Finite Group Databasebeta

← #10847387

Subgroups — #10847387

Order 768.

1474 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
1C2×#5725C211yesmaximal, characteristic, complemented
2C2×#4775C212yesmaximal, complemented
3#76475C211yesmaximal, characteristic, complemented
4C3×#2023C211yesmaximal, characteristic, complemented
5#71782C211yesmaximal, characteristic, complemented
6#70208C212yesmaximal, complemented
7#69658C211yesmaximal, characteristic, complemented
8#68894C212yesmaximal, complemented
9#68883C212yesmaximal, complemented
10D3×#336C211yesmaximal, characteristic, complemented
11#68629C211yesmaximal, characteristic, complemented
12#25496—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#947—21no—
16C2×#938C2211yescharacteristic
17C22×#213C2212yes—
18C22×D3×C8C2212yes—
19#5725—28no—
20C2×#895—21no—
21C2×#895C2212yes—
22C2×#893—21no—
23C2×#893C2212yes—
24C2×#883—21no—
25C2×#874C2211yescharacteristic
26C2×C42×D3C2211yescharacteristic
27C2×C3×#102C2212yes—
28C3×#390C2211yescharacteristic
29#5221C2211yescharacteristic
30C2×#821C2212yes—
31C2×#821C2212yes—
32#4987C2211yescharacteristic
33#4984C2211yescharacteristic
34C2×#809C2212yes—
35C2×#807C2212yes—
36#4956—21no—
37#4951—21no—
38#4945—21no—
39#4928C2211yescharacteristic
40#4775—28no—
41C3×#340C2212yes—
42C3×#336C2211yescharacteristic
43#4550C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#466—32no—
48#2854—31no—
49#2057—31no—
50#2041—31no—
51#2023D311yescharacteristic, complemented
52#2011—31no—
53#1893—32no—
54#1879—32no—
55#1862—32no—
56C2×#336—31no—
57C22×C3×Q8—21no—
58C22×C3×D4C2311yescharacteristic
59C22×#67—21no—
60C2×#239—28no—
61C2×#239—44no—
62C2×#239—44no—
63C22×D12—22no—
64C22×D12—21no—
65C22×D12—21no—
66C22×D12C2312yes—
67C22×C4×D3—22no—
68C22×C4×D3D411yescharacteristic
69C22×C4×D3D411yescharacteristic
70C22×C4×D3C2311yescharacteristic
71C22×C3×C8C2312yes—
72C3×C4⋊Q8—24no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C42C2311yescharacteristic
79C2×#217—21no—
80C2×#217—21no—
81C2×#217—21no—
82#947—28no—
83#938—24no—
84C2×#213—24no—
85C2×#213—24no—
86C2×#213—42no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3C2311yescharacteristic
92C2×C4⋊Dic3C2311yescharacteristic
93C2×Dic3⋊C4—21no—
94C2×C4×Dic3D411yescharacteristic
95C2×C4×Dic3D411yescharacteristic
96C2×C4×Dic3C2311yescharacteristic
97C22×C3⋊C8C2312yes—
98C2×D3×C8—24no—
99C2×D3×C8—24no—
100C2×D3×C8—42no—
101#895—28no—
102#895—24no—
103#895D418yes—
104#893—28no—
105#893—24no—
106#893—24no—
107#883—44no—
108#874—24no—
109C42×D3—24no—
110C3×#102—24no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113C3⋊#95—21no—
114#821—24no—
115#821—42no—
116#809—24no—
117#807—24no—
118C22×#41—61no—
119C22×D8—32no—
120C23×C8—32no—
121#532—68no—
122C2×C4⋊Q8C2×D311yescharacteristic
123C2×#127—31no—
124C2×#124—61no—
125C2×#123—61no—
126C2×#121—31no—
127C2×#121—32no—
128C2×#121—61no—
129C2×C4×D4—32no—
130C2×C4×D4—61no—
131C22×C42—31no—
132#466—68no—
133C2×#107—32no—
134C2×#102—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×#102C2×D312yes—
138#390—31no—
139#390C2×D311yescharacteristic
140#389—31no—
141#389—31no—
142#388—61no—
143#388—61no—
144#386—61no—
145#380—31no—
146#340—32no—
147#340C2×D312yes—
148#336—32no—
149#336—31no—
150#336C2×D311yescharacteristic
151C24×C3—21no—
152C23×D3—22no—
153C23×D3C2×D411yescharacteristic
154C2×C3×Q8—24no—
155C2×C3×Q8—42no—
156C2×C3×Q8—42no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C2×#67—42no—
167C2×#67—42no—
168C2×#67—42no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3C2×D411yescharacteristic
172C22×Dic3C2×D411yescharacteristic
173C22×Dic3C2×D411yescharacteristic
174#239—432no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—22no—
179C2×D12—24no—
180C2×D12—24no—
181C2×D12—28no—
182C2×D12—22no—
183C2×D12—42no—
184C2×D12—42no—
185C2×D12—44no—
186C2×D12—44no—
187C2×D12—42no—
188C2×D12—42no—
189C2×D12C2×D414yes—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—21no—
194C2×C4×D3—21no—
195C2×C4×D3—28no—
196C2×C4×D3—22no—
197C2×C4×D3—22no—
198C2×C4×D3—41no—
199C2×C4×D3—44no—
200C2×C4×D3—44no—
201C2×C4×D3D814yescomplemented
202C2×C4×D3C2×D414yes—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C3×C4⋊C4—24no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4C2×D414yes—
210C3×C22⋊C4—42no—
211C3×C22⋊C4—42no—
212C3×C42—22no—
213#217—42no—
214#217—42no—
215#217—42no—
216#213—48no—
217(C2×D3)⋊C4—28no—
218(C2×D3)⋊C4—28no—
219(C2×D3)⋊C4—42no—
220C4⋊Dic3—22no—
221C4⋊Dic3—21no—
222C4⋊Dic3—21no—
223C4⋊Dic3—42no—
224Dic3⋊C4—42no—
225C4×Dic3—22no—
226C4×Dic3—22no—
227C4×Dic3—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8#4114yescomplemented
231D3×C8—48no—
232C2×#41—68no—
233C2×#41—124no—
234C2×#41—124no—
235C22×Q8—21no—
236C22×D4—32no—
237C22×D4—62no—
238C22×D4—61no—
239C22×D4—61no—
240C22×D4—61no—
241C22×D4C22×D311yescharacteristic
242C23×C4—31no—
243C23×C4—31no—
244C23×C4—31no—
245C23×C4—62no—
246C2×D8—64no—
247C2×D8—64no—
248C2×D8—122no—
249C22×C8—32no—
250C22×C8—64no—
251C22×C8—64no—
252C22×C8—122no—
253C22×C8C22×D312yes—
254C4⋊Q8—24no—
255#127—64no—
256#124—68no—
257#123—124no—
258#121—38no—
259#121—64no—
260#121—68no—
261#121—64no—
262C4×D4—64no—
263C4×D4—68no—
264C4×D4—64no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C4⋊C4C22×D312yes—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
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×C42C22×D311yescharacteristic
285#107—64no—
286#102—24no—
287#102—64no—
288#102—64no—
289#102—122no—
290#95—61no—
291#95—61no—
292#95—61no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3C22×D411yescharacteristic
301C22×D3—21no—
302C22×D3—21no—
303C22×D3—21no—
304C22×D3—21no—
305C22×D3—24no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—42no—
312C22×D3#12012yes—
313C22×D3C2×D812yes—
314C22×D3AGL(1,ℤ/8)12yes—
315C3×Q8—48no—
316C3×D4—42no—
317C3×D4—42no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—24no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4C2×D812yes—
329C2×C3×C4#13512yes—
330C2×C3×C4C22×D412yes—
331C2×C3×C4C2×#4112yes—
332C2×C3×C4H(5,𝔽2)11yescharacteristic
333C2×C3×C42−1+411yescharacteristic
334#67—84no—
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—21no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3#12012yes—
349C2×Dic3C2×D812yes—
350D12—24no—
351D12—24no—
352D12—48no—
353D12—48no—
354D12—416no—
355D12—42no—
356D12—42no—
357C4×D3—28no—
358C4×D3—24no—
359C4×D3—24no—
360C4×D3—416no—
361C4×D3—44no—
362C3×C8—42no—
363C3⋊C8—24no—
364C24—21no—
365C24—31no—
366C24—62no—
367#41—1232no—
368C2×Q8—24no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—34no—
376C2×D4—68no—
377C2×D4—64no—
378C2×D4—64no—
379C2×D4—62no—
380C2×D4—62no—
381C2×D4—62no—
382C2×D4—62no—
383C2×D4—62no—
384C2×D4—122no—
385C2×D4—122no—
386C2×D4—124no—
387C2×D4—124no—
388C2×D4—122no—
389C2×D4—122no—
390C2×D4—122no—
391C2×D4—122no—
392C2×D4—122no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—34no—
398C22×C4—34no—
399C22×C4—31no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—68no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—61no—
407C22×C4—62no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—121no—
411C22×C4—124no—
412C22×C4—124no—
413C22×C4D3×D411yescharacteristic
414C22×C4D3×D411yescharacteristic
415C22×C4C23×D311yescharacteristic
416D8—128no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—34no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—128no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—62no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—122no—
431C4⋊C4—122no—
432C4⋊C4D3×D414yes—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—68no—
436C22⋊C4—68no—
437C22⋊C4—122no—
438C22⋊C4—122no—
439C22⋊C4—122no—
440C22⋊C4—122no—
441C42—22no—
442C42—62no—
443C42—62no—
444C42—62no—
445C22×C3—81no—
446C22×C3—81no—
447C22×C3—81no—
448C22×C3—81no—
449C22×C3—81no—
450C22×C3—81no—
451C22×C3—81no—
452C22×C3C2×#12011yescharacteristic
453C22×C3#53211yescharacteristic
454C22×C3#53711yescharacteristic
455C22×C3C22×D811yescharacteristic
456C22×C3C2×#13511yescharacteristic
457C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
458C22×C3#57511yescharacteristic
459C2×D3—22no—
460C2×D3—22no—
461C2×D3—22no—
462C2×D3—22no—
463C2×D3—22no—
464C2×D3—22no—
465C2×D3—44no—
466C2×D3—44no—
467C2×D3—44no—
468C2×D3—82no—
469C2×D3—82no—
470C2×D3—82no—
471C2×D3—82no—
472C2×D3#44714yes—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—44no—
476C3×C4—81no—
477C3×C4#54612yes—
478C3×C4#54712yes—
479Dic3—22no—
480Dic3—22no—
481Dic3—41no—
482Dic3—81no—
483Dic3—81no—
484C23—32no—
485C23—32no—
486C23—32no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—61no—
495C23—61no—
496C23—61no—
497C23—61no—
498C23—64no—
499C23—122no—
500C23—122no—
501C23—122no—
502C23—122no—
503C23—122no—
504C23—122no—
505C23C2×D3×D411yescharacteristic
506Q8—48no—
507D4—42no—
508D4—42no—
509D4—64no—
510D4—64no—
511D4—128no—
512D4—128no—
513D4—1216no—
514D4—122no—
515D4—122no—
516D4—244no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—24no—
522C2×C4—32no—
523C2×C4—32no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—68no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—64no—
535C2×C4—64no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—124no—
540C2×C4—1216no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4D3×D812yes—
548C2×C4#91612yes—
549C2×C4C2×D3×D412yes—
550C2×C4#100711yescharacteristic
551C2×C4D3×#4112yes—
552C2×C4#100911yescharacteristic
553C8—42no—
554C8—64no—
555C2×C3—161no—
556C2×C3#243011yescharacteristic
557C2×C3C2×#44711yescharacteristic
558C2×C3#300211yescharacteristic
559C2×C3#316911yescharacteristic
560C2×C3#318011yescharacteristic
561C2×C3#324511yescharacteristic
562C2×C3#324711yescharacteristic
563D3—24no—
564D3—84no—
565C22—34no—
566C22—62no—
567C22—62no—
568C22—62no—
569C22—62no—
570C22—62no—
571C22—62no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—124no—
580C22—124no—
581C22—124no—
582C22—242no—
583C22—242no—
584C22—242no—
585C22—242no—
586C22D3×#12011yescharacteristic
587C22#562911yescharacteristic
588C22#566811yescharacteristic
589C22C2×D3×D811yescharacteristic
590C22C2×#91611yescharacteristic
591C22D3×AGL(1,ℤ/8)11yescharacteristic
592C22#577711yescharacteristic
593C4—22no—
594C4—22no—
595C4—44no—
596C4—62no—
597C4—62no—
598C4—81no—
599C4—121no—
600C4—241no—
601C4—241no—
602C4#563112yes—
603C4#567612yes—
604C3#2549611yescharacteristic, complemented, Sylow
605C2—64no—
606C2—161no—
607C2—244no—
608C2#7686411yescharacteristic
609C2D3×#44711yescharacteristic
610C2#8003811yescharacteristic
611C2#8097511yescharacteristic
612C2#8107111yescharacteristic
613C2#8155711yescharacteristic
614C2#8166811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2023C2
socleC23×C3C22×D4
hypercenter#2023D3
hypocenterC3#25496
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×#4775C2semidirect
3. #76475C2semidirect
4. C3×#2023C2semidirect
5. #71782C2semidirect
6. #70208C2semidirect
7. #69658C2semidirect
8. #68894C2semidirect
9. #68883C2semidirect
10. D3×#336C2semidirect
11. #68629C2semidirect
51. #2023D3semidirect
201. C2×C4×D3D8semidirect
230. C2×C3⋊C8#41semidirect
604. C3#25496semidirect

Maximal subgroups

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

subgroupindex
C2×#57252
C2×#47752
#764752
C3×#20232
#717822
#702082
#696582
#688942
#688832
D3×#3362
#686292
#254963