Finite Group Databasebeta

← #10822303

Subgroups — #10822303

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×#135C211yesmaximal, characteristic, complemented
2C2×#5830C211yesmaximal, characteristic, complemented
3C22×#949C211yesmaximal, characteristic, complemented
4#84087C211yesmaximal, characteristic, complemented
5C2×#5232C211yesmaximal, characteristic, complemented
6C22×#821C211yesmaximal, characteristic, complemented
7#72072C212yesmaximal, complemented
8#72056C212yesmaximal, complemented
9#72055C212yesmaximal, complemented
10#72044C212yesmaximal, complemented
11#71965C211yesmaximal, characteristic, complemented
12#23426—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×#41C2211yescharacteristic
15C23×D12C2211yescharacteristic
16C2×C3×#135—24no—
17C2×C3×#135—24no—
18C2×C3×#135—24no—
19C22×C3×Q16C2211yescharacteristic, complemented
20C22×C3×SD16C2211yescharacteristic, complemented
21C22×C3×SD16C2211yescharacteristic, complemented
22C22×C3×D8C2211yescharacteristic, complemented
23C23×C3×C8C2211yescharacteristic
24#5830—24no—
25#5830—24no—
26C2×#949—24no—
27C2×#949—24no—
28C2×#949—24no—
29C2×#947C2212yes—
30C22×#215C2212yes—
31C2×#940C2212yes—
32C2×#937—22no—
33C22×#213C2212yes—
34C2×#930—22no—
35C2×#926C2211yescharacteristic
36C22×C4⋊Dic3C2211yescharacteristic
37C22×#208C2211yescharacteristic
38#5232—24no—
39#5229C2212yescomplemented
40C2×#835C2212yescomplemented
41#5221C2212yescomplemented
42#5220—22no—
43C2×#832C2212yescomplemented
44#5212—22no—
45C2×#821—24no—
46C2×#821C2212yes—
47C2×#821C2212yes—
48#4550C2212yescomplemented
49#4550C2212yescomplemented
50C22×AGL(1,ℤ/8)—31no—
51C22×#135D311yescharacteristic, complemented
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—
61C24×D3—21no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41—22no—
65C2×C3×#41—22no—
66C2×C3×#41—22no—
67C2×C3×#41—22no—
68C2×C3×#41—22no—
69C2×C3×#41—22no—
70C2×C3×#41—22no—
71C2×C3×#41—22no—
72C2×C3×#41D414yes—
73C2×C3×#41D414yescomplemented
74C22×C3×Q8C2311yescharacteristic
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4—21no—
77C22×C3×D4—21no—
78C22×C3×D4C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4—21no—
81C23×C3×C4—21no—
82C23×C3×C4C2311yescharacteristic
83C22×#67—24no—
84C23×Dic3—21no—
85C22×D12—22no—
86C22×D12—22no—
87C22×D12—22no—
88C22×D12—22no—
89C22×D12—21no—
90C22×D12—22no—
91C22×D12D414yes—
92C22×D12C2312yes—
93C3×#135—232no—
94C2×C3×Q16—22no—
95C2×C3×Q16—22no—
96C2×C3×Q16—22no—
97C2×C3×SD16—22no—
98C2×C3×SD16—22no—
99C2×C3×SD16—22no—
100C2×C3×SD16—22no—
101C2×C3×SD16—22no—
102C2×C3×SD16—22no—
103C2×C3×D8—22no—
104C2×C3×D8—22no—
105C2×C3×D8—22no—
106C22×C3×C8—22no—
107C22×C3×C8—22no—
108C22×C3×C8D414yescomplemented
109C22×C3×C8C2311yescharacteristic
110C22×C3×C8C2311yescharacteristic
111C2×#217—24no—
112C2×#217D412yes—
113#949—232no—
114#947—24no—
115#947—24no—
116C2×#215—24no—
117C2×#215—24no—
118C2×#215—24no—
119#940—24no—
120#940—24no—
121#937—216no—
122C2×#213—24no—
123C2×#213—24no—
124C2×#213—24no—
125#930—216no—
126C2×(C2×D3)⋊C4—24no—
127#926—22no—
128#926—22no—
129C2×C4⋊Dic3—22no—
130C2×C4⋊Dic3—22no—
131C2×C4⋊Dic3—21no—
132C2×C4⋊Dic3D412yes—
133C2×C4⋊Dic3C2312yes—
134C2×C4×Dic3C2312yes—
135C2×#208—22no—
136C2×#208—22no—
137C2×#208D414yescomplemented
138C22×C3⋊C8C2312yes—
139#835—24no—
140#832—24no—
141C3⋊#95—24no—
142#821—24no—
143#821—24no—
144C22×#41C2×D311yescharacteristic
145C22×#41C2×D311yescharacteristic
146C23×D4—31no—
147C2×AGL(1,ℤ/8)—64no—
148C2×AGL(1,ℤ/8)—64no—
149C2×AGL(1,ℤ/8)—64no—
150C2×#135—24no—
151C2×#135—24no—
152C2×#135—24no—
153C22×Q16C2×D311yescharacteristic, complemented
154C22×SD16—32no—
155C22×SD16C2×D311yescharacteristic, complemented
156C22×SD16C2×D311yescharacteristic, complemented
157C22×D8—32no—
158C22×D8C2×D311yescharacteristic, complemented
159C22×M16—31no—
160C23×C8C2×D311yescharacteristic
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—21no—
182C24×C3—21no—
183C24×C3C2×D411yescharacteristic
184C23×D3—22no—
185C23×D3—24no—
186C23×D3—42no—
187C23×D3—42no—
188C23×D3—82no—
189C23×D3—82no—
190C3×#41—28no—
191C3×#41—28no—
192C3×#41—28no—
193C3×#41—28no—
194C3×#41—44no—
195C3×#41—44no—
196C3×#41—44no—
197C3×#41—44no—
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×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8C2×D412yes—
209C2×C3×Q8C2×D412yes—
210C2×C3×D4—24no—
211C2×C3×D4—24no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—42no—
223C2×C3×D4—42no—
224C2×C3×D4—42no—
225C2×C3×D4—42no—
226C2×C3×D4C2×D412yes—
227C2×C3×D4C2×D412yes—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4—24no—
231C22×C3×C4—24no—
232C22×C3×C4—42no—
233C22×C3×C4—42no—
234C22×C3×C4—42no—
235C22×C3×C4—42no—
236C22×C3×C4C2×D412yes—
237C22×C3×C4C2×D411yescharacteristic
238C22×C3×C4C2×D412yes—
239C22×C3×C4C2×D412yes—
240C22×C3×C4C2×D412yes—
241C22×C3×C4C2×D412yes—
242C22×C3×C4#4112yes—
243C22×C3×C4C2411yescharacteristic
244C2×#67—216no—
245C2×#67—84no—
246C2×#67—84no—
247C22×Dic3—22no—
248C22×Dic3—42no—
249C22×Dic3—82no—
250C22×Dic3C2×D414yes—
251C2×D12—28no—
252C2×D12—28no—
253C2×D12—22no—
254C2×D12—22no—
255C2×D12—22no—
256C2×D12—22no—
257C2×D12—24no—
258C2×D12—22no—
259C2×D12—44no—
260C2×D12—44no—
261C2×D12—42no—
262C2×D12—42no—
263C2×D12C2×D414yes—
264C3×Q16—28no—
265C3×SD16—28no—
266C3×SD16—28no—
267C3×D8—28no—
268C2×C3×C8—28no—
269C2×C3×C8—21no—
270C2×C3×C8—21no—
271C2×C3×C8—21no—
272C2×C3×C8—21no—
273C2×C3×C8C2×D412yes—
274C2×C3×C8C2×D412yes—
275#217—24no—
276#217—84no—
277#215—216no—
278#213—216no—
279(C2×D3)⋊C4—216no—
280C4⋊Dic3—24no—
281C4⋊Dic3—22no—
282C4⋊Dic3—22no—
283C4⋊Dic3—42no—
284C4×Dic3—22no—
285C4×Dic3—22no—
286#208—28no—
287C2×C3⋊C8—22no—
288C2×C3⋊C8—22no—
289C2×C3⋊C8C2×D414yes—
290C25—61no—
291C2×#41—22no—
292C2×#41—22no—
293C2×#41—22no—
294C2×#41—22no—
295C2×#41—22no—
296C2×#41—22no—
297C2×#41—22no—
298C2×#41—22no—
299C2×#41—22no—
300C2×#41—22no—
301C2×#41#6714yes—
302C2×#41#6714yescomplemented
303C22×Q8C22×D311yescharacteristic
304C22×Q8C22×D311yescharacteristic
305C22×D4—21no—
306C22×D4—21no—
307C22×D4—34no—
308C22×D4—32no—
309C22×D4—64no—
310C22×D4—62no—
311C22×D4—61no—
312C22×D4—62no—
313C22×D4—62no—
314C22×D4—62no—
315C22×D4—62no—
316C22×D4C22×D311yescharacteristic
317C22×D4C22×D311yescharacteristic
318C23×C4—21no—
319C23×C4—21no—
320C23×C4—61no—
321C23×C4C22×D311yescharacteristic
322AGL(1,ℤ/8)—632no—
323#135—232no—
324C2×Q16—22no—
325C2×Q16—22no—
326C2×Q16—22no—
327C2×SD16—22no—
328C2×SD16—22no—
329C2×SD16—22no—
330C2×SD16—22no—
331C2×SD16—22no—
332C2×SD16—22no—
333C2×SD16—64no—
334C2×SD16—64no—
335C2×SD16—64no—
336C2×D8—22no—
337C2×D8—22no—
338C2×D8—22no—
339C2×D8—64no—
340C2×D8—64no—
341C2×D8—64no—
342C2×M16—34no—
343C2×M16—62no—
344C2×M16—62no—
345C22×C8—22no—
346C22×C8—22no—
347C22×C8—32no—
348C22×C8#6714yescomplemented
349C22×C8C22×D311yescharacteristic
350C22×C8C22×D311yescharacteristic
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—22no—
376C23×C3—22no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3—41no—
380C23×C3—41no—
381C23×C3—42no—
382C23×C3—42no—
383C23×C3—42no—
384C23×C3—42no—
385C23×C3#12012yes—
386C23×C3#12112yes—
387C23×C3#12712yes—
388C23×C3C22×D411yescharacteristic
389C22×D3—24no—
390C22×D3—42no—
391C22×D3—42no—
392C22×D3—44no—
393C22×D3—44no—
394C22×D3—48no—
395C22×D3—88no—
396C22×D3—84no—
397C22×D3—84no—
398C22×D3—88no—
399C3×Q8—22no—
400C3×Q8—22no—
401C3×Q8—22no—
402C3×Q8—22no—
403C3×Q8—41no—
404C3×Q8—41no—
405C3×Q8—41no—
406C3×Q8—41no—
407C3×D4—22no—
408C3×D4—22no—
409C3×D4—22no—
410C3×D4—22no—
411C3×D4—48no—
412C3×D4—48no—
413C3×D4—41no—
414C3×D4—41no—
415C3×D4—41no—
416C3×D4—41no—
417C2×C3×C4—22no—
418C2×C3×C4—22no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
422C2×C3×C4—41no—
423C2×C3×C4—48no—
424C2×C3×C4—48no—
425C2×C3×C4#12014yes—
426C2×C3×C4#12012yes—
427C2×C3×C4#12014yes—
428C2×C3×C4#12114yes—
429C2×C3×C4#12114yes—
430C2×C3×C4#12112yes—
431C2×C3×C4#12712yes—
432C2×C3×C4C22×D411yescharacteristic
433C2×C3×C4C22×D411yescharacteristic
434C2×C3×C4C22×D411yescharacteristic
435C2×C3×C4C22×D411yescharacteristic
436C2×C3×C4C22×D411yescharacteristic
437C2×C3×C4C2×#4111yescharacteristic
438#67—816no—
439C2×Dic3—24no—
440C2×Dic3—24no—
441C2×Dic3—24no—
442C2×Dic3—42no—
443C2×Dic3—44no—
444C2×Dic3—84no—
445D12—24no—
446D12—24no—
447D12—42no—
448D12—42no—
449D12—48no—
450C3×C8—22no—
451C3×C8—22no—
452C3⋊C8—24no—
453C24—21no—
454C24—21no—
455C24—62no—
456C24—64no—
457C24—122no—
458C24—122no—
459C24—242no—
460C24—242no—
461C24D3×D411yescharacteristic
462#41—28no—
463#41—28no—
464#41—28no—
465#41—28no—
466#41—44no—
467#41—44no—
468#41—44no—
469#41—44no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8C2×#6712yes—
481C2×Q8C2×#6712yes—
482C2×D4—24no—
483C2×D4—24no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—34no—
495C2×D4—42no—
496C2×D4—42no—
497C2×D4—42no—
498C2×D4—42no—
499C2×D4—616no—
500C2×D4—68no—
501C2×D4—64no—
502C2×D4—68no—
503C2×D4—62no—
504C2×D4—62no—
505C2×D4—62no—
506C2×D4—62no—
507C2×D4—62no—
508C2×D4—122no—
509C2×D4—124no—
510C2×D4—122no—
511C2×D4—124no—
512C2×D4—244no—
513C2×D4—244no—
514C2×D4C2×#6712yes—
515C2×D4C2×#6712yes—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—24no—
519C22×C4—24no—
520C22×C4—34no—
521C22×C4—42no—
522C22×C4—42no—
523C22×C4—42no—
524C22×C4—42no—
525C22×C4—62no—
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—28no—
537SD16—28no—
538SD16—28no—
539SD16—616no—
540D8—28no—
541D8—616no—
542M16—68no—
543C2×C8—28no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—34no—
549C2×C8—62no—
550C2×C8—62no—
551C2×C8C2×#6712yes—
552C2×C8C2×#6712yes—
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—42no—
568C22×C3—44no—
569C22×C3—44no—
570C22×C3—81no—
571C22×C3—81no—
572C22×C3—81no—
573C22×C3—81no—
574C22×C3—82no—
575C22×C3—82no—
576C22×C3#39214yes—
577C22×C3C2×#12011yescharacteristic
578C22×C3C2×#12111yescharacteristic
579C22×C3C2×#12711yescharacteristic
580C22×C3#57612yes—
581C22×C3#57712yes—
582C2×D3—44no—
583C2×D3—48no—
584C2×D3—816no—
585C2×D3—82no—
586C2×D3—82no—
587C2×D3—84no—
588C3×C4—42no—
589C3×C4—42no—
590C3×C4#39214yes—
591C3×C4C2×#12011yescharacteristic
592C3×C4C2×#12011yescharacteristic
593C3×C4C2×#12111yescharacteristic
594C3×C4C2×#12111yescharacteristic
595Dic3—44no—
596Dic3—82no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—41no—
606C23—41no—
607C23—41no—
608C23—41no—
609C23—64no—
610C23—128no—
611C23—124no—
612C23—124no—
613C23—122no—
614C23—122no—
615C23—248no—
616C23—248no—
617C23—244no—
618C23—244no—
619C23#93712yes—
620C23#93912yes—
621C23#94012yes—
622C23C2×D3×D411yescharacteristic
623Q8—22no—
624Q8—22no—
625Q8—22no—
626Q8—22no—
627Q8—41no—
628Q8—41no—
629Q8—41no—
630Q8—41no—
631D4—22no—
632D4—22no—
633D4—22no—
634D4—22no—
635D4—48no—
636D4—48no—
637D4—41no—
638D4—41no—
639D4—41no—
640D4—41no—
641D4—64no—
642D4—64no—
643D4—128no—
644D4—122no—
645D4—122no—
646D4—2416no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—48no—
654C2×C4—48no—
655C2×C4—64no—
656C2×C4—64no—
657C2×C4—64no—
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—22no—
675C8—22no—
676C8—64no—
677C2×C3—24no—
678C2×C3—82no—
679C2×C3—82no—
680C2×C3C2×#39211yescharacteristic
681C2×C3#301212yes—
682C2×C3#304512yes—
683C2×C3#315411yescharacteristic
684C2×C3#315511yescharacteristic
685D3—84no—
686C22—24no—
687C22—22no—
688C22—22no—
689C22—24no—
690C22—42no—
691C22—42no—
692C22—44no—
693C22—44no—
694C22—82no—
695C22—81no—
696C22—81no—
697C22—81no—
698C22—81no—
699C22—82no—
700C22—124no—
701C22—128no—
702C22—2416no—
703C22—244no—
704C22—242no—
705C22—242no—
706C22#522314yes—
707C22#576912yes—
708C22#577012yes—
709C22C2×#93711yescharacteristic
710C22C2×#93911yescharacteristic
711C22C2×#94011yescharacteristic
712C4—42no—
713C4—42no—
714C4—124no—
715C4—242no—
716C4#522314yes—
717C4C2×#93711yescharacteristic
718C4C2×#93811yescharacteristic
719C4C2×#93911yescharacteristic
720C4C2×#95311yescharacteristic
721C3#2342611yescharacteristic, complemented, Sylow
722C2—24no—
723C2—82no—
724C2—82no—
725C2—244no—
726C2#8390112yes—
727C2#8390511yescharacteristic
728C2#8390611yescharacteristic
729C2#8391212yes—
730C2C2×#522311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#135C2
socleC23×C3C22×D4
hypercenterC22×#135D3
hypocenterC3#23426
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#135

Complements

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

NH ≅ G/Ntype
1. C22×C3×#135C2semidirect
2. C2×#5830C2semidirect
3. C22×#949C2semidirect
4. #84087C2semidirect
5. C2×#5232C2semidirect
6. C22×#821C2semidirect
7. #72072C2semidirect
8. #72056C2semidirect
9. #72055C2semidirect
10. #72044C2semidirect
11. #71965C2semidirect
19. C22×C3×Q16C22semidirect
20. C22×C3×SD16C22semidirect
21. C22×C3×SD16C22semidirect
22. C22×C3×D8C22semidirect
39. #5229C22semidirect
40. C2×#835C22semidirect
41. #5221C22semidirect
43. C2×#832C22semidirect
48. #4550C22semidirect
49. #4550C22semidirect
51. C22×#135D3semidirect
73. C2×C3×#41D4semidirect
108. C22×C3×C8D4semidirect
137. C2×#208D4semidirect
153. C22×Q16C2×D3semidirect
155. C22×SD16C2×D3semidirect
156. C22×SD16C2×D3semidirect
158. C22×D8C2×D3semidirect
302. C2×#41#67semidirect
348. C22×C8#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×#1352
C2×#58302
C22×#9492
#840872
C2×#52322
C22×#8212
#720722
#720562
#720552
#720442
#719652
#234263