Finite Group Databasebeta

← #10848766

Subgroups — #10848766

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×#5680C211yesmaximal, characteristic, complemented
2C2×C3×#496C211yesmaximal, characteristic, complemented
3#83192C211yesmaximal, characteristic, complemented
4C2×#5071C211yesmaximal, characteristic, complemented
5#72054C212yesmaximal, complemented
6#72048C212yesmaximal, complemented
7#72043C212yesmaximal, complemented
8#72037C212yesmaximal, complemented
9#71726C211yesmaximal, characteristic, complemented
10#69827C211yesmaximal, characteristic, complemented
11#69807C211yesmaximal, characteristic, complemented
12#25605—31nomaximal, Sylow
13C22×C3×SD16C2212yes—
14C22×C3×D8C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124C2211yescharacteristic
17C2×#946C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×#938—21no—
20C2×#938—21no—
21C2×#938C2211yescharacteristic
22C2×#938C2212yes—
23C2×D4×Dic3C2211yescharacteristic
24#5680—24no—
25#5680—24no—
26C2×#883—21no—
27C2×#67⋊C4—21no—
28C2×#879—21no—
29C2×#877—21no—
30C2×C42×D3C2211yescharacteristic
31C2×#870C2211yescharacteristic
32C3×#496—24no—
33C2×C3×#97C2211yescharacteristic
34C2×#835C2212yes—
35#5221—21no—
36#5221—21no—
37#5221C2212yes—
38C2×#832C2212yes—
39C2×#832C2212yes—
40C2×#832C2212yes—
41C2×(C2×D3)⋊C8C2212yes—
42#5071—24no—
43#4935C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4551C2212yes—
46#4551C2212yes—
47#4550C2212yes—
48C2×#529—31no—
49C2×#496D311yescharacteristic, complemented
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×D4—21no—
62C22×C3×D4C2312yes—
63C22×C3×D4C2311yescharacteristic
64C22×#67—21no—
65C22×#67—22no—
66C23×Dic3—21no—
67C22×C4×D3D411yescharacteristic
68C22×C4×D3D411yescharacteristic
69C22×C4×D3C2311yescharacteristic
70C2×C3×SD16—24no—
71C2×C3×SD16—24no—
72C2×C3×SD16—42no—
73C2×C3×D8—24no—
74C2×C3×D8—24no—
75C2×C3×D8—42no—
76C22×C3×C8C2312yes—
77C3×#127—22no—
78C3×#127—22no—
79C3×#124—22no—
80C3×#124D414yescomplemented
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C42C2311yescharacteristic
84C2×#217—21no—
85C2×#217—21no—
86C2×#217—22no—
87C2×#217—22no—
88C2×#217—22no—
89#946—22no—
90#946—22no—
91Q8×Dic3—22no—
92Q8×Dic3—22no—
93#938—22no—
94#938—22no—
95#938—42no—
96#938—42no—
97#938—44no—
98#938—44no—
99D4×Dic3—22no—
100D4×Dic3—22no—
101C2×(C2×D3)⋊C4—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2312yes—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3D411yescharacteristic
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2312yes—
116#883—28no—
117#67⋊C4—28no—
118#879—28no—
119#877—28no—
120C42×D3—22no—
121C42×D3—22no—
122#870—22no—
123#870D414yescomplemented
124C3×#97—22no—
125#835—42no—
126#832—24no—
127#832—24no—
128#832—42no—
129(C2×D3)⋊C8—24no—
130#803—22no—
131C22×SD16C2×D312yes—
132C22×D8C2×D312yes—
133C2×#127C2×D311yescharacteristic
134#529—64no—
135#529—64no—
136C2×#126—61no—
137C2×#125—31no—
138C2×#124C2×D311yescharacteristic
139C2×#123—61no—
140C2×C22⋊Q8—31no—
141C2×#121—31no—
142C2×#121—32no—
143C2×#121—61no—
144C2×#121—61no—
145C2×C4×Q8—31no—
146C2×C4×D4—31no—
147C2×C4×D4—61no—
148C2×#117—61no—
149C22×C42—31no—
150#496—24no—
151#490—64no—
152C2×#103—32no—
153C2×#102—32no—
154C2×#102—32no—
155C2×#102—32no—
156C2×C22⋊C8—32no—
157C2×#97—31no—
158C2×#97C2×D311yescharacteristic
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×C3—22no—
168C23×D3C2×D411yescharacteristic
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8C2×D412yes—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—42no—
183C2×C3×D4—42no—
184C2×C3×D4—42no—
185C2×C3×D4—42no—
186C2×C3×D4—42no—
187C2×C3×D4—42no—
188C2×C3×D4C2×D412yes—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C2×#67—24no—
194C2×#67—42no—
195C2×#67—42no—
196C2×#67—44no—
197C2×#67—44no—
198C2×#67—44no—
199C22×Dic3—24no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—22no—
203C22×Dic3—42no—
204C22×Dic3—42no—
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C2×C4×D3—22no—
209C2×C4×D3—22no—
210C2×C4×D3—22no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—22no—
214C2×C4×D3—22no—
215C2×C4×D3—22no—
216C2×C4×D3—41no—
217C2×C4×D3—41no—
218C3×SD16—48no—
219C3×D8—48no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8#4114yes—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—24no—
225C3×C42C2×D412yes—
226C3×C42C2×D412yes—
227#217—24no—
228#217—24no—
229#217—44no—
230#217—44no—
231#217—44no—
232(C2×D3)⋊C4—24no—
233C4⋊Dic3—24no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—22no—
237C4⋊Dic3—22no—
238C4⋊Dic3—42no—
239C4⋊Dic3—42no—
240C4⋊Dic3C2×D412yes—
241C4⋊Dic3C2×D412yes—
242Dic3⋊C4—24no—
243Dic3⋊C4—24no—
244Dic3⋊C4—24no—
245C4×Dic3—24no—
246C4×Dic3—22no—
247C4×Dic3—22no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C22×Q8C22×D311yescharacteristic
254C22×D4—21no—
255C22×D4—21no—
256C22×D4—61no—
257C22×D4—62no—
258C22×D4C22×D312yes—
259C22×D4C22×D311yescharacteristic
260C23×C4—31no—
261C23×C4—31no—
262C23×C4—31no—
263C23×C4—61no—
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—42no—
267C2×D8—24no—
268C2×D8—24no—
269C2×D8—42no—
270C22×C8—32no—
271C22×C8C22×D312yes—
272#127—22no—
273#127—22no—
274#126—68no—
275#125—34no—
276#125—62no—
277#124—22no—
278#124#6714yescomplemented
279#123—68no—
280C22⋊Q8—62no—
281C22⋊Q8—62no—
282#121—62no—
283#121—62no—
284#121—122no—
285#121—122no—
286#121—124no—
287#121—124no—
288C4×Q8—62no—
289C4×Q8—62no—
290C4×D4—68no—
291C4×D4—62no—
292C4×D4—62no—
293#117—68no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—32no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—62no—
309C2×C22⋊C4—62no—
310C2×C22⋊C4—62no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—61no—
315C2×C42—62no—
316C2×C42—62no—
317C2×C42C22×D311yescharacteristic
318#103—122no—
319#102—64no—
320#102—64no—
321#102—122no—
322C22⋊C8—64no—
323#97—22no—
324#97—62no—
325C23×C3—22no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3C22×D411yescharacteristic
340C22×D3—21no—
341C22×D3—21no—
342C22×D3—21no—
343C22×D3—21no—
344C22×D3—21no—
345C22×D3—21no—
346C22×D3AGL(1,ℤ/8)12yes—
347C3×Q8—24no—
348C3×Q8—42no—
349C3×D4—24no—
350C3×D4—42no—
351C3×D4—42no—
352C3×D4—42no—
353C3×D4—42no—
354C3×D4—42no—
355C3×D4—84no—
356C3×D4—84no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4#12012yes—
365C2×C3×C4#12012yes—
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C2×#4112yes—
369C2×C3×C4H(5,𝔽2)12yes—
370#67—48no—
371#67—88no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—48no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—42no—
390C2×Dic3AGL(1,ℤ/8)12yes—
391C4×D3—44no—
392C4×D3—44no—
393C4×D3—42no—
394C4×D3—42no—
395C3×C8—24no—
396C3⋊C8—42no—
397C24—21no—
398C24—22no—
399C24—31no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8C2×#6712yes—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—64no—
420C2×D4—122no—
421C2×D4—122no—
422C2×D4—124no—
423C2×D4—124no—
424C2×D4—124no—
425C2×D4C2×#6712yes—
426C22×C4—21no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—64no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—61no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—61no—
441C22×C4—62no—
442C22×C4—121no—
443C22×C4—121no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4D3×D411yescharacteristic
447C22×C4D3×D411yescharacteristic
448C22×C4C23×D311yescharacteristic
449SD16—48no—
450D8—48no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—62no—
454C2×C8—62no—
455C2×C8—62no—
456C2×C8#23714yes—
457C4⋊C4—32no—
458C4⋊C4—32no—
459C4⋊C4—64no—
460C4⋊C4—64no—
461C4⋊C4—64no—
462C4⋊C4—64no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—62no—
466C4⋊C4—62no—
467C4⋊C4—122no—
468C4⋊C4—122no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—64no—
472C22⋊C4—64no—
473C22⋊C4—64no—
474C22⋊C4—124no—
475C22⋊C4—124no—
476C22⋊C4—124no—
477C42—64no—
478C42—62no—
479C42—62no—
480C42—61no—
481C42—61no—
482C42D3×D412yes—
483C42C2×#6712yes—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3—82no—
492C22×C3—82no—
493C22×C3—82no—
494C22×C3—82no—
495C22×C3—82no—
496C22×C3—82no—
497C22×C3—82no—
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—22no—
506C2×D3—22no—
507C2×D3—22no—
508C2×D3—41no—
509C2×D3—41no—
510C2×D3—41no—
511C2×D3—41no—
512C3×C4—22no—
513C3×C4—22no—
514C3×C4—42no—
515C3×C4#54612yes—
516C3×C4#55012yes—
517Dic3—22no—
518Dic3—41no—
519Dic3—41no—
520Dic3—42no—
521Dic3—42no—
522Dic3—82no—
523C23—22no—
524C23—32no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—61no—
539C23—61no—
540C23—61no—
541C23—61no—
542C23—61no—
543C23—61no—
544C23C2×D3×D411yescharacteristic
545Q8—24no—
546Q8—42no—
547D4—24no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—84no—
554D4—84no—
555D4—128no—
556D4—248no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—32no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—124no—
576C2×C4—124no—
577C2×C4—128no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—122no—
586C2×C4—122no—
587C2×C4#93712yes—
588C2×C4#93712yes—
589C2×C4C2×D3×D411yescharacteristic
590C2×C4C2×#23712yes—
591C2×C4#100412yes—
592C2×C4C22×#6711yescharacteristic
593C8—24no—
594C8—122no—
595C2×C3—82no—
596C2×C3—162no—
597C2×C3#243111yescharacteristic
598C2×C3#300611yescharacteristic
599C2×C3#300611yescharacteristic
600C2×C3#317111yescharacteristic
601C2×C3#318011yescharacteristic
602C2×C3#321711yescharacteristic
603C2×C3#321911yescharacteristic
604D3—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—62no—
609C22—62no—
610C22—62no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—121no—
623C22—121no—
624C22—121no—
625C22—121no—
626C22#571511yescharacteristic
627C22#571611yescharacteristic
628C22D3×AGL(1,ℤ/8)11yescharacteristic
629C22D3×AGL(1,ℤ/8)11yescharacteristic
630C22#577311yescharacteristic
631C22#577311yescharacteristic
632C22C2×#93711yescharacteristic
633C4—22no—
634C4—22no—
635C4—42no—
636C4—62no—
637C4—121no—
638C4—122no—
639C4—122no—
640C4—121no—
641C4—242no—
642C4#571812yes—
643C4#580612yes—
644C3#2560511yescharacteristic, complemented, Sylow
645C2—82no—
646C2—122no—
647C2—162no—
648C2#8183611yescharacteristic
649C2#8183811yescharacteristic
650C2#8185111yescharacteristic
651C2#8185211yescharacteristic
652C2#8318711yescharacteristic
653C2#8394611yescharacteristic
654C2#8394611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#496C2
socleC23×C3C22×D4
hypercenterC2×#496D3
hypocenterC3#25605
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#496

Complements

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

NH ≅ G/Ntype
1. C2×#5680C2semidirect
2. C2×C3×#496C2semidirect
3. #83192C2semidirect
4. C2×#5071C2semidirect
5. #72054C2semidirect
6. #72048C2semidirect
7. #72043C2semidirect
8. #72037C2semidirect
9. #71726C2semidirect
10. #69827C2semidirect
11. #69807C2semidirect
49. C2×#496D3semidirect
80. C3×#124D4semidirect
123. #870D4semidirect
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×#56802
C2×C3×#4962
#831922
C2×#50712
#720542
#720482
#720432
#720372
#717262
#698272
#698072
#256053