Finite Group Databasebeta

← #10944274

Subgroups — #10944274

Order 768.

1658 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×M16C211yesmaximal, characteristic, complemented
2C2×C3×#469C211yesmaximal, characteristic, complemented
3C2×#5159C211yesmaximal, characteristic, complemented
4C2×#5157C211yesmaximal, characteristic, complemented
5#82723C212yesmaximal, complemented
6#71988C211yesmaximal, characteristic, complemented
7#71682C212yesmaximal, complemented
8#70578C211yesmaximal, characteristic, complemented
9#70577C211yesmaximal, characteristic, complemented
10#70203C212yesmaximal, complemented
11#70202C212yesmaximal, complemented
12#32936—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C22×C3×D8C2211yescharacteristic, complemented
15C22×C3×M16C2211yescharacteristic
16C2×C3×#121C2212yes—
17C2×#938C2212yes—
18C2×#938C2212yes—
19C2×#937—22no—
20C22×#208C2211yescharacteristic
21C2×D3×M16—24no—
22C2×D3×M16—24no—
23C2×D3×M16—24no—
24C22×ΓL(1,𝔽25)C2211yescharacteristic
25C22×ΓL(1,𝔽25)C2211yescharacteristic
26C22×D3×C8C2211yescharacteristic
27C22×D3×C8C2211yescharacteristic
28C3×#469—24no—
29C2×C3×#106C2211yescharacteristic, complemented
30C2×C3×#102C2212yescomplemented
31#5220—22no—
32C2×#832C2212yescomplemented
33C2×#832C2212yescomplemented
34#5212C2212yes—
35#5159—24no—
36#5157—24no—
37C2×#818C2211yescharacteristic, complemented
38C2×#817C2211yescharacteristic, complemented
39#4987C2212yes—
40#4987C2212yes—
41#4956—22no—
42#4956—22no—
43#4551C2212yescomplemented
44#4529C2212yescomplemented
45#4529C2212yescomplemented
46C23×M16—31no—
47C2×#469—31no—
48C2×#469D311yescharacteristic, complemented
49C2×#466—31no—
50#2847—32no—
51#1895—32no—
52#1895—32no—
53#1893—32no—
54#1809—31no—
55#1809—31no—
56#1807—31no—
57C24×D3D411yescharacteristic
58C22×C3×D4—22no—
59C22×C3×D4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C22×#67—22no—
62C22×#67—22no—
63C23×Dic3D411yescharacteristic
64C22×C4×D3—22no—
65C22×C4×D3—22no—
66C22×C4×D3—22no—
67C22×C4×D3—22no—
68C22×C4×D3—22no—
69C22×C4×D3—22no—
70C22×C4×D3D411yescharacteristic
71C22×C4×D3D411yescharacteristic
72C22×C4×D3C2311yescharacteristic
73C22×C4×D3C2311yescharacteristic
74C2×C3×D8—22no—
75C2×C3×D8—22no—
76C2×C3×D8—22no—
77C2×C3×M16—22no—
78C2×C3×M16—22no—
79C2×C3×M16D414yescomplemented
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×#121—24no—
83C3×#121—24no—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C22⋊C4—22no—
86C2×#217—22no—
87C2×#217—22no—
88C2×#217—22no—
89#938—24no—
90#938—24no—
91#938—24no—
92#938—24no—
93#937—48no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—22no—
96C2×C4⋊Dic3C2312yes—
97C2×C4⋊Dic3C2312yes—
98C2×#208—22no—
99C2×#208—22no—
100C2×#208—22no—
101C22×C3⋊C8C2311yescharacteristic
102C22×C3⋊C8C2311yescharacteristic
103D3×M16—232no—
104C2×ΓL(1,𝔽25)—22no—
105C2×ΓL(1,𝔽25)—22no—
106C2×ΓL(1,𝔽25)—22no—
107C2×ΓL(1,𝔽25)—22no—
108C2×ΓL(1,𝔽25)—22no—
109C2×ΓL(1,𝔽25)D414yescomplemented
110C2×D3×C8—22no—
111C2×D3×C8—22no—
112C2×D3×C8—22no—
113C2×D3×C8—22no—
114C2×D3×C8—22no—
115C2×D3×C8D414yescomplemented
116C3×#106—22no—
117C3×#102—24no—
118#832—24no—
119#832—24no—
120C3⋊#95—22no—
121C3⋊#95—22no—
122C3⋊#95—22no—
123#818—22no—
124#817—22no—
125C24×C4—31no—
126C22×D8C2×D311yescharacteristic, complemented
127C22×M16—31no—
128C22×M16—31no—
129C22×M16—31no—
130C22×M16—64no—
131C22×M16—64no—
132C22×M16—64no—
133C22×M16C2×D311yescharacteristic
134C23×C8—31no—
135C23×C8—31no—
136C2×#121—32no—
137C2×#121—32no—
138C2×#121C2×D312yes—
139C2×#120—62no—
140#469—24no—
141#469—64no—
142#466—64no—
143C2×#107—31no—
144C2×#106—31no—
145C2×#106C2×D311yescharacteristic, complemented
146C2×#102—32no—
147C2×#102—32no—
148C2×#102C2×D312yescomplemented
149#386—62no—
150#386—62no—
151#386—62no—
152#380—32no—
153#380—32no—
154#380—32no—
155#340—32no—
156#340—32no—
157#340—32no—
158C24×C3—22no—
159C24×C3C2×D411yescharacteristic
160C23×D3—22no—
161C23×D3—22no—
162C23×D3—22no—
163C23×D3—24no—
164C23×D3—42no—
165C23×D3C2×D411yescharacteristic
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—22no—
173C2×C3×D4—44no—
174C2×C3×D4—44no—
175C2×C3×D4—44no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D412yes—
183C22×C3×C4#4112yes—
184C22×C3×C4C2411yescharacteristic
185C2×#67—44no—
186C2×#67—44no—
187C2×#67—44no—
188C2×#67—44no—
189C2×#67—44no—
190C2×#67—44no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C2×C4×D3—28no—
199C2×C4×D3—28no—
200C2×C4×D3—28no—
201C2×C4×D3—28no—
202C2×C4×D3—22no—
203C2×C4×D3—22no—
204C2×C4×D3—22no—
205C2×C4×D3—22no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—21no—
214C2×C4×D3—22no—
215C2×C4×D3—22no—
216C2×C4×D3C2×D412yes—
217C2×C4×D3C2×D412yes—
218C2×C4×D3#4112yes—
219C2×C4×D3#4112yes—
220C3×D8—44no—
221C3×M16—28no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8C2×D412yes—
225C2×C3×C8C2×D412yes—
226C2×C3×C8C2×D412yes—
227C2×C3×C8C2×D412yes—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C22⋊C4—44no—
231#217—44no—
232#217—44no—
233#217—44no—
234(C2×D3)⋊C4—44no—
235(C2×D3)⋊C4—44no—
236C4⋊Dic3—22no—
237C4⋊Dic3—22no—
238C4⋊Dic3—22no—
239C4⋊Dic3—22no—
240#208—28no—
241C2×C3⋊C8—21no—
242C2×C3⋊C8—21no—
243C2×C3⋊C8—21no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8C2×D412yes—
246C2×C3⋊C8C2×D412yes—
247ΓL(1,𝔽25)—28no—
248ΓL(1,𝔽25)—28no—
249D3×C8—28no—
250D3×C8—28no—
251C25—31no—
252C22×D4—22no—
253C22×D4—62no—
254C22×D4—62no—
255C22×D4C22×D312yes—
256C23×C4—31no—
257C23×C4—31no—
258C23×C4—31no—
259C23×C4—31no—
260C23×C4—31no—
261C23×C4—62no—
262C23×C4—62no—
263C23×C4—62no—
264C23×C4—62no—
265C23×C4—62no—
266C23×C4—62no—
267C23×C4C22×D311yescharacteristic
268C2×D8—22no—
269C2×D8—22no—
270C2×D8—22no—
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—34no—
274C2×M16—62no—
275C2×M16—62no—
276C2×M16—62no—
277C2×M16—62no—
278C2×M16—632no—
279C2×M16—62no—
280C2×M16—62no—
281C2×M16—62no—
282C2×M16—62no—
283C2×M16#6714yescomplemented
284C22×C8—31no—
285C22×C8—31no—
286C22×C8—34no—
287C22×C8—62no—
288C22×C8—62no—
289C22×C8—62no—
290C22×C8—62no—
291C22×C8—62no—
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294#121—24no—
295#121—24no—
296#121—64no—
297#121—64no—
298#121—64no—
299#121—64no—
300#120—128no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4C22×D312yes—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—62no—
306C2×C22⋊C4—62no—
307C2×C22⋊C4—62no—
308C2×C22⋊C4—62no—
309C2×C22⋊C4—62no—
310#107—62no—
311#106—22no—
312#106—62no—
313#102—24no—
314#102—64no—
315#102—64no—
316#95—62no—
317#95—62no—
318#95—62no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3#12112yes—
331C23×C3C22×D411yescharacteristic
332C22×D3—21no—
333C22×D3—21no—
334C22×D3—22no—
335C22×D3—28no—
336C22×D3—21no—
337C22×D3—21no—
338C22×D3—22no—
339C22×D3—22no—
340C22×D3—48no—
341C22×D3—48no—
342C22×D3—44no—
343C22×D3—44no—
344C22×D3#12112yes—
345C22×D3#12112yes—
346C22×D3AGL(1,ℤ/8)14yes—
347C3×D4—42no—
348C3×D4—42no—
349C3×D4—42no—
350C3×D4—42no—
351C3×D4—88no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4#12112yes—
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368#67—88no—
369#67—88no—
370C2×Dic3—28no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3#12112yes—
382C2×Dic3#12112yes—
383C4×D3—28no—
384C4×D3—28no—
385C4×D3—22no—
386C4×D3—22no—
387C4×D3—22no—
388C4×D3—22no—
389C4×D3—22no—
390C4×D3—22no—
391C4×D3—22no—
392C4×D3—22no—
393C3×C8—22no—
394C3×C8#12014yescomplemented
395C3⋊C8—22no—
396C3⋊C8—22no—
397C24—22no—
398C24—31no—
399C24—31no—
400C24—62no—
401C24—62no—
402C24—64no—
403C24—62no—
404C24—122no—
405C24D3×D411yescharacteristic
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—124no—
416C2×D4—124no—
417C2×D4—124no—
418C2×D4—124no—
419C2×D4—124no—
420C2×D4—124no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—32no—
429C22×C4—32no—
430C22×C4—32no—
431C22×C4—32no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—68no—
437C22×C4—68no—
438C22×C4—68no—
439C22×C4—68no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4D3×D411yescharacteristic
456C22×C4#23712yes—
457C22×C4C2×#6712yes—
458C22×C4C23×D311yescharacteristic
459D8—44no—
460M16—28no—
461M16—68no—
462M16—68no—
463M16—68no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—32no—
467C2×C8—32no—
468C2×C8—68no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—68no—
474C2×C8D3×D412yes—
475C2×C8D3×D412yes—
476C2×C8C2×#6712yes—
477C2×C8C2×#6712yes—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—62no—
481C4⋊C4—62no—
482C4⋊C4—62no—
483C4⋊C4—62no—
484C22⋊C4—44no—
485C22⋊C4—124no—
486C22⋊C4—124no—
487C22⋊C4—124no—
488C22⋊C4—124no—
489C22⋊C4—124no—
490C22×C3—22no—
491C22×C3—24no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—42no—
495C22×C3—82no—
496C22×C3—82no—
497C22×C3—82no—
498C22×C3—82no—
499C22×C3—82no—
500C22×C3—82no—
501C22×C3—82no—
502C22×C3C2×#12111yescharacteristic
503C22×C3C2×#12111yescharacteristic
504C22×C3C2×#12111yescharacteristic
505C22×C3C2×AGL(1,ℤ/8)12yes—
506C22×C3#57512yes—
507C2×D3—22no—
508C2×D3—22no—
509C2×D3—24no—
510C2×D3—24no—
511C2×D3—22no—
512C2×D3—24no—
513C2×D3—416no—
514C2×D3—41no—
515C2×D3—41no—
516C2×D3—42no—
517C2×D3#46914yescomplemented
518C3×C4—22no—
519C3×C4—82no—
520C3×C4C2×#12011yescharacteristic
521C3×C4#52511yescharacteristic
522C3×C4#52511yescharacteristic
523C3×C4#52511yescharacteristic
524Dic3—22no—
525Dic3—22no—
526Dic3—82no—
527Dic3—82no—
528C23—22no—
529C23—22no—
530C23—21no—
531C23—21no—
532C23—34no—
533C23—32no—
534C23—32no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—68no—
543C23—62no—
544C23—61no—
545C23—61no—
546C23—62no—
547C23—62no—
548C23—61no—
549C23—61no—
550C23—128no—
551C23—124no—
552C23—128no—
553C23—124no—
554C23#93812yes—
555C23C2×D3×D411yescharacteristic
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—88no—
561D4—248no—
562D4—248no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—32no—
570C2×C4—32no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—68no—
575C2×C4—68no—
576C2×C4—68no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—122no—
595C2×C4#93812yes—
596C2×C4C2×D3×D411yescharacteristic
597C2×C4C2×D3×D411yescharacteristic
598C2×C4C2×#23711yescharacteristic
599C2×C4D3×#4111yescharacteristic
600C2×C4D3×#4111yescharacteristic
601C2×C4C22×#6711yescharacteristic
602C8—22no—
603C8—62no—
604C8—62no—
605C8#93714yescomplemented
606C2×C3—42no—
607C2×C3—162no—
608C2×C3#284711yescharacteristic
609C2×C3C2×#46911yescharacteristic
610C2×C3#305411yescharacteristic
611C2×C3#305512yes—
612C2×C3#305512yes—
613D3—24no—
614D3—42no—
615C22—22no—
616C22—24no—
617C22—34no—
618C22—41no—
619C22—41no—
620C22—42no—
621C22—62no—
622C22—62no—
623C22—64no—
624C22—64no—
625C22—62no—
626C22—64no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—1216no—
635C22—121no—
636C22—121no—
637C22—122no—
638C22D3×#12111yescharacteristic
639C22D3×#12111yescharacteristic
640C22D3×AGL(1,ℤ/8)12yes—
641C22#577312yes—
642C22C2×#93811yescharacteristic
643C4—22no—
644C4—62no—
645C4—62no—
646C4—82no—
647C4—242no—
648C4—242no—
649C4#560611yescharacteristic
650C4#560611yescharacteristic
651C4C2×#93711yescharacteristic
652C4#582811yescharacteristic
653C3#3293611yescharacteristic, complemented, Sylow
654C2—42no—
655C2—64no—
656C2—122no—
657C2—162no—
658C2D3×#46911yescharacteristic
659C2#8086311yescharacteristic
660C2#8086412yes—
661C2#8272311yescharacteristic
662C2#8395012yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#469C2
socleC23×C3C22×D4
hypercenterC2×#469D3
hypocenterC3#32936
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×M16C2semidirect
2. C2×C3×#469C2semidirect
3. C2×#5159C2semidirect
4. C2×#5157C2semidirect
5. #82723C2semidirect
6. #71988C2semidirect
7. #71682C2semidirect
8. #70578C2semidirect
9. #70577C2semidirect
10. #70203C2semidirect
11. #70202C2semidirect
14. C22×C3×D8C22semidirect
29. C2×C3×#106C22semidirect
30. C2×C3×#102C22semidirect
32. C2×#832C22semidirect
33. C2×#832C22semidirect
37. C2×#818C22semidirect
38. C2×#817C22semidirect
43. #4551C22semidirect
44. #4529C22semidirect
45. #4529C22semidirect
48. C2×#469D3semidirect
79. C2×C3×M16D4semidirect
109. C2×ΓL(1,𝔽25)D4semidirect
115. C2×D3×C8D4semidirect
126. C22×D8C2×D3semidirect
145. C2×#106C2×D3semidirect
148. C2×#102C2×D3semidirect
283. C2×M16#67semidirect
394. C3×C8#120semidirect
517. C2×D3#469semidirect
605. C8#937semidirect
653. C3#32936semidirect

Maximal subgroups

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

subgroupindex
C22×D3×M162
C2×C3×#4692
C2×#51592
C2×#51572
#827232
#719882
#716822
#705782
#705772
#702032
#702022
#329363