Finite Group Databasebeta

← #10799100

Subgroups — #10799100

Order 768.

1678 conjugacy classes of subgroups.

The subgroup lattice has 608 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#84087C211yesmaximal, characteristic, complemented
3C2×#5233C211yesmaximal, characteristic, complemented
4#84044C211yesmaximal, characteristic, complemented
5C22×#832C211yesmaximal, characteristic, complemented
6C2×#5121C211yesmaximal, characteristic, complemented
7C2×#5122C211yesmaximal, characteristic, complemented
8#72050C212yesmaximal, complemented
9#72049C212yesmaximal, complemented
10#72039C212yesmaximal, complemented
11#72038C212yesmaximal, complemented
12#21235—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
21C2×#926C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C23×C3⋊C8C2211yescharacteristic
25#5247—22no—
26#5233—24no—
27#5229C2212yescomplemented
28C2×#835C2211yescharacteristic, complemented
29C2×#835C2211yescharacteristic, complemented
30#5221—22no—
31#5221C2212yes—
32#5221C2212yes—
33#5221C2212yescomplemented
34#5220—22no—
35C2×#832—24no—
36C2×#832C2212yes—
37C2×#832C2212yes—
38C2×#832C2211yescharacteristic, complemented
39C2×#832C2211yescharacteristic, complemented
40#5212—22no—
41#5122—24no—
42#5121—24no—
43C2×#818C2212yescomplemented
44C2×#817C2212yescomplemented
45C2×#815C2212yescomplemented
46C2×C8×Dic3C2212yescomplemented
47C22×AGL(1,ℤ/8)D311yescharacteristic
48C2×#428—31no—
49C2×#416—31no—
50C22×#102—31no—
51C2×#405—31no—
52#2311—31no—
53#2309—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—
79C3×AGL(1,ℤ/8)—232no—
80C2×C3×SD16—24no—
81C2×C3×SD16—24no—
82C2×C3×SD16C2×C418yescomplemented
83C2×C3×D8—24no—
84C2×C3×D8—24no—
85C2×C3×D8C2×C418yescomplemented
86C2×C3×M16C2×C414yes—
87C2×C3×M16D414yes—
88C2×C3×M16D414yes—
89C22×C3×C8C2312yes—
90C2×#217—21no—
91C2×#217—24no—
92C2×#217—24no—
93C2×#217—24no—
94C2×#217D412yes—
95#926—22no—
96#926—22no—
97#926—22no—
98#926—22no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3—22no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3D412yes—
103C2×C4⋊Dic3C2312yes—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4×Dic3—21no—
107C2×C4×Dic3C2312yes—
108C22×C3⋊C8—22no—
109C22×C3⋊C8—22no—
110C22×C3⋊C8D414yes—
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113#835—22no—
114#835—22no—
115#832—24no—
116#832—24no—
117#832—22no—
118#832—22no—
119C3⋊#95—24no—
120#818—24no—
121#817—24no—
122#815—24no—
123C8×Dic3—24no—
124C22×#41C2×D311yescharacteristic
125C23×D4C2×D311yescharacteristic
126C2×AGL(1,ℤ/8)—24no—
127C2×AGL(1,ℤ/8)—24no—
128C2×AGL(1,ℤ/8)Dic318yescomplemented
129C22×SD16C2×D312yes—
130C22×D8C2×D312yes—
131C22×M16C2×D311yescharacteristic
132C23×C8—31no—
133C2×#117—31no—
134C2×#117—31no—
135C22×C4⋊C4—31no—
136#428—64no—
137C2×#107—32no—
138C2×#106—32no—
139#416—64no—
140C2×#103—31no—
141C2×#103—31no—
142C2×#102—32no—
143C2×#102—32no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—64no—
147#405—64no—
148C2×#97—32no—
149C2×C4×C8—32no—
150#391—32no—
151#390—32no—
152#390—32no—
153#390—32no—
154#390—62no—
155#386—62no—
156#380—62no—
157#379—62no—
158C24×C3—22no—
159C24×C3—21no—
160C24×C3—24no—
161C24×C3—42no—
162C24×C3—42no—
163C24×C3—82no—
164C24×C3—82no—
165C24×C3C2×D411yescharacteristic
166C3×#41—28no—
167C3×#41—28no—
168C3×#41—44no—
169C3×#41—44no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8C22×C412yes—
176C2×C3×D4—24no—
177C2×C3×D4—28no—
178C2×C3×D4—28no—
179C2×C3×D4—24no—
180C2×C3×D4—22no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—21no—
188C2×C3×D4—22no—
189C2×C3×D4—21no—
190C2×C3×D4—42no—
191C2×C3×D4—42no—
192C2×C3×D4—44no—
193C2×C3×D4—42no—
194C2×C3×D4—44no—
195C2×C3×D4—42no—
196C2×C3×D4C22×C412yes—
197C2×C3×D4C22×C414yes—
198C22×C3×C4—21no—
199C22×C3×C4—24no—
200C22×C3×C4—42no—
201C22×C3×C4—42no—
202C22×C3×C4C22×C412yes—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2×D412yes—
207C22×C3×C4#4112yes—
208C22×C3×C4#4112yes—
209C22×C3×C4C2411yescharacteristic
210C22×Dic3—22no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—42no—
214C22×Dic3—82no—
215C22×Dic3C2×D414yes—
216C3×SD16—216no—
217C3×D8—216no—
218C3×M16C22⋊C4116yescomplemented
219C2×C3×C8C22×C414yes—
220C2×C3×C8C2×D414yes—
221C2×C3×C8C2×D414yes—
222#217—24no—
223#217—42no—
224#217—84no—
225#217—84no—
226#217—84no—
227C4⋊Dic3—24no—
228C4⋊Dic3—21no—
229C4⋊Dic3—22no—
230C4⋊Dic3—21no—
231C4⋊Dic3—22no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—42no—
235C4×Dic3—22no—
236C4×Dic3—22no—
237C4×Dic3—42no—
238C2×C3⋊C8—28no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—21no—
241C2×C3⋊C8—21no—
242C2×C3⋊C8—21no—
243C2×C3⋊C8C2×D412yes—
244C2×C3⋊C8C2×D412yes—
245C25—21no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41C2×Dic314yes—
252C22×Q8C22×D311yescharacteristic
253C22×D4—21no—
254C22×D4—22no—
255C22×D4—21no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4C2×Dic314yes—
261C22×D4C22×D311yescharacteristic
262C22×D4C22×D312yes—
263C23×C4—21no—
264C23×C4—61no—
265C23×C4C22×D311yescharacteristic
266AGL(1,ℤ/8)—232no—
267C2×SD16—24no—
268C2×SD16—24no—
269C2×SD16C2×Dic318yescomplemented
270C2×D8—24no—
271C2×D8—24no—
272C2×D8C2×Dic318yescomplemented
273C2×M16C2×Dic314yes—
274C2×M16#6714yes—
275C2×M16#6714yes—
276C22×C8—34no—
277C22×C8—31no—
278C22×C8—31no—
279C22×C8—62no—
280C22×C8—62no—
281C22×C8C22×D312yes—
282#117—62no—
283#117—62no—
284#117—62no—
285#117—62no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4—32no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—62no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—61no—
293C2×C22⋊C4—32no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—64no—
296C2×C22⋊C4—64no—
297C2×C22⋊C4—64no—
298C2×C42—32no—
299C2×C42—61no—
300#107—64no—
301#106—64no—
302#103—62no—
303#103—62no—
304#102—64no—
305#102—64no—
306#102—62no—
307#102—62no—
308#97—64no—
309C4×C8—64no—
310#95—64no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—24no—
315C23×C3—41no—
316C23×C3—42no—
317C23×C3—41no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—44no—
321C23×C3—44no—
322C23×C3—42no—
323C23×C3—48no—
324C23×C3—88no—
325C23×C3—84no—
326C23×C3—84no—
327C23×C3—88no—
328C23×C3C4×D412yes—
329C23×C3#12112yes—
330C23×C3#12712yes—
331C23×C3C22×D411yescharacteristic
332C3×Q8—22no—
333C3×Q8—22no—
334C3×Q8—41no—
335C3×Q8—41no—
336C3×D4—22no—
337C3×D4—24no—
338C3×D4—24no—
339C3×D4—22no—
340C3×D4—48no—
341C3×D4—42no—
342C3×D4—41no—
343C3×D4—42no—
344C3×D4—41no—
345C3×D4—48no—
346C2×C3×C4—22no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—48no—
350C2×C3×C4C2×C22⋊C414yes—
351C2×C3×C4C4×D414yes—
352C2×C3×C4C4×D412yes—
353C2×C3×C4#12114yes—
354C2×C3×C4#12112yes—
355C2×C3×C4#12414yes—
356C2×C3×C4#12712yes—
357C2×C3×C4C23×C411yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—44no—
371C2×Dic3—42no—
372C2×Dic3—84no—
373C3×C8C2×C22⋊C418yes—
374C3⋊C8—22no—
375C3⋊C8—22no—
376C24—21no—
377C24—22no—
378C24—24no—
379C24—42no—
380C24—42no—
381C24—82no—
382C24—82no—
383C24D3×D411yescharacteristic
384#41—28no—
385#41—28no—
386#41—44no—
387#41—44no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8C22×Dic312yes—
394C2×D4—24no—
395C2×D4—28no—
396C2×D4—24no—
397C2×D4—28no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4C22×Dic312yes—
415C2×D4C22×Dic314yes—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—34no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—62no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—122no—
425C22×C4—242no—
426C22×C4D3×D411yescharacteristic
427C22×C4D3×D412yes—
428C22×C4#23712yes—
429C22×C4#23712yes—
430C22×C4C22×Dic312yes—
431C22×C4C2×#6712yes—
432C22×C4C2×#6712yes—
433C22×C4C23×D311yescharacteristic
434SD16—216no—
435D8—216no—
436M16#217116yescomplemented
437C2×C8—32no—
438C2×C8—32no—
439C2×C8—68no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8C22×Dic314yes—
445C2×C8C2×#6714yes—
446C2×C8C2×#6714yes—
447C4⋊C4—64no—
448C4⋊C4—62no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—62no—
453C4⋊C4—61no—
454C4⋊C4—122no—
455C22⋊C4—64no—
456C22⋊C4—122no—
457C22⋊C4—244no—
458C22⋊C4—244no—
459C22⋊C4—244no—
460C42—62no—
461C42—62no—
462C42—122no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—24no—
466C22×C3—24no—
467C22×C3—44no—
468C22×C3—48no—
469C22×C3—42no—
470C22×C3—44no—
471C22×C3—816no—
472C22×C3—81no—
473C22×C3—82no—
474C22×C3—81no—
475C22×C3—82no—
476C22×C3—82no—
477C22×C3—84no—
478C22×C3#39014yes—
479C22×C3C2×C4×D411yescharacteristic
480C22×C3C2×#12111yescharacteristic
481C22×C3C2×#12711yescharacteristic
482C22×C3#57612yes—
483C22×C3#57712yes—
484C3×C4—42no—
485C3×C4#39014yes—
486C3×C4C22×C22⋊C411yescharacteristic
487C3×C4C2×C4×D411yescharacteristic
488C3×C4C2×#12111yescharacteristic
489C3×C4C2×#12411yescharacteristic
490Dic3—44no—
491Dic3—81no—
492Dic3—81no—
493Dic3—82no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—48no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23—41no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—88no—
508C23—84no—
509C23—84no—
510C23—88no—
511C23D4×Dic312yes—
512C23#93812yes—
513C23#94012yes—
514C23C2×D3×D411yescharacteristic
515Q8—22no—
516Q8—22no—
517Q8—41no—
518Q8—41no—
519D4—24no—
520D4—22no—
521D4—22no—
522D4—24no—
523D4—48no—
524D4—48no—
525D4—42no—
526D4—41no—
527D4—42no—
528D4—41no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—48no—
533C2×C4—64no—
534C2×C4—64no—
535C2×C4—64no—
536C2×C4—122no—
537C2×C4—124no—
538C2×C4—122no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—122no—
542C2×C4—244no—
543C2×C4D4×Dic312yes—
544C2×C4D4×Dic314yes—
545C2×C4#93614yes—
546C2×C4#93812yes—
547C2×C4#93814yes—
548C2×C4#94012yes—
549C2×C4C2×#21714yes—
550C2×C4C2×D3×D411yescharacteristic
551C2×C4C2×#23711yescharacteristic
552C2×C4C2×#23711yescharacteristic
553C2×C4C23×Dic311yescharacteristic
554C2×C4C22×#6711yescharacteristic
555C2×C4C22×#6711yescharacteristic
556C8—62no—
557C8—62no—
558C8C2×#21718yes—
559C2×C3—24no—
560C2×C3—82no—
561C2×C3—84no—
562C2×C3C2×#39011yescharacteristic
563C2×C3#294612yes—
564C2×C3#306212yes—
565C2×C3#315411yescharacteristic
566C2×C3#315511yescharacteristic
567C22—24no—
568C22—22no—
569C22—22no—
570C22—24no—
571C22—42no—
572C22—44no—
573C22—48no—
574C22—44no—
575C22—816no—
576C22—82no—
577C22—81no—
578C22—82no—
579C22—81no—
580C22—84no—
581C22—82no—
582C22#522114yes—
583C22#577412yes—
584C22#577512yes—
585C22C2×D4×Dic311yescharacteristic
586C22C2×#93811yescharacteristic
587C22C2×#94011yescharacteristic
588C4—42no—
589C4—124no—
590C4—241no—
591C4—242no—
592C4—241no—
593C4#522114yes—
594C4C2×D4×Dic311yescharacteristic
595C4C2×#93611yescharacteristic
596C4C2×#93811yescharacteristic
597C4C22×#21711yescharacteristic
598C3#2123511yescharacteristic, complemented, Sylow
599C2—24no—
600C2—84no—
601C2—82no—
602C2#8393012yes—
603C2#8393911yescharacteristic
604C2#8394011yescharacteristic
605C2#8395112yes—
606C2C2×#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#21235
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. #84087C2semidirect
3. C2×#5233C2semidirect
4. #84044C2semidirect
5. C22×#832C2semidirect
6. C2×#5121C2semidirect
7. C2×#5122C2semidirect
8. #72050C2semidirect
9. #72049C2semidirect
10. #72039C2semidirect
11. #72038C2semidirect
17. C2×C3×AGL(1,ℤ/8)C4semidirect
27. #5229C22semidirect
28. C2×#835C22semidirect
29. C2×#835C22semidirect
33. #5221C22semidirect
38. C2×#832C22semidirect
39. C2×#832C22semidirect
43. C2×#818C22semidirect
44. C2×#817C22semidirect
45. C2×#815C22semidirect
46. C2×C8×Dic3C22semidirect
82. C2×C3×SD16C2×C4semidirect
85. C2×C3×D8C2×C4semidirect
128. C2×AGL(1,ℤ/8)Dic3semidirect
218. C3×M16C22⋊C4semidirect
269. C2×SD16C2×Dic3semidirect
272. C2×D8C2×Dic3semidirect
436. M16#217semidirect
598. C3#21235semidirect

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
#840872
C2×#52332
#840442
C22×#8322
C2×#51212
C2×#51222
#720502
#720492
#720392
#720382
#212353