Finite Group Databasebeta

← #10799111

Subgroups — #10799111

Order 768.

1782 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic
2#84086C211yesmaximal, characteristic, complemented
3C2×#5232C211yesmaximal, characteristic, complemented
4C2×#5221C211yesmaximal, characteristic, complemented
5C2×#5215C211yesmaximal, characteristic, complemented
6C2×#5121C211yesmaximal, characteristic, complemented
7C2×Dic3×M16C211yesmaximal, characteristic, complemented
8#72050C212yesmaximal, complemented
9#72049C212yesmaximal, complemented
10#72039C212yesmaximal, complemented
11#72038C212yesmaximal, complemented
12#21236—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C2×C3×AGL(1,ℤ/8)—24no—
16C2×C3×AGL(1,ℤ/8)—24no—
17C2×C3×AGL(1,ℤ/8)C418yescomplemented
18C22×C3×SD16C2212yes—
19C22×C3×D8C2212yes—
20C22×C3×M16C2211yescharacteristic
21C22×#217—22no—
22C2×#926C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C22×#208C2211yescharacteristic
26#5232—24no—
27#5229C2211yescharacteristic, complemented
28#5229C2211yescharacteristic, complemented
29C2×#835C2212yescomplemented
30#5221—22no—
31#5221C2212yes—
32#5221C2212yes—
33#5221C2211yescharacteristic, complemented
34#5221C2211yescharacteristic, complemented
35#5220—22no—
36#5215—24no—
37C2×#832C2212yes—
38C2×#832C2212yes—
39C2×#832C2212yescomplemented
40C4×#217—22no—
41#5121—24no—
42Dic3×M16—24no—
43C2×#818C2212yescomplemented
44C2×#817C2212yescomplemented
45C2×#815C2212yescomplemented
46C2×C8×Dic3C2212yescomplemented
47C22×AGL(1,ℤ/8)D311yescharacteristic
48C2×#428—31no—
49C2×#417—31no—
50C2×#418—31no—
51C2×C4×M16—31no—
52#2297—31no—
53C2×#390—31no—
54#1973—32no—
55#1972—32no—
56#1970—32no—
57#1968—32no—
58C25×C3—21no—
59C2×C3×#41—22no—
60C2×C3×#41—22no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41C2×C414yes—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×D4—21no—
67C22×C3×D4—22no—
68C22×C3×D4—22no—
69C22×C3×D4—22no—
70C22×C3×D4—21no—
71C22×C3×D4—22no—
72C22×C3×D4—22no—
73C22×C3×D4C2×C414yes—
74C22×C3×D4C2311yescharacteristic
75C22×C3×D4C2312yes—
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C23×Dic3—21no—
79C23×Dic3D412yes—
80C3×AGL(1,ℤ/8)—232no—
81C2×C3×SD16—24no—
82C2×C3×SD16—24no—
83C2×C3×SD16C2×C418yescomplemented
84C2×C3×D8—24no—
85C2×C3×D8—24no—
86C2×C3×D8C2×C418yescomplemented
87C2×C3×M16C2×C414yes—
88C2×C3×M16D414yes—
89C2×C3×M16D414yes—
90C22×C3×C8C2312yes—
91C2×#217—21no—
92C2×#217—22no—
93C2×#217—22no—
94C2×#217—24no—
95C2×#217—24no—
96C2×#217—48no—
97#926—22no—
98#926—22no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3—22no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2312yes—
105C2×C4×Dic3—22no—
106C2×C4×Dic3—22no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3D412yes—
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C2×#208—22no—
112C2×#208—22no—
113C2×#208D414yes—
114C22×C3⋊C8C2312yes—
115#835—24no—
116#832—24no—
117#832—24no—
118#832—24no—
119C3⋊#95—22no—
120C3⋊#95—22no—
121#818—24no—
122#817—24no—
123#815—24no—
124C8×Dic3—24no—
125C22×#41C2×D311yescharacteristic
126C23×D4C2×D311yescharacteristic
127C2×AGL(1,ℤ/8)—24no—
128C2×AGL(1,ℤ/8)—24no—
129C2×AGL(1,ℤ/8)Dic318yescomplemented
130C22×SD16C2×D312yes—
131C22×D8C2×D312yes—
132C22×M16—31no—
133C22×M16C2×D311yescharacteristic
134C2×#117—31no—
135C22×C4⋊C4—31no—
136C22×C22⋊C4—62no—
137C22×C42—31no—
138#428—64no—
139C2×#107—32no—
140C2×#106—32no—
141#418—64no—
142#417—64no—
143C2×#103—32no—
144C2×#102—32no—
145C2×#102—32no—
146C2×#102—32no—
147C4×M16—64no—
148C2×#97—32no—
149C2×C4×C8—32no—
150#391—31no—
151#391—31no—
152#390—32no—
153#390—32no—
154#390—31no—
155#390—31no—
156#390—62no—
157#386—62no—
158C4×C22⋊C4—62no—
159C24×C3—24no—
160C24×C3—22no—
161C24×C3—21no—
162C24×C3—44no—
163C24×C3—44no—
164C24×C3—42no—
165C24×C3—42no—
166C24×C3C2×D411yescharacteristic
167C3×#41—28no—
168C3×#41—28no—
169C3×#41—44no—
170C3×#41—44no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8C22×C412yes—
177C2×C3×D4—24no—
178C2×C3×D4—28no—
179C2×C3×D4—28no—
180C2×C3×D4—24no—
181C2×C3×D4—22no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—21no—
189C2×C3×D4—22no—
190C2×C3×D4—21no—
191C2×C3×D4—42no—
192C2×C3×D4—42no—
193C2×C3×D4—44no—
194C2×C3×D4—42no—
195C2×C3×D4—44no—
196C2×C3×D4—42no—
197C2×C3×D4C22×C412yes—
198C2×C3×D4C22×C414yes—
199C22×C3×C4—21no—
200C22×C3×C4—24no—
201C22×C3×C4—42no—
202C22×C3×C4—42no—
203C22×C3×C4C22×C412yes—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2×D412yes—
207C22×C3×C4C2×D412yes—
208C22×C3×C4#4112yes—
209C22×C3×C4#4112yes—
210C22×C3×C4C2411yescharacteristic
211C22×Dic3—24no—
212C22×Dic3—24no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—22no—
216C22×Dic3—42no—
217C22×Dic3—44no—
218C22×Dic3—42no—
219C22×Dic3C2×D412yes—
220C22×Dic3C2×D412yes—
221C3×SD16—216no—
222C3×D8—216no—
223C3×M16C22⋊C4116yescomplemented
224C2×C3×C8C22×C414yes—
225C2×C3×C8C2×D414yes—
226C2×C3×C8C2×D414yes—
227#217—44no—
228#217—48no—
229#217—42no—
230#217—44no—
231#217—48no—
232C4⋊Dic3—21no—
233C4⋊Dic3—22no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—22no—
238C4⋊Dic3—42no—
239C4×Dic3—24no—
240C4×Dic3—21no—
241C4×Dic3—21no—
242C4×Dic3—21no—
243C4×Dic3—21no—
244C4×Dic3—42no—
245#208—28no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8C2×D414yes—
249C25—21no—
250C2×#41—22no—
251C2×#41—22no—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41C2×Dic314yes—
256C22×Q8C22×D311yescharacteristic
257C22×D4—21no—
258C22×D4—22no—
259C22×D4—21no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4C2×Dic314yes—
265C22×D4C22×D311yescharacteristic
266C22×D4C22×D312yes—
267C23×C4—21no—
268C23×C4—32no—
269C23×C4—61no—
270C23×C4C22×D311yescharacteristic
271AGL(1,ℤ/8)—232no—
272C2×SD16—24no—
273C2×SD16—24no—
274C2×SD16C2×Dic318yescomplemented
275C2×D8—24no—
276C2×D8—24no—
277C2×D8C2×Dic318yescomplemented
278C2×M16—34no—
279C2×M16—62no—
280C2×M16—62no—
281C2×M16C2×Dic314yes—
282C2×M16#6714yes—
283C2×M16#6714yes—
284C22×C8—32no—
285C22×C8C22×D312yes—
286#117—62no—
287#117—62no—
288C2×C4⋊C4—32no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—61no—
294C2×C22⋊C4—64no—
295C2×C22⋊C4—64no—
296C2×C22⋊C4—62no—
297C2×C22⋊C4—62no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—128no—
300C2×C42—32no—
301C2×C42—31no—
302C2×C42—31no—
303C2×C42—62no—
304C2×C42—62no—
305C2×C42—61no—
306#107—64no—
307#106—64no—
308#103—64no—
309#102—64no—
310#102—64no—
311#102—64no—
312#97—64no—
313C4×C8—64no—
314#95—62no—
315#95—62no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—24no—
320C23×C3—416no—
321C23×C3—41no—
322C23×C3—42no—
323C23×C3—41no—
324C23×C3—42no—
325C23×C3—44no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—44no—
329C23×C3—48no—
330C23×C3—84no—
331C23×C3—84no—
332C23×C3—88no—
333C23×C3C4×D412yes—
334C23×C3#12112yes—
335C23×C3#12712yes—
336C23×C3C22×D411yescharacteristic
337C3×Q8—22no—
338C3×Q8—22no—
339C3×Q8—41no—
340C3×Q8—41no—
341C3×D4—22no—
342C3×D4—24no—
343C3×D4—24no—
344C3×D4—22no—
345C3×D4—48no—
346C3×D4—42no—
347C3×D4—41no—
348C3×D4—42no—
349C3×D4—41no—
350C3×D4—48no—
351C2×C3×C4—22no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—48no—
355C2×C3×C4C2×C22⋊C414yes—
356C2×C3×C4C4×D414yes—
357C2×C3×C4C4×D412yes—
358C2×C3×C4#12114yes—
359C2×C3×C4#12112yes—
360C2×C3×C4#12414yes—
361C2×C3×C4#12712yes—
362C2×C3×C4C23×C411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—48no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—42no—
377C2×Dic3—44no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—84no—
381C2×Dic3AGL(1,ℤ/8)14yescomplemented
382C3×C8C2×C22⋊C418yescomplemented
383C3⋊C8—24no—
384C24—24no—
385C24—21no—
386C24—22no—
387C24—44no—
388C24—44no—
389C24—42no—
390C24—42no—
391C24D3×D411yescharacteristic
392#41—28no—
393#41—28no—
394#41—44no—
395#41—44no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8C22×Dic312yes—
402C2×D4—24no—
403C2×D4—28no—
404C2×D4—24no—
405C2×D4—28no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4C22×Dic312yes—
423C2×D4C22×Dic314yes—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—32no—
427C22×C4—32no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—64no—
431C22×C4—64no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—62no—
435C22×C4—122no—
436C22×C4—124no—
437C22×C4—122no—
438C22×C4D3×D411yescharacteristic
439C22×C4D3×D412yes—
440C22×C4#23712yes—
441C22×C4#23712yes—
442C22×C4C22×Dic312yes—
443C22×C4C2×#6712yes—
444C22×C4C2×#6712yes—
445C22×C4C23×D311yescharacteristic
446SD16—216no—
447D8—216no—
448M16—68no—
449M16#217116yescomplemented
450C2×C8—34no—
451C2×C8—62no—
452C2×C8—62no—
453C2×C8C22×Dic314yes—
454C2×C8C2×#6714yes—
455C2×C8C2×#6714yes—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—62no—
459C4⋊C4—61no—
460C4⋊C4—62no—
461C4⋊C4—61no—
462C4⋊C4—122no—
463C22⋊C4—128no—
464C22⋊C4—124no—
465C22⋊C4—128no—
466C22⋊C4—124no—
467C22⋊C4—122no—
468C42—64no—
469C42—61no—
470C42—61no—
471C42—61no—
472C42—61no—
473C42—122no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—24no—
477C22×C3—24no—
478C22×C3—44no—
479C22×C3—48no—
480C22×C3—42no—
481C22×C3—44no—
482C22×C3—816no—
483C22×C3—81no—
484C22×C3—82no—
485C22×C3—81no—
486C22×C3—82no—
487C22×C3—82no—
488C22×C3—84no—
489C22×C3#39014yes—
490C22×C3C2×C4×D411yescharacteristic
491C22×C3C2×#12111yescharacteristic
492C22×C3C2×#12711yescharacteristic
493C22×C3C2×AGL(1,ℤ/8)12yes—
494C22×C3#57512yes—
495C3×C4—42no—
496C3×C4#39014yes—
497C3×C4C22×C22⋊C411yescharacteristic
498C3×C4C2×C4×D411yescharacteristic
499C3×C4C2×#12111yescharacteristic
500C3×C4C2×#12411yescharacteristic
501Dic3—24no—
502Dic3—42no—
503Dic3—81no—
504Dic3—82no—
505Dic3—81no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—416no—
511C23—48no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—41no—
517C23—42no—
518C23—41no—
519C23—42no—
520C23—84no—
521C23—84no—
522C23—88no—
523C23D4×Dic312yes—
524C23#93812yes—
525C23#94012yes—
526C23C2×D3×D411yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—41no—
530Q8—41no—
531D4—24no—
532D4—22no—
533D4—22no—
534D4—24no—
535D4—48no—
536D4—48no—
537D4—42no—
538D4—41no—
539D4—42no—
540D4—41no—
541C2×C4—22no—
542C2×C4—34no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—48no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—128no—
552C2×C4—124no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—122no—
558C2×C4—244no—
559C2×C4D4×Dic312yes—
560C2×C4D4×Dic314yes—
561C2×C4#93614yes—
562C2×C4#93812yes—
563C2×C4#93814yes—
564C2×C4#94012yes—
565C2×C4C2×#21714yes—
566C2×C4C2×D3×D411yescharacteristic
567C2×C4C2×#23711yescharacteristic
568C2×C4C2×#23711yescharacteristic
569C2×C4C23×Dic311yescharacteristic
570C2×C4C22×#6711yescharacteristic
571C2×C4C22×#6711yescharacteristic
572C8—64no—
573C8C2×#21718yescomplemented
574C2×C3—24no—
575C2×C3—82no—
576C2×C3—84no—
577C2×C3C2×#39011yescharacteristic
578C2×C3C4×AGL(1,ℤ/8)12yes—
579C2×C3#305512yes—
580C2×C3#315111yescharacteristic
581C2×C3#315311yescharacteristic
582C22—24no—
583C22—22no—
584C22—22no—
585C22—24no—
586C22—42no—
587C22—44no—
588C22—48no—
589C22—44no—
590C22—816no—
591C22—82no—
592C22—81no—
593C22—82no—
594C22—81no—
595C22—84no—
596C22—82no—
597C22#522114yes—
598C22D3×AGL(1,ℤ/8)12yes—
599C22#577312yes—
600C22C2×D4×Dic311yescharacteristic
601C22C2×#93811yescharacteristic
602C22C2×#94011yescharacteristic
603C4—42no—
604C4—64no—
605C4—122no—
606C4—242no—
607C4—241no—
608C4—241no—
609C4#522114yes—
610C4C2×D4×Dic311yescharacteristic
611C4C2×#93611yescharacteristic
612C4C2×#93811yescharacteristic
613C4C22×#21711yescharacteristic
614C3#2123611yescharacteristic, complemented, Sylow
615C2—24no—
616C2—84no—
617C2—82no—
618C2Dic3×AGL(1,ℤ/8)12yes—
619C2#8393711yescharacteristic
620C2#8393811yescharacteristic
621C2#8395012yes—
622C2C2×#522111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
Fitting, prime radicalC22×C3×AGL(1,ℤ/8)C2
socleC23×C3C22×D4
cosocleC22×C3×C4C24
hypercenterC22×AGL(1,ℤ/8)D3
hypocenterC3#21236
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
2. #84086C2semidirect
3. C2×#5232C2semidirect
4. C2×#5221C2semidirect
5. C2×#5215C2semidirect
6. C2×#5121C2semidirect
7. C2×Dic3×M16C2semidirect
8. #72050C2semidirect
9. #72049C2semidirect
10. #72039C2semidirect
11. #72038C2semidirect
17. C2×C3×AGL(1,ℤ/8)C4semidirect
27. #5229C22semidirect
28. #5229C22semidirect
29. C2×#835C22semidirect
33. #5221C22semidirect
34. #5221C22semidirect
39. C2×#832C22semidirect
43. C2×#818C22semidirect
44. C2×#817C22semidirect
45. C2×#815C22semidirect
46. C2×C8×Dic3C22semidirect
83. C2×C3×SD16C2×C4semidirect
86. C2×C3×D8C2×C4semidirect
129. C2×AGL(1,ℤ/8)Dic3semidirect
223. C3×M16C22⋊C4semidirect
274. C2×SD16C2×Dic3semidirect
277. C2×D8C2×Dic3semidirect
381. C2×Dic3AGL(1,ℤ/8)semidirect
382. C3×C8C2×C22⋊C4semidirect
449. M16#217semidirect
573. C8C2×#217semidirect
614. C3#21236semidirect

Maximal subgroups

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

subgroupindex
C22×C3×AGL(1,ℤ/8)2
#840862
C2×#52322
C2×#52212
C2×#52152
C2×#51212
C2×Dic3×M162
#720502
#720492
#720392
#720382
#212363