Finite Group Databasebeta

← #10944035

Subgroups — #10944035

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×C3×C4×D4C211yesmaximal, characteristic, complemented
2C2×#5218C211yesmaximal, characteristic, complemented
3C2×#5217C211yesmaximal, characteristic
4C2×#5016C212yesmaximal, complemented
5#82077C211yesmaximal, characteristic, complemented
6#71613C212yesmaximal, complemented
7#71612C212yesmaximal, complemented
8#71605C212yesmaximal, complemented
9#71560C211yesmaximal, characteristic, complemented
10#71559C211yesmaximal, characteristic, complemented
11#71530C211yesmaximal, characteristic, complemented
12#32908—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C24×C3×C4—21no—
15C2×C3×C4×D4—28no—
16C2×C3×C4×D4—21no—
17C2×C3×C4×D4—21no—
18C2×C3×C4×D4C2212yes—
19C2×C3×C4×D4C2212yes—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4—21no—
22C22×C3×C42C2211yescharacteristic
23C22×#214C2212yes—
24C22×#213C2212yes—
25C2×#930C2211yescharacteristic
26C2×#924C2211yescharacteristic
27C2×#875C2211yescharacteristic, complemented
28#5218—24no—
29#5217—24no—
30C2×#832C2212yes—
31C2×#832C2212yes—
32#5213—21no—
33#5213—21no—
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic
36#5016—28no—
37C2×#810C2212yescomplemented
38C2×#809C2212yescomplemented
39#4940—21no—
40#4940—21no—
41#4939C2211yescharacteristic, complemented
42#4936C2211yescharacteristic, complemented
43#4929C2211yescharacteristic, complemented
44C2×C4⋊(C3⋊C8)C2212yescomplemented
45#4529C2212yescomplemented
46#4529C2212yescomplemented
47#4524C2212yescomplemented
48C22×C4×D4D311yescharacteristic, complemented
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#1797—31no—
59C25×C3—21no—
60C22×C3×D4—24no—
61C22×C3×D4—24no—
62C22×C3×D4—21no—
63C22×C3×D4—22no—
64C22×C3×D4D414yes—
65C22×C3×D4C2312yes—
66C23×C3×C4—28no—
67C23×C3×C4—21no—
68C23×C3×C4—22no—
69C23×C3×C4—22no—
70C23×C3×C4—44no—
71C23×C3×C4—44no—
72C23×C3×C4D411yescharacteristic
73C23×C3×C4D411yescharacteristic
74C23×C3×C4C2311yescharacteristic
75C22×#67—21no—
76C22×D12C2311yescharacteristic
77C22×Dic6C2311yescharacteristic
78C3×C4×D4—28no—
79C3×C4×D4—28no—
80C3×C4×D4—24no—
81C3×C4×D4—24no—
82C3×C4×D4—24no—
83C3×C4×D4D418yescomplemented
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C22⋊C4—28no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C42—24no—
91C2×C3×C42D411yescharacteristic
92C2×C3×C42D411yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#214—24no—
97C2×#214—24no—
98C2×#214—24no—
99C2×#213—24no—
100C2×#213—24no—
101C2×#213—24no—
102#930—22no—
103#930—22no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111#924—22no—
112#924—22no—
113C2×Dic3⋊C4—21no—
114C22×C3⋊C8C2312yes—
115C22×C3⋊C8C2312yes—
116#875—24no—
117#832—24no—
118#832—24no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123#830—22no—
124#830—22no—
125#810—24no—
126#809—24no—
127C4⋊(C3⋊C8)—24no—
128C23×D4C2×D311yescharacteristic
129C24×C4—21no—
130C22×SD16—32no—
131C22×D8—32no—
132C2×#124—31no—
133C2×C22⋊Q8—31no—
134C2×#121—31no—
135C2×C4×D4—28no—
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4C2×D312yes—
139C2×C4×D4C2×D312yes—
140C22×C4⋊C4C2×D311yescharacteristic
141C22×C22⋊C4—21no—
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#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#336—32no—
163C24×C3—24no—
164C24×C3—24no—
165C24×C3—24no—
166C24×C3—22no—
167C24×C3—42no—
168C24×C3—42no—
169C24×C3—42no—
170C24×C3—42no—
171C24×C3C2×D411yescharacteristic
172C23×D3—21no—
173C2×C3×D4—24no—
174C2×C3×D4—24no—
175C2×C3×D4—216no—
176C2×C3×D4—216no—
177C2×C3×D4—24no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4C2×D414yes—
182C2×C3×D4C2×D414yes—
183C2×C3×D4#4114yes—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—24no—
190C22×C3×C4—28no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—28no—
195C22×C3×C4—41no—
196C22×C3×C4—432no—
197C22×C3×C4—44no—
198C22×C3×C4—44no—
199C22×C3×C4—44no—
200C22×C3×C4—42no—
201C22×C3×C4—42no—
202C22×C3×C4—44no—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4#4112yes—
210C22×C3×C4C2411yescharacteristic
211C2×#67—42no—
212C2×#67—42no—
213C2×#67—42no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C2×D12—22no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—22no—
221C2×Dic6—22no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—22no—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4C2×D414yes—
228C3×C22⋊C4—28no—
229C3×C22⋊C4—28no—
230C3×C42—22no—
231C3×C42—22no—
232C3×C42—22no—
233C3×C42C2×D414yes—
234#217—42no—
235#214—48no—
236#213—48no—
237(C2×D3)⋊C4—42no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—42no—
241(C2×D3)⋊C4—42no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246Dic3⋊C4—42no—
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—21no—
254C22×Q8—31no—
255C22×D4—24no—
256C22×D4—24no—
257C22×D4—22no—
258C22×D4—21no—
259C22×D4—31no—
260C22×D4—61no—
261C22×D4D1214yes—
262C22×D4C22×D312yes—
263C23×C4—28no—
264C23×C4—21no—
265C23×C4—22no—
266C23×C4—22no—
267C23×C4—44no—
268C23×C4—44no—
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—64no—
281C22⋊Q8—62no—
282C22⋊Q8—62no—
283#121—62no—
284#121—62no—
285C4×D4—28no—
286C4×D4—28no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4#6718yescomplemented
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4C22×D312yes—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
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×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—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—216no—
327C23×C3—24no—
328C23×C3—24no—
329C23×C3—24no—
330C23×C3—48no—
331C23×C3—48no—
332C23×C3—48no—
333C23×C3—48no—
334C23×C3—48no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3—48no—
340C23×C3#12112yes—
341C23×C3#13512yes—
342C23×C3AGL(1,ℤ/8)12yes—
343C23×C3C22×D411yescharacteristic
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C3×D4—216no—
352C3×D4—24no—
353C3×D4—24no—
354C3×D4—24no—
355C3×D4#12118yes—
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—24no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—44no—
372C2×C3×C4—416no—
373C2×C3×C4—416no—
374C2×C3×C4—48no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4#12012yes—
378C2×C3×C4#12012yes—
379C2×C3×C4#12112yes—
380C2×C3×C4AGL(1,ℤ/8)12yes—
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D412yes—
383C2×C3×C4C2×#4112yes—
384C2×C3×C4C2×#4111yescharacteristic
385#67—84no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395D12—42no—
396D12—42no—
397Dic6—42no—
398Dic6—42no—
399C3⋊C8—42no—
400C3⋊C8—42no—
401C24—24no—
402C24—24no—
403C24—24no—
404C24—22no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24—61no—
410C24C2×#6711yescharacteristic
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8—62no—
414C2×Q8—62no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—216no—
418C2×D4—216no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—62no—
426C2×D4—62no—
427C2×D4—122no—
428C2×D4—122no—
429C2×D4—122no—
430C2×D4C2×D1214yes—
431C2×D4#23514yes—
432C2×D4C2×#6714yes—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—41no—
445C22×C4—432no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4C2×D1212yes—
456C22×C4#23512yes—
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—24no—
472C4⋊C4—22no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—122no—
478C4⋊C4C2×#6714yes—
479C22⋊C4—28no—
480C22⋊C4—28no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42C2×#6714yes—
491C22×C3—22no—
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—28no—
498C22×C3—432no—
499C22×C3—44no—
500C22×C3—44no—
501C22×C3—44no—
502C22×C3—44no—
503C22×C3—44no—
504C22×C3—44no—
505C22×C3#46314yescomplemented
506C22×C3C2×#12011yescharacteristic
507C22×C3C2×#12111yescharacteristic
508C22×C3#52511yescharacteristic
509C22×C3C2×#13511yescharacteristic
510C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
511C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
512C22×C3#57511yescharacteristic
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×C4—44no—
526C3×C4C2×#12112yes—
527C3×C4#52512yes—
528Dic3—81no—
529Dic3—81no—
530Dic3—81no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—216no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—48no—
544C23—48no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23#93012yes—
557C23#93212yes—
558C23#95012yes—
559C23C22×#6711yescharacteristic
560Q8—122no—
561Q8—122no—
562D4—216no—
563D4—24no—
564D4—24no—
565D4—24no—
566D4—122no—
567D4—122no—
568D4—244no—
569D4#93018yes—
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—24no—
585C2×C4—44no—
586C2×C4—416no—
587C2×C4—416no—
588C2×C4—48no—
589C2×C4—42no—
590C2×C4—42no—
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#93012yes—
601C2×C4#93212yes—
602C2×C4#95312yes—
603C2×C4#95312yes—
604C2×C4C22×D1211yescharacteristic
605C2×C4C2×#23512yes—
606C2×C4C2×#23511yescharacteristic
607C2×C4C22×#6712yes—
608C8—122no—
609C8—122no—
610C2×C3—24no—
611C2×C3—48no—
612C2×C3#284711yescharacteristic
613C2×C3#300211yescharacteristic
614C2×C3#300311yescharacteristic
615C2×C3C2×#46311yescharacteristic
616C2×C3#303611yescharacteristic
617C2×C3#310511yescharacteristic
618C2×C3#311411yescharacteristic
619D3—161no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—28no—
627C22—432no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—241no—
635C22—241no—
636C22—241no—
637C22—241no—
638C22—241no—
639C22—241no—
640C22—241no—
641C22#501614yescomplemented
642C22C2×#93011yescharacteristic
643C22#579111yescharacteristic
644C22C2×#93211yescharacteristic
645C22C2×#93211yescharacteristic
646C22C2×#95011yescharacteristic
647C22#582211yescharacteristic
648C22C2×#95311yescharacteristic
649C4—22no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—44no—
654C4—41no—
655C4—241no—
656C4—241no—
657C4—241no—
658C4C2×#93012yes—
659C4#579112yes—
660C3#3290811yescharacteristic, complemented, Sylow
661C2—24no—
662C2—48no—
663C2—481no—
664C2C2×#501611yescharacteristic
665C2#8230711yescharacteristic
666C2#8230911yescharacteristic
667C2#8231111yescharacteristic
668C2#8403211yescharacteristic
669C2#8403811yescharacteristic
670C2#8408011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×C4×D4C2
socleC23×C3C22×D4
prime radicalC2×#5217C2
hypercenterC22×C4×D4D3
hypocenterC3#32908
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×D4C2semidirect
2. C2×#5218C2semidirect
4. C2×#5016C2semidirect
5. #82077C2semidirect
6. #71613C2semidirect
7. #71612C2semidirect
8. #71605C2semidirect
9. #71560C2semidirect
10. #71559C2semidirect
11. #71530C2semidirect
27. C2×#875C22semidirect
37. C2×#810C22semidirect
38. C2×#809C22semidirect
41. #4939C22semidirect
42. #4936C22semidirect
43. #4929C22semidirect
44. C2×C4⋊(C3⋊C8)C22semidirect
45. #4529C22semidirect
46. #4529C22semidirect
47. #4524C22semidirect
48. C22×C4×D4D3semidirect
83. C3×C4×D4D4semidirect
290. C4×D4#67semidirect
505. C22×C3#463semidirect
641. C22#5016semidirect
660. C3#32908semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C2×#52182
C2×#52172
C2×#50162
#820772
#716132
#716122
#716052
#715602
#715592
#715302
#329083