500D3—128no—
501D3—128no—
502C22—21no—
503C22—21no—
504C22—24no—
505C22—42no—
506C22—42no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—84no—
511C22—182no—
512C22—182no—
513C22—182no—
514C22—182no—
515C22—361no—
516C22—361no—
517C22—728no—
518C22#6414612yes—
519C22C2×D4×C3⋊D311yescharacteristic
520C4—41no—
521C4—182no—
522C4—182no—
523C4—362no—
524C4#6414612yes—
525C4C2×D4×C3⋊D311yescharacteristic
526C4C22×#63811yescharacteristic
527C3#8394914yescomplemented
528C2—22no—
529C2—44no—
530C2—81no—
531C2—362no—
532C2—362no—
533C2#1059624511yescharacteristic
534C2#1059624611yescharacteristic
535C2C2×#6414611yescharacteristic
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#3012
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×#64361C2semidirect
2. C2×C32×AGL(1,ℤ/8)C2semidirect
3. #10596304C2semidirect
4. C2×#64159C2semidirect
5. C2×#63659C2semidirect
6. #10592872C2semidirect
7. #10592868C2semidirect
8. #10592857C2semidirect
9. #10592855C2semidirect
10. #10592832C2semidirect
11. #10592828C2semidirect
13. C2×C32×#41C22semidirect
25. C2×C32×SD16C22semidirect
26. C2×C32×D8C22semidirect
27. C2×C32×M16C22semidirect
29. #64155C22semidirect
30. C2×#3756C22semidirect
31. C2×#3755C22semidirect
33. #64147C22semidirect
34. C2×#3754C22semidirect
35. C2×#3753C22semidirect
37. C2×C4⋊(C3⋊Dic3)C22semidirect
39. C22×C3⋊(C3⋊C8)C22semidirect
40. #63659C22semidirect
41. #63659C22semidirect
42. #63648C22semidirect
43. #63647C22semidirect
44. #63647C22semidirect
45. #63643C22semidirect
47. C2×C3×AGL(1,ℤ/8)D3semidirect
65. C2×C32×D4D4semidirect
87. #3828D4semidirect
111. C2×C3×#41C2×D3semidirect
123. C2×C3×SD16C2×D3semidirect
124. C2×C3×D8C2×D3semidirect
125. C2×C3×M16C2×D3semidirect
192. C2×AGL(1,ℤ/8)C3⋊D3semidirect
210. C2×C3×D4#67semidirect
286. C2×#41C2×C3⋊D3semidirect
296. C2×SD16C2×C3⋊D3semidirect
298. C2×D8C2×C3⋊D3semidirect
299. C2×M16C2×C3⋊D3semidirect
396. C2×D4#638semidirect
446. C32#3012semidirect
527. C3#83949semidirect

Maximal subgroups

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

subgroupindex
C2×#643612
C2×C32×AGL(1,ℤ/8)2
#105963042
C2×#641592
C2×#636592
#105928722
#105928682
#105928572
#105928552
#105928322
#105928282
#839493