Finite Group Databasebeta

← #10944271

Subgroups — #10944271

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×#208C211yesmaximal, characteristic, complemented
2C3×#2848C211yesmaximal, characteristic, complemented
3#84032C211yesmaximal, characteristic
4C2×#5049C212yesmaximal, complemented
5C2×#5048C211yesmaximal, characteristic, complemented
6#71697C212yesmaximal, complemented
7#71696C211yesmaximal, characteristic, complemented
8#69517C212yesmaximal, complemented
9#69515C211yesmaximal, characteristic, complemented
10#69513C212yesmaximal, complemented
11#69512C211yesmaximal, characteristic, complemented
12#32935—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×C22⋊Q8C2212yes—
16C2×#953—21no—
17C22×#215C2211yescharacteristic, complemented
18C2×#930C2211yescharacteristic
19C2×#930C2212yes—
20C22×#208—28no—
21C22×#208—24no—
22C22×#208C2212yes—
23C22×#208C2212yes—
24C23×C3⋊C8C2211yescharacteristic
25C23×C3⋊C8C2211yescharacteristic
26C3×#385—22no—
27C3×#385—21no—
28C3×#380C2211yescharacteristic
29C3×#380C2212yes—
30C3×#379—21no—
31C2×#835C2212yescomplemented
32C2×#835C2211yescharacteristic, complemented
33#5213—22no—
34#5213—21no—
35#5212C2211yescharacteristic
36#5212C2212yes—
37#5049—28no—
38#5048—24no—
39C2×#809C2211yescharacteristic, complemented
40C2×#809C2212yescomplemented
41C2×#808C2211yescharacteristic, complemented
42C2×#807C2212yescomplemented
43#4529C2212yescomplemented
44#4529C2212yescomplemented
45#4529C2212yescomplemented
46C23×M16—31no—
47C2×#468—32no—
48C2×#465—31no—
49#2848D311yescharacteristic, complemented
50#2847—31no—
51#1895—32no—
52#1894—32no—
53#1893—31no—
54#1892—31no—
55#1809—32no—
56#1808—31no—
57C25×C3D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C23×C3×C4—24no—
60C23×C3×C4—22no—
61C23×C3×C4—24no—
62C23×C3×C4—22no—
63C23×C3×C4D412yes—
64C23×C3×C4D411yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2312yes—
67C22×#67—22no—
68C22×#67—21no—
69C22×D12C2311yescharacteristic
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—24no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—21no—
82#953—44no—
83C2×#217—22no—
84C2×#217—21no—
85C2×#215—22no—
86C2×#215—24no—
87#930—22no—
88#930—24no—
89#930—22no—
90#930—24no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—22no—
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×#208—232no—
96C2×#208—24no—
97C2×#208—24no—
98C2×#208—24no—
99C2×#208—24no—
100C2×#208—24no—
101C2×#208D418yescomplemented
102C22×C3⋊C8—22no—
103C22×C3⋊C8—24no—
104C22×C3⋊C8—24no—
105C22×C3⋊C8D414yescomplemented
106C22×C3⋊C8C2312yes—
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C3×#95—21no—
110C3×#95—22no—
111#835—22no—
112#835—24no—
113C3⋊#95—21no—
114C3⋊#95—22no—
115#809—22no—
116#809—24no—
117#808—22no—
118#807—24no—
119C24×C4C2×D311yescharacteristic
120C22×SD16—31no—
121C22×M16—32no—
122C22×M16—32no—
123C22×M16—64no—
124C22×M16—68no—
125C23×C8—31no—
126C23×C8—31no—
127C2×C22⋊Q8C2×D311yescharacteristic
128C2×C22⋊Q8C2×D312yes—
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—22no—
143#385—21no—
144#380—31no—
145#380—32no—
146#380C2×D311yescharacteristic
147#380C2×D312yes—
148#379—21no—
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—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C22×C3×C4—28no—
165C22×C3×C4—28no—
166C22×C3×C4—216no—
167C22×C3×C4—24no—
168C22×C3×C4—22no—
169C22×C3×C4—24no—
170C22×C3×C4—24no—
171C22×C3×C4—24no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D414yes—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D412yes—
185C22×C3×C4#4114yes—
186C22×C3×C4#4112yes—
187C22×C3×C4C2411yescharacteristic
188C2×#67—44no—
189C2×#67—44no—
190C2×#67—44no—
191C2×#67—42no—
192C2×#67—42no—
193C2×#67—42no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C2×D12—21no—
197C2×D12—22no—
198C2×D12—21no—
199C2×D12—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—42no—
205C3×C4⋊C4—44no—
206C3×C22⋊C4—44no—
207C3×C22⋊C4—44no—
208C3×C22⋊C4—42no—
209C3×C22⋊C4—42no—
210#217—44no—
211#217—42no—
212#215—44no—
213(C2×D3)⋊C4—42no—
214(C2×D3)⋊C4—44no—
215C4⋊Dic3—22no—
216C4⋊Dic3—21no—
217C4⋊Dic3—22no—
218C4⋊Dic3—21no—
219#208—216no—
220#208—216no—
221C2×C3⋊C8—28no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—28no—
226C2×C3⋊C8C2×D414yes—
227C2×C3⋊C8C2×D412yes—
228C2×C3⋊C8C2×D414yes—
229C2×C3⋊C8C2×D412yes—
230C25#6711yescharacteristic
231C22×Q8C22×D311yescharacteristic
232C22×D4—31no—
233C22×D4—62no—
234C22×D4—61no—
235C23×C4—24no—
236C23×C4—22no—
237C23×C4—24no—
238C23×C4—22no—
239C23×C4#6712yes—
240C23×C4#6711yescharacteristic
241C23×C4C22×D311yescharacteristic
242C23×C4C22×D312yes—
243C2×SD16—62no—
244C2×SD16—64no—
245C2×M16—38no—
246C2×M16—64no—
247C2×M16—64no—
248C2×M16—64no—
249C2×M16—64no—
250C2×M16—632no—
251C2×M16—64no—
252C22×C8—31no—
253C22×C8—34no—
254C22×C8—31no—
255C22×C8—32no—
256C22×C8—64no—
257C22×C8—64no—
258C22×C8—62no—
259C22⋊Q8—22no—
260C22⋊Q8—24no—
261C22⋊Q8—22no—
262C22⋊Q8—24no—
263#121—62no—
264#121—64no—
265#121—62no—
266#121—64no—
267#120—124no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—32no—
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C4⋊C4C22×D312yes—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
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—21no—
289#95—22no—
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—
315C3×Q8—41no—
316C3×Q8—41no—
317C3×Q8—42no—
318C2×C3×C4—28no—
319C2×C3×C4—216no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—41no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4#12112yes—
338C2×C3×C4#12114yes—
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D412yes—
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4C2×#4112yes—
343#67—88no—
344#67—84no—
345C2×Dic3—41no—
346C2×Dic3—42no—
347C2×Dic3—41no—
348C2×Dic3—42no—
349C2×Dic3—41no—
350C2×Dic3—42no—
351D12—41no—
352D12—41no—
353D12—42no—
354C3⋊C8—24no—
355C3⋊C8—22no—
356C3⋊C8#12014yescomplemented
357C24—24no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—42no—
362C24—61no—
363C24C2×#6712yes—
364C24C2×#6711yescharacteristic
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×D4—61no—
370C2×D4—62no—
371C2×D4—61no—
372C2×D4—62no—
373C2×D4—124no—
374C2×D4—124no—
375C2×D4—124no—
376C2×D4—122no—
377C2×D4—122no—
378C2×D4—122no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—216no—
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—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—61no—
397C22×C4—62no—
398C22×C4D3×D412yes—
399C22×C4D3×D414yes—
400C22×C4#23714yes—
401C22×C4#23712yes—
402C22×C4C2×#6711yescharacteristic
403C22×C4C2×#6712yes—
404C22×C4C23×D311yescharacteristic
405SD16—124no—
406M16—616no—
407M16—616no—
408C2×C8—32no—
409C2×C8—34no—
410C2×C8—32no—
411C2×C8—34no—
412C2×C8—68no—
413C2×C8—68no—
414C2×C8—62no—
415C2×C8—62no—
416C2×C8—62no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—61no—
424C4⋊C4—62no—
425C4⋊C4—61no—
426C4⋊C4—62no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—42no—
430C22⋊C4—42no—
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×C4—81no—
461C3×C4—81no—
462C3×C4—82no—
463C3×C4C2×#12011yescharacteristic
464C3×C4#52511yescharacteristic
465C3×C4#52512yes—
466Dic3—81no—
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—122no—
483C23—121no—
484C23—121no—
485C23—122no—
486C23#93914yes—
487C23#93912yes—
488C23#94212yes—
489C23#94912yes—
490C23C22×#6711yescharacteristic
491Q8—41no—
492Q8—42no—
493Q8—41no—
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—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—121no—
519C2×C4—122no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4#93912yes—
525C2×C4#93914yes—
526C2×C4C2×D3×D411yescharacteristic
527C2×C4C2×D3×D412yes—
528C2×C4C2×#23711yescharacteristic
529C2×C4C2×#23712yes—
530C8—34no—
531C8—62no—
532C8—64no—
533C2×C3—24no—
534C2×C3—44no—
535C2×C3#284711yescharacteristic
536C2×C3C2×#46811yescharacteristic
537C2×C3#305411yescharacteristic
538C2×C3#305512yes—
539C2×C3#305612yes—
540D3—161no—
541C22—24no—
542C22—22no—
543C22—22no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—416no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—241no—
554C22—242no—
555C22—241no—
556C22—241no—
557C22—242no—
558C22#504914yescomplemented
559C22C2×#93911yescharacteristic
560C22C2×#93912yes—
561C22C2×#94211yescharacteristic
562C22C2×#94911yescharacteristic
563C22#582212yes—
564C4—22no—
565C4—24no—
566C4—81no—
567C4—81no—
568C4—82no—
569C4—241no—
570C4—242no—
571C4#558611yescharacteristic
572C4#558612yes—
573C4D3×#12011yescharacteristic
574C3#3293511yescharacteristic, complemented, Sylow
575C2—24no—
576C2—44no—
577C2—481no—
578C2#8260111yescharacteristic
579C2C2×#504911yescharacteristic
580C2#8273212yes—
581C2#8292311yescharacteristic
582C2#8292412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2848C2
socleC23×C3C22×D4
prime radical#84032C2
hypercenter#2848D3
hypocenterC3#32935
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
1. C23×#208C2semidirect
2. C3×#2848C2semidirect
4. C2×#5049C2semidirect
5. C2×#5048C2semidirect
6. #71697C2semidirect
7. #71696C2semidirect
8. #69517C2semidirect
9. #69515C2semidirect
10. #69513C2semidirect
11. #69512C2semidirect
17. C22×#215C22semidirect
31. C2×#835C22semidirect
32. C2×#835C22semidirect
39. C2×#809C22semidirect
40. C2×#809C22semidirect
41. C2×#808C22semidirect
42. C2×#807C22semidirect
43. #4529C22semidirect
44. #4529C22semidirect
45. #4529C22semidirect
49. #2848D3semidirect
101. C2×#208D4semidirect
105. C22×C3⋊C8D4semidirect
356. C3⋊C8#120semidirect
447. C22×C3#468semidirect
558. C22#5049semidirect
574. C3#32935semidirect

Maximal subgroups

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

subgroupindex
C23×#2082
C3×#28482
#840322
C2×#50492
C2×#50482
#716972
#716962
#695172
#695152
#695132
#695122
#329353