Finite Group Databasebeta

← #615654

Subgroups — #615654

Order 512.

3958 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C212yesmaximal
2C2×#2486C212yesmaximal, complemented
3C2×C4×#121C211yesmaximal, characteristic
4C22×C4×C4⋊C4C211yesmaximal, characteristic, complemented
5#15149C212yesmaximal, complemented
6#15147C212yesmaximal, complemented
7#15081C214yesmaximal, complemented
8#14927C214yesmaximal, complemented
9#14686C211yesmaximal, characteristic, complemented
10#14685C212yesmaximal, complemented
11#14151C212yesmaximal, complemented
12#13994C214yesmaximal, complemented
13#13906C212yesmaximal, complemented
14#13872C212yesmaximal, complemented
15C22×#123C2214yes—
16C22×#121C2211yescharacteristic
17C22×C4×D4C2212yes—
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2212yes—
22C23×C42C2211yescharacteristic
23#2854—216no—
24#2486—216no—
25C2×#398C2214yes—
26C2×#397C2214yes—
27C4×#121—28no—
28C2×#390C2214yes—
29C2×#390C2211yescharacteristic
30C2×#389C2214yes—
31C2×#389C2211yescharacteristic
32C2×#389C2214yes—
33C2×#388C2218yes—
34C2×#388C2212yes—
35C2×#386C2214yes—
36C2×#386C2212yes—
37C2×#384C2212yes—
38C2×#381C2214yes—
39C2×#380C2212yes—
40C2×#380C2214yes—
41C2×#380C2212yes—
42C2×#380C2214yes—
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45C2×C4×C4⋊C4—28no—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C22×#95C2212yes—
53#1442C2218yes—
54#1441C2214yes—
55#1441C2218yes—
56#1440C2218yes—
57#1439C2214yes—
58#1439C2218yes—
59C4⋊#95C2212yes—
60C4⋊#95C2214yes—
61C4⋊#95C2214yes—
62C4⋊#95C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65#1434C2218yes—
66C22⋊#95C2214yes—
67C23×D4C2312yes—
68C23×D4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2312yes—
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C2×#123—216no—
76C2×#123—216no—
77C2×#123—48no—
78C2×#121—24no—
79C2×#121—24no—
80C2×#121D418yes—
81C2×C4×D4—28no—
82C2×C4×D4—28no—
83C2×C4×D4—28no—
84C2×C4×D4—24no—
85C2×C4×D4—28no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—28no—
90C22×C4⋊C4D418yescomplemented
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2318yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2318yes—
108C22×C22⋊C4C2318yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C42—28no—
112C22×C42—28no—
113C22×C42—28no—
114C22×C42—28no—
115C22×C42D418yes—
116C22×C42C2314yes—
117C22×C42C2314yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#398—216no—
124#397—216no—
125#390—216no—
126#390—24no—
127#389—216no—
128#389—216no—
129#389—24no—
130#388—232no—
131#388—28no—
132#386—216no—
133#386—28no—
134#384—28no—
135#381—216no—
136#380—28no—
137#380—28no—
138#380—24no—
139#380—216no—
140#380—216no—
141#380—44no—
142C4×C4⋊C4—28no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—24no—
147C4×C22⋊C4—28no—
148C2×#95—28no—
149C2×#95C2314yes—
150C2×#95C2318yes—
151C2×#95C2318yes—
152C2×#95C2318yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2318yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2318yes—
161C2×#95C2314yes—
162C25C2412yes—
163C25C2411yescharacteristic
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—22no—
171C22×D4—24no—
172C22×D4—22no—
173C22×D4—24no—
174C22×D4C2×D418yes—
175C22×D4C2×D414yes—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—42no—
199C23×C4—44no—
200C23×C4C2×D418yes—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4C2×D414yes—
206C23×C4#4114yes—
207C23×C4#4114yes—
208C23×C4C2418yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220#123—464no—
221#121—232no—
222C4×D4—432no—
223C4×D4—416no—
224C2×C4⋊C4—232no—
225C2×C4⋊C4—232no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—216no—
240C2×C4⋊C4—216no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—48no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D418yes—
256C2×C4⋊C4C2×D418yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4#4118yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—216no—
266C2×C22⋊C4—216no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4C2×D418yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—232no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—28no—
288C2×C42—28no—
289C2×C42—28no—
290C2×C42—28no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—28no—
294C2×C42—432no—
295C2×C42C2×D414yes—
296C2×C42C2×D414yes—
297C2×C42C2×D414yes—
298C2×C42C2×D414yes—
299C2×C42#4118yes—
300#95—28no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—216no—
306#95—28no—
307#95—28no—
308#95—216no—
309#95—216no—
310#95—216no—
311#95—48no—
312#95—44no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—24no—
320C24—22no—
321C24—24no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—22no—
326C24—24no—
327C24C22×D414yes—
328C24C22×D412yes—
329C24C22×D412yes—
330C24C22×D412yes—
331C24C2×#4112yes—
332C24C2×#4112yes—
333C24H(5,𝔽2)12yes—
334C242−1+412yes—
335C24C2511yescharacteristic
336C2×D4—216no—
337C2×D4—216no—
338C2×D4—216no—
339C2×D4—28no—
340C2×D4—216no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—48no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—216no—
359C22×C4—216no—
360C22×C4—28no—
361C22×C4—216no—
362C22×C4—216no—
363C22×C4—216no—
364C22×C4—216no—
365C22×C4—28no—
366C22×C4—216no—
367C22×C4—216no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—44no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—416no—
412C22×C4—416no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4#121116yes—
425C22×C4#121116yes—
426C22×C4C22×D412yes—
427C22×C4C22×D418yes—
428C22×C4C22×D414yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C2×#4118yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4114yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4114yes—
446C22×C4C2×#4114yes—
447C22×C42−1+414yes—
448C4⋊C4—216no—
449C4⋊C4—216no—
450C4⋊C4—216no—
451C4⋊C4—28no—
452C4⋊C4—216no—
453C4⋊C4—432no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—416no—
457C4⋊C4—44no—
458C4⋊C4—416no—
459C4⋊C4—88no—
460C22⋊C4—216no—
461C22⋊C4—432no—
462C22⋊C4—432no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—416no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C42—28no—
469C42—28no—
470C42—28no—
471C42—416no—
472C42—416no—
473C42—48no—
474C42#123116yescomplemented
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23C2×#12114yes—
515C23C2×#12114yes—
516C23#53714yes—
517C23D4214yes—
518C23#54714yes—
519C23#54714yes—
520C23#55514yes—
521C23C23×D411yescharacteristic
522C23C23×D411yescharacteristic
523C23C23×D411yescharacteristic
524C23C22×#4111yescharacteristic
525C23C22×#4111yescharacteristic
526C23C22×#4112yes—
527C23C22×#4112yes—
528C23C2×H(5,𝔽2)11yescharacteristic
529C23C2×2−1+412yes—
530C23C2×2−1+411yescharacteristic
531C23#58512yes—
532D4—432no—
533D4—416no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—232no—
547C2×C4—232no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—416no—
561C2×C4—432no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—432no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—816no—
595C2×C4C2×#12114yes—
596C2×C4C2×#12114yes—
597C2×C4C2×#12114yes—
598C2×C4C2×#12114yes—
599C2×C4C2×#12314yes—
600C2×C4C2×#12314yes—
601C2×C4C2×#12314yes—
602C2×C4#52514yes—
603C2×C4#52918yes—
604C2×C4#53218yes—
605C2×C4#53614yes—
606C2×C4#54718yes—
607C2×C4#55218yes—
608C22—24no—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—28no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—48no—
619C22—48no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—82no—
624C22—84no—
625C22—84no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22#253718yes—
630C22C22×#12111yescharacteristic
631C22C22×#12111yescharacteristic
632C22C22×#12311yescharacteristic
633C22C2×#52511yescharacteristic
634C22C2×#52912yes—
635C22C2×#53212yes—
636C22C2×#53611yescharacteristic
637C22C2×#53711yescharacteristic
638C22#344911yescharacteristic
639C22C2×D4211yescharacteristic
640C22C2×#54711yescharacteristic
641C22C2×#54712yes—
642C22C2×#54711yescharacteristic
643C22D4×#4112yes—
644C22C2×#55212yes—
645C22C2×#55511yescharacteristic
646C22#347512yes—
647C22#349412yes—
648C22#349512yes—
649C22#349712yes—
650C22#350712yes—
651C22#352212yes—
652C22#352312yes—
653C4—24no—
654C4—24no—
655C4—24no—
656C4—48no—
657C4—44no—
658C4—48no—
659C4—42no—
660C4—82no—
661C4—88no—
662C4—82no—
663C4#248618yescomplemented
664C4#285418yescomplemented
665C2—28no—
666C2—84no—
667C2C2×#248611yescharacteristic
668C2C2×#253711yescharacteristic
669C2#3493511yescharacteristic
670C2#3517811yescharacteristic
671C2#3524711yescharacteristic
672C2#3526311yescharacteristic
673C2#3611612yes—
674C2#3612612yes—
675C2#3617612yes—
676C2#3618612yes—
677C2C2×#285411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2486C2semidirect
4. C22×C4×C4⋊C4C2semidirect
5. #15149C2semidirect
6. #15147C2semidirect
7. #15081C2semidirect
8. #14927C2semidirect
9. #14686C2semidirect
10. #14685C2semidirect
11. #14151C2semidirect
12. #13994C2semidirect
13. #13906C2semidirect
14. #13872C2semidirect
90. C22×C4⋊C4D4semidirect
474. C42#123semidirect
663. C4#2486semidirect
664. C4#2854semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×#24862
C2×C4×#1212
C22×C4×C4⋊C42
#151492
#151472
#150812
#149272
#146862
#146852
#141512
#139942
#139062
#138722