Finite Group Databasebeta

← #10826926

Subgroups — #10826926

Order 768.

2596 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#121C211yesmaximal, characteristic, complemented
2C22×#930C211yesmaximal, characteristic, complemented
3C2×#5217C212yesmaximal, complemented
4C2×#5036C212yesmaximal, complemented
5#72126C211yesmaximal, characteristic, complemented
6#71684C214yesmaximal, complemented
7#71677C212yesmaximal, complemented
8#71560C212yesmaximal, complemented
9#23918—31nomaximal, Sylow
10C23×C3×D4—21no—
11C23×C3×D4C2211yescharacteristic, complemented
12C24×C3×C4C2211yescharacteristic
13C23×#67—21no—
14C23×D12C2211yescharacteristic, complemented
15C2×C3×#121—21no—
16C2×C3×#121—21no—
17C2×C3×#121—28no—
18C2×C3×#121C2212yes—
19C2×C3×#121C2212yescomplemented
20C22×C3×C4⋊C4C2211yescharacteristic, complemented
21C22×C3×C22⋊C4—21no—
22C22×#213C2214yescomplemented
23C2×#930—21no—
24C2×#930—21no—
25C2×#930—28no—
26C2×#930C2212yes—
27C2×#930C2212yescomplemented
28C22×(C2×D3)⋊C4—21no—
29C22×C4⋊Dic3C2211yescharacteristic, complemented
30#5217—28no—
31C2×#832C2214yescomplemented
32C2×#830C2212yescomplemented
33C2×#830C2212yescomplemented
34C2×#830C2212yescomplemented
35#5036—28no—
36C2×#809C2214yescomplemented
37#4529C2214yescomplemented
38C22×#121—31no—
39C22×#121D311yescharacteristic, complemented
40C2×#447—32no—
41C2×#447—32no—
42#1999—34no—
43#1874—32no—
44#1874—32no—
45#1779—31no—
46C25×C3—21no—
47C25×C3D411yescharacteristic
48C24×D3—21no—
49C22×C3×D4—28no—
50C22×C3×D4—22no—
51C22×C3×D4—22no—
52C22×C3×D4—24no—
53C22×C3×D4—21no—
54C22×C3×D4—24no—
55C22×C3×D4—42no—
56C22×C3×D4—44no—
57C22×C3×D4—44no—
58C22×C3×D4C2312yes—
59C23×C3×C4—21no—
60C23×C3×C4—24no—
61C23×C3×C4—24no—
62C23×C3×C4—42no—
63C23×C3×C4D411yescharacteristic
64C23×C3×C4D411yescharacteristic
65C23×C3×C4D411yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C22×#67—28no—
70C22×#67—22no—
71C22×#67—22no—
72C22×#67—44no—
73C22×#67—44no—
74C23×Dic3—21no—
75C22×D12—21no—
76C22×D12—24no—
77C22×D12—24no—
78C22×D12—42no—
79C22×D12C2312yes—
80C3×#121—28no—
81C3×#121—28no—
82C3×#121—24no—
83C3×#121—24no—
84C3×#121—24no—
85C3×#121D418yescomplemented
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—24no—
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C22⋊C4—28no—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4—22no—
92C2×#213—28no—
93C2×#213—28no—
94C2×#213—44no—
95#930—28no—
96#930—28no—
97#930—24no—
98#930—24no—
99#930—24no—
100#930D418yescomplemented
101C2×(C2×D3)⋊C4—28no—
102C2×(C2×D3)⋊C4—22no—
103C2×(C2×D3)⋊C4—22no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—24no—
106C2×C4⋊Dic3C2312yes—
107C22×C3⋊C8C2314yes—
108#832—28no—
109#830—24no—
110#830—24no—
111#830—42no—
112#809—28no—
113C23×D4—21no—
114C23×D4—31no—
115C23×D4—61no—
116C23×D4C2×D311yescharacteristic, complemented
117C24×C4C2×D311yescharacteristic
118C22×D8—34no—
119C2×#121—21no—
120C2×#121—21no—
121C2×#121—28no—
122C2×#121—32no—
123C2×#121—32no—
124C2×#121—68no—
125C2×#121—61no—
126C2×#121—61no—
127C2×#121C2×D312yes—
128C2×#121C2×D312yescomplemented
129C22×C4⋊C4—31no—
130C22×C4⋊C4C2×D311yescharacteristic, complemented
131C22×C22⋊C4—21no—
132C22×C22⋊C4—61no—
133#447—68no—
134#447—68no—
135C2×#102—34no—
136C2×#102—34no—
137C2×C22⋊C8—32no—
138C2×C22⋊C8—32no—
139C2×C22⋊C8—32no—
140#340—34no—
141C24×C3—22no—
142C24×C3—22no—
143C24×C3—22no—
144C24×C3—22no—
145C24×C3—24no—
146C24×C3—22no—
147C24×C3—41no—
148C24×C3—41no—
149C24×C3—42no—
150C24×C3—42no—
151C24×C3—42no—
152C24×C3—42no—
153C24×C3—42no—
154C24×C3—42no—
155C24×C3D814yescomplemented
156C24×C3C2×D411yescharacteristic
157C24×C3C2×D411yescharacteristic
158C24×C3C2×D411yescharacteristic
159C23×D3—24no—
160C23×D3—22no—
161C23×D3—42no—
162C23×D3—42no—
163C23×D3—42no—
164C23×D3—42no—
165C23×D3—42no—
166C23×D3—42no—
167C2×C3×D4—24no—
168C2×C3×D4—28no—
169C2×C3×D4—28no—
170C2×C3×D4—24no—
171C2×C3×D4—22no—
172C2×C3×D4—24no—
173C2×C3×D4—416no—
174C2×C3×D4—42no—
175C2×C3×D4—432no—
176C2×C3×D4—44no—
177C2×C3×D4—44no—
178C2×C3×D4—44no—
179C2×C3×D4—44no—
180C2×C3×D4—42no—
181C2×C3×D4C2×D414yescomplemented
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—24no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—41no—
198C22×C3×C4—41no—
199C22×C3×C4—41no—
200C22×C3×C4—416no—
201C22×C3×C4—42no—
202C22×C3×C4—42no—
203C22×C3×C4C2×D414yes—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D414yes—
208C22×C3×C4C2411yescharacteristic
209C2×#67—28no—
210C2×#67—28no—
211C2×#67—432no—
212C2×#67—44no—
213C2×#67—44no—
214C2×#67—44no—
215C2×#67—44no—
216C22×Dic3—24no—
217C22×Dic3—22no—
218C22×Dic3—42no—
219C22×Dic3—42no—
220C2×D12—24no—
221C2×D12—24no—
222C2×D12—24no—
223C2×D12—22no—
224C2×D12—416no—
225C2×D12—42no—
226C2×D12—42no—
227C2×D12C2×D414yescomplemented
228C3×C4⋊C4—24no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4C2×D414yescomplemented
231C3×C22⋊C4—28no—
232C3×C22⋊C4—28no—
233#213—416no—
234(C2×D3)⋊C4—28no—
235(C2×D3)⋊C4—28no—
236C4⋊Dic3—24no—
237C4⋊Dic3—22no—
238C4⋊Dic3C2×D414yescomplemented
239C2×C3⋊C8—24no—
240C2×C3⋊C8—24no—
241C2×C3⋊C8—24no—
242C25—21no—
243C25—61no—
244C25#6711yescharacteristic
245C22×D4—28no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—24no—
249C22×D4—24no—
250C22×D4—21no—
251C22×D4—32no—
252C22×D4—42no—
253C22×D4—44no—
254C22×D4—44no—
255C22×D4—64no—
256C22×D4—68no—
257C22×D4—62no—
258C22×D4—62no—
259C22×D4—61no—
260C22×D4—64no—
261C22×D4—122no—
262C22×D4—124no—
263C22×D4—124no—
264C22×D4C22×D312yes—
265C23×C4—21no—
266C23×C4—24no—
267C23×C4—24no—
268C23×C4—42no—
269C23×C4—61no—
270C23×C4#6711yescharacteristic
271C23×C4#6711yescharacteristic
272C23×C4#6711yescharacteristic
273C23×C4C22×D311yescharacteristic
274C23×C4C22×D311yescharacteristic
275C23×C4C22×D311yescharacteristic
276C2×D8—68no—
277C2×D8—68no—
278C2×D8—124no—
279C22×C8—34no—
280#121—28no—
281#121—28no—
282#121—24no—
283#121—24no—
284#121—24no—
285#121—38no—
286#121—64no—
287#121—64no—
288#121—68no—
289#121—68no—
290#121—64no—
291#121#6718yescomplemented
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—32no—
295C2×C4⋊C4—64no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4C22×D312yes—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—68no—
302C2×C22⋊C4—62no—
303C2×C22⋊C4—62no—
304#102—68no—
305#102—68no—
306C22⋊C8—64no—
307C22⋊C8—64no—
308C22⋊C8—122no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—28no—
324C23×C3—28no—
325C23×C3—24no—
326C23×C3—44no—
327C23×C3—44no—
328C23×C3—44no—
329C23×C3—44no—
330C23×C3—44no—
331C23×C3—48no—
332C23×C3—48no—
333C23×C3—48no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3—42no—
340C23×C3—84no—
341C23×C3—84no—
342C23×C3—84no—
343C23×C3—84no—
344C23×C3#12012yes—
345C23×C3#12012yes—
346C23×C3C2×D812yes—
347C23×C3C2×D812yes—
348C23×C3C2×D812yes—
349C23×C3AGL(1,ℤ/8)12yes—
350C23×C3AGL(1,ℤ/8)12yes—
351C23×C3C22×D411yescharacteristic
352C22×D3—24no—
353C22×D3—48no—
354C22×D3—48no—
355C22×D3—48no—
356C22×D3—42no—
357C22×D3—42no—
358C22×D3—42no—
359C22×D3—42no—
360C22×D3—42no—
361C22×D3—42no—
362C22×D3—84no—
363C22×D3—84no—
364C22×D3—84no—
365C22×D3—84no—
366C3×D4—28no—
367C3×D4—48no—
368C3×D4—416no—
369C3×D4—416no—
370C3×D4—44no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—28no—
376C2×C3×C4—28no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—24no—
380C2×C3×C4—21no—
381C2×C3×C4—44no—
382C2×C3×C4—44no—
383C2×C3×C4—44no—
384C2×C3×C4—48no—
385C2×C3×C4—42no—
386C2×C3×C4—42no—
387C2×C3×C4—44no—
388C2×C3×C4#12012yes—
389C2×C3×C4#12012yes—
390C2×C3×C4C22×D412yes—
391C2×C3×C4C22×D412yes—
392C2×C3×C4H(5,𝔽2)12yes—
393#67—416no—
394#67—416no—
395C2×Dic3—24no—
396C2×Dic3—48no—
397C2×Dic3—42no—
398C2×Dic3—42no—
399D12—28no—
400D12—48no—
401D12—44no—
402C3⋊C8—44no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—24no—
408C24—22no—
409C24—41no—
410C24—41no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24—64no—
418C24—62no—
419C24—122no—
420C24—122no—
421C24—122no—
422C24—122no—
423C24—122no—
424C24—122no—
425C24#21314yescomplemented
426C24C2×#6711yescharacteristic
427C24C2×#6711yescharacteristic
428C24C2×#6711yescharacteristic
429C2×D4—24no—
430C2×D4—28no—
431C2×D4—28no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—34no—
436C2×D4—416no—
437C2×D4—42no—
438C2×D4—432no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—42no—
444C2×D4—64no—
445C2×D4—68no—
446C2×D4—68no—
447C2×D4—62no—
448C2×D4—64no—
449C2×D4—64no—
450C2×D4—1216no—
451C2×D4—122no—
452C2×D4—1232no—
453C2×D4—124no—
454C2×D4—124no—
455C2×D4—124no—
456C2×D4—124no—
457C2×D4—122no—
458C2×D4C2×#6714yescomplemented
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—416no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—64no—
481C22×C4—62no—
482C22×C4—122no—
483C22×C4—122no—
484C22×C4D3×D414yes—
485C22×C4C2×#6711yescharacteristic
486C22×C4C2×#6711yescharacteristic
487C22×C4C2×#6711yescharacteristic
488C22×C4C2×#6714yes—
489C22×C4C23×D311yescharacteristic
490D8—1216no—
491C2×C8—64no—
492C2×C8—64no—
493C2×C8—64no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—34no—
497C4⋊C4—64no—
498C4⋊C4—62no—
499C4⋊C4C2×#6714yescomplemented
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—68no—
503C22⋊C4—68no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—22no—
517C22×C3—22no—
518C22×C3—22no—
519C22×C3—416no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—41no—
524C22×C3—44no—
525C22×C3—44no—
526C22×C3—44no—
527C22×C3—816no—
528C22×C3—82no—
529C22×C3—82no—
530C22×C3—82no—
531C22×C3—82no—
532C22×C3#44714yescomplemented
533C22×C3#44714yescomplemented
534C22×C3C2×#12011yescharacteristic
535C22×C3C2×#12011yescharacteristic
536C22×C3#53411yescharacteristic
537C22×C3C22×D811yescharacteristic
538C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
539C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
540C22×C3#57511yescharacteristic
541C2×D3—44no—
542C2×D3—44no—
543C2×D3—44no—
544C2×D3—816no—
545C2×D3—82no—
546C2×D3—82no—
547C2×D3—82no—
548C2×D3—82no—
549C3×C4—22no—
550C3×C4—22no—
551C3×C4—41no—
552C3×C4—44no—
553C3×C4D4214yes—
554Dic3—44no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—28no—
564C23—21no—
565C23—21no—
566C23—28no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—24no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—48no—
578C23—48no—
579C23—48no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—64no—
587C23—84no—
588C23—84no—
589C23—84no—
590C23—84no—
591C23—128no—
592C23—128no—
593C23—128no—
594C23—122no—
595C23—122no—
596C23—122no—
597C23—122no—
598C23—122no—
599C23—122no—
600C23—244no—
601C23—244no—
602C23—244no—
603C23—244no—
604C23C2×#21312yes—
605C23C2×#21312yes—
606C23C2×#21312yes—
607C23#93212yes—
608C23#93712yes—
609C23#94912yes—
610C23#95312yes—
611C23C22×#6711yescharacteristic
612D4—28no—
613D4—48no—
614D4—416no—
615D4—416no—
616D4—44no—
617D4—68no—
618D4—128no—
619D4—1216no—
620D4—1216no—
621D4—124no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—24no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—48no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—64no—
640C2×C4—128no—
641C2×C4—122no—
642C2×C4—122no—
643C2×C4#93712yes—
644C2×C4#95312yes—
645C2×C4C2×D3×D412yes—
646C2×C4#100412yes—
647C2×C4C22×#6712yes—
648C8—124no—
649C2×C3—24no—
650C2×C3—24no—
651C2×C3—42no—
652C2×C3—84no—
653C2×C3#240311yescharacteristic
654C2×C3C2×#44711yescharacteristic
655C2×C3C2×#44711yescharacteristic
656C2×C3#300611yescharacteristic
657C2×C3#300611yescharacteristic
658C2×C3#318611yescharacteristic
659C2×C3#319211yescharacteristic
660D3—84no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—416no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—816no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—124no—
690C22—124no—
691C22—124no—
692C22—2416no—
693C22—242no—
694C22—242no—
695C22—242no—
696C22—242no—
697C22#503614yescomplemented
698C22#521714yescomplemented
699C22C22×#21311yescharacteristic
700C22C2×#93211yescharacteristic
701C22C2×#93711yescharacteristic
702C22#580411yescharacteristic
703C22C2×#94911yescharacteristic
704C22#582211yescharacteristic
705C22C2×#95311yescharacteristic
706C4—22no—
707C4—22no—
708C4—44no—
709C4—41no—
710C4—124no—
711C4D4×#6714yes—
712C3#2391811yescharacteristic, complemented, Sylow
713C2—24no—
714C2—24no—
715C2—42no—
716C2—84no—
717C2—244no—
718C2C2×#503611yescharacteristic
719C2#8262411yescharacteristic
720C2#8262511yescharacteristic
721C2#8262711yescharacteristic
722C2C2×#521711yescharacteristic
723C2#8405211yescharacteristic
724C2#8408211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#121C2
socleC23×C3C22×D4
hypercenterC22×#121D3
hypocenterC3#23918
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
1. C22×C3×#121C2semidirect
2. C22×#930C2semidirect
3. C2×#5217C2semidirect
4. C2×#5036C2semidirect
5. #72126C2semidirect
6. #71684C2semidirect
7. #71677C2semidirect
8. #71560C2semidirect
11. C23×C3×D4C22semidirect
14. C23×D12C22semidirect
19. C2×C3×#121C22semidirect
20. C22×C3×C4⋊C4C22semidirect
22. C22×#213C22semidirect
27. C2×#930C22semidirect
29. C22×C4⋊Dic3C22semidirect
31. C2×#832C22semidirect
32. C2×#830C22semidirect
33. C2×#830C22semidirect
34. C2×#830C22semidirect
36. C2×#809C22semidirect
37. #4529C22semidirect
39. C22×#121D3semidirect
85. C3×#121D4semidirect
100. #930D4semidirect
116. C23×D4C2×D3semidirect
128. C2×#121C2×D3semidirect
130. C22×C4⋊C4C2×D3semidirect
155. C24×C3D8semidirect
181. C2×C3×D4C2×D4semidirect
227. C2×D12C2×D4semidirect
230. C3×C4⋊C4C2×D4semidirect
238. C4⋊Dic3C2×D4semidirect
291. #121#67semidirect
425. C24#213semidirect
458. C2×D4C2×#67semidirect
499. C4⋊C4C2×#67semidirect
532. C22×C3#447semidirect
533. C22×C3#447semidirect
697. C22#5036semidirect
698. C22#5217semidirect
712. C3#23918semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C22×#9302
C2×#52172
C2×#50362
#721262
#716842
#716772
#715602
#239183