Finite Group Databasebeta

← #219808

Subgroups — #219808

Order 512.

1658 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D42C211yesmaximal, characteristic
2#22650C211yesmaximal, characteristic, complemented
3#21264C211yesmaximal, characteristic, complemented
4C2×#1486C212yesmaximal, complemented
5#9674C211yesmaximal, characteristic, complemented
6#9638C211yesmaximal, characteristic, complemented
7#9076C212yesmaximal, complemented
8#8944C212yesmaximal, complemented
9#8563C211yesmaximal, characteristic, complemented
10#8310C212yesmaximal, complemented
11C4×D4⋊C8C211yesmaximal, characteristic, complemented
12C2×D42C411yescharacteristic
13C22×C4×D4—21no—
14C22×C4×D4C2212yes—
15C2×#436—21no—
16C2×#431C2212yescomplemented
17C2×#423C2212yescomplemented
18#2366C411yescharacteristic
19#2361—21no—
20#2361C412yes—
21#2358—22no—
22D4×C4⋊C4C411yescharacteristic
23D4×C4⋊C4C411yescharacteristic
24D4×C22⋊C4—21no—
25D4×C22⋊C4C412yes—
26C4×#127C2211yescharacteristic
27C4×#121—21no—
28C4×#121C2212yes—
29C4×#120—22no—
30C42×D4C2211yescharacteristic
31C42×D4C2211yescharacteristic
32#1954—21no—
33#1939—21no—
34#1939—21no—
35#1923—22no—
36#1851—22no—
37#1842C2211yescharacteristic, complemented
38#1797C2212yescomplemented
39C4×C4⋊C8C2211yescharacteristic, complemented
40C4×C4⋊C8C2211yescharacteristic, complemented
41C4×C22⋊C8C2212yescomplemented
42C42×C8C2211yescharacteristic, complemented
43#1486—28no—
44C2×D4⋊C8C2212yes—
45C2×D4⋊C8C2211yescharacteristic
46C2×D4⋊C8C2211yescharacteristic
47C2×D4⋊C8C2212yescomplemented
48C2×D4⋊C8C2212yescomplemented
49C2×D4⋊C8C2212yescomplemented
50C2×D4⋊C8C2212yescomplemented
51#1279C2212yescomplemented
52#1279C2211yescharacteristic, complemented
53#1279C2211yescharacteristic, complemented
54C23×D4—21no—
55C23×D4C2×C412yes—
56C24×C4—22no—
57C22×M16—22no—
58D42—24no—
59D42—24no—
60C2×#127C2×C411yescharacteristic
61C2×#121—21no—
62C2×#121C2×C412yes—
63C2×#120—22no—
64C2×C4×D4—21no—
65C2×C4×D4—22no—
66C2×C4×D4—21no—
67C2×C4×D4—22no—
68C2×C4×D4—24no—
69C2×C4×D4—24no—
70C2×C4×D4—21no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—21no—
74C2×C4×D4—42no—
75C2×C4×D4—42no—
76C2×C4×D4—44no—
77C2×C4×D4—44no—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4C2×C412yes—
80C2×C4×D4C2×C412yes—
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4C2×C411yescharacteristic
83C2×C4×D4D411yescharacteristic
84C2×C4×D4D411yescharacteristic
85C2×C4×D4C2312yes—
86C2×C4×D4C2311yescharacteristic
87C22×C4⋊C4—21no—
88C22×C4⋊C4C2×C412yes—
89C22×C22⋊C4—22no—
90C22×C22⋊C4—22no—
91C22×C42—21no—
92C22×C42C2312yes—
93#436—44no—
94#431—24no—
95#431—42no—
96#423—24no—
97#423—42no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2311yescharacteristic
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2312yes—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#390—21no—
116#390—21no—
117#390—21no—
118#390C2×C412yes—
119#390C2×C412yes—
120#389—21no—
121#389C2×C412yes—
122#388—22no—
123#388—22no—
124#386—22no—
125#386—22no—
126#386—22no—
127#386—22no—
128#386—22no—
129#385—22no—
130#385—22no—
131#385—22no—
132#384C2×C411yescharacteristic
133#380—21no—
134#380C2×C412yes—
135#379—22no—
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C4⋊C4D411yescharacteristic
140C4×C4⋊C4D411yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4C2×C412yes—
147C4×C22⋊C4D412yes—
148C4×C22⋊C4D412yes—
149C43C2311yescharacteristic
150#336—22no—
151#336—21no—
152#336—21no—
153#336—22no—
154D4⋊C8—24no—
155D4⋊C8—22no—
156D4⋊C8—24no—
157D4⋊C8—24no—
158D4⋊C8—42no—
159D4⋊C8—42no—
160D4⋊C8D414yescomplemented
161C25—22no—
162C22×D4—28no—
163C22×D4—22no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—22no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4C22⋊C411yescharacteristic
176C22×D4C22×C412yes—
177C22×D4C22×C411yescharacteristic
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—82no—
190C23×C4—82no—
191C23×C4C22×C412yes—
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C2×M16—44no—
195C2×M16—44no—
196C2×M16—82no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—21no—
203#127—22no—
204#127M1614yescomplemented
205#121—28no—
206#121—24no—
207#121—24no—
208#120—216no—
209C4×D4—24no—
210C4×D4—22no—
211C4×D4—24no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—44no—
222C4×D4—44no—
223C4×D4—42no—
224C4×D4—42no—
225C4×D4—41no—
226C4×D4—42no—
227C4×D4—44no—
228C4×D4—44no—
229C4×D4—44no—
230C4×D4—44no—
231C4×D4—44no—
232C4×D4—44no—
233C4×D4—48no—
234C4×D4—42no—
235C4×D4—42no—
236C4×D4—81no—
237C4×D4—81no—
238C4×D4M1614yescomplemented
239C4×D4D814yes—
240C4×D4SD1614yes—
241C4×D4C2×D412yes—
242C4×D4#4112yes—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C22⋊C411yescharacteristic
252C2×C4⋊C4C22⋊C411yescharacteristic
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4C22⋊C412yes—
273C2×C22⋊C4C22⋊C412yes—
274C2×C22⋊C4C22⋊C412yes—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42C22×C412yes—
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C411yescharacteristic
287C2×C42C2×D412yes—
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D412yes—
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2411yescharacteristic
294C4⋊C8—22no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—22no—
299C4⋊C8—44no—
300C4⋊C8—44no—
301C4⋊C8—42no—
302C4⋊C8—42no—
303C4⋊C8—81no—
304C4⋊C8C2×D412yescomplemented
305C22⋊C8—44no—
306C22⋊C8—44no—
307C22⋊C8—44no—
308C4×C8—22no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—22no—
313C4×C8—42no—
314C4×C8—42no—
315C4×C8—42no—
316C4×C8C2×D412yescomplemented
317#95—21no—
318#95—21no—
319#95—22no—
320#95—22no—
321#95—21no—
322#95C22⋊C412yes—
323#95C22⋊C412yes—
324#95C22⋊C412yes—
325C24—21no—
326C24—28no—
327C24—22no—
328C24—44no—
329C24—44no—
330C24—44no—
331C24—44no—
332C24—82no—
333C24—82no—
334C24C2×C22⋊C412yes—
335C2×D4—28no—
336C2×D4—28no—
337C2×D4—22no—
338C2×D4—21no—
339C2×D4—22no—
340C2×D4—21no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—21no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—22no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—432no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—48no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—81no—
361C2×D4—81no—
362C2×D4#10214yes—
363C2×D4C4×D412yes—
364C2×D4C2×M1614yes—
365C2×D4C2×M1612yes—
366C2×D4#13112yes—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—81no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4—88no—
412C22×C4—88no—
413C22×C4—88no—
414C22×C4—88no—
415C22×C4—82no—
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C412yes—
418C22×C4C2×C22⋊C412yes—
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C412yes—
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×D814yes—
424C22×C4C2×SD1614yes—
425C22×C4C23×C411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428M16—88no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—41no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—81no—
459C4⋊C4#10214yes—
460C4⋊C4C4×D412yes—
461C4⋊C4C2×M1612yes—
462C22⋊C4—28no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—42no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—42no—
471C22⋊C4—44no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C22⋊C4#10218yes—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—22no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—42no—
494C42—42no—
495C42C2×M1612yes—
496C42C2×M1612yes—
497C42C2×D812yes—
498C42C2×SD1612yes—
499C42C22×D411yescharacteristic
500C42C2×#4111yescharacteristic
501C42H(5,𝔽2)11yescharacteristic
502C42H(5,𝔽2)11yescharacteristic
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—416no—
514C23—416no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—88no—
526C23—88no—
527C23—88no—
528C23—88no—
529C23—88no—
530C23C2×#10214yes—
531C23C22×C22⋊C411yescharacteristic
532D4—22no—
533D4—24no—
534D4—22no—
535D4—24no—
536D4—22no—
537D4—28no—
538D4—416no—
539D4—416no—
540D4—42no—
541D4—42no—
542D4—48no—
543D4—84no—
544D4#43614yescomplemented
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—48no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—88no—
590C2×C4—84no—
591C2×C4—88no—
592C2×C4—88no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4#40712yes—
598C2×C4#40712yes—
599C2×C4C2×#10212yes—
600C2×C4C2×#10214yes—
601C2×C4C2×#10212yes—
602C2×C4#42112yes—
603C2×C4C2×C4×D411yescharacteristic
604C2×C4#51811yescharacteristic
605C2×C4D4211yescharacteristic
606C2×C4D4211yescharacteristic
607C2×C4#54611yescharacteristic
608C2×C4#54611yescharacteristic
609C2×C4C22×M1611yescharacteristic
610C2×C4C22×M1611yescharacteristic
611C2×C4C2×#13111yescharacteristic
612C2×C4#56811yescharacteristic
613C2×C4C22×D811yescharacteristic
614C2×C4C22×SD1611yescharacteristic
615C2×C4#57611yescharacteristic
616C2×C4#57711yescharacteristic
617C8—42no—
618C8—42no—
619C8—42no—
620C8—82no—
621C8—82no—
622C8—81no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—832no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22D4×C22⋊C411yescharacteristic
646C22C2×#40711yescharacteristic
647C22#288011yescharacteristic
648C22C22×#10211yescharacteristic
649C22#289711yescharacteristic
650C22C2×#42111yescharacteristic
651C22#290111yescharacteristic
652C4—22no—
653C4—22no—
654C4—24no—
655C4—21no—
656C4—21no—
657C4—42no—
658C4—41no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—81no—
663C4—82no—
664C4#148614yescomplemented
665C4C2×#43611yescharacteristic
666C4D4×M1611yescharacteristic
667C4#297411yescharacteristic
668C4#297511yescharacteristic
669C4D4×D811yescharacteristic
670C4D4×SD1611yescharacteristic
671C4#329411yescharacteristic
672C4#329511yescharacteristic
673C2—44no—
674C2—44no—
675C2—84no—
676C2C2×#148611yescharacteristic
677C2#1745311yescharacteristic
678C2#1750711yescharacteristic
679C2#1752211yescharacteristic
680C2#2240111yescharacteristic
681C2D4×#10211yescharacteristic
682C2#2243511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×D42C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22650C2semidirect
3. #21264C2semidirect
4. C2×#1486C2semidirect
5. #9674C2semidirect
6. #9638C2semidirect
7. #9076C2semidirect
8. #8944C2semidirect
9. #8563C2semidirect
10. #8310C2semidirect
11. C4×D4⋊C8C2semidirect
16. C2×#431C22semidirect
17. C2×#423C22semidirect
37. #1842C22semidirect
38. #1797C22semidirect
39. C4×C4⋊C8C22semidirect
40. C4×C4⋊C8C22semidirect
41. C4×C22⋊C8C22semidirect
42. C42×C8C22semidirect
47. C2×D4⋊C8C22semidirect
48. C2×D4⋊C8C22semidirect
49. C2×D4⋊C8C22semidirect
50. C2×D4⋊C8C22semidirect
51. #1279C22semidirect
52. #1279C22semidirect
53. #1279C22semidirect
160. D4⋊C8D4semidirect
204. #127M16semidirect
238. C4×D4M16semidirect
304. C4⋊C8C2×D4semidirect
316. C4×C8C2×D4semidirect
544. D4#436semidirect
664. C4#1486semidirect

Maximal subgroups

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

subgroupindex
C4×D422
#226502
#212642
C2×#14862
#96742
#96382
#90762
#89442
#85632
#83102
C4×D4⋊C82