Finite Group Databasebeta

← #10944002

Subgroups — #10944002

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×#239C211yesmaximal, characteristic, complemented
2C2×#5027C211yesmaximal, characteristic, complemented
3C2×#4843C211yesmaximal, characteristic, complemented
4C2×#4810C212yesmaximal, complemented
5#74375C211yesmaximal, characteristic, complemented
6C3×#1803C211yesmaximal, characteristic, complemented
7#70475C211yesmaximal, characteristic, complemented
8#70240C212yesmaximal, complemented
9#69584C211yesmaximal, characteristic, complemented
10#68897C212yesmaximal, complemented
11#68881C212yesmaximal, complemented
12#32904—31nomaximal, Sylow
13C22×#239C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×Q8×Dic3C2211yescharacteristic
16C22×#215C2212yes—
17C22×D24C2212yes—
18C4×#239—24no—
19C4×#239—24no—
20C2×#895—21no—
21C2×#895—21no—
22C2×#895C2212yes—
23C2×#893—21no—
24C2×#893C2211yescharacteristic
25C2×#892—21no—
26C2×#892C2211yescharacteristic
27C2×#874C2211yescharacteristic
28C2×C4×D12—21no—
29C2×C4×D12C2211yescharacteristic
30C2×C42×D3—21no—
31C2×C42×D3C2211yescharacteristic
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#103C2212yes—
34C3×#384C2211yescharacteristic
35C2×#821C2212yes—
36C2×(C2×D3)⋊C8C2212yes—
37#5027—24no—
38#4988—21no—
39#4988—21no—
40#4988C2212yes—
41C2×#809C2212yes—
42C2×#809C2212yes—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44#4843—24no—
45#4810—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#1803D311yescharacteristic, complemented
58#1802—31no—
59#1797—31no—
60C22×C3×Q8C2311yescharacteristic
61C2×#239—22no—
62C2×#239—22no—
63C2×#239—22no—
64C2×#239—22no—
65C2×#239—22no—
66C2×#239D414yescomplemented
67C22×D12—21no—
68C22×D12—22no—
69C22×D12—21no—
70C22×D12—21no—
71C22×D12C2312yes—
72C22×D12C2311yescharacteristic
73C22×C4×D3—21no—
74C22×C4×D3—21no—
75C22×C4×D3—21no—
76C22×C4×D3D411yescharacteristic
77C22×C4×D3D411yescharacteristic
78C22×C4×D3C2311yescharacteristic
79C22×C3×C8C2312yes—
80C3×C4×Q8—22no—
81C3×C4×Q8D414yescomplemented
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89Q8×Dic3—22no—
90Q8×Dic3—22no—
91C2×#215—24no—
92C2×#215—24no—
93C2×#215—24no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—22no—
99C2×(C2×D3)⋊C4—22no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3D411yescharacteristic
105C2×C4×Dic3D411yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2312yes—
108C2×D24—24no—
109C2×D24—24no—
110C2×D24—24no—
111#895—24no—
112#895—24no—
113#895—44no—
114#895—44no—
115#893—28no—
116#893—22no—
117#893—22no—
118#892—28no—
119#892—22no—
120#892—22no—
121#874—22no—
122#874—22no—
123C4×D12—28no—
124C4×D12—22no—
125C4×D12D414yescomplemented
126C42×D3—28no—
127C42×D3—22no—
128C42×D3—22no—
129C3×C4⋊C8—22no—
130C3×#103—24no—
131#821—24no—
132(C2×D3)⋊C8—24no—
133#809—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×Q8C2×D311yescharacteristic
147C2×C4×D4—31no—
148C2×C4×D4—31no—
149C2×C4×D4—61no—
150C2×C4×D4—61no—
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×#103C2×D312yes—
161C2×#102—32no—
162C2×#102—32no—
163C2×#102—32no—
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—
172C23×D3—21no—
173C23×D3—22no—
174C23×D3C2×D411yescharacteristic
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×Q8C2×D412yes—
179C2×C3×Q8C2×D412yes—
180C2×C3×Q8#4112yes—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C22×Dic3—21no—
188C22×Dic3C2×D411yescharacteristic
189C22×Dic3C2×D411yescharacteristic
190C22×Dic3C2×D411yescharacteristic
191#239—28no—
192#239—28no—
193#239—28no—
194#239—28no—
195C2×D12—24no—
196C2×D12—24no—
197C2×D12—24no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—22no—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—22no—
205C2×D12—22no—
206C2×D12—22no—
207C2×D12—42no—
208C2×D12—42no—
209C2×D12—44no—
210C2×D12—44no—
211C2×D12—44no—
212C2×D12—42no—
213C2×D12—42no—
214C2×D12—42no—
215C2×D12—42no—
216C2×D12C2×D412yes—
217C2×D12C2×D412yes—
218C2×D12#4112yes—
219C2×C4×D3—24no—
220C2×C4×D3—24no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—24no—
226C2×C4×D3—24no—
227C2×C4×D3—22no—
228C2×C4×D3—24no—
229C2×C4×D3—22no—
230C2×C4×D3—22no—
231C2×C4×D3—22no—
232C2×C4×D3—22no—
233C2×C4×D3—41no—
234C2×C4×D3—42no—
235C2×C4×D3—42no—
236C2×C4×D3—42no—
237C2×C4×D3—42no—
238C2×C4×D3C2×D412yes—
239C2×C4×D3#4112yes—
240C2×C3×C8—22no—
241C2×C3×C8—22no—
242C2×C3×C8—22no—
243C3×C4⋊C4—24no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—42no—
248C3×C4⋊C4—42no—
249C3×C4⋊C4C2×D412yes—
250C3×C42—24no—
251C3×C42C2×D412yes—
252C3×C42C2×D412yes—
253#215—48no—
254(C2×D3)⋊C4—24no—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—44no—
258(C2×D3)⋊C4—44no—
259(C2×D3)⋊C4—44no—
260C4⋊Dic3—24no—
261C4⋊Dic3—21no—
262C4⋊Dic3C2×D412yes—
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—
272D24—48no—
273C2×#41—34no—
274C2×#41—62no—
275C2×#41—62no—
276C2×#41—62no—
277C2×#41—62no—
278C2×#41—62no—
279C22×Q8C22×D311yescharacteristic
280C22×D4—31no—
281C22×D4—32no—
282C22×D4—61no—
283C22×D4—62no—
284C22×D4—61no—
285C22×D4—61no—
286C23×C4—31no—
287C23×C4—31no—
288C23×C4—31no—
289C23×C4—61no—
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—22no—
307C4×Q8—62no—
308C4×Q8—62no—
309C4×Q8D1214yescomplemented
310C4×D4—34no—
311C4×D4—68no—
312C4×D4—62no—
313C4×D4—68no—
314C4×D4—62no—
315C4×D4—62no—
316#117—62no—
317#117—68no—
318#117—62no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4C22×D312yes—
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—62no—
330C2×C22⋊C4—62no—
331C2×C22⋊C4—62no—
332C2×C42—21no—
333C2×C42—31no—
334C2×C42—31no—
335C2×C42—31no—
336C2×C42—68no—
337C2×C42—61no—
338C2×C42—61no—
339C2×C42—62no—
340C2×C42—62no—
341C2×C42C22×D311yescharacteristic
342C4⋊C8—22no—
343C4⋊C8—62no—
344#103—24no—
345#102—64no—
346#102—64no—
347#102—64no—
348C22⋊C8—64no—
349C23×C3C22×D411yescharacteristic
350C22×D3—21no—
351C22×D3—22no—
352C22×D3—22no—
353C22×D3—21no—
354C22×D3—22no—
355C22×D3—21no—
356C22×D3—21no—
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
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×Q8—22no—
372C3×Q8—22no—
373C3×Q8—22no—
374C3×Q8#12114yescomplemented
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—22no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—42no—
384C2×C3×C4—42no—
385C2×C3×C4—42no—
386C2×C3×C4#12012yes—
387C2×C3×C4#12012yes—
388C2×C3×C4#13512yes—
389C2×C3×C4AGL(1,ℤ/8)12yes—
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—22no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3#12112yes—
407C2×Dic3AGL(1,ℤ/8)12yes—
408D12—216no—
409D12—22no—
410D12—22no—
411D12—22no—
412D12—42no—
413D12—42no—
414D12—42no—
415D12—42no—
416D12—84no—
417D12—88no—
418D12—84no—
419D12#12114yescomplemented
420C4×D3—216no—
421C4×D3—22no—
422C4×D3—22no—
423C4×D3—22no—
424C4×D3—22no—
425C4×D3—44no—
426C4×D3—48no—
427C4×D3—44no—
428C4×D3—48no—
429C3×C8—42no—
430C3⋊C8—42no—
431C24—31no—
432C24—61no—
433C24—62no—
434#41—68no—
435#41—68no—
436#41—68no—
437#41—68no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8C2×D1212yes—
442C2×Q8D3×D412yes—
443C2×Q8#23912yes—
444C2×D4—32no—
445C2×D4—32no—
446C2×D4—32no—
447C2×D4—64no—
448C2×D4—64no—
449C2×D4—64no—
450C2×D4—61no—
451C2×D4—62no—
452C2×D4—62no—
453C2×D4—62no—
454C2×D4—61no—
455C2×D4—62no—
456C2×D4—61no—
457C2×D4—62no—
458C2×D4—62no—
459C2×D4—122no—
460C2×D4—122no—
461C2×D4—124no—
462C2×D4—124no—
463C2×D4—124no—
464C2×D4—122no—
465C2×D4—122no—
466C2×D4—122no—
467C2×D4—122no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—32no—
474C22×C4—32no—
475C22×C4—31no—
476C22×C4—64no—
477C22×C4—62no—
478C22×C4—64no—
479C22×C4—64no—
480C22×C4—64no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—61no—
486C22×C4—64no—
487C22×C4—62no—
488C22×C4—62no—
489C22×C4—62no—
490C22×C4—62no—
491C22×C4—121no—
492C22×C4—122no—
493C22×C4—122no—
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—24no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—32no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—64no—
515C4⋊C4—61no—
516C4⋊C4C2×D1212yes—
517C22⋊C4—64no—
518C22⋊C4—64no—
519C22⋊C4—64no—
520C22⋊C4—124no—
521C22⋊C4—124no—
522C22⋊C4—124no—
523C42—24no—
524C42—64no—
525C42—64no—
526C42—62no—
527C42—62no—
528C42—61no—
529C42—61no—
530C42C2×D1212yes—
531C42D3×D412yes—
532C22×C3C2×#12011yescharacteristic
533C22×C3C2×#12111yescharacteristic
534C22×C3#52511yescharacteristic
535C22×C3C2×#13511yescharacteristic
536C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
537C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
538C22×C3#57511yescharacteristic
539C2×D3—22no—
540C2×D3—24no—
541C2×D3—22no—
542C2×D3—22no—
543C2×D3—41no—
544C2×D3—42no—
545C2×D3—41no—
546C2×D3—42no—
547C2×D3—41no—
548C2×D3—42no—
549C2×D3—41no—
550C2×D3—42no—
551C2×D3—42no—
552C2×D3—42no—
553C2×D3—82no—
554C2×D3—82no—
555C2×D3—82no—
556C2×D3—82no—
557C2×D3—82no—
558C2×D3—82no—
559C2×D3—82no—
560C3×C4—24no—
561C3×C4—22no—
562C3×C4—42no—
563C3×C4—82no—
564C3×C4#44914yescomplemented
565C3×C4C2×#12111yescharacteristic
566C3×C4C2×#12111yescharacteristic
567C3×C4#52511yescharacteristic
568C3×C4#52511yescharacteristic
569Dic3—22no—
570Dic3—22no—
571Dic3—41no—
572Dic3—42no—
573C23—32no—
574C23—62no—
575C23—61no—
576C23—61no—
577C23—61no—
578C23—61no—
579C23—61no—
580C23—62no—
581C23—62no—
582C23—61no—
583C23—121no—
584C23—121no—
585C23—121no—
586C23—121no—
587C23—122no—
588C23—122no—
589C23—122no—
590C23—122no—
591C23—122no—
592C23—122no—
593C23—122no—
594C23C2×D3×D411yescharacteristic
595Q8—22no—
596Q8—22no—
597Q8—22no—
598Q8#89514yescomplemented
599D4—34no—
600D4—616no—
601D4—62no—
602D4—62no—
603D4—62no—
604D4—122no—
605D4—122no—
606D4—122no—
607D4—122no—
608D4—244no—
609D4—248no—
610D4—244no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—32no—
618C2×C4—32no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—616no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—62no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—124no—
638C2×C4—128no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—124no—
642C2×C4—128no—
643C2×C4#88212yes—
644C2×C4#88212yes—
645C2×C4#91412yes—
646C2×C4#91912yes—
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#284711yescharacteristic
656C2×C3C2×#44911yescharacteristic
657C2×C3#300611yescharacteristic
658C2×C3#303511yescharacteristic
659C2×C3#303811yescharacteristic
660C2×C3#310611yescharacteristic
661C2×C3#311511yescharacteristic
662D3—42no—
663D3—44no—
664D3—162no—
665C22—62no—
666C22—64no—
667C22—62no—
668C22—62no—
669C22—122no—
670C22—121no—
671C22—122no—
672C22—122no—
673C22—121no—
674C22—122no—
675C22—121no—
676C22—122no—
677C22—121no—
678C22—122no—
679C22—242no—
680C22—242no—
681C22—242no—
682C22—242no—
683C22—242no—
684C22—242no—
685C22—242no—
686C22C2×#88211yescharacteristic
687C22D3×#12111yescharacteristic
688C22#562711yescharacteristic
689C22C2×#91411yescharacteristic
690C22C2×#91911yescharacteristic
691C22D3×AGL(1,ℤ/8)11yescharacteristic
692C22#577711yescharacteristic
693C4—22no—
694C4—24no—
695C4—42no—
696C4—62no—
697C4—62no—
698C4—82no—
699C4—121no—
700C4—122no—
701C4#481014yescomplemented
702C4C2×#89511yescharacteristic
703C4#552411yescharacteristic
704C4D3×#12111yescharacteristic
705C4#562711yescharacteristic
706C3#3290411yescharacteristic, complemented, Sylow
707C2—122no—
708C2—124no—
709C2—482no—
710C2#7404811yescharacteristic
711C2C2×#481011yescharacteristic
712C2#7888411yescharacteristic
713C2#8038411yescharacteristic
714C2#8047511yescharacteristic
715C2#8099911yescharacteristic
716C2#8109311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1803C2
socleC23×C3C22×D4
hypercenter#1803D3
hypocenterC3#32904
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
1. C2×C4×#239C2semidirect
2. C2×#5027C2semidirect
3. C2×#4843C2semidirect
4. C2×#4810C2semidirect
5. #74375C2semidirect
6. C3×#1803C2semidirect
7. #70475C2semidirect
8. #70240C2semidirect
9. #69584C2semidirect
10. #68897C2semidirect
11. #68881C2semidirect
57. #1803D3semidirect
66. C2×#239D4semidirect
81. C3×C4×Q8D4semidirect
125. C4×D12D4semidirect
309. C4×Q8D12semidirect
374. C3×Q8#121semidirect
419. D12#121semidirect
564. C3×C4#449semidirect
598. Q8#895semidirect
701. C4#4810semidirect
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×#2392
C2×#50272
C2×#48432
C2×#48102
#743752
C3×#18032
#704752
#702402
#695842
#688972
#688812
#329043