Finite Group Databasebeta

← #10827572

Subgroups — #10827572

Order 768.

1418 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5641C211yesmaximal, characteristic, complemented
2C2×C3×#450C211yesmaximal, characteristic, complemented
3#84053C211yesmaximal, characteristic
4C2×#5215C211yesmaximal, characteristic, complemented
5#72054C212yesmaximal, complemented
6#72048C212yesmaximal, complemented
7#71697C211yesmaximal, characteristic, complemented
8#70201C212yesmaximal, complemented
9#70163C212yesmaximal, complemented
10#69953C211yesmaximal, characteristic, complemented
11#69926C211yesmaximal, characteristic, complemented
12#23979—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×#953—21no—
17C2×#946C2211yescharacteristic, complemented
18C2×Q8×Dic3C2211yescharacteristic, complemented
19C2×#938C2212yes—
20C2×C4×#67C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C22×#208C2211yescharacteristic
23#5641—24no—
24#5641—24no—
25C2×#898—21no—
26C2×#892C2211yescharacteristic, complemented
27C2×#888C2211yescharacteristic, complemented
28C2×#887—21no—
29C2×#885—21no—
30C2×#879—21no—
31C3×#450—24no—
32C2×C3×#102C2212yes—
33C2×C3×C22⋊C8C2211yescharacteristic
34#5247—21no—
35C2×#835C2212yescomplemented
36#5223—21no—
37#5222—21no—
38#5221C2212yes—
39#5215—24no—
40C2×#832C2212yes—
41C2×#832C2212yes—
42#5212C2211yescharacteristic
43C2×(C2×D3)⋊C8C2212yescomplemented
44#4551C2212yescomplemented
45#4551C2212yescomplemented
46#4550C2212yescomplemented
47#4529C2212yescomplemented
48C2×#556—31no—
49C2×#450D311yescharacteristic, complemented
50C2×#418—31no—
51#2408—31no—
52#2023—32no—
53#2000—32no—
54#1894—31no—
55#1880—32no—
56#1875—32no—
57#1870—31no—
58#1825—31no—
59C25×C3—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—24no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4—21no—
65C22×C3×D4—42no—
66C22×C3×D4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C22×#67—22no—
69C22×#67D411yescharacteristic
70C22×C4×D3C2311yescharacteristic
71C22×Dic6—21no—
72C2×C3×SD16—24no—
73C2×C3×SD16—24no—
74C2×C3×SD16—42no—
75C22×C3×C8C2312yes—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8D414yescomplemented
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C22⋊C4—21no—
81#953—44no—
82C2×#217—21no—
83C2×#217—21no—
84C2×#217—22no—
85C2×#217—22no—
86C2×#217—22no—
87C2×#217D411yescharacteristic
88#946—22no—
89#946—22no—
90Q8×Dic3—22no—
91Q8×Dic3—22no—
92#938—24no—
93#938—42no—
94C4×#67—22no—
95C4×#67—22no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4D411yescharacteristic
99#926—22no—
100#926D414yescomplemented
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2312yes—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4D411yescharacteristic
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3C2311yescharacteristic
112C2×#208—22no—
113C2×#208—42no—
114C22×C3⋊C8C2312yes—
115#898—28no—
116#892—22no—
117#892—22no—
118#888—22no—
119#888—22no—
120#887—28no—
121#885—28no—
122#879—28no—
123C3×#102—24no—
124C3×C22⋊C8—22no—
125#835—42no—
126#832—24no—
127#832—24no—
128C3⋊#95—21no—
129(C2×D3)⋊C8—42no—
130C23×D4C2×D311yescharacteristic
131C22×SD16C2×D312yes—
132C22×M16—31no—
133#556—64no—
134#556—64no—
135C2×C4⋊Q8—31no—
136C2×#126—61no—
137C2×#126—61no—
138C2×#124—61no—
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8C2×D311yescharacteristic
141C2×#121—32no—
142C2×#120—61no—
143C2×C4×Q8—31no—
144C2×C4×Q8—61no—
145C2×C4×D4—31no—
146C2×#117—31no—
147C2×#117—31no—
148#450—24no—
149#418—64no—
150C2×#103—32no—
151C2×#102—32no—
152C2×#102—32no—
153C2×#102C2×D312yes—
154C2×C22⋊C8—32no—
155C2×C22⋊C8C2×D311yescharacteristic
156#394—61no—
157#392—61no—
158#390—32no—
159#380—31no—
160#379—61no—
161#340—32no—
162#340—32no—
163#340—32no—
164#336—32no—
165C24×C3—22no—
166C24×C3—44no—
167C24×C3—42no—
168C24×C3—42no—
169C24×C3—42no—
170C24×C3—82no—
171C24×C3C2×D411yescharacteristic
172C23×D3C2×D411yescharacteristic
173C2×C3×Q8—21no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8C2×D412yes—
177C2×C3×D4—28no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—48no—
184C2×C3×D4—44no—
185C2×C3×D4—44no—
186C2×C3×D4—42no—
187C2×C3×D4#4114yes—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2411yescharacteristic
196C2×#67—22no—
197C2×#67—22no—
198C2×#67—41no—
199C2×#67—44no—
200C2×#67—44no—
201C2×#67—82no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—22no—
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—22no—
211C2×C4×D3—22no—
212C2×Dic6—24no—
213C2×Dic6—42no—
214C2×Dic6—42no—
215C3×SD16—48no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4C2×D412yes—
222C3×C22⋊C4—24no—
223#217—22no—
224#217—24no—
225#217—44no—
226#217—42no—
227#217—44no—
228#217—82no—
229(C2×D3)⋊C4—24no—
230(C2×D3)⋊C4—24no—
231(C2×D3)⋊C4—22no—
232C4⋊Dic3—24no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—22no—
236C4⋊Dic3—22no—
237C4⋊Dic3C2×D412yes—
238C4⋊Dic3C2×D412yes—
239Dic3⋊C4—24no—
240Dic3⋊C4—24no—
241Dic3⋊C4—22no—
242C4×Dic3—24no—
243C4×Dic3—24no—
244C4×Dic3—24no—
245C4×Dic3—21no—
246C4×Dic3—21no—
247#208—44no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C25—21no—
252C22×Q8—61no—
253C22×Q8C22×D311yescharacteristic
254C22×D4—24no—
255C22×D4—22no—
256C22×D4—21no—
257C22×D4—22no—
258C22×D4—31no—
259C22×D4—42no—
260C22×D4—62no—
261C22×D4C22×D312yes—
262C23×C4—31no—
263C23×C4C22×D311yescharacteristic
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—42no—
267C2×M16—62no—
268C2×M16—122no—
269C22×C8—32no—
270C22×C8C22×D312yes—
271C4⋊Q8—62no—
272C4⋊Q8—62no—
273#126—68no—
274#126—68no—
275#124—68no—
276C22⋊Q8—22no—
277C22⋊Q8—62no—
278C22⋊Q8—62no—
279C22⋊Q8#6714yescomplemented
280#121—64no—
281#121—122no—
282#120—124no—
283C4×Q8—68no—
284C4×Q8—62no—
285C4×Q8—62no—
286C4×D4—62no—
287C4×D4—62no—
288#117—34no—
289#117—62no—
290#117—62no—
291#117—62no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—31no—
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⋊C4C22×D311yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—31no—
303C2×C22⋊C4—31no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—62no—
308C2×C22⋊C4—62no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—62no—
311C2×C42—31no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42—61no—
315#103—122no—
316#102—24no—
317#102—64no—
318#102—64no—
319C22⋊C8—22no—
320C22⋊C8—122no—
321#95—61no—
322C23×C3—22no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—22no—
326C23×C3—24no—
327C23×C3—48no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—84no—
335C23×C3—84no—
336C23×C3—88no—
337C23×C3—88no—
338C23×C3#12012yes—
339C23×C3C22×D411yescharacteristic
340C22×D3—21no—
341C22×D3—22no—
342C22×D3—21no—
343C22×D3—21no—
344C22×D3—22no—
345C3×Q8—24no—
346C3×Q8—41no—
347C3×Q8—41no—
348C3×D4—24no—
349C3×D4—24no—
350C3×D4—48no—
351C3×D4—42no—
352C3×D4—42no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—42no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4#12012yes—
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C2×#4112yes—
366C2×C3×C4H(5,𝔽2)12yes—
367#67—44no—
368#67—88no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—22no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3AGL(1,ℤ/8)12yes—
384C4×D3—41no—
385C4×D3—41no—
386C4×D3—42no—
387Dic6—48no—
388C3×C8—42no—
389C3⋊C8—42no—
390C24—22no—
391C24—31no—
392C24—44no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—82no—
397C24D3×D411yescharacteristic
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—64no—
402C2×Q8—122no—
403C2×Q8—122no—
404C2×Q8C2×#6712yes—
405C2×D4—28no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—48no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—62no—
416C2×D4—62no—
417C2×D4—121no—
418C2×D4—124no—
419C2×D4—124no—
420C2×D4—242no—
421C2×D4#23714yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—61no—
435C22×C4—62no—
436C22×C4D3×D411yescharacteristic
437C22×C4D3×D412yes—
438C22×C4C2×#6712yes—
439C22×C4C23×D311yescharacteristic
440SD16—48no—
441M16—124no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—62no—
446C2×C8—62no—
447C2×C8—62no—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—32no—
451C4⋊C4—32no—
452C4⋊C4—64no—
453C4⋊C4—64no—
454C4⋊C4—64no—
455C4⋊C4—62no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—62no—
459C4⋊C4—62no—
460C4⋊C4C2×#6712yes—
461C22⋊C4—24no—
462C22⋊C4—64no—
463C22⋊C4—64no—
464C22⋊C4—62no—
465C22⋊C4—64no—
466C22⋊C4—62no—
467C22⋊C4—124no—
468C22⋊C4—124no—
469C22⋊C4—122no—
470C22⋊C4—242no—
471C42—64no—
472C42—64no—
473C42—64no—
474C42—61no—
475C42—61no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—44no—
482C22×C3—44no—
483C22×C3—44no—
484C22×C3—816no—
485C22×C3—82no—
486C22×C3—82no—
487C22×C3—82no—
488C22×C3—82no—
489C22×C3C2×#12011yescharacteristic
490C22×C3#54612yes—
491C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
492C22×C3#57511yescharacteristic
493C22×C3#57712yes—
494C2×D3—41no—
495C2×D3—41no—
496C2×D3—41no—
497C2×D3—42no—
498C2×D3—42no—
499C3×C4—22no—
500C3×C4—42no—
501C3×C4—42no—
502C3×C4#52511yescharacteristic
503C3×C4#53511yescharacteristic
504C3×C4#54612yes—
505Dic3—22no—
506Dic3—41no—
507Dic3—42no—
508Dic3—41no—
509Dic3—42no—
510Dic3—82no—
511C23—22no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—24no—
516C23—48no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—62no—
524C23—61no—
525C23—61no—
526C23—61no—
527C23—62no—
528C23—84no—
529C23—84no—
530C23—88no—
531C23—88no—
532C23#93712yes—
533C23C2×D3×D411yescharacteristic
534Q8—24no—
535Q8—41no—
536Q8—41no—
537Q8—128no—
538D4—24no—
539D4—24no—
540D4—48no—
541D4—42no—
542D4—42no—
543D4—124no—
544D4—248no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—32no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—61no—
560C2×C4—62no—
561C2×C4—61no—
562C2×C4—62no—
563C2×C4—122no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—122no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4#93712yes—
573C2×C4C2×D3×D411yescharacteristic
574C2×C4C2×#23712yes—
575C2×C4#100412yes—
576C2×C4C22×#6711yescharacteristic
577C8—42no—
578C8—122no—
579C2×C3—42no—
580C2×C3—84no—
581C2×C3#240411yescharacteristic
582C2×C3#300611yescharacteristic
583C2×C3#301011yescharacteristic
584C2×C3#331112yes—
585C2×C3#332412yes—
586D3—81no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—42no—
591C22—42no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—816no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—122no—
601C22—122no—
602C22—121no—
603C22—121no—
604C22—121no—
605C22#559112yes—
606C22#576212yes—
607C22D3×AGL(1,ℤ/8)11yescharacteristic
608C22#577311yescharacteristic
609C22C2×#93711yescharacteristic
610C4—22no—
611C4—42no—
612C4—42no—
613C4—62no—
614C4—121no—
615C4—122no—
616C4—121no—
617C4—122no—
618C4—242no—
619C4#558611yescharacteristic
620C4#559211yescharacteristic
621C4#580612yes—
622C3#2397911yescharacteristic, complemented, Sylow
623C2—42no—
624C2—84no—
625C2—241no—
626C2#8016212yes—
627C2#8016512yes—
628C2#8258111yescharacteristic
629C2#8386011yescharacteristic
630C2#8394611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#450C2
socleC23×C3C22×D4
prime radical#84053C2
hypercenterC2×#450D3
hypocenterC3#23979
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
1. C2×#5641C2semidirect
2. C2×C3×#450C2semidirect
4. C2×#5215C2semidirect
5. #72054C2semidirect
6. #72048C2semidirect
7. #71697C2semidirect
8. #70201C2semidirect
9. #70163C2semidirect
10. #69953C2semidirect
11. #69926C2semidirect
17. C2×#946C22semidirect
18. C2×Q8×Dic3C22semidirect
26. C2×#892C22semidirect
27. C2×#888C22semidirect
35. C2×#835C22semidirect
43. C2×(C2×D3)⋊C8C22semidirect
44. #4551C22semidirect
45. #4551C22semidirect
46. #4550C22semidirect
47. #4529C22semidirect
49. C2×#450D3semidirect
77. C3×C22⋊Q8D4semidirect
100. #926D4semidirect
279. C22⋊Q8#67semidirect
622. C3#23979semidirect

Maximal subgroups

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

subgroupindex
C2×#56412
C2×C3×#4502
#840532
C2×#52152
#720542
#720482
#716972
#702012
#701632
#699532
#699262
#239793