Finite Group Databasebeta

← #432151

Subgroups — #432151

Order 512.

1794 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C211yesmaximal, characteristic
2C2×#2043C211yesmaximal, characteristic, complemented
3C2×#2024C212yesmaximal, complemented
4C2×#1956C211yesmaximal, characteristic, complemented
5C2×#1872C212yesmaximal
6#15178C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic, complemented
8#11067C212yesmaximal, complemented
9#10970C212yesmaximal
10#10921C211yesmaximal, characteristic
11#10880C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8—21no—
15C22×C22⋊Q8C2212yes—
16C23×C4⋊C4C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2212yes—
22C22×C4×C8C2211yescharacteristic
23#2440—24no—
24#2440—24no—
25C2×#398—21no—
26C2×#395—21no—
27C2×#391C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#385—21no—
31C2×#384C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#2043—24no—
35#2024—28no—
36#1956—24no—
37#1872—28no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44#1444—21no—
45#1444—21no—
46#1439—21no—
47#1439—21no—
48C4⋊#95C2211yescharacteristic
49C23×Q8—21no—
50C23×Q8C2312yes—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—42no—
62C2×C4⋊Q8—42no—
63C2×C22⋊Q8—28no—
64C2×C22⋊Q8—24no—
65C2×C22⋊Q8—24no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—44no—
69C2×C22⋊Q8—42no—
70C2×C22⋊Q8—42no—
71C2×C22⋊Q8—44no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4—21no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4D411yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—42no—
106C2×#103—42no—
107C2×#103—42no—
108C2×C22⋊C8—24no—
109C2×C22⋊C8—24no—
110C2×C22⋊C8—42no—
111C2×C4×C8—22no—
112C2×C4×C8—42no—
113#398—28no—
114#395—28no—
115#391—24no—
116#391—42no—
117#389—22no—
118#389—22no—
119#389—22no—
120#389D414yes—
121#385—28no—
122#384—22no—
123#384D414yes—
124#380—22no—
125#380—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95D411yescharacteristic
136C2×#95D411yescharacteristic
137#340—24no—
138#340—24no—
139#340—42no—
140#340D418yes—
141#336—24no—
142#336—22no—
143C25C2×D411yescharacteristic
144C22×Q8—24no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—44no—
159C22×Q8—42no—
160C22×Q8—42no—
161C22×Q8—82no—
162C22×Q8#4114yes—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4—24no—
177C23×C4—44no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—82no—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4#4114yes—
187C23×C4C2411yescharacteristic
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8C2×D414yes—
202C4⋊Q8—28no—
203C4⋊Q8—48no—
204C4⋊Q8—44no—
205C4⋊Q8—48no—
206C4⋊Q8—44no—
207C4⋊Q8—44no—
208C22⋊Q8—216no—
209C22⋊Q8—432no—
210C22⋊Q8—48no—
211C22⋊Q8—48no—
212C22⋊Q8—48no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
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—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—48no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—82no—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D412yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4SD1614yes—
271C2×C22⋊C4Q1614yes—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280#103—48no—
281#103—48no—
282#103—48no—
283#103—48no—
284C22⋊C8—48no—
285C4×C8—44no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—22no—
290#95—22no—
291#95—44no—
292#95—44no—
293#95—42no—
294#95—42no—
295C24—21no—
296C24—21no—
297C24—22no—
298C24—22no—
299C24—21no—
300C24—21no—
301C24—22no—
302C24—22no—
303C24#12412yes—
304C24C2×SD1612yes—
305C24C2×Q1612yes—
306C24C22×D411yescharacteristic
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—48no—
314C2×Q8—48no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—88no—
338C2×Q8—84no—
339C2×Q8—88no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—48no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—48no—
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×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—84no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4#12012yes—
412C22×C4#12012yes—
413C22×C4#12412yes—
414C22×C4C2×SD1612yes—
415C22×C4C2×SD1612yes—
416C22×C4C2×Q1612yes—
417C22×C4C2×Q1612yes—
418C22×C4AGL(1,ℤ/8)12yes—
419C22×C4#13712yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C42−1+412yes—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—416no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—84no—
466C4⋊C4#12714yes—
467C22⋊C4—28no—
468C22⋊C4—28no—
469C22⋊C4—48no—
470C22⋊C4—416no—
471C42—22no—
472C42—22no—
473C42—24no—
474C42—41no—
475C42—41no—
476C42—42no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—41no—
490C23—42no—
491C23—41no—
492C23—42no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23#48714yes—
500C23C2×#12011yescharacteristic
501C23C2×#12411yescharacteristic
502C23#53611yescharacteristic
503C23D4212yes—
504C23#54612yes—
505C23C22×SD1611yescharacteristic
506C23C22×Q1611yescharacteristic
507C23C2×AGL(1,ℤ/8)11yescharacteristic
508C23C2×#13711yescharacteristic
509C23#57712yes—
510C23#57812yes—
511Q8—44no—
512Q8—44no—
513Q8—44no—
514Q8—44no—
515Q8—816no—
516Q8—82no—
517Q8—82no—
518Q8—82no—
519Q8—82no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—816no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4#45014yes—
583C2×C4#45114yes—
584C2×C4#52511yescharacteristic
585C2×C4#52512yes—
586C2×C4C2×#12711yescharacteristic
587C2×C4#53511yescharacteristic
588C2×C4#53511yescharacteristic
589C2×C4#53712yes—
590C2×C4#53812yes—
591C2×C4#53912yes—
592C2×C4D4212yes—
593C2×C4#54612yes—
594C2×C4#54612yes—
595C2×C4#54812yes—
596C2×C4#54812yes—
597C2×C4#54812yes—
598C2×C4#55112yes—
599C2×C4#56212yes—
600C8—44no—
601C8—82no—
602C22—24no—
603C22—28no—
604C22—42no—
605C22—42no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22#240411yescharacteristic
614C22#243311yescharacteristic
615C22#243611yescharacteristic
616C22#247212yes—
617C22#268112yes—
618C22C2×#45011yescharacteristic
619C22C2×#45111yescharacteristic
620C22#300911yescharacteristic
621C22#301111yescharacteristic
622C22C2×#48711yescharacteristic
623C22#313712yes—
624C22#314111yescharacteristic
625C22#322911yescharacteristic
626C22#323011yescharacteristic
627C22#324212yes—
628C22#327212yes—
629C22#327612yes—
630C22D4×SD1612yes—
631C22D4×Q1612yes—
632C22#329812yes—
633C22#329912yes—
634C22#330812yes—
635C22#331612yes—
636C4—24no—
637C4—24no—
638C4—42no—
639C4—42no—
640C4—44no—
641C4—42no—
642C4—84no—
643C4—84no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4#246312yes—
648C4#254012yes—
649C4#267612yes—
650C4#269612yes—
651C2—44no—
652C2#1545311yescharacteristic
653C2#2396111yescharacteristic
654C2#2397011yescharacteristic
655C2#2575311yescharacteristic
656C2#2576011yescharacteristic
657C2#2590211yescharacteristic
658C2#2591411yescharacteristic
659C2#2725812yes—
660C2#2727812yes—
661C2#3112012yes—
662C2#3112212yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2043C2semidirect
3. C2×#2024C2semidirect
4. C2×#1956C2semidirect
7. #11357C2semidirect
8. #11067C2semidirect
11. #10880C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
C2×#20432
C2×#20242
C2×#19562
C2×#18722
#151782
#113572
#110672
#109702
#109212
#108802