Finite Group Databasebeta

← #10944037

Subgroups — #10944037

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
2#82077C211yesmaximal, characteristic, complemented
3C2×#4844C212yesmaximal, complemented
4C2×#4731C211yesmaximal, characteristic, complemented
5C2×#4732C211yesmaximal, characteristic
6C3×#1797C211yesmaximal, characteristic, complemented
7#70480C212yesmaximal, complemented
8#69932C211yesmaximal, characteristic, complemented
9#69930C211yesmaximal, characteristic, complemented
10#68673C212yesmaximal, complemented
11#68672C212yesmaximal, complemented
12#32908—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C23×C4×D3—21no—
15C22×C3×C42C2211yescharacteristic
16C2×#930C2211yescharacteristic
17C22×(C2×D3)⋊C4—21no—
18C22×C4⋊Dic3C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C22×D24C2212yes—
21C22×#204C2212yes—
22C2×#875C2211yescharacteristic, complemented
23C2×C4×D12—21no—
24C2×C4×D12—21no—
25C2×C4×D12—28no—
26C2×C4×D12C2212yes—
27C2×C4×D12C2212yes—
28C2×C3×C4⋊C8C2212yescomplemented
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31#5213—21no—
32#5213—21no—
33C2×#821C2212yes—
34C2×#821C2212yes—
35C2×#821C2212yescomplemented
36C2×#816C2212yescomplemented
37#4940—21no—
38#4940—21no—
39#4939C2211yescharacteristic, complemented
40#4936C2211yescharacteristic, complemented
41#4929C2211yescharacteristic, complemented
42#4844—28no—
43#4732—24no—
44#4731—24no—
45C3×#336C2212yescomplemented
46#4551C2212yescomplemented
47#4551C2212yescomplemented
48C22×C4×D4—31no—
49C2×#463—32no—
50C2×#450—31no—
51C2×#447—31no—
52#2850—31no—
53#1879—32no—
54#1877—32no—
55#1874—31no—
56#1872—31no—
57#1804—32no—
58#1797D311yescharacteristic, complemented
59C24×D3—21no—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×#67—21no—
64C23×Dic3—21no—
65C22×D12—21no—
66C22×D12—22no—
67C22×D12—24no—
68C22×D12—24no—
69C22×D12D414yes—
70C22×D12C2311yescharacteristic
71C22×D12C2312yes—
72C22×C4×D3—28no—
73C22×C4×D3—22no—
74C22×C4×D3—22no—
75C22×C4×D3—44no—
76C22×C4×D3—44no—
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2312yes—
79C22×C3×C8C2312yes—
80C2×C3×C42—24no—
81C2×C3×C42D411yescharacteristic
82C2×C3×C42D411yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×#217—21no—
86#930—22no—
87#930—22no—
88C2×(C2×D3)⋊C4—28no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—24no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2312yes—
101#924—22no—
102#924—22no—
103C2×Dic3⋊C4—21no—
104C2×D24—24no—
105C2×D24—24no—
106C2×D24—24no—
107C2×#204—24no—
108C2×#204—24no—
109C2×#204—24no—
110#875—24no—
111C4×D12—28no—
112C4×D12—28no—
113C4×D12—24no—
114C4×D12—24no—
115C4×D12—24no—
116C4×D12D418yescomplemented
117C3×C4⋊C8—24no—
118C3×C22⋊C8—22no—
119C3×C22⋊C8—22no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124#821—24no—
125#821—24no—
126#821—24no—
127#816—24no—
128C23×D4—31no—
129C24×C4—61no—
130C22×SD16—32no—
131C22×D8—32no—
132C2×#124—31no—
133C2×C22⋊Q8—31no—
134C2×#121—31no—
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⋊C8C2×D312yescomplemented
147C2×#103—32no—
148C2×#102—32no—
149C2×#102—32no—
150C2×#102—32no—
151C2×C22⋊C8C2×D311yescharacteristic
152C2×C22⋊C8C2×D311yescharacteristic
153#391—31no—
154#390—31no—
155#388—61no—
156#388—61no—
157#386—61no—
158#386—61no—
159#382—31no—
160#340—32no—
161#340—32no—
162#336C2×D312yescomplemented
163C24×C3C2×D411yescharacteristic
164C23×D3—24no—
165C23×D3—24no—
166C23×D3—24no—
167C23×D3—22no—
168C23×D3—21no—
169C23×D3—42no—
170C23×D3—42no—
171C23×D3—42no—
172C23×D3—42no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—41no—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D412yes—
187C22×C3×C4#4112yes—
188C22×C3×C4C2411yescharacteristic
189C2×#67—42no—
190C2×#67—42no—
191C2×#67—42no—
192C22×Dic3—24no—
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—42no—
198C22×Dic3—42no—
199C2×D12—24no—
200C2×D12—24no—
201C2×D12—216no—
202C2×D12—24no—
203C2×D12—22no—
204C2×D12—22no—
205C2×D12—22no—
206C2×D12—216no—
207C2×D12—22no—
208C2×D12—21no—
209C2×D12—22no—
210C2×D12—21no—
211C2×D12C2×D414yes—
212C2×D12C2×D414yes—
213C2×D12#4114yes—
214C2×C4×D3—28no—
215C2×C4×D3—28no—
216C2×C4×D3—432no—
217C2×C4×D3—44no—
218C2×C4×D3—44no—
219C2×C4×D3—44no—
220C2×C4×D3—44no—
221C2×Dic6—22no—
222C2×Dic6—21no—
223C2×Dic6—22no—
224C2×Dic6—21no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C3×C42—22no—
232C3×C42—22no—
233C3×C42—22no—
234C3×C42C2×D414yes—
235#217—42no—
236(C2×D3)⋊C4—28no—
237(C2×D3)⋊C4—28no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—42no—
241(C2×D3)⋊C4—42no—
242(C2×D3)⋊C4—42no—
243C4⋊Dic3—24no—
244C4⋊Dic3—22no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3C2×D414yes—
250Dic3⋊C4—42no—
251D24—48no—
252#204—48no—
253C25—61no—
254C22×Q8—31no—
255C22×D4—34no—
256C22×D4—32no—
257C22×D4—31no—
258C22×D4—64no—
259C22×D4—64no—
260C22×D4—61no—
261C22×D4—61no—
262C22×D4—62no—
263C23×C4—68no—
264C23×C4—61no—
265C23×C4—62no—
266C23×C4—62no—
267C23×C4—124no—
268C23×C4—124no—
269C23×C4D1211yescharacteristic
270C23×C4D1211yescharacteristic
271C23×C4C22×D311yescharacteristic
272C2×SD16—64no—
273C2×SD16—64no—
274C2×SD16—64no—
275C2×D8—64no—
276C2×D8—64no—
277C2×D8—64no—
278C22×C8C22×D312yes—
279C22×C8C22×D312yes—
280#124—64no—
281C22⋊Q8—62no—
282C22⋊Q8—62no—
283#121—62no—
284#121—62no—
285C4×D4—38no—
286C4×D4—64no—
287C4×D4—64no—
288C4×D4—68no—
289C4×D4—68no—
290C4×D4—64no—
291C2×C4⋊C4—32no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—64no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C22⋊C4—68no—
298C2×C22⋊C4—62no—
299C2×C22⋊C4—62no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C42—24no—
307C2×C42D1211yescharacteristic
308C2×C42D1211yescharacteristic
309C2×C42C22×D311yescharacteristic
310C2×C42C22×D311yescharacteristic
311C4⋊C8—24no—
312#103—64no—
313#102—64no—
314#102—64no—
315#102—64no—
316C22⋊C8—22no—
317C22⋊C8—22no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3#12112yes—
327C23×C3#13512yes—
328C23×C3AGL(1,ℤ/8)12yes—
329C23×C3C22×D411yescharacteristic
330C22×D3—216no—
331C22×D3—24no—
332C22×D3—24no—
333C22×D3—24no—
334C22×D3—48no—
335C22×D3—48no—
336C22×D3—48no—
337C22×D3—48no—
338C22×D3—48no—
339C22×D3—42no—
340C22×D3—42no—
341C22×D3—42no—
342C22×D3—42no—
343C22×D3—48no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C2×C3×C4—28no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—24no—
361C2×C3×C4—24no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—44no—
366C2×C3×C4#12012yes—
367C2×C3×C4#12012yes—
368C2×C3×C4#12112yes—
369C2×C3×C4AGL(1,ℤ/8)12yes—
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D412yes—
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4112yes—
374#67—84no—
375C2×Dic3—24no—
376C2×Dic3—48no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388D12—216no—
389D12—24no—
390D12—24no—
391D12—24no—
392D12—42no—
393D12—42no—
394D12#12118yes—
395C4×D3—416no—
396C4×D3—416no—
397Dic6—42no—
398Dic6—42no—
399C3×C8—42no—
400C3×C8—42no—
401C24—64no—
402C24—64no—
403C24—64no—
404C24—62no—
405C24—61no—
406C24—122no—
407C24—122no—
408C24—122no—
409C24—122no—
410C24C2×D1211yescharacteristic
411C2×Q8—62no—
412C2×Q8—61no—
413C2×Q8—61no—
414C2×Q8—62no—
415C2×D4—34no—
416C2×D4—34no—
417C2×D4—34no—
418C2×D4—616no—
419C2×D4—64no—
420C2×D4—64no—
421C2×D4—616no—
422C2×D4—64no—
423C2×D4—62no—
424C2×D4—62no—
425C2×D4—61no—
426C2×D4—62no—
427C2×D4—61no—
428C2×D4—62no—
429C2×D4—62no—
430C2×D4—122no—
431C2×D4—122no—
432C2×D4—122no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—41no—
441C22×C4—68no—
442C22×C4—64no—
443C22×C4—62no—
444C22×C4—68no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—1232no—
449C22×C4—124no—
450C22×C4—124no—
451C22×C4—124no—
452C22×C4—122no—
453C22×C4—122no—
454C22×C4—124no—
455C22×C4C2×D1211yescharacteristic
456C22×C4C2×D1211yescharacteristic
457C22×C4C2×D1211yescharacteristic
458C22×C4C2×D1211yescharacteristic
459C22×C4C2×D1211yescharacteristic
460C22×C4D3×D412yes—
461C22×C4#23912yes—
462C22×C4C23×D311yescharacteristic
463SD16—128no—
464D8—128no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C4⋊C4—34no—
472C4⋊C4—64no—
473C4⋊C4—62no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—122no—
479C22⋊C4—68no—
480C22⋊C4—68no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
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#46314yescomplemented
498C22×C3C2×#12011yescharacteristic
499C22×C3C2×#12111yescharacteristic
500C22×C3#52511yescharacteristic
501C22×C3C2×#13511yescharacteristic
502C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
503C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
504C22×C3#57511yescharacteristic
505C2×D3—28no—
506C2×D3—432no—
507C2×D3—44no—
508C2×D3—44no—
509C2×D3—44no—
510C2×D3—44no—
511C2×D3—44no—
512C2×D3—44no—
513C2×D3—81no—
514C2×D3—81no—
515C2×D3—81no—
516C2×D3—81no—
517C2×D3—81no—
518C2×D3—81no—
519C2×D3—81no—
520C3×C4—22no—
521C3×C4—22no—
522C3×C4—22no—
523C3×C4—22no—
524C3×C4—41no—
525C3×C4C2×#12112yes—
526C3×C4#52512yes—
527Dic3—44no—
528Dic3—81no—
529Dic3—81no—
530Dic3—81no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—616no—
536C23—64no—
537C23—64no—
538C23—64no—
539C23—128no—
540C23—128no—
541C23—128no—
542C23—128no—
543C23—128no—
544C23—122no—
545C23—122no—
546C23—122no—
547C23—122no—
548C23—128no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23#89512yes—
557C23#90412yes—
558C23#90812yes—
559C23C22×D1211yescharacteristic
560Q8—122no—
561Q8—122no—
562D4—38no—
563D4—616no—
564D4—64no—
565D4—64no—
566D4—64no—
567D4—122no—
568D4—122no—
569D4—244no—
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—64no—
586C2×C4—1216no—
587C2×C4—1216no—
588C2×C4—128no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4#88212yes—
601C2×C4#88212yes—
602C2×C4#89512yes—
603C2×C4#90812yes—
604C2×C4C2×D3×D411yescharacteristic
605C2×C4C2×D3×D412yes—
606C2×C4C2×#23911yescharacteristic
607C2×C4D3×#4112yes—
608C8—42no—
609C8—42no—
610C2×C3—24no—
611C2×C3#284711yescharacteristic
612C2×C3#300211yescharacteristic
613C2×C3#300311yescharacteristic
614C2×C3C2×#46311yescharacteristic
615C2×C3#303611yescharacteristic
616C2×C3#310511yescharacteristic
617C2×C3#311411yescharacteristic
618D3—48no—
619D3—161no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—68no—
627C22—1232no—
628C22—124no—
629C22—124no—
630C22—124no—
631C22—124no—
632C22—124no—
633C22—124no—
634C22—241no—
635C22—241no—
636C22—241no—
637C22—241no—
638C22—241no—
639C22—241no—
640C22—241no—
641C22#484414yescomplemented
642C22C2×#88211yescharacteristic
643C22C2×#89511yescharacteristic
644C22#552411yescharacteristic
645C22C2×#90411yescharacteristic
646C22C2×#90811yescharacteristic
647C22C2×#90811yescharacteristic
648C22#574811yescharacteristic
649C4—22no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—41no—
654C4—124no—
655C4—241no—
656C4—241no—
657C4—241no—
658C4D3×#12112yes—
659C4#562712yes—
660C3#3290811yescharacteristic, complemented, Sylow
661C2—24no—
662C2—128no—
663C2—481no—
664C2#7404811yescharacteristic
665C2#7861011yescharacteristic
666C2#7863711yescharacteristic
667C2C2×#484411yescharacteristic
668C2#7905511yescharacteristic
669C2#7930011yescharacteristic
670C2#7934611yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1797C2
socleC23×C3C22×D4
prime radicalC2×#4732C2
hypercenter#1797D3
hypocenterC3#32908
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
1. C22×C4×D12C2semidirect
2. #82077C2semidirect
3. C2×#4844C2semidirect
4. C2×#4731C2semidirect
6. C3×#1797C2semidirect
7. #70480C2semidirect
8. #69932C2semidirect
9. #69930C2semidirect
10. #68673C2semidirect
11. #68672C2semidirect
22. C2×#875C22semidirect
28. C2×C3×C4⋊C8C22semidirect
35. C2×#821C22semidirect
36. C2×#816C22semidirect
39. #4939C22semidirect
40. #4936C22semidirect
41. #4929C22semidirect
45. C3×#336C22semidirect
46. #4551C22semidirect
47. #4551C22semidirect
58. #1797D3semidirect
116. C4×D12D4semidirect
146. C2×C4⋊C8C2×D3semidirect
162. #336C2×D3semidirect
497. C22×C3#463semidirect
641. C22#4844semidirect
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
#820772
C2×#48442
C2×#47312
C2×#47322
C3×#17972
#704802
#699322
#699302
#686732
#686722
#329083