Finite Group Databasebeta

← #358193

Subgroups — #358193

Order 512.

1964 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×SD16C211yesmaximal, characteristic
2C2×#2293C211yesmaximal, characteristic
3C2×C4×#103C211yesmaximal, characteristic, complemented
4C2×C4×#102C211yesmaximal, characteristic
5C4×#391C211yesmaximal, characteristic, complemented
6#11220C212yesmaximal, complemented
7#11189C212yesmaximal
8#11067C212yesmaximal, complemented
9#11023C212yesmaximal
10C4×#340C211yesmaximal, characteristic, complemented
11C4×#336C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#117—21no—
16C2×C4×SD16—24no—
17C2×C4×SD16—24no—
18C2×C4×SD16C418yescomplemented
19C22×#106C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×#391C2212yes—
27#2293—24no—
28#2293—24no—
29C2×#390C2212yes—
30C2×#389C2212yes—
31C2×#388—22no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4—21no—
35C2×C43C2211yescharacteristic
36C4×#103—24no—
37C4×#102—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1436—22no—
45C4⋊#95C2212yes—
46C4×#95—21no—
47C4×#95—21no—
48C23×Q8C2311yescharacteristic
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C22×SD16—24no—
52C22×SD16—22no—
53C22×SD16—22no—
54C22×SD16—24no—
55C22×SD16C2×C414yes—
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8C2×C414yes—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4C2×C414yes—
69C2×C4×D4D414yes—
70C2×C4×D4D414yes—
71C2×#117—28no—
72C2×#117—44no—
73C2×#117—44no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42D411yescharacteristic
87C22×C42D412yes—
88C22×C42D411yescharacteristic
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91C4×SD16—232no—
92C2×#106—22no—
93C2×#106—22no—
94C2×#106C2×C414yes—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—24no—
100C2×#103C2×C414yes—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102C2×C414yes—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8C2×C414yes—
113#391—24no—
114#391—24no—
115#390—24no—
116#390—24no—
117#389—24no—
118#389—24no—
119#388—216no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4D414yes—
124C4×C22⋊C4—28no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—21no—
132C43—22no—
133C43—22no—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—24no—
138#336—24no—
139#336—24no—
140C25—21no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8C22×C412yes—
150C22×Q8#4114yes—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4C22×C412yes—
159C22×D4C2×D412yes—
160C22×D4C2×D412yes—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2411yescharacteristic
178C2×SD16—216no—
179C2×SD16—216no—
180C2×SD16—28no—
181C2×SD16—28no—
182C2×SD16—28no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8C22×C412yes—
197C22×C8C22×C412yes—
198C22×C8#4114yes—
199C22×C8#4114yes—
200C4×Q8—28no—
201C4×Q8—28no—
202C4×Q8—44no—
203C4×Q8—44no—
204C4×D4—28no—
205C4×D4—28no—
206C4×D4—28no—
207C4×D4C22⋊C4116yes—
208#117—432no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4#4114yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42SD1614yescomplemented
272C2×C42C22×C412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42#4112yes—
277C2×C42#4112yes—
278#106—28no—
279#103—216no—
280#103—28no—
281#102—216no—
282#102—28no—
283C4×C8—28no—
284C4×C8—44no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—44no—
290#95—42no—
291#95—44no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—42no—
296C24—42no—
297C24—41no—
298C24—42no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—42no—
303C24C22×D411yescharacteristic
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8#11714yescomplemented
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4C2×C22⋊C414yes—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—216no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4C4×D412yes—
393C22×C4C4×D412yes—
394C22×C4#12012yes—
395C22×C4#12012yes—
396C22×C4#12112yes—
397C22×C4#12112yes—
398C22×C4C2×SD1614yes—
399C22×C4C2×SD1612yes—
400C22×C4#13514yes—
401C22×C4#13512yes—
402C22×C4AGL(1,ℤ/8)12yes—
403C22×C4#13712yes—
404C22×C4C23×C411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4112yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4111yescharacteristic
414SD16—232no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8#11714yescomplemented
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—416no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—48no—
449C4⋊C4C2×C22⋊C414yes—
450C22⋊C4—216no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C42—28no—
454C42—48no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42—42no—
459C42—48no—
460C42—48no—
461C42—44no—
462C42—84no—
463C42C2×C22⋊C414yes—
464C42C4×D414yes—
465C42#12114yes—
466C42#12314yes—
467C23—24no—
468C23—44no—
469C23—44no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23C2×C4×D411yescharacteristic
491C23C2×#12011yescharacteristic
492C23C2×#12111yescharacteristic
493C23#52512yes—
494C23#53212yes—
495C23C22×SD1611yescharacteristic
496C23C2×#13511yescharacteristic
497C23C2×#13512yes—
498C23C2×AGL(1,ℤ/8)11yescharacteristic
499C23C2×#13711yescharacteristic
500C23#57512yes—
501Q8—24no—
502Q8—24no—
503Q8—44no—
504Q8—44no—
505Q8—42no—
506Q8—42no—
507D4—24no—
508D4—24no—
509D4—44no—
510D4—44no—
511D4—42no—
512D4—42no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—432no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4#38614yes—
564C2×C4#38614yes—
565C2×C4C4×SD1614yes—
566C2×C4#44014yes—
567C2×C4#45014yes—
568C2×C4#45214yes—
569C2×C4#46114yes—
570C2×C4#46314yes—
571C2×C4C22×C22⋊C411yescharacteristic
572C2×C4C2×#11711yescharacteristic
573C2×C4C2×#11711yescharacteristic
574C2×C4C2×C4×D411yescharacteristic
575C2×C4C4×#4111yescharacteristic
576C2×C4C4×#4111yescharacteristic
577C2×C4C2×#12111yescharacteristic
578C2×C4C2×#12311yescharacteristic
579C2×C4#52512yes—
580C2×C4#52512yes—
581C2×C4#52512yes—
582C2×C4#52512yes—
583C2×C4#52912yes—
584C2×C4#52912yes—
585C2×C4#52912yes—
586C2×C4#52912yes—
587C2×C4#53212yes—
588C2×C4#53212yes—
589C8—24no—
590C8—44no—
591C8—42no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—84no—
596C22—84no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—84no—
601C22—82no—
602C22—84no—
603C22C2×#38611yescharacteristic
604C22#229011yescharacteristic
605C22#229311yescharacteristic
606C22#238812yes—
607C22#239212yes—
608C22C2×C4×SD1611yescharacteristic
609C22C4×#13511yescharacteristic
610C22C2×#44011yescharacteristic
611C22#294311yescharacteristic
612C22C2×#45011yescharacteristic
613C22C2×#45211yescharacteristic
614C22#300212yes—
615C22C2×#46111yescharacteristic
616C22C2×#46311yescharacteristic
617C22#303512yes—
618C22#310212yes—
619C22#310312yes—
620C22#310612yes—
621C22#310812yes—
622C22#316712yes—
623C22#316812yes—
624C22#317112yes—
625C22#317212yes—
626C4—24no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—44no—
631C4—84no—
632C4—84no—
633C4—82no—
634C4#187718yes—
635C4#228011yescharacteristic
636C4#229011yescharacteristic
637C4#229311yescharacteristic
638C4#229711yescharacteristic
639C4#239212yes—
640C4#239512yes—
641C2—84no—
642C2#1383911yescharacteristic
643C2C2×#187711yescharacteristic
644C2#2092411yescharacteristic
645C2#2111511yescharacteristic
646C2#2112011yescharacteristic
647C2#2127411yescharacteristic
648C2#2127911yescharacteristic
649C2#2357012yes—
650C2#2357212yes—
651C2#2367712yes—
652C2#2368012yes—
subgroupG/H
centerC23×C4C2×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC24C22×D4
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×C4×#103C2semidirect
5. C4×#391C2semidirect
6. #11220C2semidirect
8. #11067C2semidirect
10. C4×#340C2semidirect
11. C4×#336C2semidirect
18. C2×C4×SD16C4semidirect
271. C2×C42SD16semidirect
321. C2×Q8#117semidirect
431. C2×C8#117semidirect

Maximal subgroups

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

subgroupindex
C22×C4×SD162
C2×#22932
C2×C4×#1032
C2×C4×#1022
C4×#3912
#112202
#111892
#110672
#110232
C4×#3402
C4×#3362