Finite Group Databasebeta

← #10943978

Subgroups — #10943978

Order 768.

2008 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#447C211yesmaximal, characteristic, complemented
3C2×#5217C211yesmaximal, characteristic, complemented
4C2×#5150C212yesmaximal, complemented
5#82700C211yesmaximal, characteristic, complemented
6#72037C212yesmaximal, complemented
7#71677C211yesmaximal, characteristic, complemented
8#70172C212yesmaximal, complemented
9#70153C212yesmaximal, complemented
10#69932C211yesmaximal, characteristic, complemented
11#69908C211yesmaximal, characteristic, complemented
12#32902—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C24×Dic3—21no—
15C22×C3×D8C2212yes—
16C2×C3×#121C2211yescharacteristic
17C22×#217—21no—
18C2×#940C2211yescharacteristic, complemented
19C2×#939—21no—
20C2×#939—21no—
21C2×D4×Dic3—28no—
22C2×D4×Dic3—21no—
23C2×D4×Dic3—21no—
24C2×D4×Dic3C2212yes—
25C2×D4×Dic3C2212yes—
26C22×#213C2212yes—
27C2×#930C2211yescharacteristic
28C22×C4⋊Dic3C2211yescharacteristic
29C22×C4×Dic3C2211yescharacteristic
30C3×#447—24no—
31C2×C3×#102C2212yes—
32C2×C3×C22⋊C8C2211yescharacteristic
33#5221C2211yescharacteristic, complemented
34#5217—24no—
35C2×#832C2212yes—
36C2×#832C2212yescomplemented
37C2×#830C2211yescharacteristic
38#5150—28no—
39C2×#821C2212yescomplemented
40C2×Dic3⋊C8C2212yescomplemented
41#4988C2211yescharacteristic, complemented
42#4975C2211yescharacteristic, complemented
43#4957—21no—
44#4957—21no—
45#4551C2212yescomplemented
46#4550C2212yescomplemented
47#4529C2212yescomplemented
48C22×C4×D4—31no—
49C2×#459—32no—
50C2×#447—31no—
51C2×#447D311yescharacteristic, complemented
52#2856—31no—
53#1879—32no—
54#1879—32no—
55#1874—31no—
56#1874—31no—
57#1802—32no—
58#1797—31no—
59C25×C3—21no—
60C22×C3×D4—24no—
61C22×C3×D4—24no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4—22no—
65C22×C3×D4D414yescomplemented
66C22×C3×D4C2311yescharacteristic
67C22×C3×D4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C22×#67—21no—
70C22×#67—21no—
71C22×#67—21no—
72C23×Dic3—28no—
73C23×Dic3—21no—
74C23×Dic3—22no—
75C23×Dic3—22no—
76C23×Dic3—44no—
77C23×Dic3—44no—
78C23×Dic3D411yescharacteristic
79C23×Dic3D411yescharacteristic
80C22×D12C2311yescharacteristic
81C2×C3×D8—24no—
82C2×C3×D8—24no—
83C2×C3×D8—24no—
84C22×C3×C8C2312yes—
85C3×#121—22no—
86C3×#121—22no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×#217—28no—
90C2×#217—22no—
91C2×#217—22no—
92C2×#217—21no—
93C2×#217—21no—
94#940—24no—
95#939—44no—
96#939—44no—
97D4×Dic3—28no—
98D4×Dic3—28no—
99D4×Dic3—24no—
100D4×Dic3—24no—
101D4×Dic3—24no—
102D4×Dic3D418yescomplemented
103C2×#213—24no—
104C2×#213—24no—
105C2×#213—24no—
106#930—22no—
107#930—22no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—24no—
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2312yes—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×C4×Dic3—24no—
118C2×C4×Dic3D411yescharacteristic
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2312yes—
123C3×#102—24no—
124C3×C22⋊C8—22no—
125#832—24no—
126#832—24no—
127#830—22no—
128#821—24no—
129Dic3⋊C8—24no—
130C23×D4C2×D311yescharacteristic
131C24×C4—61no—
132C22×D8—32no—
133C22×D8C2×D312yes—
134C2×#127—31no—
135C2×#121—31no—
136C2×#121—61no—
137C2×#121—61no—
138C2×#121C2×D311yescharacteristic
139C2×C4×D4—32no—
140C2×C4×D4—32no—
141C2×C4×D4—61no—
142C2×C4×D4—61no—
143C2×C4×D4—68no—
144C22×C4⋊C4—31no—
145C22×C22⋊C4—61no—
146C22×C42—31no—
147#459—68no—
148#447—24no—
149#447—64no—
150C2×C4⋊C8—32no—
151C2×#102—32no—
152C2×#102—32no—
153C2×#102—32no—
154C2×#102C2×D312yes—
155C2×C22⋊C8—31no—
156C2×C22⋊C8C2×D311yescharacteristic
157#390—31no—
158#390—31no—
159#390—61no—
160#390—61no—
161#384—31no—
162#340—32no—
163#340—32no—
164#336—32no—
165C24×C3—24no—
166C24×C3—24no—
167C24×C3—24no—
168C24×C3—22no—
169C24×C3—21no—
170C24×C3—42no—
171C24×C3—42no—
172C24×C3—42no—
173C24×C3—42no—
174C24×C3C2×D411yescharacteristic
175C23×D3—21no—
176C2×C3×D4—24no—
177C2×C3×D4—24no—
178C2×C3×D4—216no—
179C2×C3×D4—216no—
180C2×C3×D4—24no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—21no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—21no—
187C2×C3×D4—22no—
188C2×C3×D4—42no—
189C2×C3×D4—42no—
190C2×C3×D4—42no—
191C2×C3×D4C2×D414yes—
192C2×C3×D4C2×D414yes—
193C2×C3×D4#4114yes—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—21no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D412yes—
199C22×C3×C4#4112yes—
200C22×C3×C4C2411yescharacteristic
201C2×#67—42no—
202C2×#67—42no—
203C2×#67—42no—
204C2×#67—42no—
205C2×#67—42no—
206C2×#67—42no—
207C2×#67—42no—
208C2×#67—42no—
209C2×#67—42no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3—24no—
213C22×Dic3—28no—
214C22×Dic3—22no—
215C22×Dic3—22no—
216C22×Dic3—22no—
217C22×Dic3—28no—
218C22×Dic3—21no—
219C22×Dic3—41no—
220C22×Dic3—44no—
221C22×Dic3—432no—
222C22×Dic3—42no—
223C22×Dic3—42no—
224C22×Dic3—44no—
225C22×Dic3—44no—
226C22×Dic3—44no—
227C22×Dic3D814yescomplemented
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×D12—22no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—22no—
236C3×D8—48no—
237C2×C3×C8—22no—
238C2×C3×C8—22no—
239C2×C3×C8—22no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C22⋊C4—42no—
243#217—28no—
244#217—28no—
245#217—42no—
246#217—42no—
247#213—48no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251C4⋊Dic3—24no—
252C4⋊Dic3—22no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3C2×D414yes—
256Dic3⋊C4—42no—
257Dic3⋊C4—42no—
258C4×Dic3—22no—
259C4×Dic3—22no—
260C4×Dic3—22no—
261C4×Dic3C2×D414yes—
262C2×C3⋊C8—22no—
263C2×C3⋊C8—22no—
264C2×C3⋊C8—22no—
265C25—21no—
266C22×D4—24no—
267C22×D4—24no—
268C22×D4—21no—
269C22×D4—21no—
270C22×D4—22no—
271C22×D4—31no—
272C22×D4—61no—
273C22×D4—61no—
274C22×D4—61no—
275C22×D4#6714yescomplemented
276C22×D4C22×D311yescharacteristic
277C22×D4C22×D312yes—
278C23×C4—31no—
279C23×C4—31no—
280C23×C4—62no—
281C23×C4—62no—
282C23×C4—68no—
283C23×C4—61no—
284C23×C4—124no—
285C23×C4—124no—
286C23×C4C22×D311yescharacteristic
287C2×D8—24no—
288C2×D8—24no—
289C2×D8—24no—
290C2×D8—64no—
291C2×D8—64no—
292C2×D8—64no—
293C22×C8—32no—
294C22×C8C22×D312yes—
295#127—64no—
296#121—22no—
297#121—22no—
298#121—62no—
299#121—62no—
300#121—124no—
301#121—124no—
302C4×D4—38no—
303C4×D4—64no—
304C4×D4—64no—
305C4×D4—68no—
306C4×D4—68no—
307C4×D4—64no—
308C2×C4⋊C4—32no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—64no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—62no—
317C2×C22⋊C4—62no—
318C2×C22⋊C4—68no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C42—31no—
325C2×C42—31no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—64no—
329C4⋊C8—64no—
330#102—24no—
331#102—64no—
332#102—64no—
333#102—64no—
334C22⋊C8—22no—
335C22⋊C8—62no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—216no—
341C23×C3—24no—
342C23×C3—24no—
343C23×C3—24no—
344C23×C3—48no—
345C23×C3—48no—
346C23×C3—48no—
347C23×C3—48no—
348C23×C3—48no—
349C23×C3—42no—
350C23×C3—42no—
351C23×C3—42no—
352C23×C3—42no—
353C23×C3—48no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3#12112yes—
362C23×C3C2×D812yes—
363C23×C3AGL(1,ℤ/8)12yes—
364C23×C3C22×D411yescharacteristic
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C22×D3—41no—
372C3×D4—216no—
373C3×D4—24no—
374C3×D4—24no—
375C3×D4—24no—
376C3×D4—42no—
377C3×D4—42no—
378C3×D4—84no—
379C3×D4#12118yescomplemented
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—24no—
383C2×C3×C4—24no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4#12112yes—
388C2×C3×C4C22×D412yes—
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C2×#4112yes—
391C2×C3×C4C2×#4111yescharacteristic
392#67—84no—
393#67—84no—
394#67—84no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—28no—
400C2×Dic3—21no—
401C2×Dic3—24no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—44no—
405C2×Dic3—416no—
406C2×Dic3—48no—
407C2×Dic3—42no—
408C2×Dic3—42no—
409C2×Dic3—416no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3#12012yes—
414C2×Dic3#12012yes—
415C2×Dic3C2×D812yes—
416C2×Dic3C2×D812yes—
417C2×Dic3AGL(1,ℤ/8)12yes—
418D12—42no—
419D12—42no—
420C3×C8—42no—
421C3⋊C8—42no—
422C24—24no—
423C24—24no—
424C24—24no—
425C24—22no—
426C24—21no—
427C24—42no—
428C24—42no—
429C24—42no—
430C24—42no—
431C24—61no—
432C24D3×D411yescharacteristic
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—216no—
436C2×D4—216no—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—21no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—62no—
449C2×D4—61no—
450C2×D4—62no—
451C2×D4—61no—
452C2×D4—122no—
453C2×D4—122no—
454C2×D4—122no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4—122no—
458C2×D4—122no—
459C2×D4—122no—
460C2×D4—122no—
461C2×D4D3×D414yes—
462C2×D4#23714yes—
463C2×D4C2×#6714yes—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—31no—
468C22×C4—34no—
469C22×C4—31no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—62no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4—68no—
476C22×C4—64no—
477C22×C4—68no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—61no—
481C22×C4—124no—
482C22×C4—124no—
483C22×C4—124no—
484C22×C4—121no—
485C22×C4—1232no—
486C22×C4—124no—
487C22×C4—122no—
488C22×C4—122no—
489C22×C4D3×D411yescharacteristic
490C22×C4#23712yes—
491C22×C4C2×#6712yes—
492C22×C4C23×D311yescharacteristic
493D8—48no—
494D8—128no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—62no—
499C2×C8—62no—
500C2×C8—62no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—34no—
504C4⋊C4—64no—
505C4⋊C4—62no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—122no—
509C4⋊C4—122no—
510C22⋊C4—42no—
511C22⋊C4—68no—
512C22⋊C4—68no—
513C22⋊C4—122no—
514C22⋊C4—122no—
515C22⋊C4—122no—
516C22⋊C4—122no—
517C22⋊C4—122no—
518C42—34no—
519C42—62no—
520C42—62no—
521C42—62no—
522C22×C3—22no—
523C22×C3—22no—
524C22×C3—22no—
525C22×C3—22no—
526C22×C3—22no—
527C22×C3—22no—
528C22×C3—28no—
529C22×C3—432no—
530C22×C3—44no—
531C22×C3—44no—
532C22×C3—44no—
533C22×C3—44no—
534C22×C3—44no—
535C22×C3—44no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3—81no—
539C22×C3—81no—
540C22×C3—81no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3#45914yescomplemented
544C22×C3C2×#12011yescharacteristic
545C22×C3C2×#12111yescharacteristic
546C22×C3#52511yescharacteristic
547C22×C3C22×D811yescharacteristic
548C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
549C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
550C22×C3#57511yescharacteristic
551C2×D3—81no—
552C2×D3—81no—
553C2×D3—81no—
554C2×D3—81no—
555C2×D3—81no—
556C2×D3—81no—
557C2×D3—81no—
558C3×C4—22no—
559C3×C4—81no—
560C3×C4C2×#12112yes—
561C3×C4#52512yes—
562Dic3—22no—
563Dic3—22no—
564Dic3—44no—
565Dic3—41no—
566Dic3—81no—
567Dic3#44714yescomplemented
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—216no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—48no—
577C23—48no—
578C23—48no—
579C23—48no—
580C23—48no—
581C23—48no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—121no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23—121no—
600C23D3×D812yes—
601C23#91112yes—
602C23#93912yes—
603C23C2×D3×D411yescharacteristic
604D4—216no—
605D4—24no—
606D4—24no—
607D4—24no—
608D4—42no—
609D4—42no—
610D4—84no—
611D4—122no—
612D4—122no—
613D4—244no—
614D4—244no—
615D4—244no—
616D4#93918yescomplemented
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—32no—
622C2×C4—32no—
623C2×C4—32no—
624C2×C4—32no—
625C2×C4—32no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—68no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—64no—
638C2×C4—1216no—
639C2×C4—124no—
640C2×C4—1216no—
641C2×C4—128no—
642C2×C4—122no—
643C2×C4—122no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4#93912yes—
648C2×C4C2×D3×D412yes—
649C2×C4C2×#23712yes—
650C2×C4C2×#23711yescharacteristic
651C2×C4C22×#6711yescharacteristic
652C8—42no—
653C8—122no—
654C2×C3—24no—
655C2×C3—48no—
656C2×C3—161no—
657C2×C3#284711yescharacteristic
658C2×C3C2×#44711yescharacteristic
659C2×C3#300611yescharacteristic
660C2×C3C2×#45911yescharacteristic
661C2×C3#303611yescharacteristic
662C2×C3#310011yescharacteristic
663C2×C3#310911yescharacteristic
664D3—161no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—28no—
672C22—432no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—241no—
687C22—241no—
688C22—241no—
689C22—241no—
690C22—241no—
691C22—241no—
692C22—241no—
693C22#515014yescomplemented
694C22#558611yescharacteristic
695C22D3×#12011yescharacteristic
696C22C2×D3×D811yescharacteristic
697C22C2×#91111yescharacteristic
698C22D3×AGL(1,ℤ/8)11yescharacteristic
699C22#577311yescharacteristic
700C22C2×#93911yescharacteristic
701C4—22no—
702C4—34no—
703C4—62no—
704C4—62no—
705C4—81no—
706C4—124no—
707C4—121no—
708C4—241no—
709C4#558612yes—
710C4C2×#93912yes—
711C3#3290211yescharacteristic, complemented, Sylow
712C2—24no—
713C2—48no—
714C2—161no—
715C2—481no—
716C2#7998811yescharacteristic
717C2#7999011yescharacteristic
718C2D3×#44711yescharacteristic
719C2#8000311yescharacteristic
720C2#8260111yescharacteristic
721C2C2×#515011yescharacteristic
722C2#8393111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#447C2
socleC23×C3C22×D4
hypercenterC2×#447D3
hypocenterC3#32902
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
1. C22×D4×Dic3C2semidirect
2. C2×C3×#447C2semidirect
3. C2×#5217C2semidirect
4. C2×#5150C2semidirect
5. #82700C2semidirect
6. #72037C2semidirect
7. #71677C2semidirect
8. #70172C2semidirect
9. #70153C2semidirect
10. #69932C2semidirect
11. #69908C2semidirect
18. C2×#940C22semidirect
33. #5221C22semidirect
36. C2×#832C22semidirect
39. C2×#821C22semidirect
40. C2×Dic3⋊C8C22semidirect
41. #4988C22semidirect
42. #4975C22semidirect
45. #4551C22semidirect
46. #4550C22semidirect
47. #4529C22semidirect
51. C2×#447D3semidirect
65. C22×C3×D4D4semidirect
102. D4×Dic3D4semidirect
227. C22×Dic3D8semidirect
275. C22×D4#67semidirect
379. C3×D4#121semidirect
543. C22×C3#459semidirect
567. Dic3#447semidirect
616. D4#939semidirect
693. C22#5150semidirect
711. C3#32902semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic32
C2×C3×#4472
C2×#52172
C2×#51502
#827002
#720372
#716772
#701722
#701532
#699322
#699082
#329023