Finite Group Databasebeta

← #10944031

Subgroups — #10944031

Order 768.

1894 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D12C211yesmaximal, characteristic, complemented
2C3×#2850C211yesmaximal, characteristic, complemented
3C2×#5057C212yesmaximal, complemented
4C2×#5046C211yesmaximal, characteristic, complemented
5C2×#5036C211yesmaximal, characteristic
6#71692C211yesmaximal, characteristic, complemented
7#71677C211yesmaximal, characteristic, complemented
8#71530C211yesmaximal, characteristic, complemented
9#69687C212yesmaximal, complemented
10#69657C212yesmaximal, complemented
11#69556C212yesmaximal, complemented
12#32908—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C23×C4×D3—21no—
15C2×C3×#124C2211yescharacteristic, complemented
16C2×C3×C22⋊Q8C2211yescharacteristic
17C2×C3×#121C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C22×#215C2212yes—
20C22×#213C2212yes—
21C22×(C2×D3)⋊C4—21no—
22C22×C4⋊Dic3C2211yescharacteristic
23C2×C4×D12—21no—
24C2×C4×D12—21no—
25C2×C4×D12—28no—
26C2×C4×D12C2212yes—
27C2×C4×D12C2212yes—
28C3×#391C2211yescharacteristic, complemented
29C3×#390C2211yescharacteristic, complemented
30C3×#388—21no—
31C3×#388—21no—
32C3×#386—21no—
33C3×#386—21no—
34C3×#382C2211yescharacteristic, complemented
35C2×#835C2212yescomplemented
36C2×#832C2212yescomplemented
37C2×#830C2211yescharacteristic
38C2×#830C2211yescharacteristic
39#5057—28no—
40#5046—24no—
41#5036—24no—
42C2×#809C2212yes—
43C2×#809C2212yes—
44C2×C4⋊(C3⋊C8)C2212yescomplemented
45#4529C2212yescomplemented
46#4529C2212yescomplemented
47#4524C2212yescomplemented
48C22×C4×D4—31no—
49C2×#463—32no—
50C2×#450—31no—
51C2×#447—31no—
52#2850D311yescharacteristic, complemented
53#1879—32no—
54#1877—32no—
55#1874—31no—
56#1872—31no—
57#1804—32no—
58#1797—31no—
59C24×D3—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4D411yescharacteristic
64C23×C3×C4D411yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×Dic3—21no—
67C22×D12—21no—
68C22×D12—22no—
69C22×D12—24no—
70C22×D12—24no—
71C22×D12D414yes—
72C22×D12C2312yes—
73C22×C4×D3—28no—
74C22×C4×D3—22no—
75C22×C4×D3—22no—
76C22×C4×D3—44no—
77C22×C4×D3—44no—
78C3×#124—24no—
79C3×C22⋊Q8—22no—
80C3×C22⋊Q8—22no—
81C3×#121—22no—
82C3×#121—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42—24no—
93C2×C3×C42D411yescharacteristic
94C2×C3×C42D411yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×#215—24no—
98C2×#215—24no—
99C2×#215—24no—
100C2×#213—24no—
101C2×#213—24no—
102C2×#213—24no—
103C2×(C2×D3)⋊C4—28no—
104C2×(C2×D3)⋊C4—22no—
105C2×(C2×D3)⋊C4—22no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—24no—
108C2×C4⋊Dic3C2312yes—
109C22×C3⋊C8C2312yes—
110C22×C3⋊C8C2312yes—
111C4×D12—28no—
112C4×D12—28no—
113C4×D12—24no—
114C4×D12—24no—
115C4×D12—24no—
116C4×D12D418yescomplemented
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121#835—24no—
122#832—24no—
123#830—22no—
124#830—22no—
125#809—24no—
126#809—24no—
127C4⋊(C3⋊C8)—24no—
128C23×D4—31no—
129C24×C4—61no—
130C22×SD16—32no—
131C22×D8—32no—
132C2×#124C2×D311yescharacteristic, complemented
133C2×C22⋊Q8C2×D311yescharacteristic
134C2×#121C2×D311yescharacteristic
135C2×C4×D4—32no—
136C2×C4×D4—32no—
137C2×C4×D4—68no—
138C2×C4×D4—61no—
139C2×C4×D4—61no—
140C22×C4⋊C4—31no—
141C22×C22⋊C4—61no—
142C22×C42C2×D311yescharacteristic
143#463—68no—
144#450—64no—
145#447—64no—
146C2×C4⋊C8—32no—
147C2×#103—32no—
148C2×#102—32no—
149C2×#102—32no—
150C2×#102—32no—
151C2×C22⋊C8—31no—
152C2×C22⋊C8—31no—
153#391C2×D311yescharacteristic, complemented
154#390C2×D311yescharacteristic, complemented
155#388—21no—
156#388—21no—
157#386—21no—
158#386—21no—
159#382C2×D311yescharacteristic, complemented
160#340—32no—
161#340—32no—
162#336—32no—
163C24×C3—21no—
164C24×C3C2×D411yescharacteristic
165C23×D3—24no—
166C23×D3—24no—
167C23×D3—24no—
168C23×D3—22no—
169C23×D3—42no—
170C23×D3—42no—
171C23×D3—42no—
172C23×D3—42no—
173C2×C3×Q8—22no—
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
176C2×C3×Q8—21no—
177C2×C3×D4—22no—
178C2×C3×D4—21no—
179C2×C3×D4—22no—
180C2×C3×D4—21no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C2×C3×D4—42no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—41no—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4#4112yes—
202C22×C3×C4C2411yescharacteristic
203C22×Dic3—24no—
204C22×Dic3—22no—
205C22×Dic3—42no—
206C22×Dic3—42no—
207C2×D12—24no—
208C2×D12—24no—
209C2×D12—216no—
210C2×D12—24no—
211C2×D12—22no—
212C2×D12—22no—
213C2×D12—22no—
214C2×D12—216no—
215C2×D12C2×D414yes—
216C2×D12C2×D414yes—
217C2×D12#4114yes—
218C2×C4×D3—28no—
219C2×C4×D3—28no—
220C2×C4×D3—432no—
221C2×C4×D3—44no—
222C2×C4×D3—44no—
223C2×C4×D3—44no—
224C2×C4×D3—44no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—42no—
230C3×C22⋊C4—42no—
231C3×C22⋊C4—42no—
232C3×C22⋊C4—42no—
233C3×C22⋊C4—42no—
234C3×C22⋊C4—42no—
235C3×C22⋊C4—42no—
236C3×C42—22no—
237C3×C42—22no—
238C3×C42—22no—
239C3×C42C2×D414yes—
240#215—48no—
241#213—48no—
242(C2×D3)⋊C4—28no—
243(C2×D3)⋊C4—28no—
244C4⋊Dic3—24no—
245C4⋊Dic3—22no—
246C4⋊Dic3C2×D414yes—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C25—61no—
254C22×Q8C22×D311yescharacteristic
255C22×D4—21no—
256C22×D4—34no—
257C22×D4—32no—
258C22×D4—64no—
259C22×D4—64no—
260C22×D4—62no—
261C22×D4—61no—
262C22×D4C22×D311yescharacteristic
263C23×C4—68no—
264C23×C4—61no—
265C23×C4—62no—
266C23×C4—62no—
267C23×C4—124no—
268C23×C4—124no—
269C23×C4#6711yescharacteristic
270C23×C4#6711yescharacteristic
271C23×C4C22×D311yescharacteristic
272C2×SD16—64no—
273C2×SD16—64no—
274C2×SD16—64no—
275C2×D8—64no—
276C2×D8—64no—
277C2×D8—64no—
278C22×C8—32no—
279C22×C8—32no—
280#124—24no—
281C22⋊Q8—22no—
282C22⋊Q8—22no—
283#121—22no—
284#121—22no—
285C4×D4—38no—
286C4×D4—64no—
287C4×D4—64no—
288C4×D4—68no—
289C4×D4—68no—
290C4×D4—64no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—32no—
293C2×C4⋊C4—64no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—68no—
304C2×C22⋊C4—62no—
305C2×C22⋊C4—62no—
306C2×C42—24no—
307C2×C42#6711yescharacteristic
308C2×C42#6711yescharacteristic
309C2×C42C22×D311yescharacteristic
310C2×C42C22×D311yescharacteristic
311C4⋊C8—64no—
312#103—64no—
313#102—64no—
314#102—64no—
315#102—64no—
316C22⋊C8—62no—
317C22⋊C8—62no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3#12112yes—
334C23×C3#13512yes—
335C23×C3AGL(1,ℤ/8)12yes—
336C23×C3C22×D411yescharacteristic
337C22×D3—216no—
338C22×D3—24no—
339C22×D3—24no—
340C22×D3—24no—
341C22×D3—48no—
342C22×D3—48no—
343C22×D3—48no—
344C22×D3—48no—
345C22×D3—48no—
346C22×D3—42no—
347C22×D3—42no—
348C22×D3—42no—
349C22×D3—42no—
350C22×D3—48no—
351C3×Q8—42no—
352C3×Q8—42no—
353C3×D4—42no—
354C3×D4—42no—
355C3×D4—84no—
356C2×C3×C4—28no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—44no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4#12012yes—
381C2×C3×C4#12012yes—
382C2×C3×C4#12112yes—
383C2×C3×C4AGL(1,ℤ/8)12yes—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C22×D412yes—
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4112yes—
388C2×Dic3—24no—
389C2×Dic3—48no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392D12—216no—
393D12—24no—
394D12—24no—
395D12—24no—
396D12#12118yes—
397C4×D3—416no—
398C4×D3—416no—
399C3⋊C8—42no—
400C3⋊C8—42no—
401C24—21no—
402C24—64no—
403C24—64no—
404C24—64no—
405C24—62no—
406C24—122no—
407C24—122no—
408C24—122no—
409C24—122no—
410C24C2×#6711yescharacteristic
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—34no—
420C2×D4—34no—
421C2×D4—34no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—616no—
426C2×D4—64no—
427C2×D4—64no—
428C2×D4—616no—
429C2×D4—64no—
430C2×D4—62no—
431C2×D4—62no—
432C2×D4—62no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—41no—
444C22×C4—68no—
445C22×C4—64no—
446C22×C4—62no—
447C22×C4—68no—
448C22×C4—1232no—
449C22×C4—124no—
450C22×C4—124no—
451C22×C4—124no—
452C22×C4—122no—
453C22×C4—122no—
454C22×C4—124no—
455C22×C4D3×D412yes—
456C22×C4#23712yes—
457C22×C4C2×#6711yescharacteristic
458C22×C4C2×#6711yescharacteristic
459C22×C4C2×#6711yescharacteristic
460C22×C4C2×#6711yescharacteristic
461C22×C4C2×#6711yescharacteristic
462C22×C4C23×D311yescharacteristic
463SD16—128no—
464D8—128no—
465C2×C8—62no—
466C2×C8—62no—
467C2×C8—62no—
468C2×C8—62no—
469C2×C8—62no—
470C2×C8—62no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—34no—
476C4⋊C4—42no—
477C4⋊C4—64no—
478C4⋊C4—62no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—68no—
486C22⋊C4—68no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42D3×D414yes—
491C22×C3—22no—
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3—81no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3#46314yescomplemented
505C22×C3C2×#12011yescharacteristic
506C22×C3C2×#12111yescharacteristic
507C22×C3#52511yescharacteristic
508C22×C3C2×#13511yescharacteristic
509C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
510C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
511C22×C3#57511yescharacteristic
512C2×D3—28no—
513C2×D3—432no—
514C2×D3—44no—
515C2×D3—44no—
516C2×D3—44no—
517C2×D3—44no—
518C2×D3—44no—
519C2×D3—44no—
520C3×C4—22no—
521C3×C4—22no—
522C3×C4—22no—
523C3×C4—22no—
524C3×C4—41no—
525C3×C4—81no—
526C3×C4—81no—
527C3×C4—81no—
528C3×C4C2×#12112yes—
529C3×C4#52512yes—
530Dic3—44no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—616no—
543C23—64no—
544C23—64no—
545C23—64no—
546C23—128no—
547C23—128no—
548C23—128no—
549C23—128no—
550C23—128no—
551C23—122no—
552C23—122no—
553C23—122no—
554C23—122no—
555C23—128no—
556C23#93812yes—
557C23#94912yes—
558C23#95012yes—
559C23C22×#6711yescharacteristic
560Q8—42no—
561Q8—42no—
562D4—38no—
563D4—42no—
564D4—42no—
565D4—616no—
566D4—64no—
567D4—64no—
568D4—64no—
569D4—84no—
570C2×C4—28no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—64no—
595C2×C4—1216no—
596C2×C4—1216no—
597C2×C4—128no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4#93712yes—
601C2×C4#93712yes—
602C2×C4#93812yes—
603C2×C4#94912yes—
604C2×C4C2×D3×D411yescharacteristic
605C2×C4C2×D3×D412yes—
606C2×C4C2×#23711yescharacteristic
607C2×C4D3×#4112yes—
608C8—122no—
609C8—122no—
610C2×C3—24no—
611C2×C3—161no—
612C2×C3#284711yescharacteristic
613C2×C3#300211yescharacteristic
614C2×C3#300311yescharacteristic
615C2×C3C2×#46311yescharacteristic
616C2×C3#303611yescharacteristic
617C2×C3#310511yescharacteristic
618C2×C3#311411yescharacteristic
619D3—48no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—68no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—1232no—
635C22—124no—
636C22—124no—
637C22—124no—
638C22—124no—
639C22—124no—
640C22—124no—
641C22#505714yescomplemented
642C22C2×#93711yescharacteristic
643C22C2×#93811yescharacteristic
644C22C2×#94911yescharacteristic
645C22C2×#94911yescharacteristic
646C22C2×#95011yescharacteristic
647C22#582211yescharacteristic
648C22#582811yescharacteristic
649C4—22no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—41no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4—124no—
658C4D3×#12112yes—
659C4#560612yes—
660C3#3290811yescharacteristic, complemented, Sylow
661C2—24no—
662C2—128no—
663C2—161no—
664C2#8272311yescharacteristic
665C2#8292611yescharacteristic
666C2#8292711yescharacteristic
667C2C2×#505711yescharacteristic
668C2#8314111yescharacteristic
669C2#8314311yescharacteristic
670C2#8329711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2850C2
socleC23×C3C22×D4
prime radicalC2×#5036C2
hypercenter#2850D3
hypocenterC3#32908
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2850

Complements

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

NH ≅ G/Ntype
1. C22×C4×D12C2semidirect
2. C3×#2850C2semidirect
3. C2×#5057C2semidirect
4. C2×#5046C2semidirect
6. #71692C2semidirect
7. #71677C2semidirect
8. #71530C2semidirect
9. #69687C2semidirect
10. #69657C2semidirect
11. #69556C2semidirect
15. C2×C3×#124C22semidirect
28. C3×#391C22semidirect
29. C3×#390C22semidirect
34. C3×#382C22semidirect
35. C2×#835C22semidirect
36. C2×#832C22semidirect
44. C2×C4⋊(C3⋊C8)C22semidirect
45. #4529C22semidirect
46. #4529C22semidirect
47. #4524C22semidirect
52. #2850D3semidirect
116. C4×D12D4semidirect
132. C2×#124C2×D3semidirect
153. #391C2×D3semidirect
154. #390C2×D3semidirect
159. #382C2×D3semidirect
504. C22×C3#463semidirect
641. C22#5057semidirect
660. C3#32908semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D122
C3×#28502
C2×#50572
C2×#50462
C2×#50362
#716922
#716772
#715302
#696872
#696572
#695562
#329083