Finite Group Databasebeta

← C2×#71522 #10949075

Subgroups — C2×#71522 #10949075

Order 768.

1128 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5768C211yesmaximal, characteristic
2C2×C3×#510C211yesmaximal, characteristic, complemented
3C2×#5189C211yesmaximal, characteristic, complemented
4C2×#5179C211yesmaximal, characteristic, complemented
5C2×#4566C211yesmaximal, characteristic, complemented
6#71522C218yesmaximal, direct product factor
7C2×#4518C211yesmaximal, characteristic, complemented
8C2×#4514C211yesmaximal, characteristic, complemented
9C2×#2191—31nomaximal, Sylow
10C2×D3×#41—21no—
11C2×C3×#135C2211yescharacteristic
12C22×C3×Q16C2211yescharacteristic, complemented
13C2×#950—21no—
14C2×#946—21no—
15#5768—22no—
16#5768—22no—
17#5768—22no—
18#5768C2214yes—
19C2×#918—21no—
20C2×#915—21no—
21C2×#914—21no—
22C2×#911—21no—
23C2×#904C2211yescharacteristic
24C22×ΓL(1,𝔽25)C2211yescharacteristic
25C3×#510—22no—
26C3×#510C2214yescomplemented
27C2×C3×Q32—21no—
28C2×C3×SD32—21no—
29C2×C3×M32C2211yescharacteristic, complemented
30C2×#835—21no—
31#5189—22no—
32#5189C2214yescomplemented
33C2×#829—21no—
34#5179—22no—
35#5179C2214yescomplemented
36C2×#828—21no—
37C2×#817C2211yescharacteristic, complemented
38C2×#812C2211yescharacteristic, complemented
39#4566C2214yescomplemented
40#4518C2214yescomplemented
41#4514C2214yescomplemented
42C2×#575—31no—
43C2×#510—31no—
44C2×#510D311yescharacteristic, complemented
45C2×#470—31no—
46#2191—38no—
47C2×#360—31no—
48C2×#360—31no—
49C2×#349—31no—
50C2×C3×#41—21no—
51C22×C3×Q8—21no—
52D3×#41—24no—
53D3×#41—24no—
54D3×#41—42no—
55D3×#41—42no—
56C2×#239—21no—
57C2×D3×Q8—21no—
58C2×#237—21no—
59C2×#237—41no—
60C2×D3×D4—21no—
61C2×D3×D4—41no—
62C2×#235—21no—
63C2×#235—41no—
64C22×C4×D3—41no—
65C22×C4×D3D411yescharacteristic
66C3×#135—21no—
67C3×#135—21no—
68C3×#135D412yes—
69C3×#135C2312yes—
70C2×C3×Q16—21no—
71C2×C3×Q16—21no—
72C2×C3×Q16—21no—
73C2×C3×Q16—21no—
74C2×C3×Q16—21no—
75C2×C3×Q16D411yescharacteristic
76C2×C3×Q16C2312yescomplemented
77C2×C3×SD16—21no—
78C2×C3×D8D411yescharacteristic
79C22×C3×C8C2311yescharacteristic
80#950—24no—
81#950—24no—
82#946—24no—
83#946—42no—
84C2×#216—21no—
85C2×#215—21no—
86C2×#214—21no—
87C2×#213—21no—
88C2×(C2×D3)⋊C4—41no—
89C2×C4⋊Dic3—21no—
90C2×Dic3⋊C4—41no—
91C22×C3⋊C8D411yescharacteristic
92#918—24no—
93#918—24no—
94#915—24no—
95#915—24no—
96#914—24no—
97#914—24no—
98#911—24no—
99#911—24no—
100C2×Dic12D411yescharacteristic
101#904—21no—
102#904—21no—
103#904D412yes—
104#904C2312yes—
105C2×D24D411yescharacteristic
106C2×#204—21no—
107C2×ΓL(1,𝔽25)—21no—
108C2×ΓL(1,𝔽25)—21no—
109C2×ΓL(1,𝔽25)—21no—
110C2×ΓL(1,𝔽25)—21no—
111C2×ΓL(1,𝔽25)D412yes—
112C2×ΓL(1,𝔽25)C2312yes—
113C3×Q32—24no—
114C3×SD32—24no—
115C3×M32—21no—
116C3×M32C2312yescomplemented
117C2×C3×C16—21no—
118#835—24no—
119#829—24no—
120#828—24no—
121#817—21no—
122#817C2312yescomplemented
123#812—21no—
124#812C2312yescomplemented
125C2×C3⋊C16—21no—
126C22×#41—61no—
127#575—34no—
128#575—62no—
129#575—62no—
130#575—62no—
131C2×#137—61no—
132C2×#137—61no—
133C2×AGL(1,ℤ/8)—61no—
134C2×AGL(1,ℤ/8)—61no—
135C2×#135—31no—
136C2×#135—61no—
137C2×#135C2×D311yescharacteristic
138C22×Q16C2×D311yescharacteristic, complemented
139C22×M16—31no—
140C2×C22⋊Q8—61no—
141#510—22no—
142#510—34no—
143#510—62no—
144#510C2×D314yescomplemented
145C2×Q32—21no—
146C2×Q32—61no—
147C2×SD32—21no—
148C2×SD32—61no—
149C2×M32—31no—
150C2×M32C2×D311yescharacteristic, complemented
151#470—34no—
152#470—62no—
153C2×#106—31no—
154C2×#103—61no—
155#360—34no—
156#360—34no—
157#349—34no—
158C23×D3—41no—
159C23×D3D811yescharacteristic
160C3×#41—22no—
161C3×#41—22no—
162C3×#41—41no—
163C3×#41—41no—
164C2×C3×Q8—21no—
165C2×C3×Q8—22no—
166C2×C3×Q8—41no—
167C2×C3×Q8—41no—
168C2×C3×Q8—41no—
169C2×C3×Q8—41no—
170C2×C3×Q8—41no—
171C2×C3×D4—21no—
172C2×C3×D4—41no—
173C22×C3×C4—41no—
174C22×C3×C4—41no—
175C22×C3×C4C2×D411yescharacteristic
176C2×#67—41no—
177C2×#67—41no—
178C2×#67—41no—
179C22×Dic3—41no—
180C22×Dic3—41no—
181C22×Dic3D811yescharacteristic
182#239—24no—
183#239—42no—
184D3×Q8—24no—
185D3×Q8—42no—
186#237—24no—
187#237—42no—
188#237—44no—
189#237—44no—
190D3×D4—24no—
191D3×D4—42no—
192D3×D4—44no—
193D3×D4—44no—
194#235—22no—
195#235—22no—
196#235—44no—
197#235—44no—
198#235—41no—
199#235—41no—
200C2×D12—21no—
201C2×D12—41no—
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—44no—
207C2×C4×D3—44no—
208C2×C4×D3—41no—
209C2×C4×D3—41no—
210C2×C4×D3—41no—
211C2×C4×D3D812yes—
212C2×C4×D3C2×D412yes—
213C2×Dic6—21no—
214C2×Dic6—41no—
215C3×Q16—21no—
216C3×Q16—22no—
217C3×Q16—22no—
218C3×Q16—41no—
219C3×Q16—41no—
220C3×Q16C2×D412yes—
221C3×SD16—22no—
222C3×SD16—22no—
223C3×D8—21no—
224C3×D8C2×D412yes—
225C2×C3×C8C2×D411yescharacteristic
226C2×C3×C8C2×D411yescharacteristic
227C2×C3×C8C2×D411yescharacteristic
228C2×C3×C8C2×D411yescharacteristic
229C2×C3×C8C2×D411yescharacteristic
230C2×C3×C8C2411yescharacteristic
231#216—24no—
232#215—24no—
233#214—24no—
234#213—24no—
235(C2×D3)⋊C4—44no—
236C4⋊Dic3—22no—
237C4⋊Dic3—41no—
238Dic3⋊C4—44no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—21no—
241C2×C3⋊C8C2×D412yes—
242Dic12—21no—
243Dic12C2×D412yes—
244D24—21no—
245D24C2×D412yes—
246#204—22no—
247#204—22no—
248ΓL(1,𝔽25)—22no—
249ΓL(1,𝔽25)—22no—
250ΓL(1,𝔽25)—22no—
251ΓL(1,𝔽25)—22no—
252C3×C16—22no—
253C3⋊C16—22no—
254C2×#41—21no—
255C2×#41—64no—
256C2×#41—61no—
257C2×#41—61no—
258C2×#41—64no—
259C2×#41—61no—
260C2×#41—121no—
261C2×#41—122no—
262C2×#41—121no—
263C2×#41—122no—
264C22×Q8—21no—
265C22×Q8—61no—
266C22×D4—61no—
267C22×D4—121no—
268C23×C4—31no—
269C23×C4—121no—
270#137—64no—
271#137—64no—
272#137—64no—
273#137—64no—
274AGL(1,ℤ/8)—64no—
275AGL(1,ℤ/8)—64no—
276AGL(1,ℤ/8)—64no—
277AGL(1,ℤ/8)—64no—
278#135—21no—
279#135—21no—
280#135—32no—
281#135—32no—
282#135—64no—
283#135—64no—
284#135—61no—
285#135—61no—
286#135#6712yes—
287#135C22×D312yes—
288C2×Q16—21no—
289C2×Q16—21no—
290C2×Q16—21no—
291C2×Q16—21no—
292C2×Q16—21no—
293C2×Q16—31no—
294C2×Q16—61no—
295C2×Q16#6711yescharacteristic
296C2×Q16C22×D312yescomplemented
297C2×SD16—21no—
298C2×SD16—61no—
299C2×SD16—61no—
300C2×SD16—61no—
301C2×D8—31no—
302C2×D8—61no—
303C2×D8#6711yescharacteristic
304C2×M16—32no—
305C2×M16—32no—
306C2×M16—61no—
307C2×M16—61no—
308C2×M16—61no—
309C2×M16—61no—
310C22×C8—31no—
311C22×C8C22×D311yescharacteristic
312C22⋊Q8—64no—
313C22⋊Q8—122no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—121no—
316C2×C22⋊C4—121no—
317Q32—24no—
318Q32—64no—
319SD32—24no—
320SD32—64no—
321M32—21no—
322M32—32no—
323M32—61no—
324M32C22×D312yescomplemented
325C2×C16—21no—
326C2×C16—61no—
327#106—32no—
328#106—61no—
329#103—64no—
330C23×C3—41no—
331C23×C3C2×D811yescharacteristic
332C22×D3—21no—
333C22×D3—21no—
334C22×D3—44no—
335C22×D3—42no—
336C22×D3—41no—
337C22×D3—84no—
338C22×D3C2×D812yes—
339C3×Q8—22no—
340C3×Q8—41no—
341C3×Q8—42no—
342C3×Q8—42no—
343C3×Q8—81no—
344C3×Q8—81no—
345C3×D4—22no—
346C3×D4—42no—
347C3×D4—42no—
348C3×D4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—81no—
354C2×C3×C4—81no—
355C2×C3×C4#12011yescharacteristic
356C2×C3×C4#12011yescharacteristic
357C2×C3×C4C2×D811yescharacteristic
358C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
359C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361#67—44no—
362#67—44no—
363#67—42no—
364#67—42no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—44no—
368C2×Dic3—41no—
369C2×Dic3—42no—
370C2×Dic3—82no—
371C2×Dic3C2×D812yes—
372D12—22no—
373D12—44no—
374D12—41no—
375C4×D3—22no—
376C4×D3—22no—
377C4×D3—22no—
378C4×D3—22no—
379C4×D3—44no—
380C4×D3—44no—
381C4×D3—42no—
382C4×D3—42no—
383Dic6—22no—
384Dic6—44no—
385Dic6—41no—
386C3×C8#12011yescharacteristic
387C3×C8#12011yescharacteristic
388C3×C8C22×D411yescharacteristic
389C3×C8C22×D411yescharacteristic
390C3⋊C8—22no—
391C24—31no—
392C24—121no—
393#41—22no—
394#41—22no—
395#41—41no—
396#41—41no—
397#41—64no—
398#41—64no—
399#41—62no—
400#41—62no—
401#41—124no—
402#41—124no—
403#41—124no—
404#41—124no—
405#41—122no—
406#41—122no—
407#41—121no—
408#41—121no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—41no—
412C2×Q8—41no—
413C2×Q8—41no—
414C2×Q8—41no—
415C2×Q8—41no—
416C2×Q8—64no—
417C2×Q8—61no—
418C2×Q8—121no—
419C2×Q8—122no—
420C2×D4—21no—
421C2×D4—41no—
422C2×D4—64no—
423C2×D4—61no—
424C2×D4—121no—
425C2×D4—122no—
426C2×D4—121no—
427C2×D4—124no—
428C2×D4—124no—
429C2×D4—121no—
430C2×D4—121no—
431C22×C4—31no—
432C22×C4—32no—
433C22×C4—32no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—121no—
441C22×C4—124no—
442C22×C4—124no—
443C22×C4—121no—
444C22×C4—121no—
445C22×C4—121no—
446C22×C4—121no—
447C22×C4D3×D411yescharacteristic
448Q16—21no—
449Q16—22no—
450Q16—22no—
451Q16—32no—
452Q16—41no—
453Q16—41no—
454Q16—64no—
455Q16—61no—
456Q16C2×#6712yes—
457SD16—22no—
458SD16—22no—
459SD16—64no—
460SD16—64no—
461SD16—62no—
462SD16—62no—
463D8—21no—
464D8—32no—
465D8—64no—
466D8—61no—
467D8C2×#6712yes—
468M16—62no—
469M16—62no—
470M16—62no—
471M16—62no—
472C2×C8—32no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8D3×D411yescharacteristic
476C2×C8D3×D411yescharacteristic
477C2×C8D3×D411yescharacteristic
478C2×C8C2×#6711yescharacteristic
479C2×C8C2×#6711yescharacteristic
480C2×C8C23×D311yescharacteristic
481C4⋊C4—62no—
482C4⋊C4—124no—
483C4⋊C4—121no—
484C22⋊C4—124no—
485C16—22no—
486C16—62no—
487C22×C3—22no—
488C22×C3—42no—
489C22×C3—82no—
490C22×C3#44711yescharacteristic
491C22×C3#44711yescharacteristic
492C22×C3C22×D811yescharacteristic
493C2×D3—22no—
494C2×D3—42no—
495C2×D3—42no—
496C2×D3—42no—
497C2×D3—81no—
498C2×D3—88no—
499C2×D3—82no—
500C3×C4—42no—
501C3×C4—82no—
502C3×C4#44711yescharacteristic
503C3×C4#44711yescharacteristic
504C3×C4C2×#12011yescharacteristic
505C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
506Dic3—22no—
507Dic3—42no—
508Dic3—82no—
509C23—32no—
510C23—41no—
511C23—61no—
512C23—61no—
513C23—122no—
514C23—124no—
515C23—121no—
516C23—244no—
517C23D3×D811yescharacteristic
518Q8—22no—
519Q8—41no—
520Q8—42no—
521Q8—42no—
522Q8—62no—
523Q8—81no—
524Q8—81no—
525Q8—124no—
526Q8—121no—
527D4—22no—
528D4—42no—
529D4—42no—
530D4—41no—
531D4—62no—
532D4—124no—
533D4—124no—
534D4—124no—
535D4—122no—
536D4—122no—
537D4—121no—
538C2×C4—32no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—124no—
552C2×C4—124no—
553C2×C4—124no—
554C2×C4—121no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—242no—
559C2×C4D3×D811yescharacteristic
560C2×C4#91111yescharacteristic
561C2×C4#91111yescharacteristic
562C2×C4#93711yescharacteristic
563C2×C4#93711yescharacteristic
564C2×C4C2×D3×D411yescharacteristic
565C8—62no—
566C8#93711yescharacteristic
567C8#93711yescharacteristic
568C8C2×D3×D411yescharacteristic
569C8C22×#6711yescharacteristic
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—82no—
573C2×C3#219112yescomplemented
574C2×C3C2×#44711yescharacteristic
575D3—42no—
576D3—82no—
577C22—22no—
578C22—42no—
579C22—62no—
580C22—82no—
581C22—122no—
582C22—122no—
583C22—122no—
584C22—241no—
585C22—248no—
586C22—242no—
587C22#515611yescharacteristic
588C22#515611yescharacteristic
589C22C2×D3×D811yescharacteristic
590C4—42no—
591C4—62no—
592C4—82no—
593C4—122no—
594C4—242no—
595C4#515611yescharacteristic
596C4#515611yescharacteristic
597C4C2×#91111yescharacteristic
598C4C2×#93711yescharacteristic
599C3C2×#219111yescharacteristic, complemented, Sylow
600C2—21no—
601C2—21no—
602C2—82no—
603C2—122no—
604C2—242no—
605C2#7152212yesdirect product factor
606C2C2×#515611yescharacteristic
subgroupG/H
centerC22#5156
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC2×C3×#510C2
socleC22×C3#447
prime radicalC2×#5768C2
hypercenterC2×#510D3
hypocenterC3C2×#2191
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#510

Complements

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

NH ≅ G/Ntype
2. C2×C3×#510C2semidirect
3. C2×#5189C2semidirect
4. C2×#5179C2semidirect
5. C2×#4566C2semidirect
6. #71522C2direct (N × H)
7. C2×#4518C2semidirect
8. C2×#4514C2semidirect
12. C22×C3×Q16C22semidirect
26. C3×#510C22semidirect
29. C2×C3×M32C22semidirect
32. #5189C22semidirect
35. #5179C22semidirect
37. C2×#817C22semidirect
38. C2×#812C22semidirect
39. #4566C22semidirect
40. #4518C22semidirect
41. #4514C22semidirect
44. C2×#510D3semidirect
76. C2×C3×Q16C23semidirect
116. C3×M32C23semidirect
122. #817C23semidirect
124. #812C23semidirect
138. C22×Q16C2×D3semidirect
144. #510C2×D3semidirect
150. C2×M32C2×D3semidirect
296. C2×Q16C22×D3semidirect
324. M32C22×D3semidirect
573. C2×C3#2191semidirect
599. C3C2×#2191semidirect
605. C2#71522direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57682
C2×C3×#5102
C2×#51892
C2×#51792
C2×#45662
#715222
C2×#45182
C2×#45142
C2×#21913