Finite Group Databasebeta

← #359990

Subgroups — #359990

Order 512.

2302 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#431C211yesmaximal, characteristic, complemented
2C2×C42×D4C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic, complemented
4C2×#1797C212yesmaximal, complemented
5#11147C212yesmaximal, complemented
6#10964C212yesmaximal
7#10921C212yesmaximal, complemented
8#10874C212yesmaximal, complemented
9#10832C212yesmaximal, complemented
10C22×C4×D4C411yescharacteristic
11C22×C4×D4C411yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C2×#431—24no—
16C2×#431—28no—
17C22×C4⋊C8C2212yes—
18C22×C4⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2214yes—
21C22×C22⋊C8C2212yes—
22C22×C4×C8C2212yes—
23C42×D4—24no—
24C42×D4—24no—
25C2×C4×C4⋊C4C411yescharacteristic
26C2×C4×C4⋊C4C411yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C411yescharacteristic
29C2×C4×C22⋊C4C411yescharacteristic
30C2×C4×C22⋊C4C411yescharacteristic
31C2×C4×C22⋊C4C411yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C43C2211yescharacteristic
35#1797—24no—
36#1797—28no—
37C2×#336C2214yes—
38C2×#336C2214yes—
39C2×#336C2212yes—
40C2×#336C2214yes—
41C23×D4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2314yes—
49C23×C8C2314yes—
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—24no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4C818yescomplemented
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C42—24no—
70C22×C42—24no—
71C22×C42—24no—
72C22×C42—22no—
73C22×C42—22no—
74C22×C42—24no—
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42D414yes—
84C22×C42D414yes—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#431—232no—
91C2×C4⋊C8—24no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8—24no—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—24no—
96C2×C4⋊C8D418yescomplemented
97C2×C22⋊C8—24no—
98C2×C22⋊C8—28no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—28no—
101C2×C22⋊C8—28no—
102C2×C22⋊C8—44no—
103C2×C22⋊C8—44no—
104C2×C4×C8—28no—
105C2×C4×C8—24no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C22⋊C4—24no—
111C4×C22⋊C4—24no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—24no—
115C4×C22⋊C4—24no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C2×#95C2×C411yescharacteristic
119C2×#95C2×C411yescharacteristic
120C2×#95C2×C411yescharacteristic
121C2×#95C2×C411yescharacteristic
122C2×#95C2×C411yescharacteristic
123C2×#95C2×C411yescharacteristic
124C43—22no—
125C43—22no—
126#336—28no—
127#336—24no—
128#336—28no—
129#336—44no—
130C25C22×C411yescharacteristic
131C25C22×C411yescharacteristic
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4C2×C814yes—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4C22⋊C414yes—
168C23×C4C22⋊C414yes—
169C23×C4C2×C814yes—
170C23×C4C2×C814yes—
171C23×C4M1614yes—
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4#4114yes—
187C23×C4C2411yescharacteristic
188C22×C8—24no—
189C22×C8—24no—
190C22×C8—24no—
191C22×C8—24no—
192C22×C8—24no—
193C22×C8—24no—
194C22×C8—24no—
195C22×C8—24no—
196C22×C8—24no—
197C22×C8—24no—
198C22×C8—24no—
199C22×C8—24no—
200C22×C8—24no—
201C22×C8C2×D418yes—
202C4×D4—232no—
203C4×D4—216no—
204C4×D4—216no—
205C4×D4—232no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4C2×C814yes—
214C2×C4⋊C4M1614yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4C2×C814yes—
231C2×C22⋊C4C2×C814yes—
232C2×C22⋊C4M1614yes—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—28no—
246C2×C42—216no—
247C2×C42—28no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—42no—
264C2×C42—44no—
265C2×C42—48no—
266C2×C42—44no—
267C2×C42C22⋊C414yes—
268C2×C42C22⋊C414yes—
269C2×C42C2×C814yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D414yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42#4114yes—
276C2×C42#4114yes—
277C4⋊C8—216no—
278C4⋊C8—48no—
279C22⋊C8—216no—
280C22⋊C8—416no—
281C22⋊C8—48no—
282C4×C8—216no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24C2×C22⋊C412yes—
306C24#11712yes—
307C24C22×C812yes—
308C24C22×C812yes—
309C24C2×M1612yes—
310C24C2×M1612yes—
311C24C23×C411yescharacteristic
312C2×D4—28no—
313C2×D4—28no—
314C2×D4—28no—
315C2×D4—28no—
316C2×D4—28no—
317C2×D4—28no—
318C2×D4—28no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—216no—
333C22×C4—216no—
334C22×C4—28no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—216no—
381C22×C4—44no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—44no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—48no—
409C22×C4C22⋊C8116yescomplemented
410C22×C4C2×C22⋊C412yes—
411C22×C4C2×C22⋊C412yes—
412C22×C4C2×C22⋊C412yes—
413C22×C4C2×C22⋊C412yes—
414C22×C4C2×C22⋊C412yes—
415C22×C4#11712yes—
416C22×C4#11712yes—
417C22×C4#11712yes—
418C22×C4C4×D414yes—
419C22×C4C4×D414yes—
420C22×C4C22×C812yes—
421C22×C4C22×C812yes—
422C22×C4C22×C812yes—
423C22×C4C22×C812yes—
424C22×C4C22×C812yes—
425C22×C4C2×M1612yes—
426C22×C4C2×M1612yes—
427C22×C4C2×M1612yes—
428C22×C4C2×M1612yes—
429C22×C4C2×M1612yes—
430C22×C4C2×M1612yes—
431C22×C4C2×M1612yes—
432C22×C4#13114yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C42−1+412yes—
441C2×C8—28no—
442C2×C8—28no—
443C2×C8—28no—
444C2×C8—44no—
445C2×C8—44no—
446C2×C8—44no—
447C2×C8—44no—
448C2×C8—44no—
449C2×C8—44no—
450C2×C8—44no—
451C2×C8—44no—
452C2×C8—44no—
453C2×C8—44no—
454C2×C8—44no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C22⋊C4—28no—
459C22⋊C4—28no—
460C22⋊C4—28no—
461C22⋊C4—28no—
462C22⋊C4—28no—
463C22⋊C4—28no—
464C42—28no—
465C42—22no—
466C42—24no—
467C42—22no—
468C42—24no—
469C42—24no—
470C42—28no—
471C42—28no—
472C42—28no—
473C42—44no—
474C42—41no—
475C42—42no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—42no—
480C42—44no—
481C42—44no—
482C42—44no—
483C42#12014yes—
484C42#12118yes—
485C42#12314yes—
486C23—22no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23C2×C22⋊C814yes—
531C23#43114yescomplemented
532C23C22×C22⋊C411yescharacteristic
533C23C2×#11711yescharacteristic
534C23C2×#11711yescharacteristic
535C23C2×C4×D412yes—
536C23#52012yes—
537C23C23×C811yescharacteristic
538C23C22×M1611yescharacteristic
539C23C22×M1611yescharacteristic
540C23C22×M1611yescharacteristic
541C23C2×#13112yes—
542C23#56812yes—
543D4—232no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—232no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—416no—
579C2×C4—416no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—416no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4#38614yes—
610C2×C4#38614yes—
611C2×C4C2×C22⋊C814yes—
612C2×C4C2×C22⋊C814yes—
613C2×C4#40714yes—
614C2×C4#40814yes—
615C2×C4#43114yescomplemented
616C2×C4C8×D418yes—
617C2×C4#43518yes—
618C2×C4C2×#12011yescharacteristic
619C2×C4C2×#12112yes—
620C2×C4C2×#12311yescharacteristic
621C2×C4#52512yes—
622C2×C4#52512yes—
623C2×C4#53511yescharacteristic
624C2×C4#53611yescharacteristic
625C2×C4#53712yes—
626C2×C4#54612yes—
627C2×C4#54712yes—
628C2×C4#54812yes—
629C2×C4#55012yes—
630C2×C4#55212yes—
631C2×C4#55312yes—
632C2×C4#55512yes—
633C2×C4#55612yes—
634C8—48no—
635C8—84no—
636C22—24no—
637C22—24no—
638C22—28no—
639C22—24no—
640C22—24no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—42no—
657C22—42no—
658C22#228011yescharacteristic
659C22C2×#38611yescharacteristic
660C22#231111yescharacteristic
661C22#235212yes—
662C22#235412yes—
663C22C22×C22⋊C811yescharacteristic
664C22C2×#40711yescharacteristic
665C22C2×#40811yescharacteristic
666C22#287911yescharacteristic
667C22C2×#43111yescharacteristic
668C22C2×#43111yescharacteristic
669C22#292011yescharacteristic
670C22#292011yescharacteristic
671C22#292512yes—
672C22#292512yes—
673C22C2×C8×D412yes—
674C22C2×#43512yes—
675C22D4×M1612yes—
676C22D4×M1612yes—
677C22#298812yes—
678C22#298912yes—
679C22#299012yes—
680C22#299112yes—
681C4—24no—
682C4—24no—
683C4—24no—
684C4—24no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4#186918yes—
693C4#240712yes—
694C4#241412yes—
695C4#241512yes—
696C4#248512yes—
697C2—44no—
698C2—44no—
699C2#1387111yescharacteristic
700C2#2071411yescharacteristic
701C2#2072111yescharacteristic
702C2C2×#186911yescharacteristic
703C2#2089811yescharacteristic
704C2#2184711yescharacteristic
705C2#2184911yescharacteristic
706C2#2274812yes—
707C2#2274912yes—
708C2#2279512yes—
709C2#2279712yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#431C2semidirect
3. C2×#1797C2semidirect
4. C2×#1797C2semidirect
5. #11147C2semidirect
7. #10921C2semidirect
8. #10874C2semidirect
9. #10832C2semidirect
59. C2×C4×D4C8semidirect
96. C2×C4⋊C8D4semidirect
409. C22×C4C22⋊C8semidirect
531. C23#431semidirect
615. C2×C4#431semidirect

Maximal subgroups

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

subgroupindex
C22×#4312
C2×C42×D42
C2×#17972
#111472
#109642
#109212
#108742
#108322