Finite Group Databasebeta

← #10848765

Subgroups — #10848765

Order 768.

1318 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic, complemented
2C2×#5067C211yesmaximal, characteristic, complemented
3C2×#5058C211yesmaximal, characteristic, complemented
4#82081C211yesmaximal, characteristic, complemented
5#71741C211yesmaximal, characteristic, complemented
6#71726C211yesmaximal, characteristic, complemented
7#71699C212yesmaximal, complemented
8#71684C212yesmaximal, complemented
9#71633C212yesmaximal, complemented
10#71613C212yesmaximal, complemented
11#71529C211yesmaximal, characteristic, complemented
12#25605—31nomaximal, Sylow
13C3×#529—24no—
14C3×#529—24no—
15C2×C3×#126—21no—
16C2×C3×#125C2211yescharacteristic
17C2×C3×#124C2211yescharacteristic
18C2×C3×#123—21no—
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×#121C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4—21no—
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×#117—21no—
25C22×C3×C42C2211yescharacteristic
26C22×#215C2212yes—
27C22×#213C2212yes—
28C2×#930—21no—
29C2×#930—21no—
30C2×#930C2212yes—
31C2×#874C2211yescharacteristic
32C2×#835C2212yes—
33C2×#832C2212yes—
34C2×#830C2212yes—
35#5067—24no—
36#5058—24no—
37C2×#809C2212yes—
38C2×#809C2212yes—
39#4939—21no—
40#4939—21no—
41#4939C2212yes—
42#4935C2211yescharacteristic
43C2×#803C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#529D311yescharacteristic, complemented
49C2×#496—31no—
50C2×#490—31no—
51#2856—31no—
52#2000—32no—
53#1999—32no—
54#1992—31no—
55#1959—31no—
56#1880—32no—
57#1879—32no—
58#1749—31no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4D411yescharacteristic
64C23×C3×C4D411yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C22×#67—22no—
67C22×D12—21no—
68C22×D12—21no—
69C22×D12C2312yes—
70C3×#126—28no—
71C3×#125—22no—
72C3×#125D414yescomplemented
73C3×#124—22no—
74C3×#124D414yescomplemented
75C3×#123—28no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×#121—22no—
79C3×#121—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×D4—28no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×#117—28no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×C3×C42—22no—
99C2×C3×C42—22no—
100C2×C3×C42D411yescharacteristic
101C2×C3×C42D411yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×#215—24no—
105C2×#215—24no—
106C2×#215—42no—
107C2×#213—24no—
108C2×#213—24no—
109C2×#213—42no—
110#930—42no—
111#930—42no—
112#930—44no—
113#930—44no—
114C2×(C2×D3)⋊C4—22no—
115C2×(C2×D3)⋊C4—22no—
116C2×(C2×D3)⋊C4—22no—
117C2×C4⋊Dic3—21no—
118C2×C4⋊Dic3—21no—
119C2×C4⋊Dic3C2312yes—
120C22×C3⋊C8C2312yes—
121C22×C3⋊C8C2312yes—
122#874—22no—
123#874—22no—
124#835—42no—
125#832—42no—
126#830—24no—
127#809—24no—
128#809—24no—
129#803—22no—
130#803—22no—
131C22×SD16—32no—
132C22×D8—32no—
133C2×#127—31no—
134#529—24no—
135#529—24no—
136C2×#126—21no—
137C2×#125C2×D311yescharacteristic
138C2×#124C2×D311yescharacteristic
139C2×#123—21no—
140C2×C22⋊Q8C2×D311yescharacteristic
141C2×#121—32no—
142C2×#121—61no—
143C2×#121—61no—
144C2×#121C2×D311yescharacteristic
145C2×C4×Q8C2×D311yescharacteristic
146C2×C4×D4—21no—
147C2×C4×D4C2×D311yescharacteristic
148C2×#117—21no—
149C22×C42C2×D311yescharacteristic
150#496—64no—
151#490—64no—
152C2×#103—32no—
153C2×#102—32no—
154C2×#102—32no—
155C2×#102—32no—
156C2×C22⋊C8—32no—
157C2×#97—31no—
158C2×#97—31no—
159#390—32no—
160#390—61no—
161#390—61no—
162#384—31no—
163#340—32no—
164#340—32no—
165#336—32no—
166C24×C3—21no—
167C24×C3C2×D411yescharacteristic
168C23×D3—22no—
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8C2×D412yes—
173C2×C3×D4—24no—
174C2×C3×D4—21no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—42no—
178C2×C3×D4—42no—
179C2×C3×D4C2×D412yes—
180C22×C3×C4—24no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—41no—
193C22×C3×C4—41no—
194C22×C3×C4—42no—
195C22×C3×C4—42no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2411yescharacteristic
202C2×#67—44no—
203C2×#67—44no—
204C2×#67—44no—
205C22×Dic3—22no—
206C2×D12—22no—
207C2×D12—22no—
208C2×D12—22no—
209C2×D12—22no—
210C2×D12—22no—
211C2×D12—22no—
212C2×D12—42no—
213C2×D12—42no—
214C2×D12—42no—
215C2×D12—42no—
216C2×D12—42no—
217C2×D12—42no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4C2×D412yes—
225C3×C4⋊C4C2×D412yes—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—24no—
231C3×C42—24no—
232C3×C42—22no—
233C3×C42—22no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42C2×D412yes—
237C3×C42C2×D412yes—
238#215—48no—
239#213—48no—
240(C2×D3)⋊C4—44no—
241(C2×D3)⋊C4—44no—
242(C2×D3)⋊C4—44no—
243C4⋊Dic3—22no—
244C4⋊Dic3—22no—
245C4⋊Dic3—42no—
246C4⋊Dic3—42no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8#4114yes—
253C22×Q8C22×D311yescharacteristic
254C22×D4—21no—
255C22×D4—32no—
256C22×D4—62no—
257C22×D4—61no—
258C22×D4—61no—
259C22×D4C22×D311yescharacteristic
260C23×C4—21no—
261C23×C4#6711yescharacteristic
262C23×C4#6711yescharacteristic
263C23×C4C22×D311yescharacteristic
264C2×SD16—64no—
265C2×SD16—64no—
266C2×SD16—122no—
267C2×D8—64no—
268C2×D8—64no—
269C2×D8—122no—
270C22×C8—32no—
271C22×C8—32no—
272#127—62no—
273#127—62no—
274#126—28no—
275#125—22no—
276#125#6714yescomplemented
277#124—22no—
278#124#6714yescomplemented
279#123—28no—
280C22⋊Q8—22no—
281C22⋊Q8—22no—
282#121—22no—
283#121—22no—
284#121—122no—
285#121—122no—
286#121—124no—
287#121—124no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×D4—28no—
291C4×D4—22no—
292C4×D4—22no—
293#117—28no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—32no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—62no—
309C2×C22⋊C4—62no—
310C2×C22⋊C4—62no—
311C2×C42—21no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42#6711yescharacteristic
315C2×C42#6711yescharacteristic
316C2×C42C22×D311yescharacteristic
317C2×C42C22×D311yescharacteristic
318#103—122no—
319#102—64no—
320#102—64no—
321#102—122no—
322C22⋊C8—64no—
323#97—62no—
324#97—62no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—22no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3AGL(1,ℤ/8)12yes—
339C23×C3C22×D411yescharacteristic
340C22×D3—42no—
341C22×D3—42no—
342C22×D3—42no—
343C22×D3—42no—
344C22×D3—42no—
345C22×D3—42no—
346C22×D3—42no—
347C3×Q8—24no—
348C3×Q8—42no—
349C3×D4—24no—
350C3×D4—48no—
351C3×D4—42no—
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—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—44no—
368C2×C3×C4—44no—
369C2×C3×C4—48no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4#12012yes—
379C2×C3×C4#12012yes—
380C2×C3×C4AGL(1,ℤ/8)12yes—
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4112yes—
384C2×C3×C4H(5,𝔽2)12yes—
385#67—88no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389D12—42no—
390D12—42no—
391D12—42no—
392D12—42no—
393D12—84no—
394D12—84no—
395C3⋊C8—24no—
396C3⋊C8—42no—
397C24—21no—
398C24—62no—
399C24C2×#6711yescharacteristic
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8C2×#6712yes—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4—122no—
417C2×D4—122no—
418C2×D4—122no—
419C2×D4—122no—
420C2×D4—124no—
421C2×D4—124no—
422C2×D4—124no—
423C2×D4—122no—
424C2×D4—122no—
425C2×D4C2×#6712yes—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—62no—
443C22×C4C2×#6711yescharacteristic
444C22×C4C2×#6711yescharacteristic
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C22×C4C2×#6711yescharacteristic
448C22×C4C23×D311yescharacteristic
449SD16—128no—
450D8—128no—
451C2×C8—34no—
452C2×C8—62no—
453C2×C8—62no—
454C2×C8—62no—
455C2×C8—62no—
456C2×C8—62no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—62no—
464C4⋊C4—62no—
465C4⋊C4—122no—
466C4⋊C4—122no—
467C4⋊C4C2×#6712yes—
468C4⋊C4C2×#6712yes—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—124no—
475C22⋊C4—124no—
476C22⋊C4—124no—
477C42—24no—
478C42—22no—
479C42—22no—
480C42—21no—
481C42—21no—
482C42C2×#6712yes—
483C42C2×#6712yes—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—81no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3—81no—
498C22×C3C2×#12011yescharacteristic
499C22×C3#53211yescharacteristic
500C22×C3#53511yescharacteristic
501C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
502C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
503C22×C3#57511yescharacteristic
504C22×C3#57511yescharacteristic
505C2×D3—82no—
506C2×D3—82no—
507C2×D3—82no—
508C2×D3—82no—
509C2×D3—82no—
510C2×D3—82no—
511C2×D3—82no—
512C3×C4—22no—
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—41no—
516C3×C4—42no—
517C3×C4—41no—
518C3×C4—42no—
519C3×C4—42no—
520C3×C4#54612yes—
521C3×C4#55012yes—
522Dic3—82no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—22no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—122no—
537C23—122no—
538C23—122no—
539C23—122no—
540C23—122no—
541C23—122no—
542C23—122no—
543C23#94912yes—
544C23C22×#6711yescharacteristic
545Q8—24no—
546Q8—42no—
547D4—24no—
548D4—48no—
549D4—42no—
550D4—122no—
551D4—122no—
552D4—122no—
553D4—122no—
554D4—244no—
555D4—244no—
556D4—248no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—122no—
586C2×C4#94912yes—
587C2×C4#95312yes—
588C2×C4#95312yes—
589C2×C4D3×#4112yes—
590C2×C4#100912yes—
591C2×C4C22×#6711yescharacteristic
592C2×C4C22×#6711yescharacteristic
593C8—64no—
594C8—122no—
595C2×C3—42no—
596C2×C3—82no—
597C2×C3#243111yescharacteristic
598C2×C3#300611yescharacteristic
599C2×C3#300611yescharacteristic
600C2×C3#317111yescharacteristic
601C2×C3#318011yescharacteristic
602C2×C3#321711yescharacteristic
603C2×C3#321911yescharacteristic
604D3—162no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—242no—
620C22—242no—
621C22—242no—
622C22—242no—
623C22—242no—
624C22—242no—
625C22—242no—
626C22C2×#94911yescharacteristic
627C22C2×#94911yescharacteristic
628C22#582211yescharacteristic
629C22#582211yescharacteristic
630C22#583011yescharacteristic
631C22#583211yescharacteristic
632C22C2×#95311yescharacteristic
633C4—22no—
634C4—22no—
635C4—22no—
636C4—41no—
637C4—42no—
638C4—42no—
639C4—41no—
640C4—42no—
641C4—242no—
642C4#582912yes—
643C4#583412yes—
644C3#2560511yescharacteristic, complemented, Sylow
645C2—42no—
646C2—82no—
647C2—482no—
648C2#8313511yescharacteristic
649C2#8313711yescharacteristic
650C2#8314611yescharacteristic
651C2#8314711yescharacteristic
652C2#8408211yescharacteristic
653C2#8408211yescharacteristic
654C2#8409211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#529C2
socleC23×C3C22×D4
hypercenterC2×#529D3
hypocenterC3#25605
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
1. C2×C3×#529C2semidirect
2. C2×#5067C2semidirect
3. C2×#5058C2semidirect
4. #82081C2semidirect
5. #71741C2semidirect
6. #71726C2semidirect
7. #71699C2semidirect
8. #71684C2semidirect
9. #71633C2semidirect
10. #71613C2semidirect
11. #71529C2semidirect
48. C2×#529D3semidirect
72. C3×#125D4semidirect
74. C3×#124D4semidirect
276. #125#67semidirect
278. #124#67semidirect
644. C3#25605semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C2×#50672
C2×#50582
#820812
#717412
#717262
#716992
#716842
#716332
#716132
#715292
#256053