Finite Group Databasebeta

← #10944027

Subgroups — #10944027

Order 768.

1468 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#237C211yesmaximal, characteristic
2C2×#5019C211yesmaximal, characteristic, complemented
3C2×#4849C211yesmaximal, characteristic, complemented
4C2×#4783C212yesmaximal
5#74376C211yesmaximal, characteristic, complemented
6C3×#1802C211yesmaximal, characteristic, complemented
7#70475C211yesmaximal, characteristic, complemented
8#70171C212yesmaximal, complemented
9#69588C211yesmaximal, characteristic, complemented
10#68908C212yesmaximal, complemented
11#68879C212yesmaximal, complemented
12#32907—31nomaximal, Sylow
13C22×#237C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×D4×Dic3C2211yescharacteristic
16C22×#214C2212yes—
17C2×C4×#67—21no—
18C22×C4×Dic3—21no—
19C22×Dic12C2212yes—
20C4×#237—24no—
21C4×#237—24no—
22C2×#897C2212yes—
23C2×#892C2211yescharacteristic
24C2×#887C2211yescharacteristic
25C2×#67⋊C4—21no—
26C2×#877—21no—
27C2×C42×D3C2211yescharacteristic
28C2×#869C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#102C2212yes—
32C3×#384C2211yescharacteristic
33C2×(C2×D3)⋊C8C2212yes—
34C2×#816C2212yes—
35#5019—24no—
36#4987—21no—
37#4987—21no—
38#4984—21no—
39#4984—21no—
40#4973C2212yes—
41C2×#810C2212yes—
42C2×#810C2212yes—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44#4849—24no—
45#4783—28no—
46C3×#340C2212yes—
47#4550C2212yes—
48#4529C2212yes—
49C2×C4×#41—31no—
50C2×#462—31no—
51C2×#461—31no—
52C2×#452—32no—
53#2857—31no—
54#1878—32no—
55#1877—32no—
56#1873—32no—
57#1803—31no—
58#1802D311yescharacteristic, complemented
59#1797—31no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C22×#67—21no—
63C23×Dic3—21no—
64C23×Dic3—21no—
65C2×#237—22no—
66C2×#237—22no—
67C2×#237—22no—
68C2×#237—22no—
69C2×#237—22no—
70C2×#237D414yes—
71C22×C4×D3D411yescharacteristic
72C22×C4×D3D411yescharacteristic
73C22×C4×D3C2311yescharacteristic
74C22×Dic6C2312yes—
75C22×Dic6C2311yescharacteristic
76C22×C3×C8C2312yes—
77C3×C4×D4—22no—
78C3×C4×D4D414yes—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—21no—
84C2×C3×C42C2311yescharacteristic
85C2×#217—21no—
86D4×Dic3—22no—
87D4×Dic3—22no—
88C2×#214—24no—
89C2×#214—24no—
90C2×#214—24no—
91C4×#67—28no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—22no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×C4×Dic3—28no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3D411yescharacteristic
105C2×C4×Dic3D411yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2312yes—
108C2×Dic12—24no—
109C2×Dic12—24no—
110C2×Dic12—24no—
111#897—24no—
112#897—24no—
113#892—22no—
114#892—22no—
115#887—22no—
116#887—22no—
117#67⋊C4—28no—
118#877—28no—
119C42×D3—22no—
120C42×D3—22no—
121#869—22no—
122#869—22no—
123C4×Dic6—22no—
124C4×Dic6D414yes—
125C3×C4⋊C8—22no—
126C3×#102—24no—
127C3⋊#95—22no—
128C3⋊#95—22no—
129(C2×D3)⋊C8—24no—
130#816—24no—
131#810—24no—
132#810—24no—
133C4⋊(C3⋊C8)—22no—
134C22×#41—31no—
135C22×Q16—32no—
136C22×SD16—32no—
137C2×C4⋊Q8—31no—
138C2×C22⋊Q8—32no—
139C4×#41—64no—
140C4×#41—64no—
141C2×C4×Q8—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×C4×D4—61no—
145C2×C4×D4—61no—
146C2×C4×D4C2×D311yescharacteristic
147C2×#117—31no—
148C2×#117—61no—
149C22×C42—31no—
150C22×C42—61no—
151#462—64no—
152#461—64no—
153#452—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8C2×D311yescharacteristic
156C2×#103—32no—
157C2×#103—32no—
158C2×#103—32no—
159C2×#102C2×D312yes—
160C2×C22⋊C8—32no—
161#391—32no—
162#389—61no—
163#389—61no—
164#384C2×D311yescharacteristic
165#380—61no—
166#380—61no—
167#340—32no—
168#340C2×D312yes—
169#336—32no—
170C24×C3—21no—
171C23×D3C2×D411yescharacteristic
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4C2×D412yes—
176C2×C3×D4C2×D412yes—
177C2×C3×D4#4112yes—
178C22×C3×C4—21no—
179C22×C3×C4—24no—
180C22×C3×C4—22no—
181C22×C3×C4—42no—
182C22×C3×C4—42no—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C2×#67—24no—
187C2×#67—24no—
188C2×#67—24no—
189C22×Dic3—24no—
190C22×Dic3—24no—
191C22×Dic3—21no—
192C22×Dic3—24no—
193C22×Dic3—24no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3—42no—
197C22×Dic3—42no—
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201#237—28no—
202#237—28no—
203#237—28no—
204#237—28no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—22no—
210C2×C4×D3—22no—
211C2×C4×D3—22no—
212C2×C4×D3—22no—
213C2×C4×D3—22no—
214C2×C4×D3—41no—
215C2×C4×D3C2×D412yes—
216C2×C4×D3#4112yes—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—22no—
221C2×Dic6—22no—
222C2×Dic6—22no—
223C2×Dic6—22no—
224C2×Dic6—22no—
225C2×Dic6—22no—
226C2×Dic6C2×D412yes—
227C2×Dic6C2×D412yes—
228C2×Dic6#4112yes—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8—22no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4C2×D412yes—
238C3×C22⋊C4—24no—
239C3×C42C2×D412yes—
240C3×C42C2×D412yes—
241#217—24no—
242#214—48no—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—44no—
245C4⋊Dic3—21no—
246C4⋊Dic3—42no—
247C4⋊Dic3—42no—
248C4⋊Dic3—44no—
249C4⋊Dic3C2×D412yes—
250Dic3⋊C4—24no—
251Dic3⋊C4—24no—
252C4×Dic3—24no—
253C4×Dic3—24no—
254C4×Dic3—24no—
255C4×Dic3—22no—
256C4×Dic3—22no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C2×C3⋊C8—22no—
260C2×C3⋊C8—22no—
261C2×C3⋊C8—22no—
262Dic12—48no—
263C2×#41—34no—
264C2×#41—62no—
265C2×#41—62no—
266C2×#41—62no—
267C2×#41—62no—
268C2×#41—62no—
269C22×Q8—31no—
270C22×Q8—32no—
271C22×D4—61no—
272C22×D4C22×D311yescharacteristic
273C23×C4—21no—
274C23×C4—31no—
275C23×C4—31no—
276C23×C4—31no—
277C23×C4—61no—
278C23×C4—61no—
279C2×Q16—64no—
280C2×Q16—64no—
281C2×Q16—64no—
282C2×SD16—64no—
283C2×SD16—64no—
284C2×SD16—64no—
285C22×C8—32no—
286C22×C8C22×D312yes—
287C4⋊Q8—62no—
288C4⋊Q8—62no—
289C22⋊Q8—64no—
290C22⋊Q8—64no—
291C4×Q8—34no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×D4—22no—
296C4×D4—68no—
297C4×D4—62no—
298C4×D4—68no—
299C4×D4—62no—
300C4×D4D1214yes—
301#117—68no—
302#117—62no—
303#117—62no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—62no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4C22×D312yes—
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—62no—
318C2×C42—31no—
319C2×C42—31no—
320C2×C42—31no—
321C2×C42—61no—
322C2×C42—61no—
323C2×C42—68no—
324C2×C42—61no—
325C2×C42—62no—
326C2×C42—62no—
327C2×C42C22×D311yescharacteristic
328C4⋊C8—22no—
329C4⋊C8—62no—
330#103—64no—
331#103—64no—
332#103—64no—
333#102—24no—
334C22⋊C8—64no—
335#95—62no—
336#95—62no—
337C23×C3—22no—
338C23×C3—22no—
339C23×C3—22no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3—41no—
343C23×C3—41no—
344C23×C3C22×D411yescharacteristic
345C22×D3—21no—
346C22×D3—21no—
347C22×D3—21no—
348C22×D3—21no—
349C22×D3—21no—
350C22×D3—21no—
351C22×D3#12112yes—
352C3×D4—22no—
353C3×D4—22no—
354C3×D4—22no—
355C3×D4#12114yes—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—48no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4#12012yes—
366C2×C3×C4#12012yes—
367C2×C3×C4#13512yes—
368C2×C3×C4#13712yes—
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375#67—216no—
376C2×Dic3—216no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—22no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—48no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3#12112yes—
395C2×Dic3#13712yes—
396C4×D3—22no—
397C4×D3—22no—
398C4×D3—22no—
399C4×D3—22no—
400C4×D3—44no—
401C4×D3—44no—
402Dic6—22no—
403Dic6—22no—
404Dic6—22no—
405Dic6—42no—
406Dic6—42no—
407Dic6—42no—
408Dic6—42no—
409Dic6#12114yes—
410C3×C8—42no—
411C3⋊C8—42no—
412C24—21no—
413C24—31no—
414#41—68no—
415#41—68no—
416#41—68no—
417#41—68no—
418C2×Q8—32no—
419C2×Q8—32no—
420C2×Q8—32no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—62no—
425C2×Q8—62no—
426C2×Q8—62no—
427C2×Q8—62no—
428C2×Q8—62no—
429C2×Q8—62no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—64no—
434C2×D4—64no—
435C2×D4—64no—
436C2×D4C2×D1212yes—
437C2×D4D3×D412yes—
438C2×D4#23912yes—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—31no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—62no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4—64no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—64no—
459C22×C4—61no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—121no—
466C22×C4—122no—
467C22×C4—122no—
468C22×C4D3×D411yescharacteristic
469C22×C4D3×D411yescharacteristic
470C22×C4C23×D311yescharacteristic
471Q16—128no—
472SD16—128no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—62no—
477C2×C8—62no—
478C2×C8—62no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—32no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—64no—
486C4⋊C4—64no—
487C4⋊C4—61no—
488C4⋊C4—124no—
489C4⋊C4—122no—
490C4⋊C4—122no—
491C4⋊C4C2×D1212yes—
492C22⋊C4—24no—
493C22⋊C4—64no—
494C22⋊C4—64no—
495C22⋊C4—124no—
496C42—64no—
497C42—64no—
498C42—62no—
499C42—62no—
500C42—64no—
501C42—61no—
502C42—61no—
503C42C2×D1212yes—
504C42D3×D412yes—
505C22×C3—24no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3C2×#12011yescharacteristic
513C22×C3C2×#12111yescharacteristic
514C22×C3#52511yescharacteristic
515C22×C3C2×#13511yescharacteristic
516C22×C3C2×#13711yescharacteristic
517C22×C3C2×#13711yescharacteristic
518C22×C3#57511yescharacteristic
519C2×D3—22no—
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—41no—
523C2×D3—41no—
524C2×D3—41no—
525C2×D3—41no—
526C3×C4—22no—
527C3×C4—42no—
528C3×C4—82no—
529C3×C4#45214yescomplemented
530C3×C4C2×#12111yescharacteristic
531C3×C4C2×#12111yescharacteristic
532C3×C4#52511yescharacteristic
533C3×C4#52511yescharacteristic
534Dic3—24no—
535Dic3—22no—
536Dic3—22no—
537Dic3—41no—
538Dic3—42no—
539Dic3—82no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—32no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23C2×D3×D411yescharacteristic
555Q8—34no—
556Q8—62no—
557Q8—62no—
558Q8—62no—
559Q8—122no—
560Q8—122no—
561Q8—122no—
562Q8—122no—
563D4—22no—
564D4—22no—
565D4—22no—
566D4—616no—
567D4#89514yes—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—32no—
572C2×C4—32no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—616no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—62no—
583C2×C4—61no—
584C2×C4—62no—
585C2×C4—61no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—124no—
596C2×C4—128no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—124no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4#88212yes—
604C2×C4#88212yes—
605C2×C4#91212yes—
606C2×C4#91512yes—
607C2×C4C22×D1211yescharacteristic
608C2×C4C2×D3×D411yescharacteristic
609C2×C4C2×D3×D411yescharacteristic
610C2×C4C2×#23911yescharacteristic
611C2×C4D3×#4111yescharacteristic
612C2×C4D3×#4111yescharacteristic
613C8—42no—
614C8—122no—
615C2×C3—44no—
616C2×C3#284711yescharacteristic
617C2×C3C2×#45211yescharacteristic
618C2×C3#300711yescharacteristic
619C2×C3#303511yescharacteristic
620C2×C3#303911yescharacteristic
621C2×C3#310711yescharacteristic
622C2×C3#311611yescharacteristic
623D3—42no—
624C22—24no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—62no—
632C22—62no—
633C22—62no—
634C22—121no—
635C22—121no—
636C22—121no—
637C22—121no—
638C22C2×#88211yescharacteristic
639C22D3×#12111yescharacteristic
640C22#562711yescharacteristic
641C22C2×#91211yescharacteristic
642C22C2×#91511yescharacteristic
643C22#577311yescharacteristic
644C22D3×#13711yescharacteristic
645C4—22no—
646C4—42no—
647C4—62no—
648C4—62no—
649C4—64no—
650C4—82no—
651C4—122no—
652C4—121no—
653C4—242no—
654C4#478314yescomplemented
655C4C2×#89511yescharacteristic
656C4#552411yescharacteristic
657C4D3×#12111yescharacteristic
658C4#562711yescharacteristic
659C3#3290711yescharacteristic, complemented, Sylow
660C2—44no—
661C2—122no—
662C2#7404811yescharacteristic
663C2C2×#478311yescharacteristic
664C2#7888511yescharacteristic
665C2#8033111yescharacteristic
666C2#8042911yescharacteristic
667C2#8094911yescharacteristic
668C2#8104711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1802C2
socleC23×C3C22×D4
prime radicalC22×#237C22
hypercenter#1802D3
hypocenterC3#32907
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#5019C2semidirect
3. C2×#4849C2semidirect
5. #74376C2semidirect
6. C3×#1802C2semidirect
7. #70475C2semidirect
8. #70171C2semidirect
9. #69588C2semidirect
10. #68908C2semidirect
11. #68879C2semidirect
58. #1802D3semidirect
529. C3×C4#452semidirect
654. C4#4783semidirect
659. C3#32907semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2372
C2×#50192
C2×#48492
C2×#47832
#743762
C3×#18022
#704752
#701712
#695882
#689082
#688792
#329073