Finite Group Databasebeta

← #216110

Subgroups — #216110

Order 512.

1128 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#529C211yesmaximal, characteristic
2C2×#1842C211yesmaximal, characteristic, complemented
3C4×#432C211yesmaximal, characteristic, complemented
4#9876C212yesmaximal, complemented
5#9646C212yesmaximal, complemented
6#8939C212yesmaximal
7#8937C212yesmaximal, complemented
8#8564C212yesmaximal, complemented
9#8563C212yesmaximal, complemented
10C2×#529C411yescharacteristic
11C2×#432—21no—
12C22×C4⋊C8—21no—
13C22×C4⋊C8—21no—
14C22×C4⋊C8C2212yes—
15C4×#125C411yescharacteristic
16C4×#125C2211yescharacteristic
17C4×#126—21no—
18C4×#126—21no—
19C4×#124C411yescharacteristic
20C4×#124C2211yescharacteristic
21C4×C22⋊Q8C411yescharacteristic
22C4×C22⋊Q8C2211yescharacteristic
23C4×#123—21no—
24C4×#123—21no—
25C4×#121C411yescharacteristic
26C4×#121C2211yescharacteristic
27#2298—21no—
28#2295C411yescharacteristic
29#2293C411yescharacteristic
30#2290—21no—
31#2283C411yescharacteristic
32C42×Q8C2211yescharacteristic
33C42×D4—21no—
34C42×D4C2211yescharacteristic
35C4×#117—21no—
36C2×C43C2211yescharacteristic
37#1842—21no—
38#1842—21no—
39#1842C2212yes—
40#1842C2212yes—
41#1767—21no—
42C4×C4⋊C8C2212yes—
43#1749—21no—
44C4×C22⋊C8C2212yes—
45#1745—21no—
46C4×#97C2212yes—
47C2×Q8⋊C8C2214yes—
48C2×D4⋊C8C2214yes—
49#1279C2214yes—
50#1279C2214yes—
51C23×C8—22no—
52#529—24no—
53#529—24no—
54C2×#126—21no—
55C2×#125C2×C411yescharacteristic
56C2×#124C2×C411yescharacteristic
57C2×#123—21no—
58C2×C22⋊Q8C2×C411yescharacteristic
59C2×#121C2×C411yescharacteristic
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2311yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C2×#117—21no—
71C22×C42—21no—
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79#432—28no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—44no—
87C2×C4⋊C8—44no—
88C2×C4⋊C8—44no—
89C2×C4⋊C8C2314yes—
90C2×C22⋊C8—22no—
91C2×#97—22no—
92C2×#97—22no—
93C2×C4×C8C2314yes—
94#391C2×C411yescharacteristic
95#391C2×C411yescharacteristic
96#390C2×C411yescharacteristic
97#390C2×C411yescharacteristic
98#389C2×C411yescharacteristic
99#389C2×C411yescharacteristic
100#389C2×C411yescharacteristic
101#389C2×C411yescharacteristic
102#388—21no—
103#388—21no—
104#388—21no—
105#388—21no—
106#388—21no—
107#388—21no—
108#388—21no—
109#388—21no—
110#387—21no—
111#387—21no—
112#387—21no—
113#387—21no—
114#387—21no—
115#387—21no—
116#387—21no—
117#387—21no—
118#386—21no—
119#386—21no—
120#385—21no—
121#385—21no—
122#383—21no—
123#382C2×C411yescharacteristic
124#382C2×C411yescharacteristic
125#381—21no—
126#380C2×C411yescharacteristic
127#380C2×C411yescharacteristic
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4C2×C411yescharacteristic
141C4×C4⋊C4C2×C411yescharacteristic
142C4×C4⋊C4C2311yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157#376—21no—
158#376—21no—
159C43—21no—
160C43C2×C411yescharacteristic
161C43C2×C411yescharacteristic
162C43D411yescharacteristic
163C43D411yescharacteristic
164C43D411yescharacteristic
165C43D411yescharacteristic
166C43C2311yescharacteristic
167C43C2311yescharacteristic
168#336—22no—
169#336—22no—
170Q8⋊C8—44no—
171D4⋊C8—44no—
172C22×Q8C22×C411yescharacteristic
173C22×D4—21no—
174C22×D4C22×C411yescharacteristic
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C22×C8—24no—
186C22×C8—24no—
187C22×C8—44no—
188C22×C8—44no—
189C22×C8—44no—
190#126—28no—
191#125—22no—
192#125—22no—
193#124—22no—
194#124—22no—
195#123—28no—
196C22⋊Q8—22no—
197C22⋊Q8M1614yes—
198#121—22no—
199#121M1614yes—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—24no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×D4—28no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—24no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—44no—
212C4×D4—42no—
213C4×D4—42no—
214#117—28no—
215#117—44no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C42—24no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C22⋊C411yescharacteristic
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C411yescharacteristic
281C2×C42C22×C411yescharacteristic
282C2×C42C22×C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2411yescharacteristic
295C4⋊C8—28no—
296C4⋊C8—24no—
297C4⋊C8—24no—
298C4⋊C8—44no—
299C4⋊C8—44no—
300C4⋊C8—44no—
301C4⋊C8—44no—
302C4⋊C8—44no—
303C22⋊C8—28no—
304#97—28no—
305#97—44no—
306C4×C8—24no—
307C4×C8—24no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328C24—21no—
329C24C2×C22⋊C411yescharacteristic
330C2×Q8—21no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8C2×M1612yes—
334C2×D4—24no—
335C2×D4—22no—
336C2×D4—21no—
337C2×D4—22no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4C2×M1612yes—
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—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4#10414yes—
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×M1612yes—
393C22×C4C2×M1612yes—
394C22×C4AGL(1,ℤ/8)12yes—
395C22×C4#13712yes—
396C22×C4C23×C411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C2×C8—28no—
400C2×C8—44no—
401C2×C8—44no—
402C2×C8—44no—
403C2×C8—44no—
404C2×C8—44no—
405C2×C8—88no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4C2×M1612yes—
423C4⋊C4C2×M1612yes—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C42—24no—
438C42—22no—
439C42—21no—
440C42—21no—
441C42—22no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—22no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—41no—
468C42—41no—
469C42—42no—
470C42—41no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—42no—
477C42—42no—
478C42—42no—
479C42—42no—
480C42#13114yes—
481C42AGL(1,ℤ/8)12yes—
482C42#13712yes—
483C42H(5,𝔽2)12yes—
484C422−1+412yes—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23#40712yes—
497C23#41712yes—
498C23C2×#10412yes—
499C23C22×C22⋊C411yescharacteristic
500Q8—24no—
501Q8—42no—
502D4—24no—
503D4—48no—
504D4—42no—
505C2×C4—28no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—84no—
569C2×C4#40712yes—
570C2×C4#40812yes—
571C2×C4#40812yes—
572C2×C4#41712yes—
573C2×C4#41712yes—
574C2×C4#41712yes—
575C2×C4C2×#10412yes—
576C2×C4C2×#10412yes—
577C2×C4#42112yes—
578C2×C4#52012yes—
579C2×C4#53511yescharacteristic
580C2×C4#53611yescharacteristic
581C2×C4#53712yes—
582C2×C4C22×M1611yescharacteristic
583C2×C4C22×M1611yescharacteristic
584C2×C4C2×#13112yes—
585C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
586C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
587C2×C4C2×#13711yescharacteristic
588C2×C4C2×#13711yescharacteristic
589C8—48no—
590C8—84no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22#148914yes—
605C22#231111yescharacteristic
606C22C2×#40711yescharacteristic
607C22C2×#40811yescharacteristic
608C22C2×#41711yescharacteristic
609C22C2×#41711yescharacteristic
610C22C22×#10411yescharacteristic
611C22C2×#42111yescharacteristic
612C4—22no—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—41no—
620C4—41no—
621C4—42no—
622C4—41no—
623C4—42no—
624C4—42no—
625C4—41no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4#148314yes—
631C4#299012yes—
632C4#299012yes—
633C4#321711yescharacteristic
634C4#321811yescharacteristic
635C4#323811yescharacteristic
636C4#323911yescharacteristic
637C2—24no—
638C2—82no—
639C2C2×#148911yescharacteristic
640C2#1745111yescharacteristic
641C2C2×#148311yescharacteristic
642C2#1746611yescharacteristic
643C2#2185111yescharacteristic
644C2#2185611yescharacteristic
645C2#2186711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1842C2semidirect
3. C4×#432C2semidirect
4. #9876C2semidirect
5. #9646C2semidirect
7. #8937C2semidirect
8. #8564C2semidirect
9. #8563C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5292
C2×#18422
C4×#4322
#98762
#96462
#89392
#89372
#85642
#85632