Finite Group Databasebeta

← #10944268

Subgroups — #10944268

Order 768.

1580 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×M16C211yesmaximal, characteristic, complemented
2#84032C211yesmaximal, characteristic
3#84031C211yesmaximal, characteristic, complemented
4C2×#5134C212yesmaximal, complemented
5C2×#5114C211yesmaximal, characteristic, complemented
6#71990C212yesmaximal, complemented
7#71989C212yesmaximal, complemented
8#71988C212yesmaximal, complemented
9#71973C211yesmaximal, characteristic, complemented
10#71972C211yesmaximal, characteristic, complemented
11#71971C211yesmaximal, characteristic, complemented
12#32935—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C22×C3×M16—28no—
15C22×C3×M16—24no—
16C22×C3×M16C2212yes—
17C22×C3×M16C2212yes—
18C23×C3×C8C2211yescharacteristic
19C23×C3×C8C2211yescharacteristic
20C2×#953—21no—
21C2×#930C2211yescharacteristic
22C2×#930C2212yes—
23C2×#924C2211yescharacteristic
24C2×#924C2212yes—
25C22×#204C2211yescharacteristic, complemented
26#5247—21no—
27#5213—22no—
28#5213—21no—
29#5212C2211yescharacteristic
30#5212C2211yescharacteristic
31#5212C2212yes—
32#5212C2212yes—
33#5210—22no—
34#5210—21no—
35#5134—28no—
36#5114—24no—
37C2×#821C2211yescharacteristic, complemented
38C2×#821C2212yescomplemented
39C2×#818C2212yescomplemented
40C2×#817C2211yescharacteristic, complemented
41C2×#816C2211yescharacteristic, complemented
42C2×#816C2212yescomplemented
43#4551C2212yescomplemented
44#4551C2212yescomplemented
45#4551C2212yescomplemented
46C23×M16D311yescharacteristic, complemented
47C2×#468—32no—
48C2×#465—31no—
49#2848—31no—
50#2847—31no—
51#1895—32no—
52#1894—32no—
53#1893—31no—
54#1892—31no—
55#1809—32no—
56#1808—31no—
57C25×C3D411yescharacteristic
58C23×C3×C4—24no—
59C23×C3×C4—22no—
60C23×C3×C4—24no—
61C23×C3×C4—22no—
62C23×C3×C4D412yes—
63C23×C3×C4D411yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2312yes—
66C22×#67—22no—
67C22×#67—21no—
68C22×D12C2311yescharacteristic
69C22×Dic6C2311yescharacteristic
70C2×C3×M16—232no—
71C2×C3×M16—24no—
72C2×C3×M16—24no—
73C2×C3×M16—24no—
74C2×C3×M16—24no—
75C2×C3×M16—24no—
76C2×C3×M16D418yescomplemented
77C22×C3×C8—22no—
78C22×C3×C8—24no—
79C22×C3×C8—24no—
80C22×C3×C8D414yescomplemented
81C22×C3×C8C2312yes—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84#953—44no—
85C2×#217—22no—
86C2×#217—21no—
87C2×#217—22no—
88C2×#217—21no—
89C2×#217—22no—
90C2×#217—21no—
91#930—24no—
92#930—22no—
93#930—22no—
94#930—24no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—22no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2312yes—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2312yes—
101#924—24no—
102#924—22no—
103#924—22no—
104#924—24no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—22no—
107C2×#204—24no—
108C2×#204—22no—
109C3⋊#95—21no—
110C3⋊#95—22no—
111C3⋊#95—21no—
112C3⋊#95—22no—
113#821—24no—
114#821—22no—
115#818—24no—
116#817—22no—
117#816—24no—
118#816—22no—
119C24×C4C2×D311yescharacteristic
120C22×SD16—31no—
121C22×M16—28no—
122C22×M16—24no—
123C22×M16C2×D312yes—
124C22×M16C2×D312yes—
125C23×C8C2×D311yescharacteristic
126C23×C8C2×D311yescharacteristic
127C2×C22⋊Q8—31no—
128C2×C22⋊Q8—32no—
129C2×#121—31no—
130C2×#121—32no—
131C2×#120—61no—
132#468—68no—
133#465—64no—
134C2×#107—32no—
135C2×#106—31no—
136C2×#103—32no—
137C2×#103—31no—
138C2×#102—32no—
139C2×#102—31no—
140#386—62no—
141#386—61no—
142#385—62no—
143#385—61no—
144#380—31no—
145#380—31no—
146#380—32no—
147#380—32no—
148#379—61no—
149#340—32no—
150#340—32no—
151#340—32no—
152C24×C3—24no—
153C24×C3—22no—
154C24×C3—22no—
155C24×C3—22no—
156C24×C3—42no—
157C24×C3C2×D412yes—
158C24×C3C2×D411yescharacteristic
159C23×D3—21no—
160C22×C3×C4—28no—
161C22×C3×C4—28no—
162C22×C3×C4—216no—
163C22×C3×C4—24no—
164C22×C3×C4—22no—
165C22×C3×C4—24no—
166C22×C3×C4—24no—
167C22×C3×C4—24no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D414yes—
178C22×C3×C4#4114yes—
179C22×C3×C4#4112yes—
180C22×C3×C4C2411yescharacteristic
181C2×#67—44no—
182C2×#67—44no—
183C2×#67—44no—
184C2×#67—42no—
185C2×#67—42no—
186C2×#67—42no—
187C22×Dic3—22no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—21no—
192C2×D12—22no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12—22no—
196C2×Dic6—22no—
197C2×Dic6—21no—
198C2×Dic6—21no—
199C2×Dic6—22no—
200C3×M16—216no—
201C3×M16—216no—
202C2×C3×C8—28no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—28no—
207C2×C3×C8C2×D414yes—
208C2×C3×C8C2×D412yes—
209C2×C3×C8C2×D412yes—
210C2×C3×C8C2×D414yes—
211#217—44no—
212#217—44no—
213#217—44no—
214#217—42no—
215#217—42no—
216#217—42no—
217(C2×D3)⋊C4—44no—
218(C2×D3)⋊C4—42no—
219C4⋊Dic3—21no—
220C4⋊Dic3—22no—
221C4⋊Dic3—21no—
222C4⋊Dic3—22no—
223C4⋊Dic3—21no—
224C4⋊Dic3—22no—
225C4⋊Dic3—21no—
226C4⋊Dic3—22no—
227Dic3⋊C4—44no—
228Dic3⋊C4—42no—
229#204—44no—
230C25D1211yescharacteristic
231C22×Q8—31no—
232C22×D4—31no—
233C22×D4—62no—
234C22×D4—61no—
235C23×C4—24no—
236C23×C4—22no—
237C23×C4—24no—
238C23×C4—22no—
239C23×C4D1212yes—
240C23×C4D1211yescharacteristic
241C23×C4C22×D311yescharacteristic
242C23×C4C22×D312yes—
243C2×SD16—62no—
244C2×SD16—64no—
245C2×M16—232no—
246C2×M16—24no—
247C2×M16—24no—
248C2×M16—24no—
249C2×M16—24no—
250C2×M16—24no—
251C2×M16#6718yescomplemented
252C22×C8—22no—
253C22×C8—24no—
254C22×C8—24no—
255C22×C8#6714yescomplemented
256C22×C8C22×D312yes—
257C22×C8C22×D311yescharacteristic
258C22×C8C22×D311yescharacteristic
259C22⋊Q8—62no—
260C22⋊Q8—64no—
261C22⋊Q8—62no—
262C22⋊Q8—64no—
263#121—62no—
264#121—64no—
265#121—62no—
266#121—64no—
267#120—124no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—32no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—32no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—62no—
282#107—64no—
283#106—62no—
284#103—62no—
285#103—64no—
286#102—62no—
287#102—64no—
288#95—61no—
289#95—62no—
290#95—61no—
291#95—62no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—28no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—48no—
302C23×C3—48no—
303C23×C3—44no—
304C23×C3—44no—
305C23×C3#12114yes—
306C23×C3#12112yes—
307C23×C3AGL(1,ℤ/8)12yes—
308C23×C3#13712yes—
309C23×C3C22×D411yescharacteristic
310C22×D3—41no—
311C22×D3—42no—
312C22×D3—41no—
313C22×D3—41no—
314C22×D3—42no—
315C2×C3×C4—28no—
316C2×C3×C4—216no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4#12112yes—
328C2×C3×C4#12114yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D412yes—
331C2×C3×C4C2×#4111yescharacteristic
332C2×C3×C4C2×#4112yes—
333#67—88no—
334#67—84no—
335C2×Dic3—41no—
336C2×Dic3—42no—
337C2×Dic3—41no—
338C2×Dic3—42no—
339C2×Dic3—41no—
340C2×Dic3—42no—
341C2×Dic3—41no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—41no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348D12—41no—
349D12—42no—
350D12—41no—
351Dic6—41no—
352Dic6—42no—
353Dic6—41no—
354C3×C8—24no—
355C3×C8—22no—
356C3×C8#12014yescomplemented
357C24—24no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—42no—
362C24—61no—
363C24C2×D1212yes—
364C24C2×D1211yescharacteristic
365C2×Q8—61no—
366C2×Q8—62no—
367C2×Q8—61no—
368C2×Q8—62no—
369C2×D4—61no—
370C2×D4—62no—
371C2×D4—61no—
372C2×D4—62no—
373C2×D4—124no—
374C2×D4—124no—
375C2×D4—122no—
376C2×D4—122no—
377C2×D4—124no—
378C2×D4—122no—
379C22×C4—28no—
380C22×C4—216no—
381C22×C4—28no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—62no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—62no—
397C22×C4—61no—
398C22×C4C2×D1211yescharacteristic
399C22×C4C2×D1212yes—
400C22×C4#23514yes—
401C22×C4#23512yes—
402C22×C4C2×#6712yes—
403C22×C4C2×#6714yes—
404C22×C4C23×D311yescharacteristic
405SD16—124no—
406M16—216no—
407M16—216no—
408C2×C8—28no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—28no—
413C2×C8C2×#6714yes—
414C2×C8C2×#6712yes—
415C2×C8C2×#6712yes—
416C2×C8C2×#6714yes—
417C4⋊C4—61no—
418C4⋊C4—62no—
419C4⋊C4—61no—
420C4⋊C4—62no—
421C4⋊C4—61no—
422C4⋊C4—62no—
423C4⋊C4—61no—
424C4⋊C4—62no—
425C4⋊C4—122no—
426C4⋊C4—124no—
427C22⋊C4—124no—
428C22⋊C4—124no—
429C22⋊C4—122no—
430C22⋊C4—122no—
431C22⋊C4—124no—
432C22⋊C4—122no—
433C22⋊C4—122no—
434C22⋊C4—124no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—24no—
438C22×C3—24no—
439C22×C3—24no—
440C22×C3—24no—
441C22×C3—24no—
442C22×C3—416no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3#46814yescomplemented
448C22×C3C2×#12111yescharacteristic
449C22×C3C2×#12112yes—
450C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
451C22×C3C2×#13711yescharacteristic
452C22×C3#57512yes—
453C2×D3—81no—
454C2×D3—81no—
455C2×D3—81no—
456C2×D3—82no—
457C2×D3—82no—
458C3×C4—22no—
459C3×C4—24no—
460C3×C4C2×#12011yescharacteristic
461C3×C4#52511yescharacteristic
462C3×C4#52512yes—
463Dic3—81no—
464Dic3—81no—
465Dic3—81no—
466Dic3—82no—
467Dic3—82no—
468C23—22no—
469C23—22no—
470C23—28no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—48no—
478C23—48no—
479C23—44no—
480C23—44no—
481C23—121no—
482C23—121no—
483C23—122no—
484C23—121no—
485C23—122no—
486C23#90812yes—
487C23#90912yes—
488C23#93014yes—
489C23#93012yes—
490C23C22×D1211yescharacteristic
491Q8—121no—
492Q8—122no—
493Q8—121no—
494D4—121no—
495D4—122no—
496D4—121no—
497D4—248no—
498D4—244no—
499C2×C4—28no—
500C2×C4—216no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—121no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—121no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—121no—
519C2×C4—122no—
520C2×C4—121no—
521C2×C4—122no—
522C2×C4—121no—
523C2×C4—122no—
524C2×C4#93012yes—
525C2×C4#93014yes—
526C2×C4C2×#23511yescharacteristic
527C2×C4C2×#23512yes—
528C2×C4C22×#6711yescharacteristic
529C2×C4C22×#6712yes—
530C8—24no—
531C8—22no—
532C8#95314yescomplemented
533C2×C3—24no—
534C2×C3—44no—
535C2×C3#284711yescharacteristic
536C2×C3C2×#46811yescharacteristic
537C2×C3#305411yescharacteristic
538C2×C3#305512yes—
539C2×C3#305612yes—
540D3—161no—
541C22—22no—
542C22—24no—
543C22—24no—
544C22—22no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—416no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—241no—
554C22—241no—
555C22—242no—
556C22—242no—
557C22—241no—
558C22#513414yescomplemented
559C22C2×#90811yescharacteristic
560C22C2×#90911yescharacteristic
561C22#574812yes—
562C22C2×#93011yescharacteristic
563C22C2×#93012yes—
564C4—22no—
565C4—24no—
566C4—241no—
567C4—241no—
568C4—242no—
569C4—242no—
570C4—241no—
571C4#579112yes—
572C4#579111yescharacteristic
573C4C2×#95311yescharacteristic
574C3#3293511yescharacteristic, complemented, Sylow
575C2—24no—
576C2—44no—
577C2—481no—
578C2C2×#513411yescharacteristic
579C2#8377111yescharacteristic
580C2#8377212yes—
581C2#8377612yes—
582C2#8403211yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC23×C3×M16C2
socleC23×C3C22×D4
prime radical#84032C2
hypercenterC23×M16D3
hypocenterC3#32935
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
1. C23×C3×M16C2semidirect
3. #84031C2semidirect
4. C2×#5134C2semidirect
5. C2×#5114C2semidirect
6. #71990C2semidirect
7. #71989C2semidirect
8. #71988C2semidirect
9. #71973C2semidirect
10. #71972C2semidirect
11. #71971C2semidirect
25. C22×#204C22semidirect
37. C2×#821C22semidirect
38. C2×#821C22semidirect
39. C2×#818C22semidirect
40. C2×#817C22semidirect
41. C2×#816C22semidirect
42. C2×#816C22semidirect
43. #4551C22semidirect
44. #4551C22semidirect
45. #4551C22semidirect
46. C23×M16D3semidirect
76. C2×C3×M16D4semidirect
80. C22×C3×C8D4semidirect
251. C2×M16#67semidirect
255. C22×C8#67semidirect
356. C3×C8#120semidirect
447. C22×C3#468semidirect
532. C8#953semidirect
558. C22#5134semidirect
574. C3#32935semidirect

Maximal subgroups

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

subgroupindex
C23×C3×M162
#840322
#840312
C2×#51342
C2×#51142
#719902
#719892
#719882
#719732
#719722
#719712
#329353