Finite Group Databasebeta

← #10822301

Subgroups — #10822301

Order 768.

2178 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2C22×#950C211yesmaximal, characteristic, complemented
3C2×#5805C211yesmaximal, characteristic, complemented
4#84087C211yesmaximal, characteristic, complemented
5C22×#832C211yesmaximal, characteristic, complemented
6C2×#5133C211yesmaximal, characteristic, complemented
7#72058C212yesmaximal, complemented
8#72056C212yesmaximal, complemented
9#72045C212yesmaximal, complemented
10#72044C212yesmaximal, complemented
11#71980C211yesmaximal, characteristic, complemented
12#23426—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×#235C2211yescharacteristic
16C2×C3×AGL(1,ℤ/8)—24no—
17C2×C3×AGL(1,ℤ/8)—24no—
18C2×C3×AGL(1,ℤ/8)—24no—
19C22×C3×SD16C2212yes—
20C22×C3×D8C2212yes—
21C22×C3×M16C2211yescharacteristic
22C2×#953—22no—
23C2×#950—24no—
24C2×#950—24no—
25C2×#950—24no—
26C22×#216C2211yescharacteristic, complemented
27C22×#215C2211yescharacteristic, complemented
28#5805—24no—
29#5805—24no—
30C2×#940C2212yes—
31C2×#938—22no—
32C2×#936C2212yes—
33C22×#214C2211yescharacteristic, complemented
34C22×#213C2211yescharacteristic, complemented
35C2×#926C2211yescharacteristic
36C22×C4⋊Dic3C2211yescharacteristic
37C23×C3⋊C8C2211yescharacteristic
38#5229C2212yescomplemented
39#5221C2212yescomplemented
40#5220—22no—
41C2×#832—24no—
42C2×#832C2212yes—
43C2×#832C2212yes—
44#5212—22no—
45#5133—24no—
46C2×#821C2212yescomplemented
47C2×#816C2212yescomplemented
48#4550C2212yescomplemented
49#4550C2212yescomplemented
50C22×AGL(1,ℤ/8)D311yescharacteristic, complemented
51C22×#135—31no—
52C2×#535—31no—
53C2×#417—31no—
54C22×#102—31no—
55#2311—31no—
56#2015—32no—
57#2014—32no—
58#2012—32no—
59#2011—32no—
60#1729—31no—
61C25×C3—21no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41—22no—
65C2×C3×#41—22no—
66C2×C3×#41—22no—
67C2×C3×#41D414yescomplemented
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4—21no—
70C22×C3×D4—22no—
71C22×C3×D4—22no—
72C22×C3×D4—22no—
73C22×C3×D4—21no—
74C22×C3×D4—22no—
75C22×C3×D4—22no—
76C22×C3×D4D414yes—
77C22×C3×D4C2311yescharacteristic
78C22×C3×D4C2312yes—
79C23×C3×C4—21no—
80C23×C3×C4C2311yescharacteristic
81C22×#67—21no—
82C22×#67—24no—
83C23×Dic3—21no—
84C2×#235—22no—
85C2×#235—22no—
86C2×#235—22no—
87C2×#235—22no—
88C2×#235—22no—
89C2×#235D414yes—
90C22×D12C2311yescharacteristic
91C22×C4×D3—21no—
92C22×Dic6C2311yescharacteristic
93C3×AGL(1,ℤ/8)—232no—
94C2×C3×SD16—24no—
95C2×C3×SD16—24no—
96C2×C3×SD16—24no—
97C2×C3×D8—24no—
98C2×C3×D8—24no—
99C2×C3×D8—24no—
100C2×C3×M16—22no—
101C2×C3×M16—22no—
102C2×C3×M16D414yescomplemented
103C22×C3×C8C2312yes—
104#953—216no—
105C2×#217—24no—
106C2×#217—24no—
107C2×#217D412yes—
108#950—232no—
109C2×#216—22no—
110C2×#216—22no—
111C2×#216—22no—
112C2×#215—22no—
113C2×#215—22no—
114C2×#215—22no—
115#940—24no—
116#940—24no—
117#938—216no—
118#936—24no—
119#936—24no—
120C2×#214—22no—
121C2×#214—22no—
122C2×#214—22no—
123C2×#213—22no—
124C2×#213—22no—
125C2×#213—22no—
126#926—22no—
127#926—22no—
128C2×C4⋊Dic3—22no—
129C2×C4⋊Dic3—22no—
130C2×C4⋊Dic3—21no—
131C2×C4⋊Dic3D412yes—
132C2×C4⋊Dic3C2312yes—
133C2×C4×Dic3C2312yes—
134C22×C3⋊C8—22no—
135C22×C3⋊C8—22no—
136C22×C3⋊C8D414yescomplemented
137C22×C3⋊C8C2311yescharacteristic
138C22×C3⋊C8C2311yescharacteristic
139#832—24no—
140#832—24no—
141C3⋊#95—24no—
142#821—24no—
143#816—24no—
144C22×#41—31no—
145C22×#41C2×D311yescharacteristic
146C23×D4C2×D311yescharacteristic
147C2×AGL(1,ℤ/8)—24no—
148C2×AGL(1,ℤ/8)—24no—
149C2×AGL(1,ℤ/8)—24no—
150C2×#135—64no—
151C2×#135—64no—
152C2×#135—64no—
153C22×Q16—31no—
154C22×SD16—31no—
155C22×SD16—31no—
156C22×SD16C2×D312yes—
157C22×D8—31no—
158C22×D8C2×D312yes—
159C22×M16C2×D311yescharacteristic
160C23×C8—31no—
161#535—64no—
162#535—64no—
163C2×#127—32no—
164C2×#124—32no—
165C2×#121—62no—
166C2×#120—62no—
167C2×#117—31no—
168C22×C4⋊C4—31no—
169#417—64no—
170C2×#103—32no—
171C2×#102—32no—
172C2×#102—32no—
173C2×#102—32no—
174C2×#102—64no—
175#391—32no—
176#390—32no—
177#386—62no—
178#380—62no—
179#336—32no—
180#336—32no—
181C24×C3—22no—
182C24×C3—21no—
183C24×C3—24no—
184C24×C3—42no—
185C24×C3—42no—
186C24×C3—82no—
187C24×C3—82no—
188C24×C3C2×D411yescharacteristic
189C23×D3—21no—
190C3×#41—28no—
191C3×#41—28no—
192C3×#41—44no—
193C3×#41—44no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8C2×D412yes—
200C2×C3×D4—24no—
201C2×C3×D4—28no—
202C2×C3×D4—24no—
203C2×C3×D4—28no—
204C2×C3×D4—22no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—22no—
210C2×C3×D4—22no—
211C2×C3×D4—22no—
212C2×C3×D4—22no—
213C2×C3×D4—21no—
214C2×C3×D4—42no—
215C2×C3×D4—42no—
216C2×C3×D4—44no—
217C2×C3×D4—44no—
218C2×C3×D4—42no—
219C2×C3×D4—42no—
220C2×C3×D4C2×D412yes—
221C2×C3×D4C2×D414yes—
222C22×C3×C4—21no—
223C22×C3×C4—24no—
224C22×C3×C4—42no—
225C22×C3×C4—42no—
226C22×C3×C4C2×D412yes—
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D412yes—
229C22×C3×C4C2×D412yes—
230C22×C3×C4C2×D412yes—
231C22×C3×C4C2×D412yes—
232C22×C3×C4#4112yes—
233C22×C3×C4C2411yescharacteristic
234C2×#67—24no—
235C2×#67—216no—
236C2×#67—42no—
237C2×#67—42no—
238C2×#67—84no—
239C2×#67—84no—
240C22×Dic3—21no—
241C22×Dic3—22no—
242C22×Dic3—42no—
243C22×Dic3—82no—
244C22×Dic3C2×D414yes—
245#235—28no—
246#235—28no—
247#235—44no—
248#235—44no—
249C2×D12—21no—
250C2×D12—21no—
251C2×D12—21no—
252C2×D12—21no—
253C2×D12—21no—
254C2×D12C2×D412yes—
255C2×C4×D3—24no—
256C2×C4×D3—42no—
257C2×C4×D3—42no—
258C2×Dic6—21no—
259C2×Dic6—21no—
260C2×Dic6—21no—
261C2×Dic6—21no—
262C2×Dic6—21no—
263C2×Dic6C2×D412yes—
264C3×SD16—216no—
265C3×D8—216no—
266C3×M16—28no—
267C2×C3×C8—22no—
268C2×C3×C8—22no—
269C2×C3×C8C2×D414yes—
270#217—216no—
271#217—24no—
272#217—84no—
273#216—28no—
274#215—28no—
275#214—28no—
276#213—28no—
277C4⋊Dic3—24no—
278C4⋊Dic3—22no—
279C4⋊Dic3—22no—
280C4⋊Dic3—42no—
281C4×Dic3—22no—
282C4×Dic3—22no—
283C2×C3⋊C8—28no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8C2×D412yes—
289C2×C3⋊C8C2×D412yes—
290C25—21no—
291C2×#41—22no—
292C2×#41—22no—
293C2×#41—22no—
294C2×#41—22no—
295C2×#41—22no—
296C2×#41—34no—
297C2×#41—62no—
298C2×#41—62no—
299C2×#41—62no—
300C2×#41—62no—
301C2×#41—62no—
302C2×#41#6714yescomplemented
303C22×Q8—31no—
304C22×Q8C22×D311yescharacteristic
305C22×D4—21no—
306C22×D4—22no—
307C22×D4—21no—
308C22×D4—22no—
309C22×D4—22no—
310C22×D4—22no—
311C22×D4—22no—
312C22×D4—31no—
313C22×D4—64no—
314C22×D4—61no—
315C22×D4#6714yes—
316C22×D4C22×D312yes—
317C22×D4C22×D311yescharacteristic
318C23×C4—21no—
319C23×C4—61no—
320C23×C4—61no—
321C23×C4C22×D311yescharacteristic
322AGL(1,ℤ/8)—232no—
323#135—632no—
324C2×Q16—62no—
325C2×Q16—62no—
326C2×Q16—62no—
327C2×SD16—24no—
328C2×SD16—24no—
329C2×SD16—24no—
330C2×SD16—62no—
331C2×SD16—62no—
332C2×SD16—62no—
333C2×SD16—62no—
334C2×SD16—62no—
335C2×SD16—62no—
336C2×D8—24no—
337C2×D8—24no—
338C2×D8—24no—
339C2×D8—62no—
340C2×D8—62no—
341C2×D8—62no—
342C2×M16—22no—
343C2×M16—22no—
344C2×M16#6714yescomplemented
345C22×C8—34no—
346C22×C8—31no—
347C22×C8—31no—
348C22×C8—62no—
349C22×C8—62no—
350C22×C8C22×D312yes—
351#127—64no—
352#127—64no—
353#124—64no—
354#124—64no—
355#121—616no—
356#120—616no—
357#117—62no—
358#117—62no—
359C2×C4⋊C4—32no—
360C2×C4⋊C4—32no—
361C2×C4⋊C4—62no—
362C2×C4⋊C4—62no—
363C2×C4⋊C4—61no—
364C2×C22⋊C4—32no—
365C2×C22⋊C4—64no—
366C2×C22⋊C4—64no—
367C2×C42—32no—
368#103—64no—
369#102—64no—
370#102—64no—
371#102—64no—
372#95—64no—
373C23×C3—22no—
374C23×C3—22no—
375C23×C3—24no—
376C23×C3—22no—
377C23×C3—41no—
378C23×C3—42no—
379C23×C3—41no—
380C23×C3—42no—
381C23×C3—42no—
382C23×C3—44no—
383C23×C3—44no—
384C23×C3—42no—
385C23×C3—48no—
386C23×C3—88no—
387C23×C3—84no—
388C23×C3—84no—
389C23×C3—88no—
390C23×C3#12012yes—
391C23×C3#12112yes—
392C23×C3#12712yes—
393C23×C3C22×D411yescharacteristic
394C22×D3—22no—
395C22×D3—41no—
396C22×D3—41no—
397C22×D3—42no—
398C22×D3—42no—
399C3×Q8—22no—
400C3×Q8—22no—
401C3×Q8—41no—
402C3×Q8—41no—
403C3×D4—22no—
404C3×D4—22no—
405C3×D4—24no—
406C3×D4—24no—
407C3×D4—48no—
408C3×D4—42no—
409C3×D4—42no—
410C3×D4—41no—
411C3×D4—41no—
412C3×D4—48no—
413C2×C3×C4—22no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—48no—
417C2×C3×C4#12014yes—
418C2×C3×C4#12012yes—
419C2×C3×C4#12014yes—
420C2×C3×C4#12114yes—
421C2×C3×C4#12114yes—
422C2×C3×C4#12112yes—
423C2×C3×C4#12712yes—
424C2×C3×C4C22×D411yescharacteristic
425C2×C3×C4C22×D411yescharacteristic
426C2×C3×C4C22×D411yescharacteristic
427C2×C3×C4C22×D411yescharacteristic
428C2×C3×C4C22×D411yescharacteristic
429C2×C3×C4C2×#4111yescharacteristic
430#67—48no—
431#67—816no—
432C2×Dic3—24no—
433C2×Dic3—22no—
434C2×Dic3—24no—
435C2×Dic3—24no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—42no—
439C2×Dic3—44no—
440C2×Dic3—84no—
441D12—22no—
442D12—22no—
443D12—41no—
444D12—41no—
445C4×D3—48no—
446Dic6—22no—
447Dic6—22no—
448Dic6—41no—
449Dic6—41no—
450C3×C8—24no—
451C3⋊C8—22no—
452C3⋊C8—22no—
453C24—21no—
454C24—22no—
455C24—24no—
456C24—42no—
457C24—42no—
458C24—61no—
459C24—82no—
460C24—82no—
461C24D3×D411yescharacteristic
462#41—28no—
463#41—28no—
464#41—44no—
465#41—44no—
466#41—68no—
467#41—68no—
468#41—124no—
469#41—124no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—32no—
476C2×Q8—61no—
477C2×Q8—61no—
478C2×Q8—61no—
479C2×Q8—61no—
480C2×Q8—61no—
481C2×Q8C2×#6712yes—
482C2×D4—24no—
483C2×D4—24no—
484C2×D4—28no—
485C2×D4—28no—
486C2×D4—22no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—22no—
495C2×D4—22no—
496C2×D4—32no—
497C2×D4—42no—
498C2×D4—42no—
499C2×D4—44no—
500C2×D4—44no—
501C2×D4—42no—
502C2×D4—42no—
503C2×D4—64no—
504C2×D4—616no—
505C2×D4—61no—
506C2×D4—61no—
507C2×D4—61no—
508C2×D4—61no—
509C2×D4—61no—
510C2×D4—122no—
511C2×D4—122no—
512C2×D4—244no—
513C2×D4—244no—
514C2×D4C2×#6714yes—
515C2×D4C2×#6712yes—
516C22×C4—21no—
517C22×C4—24no—
518C22×C4—34no—
519C22×C4—42no—
520C22×C4—42no—
521C22×C4—61no—
522C22×C4—64no—
523C22×C4—62no—
524C22×C4—122no—
525C22×C4—122no—
526C22×C4—122no—
527C22×C4—242no—
528C22×C4D3×D412yes—
529C22×C4D3×D411yescharacteristic
530C22×C4D3×D412yes—
531C22×C4#23712yes—
532C22×C4C2×#6712yes—
533C22×C4C2×#6712yes—
534C22×C4C2×#6712yes—
535C22×C4C23×D311yescharacteristic
536Q16—68no—
537SD16—216no—
538SD16—68no—
539SD16—68no—
540D8—216no—
541D8—68no—
542M16—28no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—32no—
546C2×C8—32no—
547C2×C8—68no—
548C2×C8—61no—
549C2×C8—61no—
550C2×C8—61no—
551C2×C8—61no—
552C2×C8C2×#6714yes—
553C4⋊C4—64no—
554C4⋊C4—62no—
555C4⋊C4—62no—
556C4⋊C4—122no—
557C22⋊C4—616no—
558C22⋊C4—64no—
559C22⋊C4—244no—
560C42—62no—
561C42—62no—
562C22×C3—22no—
563C22×C3—22no—
564C22×C3—24no—
565C22×C3—24no—
566C22×C3—42no—
567C22×C3—44no—
568C22×C3—48no—
569C22×C3—44no—
570C22×C3—816no—
571C22×C3—81no—
572C22×C3—82no—
573C22×C3—81no—
574C22×C3—82no—
575C22×C3—82no—
576C22×C3—84no—
577C22×C3#39214yes—
578C22×C3C2×#12011yescharacteristic
579C22×C3C2×#12111yescharacteristic
580C22×C3C2×#12711yescharacteristic
581C22×C3#57612yes—
582C22×C3#57712yes—
583C2×D3—42no—
584C2×D3—44no—
585C2×D3—81no—
586C2×D3—81no—
587C2×D3—82no—
588C3×C4—42no—
589C3×C4#39214yes—
590C3×C4C2×#12011yescharacteristic
591C3×C4C2×#12011yescharacteristic
592C3×C4C2×#12111yescharacteristic
593C3×C4C2×#12111yescharacteristic
594Dic3—42no—
595Dic3—44no—
596Dic3—82no—
597C23—22no—
598C23—22no—
599C23—24no—
600C23—22no—
601C23—48no—
602C23—42no—
603C23—44no—
604C23—44no—
605C23—42no—
606C23—41no—
607C23—42no—
608C23—41no—
609C23—42no—
610C23—62no—
611C23—88no—
612C23—88no—
613C23—84no—
614C23—84no—
615C23—122no—
616C23—122no—
617C23—121no—
618C23—121no—
619C23#93712yes—
620C23#93912yes—
621C23#94012yes—
622C23C2×D3×D411yescharacteristic
623Q8—22no—
624Q8—22no—
625Q8—41no—
626Q8—41no—
627Q8—62no—
628Q8—62no—
629Q8—121no—
630Q8—121no—
631D4—22no—
632D4—24no—
633D4—22no—
634D4—24no—
635D4—48no—
636D4—48no—
637D4—42no—
638D4—42no—
639D4—41no—
640D4—41no—
641D4—62no—
642D4—62no—
643D4—128no—
644D4—121no—
645D4—121no—
646D4—2416no—
647C2×C4—22no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—48no—
651C2×C4—64no—
652C2×C4—64no—
653C2×C4—64no—
654C2×C4—62no—
655C2×C4—121no—
656C2×C4—121no—
657C2×C4—128no—
658C2×C4—124no—
659C2×C4—122no—
660C2×C4—244no—
661C2×C4#93712yes—
662C2×C4#93714yes—
663C2×C4#93814yes—
664C2×C4#93912yes—
665C2×C4#93914yes—
666C2×C4#94012yes—
667C2×C4#95314yes—
668C2×C4C2×D3×D411yescharacteristic
669C2×C4C2×D3×D411yescharacteristic
670C2×C4C2×#23711yescharacteristic
671C2×C4C22×#6711yescharacteristic
672C2×C4C22×#6711yescharacteristic
673C2×C4C22×#6711yescharacteristic
674C8—24no—
675C8—62no—
676C8—62no—
677C2×C3—24no—
678C2×C3—82no—
679C2×C3—84no—
680C2×C3C2×#39211yescharacteristic
681C2×C3#301212yes—
682C2×C3#304512yes—
683C2×C3#315411yescharacteristic
684C2×C3#315511yescharacteristic
685D3—82no—
686C22—24no—
687C22—22no—
688C22—22no—
689C22—24no—
690C22—42no—
691C22—44no—
692C22—44no—
693C22—48no—
694C22—816no—
695C22—84no—
696C22—82no—
697C22—81no—
698C22—82no—
699C22—81no—
700C22—82no—
701C22—122no—
702C22—124no—
703C22—242no—
704C22—241no—
705C22—241no—
706C22#522314yes—
707C22#577412yes—
708C22#577512yes—
709C22C2×#93711yescharacteristic
710C22C2×#93911yescharacteristic
711C22C2×#94011yescharacteristic
712C4—42no—
713C4—122no—
714C4—124no—
715C4—242no—
716C4#522314yes—
717C4C2×#93711yescharacteristic
718C4C2×#93811yescharacteristic
719C4C2×#93911yescharacteristic
720C4C2×#95311yescharacteristic
721C3#2342611yescharacteristic, complemented, Sylow
722C2—24no—
723C2—84no—
724C2—82no—
725C2—242no—
726C2#8393612yes—
727C2#8393911yescharacteristic
728C2#8394011yescharacteristic
729C2#8394912yes—
730C2C2×#522311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×AGL(1,ℤ/8)C2
socleC23×C3C22×D4
hypercenterC22×AGL(1,ℤ/8)D3
hypocenterC3#23426
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×C3×AGL(1,ℤ/8)C2semidirect
2. C22×#950C2semidirect
3. C2×#5805C2semidirect
4. #84087C2semidirect
5. C22×#832C2semidirect
6. C2×#5133C2semidirect
7. #72058C2semidirect
8. #72056C2semidirect
9. #72045C2semidirect
10. #72044C2semidirect
11. #71980C2semidirect
26. C22×#216C22semidirect
27. C22×#215C22semidirect
33. C22×#214C22semidirect
34. C22×#213C22semidirect
38. #5229C22semidirect
39. #5221C22semidirect
46. C2×#821C22semidirect
47. C2×#816C22semidirect
48. #4550C22semidirect
49. #4550C22semidirect
50. C22×AGL(1,ℤ/8)D3semidirect
67. C2×C3×#41D4semidirect
102. C2×C3×M16D4semidirect
136. C22×C3⋊C8D4semidirect
302. C2×#41#67semidirect
344. C2×M16#67semidirect
721. C3#23426semidirect

Maximal subgroups

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

subgroupindex
C22×C3×AGL(1,ℤ/8)2
C22×#9502
C2×#58052
#840872
C22×#8322
C2×#51332
#720582
#720562
#720452
#720442
#719802
#234263