Finite Group Databasebeta

← #411671

Subgroups — #411671

Order 512.

2000 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#462C211yesmaximal, characteristic
2C22×C4×Q16C211yesmaximal, characteristic
3C2×#2546C211yesmaximal, characteristic
4C2×#2016C212yesmaximal
5C2×#1956C211yesmaximal, characteristic
6C2×#1803C211yesmaximal, characteristic
7#13926C211yesmaximal, characteristic
8#11124C212yesmaximal
9#11024C212yesmaximal
10#11010C212yesmaximal
11#10834C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C23×Q16C2212yes—
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4⋊Q8C2212yes—
16C22×C4×Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#462—24no—
19C2×#462—24no—
20C2×#462—24no—
21C2×C4×Q16—24no—
22C2×C4×Q16—24no—
23C2×C4×Q16—24no—
24C22×#107C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2212yes—
28C22×#103C2211yescharacteristic
29C22×#103C2212yes—
30C22×#103C2212yes—
31C22×#103C2212yes—
32C22×C4×C8C2211yescharacteristic
33#2546—24no—
34#2546—24no—
35C2×#398—22no—
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#389—22no—
39C2×#384C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41#2016—28no—
42#1956—24no—
43#1803—24no—
44C2×#340C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47#1441—22no—
48C4⋊#95—22no—
49C4⋊#95C2212yes—
50C23×Q8C2312yes—
51C23×Q8C2311yescharacteristic
52C23×Q8C2311yescharacteristic
53C22×Q16—22no—
54C22×Q16—22no—
55C22×Q16—22no—
56C22×Q16—22no—
57C22×Q16—24no—
58C22×Q16—24no—
59C22×Q16—24no—
60C22×Q16—24no—
61C22×Q16—22no—
62C22×Q16—24no—
63C22×Q16—22no—
64C22×Q16—42no—
65C22×Q16—42no—
66C22×Q16D414yes—
67C23×C8C2311yescharacteristic
68C23×C8C2312yes—
69C23×C8C2311yescharacteristic
70C2×C4⋊Q8—24no—
71C2×C4⋊Q8—24no—
72C2×C4⋊Q8—24no—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8—22no—
76C2×C4⋊Q8—22no—
77C2×C4⋊Q8—22no—
78C2×C4⋊Q8—22no—
79C2×C4⋊Q8—42no—
80C2×C4⋊Q8—42no—
81C2×C4⋊Q8D414yes—
82C2×C4×Q8—22no—
83C2×C4×Q8—22no—
84C2×C4×Q8—22no—
85C2×C4×Q8—22no—
86C2×C4×Q8—22no—
87C2×C4×Q8—22no—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8D414yes—
93C2×C4×Q8D414yes—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4D412yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C42—21no—
106C22×C42—21no—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109#462—232no—
110C4×Q16—232no—
111C2×#107—22no—
112C2×#107—22no—
113C2×#107—22no—
114C2×C4⋊C8—22no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8D414yes—
117C2×#103—22no—
118C2×#103—22no—
119C2×#103—24no—
120C2×#103—22no—
121C2×#103—22no—
122C2×#103—24no—
123C2×#103—22no—
124C2×#103—22no—
125C2×#103—24no—
126C2×#103—24no—
127C2×#103—24no—
128C2×#103—24no—
129C2×#103—24no—
130C2×#103—24no—
131C2×#103—24no—
132C2×#103—42no—
133C2×#103—42no—
134C2×#103—42no—
135C2×C4×C8—22no—
136C2×C4×C8—22no—
137C2×C4×C8D414yes—
138#398—216no—
139#391—24no—
140#391—24no—
141#391—24no—
142#391—42no—
143#389—216no—
144#384—22no—
145#384—22no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4—22no—
148C2×#95—24no—
149C2×#95—24no—
150C2×#95—24no—
151C2×#95D412yes—
152#340—24no—
153#336—24no—
154#336—42no—
155C22×Q8—21no—
156C22×Q8—22no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—21no—
183C22×Q8—21no—
184C22×Q8—21no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—21no—
189C22×Q8—22no—
190C22×Q8C2×D414yes—
191C22×Q8C2×D412yes—
192C22×Q8C2×D412yes—
193C22×Q8C2×D412yes—
194C22×Q8C2×D412yes—
195C22×Q8#4112yes—
196C22×Q8#4112yes—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—22no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—22no—
203C23×C4C2×D414yes—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2411yescharacteristic
207C2×Q16—28no—
208C2×Q16—28no—
209C2×Q16—28no—
210C2×Q16—28no—
211C2×Q16—28no—
212C2×Q16—28no—
213C2×Q16—216no—
214C2×Q16—28no—
215C2×Q16—48no—
216C2×Q16—48no—
217C2×Q16—48no—
218C2×Q16—48no—
219C2×Q16—48no—
220C2×Q16—48no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8C2×D414yes—
238C22×C8C2×D412yes—
239C22×C8C2×D412yes—
240C22×C8C2×D412yes—
241C22×C8#4112yes—
242C4⋊Q8—28no—
243C4⋊Q8—28no—
244C4⋊Q8—48no—
245C4⋊Q8—48no—
246C4⋊Q8—48no—
247C4⋊Q8—48no—
248C4⋊Q8—44no—
249C4⋊Q8—44no—
250C4×Q8—28no—
251C4×Q8—28no—
252C4×Q8—28no—
253C4×Q8—28no—
254C4×Q8—44no—
255C4×Q8—44no—
256C4×Q8—44no—
257C4×Q8—44no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—216no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—48no—
282C2×C4⋊C4—48no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4Q1618yescomplemented
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42#4112yes—
316#107—28no—
317C4⋊C8—28no—
318#103—28no—
319#103—28no—
320#103—216no—
321#103—48no—
322#103—48no—
323#103—48no—
324C4×C8—28no—
325#95—216no—
326#95—24no—
327#95—48no—
328#95—84no—
329C24C22×D411yescharacteristic
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—24no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—24no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—24no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—22no—
363C2×Q8—24no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—41no—
367C2×Q8—42no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×Q8—41no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8—42no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—41no—
383C2×Q8—42no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—41no—
387C2×Q8—41no—
388C2×Q8—41no—
389C2×Q8—41no—
390C2×Q8—41no—
391C2×Q8—41no—
392C2×Q8—41no—
393C2×Q8—41no—
394C2×Q8—41no—
395C2×Q8—41no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8#12114yes—
413C2×Q8#12114yes—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4#12012yes—
459C22×C4#12012yes—
460C22×C4#12112yes—
461C22×C4#12112yes—
462C22×C4#12712yes—
463C22×C4#12712yes—
464C22×C4C2×Q1618yes—
465C22×C4C2×Q1614yes—
466C22×C4#13714yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C2×#4111yescharacteristic
474C22×C4C2×#4111yescharacteristic
475C22×C4C2×#4111yescharacteristic
476C22×C4C2×#4111yescharacteristic
477C22×C4H(5,𝔽2)11yescharacteristic
478C22×C4H(5,𝔽2)11yescharacteristic
479C22×C4H(5,𝔽2)11yescharacteristic
480C22×C42−1+411yescharacteristic
481Q16—232no—
482Q16—432no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—42no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—42no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—42no—
502C2×C8—42no—
503C2×C8#12114yes—
504C4⋊C4—216no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—432no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—48no—
529C42—48no—
530C42#12014yes—
531C42#12014yes—
532C42#12114yes—
533C42#12114yes—
534C23C2×#12011yescharacteristic
535C23C2×#12111yescharacteristic
536C23C2×#12711yescharacteristic
537C23D4211yescharacteristic
538C23#54611yescharacteristic
539C23#54811yescharacteristic
540C23#55011yescharacteristic
541C23C22×Q1612yes—
542C23C2×#13712yes—
543C23#57712yes—
544C23#57812yes—
545Q8—24no—
546Q8—24no—
547Q8—24no—
548Q8—24no—
549Q8—42no—
550Q8—42no—
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—42no—
557Q8—42no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—44no—
562Q8—44no—
563Q8—44no—
564Q8—44no—
565Q8—82no—
566Q8—82no—
567Q8—82no—
568Q8—82no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—84no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—84no—
615C2×C4—82no—
616C2×C4#39214yes—
617C2×C4#39214yes—
618C2×C4#45118yes—
619C2×C4#46218yes—
620C2×C4#49414yes—
621C2×C4#49714yes—
622C2×C4C2×#12011yescharacteristic
623C2×C4C2×#12011yescharacteristic
624C2×C4C2×#12111yescharacteristic
625C2×C4C2×#12111yescharacteristic
626C2×C4C2×#12111yescharacteristic
627C2×C4C2×#12111yescharacteristic
628C2×C4C2×#12111yescharacteristic
629C2×C4#52511yescharacteristic
630C2×C4#52511yescharacteristic
631C2×C4#52511yescharacteristic
632C2×C4#53511yescharacteristic
633C2×C4#53511yescharacteristic
634C2×C4#53511yescharacteristic
635C2×C4#53711yescharacteristic
636C2×C4#53711yescharacteristic
637C2×C4#53711yescharacteristic
638C2×C4D4211yescharacteristic
639C2×C4D4211yescharacteristic
640C2×C4#54611yescharacteristic
641C2×C4#54611yescharacteristic
642C2×C4#54611yescharacteristic
643C2×C4#54711yescharacteristic
644C2×C4#54711yescharacteristic
645C2×C4#54811yescharacteristic
646C2×C4#55011yescharacteristic
647C2×C4#55311yescharacteristic
648C2×C4#55311yescharacteristic
649C2×C4#55511yescharacteristic
650C8—24no—
651C8—42no—
652C8—44no—
653C22C2×#39211yescharacteristic
654C22#240711yescharacteristic
655C22#240811yescharacteristic
656C22#241411yescharacteristic
657C22#241611yescharacteristic
658C22#254011yescharacteristic
659C22#254111yescharacteristic
660C22C2×#45112yes—
661C22#301312yes—
662C22C2×#46212yes—
663C22#304612yes—
664C22C2×#49411yescharacteristic
665C22C2×#49711yescharacteristic
666C22#315411yescharacteristic
667C22#315611yescharacteristic
668C22#327812yes—
669C22D4×Q1612yes—
670C22#329912yes—
671C22#332812yes—
672C22#336312yes—
673C22#337412yes—
674C22#339512yes—
675C22#340012yes—
676C4—24no—
677C4—44no—
678C4—44no—
679C4—48no—
680C4—84no—
681C4—82no—
682C4—82no—
683C4—162no—
684C4#201618yescomplemented
685C4#240711yescharacteristic
686C4#240811yescharacteristic
687C4#241411yescharacteristic
688C4#241611yescharacteristic
689C4#253511yescharacteristic
690C4#253711yescharacteristic
691C4#261511yescharacteristic
692C4#284711yescharacteristic
693C2#1512311yescharacteristic
694C2C2×#201611yescharacteristic
695C2#2342711yescharacteristic
696C2#2419112yes—
697C2#2424312yes—
698C2#2463212yes—
699C2#2473212yes—
700C2#2840111yescharacteristic
701C2#2841711yescharacteristic
702C2#2846211yescharacteristic
703C2#2847311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
295. C2×C4⋊C4Q16semidirect
684. C4#2016semidirect

Maximal subgroups

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

subgroupindex
C22×#4622
C22×C4×Q162
C2×#25462
C2×#20162
C2×#19562
C2×#18032
#139262
#111242
#110242
#110102
#108342