Finite Group Databasebeta

← #6512754

Subgroups — #6512754

Order 512.

3244 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3503C211yesmaximal, characteristic
2C2×#3104C211yesmaximal, characteristic, complemented
3C22×#462C211yesmaximal, characteristic, complemented
4#34952C211yesmaximal, characteristic
5#24243C214yesmaximal, complemented
6#24212C214yesmaximal, complemented
7#24191C214yesmaximal, complemented
8#24155C214yesmaximal, complemented
9#24042C214yesmaximal, complemented
10#23856C214yesmaximal, complemented
11#21885C211yesmaximal, characteristic, complemented
12#21847C211yesmaximal, characteristic
13#21804C211yesmaximal, characteristic, complemented
14C23×Q16C2212yes—
15#3503—28no—
16#3503—28no—
17C2×Q82—21no—
18C2×#557C2212yes—
19C2×#554—22no—
20C2×#554—21no—
21C2×D4×Q8C2214yes—
22C2×D4×Q8C2212yes—
23C2×#544—22no—
24C2×#544—21no—
25C2×#533C2212yes—
26C2×#533C2211yescharacteristic
27C22×C4⋊Q8C2212yes—
28C22×C4⋊Q8C2211yescharacteristic
29C2×#520—21no—
30C2×C4×#41C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32#3104—28no—
33C2×#484C2212yes—
34C2×#477C2212yes—
35C2×#462—28no—
36C2×#462C2214yes—
37C2×#462C2212yes—
38C2×#462C2214yes—
39C2×#451C2212yes—
40C2×#451C2218yes—
41C2×C4×Q16C2214yes—
42C2×#431C2211yescharacteristic
43C22×C4⋊C8C2211yescharacteristic
44C22×#103C2212yes—
45#2582C2212yes—
46#2581—21no—
47#2580—21no—
48#2579—22no—
49#2538C2212yes—
50#2487C2212yes—
51#2409C2212yes—
52#2396—22no—
53C4×C22⋊Q8C2214yes—
54#2297—21no—
55#2296—21no—
56#2285C2211yescharacteristic
57#2043C2218yes—
58#2036C2218yes—
59#2016C2218yes—
60#2001C2218yes—
61#1987C2212yes—
62#1984C2212yes—
63#1981C2214yes—
64#1981C2212yes—
65#1956C2214yes—
66#1954C2212yes—
67#1892C2214yes—
68#1878C2218yes—
69#1869C2214yes—
70#1863C2218yes—
71#1803C2214yes—
72#1797C2214yes—
73#1787C2214yes—
74#1787C2218yes—
75C22×#41D411yescharacteristic
76C23×Q8C2311yescharacteristic
77C23×Q8C2314yes—
78C23×Q8C2312yes—
79C22×Q16—28no—
80C22×Q16—48no—
81C22×Q16C2318yes—
82C22×Q16C2314yes—
83C23×C8C2312yes—
84Q82—216no—
85#557—28no—
86#557—28no—
87#554—216no—
88#554—232no—
89D4×Q8—216no—
90D4×Q8—28no—
91D4×Q8—216no—
92D4×Q8—28no—
93#544—232no—
94#544—216no—
95#533—28no—
96#533—28no—
97#533—24no—
98#533D418yes—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—28no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—24no—
108C2×C4⋊Q8D418yes—
109C2×C4⋊Q8C2314yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2311yescharacteristic
114C2×#125—22no—
115C2×#125—24no—
116C2×#125—22no—
117C2×#125—21no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8—24no—
122C2×C22⋊Q8—24no—
123C2×C22⋊Q8C2314yes—
124C2×C22⋊Q8C2314yes—
125C2×C22⋊Q8C2312yes—
126#520—216no—
127C4×#41—24no—
128C4×#41—24no—
129C2×C4×Q8—22no—
130C2×C4×Q8—22no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×Q8—24no—
135C2×C4×Q8—24no—
136C2×C4×Q8D411yescharacteristic
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2314yes—
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2311yescharacteristic
142C2×C4×D4—24no—
143C2×C4×D4—22no—
144C2×C4×D4D412yes—
145C2×C4×D4D411yescharacteristic
146C2×C4×D4C2312yes—
147C2×C4×D4C2311yescharacteristic
148C2×#117—22no—
149C2×#117—21no—
150C2×#117—21no—
151C2×#117D412yes—
152C2×#117D411yescharacteristic
153C22×C4⋊C4—22no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4—22no—
156C22×C4⋊C4—21no—
157C22×C4⋊C4—21no—
158C22×C4⋊C4C2311yescharacteristic
159C22×C42—21no—
160C22×C42C2312yes—
161C22×C42C2311yescharacteristic
162#484—28no—
163#477—28no—
164#462—28no—
165#462—216no—
166#462—216no—
167#451—28no—
168#451—416no—
169C4×Q16—216no—
170#431—24no—
171C2×#107C2314yes—
172C2×C4⋊C8—24no—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2312yes—
176C2×#103—28no—
177C2×#103C2314yes—
178C2×#103C2314yes—
179C2×#103C2318yes—
180C2×#103C2318yes—
181C2×#103C2318yes—
182C2×#103C2318yes—
183C2×C22⋊C8C2312yes—
184C2×C22⋊C8C2318yes—
185C2×C4×C8C2312yes—
186#400—22no—
187#399—22no—
188#398—24no—
189#396—24no—
190#395—22no—
191#393—22no—
192#391—22no—
193#391C2314yes—
194#390—22no—
195#389—22no—
196#389—22no—
197#389C2314yes—
198#387—24no—
199#386—22no—
200#386—22no—
201#385—24no—
202#385—22no—
203#385—22no—
204#384C2312yes—
205#382—21no—
206#380—21no—
207#380C2312yes—
208#380C2314yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—24no—
211C4×C4⋊C4C2312yes—
212C4×C22⋊C4—24no—
213#340C2318yes—
214#336C2318yes—
215C2×#41—24no—
216C2×#41—24no—
217C2×#41Q1618yescomplemented
218C22×Q8—28no—
219C22×Q8—28no—
220C22×Q8—21no—
221C22×Q8—24no—
222C22×Q8—22no—
223C22×Q8—24no—
224C22×Q8—21no—
225C22×Q8—24no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—42no—
229C22×Q8—44no—
230C22×Q8—44no—
231C22×Q8—48no—
232C22×Q8—48no—
233C22×Q8C2×D418yes—
234C22×Q8C2×D411yescharacteristic
235C22×Q8C2412yes—
236C22×Q8C2414yes—
237C22×Q8C2412yes—
238C22×D4C2×D412yes—
239C22×D4C2×D411yescharacteristic
240C23×C4—24no—
241C23×C4—22no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2412yes—
249C23×C4C2411yescharacteristic
250C2×Q16—216no—
251C2×Q16—28no—
252C2×Q16—216no—
253C2×Q16—28no—
254C2×Q16—48no—
255C2×Q16—44no—
256C2×Q16—432no—
257C22×C8—24no—
258C22×C8—44no—
259C22×C8C2414yes—
260C22×C8C2414yes—
261C4⋊Q8—216no—
262C4⋊Q8—216no—
263C4⋊Q8—232no—
264C4⋊Q8—216no—
265C4⋊Q8—28no—
266C4⋊Q8—28no—
267C4⋊Q8—28no—
268C4⋊Q8—28no—
269C4⋊Q8—22no—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—22no—
273C4⋊Q8C2×D414yes—
274C4⋊Q8C2×D418yes—
275C4⋊Q8C2×D418yes—
276C4⋊Q8C2×D414yes—
277#125—216no—
278#125—232no—
279#125—216no—
280#125—28no—
281C22⋊Q8—216no—
282C22⋊Q8—232no—
283C22⋊Q8—232no—
284C22⋊Q8—232no—
285C22⋊Q8—28no—
286C22⋊Q8—28no—
287C22⋊Q8—24no—
288C22⋊Q8—28no—
289C22⋊Q8—28no—
290C22⋊Q8—44no—
291C22⋊Q8C2×D418yes—
292C4×Q8—216no—
293C4×Q8—216no—
294C4×Q8—28no—
295C4×Q8—28no—
296C4×Q8—28no—
297C4×Q8—24no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—44no—
306C4×Q8—44no—
307C4×Q8C2×D418yes—
308C4×D4—232no—
309C4×D4—216no—
310C4×D4—28no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—22no—
315C4×D4—22no—
316#117—216no—
317#117—28no—
318#117—24no—
319#117—28no—
320#117—44no—
321C2×C4⋊C4—216no—
322C2×C4⋊C4—216no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2414yes—
354C2×C4⋊C4C2412yes—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C42—24no—
364C2×C42—22no—
365C2×C42—24no—
366C2×C42—24no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—44no—
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D414yes—
376C2×C42C2×D414yes—
377C2×C42C2412yes—
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380#107—28no—
381C4⋊C8—24no—
382C4⋊C8—24no—
383C4⋊C8—24no—
384#103—216no—
385#103—28no—
386#103—28no—
387#103—48no—
388#103—48no—
389#103—48no—
390C22⋊C8—24no—
391C22⋊C8—48no—
392C4×C8—24no—
393#95—24no—
394#95—24no—
395#95—22no—
396#95—24no—
397#95—24no—
398#95—22no—
399C24C22×D412yes—
400C24C22×D411yescharacteristic
401#41—232no—
402C2×Q8—216no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—28no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—28no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—24no—
414C2×Q8—28no—
415C2×Q8—22no—
416C2×Q8—28no—
417C2×Q8—22no—
418C2×Q8—24no—
419C2×Q8—44no—
420C2×Q8—48no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—48no—
424C2×Q8—48no—
425C2×Q8—42no—
426C2×Q8—84no—
427C2×Q8—88no—
428C2×Q8—88no—
429C2×Q8—88no—
430C2×Q8—88no—
431C2×Q8—816no—
432C2×Q8C2×Q1614yescomplemented
433C2×Q8C22×D418yes—
434C2×Q8C22×D414yes—
435C2×Q8H(5,𝔽2)14yes—
436C2×D4—24no—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4C2×Q1618yes—
441C2×D4C2×Q1614yescomplemented
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—216no—
451C22×C4—28no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—82no—
465C22×C4—84no—
466C22×C4—88no—
467C22×C4#12014yes—
468C22×C4#12014yes—
469C22×C4C2×Q1614yes—
470C22×C4C2×Q1618yescomplemented
471C22×C4#13714yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C2511yescharacteristic
483Q16—416no—
484Q16—432no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—28no—
489C2×C8—24no—
490C2×C8—48no—
491C4⋊C4—216no—
492C4⋊C4—216no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—24no—
496C4⋊C4—28no—
497C4⋊C4—24no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—24no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—48no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—82no—
519C4⋊C4—84no—
520C4⋊C4C22×D414yes—
521C22⋊C4—216no—
522C22⋊C4—28no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—48no—
527C22⋊C4—44no—
528C22⋊C4—84no—
529C42—216no—
530C42—24no—
531C42—24no—
532C42—28no—
533C42—22no—
534C42—22no—
535C42—44no—
536C42—44no—
537C42—44no—
538C42#12014yes—
539C42#12014yes—
540C42C22×D412yes—
541C42C22×D412yes—
542C42C22×D412yes—
543C42C22×D412yes—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—22no—
548C23—24no—
549C23—22no—
550C23C2×#12012yes—
551C23C22×Q1614yes—
552C23C22×Q1612yes—
553C23C2×#13712yes—
554C23C23×D411yescharacteristic
555Q8—28no—
556Q8—24no—
557Q8—28no—
558Q8—28no—
559Q8—24no—
560Q8—28no—
561Q8—48no—
562Q8—48no—
563Q8—42no—
564Q8—42no—
565Q8—42no—
566Q8—42no—
567Q8—48no—
568Q8—88no—
569Q8—816no—
570D4—216no—
571D4—28no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—216no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—88no—
607C2×C4—816no—
608C2×C4—84no—
609C2×C4#451116yescomplemented
610C2×C4C2×#12012yes—
611C2×C4C2×#12012yes—
612C2×C4C2×#12012yes—
613C2×C4C2×#12012yes—
614C2×C4C2×#12012yes—
615C2×C4C22×Q1614yes—
616C2×C4C22×Q1612yes—
617C2×C4C22×Q1612yes—
618C2×C4C2×#13712yes—
619C2×C4C2×#13712yes—
620C2×C4C23×D411yescharacteristic
621C2×C4C23×D411yescharacteristic
622C2×C4C2×H(5,𝔽2)12yes—
623C2×C4#58512yes—
624C8—44no—
625C8—44no—
626C22—24no—
627C22—24no—
628C22—22no—
629C22—22no—
630C22—28no—
631C22—24no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22C2×#45114yes—
636C22C22×#12011yescharacteristic
637C22#348812yes—
638C22C23×Q1611yescharacteristic
639C22C22×#13711yescharacteristic
640C22#358612yes—
641C4—22no—
642C4—24no—
643C4—22no—
644C4—22no—
645C4—42no—
646C4—44no—
647C4—48no—
648C4—44no—
649C4—42no—
650C4—44no—
651C4—84no—
652C4—82no—
653C4C2×#45114yes—
654C4C2×#45114yes—
655C4#348411yescharacteristic
656C4#348511yescharacteristic
657C4#348812yes—
658C2—24no—
659C2—44no—
660C2#3551211yescharacteristic
661C2C22×#45111yescharacteristic
662C2#6036811yescharacteristic
663C2#6195212yes—
664C2#6196412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3104C2semidirect
3. C22×#462C2semidirect
5. #24243C2semidirect
6. #24212C2semidirect
7. #24191C2semidirect
8. #24155C2semidirect
9. #24042C2semidirect
10. #23856C2semidirect
11. #21885C2semidirect
13. #21804C2semidirect
217. C2×#41Q16semidirect
432. C2×Q8C2×Q16semidirect
441. C2×D4C2×Q16semidirect
470. C22×C4C2×Q16semidirect
609. C2×C4#451semidirect

Maximal subgroups

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

subgroupindex
C2×#35032
C2×#31042
C22×#4622
#349522
#242432
#242122
#241912
#241552
#240422
#238562
#218852
#218472
#218042