Finite Group Databasebeta

← #10823409

Subgroups — #10823409

Order 768.

1508 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5716C211yesmaximal, characteristic, complemented
2C2×#5152C212yesmaximal, complemented
3#76291C211yesmaximal, characteristic, complemented
4C3×#2011C211yesmaximal, characteristic, complemented
5#72043C212yesmaximal, complemented
6#71762C211yesmaximal, characteristic, complemented
7#70176C212yesmaximal, complemented
8#70152C212yesmaximal, complemented
9#69638C211yesmaximal, characteristic, complemented
10#68633C211yesmaximal, characteristic, complemented
11#68600C211yesmaximal, characteristic, complemented
12#23573—31nomaximal, Sylow
13C22×#237—21no—
14C22×C3×D8C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×#938—21no—
17C2×#938C2211yescharacteristic
18C2×#938C2212yes—
19C2×#936—21no—
20C2×D4×Dic3—21no—
21C2×D4×Dic3C2211yescharacteristic
22C2×D4×Dic3C2212yes—
23#5716—28no—
24C2×#67⋊C4—21no—
25C2×C42×D3C2211yescharacteristic
26C2×#869C2211yescharacteristic
27C2×C3×#102C2212yes—
28C3×#390C2211yescharacteristic
29C2×#832C2212yes—
30C2×#832C2212yes—
31C2×#832C2212yes—
32#5152—28no—
33C2×(C2×D3)⋊C8C2212yes—
34C2×#817C2212yes—
35C2×#815C2212yes—
36#4987C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38#4956—21no—
39#4945—21no—
40Dic3×C22⋊C4—21no—
41#4928C2211yescharacteristic
42C3×#336C2211yescharacteristic
43#4551C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#442—32no—
48C4×#121—31no—
49#2058—31no—
50#2041—31no—
51#2034—32no—
52#2023—31no—
53#2011D311yescharacteristic, complemented
54#1999—32no—
55#1762—32no—
56#1749—31no—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C22×C3×D4C2311yescharacteristic
61C22×#67—22no—
62C22×#67—21no—
63C23×Dic3—22no—
64C23×Dic3—21no—
65C2×#237—28no—
66C2×#237—44no—
67C2×#237—44no—
68C22×C4×D3D411yescharacteristic
69C22×C4×D3D411yescharacteristic
70C22×C4×D3C2311yescharacteristic
71C22×Dic6—21no—
72C2×C3×D8—24no—
73C2×C3×D8—24no—
74C2×C3×D8—24no—
75C22×C3×C8C2312yes—
76C3×#127—24no—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C42C2311yescharacteristic
81C2×#217—21no—
82C2×#217—22no—
83C2×#217—22no—
84C2×#217—21no—
85#938—28no—
86#938—24no—
87#938—22no—
88#938—24no—
89#938—22no—
90#936—28no—
91D4×Dic3—28no—
92D4×Dic3—24no—
93D4×Dic3—24no—
94D4×Dic3D418yescomplemented
95C2×(C2×D3)⋊C4—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2312yes—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3D411yescharacteristic
103C2×C4×Dic3D411yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2312yes—
106#67⋊C4—44no—
107C42×D3—24no—
108#869—24no—
109C3×#102—24no—
110#832—24no—
111#832—24no—
112#832—24no—
113C3⋊#95—21no—
114C3⋊#95—21no—
115C3⋊#95—21no—
116(C2×D3)⋊C8—24no—
117#817—24no—
118#815—24no—
119C22×#41—61no—
120C22×D8C2×D312yes—
121#532—68no—
122C2×C4⋊Q8—31no—
123C2×#127C2×D311yescharacteristic
124C2×#124—61no—
125C2×#121—32no—
126C2×#121—31no—
127C2×#121—61no—
128C2×C4×D4—32no—
129C2×C4×D4—31no—
130C2×C4×D4—61no—
131C2×C4×D4—61no—
132C22×C42—31no—
133#442—68no—
134C2×#106—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×#102—32no—
138C2×#102C2×D312yes—
139C2×C22⋊C8—32no—
140C2×#97—32no—
141#390C2×D311yescharacteristic
142#389—31no—
143#388—61no—
144#386—61no—
145#380—31no—
146C4×C4⋊C4—31no—
147C4×C22⋊C4—61no—
148#340—32no—
149#340—32no—
150#336—31no—
151#336C2×D311yescharacteristic
152C24×C3—21no—
153C24×C3—22no—
154C23×D3C2×D411yescharacteristic
155C2×C3×D4—24no—
156C2×C3×D4—24no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—21no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4—42no—
168C2×C3×D4—42no—
169C2×C3×D4C2×D414yes—
170C2×C3×D4#4112yescomplemented
171C22×C3×C4—21no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C2×#67—28no—
176C2×#67—24no—
177C2×#67—42no—
178C2×#67—44no—
179C2×#67—44no—
180C2×#67—42no—
181C22×Dic3—21no—
182C22×Dic3—22no—
183C22×Dic3—28no—
184C22×Dic3—21no—
185C22×Dic3—44no—
186C22×Dic3—44no—
187C22×Dic3—42no—
188C22×Dic3—42no—
189C22×Dic3—42no—
190C22×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193#237—432no—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3—22no—
197C2×C4×D3—22no—
198C2×C4×D3—22no—
199C2×C4×D3—22no—
200C2×C4×D3—21no—
201C2×C4×D3—21no—
202C2×C4×D3—22no—
203C2×C4×D3—41no—
204C2×Dic6—24no—
205C2×Dic6—42no—
206C2×Dic6—42no—
207C3×D8—216no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8#4114yescomplemented
211C3×C4⋊C4—22no—
212C3×C22⋊C4—42no—
213C3×C22⋊C4—42no—
214C3×C42—22no—
215#217—28no—
216#217—24no—
217#217—28no—
218#217—42no—
219(C2×D3)⋊C4—42no—
220C4⋊Dic3—24no—
221C4⋊Dic3—22no—
222C4⋊Dic3—21no—
223C4⋊Dic3—22no—
224C4⋊Dic3C2×D414yes—
225C4⋊Dic3#4112yescomplemented
226Dic3⋊C4—42no—
227C4×Dic3—22no—
228C4×Dic3—22no—
229C4×Dic3—42no—
230C4×Dic3C2×D414yes—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×#41—68no—
235C2×#41—124no—
236C2×#41—124no—
237C22×Q8—61no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4—61no—
241C22×D4—62no—
242C22×D4C22×D312yes—
243C22×D4C22×D311yescharacteristic
244C23×C4—31no—
245C23×C4—31no—
246C23×C4—31no—
247C23×C4—62no—
248C23×C4—61no—
249C2×D8—24no—
250C2×D8—24no—
251C2×D8—24no—
252C22×C8—32no—
253C22×C8C22×D312yes—
254C4⋊Q8—64no—
255#127—24no—
256#124—68no—
257#121—64no—
258#121—62no—
259#121—68no—
260#121—64no—
261#121—62no—
262C4×D4—38no—
263C4×D4—64no—
264C4×D4—64no—
265C4×D4—68no—
266C4×D4—124no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C42—31no—
281C2×C42—31no—
282C2×C42—31no—
283C2×C42—64no—
284C2×C42—61no—
285C2×C42C22×D311yescharacteristic
286#106—64no—
287#102—24no—
288#102—64no—
289#102—64no—
290#102—64no—
291C22⋊C8—64no—
292#97—64no—
293#95—61no—
294#95—61no—
295#95—61no—
296C23×C3—24no—
297C23×C3—22no—
298C23×C3—41no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—41no—
307C23×C3—42no—
308C23×C3—41no—
309C23×C3—42no—
310C23×C3C22×D411yescharacteristic
311C22×D3—21no—
312C22×D3—21no—
313C22×D3—21no—
314C22×D3—21no—
315C22×D3—21no—
316C22×D3—21no—
317C22×D3AGL(1,ℤ/8)12yes—
318C3×D4—24no—
319C3×D4—24no—
320C3×D4—24no—
321C3×D4—48no—
322C3×D4—48no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—42no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4AGL(1,ℤ/8)12yes—
334C2×C3×C4AGL(1,ℤ/8)12yes—
335C2×C3×C4C22×D412yes—
336C2×C3×C4C2×#4111yescharacteristic
337C2×C3×C4C2×#4112yes—
338C2×C3×C4C2×#4111yescharacteristic
339#67—416no—
340#67—48no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—24no—
347C2×Dic3—22no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—416no—
351C2×Dic3—41no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—84no—
359C2×Dic3#12012yes—
360C2×Dic3#12012yes—
361C4×D3—28no—
362C4×D3—44no—
363C4×D3—42no—
364C4×D3—42no—
365Dic6—48no—
366C3×C8—24no—
367C3⋊C8—42no—
368C24—21no—
369C24—22no—
370C24—31no—
371#41—1232no—
372C2×Q8—64no—
373C2×Q8—122no—
374C2×Q8—122no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—68no—
390C2×D4—64no—
391C2×D4—124no—
392C2×D4—122no—
393C2×D4—122no—
394C2×D4—124no—
395C2×D4D3×D414yes—
396C2×D4#23712yescomplemented
397C22×C4—21no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—68no—
405C22×C4—62no—
406C22×C4—61no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—121no—
415C22×C4—122no—
416C22×C4—124no—
417C22×C4—124no—
418C22×C4—122no—
419C22×C4—122no—
420C22×C4D3×D411yescharacteristic
421C22×C4D3×D411yescharacteristic
422C22×C4C23×D311yescharacteristic
423D8—216no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—62no—
427C2×C8—62no—
428C2×C8—62no—
429C2×C8#23714yescomplemented
430C4⋊C4—22no—
431C4⋊C4—34no—
432C4⋊C4—32no—
433C4⋊C4—64no—
434C4⋊C4—62no—
435C4⋊C4—61no—
436C4⋊C4—62no—
437C4⋊C4—122no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—68no—
441C22⋊C4—68no—
442C22⋊C4—64no—
443C22⋊C4—122no—
444C22⋊C4—122no—
445C42—22no—
446C42—34no—
447C42—62no—
448C42—62no—
449C42—122no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—44no—
453C22×C3—42no—
454C22×C3—44no—
455C22×C3—44no—
456C22×C3—82no—
457C22×C3—82no—
458C22×C3—82no—
459C22×C3—81no—
460C22×C3—81no—
461C22×C3—82no—
462C22×C3—81no—
463C22×C3—81no—
464C22×C3C2×#12011yescharacteristic
465C22×C3#52511yescharacteristic
466C22×C3#53211yescharacteristic
467C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
468C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
469C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
470C22×C3#57511yescharacteristic
471C2×D3—22no—
472C2×D3—22no—
473C2×D3—22no—
474C2×D3—41no—
475C2×D3—41no—
476C2×D3—41no—
477C2×D3—41no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—81no—
481C3×C4#52512yes—
482C3×C4#53212yes—
483Dic3—22no—
484Dic3—22no—
485Dic3—42no—
486Dic3—41no—
487Dic3—44no—
488Dic3—81no—
489C23—24no—
490C23—22no—
491C23—32no—
492C23—42no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—41no—
503C23—42no—
504C23—61no—
505C23—61no—
506C23—61no—
507C23—61no—
508C23—61no—
509C23—61no—
510C23C2×D3×D411yescharacteristic
511Q8—128no—
512D4—24no—
513D4—24no—
514D4—24no—
515D4—48no—
516D4—48no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—1216no—
521D4—128no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—32no—
527C2×C4—32no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—68no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—64no—
539C2×C4—61no—
540C2×C4—62no—
541C2×C4—124no—
542C2×C4—1216no—
543C2×C4—121no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4—121no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—244no—
553C2×C4#91112yes—
554C2×C4#91112yes—
555C2×C4C2×D3×D412yes—
556C2×C4C2×#23711yescharacteristic
557C2×C4C2×#23712yes—
558C2×C4D3×#4111yescharacteristic
559C8—24no—
560C8—122no—
561C2×C3—82no—
562C2×C3—84no—
563C2×C3#238811yescharacteristic
564C2×C3#300311yescharacteristic
565C2×C3#300611yescharacteristic
566C2×C3#310911yescharacteristic
567C2×C3#311511yescharacteristic
568C2×C3#318011yescharacteristic
569C2×C3#318411yescharacteristic
570D3—42no—
571C22—44no—
572C22—42no—
573C22—44no—
574C22—44no—
575C22—42no—
576C22—42no—
577C22—62no—
578C22—62no—
579C22—62no—
580C22—81no—
581C22—82no—
582C22—81no—
583C22—81no—
584C22—82no—
585C22—81no—
586C22—82no—
587C22—82no—
588C22—121no—
589C22—121no—
590C22—121no—
591C22—121no—
592C22D3×#12011yescharacteristic
593C22#560611yescharacteristic
594C22#571611yescharacteristic
595C22C2×#91111yescharacteristic
596C22C2×#91111yescharacteristic
597C22D3×AGL(1,ℤ/8)11yescharacteristic
598C22#577311yescharacteristic
599C4—22no—
600C4—22no—
601C4—62no—
602C4—62no—
603C4—81no—
604C4—121no—
605C4—124no—
606C4—122no—
607C4—241no—
608C4#558612yes—
609C4#559612yes—
610C3#2357311yescharacteristic, complemented, Sylow
611C2—84no—
612C2—82no—
613C2—122no—
614C2#7666511yescharacteristic
615C2#8000211yescharacteristic
616C2#8000311yescharacteristic
617C2#8031411yescharacteristic
618C2#8031511yescharacteristic
619C2#8177711yescharacteristic
620C2#8183811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2011C2
socleC23×C3C22×D4
hypercenter#2011D3
hypocenterC3#23573
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2011

Complements

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

NH ≅ G/Ntype
1. C2×#5716C2semidirect
2. C2×#5152C2semidirect
3. #76291C2semidirect
4. C3×#2011C2semidirect
5. #72043C2semidirect
6. #71762C2semidirect
7. #70176C2semidirect
8. #70152C2semidirect
9. #69638C2semidirect
10. #68633C2semidirect
11. #68600C2semidirect
53. #2011D3semidirect
94. D4×Dic3D4semidirect
170. C2×C3×D4#41semidirect
210. C2×C3×C8#41semidirect
225. C4⋊Dic3#41semidirect
396. C2×D4#237semidirect
429. C2×C8#237semidirect
610. C3#23573semidirect

Maximal subgroups

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

subgroupindex
C2×#57162
C2×#51522
#762912
C3×#20112
#720432
#717622
#701762
#701522
#696382
#686332
#686002
#235733