Finite Group Databasebeta

← #10944276

Subgroups — #10944276

Order 768.

1658 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#208C211yesmaximal, characteristic, complemented
2C3×#2847C211yesmaximal, characteristic, complemented
3#84032C211yesmaximal, characteristic, complemented
4C2×#5039C212yesmaximal, complemented
5C2×#5038C211yesmaximal, characteristic, complemented
6#71682C212yesmaximal, complemented
7#71681C211yesmaximal, characteristic, complemented
8#69517C212yesmaximal, complemented
9#69515C211yesmaximal, characteristic, complemented
10#69513C212yesmaximal, complemented
11#69511C211yesmaximal, characteristic, complemented
12#32936—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C2×C3×#121C2211yescharacteristic
15C2×C3×#121C2212yes—
16C2×C3×#120—21no—
17C2×#953—21no—
18C22×#213C2211yescharacteristic, complemented
19C2×#930C2211yescharacteristic
20C2×#930C2212yes—
21C22×#208—28no—
22C22×#208—24no—
23C22×#208C2212yes—
24C22×#208C2212yes—
25C23×C3⋊C8C2211yescharacteristic
26C23×C3⋊C8C2211yescharacteristic
27C3×#386—22no—
28C3×#386—21no—
29C3×#380C2211yescharacteristic
30C3×#380C2212yes—
31C2×#832C2212yescomplemented
32C2×#832C2211yescharacteristic, complemented
33#5213—22no—
34#5213—21no—
35#5212C2211yescharacteristic
36#5212C2212yes—
37#5039—28no—
38#5038—24no—
39C2×#809C2212yescomplemented
40C2×#809C2211yescharacteristic, complemented
41C2×#808C2212yescomplemented
42C2×#807C2211yescharacteristic, complemented
43#4529C2212yescomplemented
44#4529C2212yescomplemented
45#4529C2212yescomplemented
46C23×M16—31no—
47C2×#469—32no—
48C2×#466—31no—
49#2847—31no—
50#2847D311yescharacteristic, complemented
51#1895—32no—
52#1895—32no—
53#1893—31no—
54#1893—31no—
55#1809—32no—
56#1807—31no—
57C25×C3D411yescharacteristic
58C22×C3×D4—22no—
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—24no—
62C23×C3×C4—22no—
63C23×C3×C4—24no—
64C23×C3×C4—22no—
65C23×C3×C4D412yes—
66C23×C3×C4D411yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2312yes—
69C22×#67—22no—
70C22×#67—21no—
71C22×D12C2311yescharacteristic
72C3×#121—22no—
73C3×#121—24no—
74C3×#121—22no—
75C3×#121—24no—
76C3×#120—44no—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—22no—
83#953—44no—
84C2×#217—22no—
85C2×#217—21no—
86C2×#213—22no—
87C2×#213—24no—
88#930—22no—
89#930—24no—
90#930—22no—
91#930—24no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—22no—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2312yes—
96C2×#208—232no—
97C2×#208—24no—
98C2×#208—24no—
99C2×#208—24no—
100C2×#208—24no—
101C2×#208—24no—
102C2×#208D418yescomplemented
103C22×C3⋊C8—22no—
104C22×C3⋊C8—24no—
105C22×C3⋊C8—24no—
106C22×C3⋊C8D414yescomplemented
107C22×C3⋊C8C2312yes—
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C3×#95—21no—
111C3×#95—22no—
112#832—22no—
113#832—24no—
114C3⋊#95—21no—
115C3⋊#95—22no—
116#809—22no—
117#809—24no—
118#808—24no—
119#807—22no—
120C24×C4C2×D311yescharacteristic
121C22×D8—31no—
122C22×M16—32no—
123C22×M16—32no—
124C22×M16—64no—
125C22×M16—68no—
126C23×C8—31no—
127C23×C8—31no—
128C2×#121—31no—
129C2×#121—32no—
130C2×#121C2×D311yescharacteristic
131C2×#121C2×D312yes—
132C2×#120—21no—
133C2×#120—61no—
134#469—68no—
135#466—64no—
136C2×#107—31no—
137C2×#106—32no—
138C2×#102—32no—
139C2×#102—31no—
140C2×#102—32no—
141C2×#102—31no—
142#386—22no—
143#386—21no—
144#386—62no—
145#386—61no—
146#380—31no—
147#380—32no—
148#380C2×D311yescharacteristic
149#380C2×D312yes—
150#340—32no—
151#340—32no—
152#340—32no—
153C24×C3—24no—
154C24×C3—22no—
155C24×C3—22no—
156C24×C3—22no—
157C24×C3—21no—
158C24×C3—42no—
159C24×C3C2×D412yes—
160C24×C3C2×D411yescharacteristic
161C23×D3—21no—
162C2×C3×D4—21no—
163C2×C3×D4—22no—
164C2×C3×D4—21no—
165C2×C3×D4—22no—
166C2×C3×D4—44no—
167C2×C3×D4—44no—
168C2×C3×D4—44no—
169C2×C3×D4—42no—
170C2×C3×D4—42no—
171C2×C3×D4—42no—
172C22×C3×C4—28no—
173C22×C3×C4—28no—
174C22×C3×C4—216no—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—24no—
178C22×C3×C4—24no—
179C22×C3×C4—24no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—22no—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D414yes—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D412yes—
192C22×C3×C4#4114yes—
193C22×C3×C4#4112yes—
194C22×C3×C4C2411yescharacteristic
195C2×#67—44no—
196C2×#67—44no—
197C2×#67—44no—
198C2×#67—42no—
199C2×#67—42no—
200C2×#67—42no—
201C22×Dic3—21no—
202C22×Dic3—22no—
203C2×D12—21no—
204C2×D12—22no—
205C2×D12—21no—
206C2×D12—22no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—22no—
211C3×C22⋊C4—44no—
212C3×C22⋊C4—42no—
213C3×C22⋊C4—42no—
214C3×C22⋊C4—44no—
215#217—44no—
216#217—42no—
217#213—44no—
218(C2×D3)⋊C4—42no—
219(C2×D3)⋊C4—44no—
220C4⋊Dic3—21no—
221C4⋊Dic3—22no—
222C4⋊Dic3—21no—
223C4⋊Dic3—22no—
224#208—216no—
225#208—216no—
226C2×C3⋊C8—28no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—28no—
231C2×C3⋊C8C2×D414yes—
232C2×C3⋊C8C2×D412yes—
233C2×C3⋊C8C2×D412yes—
234C2×C3⋊C8C2×D414yes—
235C25#6711yescharacteristic
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—31no—
239C22×D4—62no—
240C22×D4—61no—
241C22×D4C22×D311yescharacteristic
242C23×C4—24no—
243C23×C4—22no—
244C23×C4—24no—
245C23×C4—22no—
246C23×C4#6712yes—
247C23×C4#6711yescharacteristic
248C23×C4C22×D311yescharacteristic
249C23×C4C22×D312yes—
250C2×D8—62no—
251C2×D8—64no—
252C2×M16—38no—
253C2×M16—64no—
254C2×M16—64no—
255C2×M16—64no—
256C2×M16—64no—
257C2×M16—632no—
258C2×M16—64no—
259C22×C8—31no—
260C22×C8—34no—
261C22×C8—31no—
262C22×C8—32no—
263C22×C8—64no—
264C22×C8—64no—
265C22×C8—62no—
266#121—22no—
267#121—24no—
268#121—22no—
269#121—24no—
270#121—62no—
271#121—64no—
272#121—62no—
273#121—64no—
274#120—44no—
275#120—124no—
276C2×C4⋊C4—32no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4C22×D312yes—
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C22⋊C4—22no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—62no—
288#107—62no—
289#106—64no—
290#102—62no—
291#102—64no—
292#102—62no—
293#102—64no—
294#95—21no—
295#95—22no—
296#95—61no—
297#95—62no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—28no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—48no—
308C23×C3—48no—
309C23×C3—44no—
310C23×C3—44no—
311C23×C3—41no—
312C23×C3—42no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—42no—
316C23×C3#12114yes—
317C23×C3#12112yes—
318C23×C3AGL(1,ℤ/8)12yes—
319C23×C3AGL(1,ℤ/8)12yes—
320C23×C3C22×D411yescharacteristic
321C22×D3—41no—
322C22×D3—42no—
323C22×D3—41no—
324C22×D3—41no—
325C22×D3—42no—
326C3×D4—41no—
327C3×D4—41no—
328C3×D4—42no—
329C3×D4—88no—
330C3×D4—84no—
331C2×C3×C4—28no—
332C2×C3×C4—216no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—24no—
338C2×C3×C4—22no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4—42no—
349C2×C3×C4#12112yes—
350C2×C3×C4#12114yes—
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C22×D412yes—
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4112yes—
355#67—88no—
356#67—84no—
357C2×Dic3—41no—
358C2×Dic3—42no—
359C2×Dic3—41no—
360C2×Dic3—42no—
361C2×Dic3—41no—
362C2×Dic3—42no—
363D12—41no—
364D12—41no—
365D12—42no—
366C3⋊C8—24no—
367C3⋊C8—22no—
368C3⋊C8#12014yescomplemented
369C24—24no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—21no—
374C24—42no—
375C24—61no—
376C24C2×#6712yes—
377C24C2×#6711yescharacteristic
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—61no—
389C2×D4—62no—
390C2×D4—61no—
391C2×D4—62no—
392C2×D4—124no—
393C2×D4—124no—
394C2×D4—122no—
395C2×D4—124no—
396C2×D4—122no—
397C2×D4—122no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—216no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—61no—
415C22×C4—62no—
416C22×C4D3×D412yes—
417C22×C4D3×D414yes—
418C22×C4#23714yes—
419C22×C4#23712yes—
420C22×C4C2×#6711yescharacteristic
421C22×C4C2×#6712yes—
422C22×C4C23×D311yescharacteristic
423D8—124no—
424M16—616no—
425M16—616no—
426C2×C8—34no—
427C2×C8—32no—
428C2×C8—32no—
429C2×C8—34no—
430C2×C8—68no—
431C2×C8—68no—
432C2×C8—62no—
433C2×C8—62no—
434C2×C8—62no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—62no—
440C4⋊C4—61no—
441C4⋊C4—62no—
442C4⋊C4—61no—
443C22⋊C4—44no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—44no—
447C22⋊C4—124no—
448C22⋊C4—122no—
449C22⋊C4—122no—
450C22⋊C4—124no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—24no—
454C22×C3—24no—
455C22×C3—24no—
456C22×C3—24no—
457C22×C3—24no—
458C22×C3—416no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—81no—
464C22×C3—81no—
465C22×C3—81no—
466C22×C3—82no—
467C22×C3—82no—
468C22×C3#46914yescomplemented
469C22×C3C2×#12111yescharacteristic
470C22×C3C2×#12112yes—
471C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
472C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
473C22×C3#57512yes—
474C2×D3—81no—
475C2×D3—81no—
476C2×D3—81no—
477C2×D3—82no—
478C2×D3—82no—
479C3×C4—22no—
480C3×C4—24no—
481C3×C4—81no—
482C3×C4—82no—
483C3×C4C2×#12011yescharacteristic
484C3×C4#52511yescharacteristic
485C3×C4#52512yes—
486Dic3—81no—
487Dic3—82no—
488C23—22no—
489C23—22no—
490C23—28no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—48no—
498C23—48no—
499C23—44no—
500C23—44no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—41no—
505C23—42no—
506C23—121no—
507C23—121no—
508C23—122no—
509C23—121no—
510C23—122no—
511C23#93212yes—
512C23#93914yes—
513C23#93912yes—
514C23#94912yes—
515C23C22×#6711yescharacteristic
516D4—41no—
517D4—42no—
518D4—41no—
519D4—88no—
520D4—84no—
521D4—121no—
522D4—122no—
523D4—121no—
524D4—248no—
525D4—244no—
526C2×C4—28no—
527C2×C4—216no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—122no—
547C2×C4—121no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4#93912yes—
551C2×C4#93914yes—
552C2×C4C2×D3×D411yescharacteristic
553C2×C4C2×D3×D412yes—
554C2×C4C2×#23711yescharacteristic
555C2×C4C2×#23712yes—
556C8—34no—
557C8—62no—
558C8—64no—
559C2×C3—24no—
560C2×C3—44no—
561C2×C3—161no—
562C2×C3#284711yescharacteristic
563C2×C3C2×#46911yescharacteristic
564C2×C3#305411yescharacteristic
565C2×C3#305512yes—
566C2×C3#305512yes—
567D3—161no—
568C22—22no—
569C22—24no—
570C22—22no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—416no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—81no—
581C22—81no—
582C22—82no—
583C22—81no—
584C22—82no—
585C22—241no—
586C22—241no—
587C22—242no—
588C22—241no—
589C22—242no—
590C22#503914yescomplemented
591C22C2×#93211yescharacteristic
592C22C2×#93911yescharacteristic
593C22C2×#93912yes—
594C22C2×#94911yescharacteristic
595C22#582212yes—
596C4—22no—
597C4—24no—
598C4—81no—
599C4—82no—
600C4—241no—
601C4—242no—
602C4#558611yescharacteristic
603C4#558612yes—
604C4D3×#12011yescharacteristic
605C3#3293611yescharacteristic, complemented, Sylow
606C2—24no—
607C2—44no—
608C2—161no—
609C2—481no—
610C2#8260111yescharacteristic
611C2C2×#503911yescharacteristic
612C2#8262112yes—
613C2#8292311yescharacteristic
614C2#8292412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2847C2
socleC23×C3C22×D4
hypercenter#2847D3
hypocenterC3#32936
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2847

Complements

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

NH ≅ G/Ntype
1. C23×#208C2semidirect
2. C3×#2847C2semidirect
3. #84032C2semidirect
4. C2×#5039C2semidirect
5. C2×#5038C2semidirect
6. #71682C2semidirect
7. #71681C2semidirect
8. #69517C2semidirect
9. #69515C2semidirect
10. #69513C2semidirect
11. #69511C2semidirect
18. C22×#213C22semidirect
31. C2×#832C22semidirect
32. C2×#832C22semidirect
39. C2×#809C22semidirect
40. C2×#809C22semidirect
41. C2×#808C22semidirect
42. C2×#807C22semidirect
43. #4529C22semidirect
44. #4529C22semidirect
45. #4529C22semidirect
50. #2847D3semidirect
102. C2×#208D4semidirect
106. C22×C3⋊C8D4semidirect
368. C3⋊C8#120semidirect
468. C22×C3#469semidirect
590. C22#5039semidirect
605. C3#32936semidirect

Maximal subgroups

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

subgroupindex
C23×#2082
C3×#28472
#840322
C2×#50392
C2×#50382
#716822
#716812
#695172
#695152
#695132
#695112
#329363