Finite Group Databasebeta

← #10824465

Subgroups — #10824465

Order 768.

868 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5184C211yesmaximal, characteristic, complemented
2C2×#4817C211yesmaximal, characteristic
3#76480C211yesmaximal, characteristic
4#76477C211yesmaximal, characteristic, complemented
5C3×#2056C211yesmaximal, characteristic, complemented
6#72072C211yesmaximal, characteristic
7#70483C211yesmaximal, characteristic
8#70441C211yesmaximal, characteristic, complemented
9#70279C211yesmaximal, characteristic
10#70234C211yesmaximal, characteristic, complemented
11#70220C211yesmaximal, characteristic, complemented
12#69683C211yesmaximal, characteristic, complemented
13#69680C211yesmaximal, characteristic
14#68836C211yesmaximal, characteristic, complemented
15#68782C211yesmaximal, characteristic
16#23687—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C22×#215C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#391C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32#5184—24no—
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×#815C2211yescharacteristic
37C2×#815C2211yescharacteristic
38#4977C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40#4972C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#807C2211yescharacteristic
43#4939C2211yescharacteristic
44C42×Dic3C2211yescharacteristic
45#4928C2211yescharacteristic
46C2×C4⋊(C3⋊C8)C2211yescharacteristic
47#4817—24no—
48#4670—21no—
49#4670—21no—
50#4666—21no—
51#4666—21no—
52#4654—21no—
53#4653—21no—
54#4649—21no—
55#4648—21no—
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#441—31no—
63C2×#440—31no—
64C4×#124—31no—
65#2295—31no—
66#2085—31no—
67#2074—31no—
68#2068—31no—
69#2065—31no—
70#2056D311yescharacteristic, complemented
71#2055—31no—
72#2042—31no—
73#2015—31no—
74#1767—31no—
75#1763—31no—
76#1762—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C2×C3×Q16—22no—
82C2×C3×Q16—22no—
83C2×C3×Q16—22no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89#947—22no—
90#947—22no—
91Q8×Dic3—22no—
92Q8×Dic3—22no—
93Q8×Dic3—22no—
94Q8×Dic3D414yes—
95C2×#215—22no—
96C2×#215—22no—
97C2×#215—22no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3D411yescharacteristic
116C2×C4×Dic3D411yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#893—22no—
122#893—22no—
123C3×C4⋊C8—22no—
124C3×#103—22no—
125C3×#103—22no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130#835—22no—
131#835—22no—
132#835—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#821—22no—
141#821—22no—
142#817—22no—
143#815—22no—
144#815—22no—
145#809—22no—
146#807—22no—
147C4⋊(C3⋊C8)—22no—
148C22×Q16C2×D311yescharacteristic
149C22×SD16—31no—
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154#441—64no—
155#440—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103C2×D311yescharacteristic
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169C2×#97—31no—
170#391—31no—
171#391C2×D311yescharacteristic
172#391C2×D311yescharacteristic
173#390—31no—
174#389—31no—
175#389—31no—
176#388—61no—
177#388—61no—
178#387—61no—
179#387—61no—
180#382—31no—
181C4×C4⋊C4—31no—
182C4×C4⋊C4—61no—
183C4×C22⋊C4—61no—
184C4×C22⋊C4—61no—
185#376—61no—
186C43—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
191#336—31no—
192#336—31no—
193C23×D3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8C2×D412yes—
205C2×C3×Q8#4112yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12#4112yes—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—42no—
229C3×Q16—28no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8#4112yes—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C42—21no—
241C3×C42—21no—
242#215—28no—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—24no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—42no—
248C4⋊Dic3—21no—
249C4⋊Dic3—24no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—42no—
255C4⋊Dic3C2×D412yes—
256Dic3⋊C4—42no—
257Dic3⋊C4—42no—
258C4×Dic3—22no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—24no—
262C4×Dic3—41no—
263C4×Dic3—41no—
264C4×Dic3—41no—
265C4×Dic3—42no—
266C4×Dic3—42no—
267C4×Dic3C2×D412yes—
268C4×Dic3#4112yes—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8#4112yes—
275C22×Q8C22×D311yescharacteristic
276C22×Q8C22×D311yescharacteristic
277C22×D4—31no—
278C23×C4—61no—
279C2×Q16—22no—
280C2×Q16—22no—
281C2×Q16—22no—
282C2×SD16—62no—
283C2×SD16—62no—
284C2×SD16—62no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289#124—62no—
290#124—62no—
291C4×Q8—34no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×D4—62no—
296C4×D4—62no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—31no—
316C2×C42—31no—
317C2×C42—61no—
318C2×C42—61no—
319C2×C42—61no—
320C2×C42C22×D311yescharacteristic
321#107—62no—
322#106—62no—
323C4⋊C8—22no—
324C4⋊C8—62no—
325#103—22no—
326#103—22no—
327#103—62no—
328#103—62no—
329#103—62no—
330#102—62no—
331#102—62no—
332#102—62no—
333#97—62no—
334#97—62no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3C22×D411yescharacteristic
347C22×D3—22no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—41no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4#13712yes—
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—22no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3#12112yes—
406C2×Dic3#12112yes—
407D12—22no—
408D12—22no—
409D12—41no—
410D12—41no—
411C4×D3—84no—
412C3×C8—22no—
413C3×C8—41no—
414C3⋊C8—22no—
415C3⋊C8—41no—
416C24—61no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8D3×D412yes—
428C2×Q8#23712yes—
429C2×D4—32no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4D3×D411yescharacteristic
450C22×C4D3×D411yescharacteristic
451C22×C4C23×D311yescharacteristic
452Q16—28no—
453SD16—68no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—32no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8#23712yes—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—32no—
471C4⋊C4—64no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—122no—
478C4⋊C4—122no—
479C4⋊C4—122no—
480C22⋊C4—64no—
481C22⋊C4—64no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C42—21no—
486C42—21no—
487C42—32no—
488C42—32no—
489C42—64no—
490C42—62no—
491C42—61no—
492C42—61no—
493C42—121no—
494C42—121no—
495C42—121no—
496C42—122no—
497C42—122no—
498C22×C3C2×#12111yescharacteristic
499C22×C3#52511yescharacteristic
500C22×C3#53211yescharacteristic
501C22×C3C2×#13711yescharacteristic
502C22×C3#57511yescharacteristic
503C22×C3#57511yescharacteristic
504C22×C3#57511yescharacteristic
505C2×D3—42no—
506C2×D3—42no—
507C2×D3—42no—
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C3×C4—22no—
513C3×C4—42no—
514C3×C4—42no—
515C3×C4—41no—
516C3×C4—81no—
517C3×C4—81no—
518C3×C4#52511yescharacteristic
519C3×C4#52511yescharacteristic
520C3×C4#52911yescharacteristic
521C3×C4#52911yescharacteristic
522Dic3—22no—
523Dic3—22no—
524Dic3—41no—
525Dic3—41no—
526Dic3—42no—
527Dic3—81no—
528Dic3—81no—
529C23—62no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23C2×D3×D411yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—22no—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545D4—62no—
546D4—62no—
547D4—121no—
548D4—121no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—32no—
557C2×C4—32no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—62no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—244no—
588C2×C4#91812yes—
589C2×C4C2×D3×D411yescharacteristic
590C2×C4C2×#23711yescharacteristic
591C2×C4C2×#23711yescharacteristic
592C2×C4D3×#4111yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4D3×#4111yescharacteristic
595C8—22no—
596C8—41no—
597C8—62no—
598C8—121no—
599C2×C3#239211yescharacteristic
600C2×C3#303711yescharacteristic
601C2×C3#304011yescharacteristic
602C2×C3#311111yescharacteristic
603C2×C3#311611yescharacteristic
604C2×C3#318111yescharacteristic
605C2×C3#318311yescharacteristic
606D3—82no—
607C22—122no—
608C22—122no—
609C22—122no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22#559611yescharacteristic
615C22D3×#12111yescharacteristic
616C22#560611yescharacteristic
617C22C2×#91811yescharacteristic
618C22#576811yescharacteristic
619C22#577711yescharacteristic
620C22#577711yescharacteristic
621C4—22no—
622C4—42no—
623C4—42no—
624C4—41no—
625C4—62no—
626C4—62no—
627C4—81no—
628C4—81no—
629C4—121no—
630C4—121no—
631C4—122no—
632C4—241no—
633C4—241no—
634C4#558611yescharacteristic
635C4#564911yescharacteristic
636C4#565611yescharacteristic
637C4#568011yescharacteristic
638C3#2368711yescharacteristic, complemented, Sylow
639C2—242no—
640C2#7689011yescharacteristic
641C2#8045911yescharacteristic
642C2#8047611yescharacteristic
643C2#8057011yescharacteristic
644C2#8059911yescharacteristic
645C2#8074511yescharacteristic
646C2#8089211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2056C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2056D3
hypocenterC3#23687
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2056

Complements

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

NH ≅ G/Ntype
1. C2×#5184C2semidirect
4. #76477C2semidirect
5. C3×#2056C2semidirect
8. #70441C2semidirect
10. #70234C2semidirect
11. #70220C2semidirect
12. #69683C2semidirect
14. #68836C2semidirect
70. #2056D3semidirect
638. C3#23687semidirect

Maximal subgroups

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

subgroupindex
C2×#51842
C2×#48172
#764802
#764772
C3×#20562
#720722
#704832
#704412
#702792
#702342
#702202
#696832
#696802
#688362
#687822
#236873