Finite Group Databasebeta

← #10823414

Subgroups — #10823414

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×#5725C211yesmaximal, characteristic, complemented
2C2×#4785C212yesmaximal, complemented
3#76291C211yesmaximal, characteristic, complemented
4C3×#2023C211yesmaximal, characteristic, complemented
5#71783C211yesmaximal, characteristic, complemented
6#70204C212yesmaximal, complemented
7#69658C211yesmaximal, characteristic, complemented
8#68823C212yesmaximal, complemented
9#68782C212yesmaximal, complemented
10#68629C211yesmaximal, characteristic, complemented
11#68600C211yesmaximal, characteristic, complemented
12#23573—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#947—21no—
16C2×#938C2211yescharacteristic
17C2×D4×Dic3C2211yescharacteristic
18C22×#213C2212yes—
19#5725—28no—
20C2×#895—21no—
21C2×#895C2212yes—
22C2×#893—21no—
23C2×#893C2212yes—
24C2×#67⋊C4—21no—
25C2×#874C2211yescharacteristic
26C2×C42×D3C2211yescharacteristic
27C2×C3×#102C2212yes—
28C3×#390C2211yescharacteristic
29C2×#821C2212yes—
30C2×#821C2212yes—
31C2×(C2×D3)⋊C8C2212yes—
32C2×#815C2212yes—
33#4987C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35C2×#809C2212yes—
36C2×#808C2212yes—
37#4956—21no—
38#4945—21no—
39Dic3×C22⋊C4—21no—
40#4928C2211yescharacteristic
41#4785—28no—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#442—32no—
48C4×#121—31no—
49#2058—31no—
50#2041—31no—
51#2034—32no—
52#2023D311yescharacteristic, complemented
53#2011—31no—
54#1999—32no—
55#1762—32no—
56#1749—31no—
57C22×C3×Q8—21no—
58C22×C3×D4C2311yescharacteristic
59C22×#67—21no—
60C23×Dic3—21no—
61C2×#239—28no—
62C2×#239—44no—
63C2×#239—44no—
64C22×D12—22no—
65C22×D12—21no—
66C22×D12—21no—
67C22×D12C2312yes—
68C22×C4×D3—22no—
69C22×C4×D3D411yescharacteristic
70C22×C4×D3D411yescharacteristic
71C22×C4×D3C2311yescharacteristic
72C22×C3×C8C2312yes—
73C3×C4⋊Q8—24no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—21no—
79C2×C3×C42C2311yescharacteristic
80C2×#217—21no—
81C2×#217—21no—
82#947—28no—
83#938—22no—
84#938—22no—
85D4×Dic3—24no—
86C2×#213—24no—
87C2×#213—24no—
88C2×#213—24no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—21no—
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94C2×Dic3⋊C4—21no—
95C2×C4×Dic3—21no—
96C2×C4×Dic3D411yescharacteristic
97C2×C4×Dic3D411yescharacteristic
98C2×C4×Dic3C2311yescharacteristic
99C22×C3⋊C8C2312yes—
100#895—28no—
101#895—24no—
102#895—24no—
103#893—28no—
104#893—24no—
105#893D418yescomplemented
106#67⋊C4—44no—
107#874—24no—
108C42×D3—24no—
109C3×#102—24no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113#821—24no—
114#821—24no—
115(C2×D3)⋊C8—24no—
116#815—24no—
117#809—24no—
118#808—24no—
119C22×#41—61no—
120C22×D8—32no—
121#532—68no—
122C2×C4⋊Q8C2×D311yescharacteristic
123C2×#127—31no—
124C2×#124—61no—
125C2×#121—31no—
126C2×#121—32no—
127C2×#121—61no—
128C2×C4×D4—31no—
129C2×C4×D4—32no—
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#340C2×D312yes—
150#336—31no—
151#336C2×D311yescharacteristic
152C24×C3—21no—
153C23×D3—22no—
154C23×D3C2×D411yescharacteristic
155C2×C3×Q8—24no—
156C2×C3×Q8—42no—
157C2×C3×Q8—42no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—21no—
161C2×C3×D4#4112yescomplemented
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C2×#67—24no—
168C2×#67—42no—
169C2×#67—42no—
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3—42no—
173C22×Dic3—42no—
174C22×Dic3—42no—
175C22×Dic3C2×D411yescharacteristic
176C22×Dic3C2×D411yescharacteristic
177C22×Dic3C2×D411yescharacteristic
178#239—432no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—22no—
182C2×D12—22no—
183C2×D12—24no—
184C2×D12—24no—
185C2×D12—28no—
186C2×D12—22no—
187C2×D12—42no—
188C2×D12—42no—
189C2×D12—44no—
190C2×D12—44no—
191C2×D12—42no—
192C2×D12—42no—
193C2×D12C2×D414yes—
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—28no—
204C2×C4×D3—41no—
205C2×C4×D3—44no—
206C2×C4×D3—44no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4C2×D414yes—
214C3×C22⋊C4—42no—
215C3×C22⋊C4—42no—
216C3×C42—22no—
217#217—24no—
218#217—42no—
219#213—216no—
220(C2×D3)⋊C4—28no—
221(C2×D3)⋊C4—28no—
222(C2×D3)⋊C4—42no—
223C4⋊Dic3—21no—
224C4⋊Dic3—22no—
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#4114yescomplemented
234C2×#41—68no—
235C2×#41—124no—
236C2×#41—124no—
237C22×Q8—21no—
238C22×D4—32no—
239C22×D4—62no—
240C22×D4—61no—
241C22×D4—61no—
242C22×D4—61no—
243C22×D4C22×D311yescharacteristic
244C23×C4—31no—
245C23×C4—31no—
246C23×C4—31no—
247C23×C4—62no—
248C23×C4—61no—
249C2×D8—64no—
250C2×D8—64no—
251C2×D8—64no—
252C22×C8—32no—
253C22×C8C22×D312yes—
254C4⋊Q8—24no—
255#127—64no—
256#124—68no—
257#121—64no—
258#121—68no—
259#121—62no—
260#121—62no—
261#121—64no—
262C4×D4—38no—
263C4×D4—64no—
264C4×D4—64no—
265C4×D4—68no—
266C4×D4—124no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4C22×D311yescharacteristic
272C2×C4⋊C4C22×D312yes—
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—22no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3C22×D411yescharacteristic
304C22×D3—21no—
305C22×D3—21no—
306C22×D3—21no—
307C22×D3—21no—
308C22×D3—21no—
309C22×D3—21no—
310C22×D3—24no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—42no—
315C22×D3—42no—
316C22×D3—42no—
317C22×D3AGL(1,ℤ/8)12yes—
318C3×Q8—48no—
319C3×D4—24no—
320C3×D4—42no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—24no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4AGL(1,ℤ/8)12yes—
332C2×C3×C4AGL(1,ℤ/8)12yes—
333C2×C3×C4C22×D412yes—
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4C2×#4112yes—
336C2×C3×C4C2×#4111yescharacteristic
337#67—48no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—22no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—84no—
352C2×Dic3#12012yes—
353C2×Dic3#12012yes—
354D12—24no—
355D12—24no—
356D12—48no—
357D12—48no—
358D12—416no—
359D12—42no—
360D12—42no—
361C4×D3—28no—
362C4×D3—44no—
363C4×D3—416no—
364C4×D3—42no—
365C4×D3—42no—
366C3×C8—42no—
367C3⋊C8—24no—
368C24—21no—
369C24—31no—
370C24—62no—
371#41—1232no—
372C2×Q8—24no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4—34no—
379C2×D4—64no—
380C2×D4—68no—
381C2×D4—64no—
382C2×D4—64no—
383C2×D4—62no—
384C2×D4—62no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—122no—
389C2×D4—122no—
390C2×D4—124no—
391C2×D4—124no—
392C2×D4—122no—
393C2×D4—122no—
394C2×D4—122no—
395C2×D4—122no—
396C2×D4#23912yescomplemented
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—31no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—68no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—61no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—61no—
411C22×C4—62no—
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—616no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—34no—
428C2×C8—62no—
429C2×C8—62no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—32no—
434C4⋊C4—61no—
435C4⋊C4—62no—
436C4⋊C4—122no—
437C4⋊C4D3×D414yes—
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—42no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3—81no—
456C22×C3—81no—
457C22×C3C2×#12011yescharacteristic
458C22×C3#52511yescharacteristic
459C22×C3#53211yescharacteristic
460C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
461C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
462C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
463C22×C3#57511yescharacteristic
464C2×D3—22no—
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—41no—
468C2×D3—41no—
469C2×D3—41no—
470C2×D3—41no—
471C2×D3—44no—
472C2×D3—44no—
473C2×D3—44no—
474C2×D3—82no—
475C2×D3—82no—
476C2×D3—82no—
477C2×D3—82no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—44no—
481C3×C4—81no—
482C3×C4#52512yes—
483C3×C4#53212yes—
484Dic3—22no—
485Dic3—22no—
486Dic3—42no—
487Dic3—41no—
488Dic3—81no—
489C23—22no—
490C23—32no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—61no—
498C23—61no—
499C23—61no—
500C23—61no—
501C23—61no—
502C23—61no—
503C23—64no—
504C23—122no—
505C23—122no—
506C23—122no—
507C23—122no—
508C23—122no—
509C23—122no—
510C23C2×D3×D411yescharacteristic
511Q8—48no—
512D4—24no—
513D4—42no—
514D4—64no—
515D4—64no—
516D4—128no—
517D4—128no—
518D4—1216no—
519D4—128no—
520D4—122no—
521D4—122no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—32no—
528C2×C4—32no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—68no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—62no—
543C2×C4—124no—
544C2×C4—1216no—
545C2×C4—121no—
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#91412yes—
555C2×C4C2×D3×D412yes—
556C2×C4C2×#23911yescharacteristic
557C2×C4D3×#4111yescharacteristic
558C2×C4D3×#4112yes—
559C8—42no—
560C8—64no—
561C2×C3—82no—
562C2×C3#238811yescharacteristic
563C2×C3#300311yescharacteristic
564C2×C3#300611yescharacteristic
565C2×C3#310911yescharacteristic
566C2×C3#311511yescharacteristic
567C2×C3#318011yescharacteristic
568C2×C3#318411yescharacteristic
569D3—42no—
570D3—84no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—62no—
575C22—62no—
576C22—62no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—121no—
582C22—121no—
583C22—121no—
584C22—121no—
585C22—124no—
586C22—124no—
587C22—124no—
588C22—242no—
589C22—242no—
590C22—242no—
591C22—242no—
592C22D3×#12011yescharacteristic
593C22#562711yescharacteristic
594C22#566811yescharacteristic
595C22C2×#91111yescharacteristic
596C22C2×#91411yescharacteristic
597C22D3×AGL(1,ℤ/8)11yescharacteristic
598C22#577711yescharacteristic
599C4—22no—
600C4—22no—
601C4—44no—
602C4—62no—
603C4—62no—
604C4—81no—
605C4—122no—
606C4—121no—
607C4—241no—
608C4#562712yes—
609C4#566812yes—
610C3#2357311yescharacteristic, complemented, Sylow
611C2—82no—
612C2—122no—
613C2—244no—
614C2#7677711yescharacteristic
615C2#8000211yescharacteristic
616C2#8004011yescharacteristic
617C2#8095111yescharacteristic
618C2#8104611yescharacteristic
619C2#8156011yescharacteristic
620C2#8166811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2023C2
socleC23×C3C22×D4
hypercenter#2023D3
hypocenterC3#23573
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. C2×#5725C2semidirect
2. C2×#4785C2semidirect
3. #76291C2semidirect
4. C3×#2023C2semidirect
5. #71783C2semidirect
6. #70204C2semidirect
7. #69658C2semidirect
8. #68823C2semidirect
9. #68782C2semidirect
10. #68629C2semidirect
11. #68600C2semidirect
52. #2023D3semidirect
105. #893D4semidirect
161. C2×C3×D4#41semidirect
225. C4⋊Dic3#41semidirect
233. C2×C3⋊C8#41semidirect
396. C2×D4#239semidirect
610. C3#23573semidirect

Maximal subgroups

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

subgroupindex
C2×#57252
C2×#47852
#762912
C3×#20232
#717832
#702042
#696582
#688232
#687822
#686292
#686002
#235733