Finite Group Databasebeta

← #364669

Subgroups — #364669

Order 512.

2142 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×D4C211yesmaximal, characteristic
2C2×#2486C211yesmaximal, characteristic
3C2×#1823C211yesmaximal, characteristic, complemented
4C22×#340C211yesmaximal, characteristic, complemented
5#13872C211yesmaximal, characteristic
6#11227C212yesmaximal, complemented
7#11199C212yesmaximal, complemented
8#10880C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic, complemented
10#10721C212yesmaximal, complemented
11#10721C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#123—21no—
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×C8×D4—24no—
19C2×C8×D4—24no—
20C2×C8×D4—24no—
21C22×C4⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2486—24no—
26#2486C418yescomplemented
27C2×#398—21no—
28C2×#397—21no—
29C2×#389C2212yes—
30C2×#386—21no—
31C2×#381—21no—
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×#380C2212yes—
36#1823—24no—
37C2×#340—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46#1441—21no—
47#1439—21no—
48C4⋊#95C2212yes—
49C22⋊#95—21no—
50C23×D4D411yescharacteristic
51C24×C4—21no—
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C24×C4C2311yescharacteristic
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—24no—
61C23×C8—24no—
62C23×C8—42no—
63C23×C8—42no—
64C23×C8D414yes—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C2×#123—28no—
70C2×#123—44no—
71C2×#123—44no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4C2×C414yes—
77C2×C4×D4D414yes—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4C2×C414yes—
91C22×C4⋊C4D411yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4—22no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4—22no—
100C22×C22⋊C4D411yescharacteristic
101C22×C22⋊C4D411yescharacteristic
102C22×C42C2311yescharacteristic
103C8×D4—232no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8D414yes—
113C2×C4×C8—22no—
114C2×C4×C8—22no—
115C2×C4×C8—22no—
116#398—28no—
117#397—28no—
118#389—24no—
119#389C2×C418yescomplemented
120#386—44no—
121#381—28no—
122#380—22no—
123#380—24no—
124#380—22no—
125#380—22no—
126#380—24no—
127#380—42no—
128#380C2×C414yes—
129#380C2×C418yescomplemented
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—21no—
136C2×#95—22no—
137#340—24no—
138#340—24no—
139#340—24no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—42no—
144#340—42no—
145C25C2×D411yescharacteristic
146C25C2×D411yescharacteristic
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4C22⋊C414yes—
152C22×D4D814yes—
153C22×D4SD1614yes—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—44no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—82no—
182C23×C4C22×C412yes—
183C23×C4C22×C412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D412yes—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D412yes—
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4#4112yes—
193C23×C4#4112yes—
194C23×C4C2411yescharacteristic
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—22no—
198C22×C8—28no—
199C22×C8—28no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—28no—
213C22×C8—44no—
214C22×C8—48no—
215C22×C8—48no—
216C22×C8—48no—
217C22×C8—48no—
218C22×C8—44no—
219C22×C8—44no—
220C22×C8C2×D412yes—
221C22×C8C2×D412yes—
222C22×C8C2×D412yes—
223C22×C8C2×D412yes—
224C22×C8#4114yes—
225#123—432no—
226C4×D4—28no—
227C4×D4—28no—
228C4×D4—28no—
229C4×D4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
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—48no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4C22⋊C414yes—
269C2×C4⋊C4C22×C414yes—
270C2×C4⋊C4C22×C414yes—
271C2×C4⋊C4#4114yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—82no—
288C2×C22⋊C4C22⋊C414yes—
289C2×C22⋊C4C22⋊C414yes—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42C22×C412yes—
295C2×C42C2×D412yes—
296C4⋊C8—28no—
297C22⋊C8—28no—
298C22⋊C8—28no—
299C4×C8—28no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—44no—
304#95—42no—
305#95—44no—
306#95—82no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—22no—
315C24—21no—
316C24—21no—
317C24—22no—
318C24—22no—
319C24C2×C22⋊C412yes—
320C24C2×C22⋊C412yes—
321C24#12112yes—
322C24#12112yes—
323C24C2×D812yes—
324C24C2×D812yes—
325C24C2×SD1612yes—
326C24C2×SD1612yes—
327C24#13512yes—
328C24#13512yes—
329C24C22×D411yescharacteristic
330C2×D4—28no—
331C2×D4—28no—
332C2×D4—28no—
333C2×D4—28no—
334C2×D4—28no—
335C2×D4—28no—
336C2×D4#102116yes—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—48no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—88no—
416C22×C4—88no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4C2×C22⋊C412yes—
424C22×C4C2×C22⋊C412yes—
425C22×C4C2×C22⋊C412yes—
426C22×C4C2×C22⋊C412yes—
427C22×C4C4×D414yes—
428C22×C4C4×D414yes—
429C22×C4#12014yes—
430C22×C4#12112yes—
431C22×C4#12112yes—
432C22×C4#12314yes—
433C22×C4C2×D812yes—
434C22×C4C2×SD1612yes—
435C22×C4C23×C411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C42−1+412yes—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—416no—
457C2×C8—416no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8#12314yescomplemented
467C4⋊C4—28no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—48no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—48no—
482C4⋊C4—416no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C22⋊C4—28no—
488C22⋊C4—28no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C22⋊C4—88no—
492C22⋊C4—84no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23#38614yes—
536C23C2×#10214yes—
537C23C2×#10214yes—
538C23#41614yes—
539C23#46514yes—
540C23#46514yes—
541C23#46614yes—
542C23#46614yes—
543C23C22×C22⋊C411yescharacteristic
544C23C2×#12111yescharacteristic
545C23C2×#12111yescharacteristic
546C23#54612yes—
547C23#54712yes—
548C23C22×D811yescharacteristic
549C23C22×SD1611yescharacteristic
550C23C2×#13511yescharacteristic
551C23C2×#13511yescharacteristic
552C23#57712yes—
553C23#57812yes—
554D4—232no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—816no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—816no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4#38614yes—
618C2×C4C2×#10214yes—
619C2×C4C2×C4×D411yescharacteristic
620C2×C4C2×C4×D411yescharacteristic
621C2×C4C4×#4112yes—
622C2×C4#52012yes—
623C2×C4C2×#12011yescharacteristic
624C2×C4C2×#12311yescharacteristic
625C2×C4C2×#12311yescharacteristic
626C2×C4#52511yescharacteristic
627C2×C4#52912yes—
628C2×C4#53411yescharacteristic
629C2×C4#53611yescharacteristic
630C2×C4#53912yes—
631C2×C4#54612yes—
632C2×C4#54612yes—
633C2×C4#54712yes—
634C2×C4#54712yes—
635C2×C4#55112yes—
636C2×C4#55212yes—
637C8—24no—
638C8—42no—
639C8—42no—
640C8—42no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—28no—
645C22—28no—
646C22—24no—
647C22—24no—
648C22—28no—
649C22—28no—
650C22—24no—
651C22—28no—
652C22—24no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—44no—
658C22—42no—
659C22—42no—
660C22—44no—
661C22—42no—
662C22#189318yes—
663C22C2×#38611yescharacteristic
664C22#233912yes—
665C22#242512yes—
666C22#248611yescharacteristic
667C22#253411yescharacteristic
668C22C22×#10211yescharacteristic
669C22C2×#41611yescharacteristic
670C22#289812yes—
671C22C2×#46511yescharacteristic
672C22C2×#46511yescharacteristic
673C22C2×#46611yescharacteristic
674C22C2×#46611yescharacteristic
675C22#305711yescharacteristic
676C22#305911yescharacteristic
677C22#306312yes—
678C22#329412yes—
679C22#329612yes—
680C22#329812yes—
681C22#330012yes—
682C22#333412yes—
683C22#333512yes—
684C22#333612yes—
685C22#333712yes—
686C4—24no—
687C4—24no—
688C4—42no—
689C4—42no—
690C4—84no—
691C4—84no—
692C4—84no—
693C4—162no—
694C4C4×#12311yescharacteristic
695C4#232211yescharacteristic
696C4#235912yes—
697C4#240511yescharacteristic
698C4#240611yescharacteristic
699C4#241912yes—
700C2—28no—
701C2—44no—
702C2#1396811yescharacteristic
703C2C2×#189311yescharacteristic
704C2#2097411yescharacteristic
705C2#2248512yes—
706C2#2248612yes—
707C2#2523212yes—
708C2#2523612yes—
709C2#2765811yescharacteristic
710C2#2766111yescharacteristic
711C2#2809411yescharacteristic
712C2#2809811yescharacteristic
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×#1823C2semidirect
4. C22×#340C2semidirect
6. #11227C2semidirect
7. #11199C2semidirect
8. #10880C2semidirect
9. #10730C2semidirect
10. #10721C2semidirect
11. #10721C2semidirect
26. #2486C4semidirect
119. #389C2×C4semidirect
129. #380C2×C4semidirect
466. C2×C8#123semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D42
C2×#24862
C2×#18232
C22×#3402
#138722
#112272
#111992
#108802
#107302
#107212