Finite Group Databasebeta

← #10826931

Subgroups — #10826931

Order 768.

2252 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×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
2C22×#930C211yesmaximal, characteristic
3C2×#5227C212yesmaximal, complemented
4C2×#5046C212yesmaximal, complemented
5#72126C211yesmaximal, characteristic, complemented
6#71699C214yesmaximal, complemented
7#71692C212yesmaximal, complemented
8#71560C212yesmaximal, complemented
9#23919—31nomaximal, Sylow
10C23×C3×Q8C2211yescharacteristic, complemented
11C24×C3×C4C2211yescharacteristic
12C23×#67—21no—
13C23×D12C2211yescharacteristic
14C2×C3×C22⋊Q8—21no—
15C2×C3×C22⋊Q8—21no—
16C2×C3×C22⋊Q8—28no—
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×C22⋊Q8C2212yescomplemented
19C22×C3×C4⋊C4—21no—
20C22×C3×C4⋊C4C2211yescharacteristic, complemented
21C22×C3×C22⋊C4—21no—
22C22×#215C2214yescomplemented
23C2×#930—21no—
24C2×#930—21no—
25C2×#930—28no—
26C2×#930C2212yes—
27C2×#930C2212yes—
28C22×(C2×D3)⋊C4—21no—
29C22×C4⋊Dic3C2211yescharacteristic
30#5227—28no—
31C2×#835C2214yescomplemented
32C2×#830C2212yes—
33C2×#830C2212yescomplemented
34C2×#830C2212yescomplemented
35#5046—28no—
36C2×#809C2214yescomplemented
37#4529C2214yescomplemented
38C22×C22⋊Q8D311yescharacteristic, complemented
39C22×#121—31no—
40C2×#448—32no—
41C2×#450—32no—
42#2000—34no—
43#1874—32no—
44#1872—32no—
45#1779—31no—
46C25×C3D411yescharacteristic
47C24×D3—21no—
48C22×C3×Q8—24no—
49C22×C3×Q8—21no—
50C22×C3×Q8—24no—
51C22×C3×Q8—42no—
52C22×C3×Q8C2312yes—
53C23×C3×C4—21no—
54C23×C3×C4—21no—
55C23×C3×C4—24no—
56C23×C3×C4—24no—
57C23×C3×C4—42no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C22×#67—28no—
65C22×#67—22no—
66C22×#67—22no—
67C22×#67—44no—
68C22×#67—44no—
69C23×Dic3—21no—
70C22×D12—21no—
71C22×D12—24no—
72C22×D12—24no—
73C22×D12—42no—
74C22×D12C2312yes—
75C3×C22⋊Q8—28no—
76C3×C22⋊Q8—28no—
77C3×C22⋊Q8—24no—
78C3×C22⋊Q8—24no—
79C3×C22⋊Q8—24no—
80C3×C22⋊Q8D418yescomplemented
81C2×C3×C4⋊C4—28no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—22no—
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×#215—28no—
91C2×#215—28no—
92C2×#215—44no—
93#930—28no—
94#930—28no—
95#930—24no—
96#930—24no—
97#930—24no—
98#930D418yes—
99C2×(C2×D3)⋊C4—28no—
100C2×(C2×D3)⋊C4—22no—
101C2×(C2×D3)⋊C4—22no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—24no—
104C2×C4⋊Dic3C2312yes—
105C22×C3⋊C8C2314yes—
106#835—28no—
107#830—24no—
108#830—24no—
109#830—42no—
110#809—28no—
111C23×Q8C2×D311yescharacteristic, complemented
112C23×D4—31no—
113C23×D4—61no—
114C24×C4C2×D311yescharacteristic
115C22×SD16—34no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—28no—
119C2×C22⋊Q8C2×D312yes—
120C2×C22⋊Q8C2×D312yescomplemented
121C2×#121—32no—
122C2×#121—32no—
123C2×#121—68no—
124C2×#121—61no—
125C2×#121—61no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4C2×D311yescharacteristic, complemented
129C22×C22⋊C4—21no—
130C22×C22⋊C4—61no—
131#450—68no—
132#448—68no—
133C2×#103—34no—
134C2×#102—34no—
135C2×C22⋊C8—32no—
136C2×C22⋊C8—32no—
137C2×C22⋊C8—32no—
138#340—34no—
139C24×C3—22no—
140C24×C3—22no—
141C24×C3—22no—
142C24×C3—22no—
143C24×C3—41no—
144C24×C3—41no—
145C24×C3SD1614yescomplemented
146C24×C3C2×D411yescharacteristic
147C24×C3C2×D411yescharacteristic
148C24×C3C2×D411yescharacteristic
149C23×D3—24no—
150C23×D3—22no—
151C23×D3—42no—
152C23×D3—42no—
153C23×D3—42no—
154C23×D3—42no—
155C23×D3—42no—
156C23×D3—42no—
157C2×C3×Q8—24no—
158C2×C3×Q8—22no—
159C2×C3×Q8—24no—
160C2×C3×Q8—24no—
161C2×C3×Q8—416no—
162C2×C3×Q8—42no—
163C2×C3×Q8—42no—
164C2×C3×Q8C2×D414yescomplemented
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—24no—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
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—22no—
181C22×C3×C4—22no—
182C22×C3×C4—41no—
183C22×C3×C4—41no—
184C22×C3×C4—41no—
185C22×C3×C4—416no—
186C22×C3×C4—42no—
187C22×C3×C4—42no—
188C22×C3×C4—42no—
189C22×C3×C4—42no—
190C22×C3×C4C2×D414yes—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D414yes—
195C22×C3×C4C2411yescharacteristic
196C2×#67—28no—
197C2×#67—28no—
198C2×#67—432no—
199C2×#67—44no—
200C2×#67—44no—
201C2×#67—44no—
202C2×#67—44no—
203C22×Dic3—24no—
204C22×Dic3—22no—
205C22×Dic3—42no—
206C22×Dic3—42no—
207C2×D12—24no—
208C2×D12—24no—
209C2×D12—24no—
210C2×D12—22no—
211C2×D12—416no—
212C2×D12—42no—
213C2×D12—42no—
214C2×D12C2×D414yes—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—28no—
217C3×C4⋊C4—28no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4C2×D414yescomplemented
220C3×C22⋊C4—28no—
221C3×C22⋊C4—28no—
222#215—416no—
223(C2×D3)⋊C4—28no—
224(C2×D3)⋊C4—28no—
225C4⋊Dic3—24no—
226C4⋊Dic3—22no—
227C4⋊Dic3C2×D414yes—
228C2×C3⋊C8—24no—
229C2×C3⋊C8—24no—
230C2×C3⋊C8—24no—
231C25—61no—
232C25#6711yescharacteristic
233C22×Q8—24no—
234C22×Q8—21no—
235C22×Q8—24no—
236C22×Q8—42no—
237C22×Q8C22×D312yes—
238C22×D4—32no—
239C22×D4—64no—
240C22×D4—68no—
241C22×D4—62no—
242C22×D4—62no—
243C22×D4—61no—
244C22×D4—64no—
245C22×D4—122no—
246C22×D4—124no—
247C22×D4—124no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—24no—
251C23×C4—24no—
252C23×C4—42no—
253C23×C4—61no—
254C23×C4#6711yescharacteristic
255C23×C4#6711yescharacteristic
256C23×C4#6711yescharacteristic
257C23×C4C22×D311yescharacteristic
258C23×C4C22×D311yescharacteristic
259C23×C4C22×D311yescharacteristic
260C2×SD16—68no—
261C2×SD16—68no—
262C2×SD16—124no—
263C22×C8—34no—
264C22⋊Q8—28no—
265C22⋊Q8—28no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8#6718yescomplemented
270#121—38no—
271#121—64no—
272#121—64no—
273#121—68no—
274#121—68no—
275#121—64no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—32no—
282C2×C4⋊C4—64no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4C22×D312yes—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—68no—
289C2×C22⋊C4—62no—
290C2×C22⋊C4—62no—
291#103—68no—
292#102—68no—
293C22⋊C8—64no—
294C22⋊C8—64no—
295C22⋊C8—122no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—28no—
311C23×C3—28no—
312C23×C3—44no—
313C23×C3—44no—
314C23×C3—44no—
315C23×C3—44no—
316C23×C3—44no—
317C23×C3#12012yes—
318C23×C3#12012yes—
319C23×C3C2×SD1612yes—
320C23×C3C2×SD1612yes—
321C23×C3C2×SD1612yes—
322C23×C3AGL(1,ℤ/8)12yes—
323C23×C3#13712yes—
324C23×C3C22×D411yescharacteristic
325C22×D3—24no—
326C22×D3—48no—
327C22×D3—48no—
328C22×D3—48no—
329C22×D3—42no—
330C22×D3—42no—
331C22×D3—42no—
332C22×D3—42no—
333C22×D3—42no—
334C22×D3—42no—
335C22×D3—84no—
336C22×D3—84no—
337C22×D3—84no—
338C22×D3—84no—
339C3×Q8—28no—
340C3×Q8—48no—
341C3×Q8—44no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—28no—
347C2×C3×C4—28no—
348C2×C3×C4—21no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—44no—
354C2×C3×C4—44no—
355C2×C3×C4—44no—
356C2×C3×C4—48no—
357C2×C3×C4—48no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—44no—
363C2×C3×C4#12012yes—
364C2×C3×C4#12012yes—
365C2×C3×C4C22×D412yes—
366C2×C3×C4C22×D412yes—
367C2×C3×C4H(5,𝔽2)12yes—
368#67—416no—
369#67—416no—
370C2×Dic3—24no—
371C2×Dic3—48no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374D12—28no—
375D12—48no—
376D12—44no—
377C3⋊C8—44no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—41no—
383C24—41no—
384C24—64no—
385C24—62no—
386C24—122no—
387C24—122no—
388C24—122no—
389C24—122no—
390C24—122no—
391C24—122no—
392C24#21514yescomplemented
393C24C2×#6711yescharacteristic
394C24C2×#6711yescharacteristic
395C24C2×#6711yescharacteristic
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—416no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8C2×#6714yescomplemented
404C2×D4—34no—
405C2×D4—64no—
406C2×D4—68no—
407C2×D4—68no—
408C2×D4—62no—
409C2×D4—64no—
410C2×D4—64no—
411C2×D4—1216no—
412C2×D4—122no—
413C2×D4—1232no—
414C2×D4—124no—
415C2×D4—124no—
416C2×D4—124no—
417C2×D4—124no—
418C2×D4—122no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—416no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—64no—
445C22×C4—62no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4D3×D414yes—
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6714yes—
453C22×C4C23×D311yescharacteristic
454SD16—1216no—
455C2×C8—64no—
456C2×C8—64no—
457C2×C8—64no—
458C4⋊C4—24no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—22no—
462C4⋊C4—34no—
463C4⋊C4—64no—
464C4⋊C4—62no—
465C4⋊C4C2×#6714yescomplemented
466C22⋊C4—28no—
467C22⋊C4—28no—
468C22⋊C4—68no—
469C22⋊C4—68no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—416no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3#44814yescomplemented
491C22×C3#45014yescomplemented
492C22×C3C2×#12011yescharacteristic
493C22×C3C2×#12011yescharacteristic
494C22×C3#53411yescharacteristic
495C22×C3C22×SD1611yescharacteristic
496C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
497C22×C3C2×#13711yescharacteristic
498C22×C3#57511yescharacteristic
499C2×D3—44no—
500C2×D3—44no—
501C2×D3—44no—
502C2×D3—816no—
503C2×D3—82no—
504C2×D3—82no—
505C2×D3—82no—
506C2×D3—82no—
507C3×C4—22no—
508C3×C4—22no—
509C3×C4—41no—
510C3×C4—44no—
511C3×C4—44no—
512C3×C4D4214yes—
513Dic3—44no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—28no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—28no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—64no—
536C23—128no—
537C23—128no—
538C23—128no—
539C23—122no—
540C23—122no—
541C23—122no—
542C23—122no—
543C23—122no—
544C23—122no—
545C23—244no—
546C23—244no—
547C23—244no—
548C23—244no—
549C23#93712yes—
550C23C2×#21512yes—
551C23C2×#21512yes—
552C23C2×#21512yes—
553C23#94212yes—
554C23#94912yes—
555C23#95312yes—
556C23C22×#6711yescharacteristic
557Q8—28no—
558Q8—48no—
559Q8—44no—
560D4—68no—
561D4—128no—
562D4—1216no—
563D4—1216no—
564D4—124no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—64no—
587C2×C4—128no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4#93712yes—
591C2×C4#95312yes—
592C2×C4C2×D3×D412yes—
593C2×C4#100412yes—
594C2×C4C22×#6712yes—
595C8—124no—
596C2×C3—24no—
597C2×C3—24no—
598C2×C3—42no—
599C2×C3#240311yescharacteristic
600C2×C3C2×#45011yescharacteristic
601C2×C3C2×#44811yescharacteristic
602C2×C3#300611yescharacteristic
603C2×C3#300711yescharacteristic
604C2×C3#318711yescharacteristic
605C2×C3#319311yescharacteristic
606D3—84no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—416no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—124no—
628C22—124no—
629C22—124no—
630C22—2416no—
631C22—242no—
632C22—242no—
633C22—242no—
634C22—242no—
635C22#504614yescomplemented
636C22#522714yescomplemented
637C22C2×#93711yescharacteristic
638C22#580411yescharacteristic
639C22C22×#21511yescharacteristic
640C22C2×#94211yescharacteristic
641C22C2×#94911yescharacteristic
642C22#582211yescharacteristic
643C22C2×#95311yescharacteristic
644C4—22no—
645C4—22no—
646C4—41no—
647C4—44no—
648C4—44no—
649C4—124no—
650C4D4×#6714yes—
651C3#2391911yescharacteristic, complemented, Sylow
652C2—24no—
653C2—24no—
654C2—42no—
655C2—244no—
656C2C2×#504611yescharacteristic
657C2#8273511yescharacteristic
658C2#8273611yescharacteristic
659C2#8273811yescharacteristic
660C2#8405211yescharacteristic
661C2C2×#522711yescharacteristic
662C2#8408211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×C22⋊Q8C2
socleC23×C3C22×D4
prime radicalC22×#930C2
hypercenterC22×C22⋊Q8D3
hypocenterC3#23919
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C22⋊Q8C2semidirect
3. C2×#5227C2semidirect
4. C2×#5046C2semidirect
5. #72126C2semidirect
6. #71699C2semidirect
7. #71692C2semidirect
8. #71560C2semidirect
10. C23×C3×Q8C22semidirect
18. C2×C3×C22⋊Q8C22semidirect
20. C22×C3×C4⋊C4C22semidirect
22. C22×#215C22semidirect
31. C2×#835C22semidirect
33. C2×#830C22semidirect
34. C2×#830C22semidirect
36. C2×#809C22semidirect
37. #4529C22semidirect
38. C22×C22⋊Q8D3semidirect
80. C3×C22⋊Q8D4semidirect
111. C23×Q8C2×D3semidirect
120. C2×C22⋊Q8C2×D3semidirect
128. C22×C4⋊C4C2×D3semidirect
145. C24×C3SD16semidirect
164. C2×C3×Q8C2×D4semidirect
219. C3×C4⋊C4C2×D4semidirect
269. C22⋊Q8#67semidirect
392. C24#215semidirect
403. C2×Q8C2×#67semidirect
465. C4⋊C4C2×#67semidirect
490. C22×C3#448semidirect
491. C22×C3#450semidirect
635. C22#5046semidirect
636. C22#5227semidirect
651. C3#23919semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C22×#9302
C2×#52272
C2×#50462
#721262
#716992
#716922
#715602
#239193