Finite Group Databasebeta

← #196523

Subgroups — #196523

Order 512.

1890 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×M16C211yesmaximal, characteristic, complemented
2C2×#2850C211yesmaximal, characteristic
3#20715C211yesmaximal, characteristic, complemented
4C2×#1527C212yesmaximal, complemented
5#9120C214yesmaximal, complemented
6#9119C214yesmaximal, complemented
7#8033C212yesmaximal, complemented
8C23×M16—21no—
9C22×#124C2211yescharacteristic
10C22×C22⋊Q8C411yescharacteristic
11C22×#121C411yescharacteristic
12C23×C42C2211yescharacteristic
13C2×#432C2212yes—
14C2×C4×M16—21no—
15C2×C4×M16—21no—
16C2×C4×M16—28no—
17C2×C4×M16C2212yes—
18C2×C4×M16C2212yescomplemented
19C22×#97C2212yescomplemented
20C22×C4×C8—21no—
21#2850—21no—
22#2850—21no—
23#2850—21no—
24#2850—21no—
25#2850C412yes—
26#2850C412yes—
27#2850C2212yes—
28#2850C2212yes—
29C2×#391C411yescharacteristic
30C2×#390C411yescharacteristic
31C2×#388—21no—
32C2×#388—21no—
33C2×#386—21no—
34C2×#386—21no—
35C2×#382C2211yescharacteristic
36#1817—21no—
37#1797—21no—
38#1797—21no—
39#1779—21no—
40#1749C2212yescomplemented
41#1749C2212yescomplemented
42#1527—28no—
43C2×#329C2218yescomplemented
44#1295C2218yescomplemented
45C23×Q8C2×C411yescharacteristic
46C23×D4—21no—
47C23×D4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C22×M16—28no—
52C22×M16—28no—
53C22×M16—28no—
54C22×M16—22no—
55C22×M16—22no—
56C23×C8—22no—
57C2×#124—21no—
58C2×#124—21no—
59C2×#124—24no—
60C2×#124—24no—
61C2×#124C2×C412yes—
62C2×#124C2312yes—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—44no—
66C2×C22⋊Q8C2×C412yes—
67C2×C22⋊Q8C2×C412yes—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—44no—
71C2×#121C2×C412yes—
72C2×#121C2×C412yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C42—24no—
83C22×C42—24no—
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#432—24no—
97#432—24no—
98C2×C4⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C4×M16—28no—
103C4×M16—28no—
104C4×M16—24no—
105C4×M16—28no—
106C4×M16D418yescomplemented
107C2×#97—22no—
108C2×#97—28no—
109C2×#97C2314yes—
110C2×#97C2314yes—
111C2×C4×C8—28no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114#391—21no—
115#391—21no—
116#391C2×C412yes—
117#391C2×C412yes—
118#390—21no—
119#390—21no—
120#390C2×C412yes—
121#390C2×C412yes—
122#388—22no—
123#388—22no—
124#388—22no—
125#388—22no—
126#388—22no—
127#388—22no—
128#388—22no—
129#388—22no—
130#386—22no—
131#386—22no—
132#386—22no—
133#386—22no—
134#386—22no—
135#386—22no—
136#386—22no—
137#386—22no—
138#382—21no—
139#382—21no—
140#382C2×C412yes—
141#382C2312yes—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—21no—
146#336—22no—
147#336—22no—
148#329—216no—
149C25—21no—
150C25C22⋊C411yescharacteristic
151C22×Q8—21no—
152C22×Q8—24no—
153C22×Q8—42no—
154C22×Q8—42no—
155C22×Q8C22×C412yes—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—21no—
159C22×D4—24no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4—44no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4C22×C412yes—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C2×M16—264no—
198C2×M16—28no—
199C2×M16—28no—
200C2×M16—28no—
201C2×M16—28no—
202C2×M16—28no—
203C2×M16—28no—
204C22×C8—28no—
205C22×C8—28no—
206C22×C8—28no—
207C22×C8—24no—
208C22×C8—24no—
209#124—28no—
210#124—24no—
211#124—24no—
212#124—42no—
213#124—44no—
214#124—42no—
215C22⋊Q8—44no—
216C22⋊Q8—44no—
217C22⋊Q8—44no—
218C22⋊Q8—44no—
219#121—44no—
220#121—44no—
221#121—44no—
222#121—44no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C412yes—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—42no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
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×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D414yes—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42#4114yes—
284C2×C42C2411yescharacteristic
285C4⋊C8—44no—
286C22⋊C8—44no—
287C22⋊C8—44no—
288C22⋊C8—44no—
289#97—28no—
290#97—24no—
291#97—24no—
292#97—24no—
293#97C2×D418yescomplemented
294C4×C8—28no—
295C4×C8—28no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—41no—
310C24—41no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24#10414yes—
319C24C2×C22⋊C411yescharacteristic
320C24C2×C22⋊C411yescharacteristic
321C24C2×C22⋊C411yescharacteristic
322C2×Q8—24no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—88no—
329C2×D4—24no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—88no—
342C2×D4—816no—
343C22×C4—216no—
344C22×C4—216no—
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—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
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—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—416no—
400C22×C4—416no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4#10014yes—
411C22×C4#10414yes—
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C4×D414yes—
422C22×C4#12012yes—
423C22×C4#12114yes—
424C22×C4#12312yes—
425C22×C4C23×C411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428M16—232no—
429M16—232no—
430C2×C8—232no—
431C2×C8—28no—
432C2×C8—28no—
433C2×C8—28no—
434C2×C8—44no—
435C2×C8—44no—
436C2×C8—44no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—42no—
448C22⋊C4—44no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C42—22no—
457C42—22no—
458C42—21no—
459C42—21no—
460C42—22no—
461C42—22no—
462C42—21no—
463C42—21no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42C4×D414yes—
471C42#12012yes—
472C42#12114yes—
473C42#12312yes—
474C42C22×D412yes—
475C42C2×#4112yes—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—28no—
490C23—21no—
491C23—28no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—84no—
505C23—84no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23#38612yes—
512C23C2×#10012yes—
513C23C2×#10412yes—
514C23C2×#10412yes—
515C23C2×#10412yes—
516C23C2×#10412yes—
517C23#42112yes—
518C23C22×C22⋊C411yescharacteristic
519Q8—84no—
520Q8—84no—
521D4—84no—
522D4—88no—
523D4—88no—
524D4—84no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—21no—
533C2×C4—21no—
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—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4#38612yes—
583C2×C4#38612yes—
584C2×C4#38612yes—
585C2×C4C2×#10012yes—
586C2×C4C2×#10012yes—
587C2×C4#41312yes—
588C2×C4C2×#10412yes—
589C2×C4C2×#10412yes—
590C2×C4#42112yes—
591C2×C4C2×C4×D412yes—
592C2×C4C2×#12011yescharacteristic
593C2×C4C2×#12112yes—
594C2×C4C2×#12311yescharacteristic
595C8—216no—
596C8—84no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—416no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—168no—
625C22#152714yescomplemented
626C22#189614yes—
627C22C2×#38611yescharacteristic
628C22C22×#10011yescharacteristic
629C22C2×#41311yescharacteristic
630C22C22×#10411yescharacteristic
631C22C22×#10411yescharacteristic
632C22C2×#42111yescharacteristic
633C22C2×#42111yescharacteristic
634C4—22no—
635C4—22no—
636C4—22no—
637C4—21no—
638C4—21no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—41no—
645C4—41no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4#190014yes—
650C2—24no—
651C2—24no—
652C2—42no—
653C2—162no—
654C2C2×#152711yescharacteristic
655C2#1764211yescharacteristic
656C2#1764711yescharacteristic
657C2#1765011yescharacteristic
658C2#2090211yescharacteristic
659C2C2×#189611yescharacteristic
660C2C2×#190011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C4×M16C2semidirect
3. #20715C2semidirect
4. C2×#1527C2semidirect
5. #9120C2semidirect
6. #9119C2semidirect
7. #8033C2semidirect
18. C2×C4×M16C22semidirect
19. C22×#97C22semidirect
40. #1749C22semidirect
41. #1749C22semidirect
43. C2×#329C22semidirect
44. #1295C22semidirect
106. C4×M16D4semidirect
293. #97C2×D4semidirect
625. C22#1527semidirect

Maximal subgroups

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

subgroupindex
C22×C4×M162
C2×#28502
#207152
C2×#15272
#91202
#91192
#80332