Finite Group Databasebeta

← C2×#10596499 #11918775

Subgroups — C2×#10596499 #11918775

Order 1152.

2374 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64306C211yesmaximal, characteristic, complemented
2C22×#3798C211yesmaximal, characteristic, complemented
3C2×#64283C211yesmaximal, characteristic, complemented
4C2×#64239C212yesmaximal, complemented
5C2×#64236C212yesmaximal, complemented
6C2×#64226C212yesmaximal, complemented
7#10596499C2116yesmaximal, direct product factor
8C2×#64224C212yesmaximal, complemented
9C2×#64223C214yesmaximal, complemented
10C22×C4×C3⋊D3C2211yescharacteristic
11C2×C3×#235—21no—
12C22×C3×C4×D3—21no—
13C2×#3811—21no—
14C22×#627C2211yescharacteristic, complemented
15C22×#626—21no—
16C22×#625C2211yescharacteristic, complemented
17C2×#3805C2212yes—
18#64306—24no—
19#64306C2218yescomplemented
20C2×#3802C2211yescharacteristic, complemented
21C2×C4×D32—21no—
22C2×#3798—24no—
23C2×#3798—21no—
24C2×#3798—21no—
25C2×#3798C2212yes—
26C2×#3798C2218yescomplemented
27C2×#3798C2212yescomplemented
28C2×#3798C2212yescomplemented
29C2×#3797C2211yescharacteristic, complemented
30C2×#3796—21no—
31C2×#3792C2212yescomplemented
32#64283C2218yescomplemented
33C2×#3788C2212yescomplemented
34C2×#3787C2212yescomplemented
35C2×#3776C2214yescomplemented
36#64239C22116yescomplemented
37C2×#3775C2214yescomplemented
38C2×#3775C2214yescomplemented
39#64236C22116yescomplemented
40C2×#3774C2214yescomplemented
41#64226C22116yescomplemented
42#64224C22116yescomplemented
43#64223C22132yescomplemented
44C2×D3×#41—61no—
45C22×#235—61no—
46C23×C3⋊D3D411yescharacteristic
47C23×C3×D3—21no—
48C22×D32—21no—
49C22×C32×C4D411yescharacteristic
50C22×C3⋊Dic3C2311yescharacteristic
51C2×C4×C3⋊D3D411yescharacteristic
52C2×C4×C3⋊D3D411yescharacteristic
53C2×C4×C3⋊D3D414yes—
54C2×C4×C3⋊D3C2311yescharacteristic
55C2×C4×C3⋊D3C2311yescharacteristic
56C2×C4×C3⋊D3C2314yes—
57C2×C3×#67—22no—
58C22×C3×Dic3—21no—
59C3×#235—28no—
60C2×C3×D12—21no—
61C2×C3×C4×D3—22no—
62C2×C3×C4×D3—28no—
63C2×C3×C4×D3—22no—
64C2×C3×C4×D3—22no—
65C2×C3×Dic6—21no—
66#3811—28no—
67C2×#627—22no—
68C2×#627—21no—
69C2×#627C2312yes—
70C2×#627C2312yes—
71C2×#627C2314yescomplemented
72C2×#626—22no—
73C2×#626—22no—
74C2×#626—28no—
75C2×#626—22no—
76C2×#625—22no—
77C2×#625—21no—
78C2×#625C2312yes—
79C2×#625C2312yes—
80C2×#625C2314yescomplemented
81C2×#624—21no—
82#3805—24no—
83#3805C2318yes—
84C2×D3×Dic3—22no—
85#3802—22no—
86#3802C2314yescomplemented
87C4×D32—28no—
88#3798—24no—
89#3798—24no—
90#3798—28no—
91#3798—28no—
92#3798—24no—
93#3798C2318yes—
94#3798C2318yescomplemented
95#3798C2318yescomplemented
96#3797—22no—
97#3797C2314yescomplemented
98#3796—28no—
99#3792C2318yescomplemented
100C2×#622C2314yes—
101#3788C2318yescomplemented
102#3787C2318yescomplemented
103#3776C23116yescomplemented
104#3775C23116yescomplemented
105#3775C23116yescomplemented
106#3774C23116yescomplemented
107C2×#575—91nomaximal, Sylow
108C2×C3×#41—61no—
109C23×C3×C4—61no—
110C22×#67—61no—
111C22×#67—61no—
112D3×#41—68no—
113D3×#41—68no—
114C2×#239—61no—
115C2×D3×Q8—61no—
116C2×#237—62no—
117C2×#237—61no—
118C2×D3×D4—61no—
119C2×D3×D4—62no—
120C2×#235—21no—
121C2×#235—62no—
122C2×#235—62no—
123C2×#235—68no—
124C2×#235—62no—
125C2×#235—62no—
126C2×#235—68no—
127C2×#235—62no—
128C22×D12—61no—
129C22×C4×D3—21no—
130C22×C4×D3—61no—
131C22×C4×D3—61no—
132C22×C4×D3—62no—
133C22×Dic6—61no—
134C23×C32C2×D411yescharacteristic
135C22×C3⋊D3—42no—
136C22×C3⋊D3C2×D411yescharacteristic
137C22×C3⋊D3C2×D411yescharacteristic
138C22×C3⋊D3C2×D414yes—
139C22×C3×D3—22no—
140C22×C3×D3—22no—
141C22×C3×D3—24no—
142C22×C3×D3—44no—
143C2×D32—24no—
144C2×D32—44no—
145C2×C32×C4C2×D411yescharacteristic
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D414yes—
148C2×C3⋊Dic3C2×D411yescharacteristic
149C2×C3⋊Dic3C2×D412yes—
150C2×C3⋊Dic3C2411yescharacteristic
151C2×C3⋊Dic3C2412yes—
152C4×C3⋊D3C2×D414yes—
153C4×C3⋊D3C2×D412yes—
154C4×C3⋊D3C2×D412yes—
155C4×C3⋊D3C2×D414yes—
156C4×C3⋊D3C2412yes—
157C4×C3⋊D3C2412yes—
158C3×#67—28no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—24no—
162C3×D12—24no—
163C3×C4×D3—28no—
164C3×C4×D3—28no—
165C3×C4×D3—28no—
166C3×Dic6—24no—
167#627—22no—
168#627—22no—
169#627—24no—
170#627C2414yes—
171#627C2414yes—
172#626—28no—
173#626—28no—
174#626—28no—
175#625—22no—
176#625—24no—
177#625—22no—
178#625C2414yes—
179#625C2414yes—
180#624—24no—
181D3×Dic3—28no—
182#622C2418yes—
183C22×#41—91no—
184C22×#41—91no—
185#575—916no—
186C2×#137—92no—
187C2×#137—92no—
188C2×AGL(1,ℤ/8)—92no—
189C2×AGL(1,ℤ/8)—92no—
190C2×#135—94no—
191C22×M16—91no—
192C24×C3—61no—
193C23×D3—21no—
194C23×D3—61no—
195C23×D3—61no—
196C23×D3—62no—
197C3×#41—68no—
198C2×C3×Q8—61no—
199C2×C3×D4—61no—
200C2×C3×D4—62no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—62no—
204C22×C3×C4—62no—
205C22×C3×C4—68no—
206C22×C3×C4—62no—
207C22×C3×C4—61no—
208C2×#67—22no—
209C2×#67—62no—
210C2×#67—62no—
211C2×#67—64no—
212C2×#67—62no—
213C2×#67—62no—
214C2×#67—68no—
215C2×#67—62no—
216C2×#67—62no—
217C2×#67—64no—
218C22×Dic3—21no—
219C22×Dic3—61no—
220C22×Dic3—61no—
221C22×Dic3—62no—
222#239—68no—
223D3×Q8—64no—
224D3×Q8—64no—
225#237—68no—
226#237—68no—
227#237—68no—
228D3×D4—64no—
229D3×D4—64no—
230D3×D4—616no—
231#235—28no—
232#235—68no—
233#235—68no—
234#235—68no—
235#235—68no—
236#235—616no—
237#235—68no—
238#235—616no—
239#235—68no—
240C2×D12—21no—
241C2×D12—62no—
242C2×D12—62no—
243C2×D12—68no—
244C2×D12—62no—
245C2×C4×D3—22no—
246C2×C4×D3—28no—
247C2×C4×D3—22no—
248C2×C4×D3—22no—
249C2×C4×D3—61no—
250C2×C4×D3—61no—
251C2×C4×D3—64no—
252C2×C4×D3—61no—
253C2×C4×D3—61no—
254C2×C4×D3—64no—
255C2×C4×D3—61no—
256C2×C4×D3—61no—
257C2×C4×D3—61no—
258C2×C4×D3—61no—
259C2×C4×D3—64no—
260C2×C4×D3—64no—
261C2×C4×D3—616no—
262C2×C4×D3—62no—
263C2×C4×D3—62no—
264C2×Dic6—21no—
265C2×Dic6—62no—
266C2×Dic6—62no—
267C2×Dic6—64no—
268C2×Dic6—62no—
269C2×Dic6—64no—
270C22×C32—22no—
271C22×C32C22×D412yes—
272C22×C32C22×D411yescharacteristic
273C2×C3⋊D3—22no—
274C2×C3⋊D3—22no—
275C2×C3⋊D3—44no—
276C2×C3⋊D3C22×D412yes—
277C2×C3⋊D3C22×D412yes—
278C2×C3×D3—24no—
279C2×C3×D3—24no—
280C2×C3×D3—48no—
281C2×C3×D3—44no—
282C2×C3×D3—44no—
283D32—48no—
284C32×C4C22×D412yes—
285C32×C4C22×D412yes—
286C3⋊Dic3C22×D412yes—
287C3⋊Dic3C22×D411yescharacteristic
288C3⋊Dic3C2511yescharacteristic
289C3×Dic3—24no—
290C3×Dic3—24no—
291C2×#41—61no—
292C2×#41—92no—
293C2×#41—92no—
294C2×#41—92no—
295C2×#41—98no—
296C2×#41—92no—
297C2×#41—98no—
298C2×#41—181no—
299C2×#41—184no—
300C2×#41—181no—
301C2×#41—181no—
302C2×#41—184no—
303C22×Q8—91no—
304C22×Q8—91no—
305C22×D4—91no—
306C22×D4—91no—
307C22×D4—181no—
308C22×D4—181no—
309C23×C4—61no—
310C23×C4—91no—
311C23×C4—181no—
312#137—916no—
313#137—916no—
314AGL(1,ℤ/8)—916no—
315AGL(1,ℤ/8)—916no—
316#135—932no—
317C2×Q16—94no—
318C2×SD16—94no—
319C2×SD16—94no—
320C2×D8—94no—
321C2×M16—92no—
322C2×M16—98no—
323C2×M16—92no—
324C22×C8—92no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—62no—
328C23×C3—62no—
329C23×C3—64no—
330C23×C3—124no—
331C22×D3—22no—
332C22×D3—24no—
333C22×D3—22no—
334C22×D3—44no—
335C22×D3—61no—
336C22×D3—61no—
337C22×D3—61no—
338C22×D3—61no—
339C22×D3—64no—
340C22×D3—64no—
341C22×D3—68no—
342C22×D3—128no—
343C22×D3—124no—
344C22×D3—124no—
345C3×Q8—64no—
346C3×D4—64no—
347C3×D4—68no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—68no—
355C2×C3×C4—68no—
356C2×C3×C4—68no—
357C2×C3×C4—62no—
358C2×C3×C4—64no—
359C2×C3×C4—62no—
360#67—28no—
361#67—64no—
362#67—64no—
363#67—68no—
364#67—68no—
365#67—68no—
366#67—64no—
367#67—68no—
368#67—64no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—24no—
372C2×Dic3—61no—
373C2×Dic3—62no—
374C2×Dic3—61no—
375C2×Dic3—62no—
376C2×Dic3—61no—
377C2×Dic3—61no—
378C2×Dic3—62no—
379C2×Dic3—62no—
380C2×Dic3—68no—
381D12—24no—
382D12—68no—
383D12—68no—
384D12—68no—
385C4×D3—28no—
386C4×D3—28no—
387C4×D3—28no—
388C4×D3—64no—
389C4×D3—64no—
390C4×D3—62no—
391C4×D3—64no—
392C4×D3—62no—
393C4×D3—62no—
394C4×D3—62no—
395C4×D3—64no—
396C4×D3—62no—
397C4×D3—62no—
398C4×D3—62no—
399C4×D3—62no—
400C4×D3—68no—
401C4×D3—68no—
402Dic6—24no—
403Dic6—64no—
404Dic6—64no—
405Dic6—64no—
406Dic6—68no—
407Dic6—64no—
408C2×C32—22no—
409C2×C32#57512yescomplemented
410C2×C32C23×D411yescharacteristic
411C3⋊D3—22no—
412C3⋊D3—22no—
413C3×D3—44no—
414C3×D3—44no—
415C24—61no—
416C24—91no—
417C24—181no—
418#41—68no—
419#41—98no—
420#41—98no—
421#41—98no—
422#41—98no—
423#41—188no—
424#41—184no—
425#41—184no—
426#41—188no—
427#41—184no—
428#41—184no—
429#41—188no—
430C2×Q8—61no—
431C2×Q8—92no—
432C2×Q8—92no—
433C2×Q8—94no—
434C2×Q8—94no—
435C2×Q8—181no—
436C2×Q8—182no—
437C2×Q8—182no—
438C2×D4—62no—
439C2×D4—61no—
440C2×D4—94no—
441C2×D4—92no—
442C2×D4—92no—
443C2×D4—94no—
444C2×D4—182no—
445C2×D4—182no—
446C2×D4—188no—
447C2×D4—182no—
448C2×D4—182no—
449C2×D4—181no—
450C2×D4—182no—
451C2×D4—188no—
452C22×C4—61no—
453C22×C4—62no—
454C22×C4—68no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—91no—
458C22×C4—91no—
459C22×C4—94no—
460C22×C4—91no—
461C22×C4—94no—
462C22×C4—91no—
463C22×C4—91no—
464C22×C4—188no—
465C22×C4—181no—
466C22×C4—182no—
467C22×C4GO+(4,𝔽2)11yescharacteristic
468Q16—916no—
469SD16—916no—
470SD16—916no—
471D8—916no—
472M16—98no—
473M16—98no—
474C2×C8—94no—
475C2×C8—98no—
476C22×C3—24no—
477C22×C3—22no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—22no—
481C22×C3—42no—
482C22×C3—64no—
483C22×C3—64no—
484C22×C3—124no—
485C22×C3—128no—
486C22×C3—124no—
487C2×D3—24no—
488C2×D3—24no—
489C2×D3—48no—
490C2×D3—44no—
491C2×D3—44no—
492C2×D3—62no—
493C2×D3—62no—
494C2×D3—62no—
495C2×D3—62no—
496C2×D3—64no—
497C2×D3—122no—
498C2×D3—122no—
499C2×D3—122no—
500C2×D3—1216no—
501C2×D3—128no—
502C2×D3—128no—
503C3×C4—22no—
504C3×C4—22no—
505C3×C4—22no—
506C3×C4—22no—
507C3×C4—64no—
508C3×C4—64no—
509Dic3—24no—
510Dic3—24no—
511Dic3—62no—
512Dic3—61no—
513Dic3—61no—
514Dic3—61no—
515Dic3—61no—
516Dic3—62no—
517C32C2×#57511yescharacteristic, complemented, Sylow
518C23—64no—
519C23—62no—
520C23—62no—
521C23—91no—
522C23—91no—
523C23—94no—
524C23—124no—
525C23—184no—
526C23—362no—
527C23—364no—
528C23C2×GO+(4,𝔽2)11yescharacteristic
529Q8—64no—
530Q8—94no—
531Q8—94no—
532Q8—184no—
533Q8—182no—
534Q8—182no—
535D4—68no—
536D4—64no—
537D4—94no—
538D4—94no—
539D4—188no—
540D4—188no—
541D4—188no—
542D4—184no—
543D4—182no—
544D4—182no—
545C2×C4—64no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—68no—
549C2×C4—68no—
550C2×C4—68no—
551C2×C4—91no—
552C2×C4—92no—
553C2×C4—91no—
554C2×C4—92no—
555C2×C4—92no—
556C2×C4—92no—
557C2×C4—94no—
558C2×C4—92no—
559C2×C4—94no—
560C2×C4—92no—
561C2×C4—184no—
562C2×C4—188no—
563C2×C4C2×GO+(4,𝔽2)11yescharacteristic
564C2×C4C2×GO+(4,𝔽2)14yes—
565C2×C4C2×GO+(4,𝔽2)11yescharacteristic
566C8—98no—
567C2×C3—24no—
568C2×C3—22no—
569C2×C3—22no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—42no—
573C2×C3—124no—
574C2×C3—124no—
575D3—44no—
576D3—44no—
577D3—64no—
578D3—122no—
579D3—122no—
580D3—122no—
581C22—22no—
582C22—64no—
583C22—64no—
584C22—92no—
585C22—92no—
586C22—128no—
587C22—124no—
588C22—124no—
589C22—182no—
590C22—182no—
591C22—364no—
592C22—368no—
593C22C22×GO+(4,𝔽2)12yes—
594C22C22×GO+(4,𝔽2)11yescharacteristic
595C4—64no—
596C4—64no—
597C4—91no—
598C4—92no—
599C4—91no—
600C4C22×GO+(4,𝔽2)12yes—
601C4C22×GO+(4,𝔽2)12yes—
602C3—21no—
603C3—21no—
604C2—22no—
605C2—124no—
606C2—124no—
607C2—182no—
608C2—182no—
609C2#1059649912yesdirect product factor
610C2C23×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C4C2×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C25
FrattiniC2C23×GO+(4,𝔽2)
FittingC22×C32×C4D4
socleC22×C32C22×D4
hypercenterC22×C4GO+(4,𝔽2)
hypocenterC32C2×#575
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#64306C2semidirect
2. C22×#3798C2semidirect
3. C2×#64283C2semidirect
4. C2×#64239C2semidirect
5. C2×#64236C2semidirect
6. C2×#64226C2semidirect
7. #10596499C2direct (N × H)
8. C2×#64224C2semidirect
9. C2×#64223C2semidirect
14. C22×#627C22semidirect
16. C22×#625C22semidirect
19. #64306C22semidirect
20. C2×#3802C22semidirect
26. C2×#3798C22semidirect
27. C2×#3798C22semidirect
28. C2×#3798C22semidirect
29. C2×#3797C22semidirect
31. C2×#3792C22semidirect
32. #64283C22semidirect
33. C2×#3788C22semidirect
34. C2×#3787C22semidirect
35. C2×#3776C22semidirect
36. #64239C22semidirect
37. C2×#3775C22semidirect
38. C2×#3775C22semidirect
39. #64236C22semidirect
40. C2×#3774C22semidirect
41. #64226C22semidirect
42. #64224C22semidirect
43. #64223C22semidirect
71. C2×#627C23semidirect
80. C2×#625C23semidirect
86. #3802C23semidirect
94. #3798C23semidirect
95. #3798C23semidirect
97. #3797C23semidirect
99. #3792C23semidirect
101. #3788C23semidirect
102. #3787C23semidirect
103. #3776C23semidirect
104. #3775C23semidirect
105. #3775C23semidirect
106. #3774C23semidirect
409. C2×C32#575semidirect
517. C32C2×#575semidirect
609. C2#10596499direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#643062
C22×#37982
C2×#642832
C2×#642392
C2×#642362
C2×#642262
#105964992
C2×#642242
C2×#642232
C2×#5759