Finite Group Databasebeta

← #10822326

Subgroups — #10822326

Order 768.

2282 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#914C212yesmaximal, complemented
2C2×#5725C211yesmaximal, characteristic, complemented
3C2×#4803C212yesmaximal, complemented
4D3×#390C211yesmaximal, characteristic, complemented
5C3×#2013C211yesmaximal, characteristic, complemented
6#72055C212yesmaximal, complemented
7#71791C211yesmaximal, characteristic, complemented
8#70277C212yesmaximal, complemented
9#69658C211yesmaximal, characteristic, complemented
10#68674C211yesmaximal, characteristic, complemented
11#68660C211yesmaximal, characteristic, complemented
12#23428—31nomaximal, Sylow
13C22×#239C2212yes—
14C22×D3×D4C2211yescharacteristic
15C22×C3×SD16C2212yes—
16C2×C3×C4⋊Q8C2211yescharacteristic, complemented
17C2×#947C2212yescomplemented
18C22×#215C2212yes—
19C22×#213C2212yes—
20C2×#914—28no—
21C2×#914—28no—
22C2×#914—28no—
23C22×D24C2212yes—
24C22×ΓL(1,𝔽25)C2212yes—
25#5725—28no—
26C2×#895—22no—
27C2×#893—22no—
28C2×D3×C4⋊C4C2211yescharacteristic
29C2×D3×C22⋊C4—22no—
30C2×#874C2211yescharacteristic, complemented
31C2×C42×D3C2211yescharacteristic
32C2×C3×#103C2212yescomplemented
33C3×#390C2211yescharacteristic
34C2×#835C2212yescomplemented
35#5221C2211yescharacteristic
36C2×#821C2212yescomplemented
37#4988C2211yescharacteristic
38C2×#809C2212yescomplemented
39#4957—22no—
40#4939C2211yescharacteristic
41#4803—28no—
42C3×#336C2211yescharacteristic, complemented
43#4550C2212yescomplemented
44#4524C2211yescharacteristic, complemented
45C22×AGL(1,ℤ/8)—32no—
46C2×#532—31no—
47C2×#417—32no—
48C2×#390—31no—
49#2013—31no—
50#2013D311yescharacteristic, complemented
51#2012—32no—
52#2012—32no—
53#2011—31no—
54#2011—31no—
55#1728—31no—
56C24×D3—21no—
57C22×C3×Q8C2312yes—
58C22×C3×D4C2311yescharacteristic
59C22×#67—21no—
60C2×#239—24no—
61C2×#239—24no—
62C2×#239—24no—
63C2×#239—24no—
64C2×#239—24no—
65C2×#239D418yes—
66C2×D3×D4—24no—
67C2×D3×D4—24no—
68C2×D3×D4—22no—
69C2×D3×D4D414yescomplemented
70C22×D12—22no—
71C22×D12—22no—
72C22×D12C2311yescharacteristic
73C22×D12C2312yes—
74C22×C4×D3—22no—
75C22×C4×D3—21no—
76C22×C4×D3D412yes—
77C22×C4×D3C2311yescharacteristic
78C2×C3×SD16—24no—
79C2×C3×SD16—24no—
80C2×C3×SD16—24no—
81C22×C3×C8C2312yes—
82C3×C4⋊Q8—24no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—22no—
86C2×C3×C42C2311yescharacteristic
87C2×#217—22no—
88#947—24no—
89#947—24no—
90C2×#215—24no—
91C2×#215—24no—
92C2×#215—24no—
93C2×#213—24no—
94C2×#213—24no—
95C2×#213—24no—
96C2×(C2×D3)⋊C4—24no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—22no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3D412yes—
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2312yes—
104#914—264no—
105C2×D24—24no—
106C2×D24—24no—
107C2×D24—24no—
108C2×ΓL(1,𝔽25)—24no—
109C2×ΓL(1,𝔽25)—24no—
110C2×ΓL(1,𝔽25)D418yescomplemented
111#895—216no—
112#893—216no—
113D3×C4⋊C4—24no—
114D3×C22⋊C4—48no—
115#874—24no—
116C42×D3—24no—
117C3×#103—24no—
118#835—24no—
119#821—24no—
120#809—24no—
121C22×#41—32no—
122C23×D4—31no—
123C2×AGL(1,ℤ/8)—68no—
124C2×AGL(1,ℤ/8)—68no—
125C2×AGL(1,ℤ/8)—68no—
126C22×SD16—32no—
127C22×SD16C2×D312yes—
128C22×D8—32no—
129C22×D8—32no—
130C22×M16—32no—
131#532—68no—
132C2×C4⋊Q8C2×D311yescharacteristic, complemented
133C2×#127—31no—
134C2×#124—32no—
135C2×#121—62no—
136C2×C4×D4—62no—
137C22×C4⋊C4—31no—
138C22×C22⋊C4—62no—
139C22×C42—31no—
140#417—68no—
141C2×#103—32no—
142C2×#103C2×D312yescomplemented
143C2×#102—32no—
144C2×#102—32no—
145#390—31no—
146#390—31no—
147#390—31no—
148#390—62no—
149#390C2×D311yescharacteristic
150#336—32no—
151#336—31no—
152#336C2×D311yescharacteristic, complemented
153C24×C3—21no—
154C23×D3—21no—
155C23×D3—22no—
156C23×D3—24no—
157C23×D3—44no—
158C23×D3—44no—
159C23×D3—42no—
160C23×D3—42no—
161C23×D3C2×D411yescharacteristic
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8C2×D414yes—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—21no—
171C2×C3×D4C2×D412yes—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2411yescharacteristic
176C2×#67—24no—
177C2×#67—42no—
178C2×#67—42no—
179C22×Dic3—21no—
180C22×Dic3C2×D412yes—
181C22×Dic3C2×D411yescharacteristic
182#239—216no—
183#239—216no—
184#239—48no—
185#239—48no—
186D3×D4—216no—
187D3×D4—48no—
188C2×D12—22no—
189C2×D12—22no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—28no—
195C2×D12—28no—
196C2×D12—22no—
197C2×D12—21no—
198C2×D12—44no—
199C2×D12—44no—
200C2×D12—44no—
201C2×D12—44no—
202C2×D12C2×D414yes—
203C2×D12C2×D412yes—
204C2×C4×D3—24no—
205C2×C4×D3—24no—
206C2×C4×D3—28no—
207C2×C4×D3—42no—
208C2×C4×D3—44no—
209C2×C4×D3—44no—
210C2×C4×D3—44no—
211C2×C4×D3—42no—
212C2×C4×D3C2×D412yes—
213C2×C4×D3C2×D414yes—
214C2×C4×D3C2×D414yes—
215C2×C4×D3#4112yes—
216C3×SD16—216no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8C2×D414yes—
220C3×C4⋊C4—28no—
221C3×C4⋊C4—22no—
222C3×C22⋊C4—44no—
223C3×C42—22no—
224#217—44no—
225#215—216no—
226#213—216no—
227(C2×D3)⋊C4—216no—
228(C2×D3)⋊C4—44no—
229(C2×D3)⋊C4—44no—
230C4⋊Dic3—22no—
231Dic3⋊C4—42no—
232C4×Dic3—22no—
233C4×Dic3—24no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8—22no—
236C2×C3⋊C8C2×D414yes—
237D24—216no—
238ΓL(1,𝔽25)—216no—
239C25—61no—
240C2×#41—38no—
241C2×#41—64no—
242C2×#41—64no—
243C2×#41—64no—
244C2×#41—64no—
245C2×#41—64no—
246C22×Q8C22×D312yes—
247C22×D4—34no—
248C22×D4—32no—
249C22×D4—31no—
250C22×D4—62no—
251C22×D4—64no—
252C22×D4—64no—
253C22×D4—62no—
254C22×D4—61no—
255C22×D4—62no—
256C22×D4C22×D311yescharacteristic
257C23×C4—31no—
258C23×C4—32no—
259C23×C4—62no—
260C23×C4—61no—
261AGL(1,ℤ/8)—664no—
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—24no—
265C2×SD16—64no—
266C2×SD16—64no—
267C2×SD16—64no—
268C2×D8—64no—
269C2×D8—64no—
270C2×D8—64no—
271C2×D8—64no—
272C2×D8—64no—
273C2×D8—64no—
274C2×M16—38no—
275C2×M16—64no—
276C2×M16—64no—
277C22×C8—32no—
278C22×C8C22×D312yes—
279C4⋊Q8—24no—
280#127—64no—
281#124—64no—
282#124—64no—
283#121—616no—
284C4×D4—616no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—64no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—22no—
291C2×C22⋊C4—64no—
292C2×C22⋊C4—62no—
293C2×C22⋊C4—62no—
294C2×C22⋊C4—62no—
295C2×C22⋊C4—128no—
296C2×C42—32no—
297C2×C42—31no—
298C2×C42—64no—
299C2×C42C22×D311yescharacteristic
300#103—24no—
301#103—64no—
302#102—64no—
303#102—64no—
304C23×C3—22no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3C22×D411yescharacteristic
310C22×D3—22no—
311C22×D3—22no—
312C22×D3—24no—
313C22×D3—22no—
314C22×D3—416no—
315C22×D3—41no—
316C22×D3—42no—
317C22×D3—42no—
318C22×D3—41no—
319C22×D3—42no—
320C22×D3—44no—
321C22×D3—44no—
322C22×D3—42no—
323C22×D3—48no—
324C22×D3—84no—
325C22×D3—84no—
326C22×D3—88no—
327C22×D3#12012yes—
328C22×D3#12112yes—
329C22×D3#12712yes—
330C3×Q8—24no—
331C3×Q8—24no—
332C3×Q8—42no—
333C3×Q8—42no—
334C3×D4—24no—
335C3×D4—42no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—24no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4AGL(1,ℤ/8)14yescomplemented
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C22×D412yes—
346C2×C3×C4C22×D412yes—
347C2×C3×C4C2×#4111yescharacteristic
348#67—48no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—41no—
353C2×Dic3—42no—
354C2×Dic3#12012yes—
355C2×Dic3#12112yes—
356C2×Dic3#12712yes—
357D12—24no—
358D12—24no—
359D12—24no—
360D12—416no—
361D12—416no—
362D12—42no—
363D12—42no—
364D12—42no—
365C4×D3—48no—
366C4×D3—416no—
367C4×D3—84no—
368C4×D3#12018yes—
369C4×D3#12118yes—
370C3×C8—24no—
371C3⋊C8—24no—
372C24—21no—
373C24—31no—
374C24—64no—
375C24—61no—
376C24—62no—
377C24—122no—
378C24—122no—
379C24—124no—
380C24—124no—
381#41—616no—
382#41—616no—
383#41—128no—
384#41—128no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8D3×D414yes—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—34no—
395C2×D4—32no—
396C2×D4—68no—
397C2×D4—68no—
398C2×D4—616no—
399C2×D4—64no—
400C2×D4—62no—
401C2×D4—62no—
402C2×D4—62no—
403C2×D4—62no—
404C2×D4—62no—
405C2×D4—62no—
406C2×D4—62no—
407C2×D4—61no—
408C2×D4—124no—
409C2×D4—128no—
410C2×D4—122no—
411C2×D4—122no—
412C2×D4—124no—
413C2×D4—124no—
414C2×D4—124no—
415C2×D4D3×D412yes—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—31no—
419C22×C4—34no—
420C22×C4—34no—
421C22×C4—32no—
422C22×C4—32no—
423C22×C4—32no—
424C22×C4—68no—
425C22×C4—64no—
426C22×C4—64no—
427C22×C4—61no—
428C22×C4—122no—
429C22×C4—124no—
430C22×C4—124no—
431C22×C4—122no—
432C22×C4—124no—
433C22×C4D3×D412yes—
434C22×C4C23×D311yescharacteristic
435SD16—216no—
436SD16—616no—
437D8—616no—
438D8—616no—
439M16—616no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—34no—
443C2×C8—62no—
444C2×C8—62no—
445C2×C8D3×D414yes—
446C4⋊C4—28no—
447C4⋊C4—22no—
448C4⋊C4—62no—
449C4⋊C4—122no—
450C22⋊C4—44no—
451C22⋊C4—616no—
452C22⋊C4—124no—
453C22⋊C4—124no—
454C22⋊C4—124no—
455C42—22no—
456C42—64no—
457C42—62no—
458C22×C3—42no—
459C22×C3—44no—
460C22×C3—81no—
461C22×C3—81no—
462C22×C3—82no—
463C22×C3C2×#12011yescharacteristic
464C22×C3C2×#12111yescharacteristic
465C22×C3C2×#12711yescharacteristic
466C22×C3C2×AGL(1,ℤ/8)12yes—
467C22×C3#57512yes—
468C2×D3—22no—
469C2×D3—22no—
470C2×D3—24no—
471C2×D3—24no—
472C2×D3—44no—
473C2×D3—42no—
474C2×D3—48no—
475C2×D3—44no—
476C2×D3—816no—
477C2×D3—82no—
478C2×D3—81no—
479C2×D3—82no—
480C2×D3—81no—
481C2×D3—82no—
482C2×D3—84no—
483C2×D3#39214yes—
484C3×C4—24no—
485C3×C4—44no—
486C3×C4—81no—
487C3×C4C2×#12012yes—
488C3×C4C2×#12112yes—
489Dic3—42no—
490Dic3—81no—
491Dic3#39214yes—
492C23—22no—
493C23—32no—
494C23—32no—
495C23—32no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23—62no—
501C23—62no—
502C23—64no—
503C23—62no—
504C23—128no—
505C23—1216no—
506C23—122no—
507C23—121no—
508C23—122no—
509C23—124no—
510C23—124no—
511C23—122no—
512C23—121no—
513C23—122no—
514C23—244no—
515C23—248no—
516C23—244no—
517C23C2×D3×D411yescharacteristic
518Q8—24no—
519Q8—24no—
520Q8—42no—
521Q8—42no—
522D4—24no—
523D4—42no—
524D4—64no—
525D4—64no—
526D4—64no—
527D4—1216no—
528D4—128no—
529D4—1216no—
530D4—122no—
531D4—122no—
532D4—122no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—32no—
537C2×C4—32no—
538C2×C4—32no—
539C2×C4—38no—
540C2×C4—38no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—128no—
549C2×C4—1216no—
550C2×C4—121no—
551C2×C4—122no—
552C2×C4—244no—
553C2×C4#91414yescomplemented
554C2×C4C2×D3×D411yescharacteristic
555C2×C4C2×D3×D412yes—
556C2×C4C2×D3×D412yes—
557C2×C4D3×#4111yescharacteristic
558C8—24no—
559C8—64no—
560C2×C3—82no—
561C2×C3C2×#39211yescharacteristic
562C2×C3#300612yes—
563C2×C3#303612yes—
564C2×C3#315111yescharacteristic
565C2×C3#315311yescharacteristic
566D3—24no—
567D3—84no—
568D3—82no—
569C22—34no—
570C22—44no—
571C22—42no—
572C22—64no—
573C22—62no—
574C22—64no—
575C22—62no—
576C22—81no—
577C22—82no—
578C22—81no—
579C22—124no—
580C22—128no—
581C22—124no—
582C22—122no—
583C22—2416no—
584C22—244no—
585C22—241no—
586C22—242no—
587C22—242no—
588C22—241no—
589C22—242no—
590C22D3×#12011yescharacteristic
591C22D3×#12111yescharacteristic
592C22D3×#12711yescharacteristic
593C22C2×#91412yes—
594C22#577712yes—
595C4—24no—
596C4—34no—
597C4—44no—
598C4—81no—
599C4—122no—
600C4—241no—
601C4D3×#12012yes—
602C4D3×#12112yes—
603C3#2342811yescharacteristic, complemented, Sylow
604C2—64no—
605C2—82no—
606C2—242no—
607C2—244no—
608C2D3×#39211yescharacteristic
609C2#8004012yes—
610C2#8042812yes—
611C2#8176111yescharacteristic
612C2#8187111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2013C2
socleC23×C3C22×D4
hypercenter#2013D3
hypocenterC3#23428
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2013

Complements

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

NH ≅ G/Ntype
1. C22×#914C2semidirect
2. C2×#5725C2semidirect
3. C2×#4803C2semidirect
4. D3×#390C2semidirect
5. C3×#2013C2semidirect
6. #72055C2semidirect
7. #71791C2semidirect
8. #70277C2semidirect
9. #69658C2semidirect
10. #68674C2semidirect
11. #68660C2semidirect
16. C2×C3×C4⋊Q8C22semidirect
17. C2×#947C22semidirect
30. C2×#874C22semidirect
32. C2×C3×#103C22semidirect
34. C2×#835C22semidirect
36. C2×#821C22semidirect
38. C2×#809C22semidirect
42. C3×#336C22semidirect
43. #4550C22semidirect
44. #4524C22semidirect
50. #2013D3semidirect
69. C2×D3×D4D4semidirect
110. C2×ΓL(1,𝔽25)D4semidirect
132. C2×C4⋊Q8C2×D3semidirect
142. C2×#103C2×D3semidirect
152. #336C2×D3semidirect
343. C2×C3×C4AGL(1,ℤ/8)semidirect
553. C2×C4#914semidirect
603. C3#23428semidirect

Maximal subgroups

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

subgroupindex
C22×#9142
C2×#57252
C2×#48032
D3×#3902
C3×#20132
#720552
#717912
#702772
#696582
#686742
#686602
#234283