Finite Group Databasebeta

← #10823187

Subgroups — #10823187

Order 768.

1674 conjugacy classes of subgroups.

The subgroup lattice has 588 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×#5118C211yesmaximal, characteristic, complemented
7#72051C214yesmaximal, complemented
8#72040C214yesmaximal, complemented
9#71985C211yesmaximal, characteristic, complemented
10#23546—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C2×C3×AGL(1,ℤ/8)—24no—
14C2×C3×AGL(1,ℤ/8)—24no—
15C2×C3×AGL(1,ℤ/8)—24no—
16C22×C3×SD16C2212yes—
17C22×C3×D8C2212yes—
18C22×C3×M16C2211yescharacteristic
19C22×#217—22no—
20C2×#926C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C22×#208C2211yescharacteristic
24#5232—24no—
25#5229C2212yescomplemented
26C2×#835C2212yescomplemented
27#5221—21no—
28#5221—21no—
29#5221C2214yes—
30#5221C2212yescomplemented
31#5220—22no—
32#5215—24no—
33C2×#832C2214yes—
34C2×#832C2212yescomplemented
35C4×#217—21no—
36C4×#217—21no—
37#5118—24no—
38C2×Dic3⋊C8C2214yescomplemented
39#4551C2214yescomplemented
40C22×AGL(1,ℤ/8)D311yescharacteristic
41C2×#423—31no—
42C2×#417—31no—
43C2×#418—31no—
44#2297—31no—
45C2×#390—31no—
46#2055—34no—
47#2054—34no—
48#1736—31no—
49C25×C3—21no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41D414yes—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4—21no—
58C22×C3×D4—22no—
59C22×C3×D4—22no—
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4D414yes—
65C22×C3×D4C2311yescharacteristic
66C22×C3×D4C2312yes—
67C23×C3×C4—21no—
68C23×C3×C4C2311yescharacteristic
69C23×Dic3—21no—
70C23×Dic3D412yes—
71C3×AGL(1,ℤ/8)—232no—
72C2×C3×SD16—24no—
73C2×C3×SD16—24no—
74C2×C3×SD16—24no—
75C2×C3×D8—24no—
76C2×C3×D8—24no—
77C2×C3×D8—24no—
78C2×C3×M16—22no—
79C2×C3×M16—22no—
80C2×C3×M16D414yes—
81C22×C3×C8C2312yes—
82C2×#217—21no—
83C2×#217—22no—
84C2×#217—22no—
85C2×#217—24no—
86C2×#217—24no—
87C2×#217—48no—
88#926—22no—
89#926—22no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2312yes—
95C2×C4×Dic3—22no—
96C2×C4×Dic3—22no—
97C2×C4×Dic3—21no—
98C2×C4×Dic3D411yescharacteristic
99C2×C4×Dic3D411yescharacteristic
100C2×C4×Dic3C2312yes—
101C2×#208—22no—
102C2×#208—22no—
103C2×#208—22no—
104C22×C3⋊C8C2312yes—
105#835—24no—
106#832—28no—
107#832—24no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110Dic3⋊C8—28no—
111C22×#41C2×D311yescharacteristic
112C23×D4C2×D311yescharacteristic
113C2×AGL(1,ℤ/8)—24no—
114C2×AGL(1,ℤ/8)—24no—
115C2×AGL(1,ℤ/8)—24no—
116C22×SD16C2×D312yes—
117C22×D8C2×D312yes—
118C22×M16—31no—
119C22×M16C2×D311yescharacteristic
120C2×#117—31no—
121C22×C4⋊C4—31no—
122C22×C22⋊C4—62no—
123C22×C42—31no—
124#423—64no—
125C2×C4⋊C8—34no—
126#418—64no—
127#417—64no—
128C2×#103—32no—
129C2×#102—32no—
130C2×#102—34no—
131#391—32no—
132#390—32no—
133#390—34no—
134#390—61no—
135#390—61no—
136#386—62no—
137C4×C22⋊C4—61no—
138C4×C22⋊C4—61no—
139#340—34no—
140C24×C3—24no—
141C24×C3—22no—
142C24×C3—21no—
143C24×C3—44no—
144C24×C3—44no—
145C24×C3—42no—
146C24×C3—42no—
147C24×C3C2×D411yescharacteristic
148C3×#41—28no—
149C3×#41—28no—
150C3×#41—44no—
151C3×#41—44no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8C2×D412yes—
158C2×C3×D4—24no—
159C2×C3×D4—28no—
160C2×C3×D4—24no—
161C2×C3×D4—28no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—21no—
172C2×C3×D4—42no—
173C2×C3×D4—42no—
174C2×C3×D4—42no—
175C2×C3×D4—42no—
176C2×C3×D4—44no—
177C2×C3×D4—44no—
178C2×C3×D4C2×D412yes—
179C2×C3×D4C2×D414yes—
180C22×C3×C4—21no—
181C22×C3×C4—24no—
182C22×C3×C4—42no—
183C22×C3×C4—42no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4#4112yes—
189C22×C3×C4#4112yes—
190C22×C3×C4#4112yes—
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—24no—
193C22×Dic3—24no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—42no—
197C22×Dic3—44no—
198C22×Dic3—42no—
199C22×Dic3C2×D412yes—
200C22×Dic3C2×D412yes—
201C3×SD16—216no—
202C3×D8—216no—
203C3×M16—28no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8C2×D414yes—
207#217—44no—
208#217—48no—
209#217—42no—
210#217—44no—
211#217—48no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—42no—
217C4×Dic3—22no—
218C4×Dic3—22no—
219C4×Dic3—22no—
220C4×Dic3—22no—
221C4×Dic3—42no—
222#208—44no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C25—21no—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—22no—
230C2×#41—22no—
231C2×#41—22no—
232C2×#41#6714yes—
233C22×Q8C22×D311yescharacteristic
234C22×D4—21no—
235C22×D4—21no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4#6714yes—
242C22×D4C22×D311yescharacteristic
243C22×D4C22×D312yes—
244C23×C4—21no—
245C23×C4—32no—
246C23×C4—61no—
247C23×C4C22×D311yescharacteristic
248AGL(1,ℤ/8)—232no—
249C2×SD16—24no—
250C2×SD16—24no—
251C2×SD16—24no—
252C2×D8—24no—
253C2×D8—24no—
254C2×D8—24no—
255C2×M16—22no—
256C2×M16—22no—
257C2×M16—62no—
258C2×M16—62no—
259C2×M16—62no—
260C2×M16#6714yes—
261C22×C8—32no—
262C22×C8C22×D312yes—
263#117—62no—
264#117—62no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—61no—
270C2×C22⋊C4—64no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—128no—
276C2×C42—31no—
277C2×C42—31no—
278C2×C42—32no—
279C2×C42—62no—
280C2×C42—62no—
281C2×C42—61no—
282C4⋊C8—68no—
283#103—64no—
284#102—64no—
285#102—68no—
286#95—62no—
287#95—62no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—24no—
292C23×C3—416no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—44no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—44no—
301C23×C3—48no—
302C23×C3—84no—
303C23×C3—84no—
304C23×C3—88no—
305C23×C3#12112yes—
306C23×C3#12112yes—
307C23×C3#12112yes—
308C23×C3C22×D411yescharacteristic
309C3×Q8—22no—
310C3×Q8—22no—
311C3×Q8—41no—
312C3×Q8—41no—
313C3×D4—22no—
314C3×D4—24no—
315C3×D4—24no—
316C3×D4—22no—
317C3×D4—48no—
318C3×D4—42no—
319C3×D4—41no—
320C3×D4—42no—
321C3×D4—41no—
322C3×D4—48no—
323C2×C3×C4—22no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—48no—
327C2×C3×C4#12014yes—
328C2×C3×C4#12112yes—
329C2×C3×C4#12112yes—
330C2×C3×C4#12112yes—
331C2×C3×C4#12314yes—
332C2×C3×C4#12314yes—
333C2×C3×C4#12414yes—
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—48no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—44no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—84no—
352C2×Dic3AGL(1,ℤ/8)14yescomplemented
353C3×C8—24no—
354C3⋊C8—42no—
355C24—24no—
356C24—21no—
357C24—22no—
358C24—44no—
359C24—44no—
360C24—42no—
361C24—42no—
362C24D3×D411yescharacteristic
363#41—28no—
364#41—28no—
365#41—44no—
366#41—44no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8C2×#6712yes—
373C2×D4—24no—
374C2×D4—28no—
375C2×D4—24no—
376C2×D4—28no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4C2×#6712yes—
394C2×D4C2×#6714yes—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—32no—
398C22×C4—32no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—64no—
402C22×C4—64no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—122no—
406C22×C4—124no—
407C22×C4—122no—
408C22×C4D3×D411yescharacteristic
409C22×C4#23712yes—
410C22×C4#23712yes—
411C22×C4#23712yes—
412C22×C4C2×#6712yes—
413C22×C4C2×#6712yes—
414C22×C4C2×#6712yes—
415C22×C4C23×D311yescharacteristic
416SD16—216no—
417D8—216no—
418M16—28no—
419M16—124no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—62no—
423C2×C8—62no—
424C2×C8—62no—
425C2×C8C2×#6714yes—
426C4⋊C4—62no—
427C4⋊C4—62no—
428C4⋊C4—62no—
429C4⋊C4—62no—
430C4⋊C4—122no—
431C22⋊C4—128no—
432C22⋊C4—124no—
433C22⋊C4—128no—
434C22⋊C4—124no—
435C22⋊C4—122no—
436C42—62no—
437C42—62no—
438C42—62no—
439C42—62no—
440C42—122no—
441C22×C3—24no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—24no—
445C22×C3—42no—
446C22×C3—44no—
447C22×C3—48no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—816no—
451C22×C3—82no—
452C22×C3—82no—
453C22×C3—82no—
454C22×C3—82no—
455C22×C3—84no—
456C22×C3#39414yes—
457C22×C3C2×#12111yescharacteristic
458C22×C3C2×#12111yescharacteristic
459C22×C3C2×#12111yescharacteristic
460C22×C3C2×AGL(1,ℤ/8)12yes—
461C22×C3#57512yes—
462C3×C4—42no—
463C3×C4#39414yes—
464C3×C4C2×#12011yescharacteristic
465C3×C4C2×#12311yescharacteristic
466C3×C4C2×#12311yescharacteristic
467C3×C4C2×#12411yescharacteristic
468Dic3—24no—
469Dic3—42no—
470Dic3—82no—
471Dic3—82no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—416no—
477C23—48no—
478C23—44no—
479C23—42no—
480C23—42no—
481C23—44no—
482C23—41no—
483C23—42no—
484C23—41no—
485C23—42no—
486C23—84no—
487C23—84no—
488C23—88no—
489C23#93812yes—
490C23#93912yes—
491C23#93912yes—
492C23C2×D3×D411yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497D4—24no—
498D4—22no—
499D4—22no—
500D4—24no—
501D4—48no—
502D4—48no—
503D4—42no—
504D4—41no—
505D4—42no—
506D4—41no—
507C2×C4—22no—
508C2×C4—34no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—48no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—128no—
518C2×C4—124no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—244no—
524C2×C4#93514yes—
525C2×C4#93514yes—
526C2×C4#93614yes—
527C2×C4#93812yes—
528C2×C4#93912yes—
529C2×C4#93912yes—
530C2×C4#95314yes—
531C2×C4C2×#23711yescharacteristic
532C2×C4C2×#23711yescharacteristic
533C2×C4C2×#23711yescharacteristic
534C2×C4C22×#6711yescharacteristic
535C2×C4C22×#6711yescharacteristic
536C2×C4C22×#6711yescharacteristic
537C8—24no—
538C8—122no—
539C2×C3—24no—
540C2×C3—82no—
541C2×C3—84no—
542C2×C3C2×#39411yescharacteristic
543C2×C3#303612yes—
544C2×C3#303811yescharacteristic
545C2×C3#304011yescharacteristic
546C2×C3#305512yes—
547C22—24no—
548C22—22no—
549C22—22no—
550C22—24no—
551C22—42no—
552C22—44no—
553C22—48no—
554C22—42no—
555C22—42no—
556C22—816no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—84no—
561C22—82no—
562C22#522214yes—
563C22D3×AGL(1,ℤ/8)12yes—
564C22#577312yes—
565C22C2×#93811yescharacteristic
566C22C2×#93911yescharacteristic
567C22C2×#93911yescharacteristic
568C4—42no—
569C4—64no—
570C4—122no—
571C4—242no—
572C4—242no—
573C4#522214yes—
574C4C2×#93511yescharacteristic
575C4C2×#93511yescharacteristic
576C4C2×#93611yescharacteristic
577C4C2×#95311yescharacteristic
578C3#2354611yescharacteristic, complemented, Sylow
579C2—24no—
580C2—84no—
581C2—82no—
582C2#8393112yes—
583C2#8393211yescharacteristic
584C2#8393311yescharacteristic
585C2#8395012yes—
586C2C2×#522211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC22×C3×AGL(1,ℤ/8)C2
socleC23×C3C22×D4
hypercenterC22×AGL(1,ℤ/8)D3
hypocenterC3#23546
LCSG ▷ C22×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×#5118C2semidirect
7. #72051C2semidirect
8. #72040C2semidirect
9. #71985C2semidirect
25. #5229C22semidirect
26. C2×#835C22semidirect
30. #5221C22semidirect
34. C2×#832C22semidirect
38. C2×Dic3⋊C8C22semidirect
39. #4551C22semidirect
352. C2×Dic3AGL(1,ℤ/8)semidirect
578. C3#23546semidirect

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×#51182
#720512
#720402
#719852
#235463