Finite Group Databasebeta

← #10944015

Subgroups — #10944015

Order 768.

1436 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#887C211yesmaximal, characteristic
2#82811C211yesmaximal, characteristic
3C2×#5050C211yesmaximal, characteristic
4C2×#4795C212yesmaximal, complemented
5C2×#4739C211yesmaximal, characteristic
6C3×#1872C211yesmaximal, characteristic
7#71559C211yesmaximal, characteristic
8#70281C212yesmaximal, complemented
9#69908C211yesmaximal, characteristic
10#68908C212yesmaximal, complemented
11#68904C212yesmaximal, complemented
12#32906—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×#944C2211yescharacteristic
17C22×#216C2212yes—
18C2×#924C2211yescharacteristic
19C22×Dic3⋊C4—21no—
20C22×C4×Dic3—21no—
21C22×C4×Dic3C2211yescharacteristic
22C22×Dic12C2212yes—
23C2×#887—21no—
24C2×#887—21no—
25C2×#887—28no—
26C2×#887C2212yes—
27C2×#887C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×C22⋊C8C2211yescharacteristic
30#5229C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#816C2212yes—
33C2×Dic3⋊C8C2212yes—
34#5050—24no—
35#4976—21no—
36#4976—21no—
37#4975C2211yescharacteristic
38#4973C2211yescharacteristic
39C2×#810C2212yes—
40C2×#810C2212yes—
41#4942—21no—
42#4942—21no—
43#4795—28no—
44#4739—24no—
45C3×#340C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C22×C4×Q8—31no—
49C2×#462—32no—
50C2×#451—31no—
51C2×#451—31no—
52#2857—31no—
53#1878—32no—
54#1878—32no—
55#1872—31no—
56#1872D311yescharacteristic
57#1803—32no—
58#1797—31no—
59C22×C3×Q8C2311yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C23×Dic3—21no—
63C23×Dic3D411yescharacteristic
64C23×Dic3D411yescharacteristic
65C22×Dic6—21no—
66C22×Dic6—22no—
67C22×Dic6—24no—
68C22×Dic6—24no—
69C22×Dic6D414yes—
70C22×Dic6C2311yescharacteristic
71C22×Dic6C2312yes—
72C22×C3×C8C2312yes—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C22⋊C4—21no—
81C2×#217—21no—
82#944—24no—
83C2×#216—24no—
84C2×#216—24no—
85C2×#216—24no—
86C2×C4⋊Dic3C2311yescharacteristic
87#924—22no—
88#924—22no—
89C2×Dic3⋊C4—28no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—21no—
94C2×Dic3⋊C4—21no—
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×C4×Dic3—28no—
98C2×C4×Dic3—22no—
99C2×C4×Dic3—22no—
100C2×C4×Dic3—24no—
101C2×C4×Dic3D411yescharacteristic
102C2×C4×Dic3D411yescharacteristic
103C2×C4×Dic3C2311yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2312yes—
106C2×Dic12—24no—
107C2×Dic12—24no—
108C2×Dic12—24no—
109#887—28no—
110#887—28no—
111#887—24no—
112#887—24no—
113#887—24no—
114#887D418yes—
115C3×#103—24no—
116C3×C22⋊C8—22no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121#830—22no—
122#816—24no—
123Dic3⋊C8—24no—
124#810—24no—
125#810—24no—
126C23×Q8—31no—
127C22×Q16—32no—
128C22×Q16—32no—
129C2×C4⋊Q8—31no—
130C2×C22⋊Q8—31no—
131C2×C22⋊Q8C2×D311yescharacteristic
132C2×C4×Q8—32no—
133C2×C4×Q8—32no—
134C2×C4×Q8—61no—
135C2×C4×Q8—61no—
136C2×C4×Q8—68no—
137C22×C4⋊C4—61no—
138C22×C4⋊C4C2×D311yescharacteristic
139C22×C42—31no—
140C22×C42—61no—
141#462—68no—
142#451—64no—
143#451—64no—
144C2×C4⋊C8—32no—
145C2×#103—32no—
146C2×#103—32no—
147C2×#103—32no—
148C2×#103C2×D312yes—
149C2×C22⋊C8—31no—
150C2×C22⋊C8C2×D311yescharacteristic
151#391—31no—
152#391—31no—
153#389—61no—
154#389—61no—
155#384—31no—
156#380—61no—
157#380—61no—
158#340—32no—
159#340C2×D312yes—
160#336—32no—
161C24×C3C2×D411yescharacteristic
162C2×C3×Q8—22no—
163C2×C3×Q8—21no—
164C2×C3×Q8—22no—
165C2×C3×Q8—21no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D412yes—
176C22×C3×C4#4112yes—
177C22×C3×C4C2411yescharacteristic
178C22×Dic3—24no—
179C22×Dic3—24no—
180C22×Dic3—24no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—41no—
189C22×Dic3Q1614yescomplemented
190C22×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C2×Dic6—24no—
195C2×Dic6—24no—
196C2×Dic6—216no—
197C2×Dic6—24no—
198C2×Dic6—22no—
199C2×Dic6—22no—
200C2×Dic6—22no—
201C2×Dic6—216no—
202C2×Dic6—22no—
203C2×Dic6—21no—
204C2×Dic6—22no—
205C2×Dic6—21no—
206C2×Dic6C2×D414yes—
207C2×Dic6C2×D414yes—
208C2×Dic6#4114yes—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—42no—
217C3×C4⋊C4C2×D414yes—
218C3×C22⋊C4—42no—
219#217—42no—
220#216—48no—
221C4⋊Dic3—21no—
222C4⋊Dic3—21no—
223Dic3⋊C4—28no—
224Dic3⋊C4—28no—
225Dic3⋊C4—42no—
226Dic3⋊C4—42no—
227Dic3⋊C4—42no—
228Dic3⋊C4—42no—
229Dic3⋊C4—42no—
230C4×Dic3—28no—
231C4×Dic3—22no—
232C4×Dic3—22no—
233C4×Dic3—22no—
234C4×Dic3—28no—
235C4×Dic3C2×D414yes—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239Dic12—48no—
240C22×Q8—34no—
241C22×Q8—32no—
242C22×Q8—31no—
243C22×Q8—64no—
244C22×Q8—64no—
245C22×Q8—62no—
246C22×Q8—61no—
247C22×Q8C22×D311yescharacteristic
248C23×C4—21no—
249C23×C4—31no—
250C23×C4—31no—
251C23×C4—61no—
252C23×C4C22×D311yescharacteristic
253C2×Q16—64no—
254C2×Q16—64no—
255C2×Q16—64no—
256C2×Q16—64no—
257C2×Q16—64no—
258C2×Q16—64no—
259C22×C8—32no—
260C22×C8C22×D312yes—
261C4⋊Q8—64no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8—62no—
265C22⋊Q8—62no—
266C4×Q8—38no—
267C4×Q8—64no—
268C4×Q8—64no—
269C4×Q8—68no—
270C4×Q8—68no—
271C4×Q8—64no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—62no—
277C2×C4⋊C4—62no—
278C2×C4⋊C4—68no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C4⋊C4C22×D312yes—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—61no—
288C2×C42—31no—
289C2×C42—31no—
290C2×C42—31no—
291C2×C42—31no—
292C2×C42—64no—
293C2×C42—62no—
294C2×C42—62no—
295C2×C42—68no—
296C4⋊C8—64no—
297#103—24no—
298#103—64no—
299#103—64no—
300#103—64no—
301C22⋊C8—22no—
302C22⋊C8—62no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3#12112yes—
312C23×C3C2×Q1612yes—
313C23×C3#13712yes—
314C23×C3C22×D411yescharacteristic
315C3×Q8—42no—
316C3×Q8—42no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—48no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4#12112yes—
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C22×D412yes—
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4C2×#4112yes—
336C2×Dic3—216no—
337C2×Dic3—21no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—28no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—44no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3#12012yes—
356C2×Dic3#12012yes—
357C2×Dic3C2×Q1612yes—
358C2×Dic3C2×Q1612yes—
359C2×Dic3#13712yes—
360Dic6—216no—
361Dic6—24no—
362Dic6—24no—
363Dic6—24no—
364Dic6—42no—
365Dic6—42no—
366Dic6#12118yes—
367C3×C8—42no—
368C3⋊C8—42no—
369C24D3×D411yescharacteristic
370C2×Q8—21no—
371C2×Q8—22no—
372C2×Q8—21no—
373C2×Q8—22no—
374C2×Q8—34no—
375C2×Q8—34no—
376C2×Q8—34no—
377C2×Q8—616no—
378C2×Q8—64no—
379C2×Q8—64no—
380C2×Q8—616no—
381C2×Q8—64no—
382C2×Q8—62no—
383C2×Q8—62no—
384C2×Q8—62no—
385C2×Q8—61no—
386C2×Q8—62no—
387C2×Q8—61no—
388C2×Q8—62no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—31no—
396C22×C4—34no—
397C22×C4—31no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—64no—
406C22×C4—64no—
407C22×C4—64no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—121no—
413C22×C4#23512yes—
414C22×C4D3×D411yescharacteristic
415C22×C4D3×D412yes—
416C22×C4C23×D311yescharacteristic
417Q16—128no—
418Q16—128no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—62no—
423C2×C8—62no—
424C2×C8—62no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—42no—
430C4⋊C4—68no—
431C4⋊C4—68no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—122no—
435C4⋊C4—122no—
436C4⋊C4—122no—
437C4⋊C4—122no—
438C4⋊C4—122no—
439C4⋊C4D3×D414yes—
440C22⋊C4—42no—
441C22⋊C4—122no—
442C42—34no—
443C42—62no—
444C42—62no—
445C42—68no—
446C42—68no—
447C42—62no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3#46214yescomplemented
455C22×C3C2×#12011yescharacteristic
456C22×C3C2×#12111yescharacteristic
457C22×C3#52511yescharacteristic
458C22×C3C22×Q1611yescharacteristic
459C22×C3C2×#13711yescharacteristic
460C22×C3C2×#13711yescharacteristic
461C22×C3#57511yescharacteristic
462C3×C4—22no—
463C3×C4—44no—
464C3×C4—81no—
465C3×C4—81no—
466C3×C4C2×#12112yes—
467C3×C4#52512yes—
468Dic3—28no—
469Dic3—22no—
470Dic3—22no—
471Dic3—41no—
472Dic3—81no—
473Dic3—81no—
474Dic3#45114yescomplemented
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23#88412yes—
480C23#91512yes—
481C23D3×Q1612yes—
482C23C2×D3×D411yescharacteristic
483Q8—38no—
484Q8—42no—
485Q8—42no—
486Q8—616no—
487Q8—64no—
488Q8—64no—
489Q8—64no—
490Q8—122no—
491Q8—122no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—32no—
498C2×C4—32no—
499C2×C4—32no—
500C2×C4—32no—
501C2×C4—32no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—68no—
512C2×C4—616no—
513C2×C4—64no—
514C2×C4—64no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—64no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—124no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4#88412yes—
531C2×C4C2×#23511yescharacteristic
532C2×C4C2×D3×D412yes—
533C2×C4C2×D3×D411yescharacteristic
534C2×C4D3×#4112yes—
535C8—42no—
536C8—122no—
537C2×C3—24no—
538C2×C3#284711yescharacteristic
539C2×C3C2×#45111yescharacteristic
540C2×C3#300711yescharacteristic
541C2×C3C2×#46211yescharacteristic
542C2×C3#303711yescharacteristic
543C2×C3#310411yescharacteristic
544C2×C3#311311yescharacteristic
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22#479514yescomplemented
552C22C2×#88411yescharacteristic
553C22D3×#12011yescharacteristic
554C22#565611yescharacteristic
555C22C2×#91511yescharacteristic
556C22C2×D3×Q1611yescharacteristic
557C22#577311yescharacteristic
558C22D3×#13711yescharacteristic
559C4—22no—
560C4—34no—
561C4—44no—
562C4—62no—
563C4—62no—
564C4—68no—
565C4—81no—
566C4—81no—
567C4—121no—
568C4—241no—
569C4—241no—
570C4#554712yes—
571C4D3×#12112yes—
572C3#3290611yescharacteristic, complemented, Sylow
573C2—24no—
574C2#7493511yescharacteristic
575C2C2×#479511yescharacteristic
576C2#7891311yescharacteristic
577C2#8014711yescharacteristic
578C2D3×#45111yescharacteristic
579C2#8141411yescharacteristic
580C2#8145211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1872C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1872D3
hypocenterC3#32906
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
4. C2×#4795C2semidirect
8. #70281C2semidirect
10. #68908C2semidirect
11. #68904C2semidirect
189. C22×Dic3Q16semidirect
454. C22×C3#462semidirect
474. Dic3#451semidirect
551. C22#4795semidirect
572. C3#32906semidirect

Maximal subgroups

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

subgroupindex
C22×#8872
#828112
C2×#50502
C2×#47952
C2×#47392
C3×#18722
#715592
#702812
#699082
#689082
#689042
#329063