Finite Group Databasebeta

← #10830940

Subgroups — #10830940

Order 768.

1776 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×D8C211yesmaximal, characteristic, complemented
2C2×C3×#469C211yesmaximal, characteristic, complemented
3C2×#5135C211yesmaximal, characteristic, complemented
4C2×#5127C211yesmaximal, characteristic, complemented
5#82640C212yesmaximal, complemented
6#72039C211yesmaximal, characteristic, complemented
7#71684C212yesmaximal, complemented
8#70577C211yesmaximal, characteristic, complemented
9#70576C211yesmaximal, characteristic, complemented
10#70212C212yesmaximal, complemented
11#70198C212yesmaximal, complemented
12#24272—31nomaximal, Sylow
13C22×D3×D4C2212yes—
14C22×C3×D8C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic, complemented
16C2×C3×#121C2212yescomplemented
17C2×#937—22no—
18C22×#213C2212yes—
19C2×#930C2212yescomplemented
20C2×C4×#67C2211yescharacteristic
21C2×D3×D8—24no—
22C2×D3×D8—24no—
23C2×D3×D8—24no—
24C22×D24C2211yescharacteristic
25C22×D3×C8C2211yescharacteristic
26C3×#469—24no—
27C2×C3×#106C2211yescharacteristic, complemented
28C2×C3×#102C2212yescomplemented
29#5223—22no—
30#5221C2212yescomplemented
31C2×#832C2212yescomplemented
32C2×#830C2211yescharacteristic, complemented
33#5135—24no—
34#5127—24no—
35C2×#821C2212yescomplemented
36C2×(C2×D3)⋊C8C2211yescharacteristic
37C2×#817C2211yescharacteristic, complemented
38C2×#815C2211yescharacteristic, complemented
39C2×Dic3⋊C8C2211yescharacteristic, complemented
40#4988C2212yescomplemented
41#4987C2212yes—
42C2×#809C2212yescomplemented
43#4959—22no—
44#4955—22no—
45#4529C2212yescomplemented
46C23×D8—31no—
47C2×#469—31no—
48C2×#469D311yescharacteristic, complemented
49C2×#435—31no—
50#2408—32no—
51#2054—32no—
52#1999—32no—
53#1972—31no—
54#1972—31no—
55#1807—31no—
56#1786—32no—
57C24×D3—22no—
58C22×C3×D4—22no—
59C22×C3×D4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C22×#67—22no—
62C22×#67—22no—
63C22×#67D411yescharacteristic
64C2×D3×D4—24no—
65C2×D3×D4—24no—
66C2×D3×D4—24no—
67C2×D3×D4—24no—
68C2×D3×D4—42no—
69C2×D3×D4—42no—
70C22×D12C2312yes—
71C22×C4×D3C2311yescharacteristic
72C2×C3×D8—22no—
73C2×C3×D8—22no—
74C2×C3×D8D414yescomplemented
75C2×C3×M16—22no—
76C2×C3×M16—22no—
77C2×C3×M16—22no—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×#121—24no—
81C3×#121—42no—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—22no—
84C2×#217—22no—
85C2×#217—22no—
86C2×#217D411yescharacteristic
87#937—48no—
88C2×#213—24no—
89C2×#213—24no—
90C2×#213—24no—
91#930—24no—
92#930—42no—
93C4×#67—22no—
94C4×#67—22no—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4—22no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—22no—
99C2×(C2×D3)⋊C4D411yescharacteristic
100C2×C4⋊Dic3C2312yes—
101C2×Dic3⋊C4D411yescharacteristic
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2311yescharacteristic
104C22×C3⋊C8C2311yescharacteristic
105D3×D8—232no—
106C2×D24—22no—
107C2×D24—22no—
108C2×D24D414yescomplemented
109C2×D3×C8—22no—
110C2×D3×C8—22no—
111C2×D3×C8D414yescomplemented
112C3×#106—22no—
113C3×#102—24no—
114#832—42no—
115C3⋊#95—22no—
116#830—22no—
117#821—24no—
118(C2×D3)⋊C8—22no—
119#817—22no—
120#815—22no—
121Dic3⋊C8—22no—
122#809—42no—
123C23×D4—32no—
124C22×D8—32no—
125C22×D8—31no—
126C22×D8—64no—
127C22×D8—64no—
128C22×D8—64no—
129C22×D8C2×D311yescharacteristic
130C22×M16C2×D311yescharacteristic, complemented
131C23×C8—31no—
132C2×#121—32no—
133C2×#121C2×D312yescomplemented
134C2×#120—62no—
135C2×C4×D4—31no—
136#469—24no—
137#469—64no—
138#435—64no—
139C2×#106—31no—
140C2×#106C2×D311yescharacteristic, complemented
141C2×C4⋊C8—31no—
142C2×#102—32no—
143C2×#102—32no—
144C2×#102—32no—
145C2×#102C2×D312yescomplemented
146C2×C22⋊C8—31no—
147C2×C22⋊C8—31no—
148C2×#97—31no—
149#394—62no—
150#392—62no—
151#390—32no—
152#390—32no—
153#380—32no—
154#379—62no—
155#340—32no—
156C24×C3—22no—
157C24×C3C2×D411yescharacteristic
158C23×D3—22no—
159C23×D3—44no—
160C23×D3—44no—
161C23×D3—44no—
162C23×D3—44no—
163C23×D3—44no—
164C23×D3—82no—
165C23×D3—82no—
166C23×D3C2×D411yescharacteristic
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—44no—
173C2×C3×D4—44no—
174C2×C3×D4—82no—
175C2×C3×D4C2×D414yes—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2411yescharacteristic
185C2×#67—22no—
186C2×#67—24no—
187C2×#67—44no—
188C2×#67—44no—
189C2×#67—44no—
190C2×#67—44no—
191C2×#67—44no—
192C2×#67—82no—
193C22×Dic3—22no—
194C22×Dic3C2×D411yescharacteristic
195C22×Dic3C2×D411yescharacteristic
196C22×Dic3C2×D411yescharacteristic
197D3×D4—216no—
198D3×D4—48no—
199D3×D4—48no—
200D3×D4—48no—
201C2×D12—22no—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—22no—
205C2×D12—22no—
206C2×D12C2×D414yes—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3C2×D412yes—
212C2×C4×D3#4112yes—
213C3×D8—28no—
214C3×M16—44no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8C2×D412yes—
219C2×C3×C8C2×D412yes—
220C2×C3×C8C2×D412yes—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C22⋊C4—44no—
224#217—22no—
225#217—44no—
226#217—82no—
227#213—216no—
228(C2×D3)⋊C4—22no—
229(C2×D3)⋊C4—44no—
230(C2×D3)⋊C4—44no—
231(C2×D3)⋊C4—44no—
232(C2×D3)⋊C4—82no—
233C4⋊Dic3—22no—
234C4⋊Dic3—22no—
235Dic3⋊C4—22no—
236C4×Dic3—21no—
237C4×Dic3—21no—
238C2×C3⋊C8—21no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—21no—
241C2×C3⋊C8—21no—
242C2×C3⋊C8—21no—
243C2×C3⋊C8C2×D412yes—
244D24—28no—
245D3×C8—28no—
246C25—62no—
247C22×D4—22no—
248C22×D4—31no—
249C22×D4—32no—
250C22×D4—64no—
251C22×D4—64no—
252C22×D4—62no—
253C22×D4—64no—
254C22×D4—64no—
255C22×D4—62no—
256C22×D4—122no—
257C22×D4—122no—
258C22×D4C22×D312yes—
259C23×C4—31no—
260C23×C4C22×D311yescharacteristic
261C2×D8—22no—
262C2×D8—22no—
263C2×D8—34no—
264C2×D8—64no—
265C2×D8—632no—
266C2×D8—64no—
267C2×D8—64no—
268C2×D8—62no—
269C2×D8—62no—
270C2×D8#6714yescomplemented
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—22no—
274C22×C8—31no—
275C22×C8—34no—
276C22×C8—31no—
277C22×C8—62no—
278C22×C8—62no—
279C22×C8C22×D311yescharacteristic
280C22×C8C22×D311yescharacteristic
281#121—24no—
282#121—42no—
283#121—64no—
284#121—122no—
285#120—128no—
286C4×D4—62no—
287C4×D4—62no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—32no—
290C2×C4⋊C4C22×D312yes—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—31no—
293C2×C22⋊C4—31no—
294C2×C22⋊C4—62no—
295C2×C22⋊C4—62no—
296C2×C22⋊C4—62no—
297C2×C22⋊C4—62no—
298C2×C22⋊C4—62no—
299C2×C22⋊C4—62no—
300C2×C42—31no—
301#106—22no—
302#106—62no—
303C4⋊C8—62no—
304#102—24no—
305#102—64no—
306#102—122no—
307#102—122no—
308C22⋊C8—62no—
309C22⋊C8—62no—
310#97—62no—
311#95—62no—
312C23×C3—21no—
313C23×C3—24no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3C22×D411yescharacteristic
325C22×D3—21no—
326C22×D3—21no—
327C22×D3—22no—
328C22×D3—24no—
329C22×D3—416no—
330C22×D3—42no—
331C22×D3—42no—
332C22×D3—42no—
333C22×D3—42no—
334C22×D3—42no—
335C22×D3—42no—
336C22×D3—88no—
337C22×D3—88no—
338C22×D3—88no—
339C22×D3—88no—
340C22×D3—88no—
341C22×D3—88no—
342C22×D3#12112yes—
343C22×D3AGL(1,ℤ/8)14yes—
344C3×D4—24no—
345C3×D4—24no—
346C3×D4—42no—
347C3×D4—42no—
348C3×D4—88no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4H(5,𝔽2)11yescharacteristic
364C2×C3×C4H(5,𝔽2)11yescharacteristic
365#67—416no—
366#67—44no—
367#67—88no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3#12112yes—
378D12—24no—
379D12—24no—
380D12—42no—
381D12—42no—
382C4×D3—22no—
383C4×D3—22no—
384C4×D3—22no—
385C4×D3—22no—
386C3×C8—41no—
387C3×C8#12014yescomplemented
388C3⋊C8—22no—
389C3⋊C8—41no—
390C24—22no—
391C24—31no—
392C24—62no—
393C24—124no—
394C24—124no—
395C24—124no—
396C24—124no—
397C24—124no—
398C24—242no—
399C24—242no—
400C24D3×D411yescharacteristic
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—34no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—616no—
410C2×D4—62no—
411C2×D4—64no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4—62no—
417C2×D4—82no—
418C2×D4—124no—
419C2×D4—128no—
420C2×D4—128no—
421C2×D4—128no—
422C2×D4—124no—
423C2×D4—124no—
424C2×D4—124no—
425C2×D4—124no—
426C2×D4—242no—
427C2×D4C2×#6714yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—31no—
436C22×C4—32no—
437C22×C4—32no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—62no—
445C22×C4D3×D411yescharacteristic
446C22×C4C23×D311yescharacteristic
447D8—28no—
448D8—616no—
449D8—68no—
450M16—44no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—32no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—68no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8D3×D412yes—
462C2×C8D3×D412yes—
463C2×C8C2×#6712yes—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—62no—
467C4⋊C4—62no—
468C4⋊C4—62no—
469C22⋊C4—44no—
470C22⋊C4—62no—
471C22⋊C4—62no—
472C22⋊C4—124no—
473C22⋊C4—124no—
474C22⋊C4—124no—
475C22⋊C4—124no—
476C22⋊C4—242no—
477C22⋊C4—242no—
478C42—61no—
479C42—61no—
480C22×C3—44no—
481C22×C3—41no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—44no—
485C22×C3—44no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—82no—
489C22×C3—82no—
490C22×C3—82no—
491C22×C3—82no—
492C22×C3C2×#12111yescharacteristic
493C22×C3D4211yescharacteristic
494C22×C3D4211yescharacteristic
495C22×C3C2×AGL(1,ℤ/8)12yes—
496C22×C3#57612yes—
497C2×D3—22no—
498C2×D3—24no—
499C2×D3—24no—
500C2×D3—22no—
501C2×D3—44no—
502C2×D3—44no—
503C2×D3—44no—
504C2×D3—832no—
505C2×D3—82no—
506C2×D3—82no—
507C2×D3—82no—
508C2×D3—82no—
509C2×D3#46914yescomplemented
510C3×C4—41no—
511C3×C4—82no—
512C3×C4C2×#12011yescharacteristic
513C3×C4#53511yescharacteristic
514C3×C4#54611yescharacteristic
515C3×C4#54611yescharacteristic
516Dic3—22no—
517Dic3—41no—
518Dic3—41no—
519Dic3—82no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—34no—
527C23—32no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—62no—
535C23—61no—
536C23—61no—
537C23—64no—
538C23—1216no—
539C23—122no—
540C23—122no—
541C23—122no—
542C23—122no—
543C23—122no—
544C23—122no—
545C23—248no—
546C23—248no—
547C23—248no—
548C23—248no—
549C23—248no—
550C23—248no—
551C23C2×D3×D411yescharacteristic
552D4—24no—
553D4—24no—
554D4—42no—
555D4—42no—
556D4—64no—
557D4—64no—
558D4—88no—
559D4—1216no—
560D4—124no—
561D4—122no—
562D4—122no—
563D4—248no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—32no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—62no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—122no—
586C2×C4—122no—
587C2×C4—122no—
588C2×C4C2×D3×D411yescharacteristic
589C2×C4C2×D3×D411yescharacteristic
590C2×C4#100411yescharacteristic
591C2×C4D3×#4111yescharacteristic
592C2×C4#100911yescharacteristic
593C2×C4C22×#6711yescharacteristic
594C8—41no—
595C8—62no—
596C8—121no—
597C8#93714yescomplemented
598C2×C3—84no—
599C2×C3—81no—
600C2×C3#240811yescharacteristic
601C2×C3C2×#46911yescharacteristic
602C2×C3#305811yescharacteristic
603C2×C3#326412yes—
604C2×C3#326412yes—
605D3—24no—
606D3—84no—
607C22—34no—
608C22—41no—
609C22—44no—
610C22—41no—
611C22—41no—
612C22—42no—
613C22—44no—
614C22—44no—
615C22—42no—
616C22—62no—
617C22—62no—
618C22—64no—
619C22—64no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—124no—
625C22—124no—
626C22—124no—
627C22—2432no—
628C22—242no—
629C22—242no—
630C22—242no—
631C22—242no—
632C22D3×#12111yescharacteristic
633C22#560911yescharacteristic
634C22D3×AGL(1,ℤ/8)12yes—
635C22#577412yes—
636C22D4×#6711yescharacteristic
637C4—41no—
638C4—62no—
639C4—82no—
640C4—121no—
641C4—121no—
642C4—242no—
643C4#560711yescharacteristic
644C4#561211yescharacteristic
645C4C2×#93711yescharacteristic
646C4#582911yescharacteristic
647C3#2427211yescharacteristic, complemented, Sylow
648C2—64no—
649C2—81no—
650C2—84no—
651C2—244no—
652C2D3×#46911yescharacteristic
653C2#8086511yescharacteristic
654C2#8088112yes—
655C2#8264011yescharacteristic
656C2#8395212yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#469C2
socleC23×C3C22×D4
hypercenterC2×#469D3
hypocenterC3#24272
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#469

Complements

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

NH ≅ G/Ntype
1. C22×D3×D8C2semidirect
2. C2×C3×#469C2semidirect
3. C2×#5135C2semidirect
4. C2×#5127C2semidirect
5. #82640C2semidirect
6. #72039C2semidirect
7. #71684C2semidirect
8. #70577C2semidirect
9. #70576C2semidirect
10. #70212C2semidirect
11. #70198C2semidirect
15. C22×C3×M16C22semidirect
16. C2×C3×#121C22semidirect
19. C2×#930C22semidirect
27. C2×C3×#106C22semidirect
28. C2×C3×#102C22semidirect
30. #5221C22semidirect
31. C2×#832C22semidirect
32. C2×#830C22semidirect
35. C2×#821C22semidirect
37. C2×#817C22semidirect
38. C2×#815C22semidirect
39. C2×Dic3⋊C8C22semidirect
40. #4988C22semidirect
42. C2×#809C22semidirect
45. #4529C22semidirect
48. C2×#469D3semidirect
74. C2×C3×D8D4semidirect
108. C2×D24D4semidirect
111. C2×D3×C8D4semidirect
130. C22×M16C2×D3semidirect
133. C2×#121C2×D3semidirect
140. C2×#106C2×D3semidirect
145. C2×#102C2×D3semidirect
270. C2×D8#67semidirect
387. C3×C8#120semidirect
509. C2×D3#469semidirect
597. C8#937semidirect
647. C3#24272semidirect

Maximal subgroups

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

subgroupindex
C22×D3×D82
C2×C3×#4692
C2×#51352
C2×#51272
#826402
#720392
#716842
#705772
#705762
#702122
#701982
#242723