Finite Group Databasebeta

← #415581

Subgroups — #415581

Order 512.

1308 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#125C211yesmaximal, characteristic
2C2×#1948C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic
4#14712C211yesmaximal, characteristic
5#11385C212yesmaximal
6#11328C212yesmaximal
7#11327C212yesmaximal
8#11325C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10849C211yesmaximal, characteristic
11#10832C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C4×#125—24no—
18C4×#125—24no—
19C2×#389—21no—
20C2×#389—21no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#389C2212yes—
24C2×#387—21no—
25C2×#387—21no—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C43C2211yescharacteristic
34#1948—24no—
35#1947—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C4⋊#95—21no—
45C4⋊#95—21no—
46C4⋊#95C2212yes—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125Q814yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C42—21no—
71C22×C42—21no—
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#107—24no—
80C2×#107—42no—
81C2×#107—42no—
82C2×#106—24no—
83C2×#106—42no—
84C2×#106—42no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8Q814yes—
90C2×C4⋊C8Q814yes—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—42no—
96#389—42no—
97#389—44no—
98#389—44no—
99#387—28no—
100#387—28no—
101#384—22no—
102#384—22no—
103#384—22no—
104#384Q814yes—
105#382—22no—
106#382—22no—
107C4×C4⋊C4—28no—
108C4×C4⋊C4—28no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4D414yes—
112C4×C4⋊C4D414yes—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C43—22no—
123C43—22no—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—42no—
129#340—42no—
130#336—24no—
131#336—42no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8C2×Q814yes—
158C22×C8C2×Q814yes—
159#125—28no—
160#125—28no—
161#125—28no—
162#125—28no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
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—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×Q814yes—
222C2×C4⋊C4C2×Q812yes—
223C2×C4⋊C4C2×Q812yes—
224C2×C4⋊C4#4112yes—
225C2×C4⋊C4#4112yes—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—41no—
249C2×C42—41no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×Q812yes—
257C2×C42C2×Q812yes—
258C2×C42C2×Q812yes—
259C2×C42C2×Q812yes—
260C2×C42#4112yes—
261C2×C42#4112yes—
262#107—48no—
263#106—48no—
264C4⋊C8—28no—
265C4⋊C8—28no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—44no—
273#95—44no—
274#95—44no—
275C24C22×D411yescharacteristic
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—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
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—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4#12012yes—
347C22×C4#12012yes—
348C22×C4C22⋊Q812yes—
349C22×C4C22⋊Q812yes—
350C22×C4C22⋊Q812yes—
351C22×C4C22⋊Q812yes—
352C22×C4C4⋊Q812yes—
353C22×C4C4⋊Q812yes—
354C22×C4#13512yes—
355C22×C4#13512yes—
356C22×C4AGL(1,ℤ/8)12yes—
357C22×C4AGL(1,ℤ/8)12yes—
358C22×C4#13712yes—
359C22×C4#13712yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×Q811yescharacteristic
363C22×C4C22×Q811yescharacteristic
364C22×C4C22×Q811yescharacteristic
365C22×C4C22×Q811yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4H(5,𝔽2)11yescharacteristic
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C4H(5,𝔽2)11yescharacteristic
373C22×C42−1+411yescharacteristic
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C4⋊C4—216no—
389C4⋊C4—216no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—48no—
403C4⋊C4—48no—
404C4⋊C4—48no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—48no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—84no—
417C4⋊C4—88no—
418C4⋊C4—84no—
419C4⋊C4C22⋊Q814yes—
420C4⋊C4C22⋊Q814yes—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—44no—
426C42—44no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—44no—
432C42—48no—
433C42—44no—
434C42—48no—
435C42C22⋊Q814yes—
436C42C22⋊Q814yes—
437C42C4⋊Q814yes—
438C42C4⋊Q814yes—
439C23C2×#12011yescharacteristic
440C23C2×C22⋊Q811yescharacteristic
441C23C2×C22⋊Q811yescharacteristic
442C23#52511yescharacteristic
443C23C2×C4⋊Q811yescharacteristic
444C23#53511yescharacteristic
445C23#53711yescharacteristic
446C23C2×#13511yescharacteristic
447C23C2×#13511yescharacteristic
448C23C2×AGL(1,ℤ/8)11yescharacteristic
449C23C2×AGL(1,ℤ/8)11yescharacteristic
450C23C2×#13711yescharacteristic
451C23C2×#13711yescharacteristic
452C23#57511yescharacteristic
453C23#57511yescharacteristic
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4#39514yes—
531C2×C4#39514yes—
532C2×C4#44914yes—
533C2×C4#45214yes—
534C2×C4#47814yes—
535C2×C4#47914yes—
536C2×C4#49914yes—
537C2×C4#50114yes—
538C2×C4C2×C22⋊Q811yescharacteristic
539C2×C4C2×C22⋊Q811yescharacteristic
540C2×C4C2×C22⋊Q811yescharacteristic
541C2×C4C2×C22⋊Q811yescharacteristic
542C2×C4#52511yescharacteristic
543C2×C4#52511yescharacteristic
544C2×C4C2×C4⋊Q811yescharacteristic
545C2×C4C2×C4⋊Q811yescharacteristic
546C2×C4#53511yescharacteristic
547C2×C4#53511yescharacteristic
548C2×C4#53711yescharacteristic
549C2×C4#53711yescharacteristic
550C2×C4#55411yescharacteristic
551C2×C4#55411yescharacteristic
552C2×C4#55411yescharacteristic
553C2×C4#55411yescharacteristic
554C2×C4#55411yescharacteristic
555C2×C4#55411yescharacteristic
556C2×C4#55711yescharacteristic
557C2×C4#55711yescharacteristic
558C2×C4#55711yescharacteristic
559C2×C4#55711yescharacteristic
560C2×C4Q8211yescharacteristic
561C2×C4Q8211yescharacteristic
562C2×C4Q8211yescharacteristic
563C2×C4#55911yescharacteristic
564C2×C4#55911yescharacteristic
565C2×C4#55911yescharacteristic
566C8—44no—
567C8—44no—
568C22C2×#39511yescharacteristic
569C22#256711yescharacteristic
570C22#258011yescharacteristic
571C22#258211yescharacteristic
572C22#258411yescharacteristic
573C22#261511yescharacteristic
574C22#284811yescharacteristic
575C22C2×#44911yescharacteristic
576C22C2×#45211yescharacteristic
577C22#300611yescharacteristic
578C22#300711yescharacteristic
579C22C2×#47811yescharacteristic
580C22C2×#47911yescharacteristic
581C22#307611yescharacteristic
582C22#307611yescharacteristic
583C22#307711yescharacteristic
584C22#307811yescharacteristic
585C22#307911yescharacteristic
586C22#307911yescharacteristic
587C22#310611yescharacteristic
588C22#310711yescharacteristic
589C22#311511yescharacteristic
590C22#311611yescharacteristic
591C22C2×#49911yescharacteristic
592C22C2×#50111yescharacteristic
593C22#316511yescharacteristic
594C22#316611yescharacteristic
595C22#321211yescharacteristic
596C22#321211yescharacteristic
597C22#321711yescharacteristic
598C22#321811yescharacteristic
599C22#324711yescharacteristic
600C22#324811yescharacteristic
601C22#325111yescharacteristic
602C22#325211yescharacteristic
603C4—24no—
604C4—42no—
605C4—44no—
606C4—42no—
607C4—44no—
608C4—42no—
609C4—42no—
610C4—84no—
611C4—82no—
612C4—82no—
613C4—162no—
614C4#205918yes—
615C4#256711yescharacteristic
616C4#256711yescharacteristic
617C4#258011yescharacteristic
618C4#258011yescharacteristic
619C4#258211yescharacteristic
620C4#258211yescharacteristic
621C4#258411yescharacteristic
622C4#258411yescharacteristic
623C2#1520211yescharacteristic
624C2C2×#205911yescharacteristic
625C2#2355411yescharacteristic
626C2#2921011yescharacteristic
627C2#2921311yescharacteristic
628C2#2925611yescharacteristic
629C2#2925711yescharacteristic
630C2#2928811yescharacteristic
631C2#2928911yescharacteristic
632C2#2931411yescharacteristic
633C2#2932111yescharacteristic
634C2#2966211yescharacteristic
635C2#2966511yescharacteristic
636C2#3295511yescharacteristic
637C2#3295811yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×C4×#1252
C2×#19482
C2×#19472
#147122
#113852
#113282
#113272
#113252
#108492
#108322