Finite Group Databasebeta

← #364896

Subgroups — #364896

Order 512.

1934 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#461C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3C2×#1960C211yesmaximal, characteristic, complemented
4C2×#1880C212yesmaximal
5C2×#1762C211yesmaximal, characteristic
6#13941C211yesmaximal, characteristic
7#11190C212yesmaximal, complemented
8#11023C212yesmaximal
9#10850C211yesmaximal, characteristic, complemented
10#10835C211yesmaximal, characteristic, complemented
11#10789C212yesmaximal, complemented
12C23×SD16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C22×C4×D4C2211yescharacteristic
16C2×#461—24no—
17C2×#461—24no—
18C2×#461C418yescomplemented
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#97C2211yescharacteristic
26#2365—24no—
27#2365—24no—
28C2×#391C2212yes—
29C2×#390C2212yes—
30C2×#389—21no—
31C2×#387—22no—
32C2×#382C2211yescharacteristic
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1960—24no—
37#1880—28no—
38#1762—24no—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1439—21no—
46#1439—21no—
47C4⋊#95C2212yes—
48C22⋊#95—22no—
49C23×Q8C2312yes—
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C22×SD16—24no—
53C22×SD16—24no—
54C22×SD16—24no—
55C22×SD16—24no—
56C22×SD16—42no—
57C22×SD16—42no—
58C22×SD16C2×C418yes—
59C23×C8C2312yes—
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8C2×C414yes—
66C2×C4⋊Q8D414yes—
67C2×C4⋊Q8D414yes—
68C2×C4×Q8—24no—
69C2×C4×Q8—24no—
70C2×C4×Q8—24no—
71C2×C4×Q8—24no—
72C2×C4×Q8—42no—
73C2×C4×Q8—42no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4C2×C414yes—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D411yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C22⋊C4—21no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—22no—
92C22×C42—22no—
93C22×C42—21no—
94C22×C42C2312yes—
95C22×C42C2311yescharacteristic
96#461—232no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8C2×C414yes—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—42no—
107C2×#103—42no—
108C2×#103C2×C418yes—
109C2×#102—22no—
110C2×#102—22no—
111C2×#102—22no—
112C2×#102—22no—
113C2×#102—22no—
114C2×#102D414yes—
115C2×#97—22no—
116C2×#97—22no—
117C2×#97—22no—
118#391—24no—
119#391—24no—
120#390—24no—
121#390—42no—
122#389—28no—
123#387—216no—
124#382—22no—
125#382—22no—
126C4×C4⋊C4—28no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4D414yes—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95D412yes—
139#340—24no—
140#340—22no—
141#340—22no—
142#340—42no—
143#336—24no—
144#336—42no—
145C25—21no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8C22×C414yes—
158C22×Q8C2×D414yes—
159C22×Q8C2×D414yes—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4C22×C412yes—
172C22×D4C2×D412yes—
173C22×D4#4112yes—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—82no—
183C23×C4—82no—
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2411yescharacteristic
189C2×SD16—216no—
190C2×SD16—216no—
191C2×SD16—48no—
192C2×SD16—48no—
193C2×SD16—48no—
194C2×SD16—48no—
195C2×SD16—48no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8C22×C414yes—
215C22×C8C2×D412yes—
216C22×C8#4112yes—
217C4⋊Q8—28no—
218C4⋊Q8—28no—
219C4⋊Q8—28no—
220C4⋊Q8C22⋊C4116yescomplemented
221C4×Q8—48no—
222C4×Q8—48no—
223C4×Q8—48no—
224C4×Q8—48no—
225C4×D4—28no—
226C4×D4—28no—
227C4×D4—44no—
228C4×D4—44no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4#4112yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—82no—
274C2×C22⋊C4—82no—
275C2×C42—28no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42C22×C412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42#4112yes—
291C2×C42#4112yes—
292C4⋊C8—28no—
293#103—216no—
294#103—48no—
295#103—48no—
296#102—28no—
297#102—28no—
298#97—44no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—42no—
304#95—42no—
305#95—82no—
306#95—82no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—41no—
311C24—41no—
312C24—41no—
313C24—41no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24C22×D411yescharacteristic
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8C2×C22⋊C418yes—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4C4×D414yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—48no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—88no—
423C22×C4—84no—
424C22×C4—84no—
425C22×C4—88no—
426C22×C4C4×D412yes—
427C22×C4C4×D412yes—
428C22×C4#12012yes—
429C22×C4#12012yes—
430C22×C4#12112yes—
431C22×C4#12112yes—
432C22×C4#13714yes—
433C22×C4C23×C411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4H(5,𝔽2)11yescharacteristic
444C22×C4H(5,𝔽2)11yescharacteristic
445C22×C4H(5,𝔽2)11yescharacteristic
446C22×C42−1+411yescharacteristic
447SD16—432no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—41no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C4⋊C4—28no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—44no—
480C4⋊C4—416no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4#12114yes—
490C22⋊C4—48no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C42—48no—
494C42—416no—
495C42—42no—
496C42—42no—
497C42—42no—
498C42—42no—
499C42C2×C22⋊C414yes—
500C42C4×D414yes—
501C42#12114yes—
502C42#12314yes—
503C23—24no—
504C23—44no—
505C23—44no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—82no—
517C23—82no—
518C23—81no—
519C23—81no—
520C23—81no—
521C23—81no—
522C23—82no—
523C23—82no—
524C23—82no—
525C23—82no—
526C23C2×C4×D411yescharacteristic
527C23C2×#12011yescharacteristic
528C23C2×#12111yescharacteristic
529C23D4211yescharacteristic
530C23#54611yescharacteristic
531C23#54811yescharacteristic
532C23#55011yescharacteristic
533C23C2×#13712yes—
534C23#57512yes—
535C23#57712yes—
536C23#57712yes—
537Q8—28no—
538Q8—28no—
539Q8—44no—
540Q8—44no—
541Q8—44no—
542Q8—44no—
543Q8—44no—
544Q8—44no—
545D4—24no—
546D4—24no—
547D4—42no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—42no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—816no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4#38614yes—
606C2×C4#38614yes—
607C2×C4C22×C22⋊C411yescharacteristic
608C2×C4C2×C4×D411yescharacteristic
609C2×C4C2×C4×D411yescharacteristic
610C2×C4C4×#4111yescharacteristic
611C2×C4#51811yescharacteristic
612C2×C4#52011yescharacteristic
613C2×C4C2×#12111yescharacteristic
614C2×C4C2×#12111yescharacteristic
615C2×C4C2×#12311yescharacteristic
616C2×C4#52511yescharacteristic
617C2×C4#52911yescharacteristic
618C2×C4#53211yescharacteristic
619C2×C4#53511yescharacteristic
620C2×C4#53711yescharacteristic
621C2×C4#54011yescharacteristic
622C2×C4#54211yescharacteristic
623C2×C4D4211yescharacteristic
624C2×C4D4211yescharacteristic
625C2×C4#54611yescharacteristic
626C2×C4#54611yescharacteristic
627C2×C4#54611yescharacteristic
628C2×C4#54611yescharacteristic
629C2×C4#54711yescharacteristic
630C2×C4#54711yescharacteristic
631C2×C4#54711yescharacteristic
632C2×C4#55311yescharacteristic
633C2×C4#55311yescharacteristic
634C2×C4#55511yescharacteristic
635C8—44no—
636C8—42no—
637C8—42no—
638C22—44no—
639C22—48no—
640C22—84no—
641C22—84no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—84no—
649C22C2×#38611yescharacteristic
650C22#236111yescharacteristic
651C22#236311yescharacteristic
652C22#240711yescharacteristic
653C22#240811yescharacteristic
654C22#242711yescharacteristic
655C22#242811yescharacteristic
656C22C4×#13712yes—
657C22#294912yes—
658C22#300411yescharacteristic
659C22#300511yescharacteristic
660C22#301011yescharacteristic
661C22#301011yescharacteristic
662C22#303712yes—
663C22#304312yes—
664C22#326612yes—
665C22#326712yes—
666C22#330712yes—
667C22#332012yes—
668C22#337212yes—
669C22#337612yes—
670C22#340012yes—
671C22#340112yes—
672C4—24no—
673C4—24no—
674C4—48no—
675C4—44no—
676C4—44no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—162no—
681C4C4×#12111yescharacteristic
682C4#232411yescharacteristic
683C4D4×C22⋊C411yescharacteristic
684C4#233911yescharacteristic
685C4#242011yescharacteristic
686C4#242211yescharacteristic
687C4#253511yescharacteristic
688C4#253711yescharacteristic
689C2—84no—
690C2#1397011yescharacteristic
691C2#2092611yescharacteristic
692C2#2093611yescharacteristic
693C2#2301212yes—
694C2#2311212yes—
695C2#2419711yescharacteristic
696C2#2419911yescharacteristic
697C2#2424911yescharacteristic
698C2#2425111yescharacteristic
699C2#2533112yes—
700C2#2537712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×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
2. C2×#2365C2semidirect
3. C2×#1960C2semidirect
7. #11190C2semidirect
9. #10850C2semidirect
10. #10835C2semidirect
11. #10789C2semidirect
18. C2×#461C4semidirect
220. C4⋊Q8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4612
C2×#23652
C2×#19602
C2×#18802
C2×#17622
#139412
#111902
#110232
#108502
#108352
#107892