Finite Group Databasebeta

← #10944238

Subgroups — #10944238

Order 768.

1786 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×D3×C8C211yesmaximal, characteristic, complemented
2#82723C211yesmaximal, characteristic, complemented
3C2×#5038C211yesmaximal, characteristic, complemented
4C2×#4775C212yesmaximal, complemented
5#74048C211yesmaximal, characteristic, complemented
6C3×#1893C211yesmaximal, characteristic, complemented
7#71973C211yesmaximal, characteristic, complemented
8#70202C212yesmaximal, complemented
9#69512C211yesmaximal, characteristic, complemented
10#68883C212yesmaximal, complemented
11#68869C212yesmaximal, complemented
12#32930—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#121C2211yescharacteristic
16C2×#938C2212yes—
17C2×#937—21no—
18C22×#213C2211yescharacteristic, complemented
19C2×#930C2211yescharacteristic
20C23×C3⋊C8C2211yescharacteristic
21C22×D3×C8—24no—
22C22×D3×C8—28no—
23C22×D3×C8C2212yes—
24C22×D3×C8C2212yes—
25C2×#895C2212yes—
26C2×#882—21no—
27C2×C3×#102C2212yescomplemented
28C3×#380C2211yescharacteristic
29#5220—21no—
30C2×#832C2211yescharacteristic, complemented
31#5212C2211yescharacteristic
32C2×#821C2212yescomplemented
33#5038—24no—
34#4987C2212yes—
35#4987C2212yes—
36C2×#809C2211yescharacteristic, complemented
37C2×#807C2211yescharacteristic, complemented
38C2×#807C2212yescomplemented
39#4958—21no—
40#4956—22no—
41#4775—28no—
42#4665—22no—
43C3×#340C2212yescomplemented
44#4551C2212yescomplemented
45#4529C2212yescomplemented
46C24×C8—31no—
47C2×#466—32no—
48C2×#466—31no—
49#2847—31no—
50#2847—31no—
51#1893—32no—
52#1893—32no—
53#1893—31no—
54#1893D311yescharacteristic, complemented
55#1808—31no—
56#1808—32no—
57C24×D3D411yescharacteristic
58C22×C3×D4—21no—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C22×#67—22no—
63C23×Dic3D411yescharacteristic
64C22×D12—22no—
65C22×D12C2311yescharacteristic
66C22×C4×D3—22no—
67C22×C4×D3—22no—
68C22×C4×D3—24no—
69C22×C4×D3—24no—
70C22×C4×D3D412yes—
71C22×C4×D3C2312yes—
72C22×C3×C8—22no—
73C22×C3×C8—24no—
74C22×C3×C8C2312yes—
75C3×#121—22no—
76C3×#121—22no—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×#217—21no—
82C2×#217—22no—
83#938—24no—
84#938—24no—
85#937—44no—
86C2×#213—24no—
87C2×#213—22no—
88#930—22no—
89#930—22no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—22no—
94C2×C4⋊Dic3C2312yes—
95C2×C4⋊Dic3C2311yescharacteristic
96C22×C3⋊C8—24no—
97C22×C3⋊C8D414yescomplemented
98C22×C3⋊C8C2311yescharacteristic
99C22×C3⋊C8C2311yescharacteristic
100C2×D3×C8—232no—
101C2×D3×C8—24no—
102C2×D3×C8—24no—
103C2×D3×C8—24no—
104C2×D3×C8—24no—
105C2×D3×C8—24no—
106C2×D3×C8D418yescomplemented
107#895—24no—
108#895—24no—
109#882—44no—
110C3×#102—24no—
111C3×#95—21no—
112#832—22no—
113C3⋊#95—21no—
114C3⋊#95—22no—
115C3⋊#95—22no—
116#821—24no—
117#809—22no—
118#807—24no—
119#807—22no—
120C24×C4—31no—
121C22×D8—31no—
122C23×C8—31no—
123C23×C8—32no—
124C23×C8—32no—
125C23×C8—64no—
126C23×C8—68no—
127C23×C8C2×D311yescharacteristic
128C2×#121—32no—
129C2×#121—31no—
130C2×#121—32no—
131C2×#121C2×D311yescharacteristic
132C2×#120—61no—
133C2×#120—61no—
134#466—68no—
135#466—64no—
136C2×#107—32no—
137C2×#107—31no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#102—31no—
141C2×#102C2×D312yescomplemented
142#386—62no—
143#386—62no—
144#386—61no—
145#386—61no—
146#380—31no—
147#380—32no—
148#380—32no—
149#380C2×D311yescharacteristic
150#340—32no—
151#340—32no—
152#340C2×D312yescomplemented
153C24×C3—21no—
154C24×C3C2×D411yescharacteristic
155C23×D3—22no—
156C23×D3—22no—
157C23×D3—24no—
158C23×D3—24no—
159C23×D3—21no—
160C23×D3D814yescomplemented
161C23×D3C2×D412yes—
162C2×C3×D4—21no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—21no—
166C2×C3×D4—42no—
167C2×C3×D4—42no—
168C2×C3×D4—42no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D412yes—
175C22×C3×C4#4112yes—
176C22×C3×C4C2411yescharacteristic
177C2×#67—42no—
178C2×#67—42no—
179C2×#67—42no—
180C2×#67—44no—
181C2×#67—44no—
182C2×#67—44no—
183C22×Dic3—22no—
184C22×Dic3—24no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3C2×D412yes—
188C2×D12—21no—
189C2×D12—22no—
190C2×D12—22no—
191C2×D12—21no—
192C2×D12—44no—
193C2×D12—44no—
194C2×D12—44no—
195C2×C4×D3—28no—
196C2×C4×D3—216no—
197C2×C4×D3—28no—
198C2×C4×D3—24no—
199C2×C4×D3—24no—
200C2×C4×D3—22no—
201C2×C4×D3—22no—
202C2×C4×D3—22no—
203C2×C4×D3—22no—
204C2×C4×D3—24no—
205C2×C4×D3C2×D414yes—
206C2×C4×D3#4114yes—
207C2×C3×C8—28no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8C2×D414yes—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—21no—
215C3×C22⋊C4—42no—
216C3×C22⋊C4—42no—
217#217—42no—
218#217—44no—
219#213—44no—
220(C2×D3)⋊C4—44no—
221(C2×D3)⋊C4—44no—
222(C2×D3)⋊C4—42no—
223(C2×D3)⋊C4—44no—
224C4⋊Dic3—22no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—28no—
230C2×C3⋊C8C2×D412yes—
231C2×C3⋊C8C2×D412yes—
232C2×C3⋊C8C2×D414yes—
233D3×C8—216no—
234D3×C8—216no—
235C25—31no—
236C22×D4—21no—
237C22×D4—31no—
238C22×D4—62no—
239C22×D4—62no—
240C22×D4—61no—
241C22×D4C22×D311yescharacteristic
242C23×C4—32no—
243C23×C4—31no—
244C23×C4—32no—
245C23×C4—64no—
246C23×C4—64no—
247C23×C4—62no—
248C23×C4—62no—
249C23×C4C22×D311yescharacteristic
250C2×D8—64no—
251C2×D8—62no—
252C22×C8—22no—
253C22×C8—24no—
254C22×C8—31no—
255C22×C8—34no—
256C22×C8—31no—
257C22×C8—38no—
258C22×C8—64no—
259C22×C8—64no—
260C22×C8—64no—
261C22×C8—64no—
262C22×C8—632no—
263C22×C8—64no—
264C22×C8—64no—
265C22×C8C22×D312yes—
266#121—22no—
267#121—22no—
268#121—64no—
269#121—64no—
270#121—64no—
271#121—64no—
272#121—62no—
273#121—62no—
274#120—124no—
275#120—124no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—32no—
278C2×C4⋊C4C22×D312yes—
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—62no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288#107—64no—
289#107—62no—
290#102—24no—
291#102—62no—
292#102—62no—
293#102—64no—
294#95—21no—
295#95—62no—
296#95—62no—
297#95—61no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—42no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—42no—
304C23×C3—41no—
305C23×C3#12112yes—
306C23×C3C2×D812yes—
307C23×C3#13512yes—
308C23×C3C22×D411yescharacteristic
309C22×D3—28no—
310C22×D3—216no—
311C22×D3—28no—
312C22×D3—22no—
313C22×D3—22no—
314C22×D3—22no—
315C22×D3—22no—
316C22×D3—28no—
317C22×D3—216no—
318C22×D3—42no—
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—42no—
322C22×D3—41no—
323C22×D3#12114yes—
324C22×D3C2×D814yes—
325C22×D3#13514yes—
326C3×D4—41no—
327C3×D4—41no—
328C3×D4—42no—
329C3×D4—84no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—24no—
334C2×C3×C4—42no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—41no—
340C2×C3×C4#12112yes—
341C2×C3×C4C22×D412yes—
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4C2×#4112yes—
345#67—84no—
346#67—88no—
347C2×Dic3—28no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—41no—
351C2×Dic3—42no—
352C2×Dic3—41no—
353C2×Dic3—42no—
354C2×Dic3—41no—
355C2×Dic3—42no—
356C2×Dic3#12114yes—
357D12—41no—
358D12—41no—
359D12—42no—
360D12—88no—
361C4×D3—216no—
362C4×D3—24no—
363C4×D3—24no—
364C4×D3—24no—
365C4×D3—24no—
366C3×C8—24no—
367C3⋊C8—22no—
368C3⋊C8#12014yescomplemented
369C24—21no—
370C24—32no—
371C24—34no—
372C24—64no—
373C24—64no—
374C24—62no—
375C24—62no—
376C24—61no—
377C24D3×D411yescharacteristic
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4—124no—
390C2×D4—124no—
391C2×D4—124no—
392C2×D4—124no—
393C2×D4—124no—
394C2×D4—124no—
395C2×D4—122no—
396C2×D4—122no—
397C2×D4—122no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—32no—
403C22×C4—34no—
404C22×C4—34no—
405C22×C4—64no—
406C22×C4—64no—
407C22×C4—68no—
408C22×C4—68no—
409C22×C4—616no—
410C22×C4—64no—
411C22×C4—62no—
412C22×C4—64no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—61no—
418C22×C4—62no—
419C22×C4#23512yes—
420C22×C4D3×D411yescharacteristic
421C22×C4D3×D412yes—
422C22×C4C23×D311yescharacteristic
423D8—124no—
424C2×C8—28no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—32no—
428C2×C8—34no—
429C2×C8—32no—
430C2×C8—616no—
431C2×C8—616no—
432C2×C8—68no—
433C2×C8—62no—
434C2×C8D3×D414yes—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—62no—
442C4⋊C4—62no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—124no—
446C22⋊C4—124no—
447C22⋊C4—124no—
448C22⋊C4—124no—
449C22⋊C4—122no—
450C22⋊C4—122no—
451C22×C3—22no—
452C22×C3—24no—
453C22×C3—22no—
454C22×C3—24no—
455C22×C3—81no—
456C22×C3—82no—
457C22×C3—81no—
458C22×C3—82no—
459C22×C3—81no—
460C22×C3#46614yescomplemented
461C22×C3C2×#12111yescharacteristic
462C22×C3C2×#12112yes—
463C22×C3C22×D811yescharacteristic
464C22×C3C2×#13511yescharacteristic
465C22×C3C2×#13512yes—
466C2×D3—232no—
467C2×D3—24no—
468C2×D3—24no—
469C2×D3—24no—
470C2×D3—24no—
471C2×D3—24no—
472C2×D3—24no—
473C2×D3—81no—
474C2×D3—82no—
475C2×D3—81no—
476C2×D3—82no—
477C2×D3—81no—
478C2×D3#46618yescomplemented
479C3×C4—22no—
480C3×C4—82no—
481C3×C4—81no—
482C3×C4C2×#12011yescharacteristic
483C3×C4#52511yescharacteristic
484C3×C4#52512yes—
485Dic3—24no—
486Dic3—82no—
487Dic3—81no—
488C23—22no—
489C23—22no—
490C23—34no—
491C23—34no—
492C23—34no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—616no—
499C23—68no—
500C23—616no—
501C23—68no—
502C23—62no—
503C23—62no—
504C23—62no—
505C23—68no—
506C23—62no—
507C23—121no—
508C23—121no—
509C23—121no—
510C23—122no—
511C23—122no—
512C23#88412yes—
513C23D3×D812yes—
514C23#91612yes—
515C23C2×D3×D411yescharacteristic
516D4—41no—
517D4—42no—
518D4—41no—
519D4—84no—
520D4—121no—
521D4—122no—
522D4—121no—
523D4—248no—
524D4—248no—
525D4—244no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—34no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—616no—
538C2×C4—68no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—121no—
546C2×C4—122no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4#88412yes—
552C2×C4C2×#23511yescharacteristic
553C2×C4C2×D3×D411yescharacteristic
554C2×C4C2×D3×D412yes—
555C2×C4D3×#4112yes—
556C8—24no—
557C8—34no—
558C8—62no—
559C2×C3—24no—
560C2×C3—161no—
561C2×C3#284711yescharacteristic
562C2×C3C2×#46612yes—
563C2×C3C2×#46611yescharacteristic
564C2×C3#305012yes—
565C2×C3#305011yescharacteristic
566D3—28no—
567D3—161no—
568C22—24no—
569C22—24no—
570C22—22no—
571C22—22no—
572C22—38no—
573C22—632no—
574C22—64no—
575C22—64no—
576C22—64no—
577C22—64no—
578C22—64no—
579C22—64no—
580C22—82no—
581C22—82no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—242no—
586C22—242no—
587C22—241no—
588C22—241no—
589C22—241no—
590C22#477514yescomplemented
591C22C2×#88411yescharacteristic
592C22D3×#12112yes—
593C22C2×D3×D811yescharacteristic
594C22C2×#91611yescharacteristic
595C22D3×#13512yes—
596C4—22no—
597C4—64no—
598C4—82no—
599C4—81no—
600C4—242no—
601C4—241no—
602C4#554712yes—
603C4D3×#12011yescharacteristic
604C4#565611yescharacteristic
605C3#3293011yescharacteristic, complemented, Sylow
606C2—24no—
607C2—68no—
608C2—161no—
609C2—481no—
610C2#7493511yescharacteristic
611C2C2×#477511yescharacteristic
612C2#7883811yescharacteristic
613C2#8065612yes—
614C2D3×#46612yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1893C2
socleC23×C3C22×D4
hypercenter#1893D3
hypocenterC3#32930
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
1. C23×D3×C8C2semidirect
2. #82723C2semidirect
3. C2×#5038C2semidirect
4. C2×#4775C2semidirect
5. #74048C2semidirect
6. C3×#1893C2semidirect
7. #71973C2semidirect
8. #70202C2semidirect
9. #69512C2semidirect
10. #68883C2semidirect
11. #68869C2semidirect
18. C22×#213C22semidirect
27. C2×C3×#102C22semidirect
30. C2×#832C22semidirect
32. C2×#821C22semidirect
36. C2×#809C22semidirect
37. C2×#807C22semidirect
38. C2×#807C22semidirect
43. C3×#340C22semidirect
44. #4551C22semidirect
45. #4529C22semidirect
54. #1893D3semidirect
97. C22×C3⋊C8D4semidirect
106. C2×D3×C8D4semidirect
141. C2×#102C2×D3semidirect
152. #340C2×D3semidirect
160. C23×D3D8semidirect
368. C3⋊C8#120semidirect
460. C22×C3#466semidirect
478. C2×D3#466semidirect
590. C22#4775semidirect
605. C3#32930semidirect

Maximal subgroups

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

subgroupindex
C23×D3×C82
#827232
C2×#50382
C2×#47752
#740482
C3×#18932
#719732
#702022
#695122
#688832
#688692
#329303