Finite Group Databasebeta

← #10828459

Subgroups — #10828459

Order 768.

1424 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5596C211yesmaximal, characteristic
2C2×C3×#481C211yesmaximal, characteristic, complemented
3C2×#4972C211yesmaximal, characteristic, complemented
4C2×#4968C211yesmaximal, characteristic, complemented
5#71677C211yesmaximal, characteristic, complemented
6#70584C212yesmaximal, complemented
7#70565C212yesmaximal, complemented
8#70211C212yesmaximal, complemented
9#70169C212yesmaximal, complemented
10#69932C211yesmaximal, characteristic, complemented
11#69910C211yesmaximal, characteristic, complemented
12#24061—31nomaximal, Sylow
13C22×#237—21no—
14C2×C3×#121C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×#940C2211yescharacteristic
17C2×#939—21no—
18C2×D4×Dic3C2211yescharacteristic
19C2×#930C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×Dic3⋊C4—21no—
22C22×C4×Dic3C2211yescharacteristic
23#5596—24no—
24#5596—24no—
25C2×#893C2211yescharacteristic
26C2×#888C2211yescharacteristic
27C2×#885—21no—
28C2×#67⋊C4—21no—
29C3×#481—24no—
30C2×C3×#106C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×C22⋊C8C2211yescharacteristic
33C2×#832C2212yes—
34C2×C3⋊#95—21no—
35C2×#830C2211yescharacteristic
36C2×#821C2212yes—
37#4972—21no—
38#4972—21no—
39#4972C2212yes—
40#4972C2212yes—
41#4968—24no—
42C2×#809C2212yes—
43C2×#808C2212yes—
44#4551C2212yes—
45#4550C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#532—31no—
49C2×#481—31no—
50C2×#481D311yescharacteristic, complemented
51C2×#389—31no—
52#2058—32no—
53#2041—32no—
54#1874—31no—
55#1874—31no—
56#1864—32no—
57#1864—32no—
58#1817—31no—
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4C2311yescharacteristic
63C22×#67—21no—
64C22×#67—21no—
65C22×#67—21no—
66C23×Dic3—21no—
67C23×Dic3—21no—
68C23×Dic3D411yescharacteristic
69C23×Dic3D411yescharacteristic
70C2×#237—28no—
71C2×#237—44no—
72C2×#237—44no—
73C22×D12C2311yescharacteristic
74C22×C4×D3—21no—
75C22×Dic6—21no—
76C22×C3×C8C2312yes—
77C3×#121—22no—
78C3×#121D414yes—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×#217—21no—
86#940—22no—
87#940—22no—
88#939—28no—
89D4×Dic3—22no—
90D4×Dic3—22no—
91#930—22no—
92#930D414yes—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—24no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2312yes—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4—44no—
103C2×C4×Dic3—22no—
104C2×C4×Dic3—22no—
105C2×C4×Dic3D411yescharacteristic
106C2×C4×Dic3D411yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2312yes—
110#893—22no—
111#893—22no—
112#888—22no—
113#888—22no—
114#885—28no—
115#67⋊C4—28no—
116C3×#106—24no—
117C3×#102—24no—
118C3×C22⋊C8—22no—
119#832—24no—
120C3⋊#95—22no—
121C3⋊#95—22no—
122#830—22no—
123#821—24no—
124#809—24no—
125#808—24no—
126C22×#41—61no—
127#532—64no—
128#532—64no—
129C2×C4⋊Q8—31no—
130C2×#127—31no—
131C2×#124—61no—
132C2×#121—31no—
133C2×#121—61no—
134C2×#121C2×D311yescharacteristic
135C2×C4×D4—31no—
136C2×C4×D4—31no—
137C2×C4×D4—61no—
138C22×C4⋊C4—31no—
139C22×C4⋊C4—61no—
140C22×C4⋊C4C2×D311yescharacteristic
141C22×C42—31no—
142#481—24no—
143#481—64no—
144C2×#106—32no—
145C2×#106C2×D312yes—
146C2×#102—32no—
147C2×#102—32no—
148C2×#102—32no—
149C2×#102C2×D312yes—
150C2×C22⋊C8—31no—
151C2×C22⋊C8C2×D311yescharacteristic
152#389—32no—
153#389—32no—
154#389—61no—
155#389—61no—
156C2×#95—61no—
157#340—32no—
158#340—32no—
159#336—32no—
160#336—32no—
161C24×C3—21no—
162C24×C3C2×D411yescharacteristic
163C23×D3—21no—
164C2×C3×D4—24no—
165C2×C3×D4—21no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—42no—
169C2×C3×D4—42no—
170C2×C3×D4C2×D412yes—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—42no—
176C22×C3×C4—42no—
177C22×C3×C4—42no—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2411yescharacteristic
182C2×#67—24no—
183C2×#67—24no—
184C2×#67—24no—
185C2×#67—42no—
186C2×#67—42no—
187C2×#67—42no—
188C2×#67—42no—
189C2×#67—42no—
190C2×#67—42no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—24no—
194C22×Dic3—24no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3—41no—
200C22×Dic3—42no—
201C22×Dic3—42no—
202C22×Dic3—42no—
203C22×Dic3—42no—
204C22×Dic3SD1614yes—
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209#237—432no—
210C2×D12—22no—
211C2×D12—22no—
212C2×D12—21no—
213C2×D12C2×D412yes—
214C2×C4×D3—24no—
215C2×C4×D3—42no—
216C2×C4×D3—42no—
217C2×Dic6—24no—
218C2×Dic6—42no—
219C2×Dic6—42no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—42no—
227C3×C4⋊C4C2×D412yes—
228C3×C22⋊C4—24no—
229#217—24no—
230(C2×D3)⋊C4—24no—
231(C2×D3)⋊C4—24no—
232C4⋊Dic3—24no—
233C4⋊Dic3—21no—
234C4⋊Dic3—22no—
235C4⋊Dic3C2×D412yes—
236C4⋊Dic3#4114yescomplemented
237Dic3⋊C4—24no—
238Dic3⋊C4—44no—
239Dic3⋊C4—44no—
240C4×Dic3—22no—
241C4×Dic3—22no—
242C4×Dic3—21no—
243C4×Dic3—21no—
244C4×Dic3—21no—
245C4×Dic3—21no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×#41—68no—
250C2×#41—124no—
251C2×#41—124no—
252C22×Q8—61no—
253C22×D4—21no—
254C22×D4—31no—
255C22×D4—61no—
256C22×D4—61no—
257C22×D4—61no—
258C22×D4C22×D311yescharacteristic
259C23×C4—21no—
260C23×C4—31no—
261C23×C4—31no—
262C23×C4—61no—
263C23×C4—61no—
264C23×C4—61no—
265C23×C4C22×D311yescharacteristic
266C22×C8—32no—
267C22×C8C22×D312yes—
268C4⋊Q8—62no—
269C4⋊Q8—62no—
270#127—62no—
271#127—62no—
272#124—68no—
273#121—22no—
274#121—34no—
275#121—68no—
276#121—62no—
277#121#6714yes—
278C4×D4—68no—
279C4×D4—62no—
280C4×D4—62no—
281C4×D4—62no—
282C4×D4—62no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—64no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—124no—
294C2×C4⋊C4C22×D312yes—
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C22⋊C4—21no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C42—31no—
301C2×C42—31no—
302C2×C42—31no—
303C2×C42—31no—
304C2×C42—62no—
305C2×C42—62no—
306#106—24no—
307#106—64no—
308#102—24no—
309#102—64no—
310#102—64no—
311#102—64no—
312C22⋊C8—22no—
313C22⋊C8—62no—
314#95—62no—
315#95—62no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—22no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3#12012yes—
328C23×C3C2×SD1612yes—
329C23×C3AGL(1,ℤ/8)12yes—
330C23×C3C22×D411yescharacteristic
331C22×D3—22no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C3×D4—24no—
339C3×D4—48no—
340C3×D4—42no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—22no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—84no—
352C2×C3×C4#12012yes—
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C2×#4112yes—
356C2×C3×C42−1+412yes—
357#67—48no—
358#67—48no—
359#67—48no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—28no—
367C2×Dic3—21no—
368C2×Dic3—22no—
369C2×Dic3—24no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—44no—
373C2×Dic3—48no—
374C2×Dic3—48no—
375C2×Dic3—41no—
376C2×Dic3—42no—
377C2×Dic3—41no—
378C2×Dic3—42no—
379C2×Dic3C2×SD1612yes—
380C2×Dic3C2×SD1612yes—
381C2×Dic3AGL(1,ℤ/8)12yes—
382D12—24no—
383D12—42no—
384C4×D3—48no—
385Dic6—48no—
386C3×C8—42no—
387C3⋊C8—42no—
388C24—21no—
389C24—61no—
390C24D3×D411yescharacteristic
391#41—1232no—
392C2×Q8—64no—
393C2×Q8—122no—
394C2×Q8—122no—
395C2×D4—24no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—32no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—64no—
403C2×D4—64no—
404C2×D4—64no—
405C2×D4—62no—
406C2×D4—62no—
407C2×D4—61no—
408C2×D4—122no—
409C2×D4—122no—
410C2×D4—122no—
411C2×D4—122no—
412C2×D4—122no—
413C2×D4—122no—
414C2×D4C2×#6712yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—34no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—64no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—64no—
432C22×C4—64no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—61no—
436C22×C4—122no—
437C22×C4—122no—
438C22×C4—122no—
439C22×C4—121no—
440C22×C4—122no—
441C22×C4—122no—
442C22×C4—122no—
443C22×C4D3×D411yescharacteristic
444C22×C4D3×D412yes—
445C22×C4C2×#6712yes—
446C22×C4C23×D311yescharacteristic
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—62no—
451C2×C8—62no—
452C2×C8—62no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—32no—
457C4⋊C4—34no—
458C4⋊C4—42no—
459C4⋊C4—64no—
460C4⋊C4—64no—
461C4⋊C4—62no—
462C4⋊C4—61no—
463C4⋊C4—124no—
464C4⋊C4—124no—
465C4⋊C4C2×#6712yes—
466C22⋊C4—24no—
467C22⋊C4—64no—
468C22⋊C4—64no—
469C22⋊C4—64no—
470C42—62no—
471C42—62no—
472C42—61no—
473C42—61no—
474C42—61no—
475C42—61no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3#45014yes—
490C22×C3C2×#12011yescharacteristic
491C22×C3#52511yescharacteristic
492C22×C3#53611yescharacteristic
493C22×C3C22×SD1611yescharacteristic
494C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
495C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
496C22×C3#57511yescharacteristic
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—42no—
500C2×D3—81no—
501C2×D3—81no—
502C2×D3—81no—
503C2×D3—81no—
504C3×C4—22no—
505C3×C4—42no—
506C3×C4—82no—
507C3×C4#54712yes—
508C3×C4#54712yes—
509Dic3—22no—
510Dic3—22no—
511Dic3—22no—
512Dic3—44no—
513Dic3—42no—
514Dic3—41no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—62no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23D3×SD1612yes—
534C23#91412yes—
535C23#93712yes—
536C23C2×D3×D411yescharacteristic
537Q8—128no—
538D4—24no—
539D4—48no—
540D4—42no—
541D4—64no—
542D4—128no—
543D4—128no—
544D4—128no—
545D4—122no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—32no—
552C2×C4—32no—
553C2×C4—32no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—68no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—64no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—62no—
571C2×C4—84no—
572C2×C4—128no—
573C2×C4—124no—
574C2×C4—128no—
575C2×C4—128no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4#93712yes—
581C2×C4C2×D3×D411yescharacteristic
582C2×C4D3×#4112yes—
583C2×C4#101012yes—
584C2×C4C22×#6711yescharacteristic
585C8—42no—
586C8—122no—
587C2×C3—24no—
588C2×C3—82no—
589C2×C3#240611yescharacteristic
590C2×C3C2×#45011yescharacteristic
591C2×C3#300611yescharacteristic
592C2×C3#310311yescharacteristic
593C2×C3#310911yescharacteristic
594C2×C3#322711yescharacteristic
595C2×C3#323411yescharacteristic
596D3—82no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—122no—
611C22—122no—
612C22—122no—
613C22—241no—
614C22—241no—
615C22—241no—
616C22—241no—
617C22#516914yes—
618C22#564811yescharacteristic
619C22#565611yescharacteristic
620C22C2×D3×SD1611yescharacteristic
621C22C2×#91411yescharacteristic
622C22D3×AGL(1,ℤ/8)11yescharacteristic
623C22#577311yescharacteristic
624C22C2×#93711yescharacteristic
625C4—22no—
626C4—42no—
627C4—62no—
628C4—62no—
629C4—62no—
630C4—82no—
631C4—122no—
632C4—124no—
633C4—121no—
634C4#565512yes—
635C4#582412yes—
636C3#2406111yescharacteristic, complemented, Sylow
637C2—24no—
638C2—82no—
639C2—242no—
640C2#8130811yescharacteristic
641C2#8131011yescharacteristic
642C2#8131411yescharacteristic
643C2#8131911yescharacteristic
644C2#8284111yescharacteristic
645C2C2×#516911yescharacteristic
646C2#8394611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#481C2
socleC23×C3C22×D4
prime radicalC2×#5596C2
hypercenterC2×#481D3
hypocenterC3#24061
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
2. C2×C3×#481C2semidirect
3. C2×#4972C2semidirect
4. C2×#4968C2semidirect
5. #71677C2semidirect
6. #70584C2semidirect
7. #70565C2semidirect
8. #70211C2semidirect
9. #70169C2semidirect
10. #69932C2semidirect
11. #69910C2semidirect
50. C2×#481D3semidirect
236. C4⋊Dic3#41semidirect
636. C3#24061semidirect

Maximal subgroups

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

subgroupindex
C2×#55962
C2×C3×#4812
C2×#49722
C2×#49682
#716772
#705842
#705652
#702112
#701692
#699322
#699102
#240613