Finite Group Databasebeta

← #10944000

Subgroups — #10944000

Order 768.

1608 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#237C211yesmaximal, characteristic, complemented
2C2×#5015C211yesmaximal, characteristic, complemented
3C2×#4848C211yesmaximal, characteristic, complemented
4C2×#4781C212yesmaximal, complemented
5#74375C211yesmaximal, characteristic, complemented
6C3×#1802C211yesmaximal, characteristic, complemented
7#70475C211yesmaximal, characteristic, complemented
8#70172C212yesmaximal, complemented
9#69588C211yesmaximal, characteristic, complemented
10#68909C212yesmaximal, complemented
11#68881C212yesmaximal, complemented
12#32904—31nomaximal, Sylow
13C22×#237C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×D4×Dic3C2211yescharacteristic
16C22×#213C2212yes—
17C2×C4×#67—21no—
18C22×C4×Dic3—21no—
19C22×#204C2212yes—
20C4×#237—24no—
21C4×#237—24no—
22C2×#895—21no—
23C2×#895—21no—
24C2×#895C2212yes—
25C2×#892C2211yescharacteristic
26C2×#887C2211yescharacteristic
27C2×#67⋊C4—21no—
28C2×#877—21no—
29C2×#874C2211yescharacteristic
30C2×C42×D3C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#102C2212yes—
34C3×#384C2211yescharacteristic
35C2×#821C2212yes—
36C2×(C2×D3)⋊C8C2212yes—
37#5015—24no—
38#4988—21no—
39#4988—21no—
40#4988C2212yes—
41C2×#810C2212yes—
42C2×#809C2212yes—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44#4848—24no—
45#4781—28no—
46C3×#340C2212yes—
47#4550C2212yes—
48#4529C2212yes—
49C2×C4×#41—31no—
50C2×#460—31no—
51C2×#459—31no—
52C2×#449—32no—
53#2856—31no—
54#1880—32no—
55#1879—32no—
56#1875—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×#237D414yescomplemented
71C22×D12—22no—
72C22×D12—21no—
73C22×D12—21no—
74C22×D12C2312yes—
75C22×C4×D3D411yescharacteristic
76C22×C4×D3D411yescharacteristic
77C22×C4×D3C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2312yes—
80C3×C4×D4—22no—
81C3×C4×D4D414yescomplemented
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89D4×Dic3—22no—
90D4×Dic3—22no—
91C2×#213—24no—
92C2×#213—24no—
93C2×#213—24no—
94C4×#67—28no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—22no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—22no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3—28no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3D411yescharacteristic
107C2×C4×Dic3D411yescharacteristic
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2312yes—
110C2×#204—24no—
111C2×#204—24no—
112C2×#204—24no—
113#895—24no—
114#895—24no—
115#895—44no—
116#895—44no—
117#892—22no—
118#892—22no—
119#887—22no—
120#887—22no—
121#67⋊C4—28no—
122#877—28no—
123#874—22no—
124#874—22no—
125C42×D3—22no—
126C42×D3—22no—
127C4×Dic6—22no—
128C4×Dic6D414yescomplemented
129C3×C4⋊C8—22no—
130C3×#102—24no—
131#821—24no—
132(C2×D3)⋊C8—24no—
133#810—24no—
134#809—24no—
135C4⋊(C3⋊C8)—22no—
136C22×#41—31no—
137C22×SD16—32no—
138C22×D8—32no—
139C2×#127—31no—
140C2×#121—32no—
141C2×#121—61no—
142C2×#121—61no—
143C4×#41—64no—
144C4×#41—64no—
145C2×C4×Q8—31no—
146C2×C4×Q8—31no—
147C2×C4×D4—31no—
148C2×C4×D4—61no—
149C2×C4×D4—61no—
150C2×C4×D4C2×D311yescharacteristic
151C2×#117—31no—
152C2×#117—61no—
153C22×C42—31no—
154C22×C42—61no—
155#460—64no—
156#459—64no—
157#449—68no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—32no—
161C2×#102—32no—
162C2×#102—32no—
163C2×#102C2×D312yes—
164C2×C22⋊C8—32no—
165#390—32no—
166#390—61no—
167#390—61no—
168#384C2×D311yescharacteristic
169#340—32no—
170#340C2×D312yes—
171#336—32no—
172C24×C3—21no—
173C23×D3—22no—
174C23×D3C2×D411yescharacteristic
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4C2×D412yes—
179C2×C3×D4C2×D412yes—
180C2×C3×D4#4112yes—
181C22×C3×C4—21no—
182C22×C3×C4—24no—
183C22×C3×C4—22no—
184C22×C3×C4—42no—
185C22×C3×C4—42no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2411yescharacteristic
189C2×#67—24no—
190C2×#67—24no—
191C2×#67—24no—
192C22×Dic3—24no—
193C22×Dic3—24no—
194C22×Dic3—21no—
195C22×Dic3—24no—
196C22×Dic3—24no—
197C22×Dic3—21no—
198C22×Dic3—42no—
199C22×Dic3—42no—
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203#237—28no—
204#237—28no—
205#237—28no—
206#237—28no—
207C2×D12—22no—
208C2×D12—22no—
209C2×D12—22no—
210C2×D12—22no—
211C2×D12—22no—
212C2×D12—22no—
213C2×D12—42no—
214C2×D12—42no—
215C2×D12—44no—
216C2×D12—44no—
217C2×D12—44no—
218C2×D12—42no—
219C2×D12—42no—
220C2×D12—42no—
221C2×D12—42no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—22no—
227C2×C4×D3—22no—
228C2×C4×D3—22no—
229C2×C4×D3—22no—
230C2×C4×D3—22no—
231C2×C4×D3—41no—
232C2×C4×D3C2×D412yes—
233C2×C4×D3#4112yes—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6C2×D412yes—
238C2×Dic6C2×D412yes—
239C2×Dic6#4112yes—
240C2×C3×C8—22no—
241C2×C3×C8—22no—
242C2×C3×C8—22no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4—42no—
248C3×C4⋊C4C2×D412yes—
249C3×C22⋊C4—24no—
250C3×C42C2×D412yes—
251C3×C42C2×D412yes—
252#217—24no—
253#213—48no—
254(C2×D3)⋊C4—24no—
255(C2×D3)⋊C4—44no—
256(C2×D3)⋊C4—44no—
257(C2×D3)⋊C4—44no—
258C4⋊Dic3—21no—
259C4⋊Dic3C2×D412yes—
260Dic3⋊C4—24no—
261Dic3⋊C4—24no—
262C4×Dic3—24no—
263C4×Dic3—24no—
264C4×Dic3—24no—
265C4×Dic3—22no—
266C4×Dic3—22no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C2×C3⋊C8—22no—
270C2×C3⋊C8—22no—
271C2×C3⋊C8—22no—
272#204—48no—
273C2×#41—34no—
274C2×#41—62no—
275C2×#41—62no—
276C2×#41—62no—
277C2×#41—62no—
278C2×#41—62no—
279C22×Q8—31no—
280C22×D4—32no—
281C22×D4—62no—
282C22×D4—61no—
283C22×D4—61no—
284C22×D4—61no—
285C22×D4C22×D311yescharacteristic
286C23×C4—21no—
287C23×C4—31no—
288C23×C4—31no—
289C23×C4—31no—
290C23×C4—61no—
291C23×C4—61no—
292C2×SD16—64no—
293C2×SD16—64no—
294C2×SD16—64no—
295C2×D8—64no—
296C2×D8—64no—
297C2×D8—64no—
298C22×C8—32no—
299C22×C8C22×D312yes—
300#127—62no—
301#127—62no—
302#121—64no—
303#121—64no—
304#121—124no—
305#121—124no—
306C4×Q8—34no—
307C4×Q8—62no—
308C4×Q8—62no—
309C4×Q8—62no—
310C4×D4—22no—
311C4×D4—68no—
312C4×D4—62no—
313C4×D4—68no—
314C4×D4—62no—
315C4×D4D1214yescomplemented
316#117—68no—
317#117—62no—
318#117—62no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4C22×D312yes—
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—62no—
330C2×C22⋊C4—62no—
331C2×C22⋊C4—62no—
332C2×C42—31no—
333C2×C42—31no—
334C2×C42—31no—
335C2×C42—61no—
336C2×C42—61no—
337C2×C42—68no—
338C2×C42—61no—
339C2×C42—62no—
340C2×C42—62no—
341C2×C42C22×D311yescharacteristic
342C4⋊C8—22no—
343C4⋊C8—62no—
344#103—64no—
345#102—24no—
346#102—64no—
347#102—64no—
348C22⋊C8—64no—
349C23×C3—22no—
350C23×C3—22no—
351C23×C3—22no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3C22×D411yescharacteristic
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—42no—
364C22×D3—42no—
365C22×D3—42no—
366C22×D3—42no—
367C22×D3—42no—
368C22×D3—42no—
369C22×D3—42no—
370C22×D3#12112yes—
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—22no—
374C3×D4#12114yescomplemented
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—48no—
381C2×C3×C4—42no—
382C2×C3×C4—42no—
383C2×C3×C4—42no—
384C2×C3×C4#12012yes—
385C2×C3×C4#12012yes—
386C2×C3×C4#13512yes—
387C2×C3×C4AGL(1,ℤ/8)12yes—
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394#67—216no—
395C2×Dic3—216no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—22no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—48no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3#12112yes—
411C2×Dic3AGL(1,ℤ/8)12yes—
412D12—42no—
413D12—42no—
414D12—42no—
415D12—42no—
416D12—84no—
417D12—88no—
418D12—84no—
419C4×D3—22no—
420C4×D3—22no—
421C4×D3—22no—
422C4×D3—22no—
423C4×D3—44no—
424C4×D3—44no—
425Dic6—22no—
426Dic6—22no—
427Dic6—22no—
428Dic6#12114yescomplemented
429C3×C8—42no—
430C3⋊C8—42no—
431C24—21no—
432C24—31no—
433C24—62no—
434#41—68no—
435#41—68no—
436#41—68no—
437#41—68no—
438C2×Q8—32no—
439C2×Q8—32no—
440C2×Q8—32no—
441C2×Q8—61no—
442C2×Q8—61no—
443C2×Q8—61no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—64no—
448C2×D4—64no—
449C2×D4—64no—
450C2×D4—62no—
451C2×D4—62no—
452C2×D4—62no—
453C2×D4—62no—
454C2×D4—62no—
455C2×D4—62no—
456C2×D4—122no—
457C2×D4—122no—
458C2×D4—124no—
459C2×D4—124no—
460C2×D4—124no—
461C2×D4—122no—
462C2×D4—122no—
463C2×D4—122no—
464C2×D4—122no—
465C2×D4C2×D1212yes—
466C2×D4D3×D412yes—
467C2×D4#23912yes—
468C22×C4—21no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—32no—
474C22×C4—32no—
475C22×C4—31no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—62no—
479C22×C4—64no—
480C22×C4—64no—
481C22×C4—64no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—61no—
486C22×C4—61no—
487C22×C4—64no—
488C22×C4—61no—
489C22×C4—62no—
490C22×C4—62no—
491C22×C4—62no—
492C22×C4—62no—
493C22×C4—121no—
494C22×C4—122no—
495C22×C4—122no—
496C22×C4D3×D411yescharacteristic
497C22×C4D3×D411yescharacteristic
498C22×C4C23×D311yescharacteristic
499SD16—128no—
500D8—128no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—62no—
505C2×C8—62no—
506C2×C8—62no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—32no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—64no—
514C4⋊C4—64no—
515C4⋊C4—61no—
516C4⋊C4C2×D1212yes—
517C22⋊C4—24no—
518C22⋊C4—64no—
519C22⋊C4—64no—
520C22⋊C4—124no—
521C22⋊C4—124no—
522C22⋊C4—124no—
523C42—64no—
524C42—64no—
525C42—62no—
526C42—62no—
527C42—64no—
528C42—61no—
529C42—61no—
530C42C2×D1212yes—
531C42D3×D412yes—
532C22×C3—24no—
533C22×C3—42no—
534C22×C3—42no—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3—42no—
538C22×C3—42no—
539C22×C3C2×#12011yescharacteristic
540C22×C3C2×#12111yescharacteristic
541C22×C3#52511yescharacteristic
542C22×C3C2×#13511yescharacteristic
543C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
544C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
545C22×C3#57511yescharacteristic
546C2×D3—22no—
547C2×D3—22no—
548C2×D3—22no—
549C2×D3—41no—
550C2×D3—41no—
551C2×D3—41no—
552C2×D3—41no—
553C2×D3—82no—
554C2×D3—82no—
555C2×D3—82no—
556C2×D3—82no—
557C2×D3—82no—
558C2×D3—82no—
559C2×D3—82no—
560C3×C4—22no—
561C3×C4—42no—
562C3×C4—82no—
563C3×C4#44914yescomplemented
564C3×C4C2×#12111yescharacteristic
565C3×C4C2×#12111yescharacteristic
566C3×C4#52511yescharacteristic
567C3×C4#52511yescharacteristic
568Dic3—24no—
569Dic3—22no—
570Dic3—22no—
571Dic3—41no—
572Dic3—42no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—32no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—61no—
582C23—61no—
583C23—61no—
584C23—61no—
585C23—61no—
586C23—61no—
587C23—122no—
588C23—122no—
589C23—122no—
590C23—122no—
591C23—122no—
592C23—122no—
593C23—122no—
594C23C2×D3×D411yescharacteristic
595Q8—34no—
596Q8—62no—
597Q8—62no—
598Q8—62no—
599D4—22no—
600D4—22no—
601D4—22no—
602D4—616no—
603D4—122no—
604D4—122no—
605D4—122no—
606D4—122no—
607D4—244no—
608D4—248no—
609D4—244no—
610D4#89514yescomplemented
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—32no—
615C2×C4—32no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—48no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—616no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—62no—
626C2×C4—61no—
627C2×C4—62no—
628C2×C4—61no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—62no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—62no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—124no—
639C2×C4—128no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—124no—
643C2×C4#88212yes—
644C2×C4#88212yes—
645C2×C4#91112yes—
646C2×C4#91612yes—
647C2×C4C22×D1211yescharacteristic
648C2×C4C2×D3×D411yescharacteristic
649C2×C4C2×D3×D411yescharacteristic
650C2×C4C2×#23911yescharacteristic
651C2×C4D3×#4111yescharacteristic
652C2×C4D3×#4111yescharacteristic
653C8—42no—
654C8—122no—
655C2×C3—44no—
656C2×C3#284711yescharacteristic
657C2×C3C2×#44911yescharacteristic
658C2×C3#300611yescharacteristic
659C2×C3#303511yescharacteristic
660C2×C3#303811yescharacteristic
661C2×C3#310611yescharacteristic
662C2×C3#311511yescharacteristic
663D3—42no—
664D3—162no—
665C22—24no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—62no—
673C22—62no—
674C22—62no—
675C22—121no—
676C22—121no—
677C22—121no—
678C22—121no—
679C22—242no—
680C22—242no—
681C22—242no—
682C22—242no—
683C22—242no—
684C22—242no—
685C22—242no—
686C22C2×#88211yescharacteristic
687C22D3×#12111yescharacteristic
688C22#562711yescharacteristic
689C22C2×#91111yescharacteristic
690C22C2×#91611yescharacteristic
691C22D3×AGL(1,ℤ/8)11yescharacteristic
692C22#577711yescharacteristic
693C4—22no—
694C4—42no—
695C4—62no—
696C4—62no—
697C4—64no—
698C4—82no—
699C4—122no—
700C4—121no—
701C4#478114yescomplemented
702C4C2×#89511yescharacteristic
703C4#552411yescharacteristic
704C4D3×#12111yescharacteristic
705C4#562711yescharacteristic
706C3#3290411yescharacteristic, complemented, Sylow
707C2—44no—
708C2—122no—
709C2—482no—
710C2#7404811yescharacteristic
711C2C2×#478111yescharacteristic
712C2#7888411yescharacteristic
713C2#8033311yescharacteristic
714C2#8042711yescharacteristic
715C2#8095111yescharacteristic
716C2#8104511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1802C2
socleC23×C3C22×D4
hypercenter#1802D3
hypocenterC3#32904
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
1. C2×C4×#237C2semidirect
2. C2×#5015C2semidirect
3. C2×#4848C2semidirect
4. C2×#4781C2semidirect
5. #74375C2semidirect
6. C3×#1802C2semidirect
7. #70475C2semidirect
8. #70172C2semidirect
9. #69588C2semidirect
10. #68909C2semidirect
11. #68881C2semidirect
58. #1802D3semidirect
70. C2×#237D4semidirect
81. C3×C4×D4D4semidirect
128. C4×Dic6D4semidirect
315. C4×D4D12semidirect
374. C3×D4#121semidirect
428. Dic6#121semidirect
563. C3×C4#449semidirect
610. D4#895semidirect
701. C4#4781semidirect
706. C3#32904semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2372
C2×#50152
C2×#48482
C2×#47812
#743752
C3×#18022
#704752
#701722
#695882
#689092
#688812
#329043