Finite Group Databasebeta

← #412487

Subgroups — #412487

Order 512.

1360 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1942C211yesmaximal, characteristic, complemented
2C2×#1942C211yesmaximal, characteristic, complemented
3C2×#1845C211yesmaximal, characteristic, complemented
4C2×#1808C211yesmaximal, characteristic
5C2×#1808C211yesmaximal, characteristic
6C2×C8×C22⋊C4C211yesmaximal, characteristic
7#15131C212yesmaximal
8#11219C212yesmaximal, complemented
9#11207C212yesmaximal, complemented
10#11096C212yesmaximal
11#10862C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#107C2212yes—
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#396—22no—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#384C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1942—24no—
29#1942—24no—
30#1845—24no—
31#1808—24no—
32#1808—24no—
33C8×C22⋊C4—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1441—22no—
42#1441—22no—
43#1439—22no—
44C4⋊#95C2212yes—
45C24×C4C2311yescharacteristic
46C23×C8—22no—
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—22no—
50C23×C8—22no—
51C23×C8—22no—
52C23×C8Q814yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4D411yescharacteristic
68C22×C22⋊C4D411yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C2×#107—22no—
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#107—22no—
79C2×#107—24no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#107—24no—
83C2×#107—42no—
84C2×#107D414yes—
85C2×#107D414yes—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8Q814yes—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8D414yes—
95#396—48no—
96#389—24no—
97#389—24no—
98#389—24no—
99#389—42no—
100#384—24no—
101#384—42no—
102#380—24no—
103#380—24no—
104#380—42no—
105#380—42no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—42no—
121#336—22no—
122#336—22no—
123C25C2×D411yescharacteristic
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×Q812yes—
146C23×C4C2×Q812yes—
147C23×C4C2411yescharacteristic
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—28no—
165C22×C8—28no—
166C22×C8—28no—
167C22×C8—44no—
168C22×C8—44no—
169C22×C8—44no—
170C22×C8—44no—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173C22×C8C2×D412yes—
174C22×C8C2×D412yes—
175C22×C8C2×Q812yes—
176C22×C8C2×Q812yes—
177C22×C8C2×Q812yes—
178C22×C8C2×Q812yes—
179C22×C8#4112yes—
180C22×C8#4112yes—
181C22×C8#4112yes—
182C22×C8#4112yes—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D414yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4D814yes—
230C2×C22⋊C4Q1614yes—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42C2×D412yes—
242C2×C42#4112yes—
243#107—28no—
244#107—28no—
245#107—48no—
246#107—44no—
247#107—44no—
248C22⋊C8—28no—
249C4×C8—28no—
250C4×C8—44no—
251#95—22no—
252#95—22no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—82no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24C4⋊Q812yes—
272C24C2×D812yes—
273C24C2×Q1612yes—
274C24C22×D411yescharacteristic
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4#12112yes—
358C22×C4#12112yes—
359C22×C4C4⋊Q812yes—
360C22×C4C2×D812yes—
361C22×C4C2×D812yes—
362C22×C4C2×Q1612yes—
363C22×C4C2×Q1612yes—
364C22×C4#13512yes—
365C22×C4#13512yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×Q811yescharacteristic
370C22×C4C22×Q811yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4H(5,𝔽2)11yescharacteristic
377C22×C4H(5,𝔽2)11yescharacteristic
378C22×C42−1+411yescharacteristic
379C22×C42−1+411yescharacteristic
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—416no—
401C2×C8#12114yes—
402C2×C8#12114yes—
403C2×C8#12114yes—
404C2×C8C22⋊Q814yes—
405C2×C8C22⋊Q814yes—
406C2×C8C22⋊Q814yes—
407C2×C8C22⋊Q814yes—
408C2×C8#12614yes—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—88no—
426C4⋊C4—88no—
427C4⋊C4—88no—
428C4⋊C4—88no—
429C22⋊C4—28no—
430C22⋊C4—28no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23#50014yes—
476C23C2×#12111yescharacteristic
477C23C2×C4⋊Q811yescharacteristic
478C23#53211yescharacteristic
479C23D4211yescharacteristic
480C23D4211yescharacteristic
481C23#54711yescharacteristic
482C23#54711yescharacteristic
483C23C22×D811yescharacteristic
484C23C22×Q1611yescharacteristic
485C23C2×#13511yescharacteristic
486C23C2×#13511yescharacteristic
487C23#57611yescharacteristic
488C23#57611yescharacteristic
489C23#57811yescharacteristic
490C23#57811yescharacteristic
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4#46614yes—
561C2×C4#46714yes—
562C2×C4C2×#12111yescharacteristic
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×#12111yescharacteristic
565C2×C4C2×C22⋊Q811yescharacteristic
566C2×C4C2×C22⋊Q811yescharacteristic
567C2×C4C2×C22⋊Q811yescharacteristic
568C2×C4C2×C22⋊Q811yescharacteristic
569C2×C4C2×#12611yescharacteristic
570C2×C4#53311yescharacteristic
571C2×C4#53311yescharacteristic
572C2×C4#53711yescharacteristic
573C2×C4#53711yescharacteristic
574C2×C4#54011yescharacteristic
575C2×C4#54111yescharacteristic
576C2×C4#54411yescharacteristic
577C2×C4#54411yescharacteristic
578C2×C4#54611yescharacteristic
579C2×C4#54711yescharacteristic
580C2×C4#54811yescharacteristic
581C2×C4D4×Q811yescharacteristic
582C2×C4D4×Q811yescharacteristic
583C2×C4#55011yescharacteristic
584C2×C4#55211yescharacteristic
585C2×C4#55311yescharacteristic
586C2×C4#55411yescharacteristic
587C2×C4#55411yescharacteristic
588C2×C4#55511yescharacteristic
589C2×C4#55611yescharacteristic
590C8—24no—
591C8—42no—
592C8—42no—
593C8#39618yescomplemented
594C22—24no—
595C22—24no—
596C22—24no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22#241611yescharacteristic
610C22#242811yescharacteristic
611C22#244011yescharacteristic
612C22#244011yescharacteristic
613C22#249611yescharacteristic
614C22#253711yescharacteristic
615C22#285711yescharacteristic
616C22C2×#46611yescharacteristic
617C22C2×#46711yescharacteristic
618C22#306111yescharacteristic
619C22#306111yescharacteristic
620C22C2×#50011yescharacteristic
621C22#316011yescharacteristic
622C22#316411yescharacteristic
623C22#316411yescharacteristic
624C22#316911yescharacteristic
625C22#317011yescharacteristic
626C22#317611yescharacteristic
627C22#317711yescharacteristic
628C22D4×D811yescharacteristic
629C22D4×D811yescharacteristic
630C22#328011yescharacteristic
631C22#328311yescharacteristic
632C22#328511yescharacteristic
633C22D4×Q1611yescharacteristic
634C22D4×Q1611yescharacteristic
635C22#328711yescharacteristic
636C22#333411yescharacteristic
637C22#333411yescharacteristic
638C22#333611yescharacteristic
639C22#333811yescharacteristic
640C22#333811yescharacteristic
641C22#333911yescharacteristic
642C22#334111yescharacteristic
643C22#334211yescharacteristic
644C4—24no—
645C4—24no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—84no—
650C4—84no—
651C4—162no—
652C4—162no—
653C4C2×#39611yescharacteristic
654C4#241711yescharacteristic
655C4#242911yescharacteristic
656C4#243911yescharacteristic
657C4#244111yescharacteristic
658C4#249711yescharacteristic
659C4#253811yescharacteristic
660C4#262911yescharacteristic
661C2—44no—
662C2#1513111yescharacteristic
663C2#2478011yescharacteristic
664C2#2478111yescharacteristic
665C2#2541411yescharacteristic
666C2#2541511yescharacteristic
667C2#2606411yescharacteristic
668C2#2606511yescharacteristic
669C2#2606511yescharacteristic
670C2#2606611yescharacteristic
671C2#2783911yescharacteristic
672C2#2784011yescharacteristic
673C2#2822711yescharacteristic
674C2#2822811yescharacteristic
675C2#3320911yescharacteristic
676C2#3321611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1942C2semidirect
2. C2×#1942C2semidirect
3. C2×#1845C2semidirect
8. #11219C2semidirect
9. #11207C2semidirect
11. #10862C2semidirect
593. C8#396semidirect

Maximal subgroups

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

subgroupindex
C2×#19422
C2×#18452
C2×#18082
C2×C8×C22⋊C42
#151312
#112192
#112072
#110962
#108622