Finite Group Databasebeta

← #369211

Subgroups — #369211

Order 512.

2144 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#435C211yesmaximal, characteristic
2C2×#2542C211yesmaximal, characteristic
3C2×#1981C211yesmaximal, characteristic, complemented
4C2×#1960C211yesmaximal, characteristic, complemented
5C2×#1894C212yesmaximal, complemented
6C2×#1892C212yesmaximal, complemented
7#13905C211yesmaximal, characteristic
8#10970C212yesmaximal, complemented
9#10970C212yesmaximal, complemented
10#10850C211yesmaximal, characteristic, complemented
11#10703C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×C22⋊Q8C2212yes—
17C22×C4×D4C2211yescharacteristic
18C2×#435—24no—
19C2×#435—24no—
20C2×#435—24no—
21C22×C4⋊C8C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×#103C2212yes—
25C22×#103C2212yes—
26C22×C22⋊C8C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28C22×#97C2211yescharacteristic
29#2542—24no—
30#2542C418yescomplemented
31C2×#399—21no—
32C2×#399—21no—
33C2×#390—21no—
34C2×#384—21no—
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40#1981—24no—
41#1960—24no—
42#1894—28no—
43#1892—28no—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48#1445—21no—
49#1438—21no—
50#1437—21no—
51#1437—21no—
52C23×Q8C2312yes—
53C23×D4D411yescharacteristic
54C24×C4C2311yescharacteristic
55C24×C4C2311yescharacteristic
56C22×M16—22no—
57C22×M16—22no—
58C22×M16—22no—
59C22×M16—24no—
60C22×M16—42no—
61C23×C8—22no—
62C23×C8—22no—
63C23×C8—22no—
64C23×C8—24no—
65C23×C8—42no—
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8C2×C414yes—
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—42no—
83C2×C22⋊Q8—42no—
84C2×C22⋊Q8—42no—
85C2×C22⋊Q8—42no—
86C2×C22⋊Q8C2×C418yes—
87C2×C22⋊Q8C2×C418yes—
88C2×C4×D4—22no—
89C2×C4×D4—22no—
90C2×C4×D4—22no—
91C2×C4×D4—22no—
92C2×C4×D4C2×C414yes—
93C2×C4×D4D414yes—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4D411yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C22⋊C4—22no—
103C22×C22⋊C4—22no—
104C22×C22⋊C4—22no—
105C22×C22⋊C4—22no—
106C22×C22⋊C4D411yescharacteristic
107C22×C22⋊C4D411yescharacteristic
108C22×C42—21no—
109C22×C42—21no—
110C22×C42C2311yescharacteristic
111#435—232no—
112C2×C4⋊C8—22no—
113C2×C4⋊C8—22no—
114C2×C4⋊C8D414yes—
115C2×#103—24no—
116C2×#103—24no—
117C2×#103—24no—
118C2×#103—24no—
119C2×#103—24no—
120C2×#103—24no—
121C2×#103—24no—
122C2×#103—24no—
123C2×#103—42no—
124C2×#103—42no—
125C2×#103—42no—
126C2×#103—42no—
127C2×C22⋊C8—22no—
128C2×C22⋊C8—22no—
129C2×C22⋊C8—22no—
130C2×C22⋊C8—22no—
131C2×C22⋊C8—22no—
132C2×C22⋊C8—22no—
133C2×#97—22no—
134C2×#97—22no—
135C2×#97D414yes—
136#399—28no—
137#399—28no—
138#390—28no—
139#384—28no—
140#382—22no—
141#382—22no—
142#380—24no—
143#380—24no—
144#380—42no—
145#380—42no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4—22no—
148C4×C4⋊C4—22no—
149C4×C4⋊C4C2×C414yes—
150C2×#95—21no—
151C2×#95—21no—
152C2×#95—22no—
153C2×#95—22no—
154C2×#95—21no—
155C2×#95—21no—
156#340—24no—
157#340—24no—
158#340—24no—
159#340D418yescomplemented
160C25C2×D411yescharacteristic
161C25C2×D411yescharacteristic
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8C22×C414yes—
176C22×D4—24no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—42no—
180C22×D4C22⋊C414yes—
181C23×C4—22no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4C22×C412yes—
202C23×C4C22×C412yes—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D412yes—
206C23×C4C2×D412yes—
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2411yescharacteristic
210C2×M16—28no—
211C2×M16—48no—
212C2×M16—48no—
213C2×M16—44no—
214C2×M16—44no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—22no—
218C22×C8—28no—
219C22×C8—22no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—22no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—48no—
232C22×C8—48no—
233C22×C8—44no—
234C22×C8—44no—
235C22×C8C2×D412yes—
236C22×C8C2×D412yes—
237C22×C8C2×D412yes—
238C22×C8C2×D414yes—
239C22×C8C2×D412yes—
240C4⋊Q8—28no—
241C4⋊Q8—28no—
242C4⋊Q8—44no—
243C4⋊Q8—44no—
244C22⋊Q8—216no—
245C22⋊Q8—216no—
246C22⋊Q8—48no—
247C22⋊Q8—48no—
248C22⋊Q8—48no—
249C22⋊Q8—48no—
250C22⋊Q8—48no—
251C22⋊Q8—48no—
252C4×D4—28no—
253C4×D4—28no—
254C4×D4—28no—
255C4×D4—28no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4C22⋊C414yes—
285C2×C4⋊C4C22×C414yes—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4#4114yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—82no—
304C2×C22⋊C4—82no—
305C2×C22⋊C4C22⋊C414yes—
306C2×C22⋊C4C22⋊C414yes—
307C2×C42—24no—
308C2×C42—44no—
309C2×C42—42no—
310C2×C42—82no—
311C2×C42C22×C412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42#4112yes—
316C2×C42#4112yes—
317C4⋊C8—28no—
318#103—48no—
319#103—48no—
320#103—48no—
321#103—48no—
322C22⋊C8—28no—
323C22⋊C8—28no—
324#97—28no—
325#95—24no—
326#95—44no—
327#95—44no—
328#95—42no—
329#95—42no—
330#95—42no—
331C24—21no—
332C24—21no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—21no—
339C24—21no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24C2×C22⋊C412yes—
347C24C2×C22⋊C412yes—
348C24#13714yes—
349C24C22×D411yescharacteristic
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—24no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×D4—28no—
379C2×D4—28no—
380C2×D4—44no—
381C2×D4—48no—
382C2×D4—48no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
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—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4C2×C22⋊C412yes—
460C22×C4C2×C22⋊C412yes—
461C22×C4C2×C22⋊C412yes—
462C22×C4C2×C22⋊C412yes—
463C22×C4#12112yes—
464C22×C4#12112yes—
465C22×C4#12112yes—
466C22×C4#12112yes—
467C22×C4C23×C411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4111yescharacteristic
473C22×C4C2×#4111yescharacteristic
474C22×C4C2×#4112yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)12yes—
477M16—416no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—416no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—42no—
495C2×C8—42no—
496C2×C8—42no—
497C2×C8—42no—
498C2×C8#12714yes—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—28no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—416no—
506C4⋊C4—416no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4C4×D418yes—
516C22⋊C4—28no—
517C22⋊C4—28no—
518C22⋊C4—416no—
519C22⋊C4—416no—
520C22⋊C4—88no—
521C22⋊C4—88no—
522C42—48no—
523C42—84no—
524C42C4×D414yes—
525C42C4×D414yes—
526C42#12014yes—
527C42#12314yes—
528C23—24no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—22no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—41no—
560C23—41no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—41no—
565C23—41no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23#41914yes—
570C23C22×C22⋊C411yescharacteristic
571C23C2×#12111yescharacteristic
572C23C2×#12111yescharacteristic
573C23D4212yes—
574C23#54612yes—
575C23C2×#13712yes—
576C23#57512yes—
577C23#57712yes—
578C23#57812yes—
579Q8—44no—
580Q8—44no—
581Q8—44no—
582Q8—44no—
583Q8—82no—
584Q8—82no—
585Q8—82no—
586Q8—82no—
587D4—416no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4#38614yes—
647C2×C4#38614yes—
648C2×C4C2×C4×D411yescharacteristic
649C2×C4C2×C4×D412yes—
650C2×C4C2×C4×D411yescharacteristic
651C2×C4#51812yes—
652C2×C4C2×#12011yescharacteristic
653C2×C4C2×#12311yescharacteristic
654C2×C4C2×#12711yescharacteristic
655C2×C4#53211yescharacteristic
656C2×C4#53212yes—
657C2×C4#53511yescharacteristic
658C2×C4#53511yescharacteristic
659C2×C4#54012yes—
660C2×C4D4212yes—
661C2×C4D4212yes—
662C2×C4#54612yes—
663C2×C4#55012yes—
664C2×C4#55012yes—
665C2×C4#55912yes—
666C8—24no—
667C8—42no—
668C8—42no—
669C8—42no—
670C22—28no—
671C22—24no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—42no—
678C22—42no—
679C22—44no—
680C22—44no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22C2×#38611yescharacteristic
689C22D4×C22⋊C412yes—
690C22#242812yes—
691C22#254111yescharacteristic
692C22#254411yescharacteristic
693C22C2×#41911yescharacteristic
694C22#289611yescharacteristic
695C22#289812yes—
696C22#305612yes—
697C22#305612yes—
698C22#305711yescharacteristic
699C22#305911yescharacteristic
700C22#306312yes—
701C22#326612yes—
702C22#326712yes—
703C22#326912yes—
704C22#327012yes—
705C22#330712yes—
706C22#331212yes—
707C22#331512yes—
708C22#332012yes—
709C4—24no—
710C4—24no—
711C4—48no—
712C4—42no—
713C4—42no—
714C4—84no—
715C4—84no—
716C4—162no—
717C4C4×#12711yescharacteristic
718C4#232811yescharacteristic
719C4#236612yes—
720C4#243111yescharacteristic
721C4#243311yescharacteristic
722C4#244512yes—
723C2—44no—
724C2—82no—
725C2#1402511yescharacteristic
726C2#2097211yescharacteristic
727C2#2097411yescharacteristic
728C2#2242012yes—
729C2#2242512yes—
730C2#2539812yes—
731C2#2540012yes—
732C2#2849212yes—
733C2#2861112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1981C2semidirect
4. C2×#1960C2semidirect
5. C2×#1894C2semidirect
6. C2×#1892C2semidirect
8. #10970C2semidirect
9. #10970C2semidirect
10. #10850C2semidirect
11. #10703C2semidirect
30. #2542C4semidirect
159. #340D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4352
C2×#25422
C2×#19812
C2×#19602
C2×#18942
C2×#18922
#139052
#109702
#108502
#107032