Finite Group Databasebeta

← #10823415

Subgroups — #10823415

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×#4869C212yesmaximal, complemented
3#75964C211yesmaximal, characteristic, complemented
4C3×#2058C211yesmaximal, characteristic, complemented
5#71792C211yesmaximal, characteristic, complemented
6#70581C212yesmaximal, complemented
7#69627C211yesmaximal, characteristic, complemented
8#68886C212yesmaximal, complemented
9#68782C212yesmaximal, complemented
10#68674C211yesmaximal, characteristic, complemented
11#68600C211yesmaximal, characteristic, complemented
12#23573—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#947—21no—
16C22×D24C2212yes—
17#5725—28no—
18C2×#895—21no—
19C2×#895C2211yescharacteristic
20C2×#895C2212yes—
21C2×#893—21no—
22C2×#893—21no—
23C2×#893C2211yescharacteristic
24C2×#893C2212yes—
25C2×#874C2211yescharacteristic
26C2×C42×D3C2211yescharacteristic
27C2×C3×#106C2212yes—
28C3×#389C2211yescharacteristic
29C2×#821C2212yes—
30C2×(C2×D3)⋊C8C2212yes—
31C2×#815C2212yes—
32#4990—21no—
33#4987C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35C2×#809C2212yes—
36C2×#809C2212yes—
37C2×#809C2212yes—
38#4939C2211yescharacteristic
39#4869—28no—
40#4666—21no—
41#4665—21no—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#442—32no—
48C4×#121—31no—
49#2058D311yescharacteristic, complemented
50#2041—31no—
51#2034—32no—
52#2023—31no—
53#2011—31no—
54#1999—32no—
55#1762—32no—
56#1749—31no—
57C22×C3×Q8—21no—
58C2×#239—28no—
59C2×#239—44no—
60C2×#239—44no—
61C22×D12—21no—
62C22×D12—22no—
63C22×D12—21no—
64C22×D12—21no—
65C22×D12C2312yes—
66C22×D12C2311yescharacteristic
67C22×C4×D3—22no—
68C22×C4×D3—21no—
69C22×C4×D3D411yescharacteristic
70C22×C4×D3D411yescharacteristic
71C22×C4×D3C2311yescharacteristic
72C22×C3×C8C2312yes—
73C3×C4⋊Q8—24no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C42C2311yescharacteristic
80#947—28no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4×Dic3—21no—
90C2×C4×Dic3D411yescharacteristic
91C2×C4×Dic3D411yescharacteristic
92C2×C4×Dic3C2311yescharacteristic
93C22×C3⋊C8C2312yes—
94C2×D24—24no—
95C2×D24—24no—
96C2×D24—24no—
97#895—28no—
98#895—24no—
99#895—22no—
100#895—24no—
101#895—22no—
102#893—28no—
103#893—24no—
104#893—24no—
105#893—44no—
106#893D418yescomplemented
107#874—24no—
108C42×D3—24no—
109C3×#106—24no—
110C3×#95—21no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113#821—24no—
114(C2×D3)⋊C8—24no—
115#815—24no—
116#809—24no—
117#809—24no—
118#809—24no—
119C22×#41—61no—
120C22×D8—32no—
121#532—68no—
122C2×C4⋊Q8C2×D311yescharacteristic
123C2×#127—31no—
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×#106C2×D312yes—
135C2×#102—32no—
136C2×#102—32no—
137C2×#102—32no—
138C2×#102—32no—
139C2×C22⋊C8—32no—
140C2×#97—32no—
141#390—31no—
142#389C2×D311yescharacteristic
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
152C23×D3—21no—
153C23×D3—22no—
154C23×D3C2×D411yescharacteristic
155C2×C3×Q8—24no—
156C2×C3×Q8—42no—
157C2×C3×Q8—42no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C22×Dic3—21no—
165C22×Dic3C2×D411yescharacteristic
166C22×Dic3C2×D411yescharacteristic
167C22×Dic3C2×D411yescharacteristic
168#239—432no—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—22no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—24no—
176C2×D12—24no—
177C2×D12—28no—
178C2×D12—24no—
179C2×D12—22no—
180C2×D12—21no—
181C2×D12—42no—
182C2×D12—42no—
183C2×D12—42no—
184C2×D12—44no—
185C2×D12—42no—
186C2×D12—44no—
187C2×D12—42no—
188C2×D12—42no—
189C2×D12C2×D414yes—
190C2×D12#4112yescomplemented
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—22no—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3—22no—
197C2×C4×D3—21no—
198C2×C4×D3—21no—
199C2×C4×D3—22no—
200C2×C4×D3—28no—
201C2×C4×D3—41no—
202C2×C4×D3—44no—
203C2×C4×D3—44no—
204C2×C4×D3—42no—
205C2×C4×D3—42no—
206C2×C4×D3—42no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8#4114yescomplemented
210C3×C4⋊C4—24no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—42no—
215C3×C4⋊C4C2×D414yes—
216C3×C4⋊C4#4112yescomplemented
217C3×C42—22no—
218(C2×D3)⋊C4—28no—
219(C2×D3)⋊C4—24no—
220(C2×D3)⋊C4—28no—
221(C2×D3)⋊C4—42no—
222(C2×D3)⋊C4—42no—
223(C2×D3)⋊C4—42no—
224(C2×D3)⋊C4—42no—
225C4⋊Dic3—22no—
226C4×Dic3—22no—
227C4×Dic3—22no—
228C4×Dic3—42no—
229C4×Dic3C2×D414yes—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233D24—216no—
234C2×#41—68no—
235C2×#41—124no—
236C2×#41—124no—
237C22×Q8—21no—
238C22×D4—32no—
239C22×D4—31no—
240C22×D4—61no—
241C22×D4—62no—
242C22×D4—61no—
243C22×D4—61no—
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—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—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C4⋊C4C22×D312yes—
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
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—24no—
287#102—64no—
288#102—64no—
289#102—64no—
290#102—64no—
291C22⋊C8—64no—
292#97—64no—
293#95—21no—
294#95—61no—
295#95—61no—
296C23×C3C22×D411yescharacteristic
297C22×D3—21no—
298C22×D3—21no—
299C22×D3—21no—
300C22×D3—21no—
301C22×D3—21no—
302C22×D3—21no—
303C22×D3—24no—
304C22×D3—22no—
305C22×D3—41no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—41no—
309C22×D3—41no—
310C22×D3—41no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—41no—
315C22×D3—41no—
316C22×D3—42no—
317C22×D3AGL(1,ℤ/8)12yes—
318C3×Q8—48no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4AGL(1,ℤ/8)12yes—
333C2×C3×C4AGL(1,ℤ/8)12yes—
334C2×C3×C4C22×D412yes—
335C2×C3×C4C2×#4112yes—
336C2×C3×C4C2×#4111yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3#12012yes—
349C2×Dic3#12012yes—
350D12—24no—
351D12—24no—
352D12—24no—
353D12—48no—
354D12—48no—
355D12—48no—
356D12—416no—
357D12—42no—
358D12—42no—
359D12—42no—
360C4×D3—28no—
361C4×D3—44no—
362C4×D3—416no—
363C4×D3—42no—
364C4×D3—42no—
365C4×D3—84no—
366C3×C8—24no—
367C3⋊C8—42no—
368C24—31no—
369C24—61no—
370C24—62no—
371#41—1232no—
372C2×Q8—24no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×D4—34no—
376C2×D4—32no—
377C2×D4—64no—
378C2×D4—68no—
379C2×D4—64no—
380C2×D4—64no—
381C2×D4—62no—
382C2×D4—62no—
383C2×D4—62no—
384C2×D4—62no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—61no—
388C2×D4—62no—
389C2×D4—122no—
390C2×D4—122no—
391C2×D4—124no—
392C2×D4—122no—
393C2×D4—122no—
394C2×D4—124no—
395C2×D4—122no—
396C2×D4—122no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—31no—
401C22×C4—31no—
402C22×C4—31no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—68no—
406C22×C4—62no—
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—616no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—62no—
427C2×C8—62no—
428C2×C8—62no—
429C2×C8#23914yescomplemented
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—42no—
435C4⋊C4—62no—
436C4⋊C4D3×D414yes—
437C4⋊C4#23912yescomplemented
438C22⋊C4—68no—
439C22⋊C4—68no—
440C22⋊C4—64no—
441C22⋊C4—122no—
442C22⋊C4—122no—
443C22⋊C4—122no—
444C22⋊C4—122no—
445C42—22no—
446C42—34no—
447C42—62no—
448C42—62no—
449C42—122no—
450C22×C3C2×#12011yescharacteristic
451C22×C3#52511yescharacteristic
452C22×C3#53211yescharacteristic
453C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
454C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
455C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
456C22×C3#57511yescharacteristic
457C2×D3—22no—
458C2×D3—22no—
459C2×D3—22no—
460C2×D3—41no—
461C2×D3—41no—
462C2×D3—41no—
463C2×D3—41no—
464C2×D3—42no—
465C2×D3—42no—
466C2×D3—44no—
467C2×D3—42no—
468C2×D3—44no—
469C2×D3—44no—
470C2×D3—82no—
471C2×D3—82no—
472C2×D3—82no—
473C2×D3—81no—
474C2×D3—81no—
475C2×D3—82no—
476C2×D3—81no—
477C2×D3—81no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—42no—
481C3×C4—44no—
482C3×C4—81no—
483C3×C4#52512yes—
484C3×C4#53212yes—
485Dic3—22no—
486Dic3—22no—
487Dic3—41no—
488Dic3—81no—
489C23—32no—
490C23—61no—
491C23—61no—
492C23—61no—
493C23—61no—
494C23—61no—
495C23—61no—
496C23—64no—
497C23—62no—
498C23—122no—
499C23—121no—
500C23—121no—
501C23—121no—
502C23—122no—
503C23—122no—
504C23—122no—
505C23—121no—
506C23—121no—
507C23—122no—
508C23—121no—
509C23—122no—
510C23C2×D3×D411yescharacteristic
511Q8—48no—
512D4—64no—
513D4—64no—
514D4—64no—
515D4—128no—
516D4—128no—
517D4—1216no—
518D4—128no—
519D4—122no—
520D4—122no—
521D4—122no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—32no—
529C2×C4—32no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—68no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—124no—
546C2×C4—1216no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—244no—
553C2×C4#91412yes—
554C2×C4#91412yes—
555C2×C4C2×D3×D412yes—
556C2×C4C2×#23911yescharacteristic
557C2×C4C2×#23912yes—
558C2×C4D3×#4111yescharacteristic
559C8—24no—
560C8—122no—
561C2×C3#238811yescharacteristic
562C2×C3#300311yescharacteristic
563C2×C3#300611yescharacteristic
564C2×C3#310911yescharacteristic
565C2×C3#311511yescharacteristic
566C2×C3#318011yescharacteristic
567C2×C3#318411yescharacteristic
568D3—42no—
569D3—82no—
570D3—84no—
571C22—62no—
572C22—62no—
573C22—62no—
574C22—121no—
575C22—121no—
576C22—121no—
577C22—121no—
578C22—124no—
579C22—122no—
580C22—124no—
581C22—122no—
582C22—122no—
583C22—124no—
584C22—241no—
585C22—242no—
586C22—241no—
587C22—242no—
588C22—242no—
589C22—241no—
590C22—241no—
591C22—242no—
592C22D3×#12011yescharacteristic
593C22#562711yescharacteristic
594C22#572511yescharacteristic
595C22C2×#91411yescharacteristic
596C22C2×#91411yescharacteristic
597C22D3×AGL(1,ℤ/8)11yescharacteristic
598C22#577711yescharacteristic
599C4—22no—
600C4—22no—
601C4—44no—
602C4—42no—
603C4—62no—
604C4—62no—
605C4—81no—
606C4—121no—
607C4—241no—
608C4#562712yes—
609C4#572512yes—
610C3#2357311yescharacteristic, complemented, Sylow
611C2—122no—
612C2—244no—
613C2—242no—
614C2#7716911yescharacteristic
615C2#8004011yescharacteristic
616C2#8014611yescharacteristic
617C2#8099911yescharacteristic
618C2#8104611yescharacteristic
619C2#8188811yescharacteristic
620C2#8196211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2058C2
socleC23×C3C22×D4
hypercenter#2058D3
hypocenterC3#23573
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2058

Complements

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

NH ≅ G/Ntype
1. C2×#5725C2semidirect
2. C2×#4869C2semidirect
3. #75964C2semidirect
4. C3×#2058C2semidirect
5. #71792C2semidirect
6. #70581C2semidirect
7. #69627C2semidirect
8. #68886C2semidirect
9. #68782C2semidirect
10. #68674C2semidirect
11. #68600C2semidirect
49. #2058D3semidirect
106. #893D4semidirect
190. C2×D12#41semidirect
209. C2×C3×C8#41semidirect
216. C3×C4⋊C4#41semidirect
429. C2×C8#239semidirect
437. C4⋊C4#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×#48692
#759642
C3×#20582
#717922
#705812
#696272
#688862
#687822
#686742
#686002
#235733