Finite Group Databasebeta

← #188814

Subgroups — #188814

Order 512.

1341 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×M16C211yesmaximal, characteristic
2#20722C211yesmaximal, characteristic
3#20661C212yesmaximal, complemented
4C2×#1799C211yesmaximal, characteristic
5#8610C214yesmaximal, complemented
6#8609C212yesmaximal, complemented
7#8608C212yesmaximal, complemented
8#7983C212yesmaximal, complemented
9C23×M16C411yescharacteristic, complemented
10C23×M16C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12D4×M16—22no—
13D4×M16—24no—
14D4×M16—22no—
15C2×#436C412yescomplemented
16C2×#435C412yescomplemented
17C2×#435C2212yes—
18C2×C8×D4C2212yes—
19C2×#428—22no—
20C2×#423C411yescharacteristic, complemented
21C22×#99—21no—
22C2×#407C411yescharacteristic, complemented
23C2×#407C2211yescharacteristic
24C2×C4×M16C411yescharacteristic, complemented
25#2280C2212yes—
26C2×#380C2212yes—
27#1819—21no—
28#1809—24no—
29#1808C2214yescomplemented
30#1807C2214yescomplemented
31#1799—21no—
32#1799—21no—
33#1799C2212yes—
34#1799C2212yes—
35#1799C2211yescharacteristic
36#1799C2211yescharacteristic
37#1799C2212yes—
38#1781—21no—
39C4×#99—21no—
40C4×#99—21no—
41C2×#342—21no—
42#1736—22no—
43#1289C2214yescomplemented
44#1289C2214yescomplemented
45C23×D4D411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—21no—
49C22×M16—22no—
50C22×M16—21no—
51C22×M16—44no—
52C22×M16—44no—
53C22×M16C2×C411yescharacteristic
54C22×M16C2×C411yescharacteristic
55C22×M16C2×C411yescharacteristic
56C22×M16C2×C411yescharacteristic
57C22×M16C2×C412yes—
58C22×M16C2×C412yescomplemented
59C22×M16D412yes—
60C22×M16D414yes—
61C22×M16D412yes—
62C23×C8C2×C412yescomplemented
63C23×C8C2312yes—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4Q811yescharacteristic
73C2×C4×D4Q811yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×#117—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4Q811yescharacteristic
79C22×C22⋊C4—22no—
80C22×C22⋊C4D411yescharacteristic
81C22×C22⋊C4Q811yescharacteristic
82C22×C42C2×C411yescharacteristic
83#436—24no—
84#436—24no—
85#435—24no—
86#435—24no—
87#435—24no—
88#435—24no—
89C8×D4—24no—
90C8×D4—24no—
91#428—48no—
92C2×#107—24no—
93C2×#106—24no—
94#423—22no—
95#423—22no—
96C2×C4⋊C8C2×C412yescomplemented
97C2×C4⋊C8C2×C412yescomplemented
98C2×#99—22no—
99C2×#99—22no—
100C2×#99—22no—
101C2×#99—21no—
102C2×#99—21no—
103C2×#99—44no—
104#407—22no—
105#407—22no—
106#407—22no—
107#407—22no—
108C2×C22⋊C8C2×C412yes—
109C2×C22⋊C8C2×C412yescomplemented
110C2×C22⋊C8C2×C412yescomplemented
111C2×C22⋊C8C2312yes—
112C4×M16—21no—
113C4×M16—22no—
114C4×M16—21no—
115C2×#97C2×C412yescomplemented
116C2×C4×C8C2×C412yescomplemented
117#381—22no—
118#380—22no—
119#380—22no—
120#380—22no—
121#380—22no—
122#380C2312yes—
123#380C2314yes—
124#380C2312yes—
125#379—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C2×#95—22no—
129#342—22no—
130#342—22no—
131#342—21no—
132#342—21no—
133#342—22no—
134#340—24no—
135#340—24no—
136C25C4⋊C411yescharacteristic
137C25C2×D411yescharacteristic
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—42no—
143C22×D4C4⋊C411yescharacteristic
144C22×D4C4⋊C411yescharacteristic
145C22×D4C2×D411yescharacteristic
146C22×D4C2×D411yescharacteristic
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4C4⋊C411yescharacteristic
157C23×C4C4⋊C411yescharacteristic
158C23×C4C4⋊C411yescharacteristic
159C23×C4C4⋊C411yescharacteristic
160C23×C4C4⋊C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×Q811yescharacteristic
169C23×C4C2×Q811yescharacteristic
170C23×C4#4112yes—
171C23×C4#4112yes—
172C23×C4C2411yescharacteristic
173C2×M16—24no—
174C2×M16—28no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—44no—
182C2×M16—48no—
183C2×M16—44no—
184C2×M16—42no—
185C2×M16—44no—
186C2×M16—44no—
187C2×M16—48no—
188C2×M16—44no—
189C2×M16—42no—
190C2×M16—42no—
191C2×M16C22⋊C411yescharacteristic
192C2×M16C22⋊C412yes—
193C2×M16C22⋊C412yes—
194C2×M16C22⋊C414yes—
195C2×M16C22⋊C412yes—
196C2×M16C22⋊C414yes—
197C2×M16C22⋊C411yescharacteristic
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—44no—
201C22×C8—44no—
202C22×C8C22×C412yes—
203C22×C8C22×C412yes—
204C22×C8C22×C412yes—
205C22×C8C22×C412yes—
206C22×C8C2×D414yes—
207C22×C8C2×D414yes—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224#117—44no—
225#117—44no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4C4⋊C411yescharacteristic
235C2×C4⋊C4C4⋊C411yescharacteristic
236C2×C4⋊C4C2×Q811yescharacteristic
237C2×C4⋊C4C2×Q811yescharacteristic
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4C4⋊C411yescharacteristic
251C2×C22⋊C4C4⋊C411yescharacteristic
252C2×C22⋊C4C4⋊C411yescharacteristic
253C2×C22⋊C4C4⋊C411yescharacteristic
254C2×C22⋊C4C2×D411yescharacteristic
255C2×C22⋊C4C2×D411yescharacteristic
256C2×C22⋊C4C2×Q811yescharacteristic
257C2×C22⋊C4C2×Q811yescharacteristic
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—42no—
263C2×C42C4⋊C411yescharacteristic
264C2×C42C4⋊C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267#107—48no—
268#106—48no—
269C4⋊C8—24no—
270C4⋊C8—24no—
271#99—44no—
272#99—44no—
273#99—42no—
274#99—44no—
275#99—42no—
276C22⋊C8—24no—
277C22⋊C8—24no—
278C22⋊C8—24no—
279C22⋊C8—24no—
280#97—22no—
281#97—22no—
282C4×C8—22no—
283C4×C8—22no—
284#95—24no—
285#95—22no—
286#95—22no—
287#95—42no—
288#95—42no—
289C24—21no—
290C24—21no—
291C24—22no—
292C24—21no—
293C24—21no—
294C24—22no—
295C24—42no—
296C24—41no—
297C24—42no—
298C24—81no—
299C24—81no—
300C24C2×C4⋊C411yescharacteristic
301C24C2×C4⋊C411yescharacteristic
302C24C2×C4⋊C411yescharacteristic
303C24C2×C4⋊C411yescharacteristic
304C24#12112yes—
305C24#12112yes—
306C24C22×D411yescharacteristic
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4AGL(1,ℤ/8)12yes—
324C2×D4#13712yes—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
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—22no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—81no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—84no—
376C22×C4C2×C22⋊C412yes—
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C4⋊C411yescharacteristic
380C22×C4C2×C4⋊C411yescharacteristic
381C22×C4C2×C4⋊C411yescharacteristic
382C22×C4C2×C4⋊C411yescharacteristic
383C22×C4C2×C4⋊C411yescharacteristic
384C22×C4C2×C4⋊C411yescharacteristic
385C22×C4C2×C4⋊C411yescharacteristic
386C22×C4C2×C4⋊C411yescharacteristic
387C22×C4#11711yescharacteristic
388C22×C4#11711yescharacteristic
389C22×C4#11712yes—
390C22×C4#12112yes—
391C22×C4#12112yes—
392C22×C4C22⋊Q812yes—
393C22×C4C22⋊Q812yes—
394C22×C4C22⋊Q812yes—
395C22×C4C22⋊Q812yes—
396C22×C4C23×C411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C22×Q811yescharacteristic
399C22×C4C2×#4112yes—
400M16—22no—
401M16—24no—
402M16—24no—
403M16—28no—
404M16—44no—
405M16—42no—
406M16—44no—
407M16—42no—
408M16—42no—
409M16—42no—
410M16—42no—
411M16—44no—
412M16#9914yescomplemented
413C2×C8—28no—
414C2×C8—24no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—42no—
418C2×C8—44no—
419C2×C8—44no—
420C2×C8—44no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8C2×C22⋊C412yes—
424C2×C8C2×C22⋊C414yes—
425C2×C8C2×C22⋊C412yes—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—84no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—44no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C22⋊C4—84no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—84no—
463C23—24no—
464C23—22no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—22no—
470C23—22no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23—22no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—41no—
484C23—42no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23—81no—
495C23—81no—
496C23—82no—
497C23—82no—
498C23—84no—
499C23—84no—
500C23—84no—
501C23#38012yes—
502C23#38011yescharacteristic
503C23#38011yescharacteristic
504C23#42812yescomplemented
505C23C22×C4⋊C411yescharacteristic
506C23C2×#12112yes—
507C23C2×AGL(1,ℤ/8)11yescharacteristic
508C23C2×#13711yescharacteristic
509C23#57512yes—
510D4—24no—
511D4—24no—
512D4—44no—
513D4—44no—
514D4—42no—
515D4—42no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
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—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—168no—
576C2×C4#38011yescharacteristic
577C2×C4#38012yes—
578C2×C4#38011yescharacteristic
579C2×C4C2×#9912yes—
580C2×C4#41012yes—
581C2×C4#42812yescomplemented
582C2×C4C22×C22⋊C411yescharacteristic
583C2×C4C2×#11711yescharacteristic
584C2×C4C2×C22⋊Q812yes—
585C8—22no—
586C8—24no—
587C8—42no—
588C8—42no—
589C8C2×#9914yescomplemented
590C22—21no—
591C22—22no—
592C22—21no—
593C22—21no—
594C22—22no—
595C22—21no—
596C22—21no—
597C22—22no—
598C22—21no—
599C22—22no—
600C22—22no—
601C22—21no—
602C22—22no—
603C22—41no—
604C22—41no—
605C22—42no—
606C22—44no—
607C22—44no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—82no—
618C22—82no—
619C22—81no—
620C22—81no—
621C22—84no—
622C22—81no—
623C22—81no—
624C22—84no—
625C22#179912yes—
626C22C2×#38011yescharacteristic
627C22C2×#42811yescharacteristic
628C22#291111yescharacteristic
629C22#305512yes—
630C22#305612yes—
631C4—21no—
632C4—21no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—162no—
642C4—162no—
643C4—164no—
644C4#179912yes—
645C4C22×#9911yescharacteristic
646C4C2×#41011yescharacteristic
647C2—22no—
648C2—22no—
649C2—22no—
650C2—21no—
651C2—21no—
652C2—42no—
653C2—41no—
654C2—42no—
655C2—81no—
656C2C2×#179911yescharacteristic
657C2#2066111yescharacteristic
658C2#2067111yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20661C2semidirect
5. #8610C2semidirect
6. #8609C2semidirect
7. #8608C2semidirect
8. #7983C2semidirect
9. C23×M16C4semidirect
15. C2×#436C4semidirect
16. C2×#435C4semidirect
20. C2×#423C4semidirect
22. C2×#407C4semidirect
24. C2×C4×M16C4semidirect
29. #1808C22semidirect
30. #1807C22semidirect
43. #1289C22semidirect
44. #1289C22semidirect
58. C22×M16C2×C4semidirect
62. C23×C8C2×C4semidirect
96. C2×C4⋊C8C2×C4semidirect
97. C2×C4⋊C8C2×C4semidirect
109. C2×C22⋊C8C2×C4semidirect
110. C2×C22⋊C8C2×C4semidirect
115. C2×#97C2×C4semidirect
116. C2×C4×C8C2×C4semidirect
412. M16#99semidirect
504. C23#428semidirect
581. C2×C4#428semidirect
589. C8C2×#99semidirect

Maximal subgroups

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

subgroupindex
C2×D4×M162
#207222
#206612
C2×#17992
#86102
#86092
#86082
#79832