Finite Group Databasebeta

← #674819

Subgroups — #674819

Order 512.

2026 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2514C212yesmaximal
2#15920C214yesmaximal
3#15582C214yesmaximal
4#15241C212yesmaximal
5#15235C212yesmaximal
6C4⋊#401C212yesmaximal
7#15139C212yesmaximal
8#14687C211yesmaximal, characteristic
9#14151C211yesmaximal, characteristic
10#14133C212yesmaximal
11#14030C212yesmaximal
12#13926C211yesmaximal, characteristic
13#13926C211yesmaximal, characteristic
14#13921C212yesmaximal
15C4⋊C42C211yesmaximal, characteristic
16C4×#391C211yesmaximal, characteristic
17C4×#389C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19#2514—216no—
20C2×#401C2214yes—
21C2×#400C2214yes—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#387C2214yes—
31C2×#384C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#383C2214yes—
34C2×#382C2212yes—
35C2×#382C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C43C2211yescharacteristic
43#1443C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1443C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1436C2212yes—
57#1436C2212yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2212yes—
61#1436C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2211yescharacteristic
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80#1434C2214yes—
81C4×#95C2212yes—
82C4×#95C2212yes—
83C4×#95C2212yes—
84C23×Q8C2311yescharacteristic
85C2×C4×Q8—24no—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2312yes—
111C22×C42C2314yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#401—216no—
118#400—216no—
119#391—28no—
120#391—28no—
121#391—24no—
122#391D418yes—
123#389—28no—
124#389—24no—
125#389—24no—
126#389—28no—
127#387—216no—
128#384—24no—
129#384—24no—
130#383—216no—
131#382—28no—
132#382—28no—
133C4×C4⋊C4—28no—
134C4×C4⋊C4—28no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—24no—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C43—24no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8C2×D414yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C4×Q8—416no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4#4114yes—
250C2×C4⋊C4#4114yes—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—28no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—28no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—44no—
278C2×C42—41no—
279C2×C42—41no—
280C2×C42—41no—
281C2×C42—41no—
282C2×C42C2×D414yes—
283C2×C42#4114yes—
284C2×C42#4114yes—
285C2×C42#4114yes—
286C2×C42#4114yes—
287#95—24no—
288#95—28no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—24no—
307#95—24no—
308#95—42no—
309#95—44no—
310#95—42no—
311#95—42no—
312#95—42no—
313#95—42no—
314#95—42no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95C2×D418yes—
321#95C2×D418yes—
322C24C2511yescharacteristic
323C2×Q8—28no—
324C2×Q8—28no—
325C2×Q8—28no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—44no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4C22×D414yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D414yes—
428C22×C4C22×D412yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C42−1+412yes—
446C22×C42−1+412yes—
447C22×C42−1+412yes—
448C22×C42−1+412yes—
449C4⋊C4—28no—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—416no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—84no—
465C4⋊C4—84no—
466C42—28no—
467C42—28no—
468C42—416no—
469C42—44no—
470C42—44no—
471C42—48no—
472C42—48no—
473C42—48no—
474C42—48no—
475C42—44no—
476C42—44no—
477C42#126116yes—
478C23C23×D411yescharacteristic
479C23C22×#4111yescharacteristic
480C23C22×#4111yescharacteristic
481C23C22×#4111yescharacteristic
482C23C22×#4111yescharacteristic
483C23C22×#4111yescharacteristic
484C23C22×#4111yescharacteristic
485C23C2×2−1+411yescharacteristic
486C23C2×2−1+411yescharacteristic
487C23C2×2−1+411yescharacteristic
488C23C2×2−1+411yescharacteristic
489C23#58511yescharacteristic
490C23#58511yescharacteristic
491C23#58511yescharacteristic
492C23#58511yescharacteristic
493Q8—416no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
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—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4C2×#12614yes—
599C2×C4C2×#12614yes—
600C2×C4C2×#12614yes—
601C2×C4#52914yes—
602C2×C4#53214yes—
603C2×C4#54114yes—
604C2×C4#54114yes—
605C2×C4#54714yes—
606C2×C4#54714yes—
607C2×C4#55214yes—
608C2×C4#55214yes—
609C2×C4#55614yes—
610C2×C4#55614yes—
611C2×C4#55614yes—
612C2×C4#55614yes—
613C22C22×#12611yescharacteristic
614C22C2×#52911yescharacteristic
615C22C2×#53211yescharacteristic
616C22C2×#54111yescharacteristic
617C22C2×#54111yescharacteristic
618C22C2×#54711yescharacteristic
619C22C2×#54711yescharacteristic
620C22D4×#4111yescharacteristic
621C22D4×#4111yescharacteristic
622C22C2×#55211yescharacteristic
623C22C2×#55211yescharacteristic
624C22C2×#55611yescharacteristic
625C22C2×#55611yescharacteristic
626C22C2×#55611yescharacteristic
627C22C2×#55611yescharacteristic
628C22#348011yescharacteristic
629C22#348011yescharacteristic
630C22#349311yescharacteristic
631C22#349311yescharacteristic
632C22#349711yescharacteristic
633C22#349711yescharacteristic
634C22#350711yescharacteristic
635C22#350711yescharacteristic
636C22#351011yescharacteristic
637C22#351011yescharacteristic
638C22#351211yescharacteristic
639C22#351211yescharacteristic
640C22#351511yescharacteristic
641C22#351511yescharacteristic
642C22#351811yescharacteristic
643C22#351811yescharacteristic
644C22#355511yescharacteristic
645C22#355511yescharacteristic
646C22#355511yescharacteristic
647C22#355511yescharacteristic
648C4—24no—
649C4—44no—
650C4—44no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—44no—
656C4—44no—
657C4—82no—
658C4—82no—
659C4—84no—
660C4—84no—
661C4—82no—
662C4—82no—
663C4#249618yes—
664C4#251418yescomplemented
665C2C2×#249611yescharacteristic
666C2C2×#251411yescharacteristic
667C2#3562911yescharacteristic
668C2#3611411yescharacteristic
669C2#3627111yescharacteristic
670C2#3627111yescharacteristic
671C2#3659011yescharacteristic
672C2#3898611yescharacteristic
673C2#3898611yescharacteristic
674C2#3937911yescharacteristic
675C2#3976211yescharacteristic
676C2#5924211yescharacteristic
677C2#5924411yescharacteristic
678C2#5925611yescharacteristic
679C2#5925811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
664. C4#2514semidirect

Maximal subgroups

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

subgroupindex
C2×#25142
#159202
#155822
#152412
#152352
C4⋊#4012
#151392
#146872
#141512
#141332
#140302
#139262
#139212
C4⋊C422
C4×#3912
C4×#3892