Finite Group Databasebeta

← #369265

Subgroups — #369265

Order 512.

1760 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#489C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3C2×#1993C211yesmaximal, characteristic, complemented
4C2×#1959C211yesmaximal, characteristic
5#14708C211yesmaximal, characteristic
6#11067C212yesmaximal, complemented
7#11066C212yesmaximal
8#10932C212yesmaximal, complemented
9#10931C212yesmaximal
10#10843C211yesmaximal, characteristic, complemented
11#10693C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×C4×Q8—21no—
15C2×#489—24no—
16C2×#489—24no—
17C2×#489C418yescomplemented
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×#97C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C4×C4⋊Q8—24no—
25C4×C4⋊Q8—24no—
26C2×#391—21no—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#389—21no—
32C2×#389C2212yes—
33C2×#384C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×C4×C4⋊C4—21no—
36C2×C4×C4⋊C4C2212yes—
37C2×C43C2211yescharacteristic
38#1993—24no—
39#1959—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1437—21no—
47#1437—21no—
48#1437—21no—
49#1437—21no—
50C23×Q8—21no—
51C23×Q8C2311yescharacteristic
52C23×D4C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8D414yes—
60C2×C4⋊Q8D414yes—
61C2×#124—22no—
62C2×#124—22no—
63C2×#124—22no—
64C2×#124—22no—
65C2×#124—22no—
66C2×#124C2×C414yes—
67C2×C4×Q8—28no—
68C2×C4×Q8—44no—
69C2×C4×Q8—44no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C42—22no—
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#489—232no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—42no—
98C2×#103C2×C418yes—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—42no—
104C2×#102C2×C418yes—
105C2×#97—22no—
106C2×#97—22no—
107C2×#97—22no—
108C2×#97—22no—
109C2×#97—22no—
110C2×#97C2×C414yes—
111#391—28no—
112#391—24no—
113#391—42no—
114#390—24no—
115#390—42no—
116#389—28no—
117#389—24no—
118#389—24no—
119#384—22no—
120#384D414yes—
121#382—22no—
122#382—22no—
123C4×C4⋊C4—28no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—24no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C43—22no—
135C43—22no—
136#340—24no—
137#340—24no—
138#340—42no—
139#340—42no—
140#336—24no—
141#336—24no—
142C25—21no—
143C22×Q8—24no—
144C22×Q8—24no—
145C22×Q8—24no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8C22×C412yes—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—21no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4C22×C412yes—
170C23×C4—22no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2411yescharacteristic
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8C22×C414yes—
195C22×C8#4114yes—
196C4⋊Q8—28no—
197C4⋊Q8—28no—
198C4⋊Q8—28no—
199C4⋊Q8C22⋊C4116yescomplemented
200#124—28no—
201#124—28no—
202#124—44no—
203#124—44no—
204C4×Q8—432no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C22×C414yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2×D414yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42—28no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—41no—
272C2×C42—41no—
273C2×C42—41no—
274C2×C42—41no—
275C2×C42—44no—
276C2×C42—44no—
277C2×C42C22×C412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42#4114yes—
282C2×C42#4112yes—
283C2×C42#4112yes—
284#103—216no—
285#103—48no—
286#102—216no—
287#102—48no—
288#97—28no—
289#97—44no—
290#95—28no—
291#95—28no—
292#95—42no—
293#95—42no—
294#95—42no—
295#95—42no—
296#95—42no—
297#95—42no—
298C24—22no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24—41no—
310C24—41no—
311C24—41no—
312C24—41no—
313C24C22×D411yescharacteristic
314C2×Q8—216no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—48no—
320C2×Q8—48no—
321C2×Q8—48no—
322C2×Q8—48no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4C4×D412yes—
419C22×C4C4×D412yes—
420C22×C4#12012yes—
421C22×C4#12012yes—
422C22×C4#12112yes—
423C22×C4#12112yes—
424C22×C4#12412yes—
425C22×C4#12412yes—
426C22×C4AGL(1,ℤ/8)12yes—
427C22×C4AGL(1,ℤ/8)12yes—
428C22×C4#13712yes—
429C22×C4#13712yes—
430C22×C4C23×C411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4111yescharacteristic
438C22×C4H(5,𝔽2)12yes—
439C22×C42−1+412yes—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—216no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—416no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4C2×C22⋊C418yes—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—84no—
473C22⋊C4—84no—
474C22⋊C4—84no—
475C22⋊C4—84no—
476C42—28no—
477C42—28no—
478C42—24no—
479C42—24no—
480C42—44no—
481C42—44no—
482C42—416no—
483C42—42no—
484C42—42no—
485C42C2×C22⋊C414yes—
486C42C4×D414yes—
487C42#12114yes—
488C42#12314yes—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—81no—
497C23—81no—
498C23—81no—
499C23—81no—
500C23—81no—
501C23—81no—
502C23—81no—
503C23—81no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23—81no—
508C23—81no—
509C23—81no—
510C23—81no—
511C23—81no—
512C23—81no—
513C23—81no—
514C23—81no—
515C23—81no—
516C23—81no—
517C23—81no—
518C23—81no—
519C23—81no—
520C23—81no—
521C23—81no—
522C23—81no—
523C23—81no—
524C23C2×C4×D411yescharacteristic
525C23C2×#12011yescharacteristic
526C23C2×#12111yescharacteristic
527C23C2×#12411yescharacteristic
528C23#53211yescharacteristic
529C23#53511yescharacteristic
530C23#53611yescharacteristic
531C23C2×AGL(1,ℤ/8)11yescharacteristic
532C23C2×AGL(1,ℤ/8)11yescharacteristic
533C23C2×#13711yescharacteristic
534C23C2×#13711yescharacteristic
535C23#57511yescharacteristic
536C23#57511yescharacteristic
537C23#57511yescharacteristic
538C23#57511yescharacteristic
539Q8—432no—
540Q8—44no—
541Q8—44no—
542Q8—82no—
543Q8—82no—
544D4—44no—
545D4—44no—
546D4—82no—
547D4—82no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—28no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4#38614yes—
623C2×C4#38614yes—
624C2×C4#48914yescomplemented
625C2×C4C22×C22⋊C411yescharacteristic
626C2×C4C2×C4×D411yescharacteristic
627C2×C4C4×#4112yes—
628C2×C4#52012yes—
629C2×C4C2×#12111yescharacteristic
630C2×C4C2×#12311yescharacteristic
631C2×C4#52912yes—
632C2×C4#54212yes—
633C2×C4#54612yes—
634C2×C4#54812yes—
635C2×C4#54812yes—
636C2×C4#54812yes—
637C2×C4#55012yes—
638C2×C4#55012yes—
639C2×C4#55612yes—
640C2×C4#55912yes—
641C8—44no—
642C8—44no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—161no—
651C22—161no—
652C22—161no—
653C22—161no—
654C22—161no—
655C22—161no—
656C22—161no—
657C22—161no—
658C22C2×#38611yescharacteristic
659C22C4×#12411yescharacteristic
660C22#232611yescharacteristic
661C22#243111yescharacteristic
662C22#243611yescharacteristic
663C22#254511yescharacteristic
664C22#254811yescharacteristic
665C22C4×AGL(1,ℤ/8)11yescharacteristic
666C22C4×AGL(1,ℤ/8)11yescharacteristic
667C22C4×#13711yescharacteristic
668C22C4×#13711yescharacteristic
669C22#300311yescharacteristic
670C22#300411yescharacteristic
671C22#300511yescharacteristic
672C22#300511yescharacteristic
673C22#303611yescharacteristic
674C22#303611yescharacteristic
675C22#303711yescharacteristic
676C22#303711yescharacteristic
677C22C2×#48911yescharacteristic
678C22#313511yescharacteristic
679C22#314011yescharacteristic
680C22#314111yescharacteristic
681C22#317111yescharacteristic
682C22#317211yescharacteristic
683C22#318011yescharacteristic
684C22#318111yescharacteristic
685C22#321311yescharacteristic
686C22#321411yescharacteristic
687C22#321511yescharacteristic
688C22#321611yescharacteristic
689C22#323811yescharacteristic
690C22#323911yescharacteristic
691C22#324011yescharacteristic
692C22#324011yescharacteristic
693C4—24no—
694C4—24no—
695C4—24no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—48no—
700C4—84no—
701C4—82no—
702C4—161no—
703C4—161no—
704C4#234112yes—
705C4#236712yes—
706C4#242612yes—
707C4#244812yes—
708C2—162no—
709C2#1402611yescharacteristic
710C2#2092511yescharacteristic
711C2#2092611yescharacteristic
712C2C4×#48911yescharacteristic
713C2#2165511yescharacteristic
714C2#2222311yescharacteristic
715C2#2222411yescharacteristic
716C2#2560211yescharacteristic
717C2#2560311yescharacteristic
718C2#2590811yescharacteristic
719C2#2590911yescharacteristic
720C2#2865811yescharacteristic
721C2#2865911yescharacteristic
722C2#2875311yescharacteristic
723C2#2875411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Q8C2semidirect
3. C2×#1993C2semidirect
6. #11067C2semidirect
8. #10932C2semidirect
10. #10843C2semidirect
11. #10693C2semidirect
17. C2×#489C4semidirect
199. C4⋊Q8C22⋊C4semidirect
624. C2×C4#489semidirect

Maximal subgroups

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

subgroupindex
C22×#4892
C2×C4×C4⋊Q82
C2×#19932
C2×#19592
#147082
#110672
#110662
#109322
#109312
#108432
#106932