Finite Group Databasebeta

← #6467119

Subgroups — #6467119

Order 512.

2138 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C212yesmaximal
2C4×#533C211yesmaximal, characteristic
3C2×C4×C4⋊Q8C211yesmaximal, characteristic
4#28284C212yesmaximal, complemented
5#28255C212yesmaximal, complemented
6#28206C212yesmaximal, complemented
7#28161C212yesmaximal, complemented
8#28121C214yesmaximal
9#27648C212yesmaximal
10#24212C212yesmaximal
11#23679C214yesmaximal, complemented
12C4×#462C214yesmaximal, complemented
13#20808C211yesmaximal, characteristic
14#20734C212yesmaximal
15C2×#533C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2212yes—
18#3104—216no—
19C2×#484C2214yes—
20C2×#477C2214yes—
21C2×#462C2214yes—
22C2×#462C2214yes—
23C2×#451C2214yes—
24C2×C4×Q16C2218yes—
25C2×#431C2212yes—
26C4×C4⋊Q8—21no—
27C4×C4⋊Q8—21no—
28C4×C4⋊Q8—21no—
29C4×C4⋊Q8—21no—
30C4×C4⋊Q8C2211yescharacteristic
31C4×C4⋊Q8C2211yescharacteristic
32C4×C4⋊Q8C2211yescharacteristic
33C4×C4⋊Q8C2212yes—
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×#125—21no—
39C4×#125—21no—
40C4×C22⋊Q8—21no—
41C4×C22⋊Q8—21no—
42C4×C22⋊Q8C2212yes—
43C4×C22⋊Q8C2212yes—
44#2296—21no—
45#2295—21no—
46#2295C2212yes—
47C2×#391C2212yes—
48C2×#389—21no—
49C2×#389—21no—
50#2285C2211yescharacteristic
51C2×#384C2211yescharacteristic
52C42×Q8—21no—
53C42×Q8C2212yes—
54C4×#117—21no—
55C2×C4×C4⋊C4—21no—
56C2×C4×C4⋊C4—21no—
57C2×C43C2211yescharacteristic
58#2083C2214yes—
59#2072C2214yes—
60#2068C2218yes—
61#2056C2218yes—
62#2043C2214yes—
63#2016C2214yes—
64#1981C2214yes—
65#1947C2214yes—
66#1928C2218yes—
67#1878C2218yes—
68#1842C2212yes—
69#1831C2212yes—
70#1824C2214yes—
71#1803C2214yes—
72#1803C2214yes—
73#1803C2214yes—
74#1797C2212yes—
75#1797C2212yes—
76#1787C2214yes—
77C4×C4⋊C8C2212yes—
78C4×C4⋊C8C2212yes—
79C4×#103C2218yes—
80C23×Q8C2312yes—
81C22×Q16C2318yes—
82#533—24no—
83#533D418yes—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×#125—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8C2312yes—
94C2×C4×Q8—21no—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×Q8—28no—
100C2×C4×Q8—28no—
101C2×C4×Q8C2314yes—
102C2×C4×Q8C2314yes—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×#117—21no—
107C2×#117—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2312yes—
114C22×C42—22no—
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#484—216no—
123#477—216no—
124#462—216no—
125#462—216no—
126#451—216no—
127C4×Q16—232no—
128#431—28no—
129C2×#107C2318yes—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2318yes—
134C2×#103C2318yes—
135C2×#103C2318yes—
136C2×#103C2318yes—
137C2×C22⋊C8C2314yes—
138C2×C4×C8C2314yes—
139#391—22no—
140#391—22no—
141#391—21no—
142#391—21no—
143#391C2314yes—
144#391C2314yes—
145#391C2312yes—
146#391C2312yes—
147#389—21no—
148#389—21no—
149#389—21no—
150#389—21no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389C2312yes—
162#389C2312yes—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#385—22no—
168#385—22no—
169#384—21no—
170#384—21no—
171#384C2311yescharacteristic
172#384C2311yescharacteristic
173#384C2312yes—
174#384C2312yes—
175#382—21no—
176#380—21no—
177#380C2312yes—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204#376—22no—
205C2×#95—22no—
206C2×#95—22no—
207C43—22no—
208C43D411yescharacteristic
209C43D411yescharacteristic
210C43D411yescharacteristic
211C43D411yescharacteristic
212C43C2311yescharacteristic
213C43C2311yescharacteristic
214C43C2311yescharacteristic
215C43C2311yescharacteristic
216#340C2318yes—
217#340C2318yes—
218#336C2314yes—
219#336C2314yes—
220C22×Q8—28no—
221C22×Q8—21no—
222C22×Q8—22no—
223C22×Q8—24no—
224C22×Q8—24no—
225C22×Q8—44no—
226C22×Q8C2412yes—
227C22×Q8C2414yes—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2411yescharacteristic
237C2×Q16—216no—
238C2×Q16—216no—
239C2×Q16—48no—
240C22×C8C2418yes—
241C4⋊Q8—28no—
242C4⋊Q8—28no—
243C4⋊Q8—28no—
244C4⋊Q8—24no—
245C4⋊Q8—22no—
246C4⋊Q8—22no—
247C4⋊Q8C2×D414yes—
248C4⋊Q8C2×D414yes—
249C4⋊Q8#4118yes—
250#125—28no—
251C22⋊Q8—28no—
252C22⋊Q8—24no—
253C22⋊Q8C2×D418yes—
254C4×Q8—28no—
255C4×Q8—24no—
256C4×Q8—28no—
257C4×Q8—28no—
258C4×Q8—28no—
259C4×Q8—28no—
260C4×Q8—28no—
261C4×Q8—44no—
262C4×Q8—44no—
263C4×Q8—44no—
264C4×Q8—48no—
265C4×Q8—48no—
266C4×Q8C2×D418yes—
267#117—28no—
268#117—44no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—48no—
320C2×C42Q1618yescomplemented
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D414yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42#4114yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—216no—
352C4⋊C8—28no—
353C4⋊C8—28no—
354C4⋊C8—44no—
355#103—216no—
356#103—216no—
357#103—216no—
358#103—48no—
359C22⋊C8—28no—
360C4×C8—28no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375C24C22×D411yescharacteristic
376C2×Q8—28no—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—416no—
387C2×Q8—416no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8C22×D414yes—
393C2×Q8C2×#4118yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4#12114yes—
421C22×C4#12114yes—
422C22×C4C2×Q1614yes—
423C22×C4C2×Q1614yes—
424C22×C4C2×Q1614yes—
425C22×C4#13714yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4112yes—
441C22×C4C2511yescharacteristic
442Q16—432no—
443C2×C8—28no—
444C2×C8—28no—
445C2×C8—28no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4C22×D414yes—
476C22⋊C4—28no—
477C22⋊C4—44no—
478C42—28no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—44no—
499C42—44no—
500C42—48no—
501C42—48no—
502C42—44no—
503C42—44no—
504C42—44no—
505C42#12114yes—
506C42#12114yes—
507C42C2×Q1614yes—
508C42C2×Q1614yes—
509C42C2×Q1614yes—
510C42C2×Q1614yes—
511C42C22×D412yes—
512C42C22×D412yes—
513C42C2×#4112yes—
514C42C2×#4112yes—
515C42C2×#4114yes—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23C2×#12112yes—
521C23C22×Q1612yes—
522C23C2×#13712yes—
523C23C23×D411yescharacteristic
524Q8—28no—
525Q8—28no—
526Q8—28no—
527Q8—48no—
528Q8—416no—
529Q8—44no—
530Q8—44no—
531Q8—44no—
532C2×C4—28no—
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—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
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—416no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—88no—
581C2×C4#462116yescomplemented
582C2×C4C2×#12112yes—
583C2×C4C2×#12112yes—
584C2×C4C2×#12112yes—
585C2×C4C2×#12112yes—
586C2×C4C2×#12112yes—
587C2×C4#52512yes—
588C2×C4#52512yes—
589C2×C4C22×Q1612yes—
590C2×C4C22×Q1612yes—
591C2×C4C22×Q1612yes—
592C2×C4C22×Q1612yes—
593C2×C4C22×Q1612yes—
594C2×C4C22×Q1612yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4#57512yes—
598C2×C4#57512yes—
599C2×C4C23×D411yescharacteristic
600C2×C4C22×#4112yes—
601C2×C4C22×#4111yescharacteristic
602C2×C4#58512yes—
603C8—48no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22C2×#46214yes—
611C22C22×#12111yescharacteristic
612C22C2×#52511yescharacteristic
613C22#344911yescharacteristic
614C22C23×Q1611yescharacteristic
615C22C22×#13711yescharacteristic
616C22C2×#57511yescharacteristic
617C22C2×#57511yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—41no—
627C4—41no—
628C4—48no—
629C4—44no—
630C4—44no—
631C4—82no—
632C4—82no—
633C4—84no—
634C4C2×#46214yes—
635C4C2×#46214yes—
636C4#310414yescomplemented
637C4D4×#4112yes—
638C4#347512yes—
639C2—24no—
640C2#3493511yescharacteristic
641C2C22×#46211yescharacteristic
642C2#6041511yescharacteristic
643C2C2×#310411yescharacteristic
644C2#6056011yescharacteristic
645C2#6094311yescharacteristic
646C2#6095211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #28284C2semidirect
5. #28255C2semidirect
6. #28206C2semidirect
7. #28161C2semidirect
11. #23679C2semidirect
12. C4×#462C2semidirect
320. C2×C42Q16semidirect
581. C2×C4#462semidirect
636. C4#3104semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
C4×#5332
C2×C4×C4⋊Q82
#282842
#282552
#282062
#281612
#281212
#276482
#242122
#236792
C4×#4622
#208082
#207342