Finite Group Databasebeta

← #6529208

Subgroups — #6529208

Order 512.

3036 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×#3503C211yesmaximal, characteristic
2C2×#3249C211yesmaximal, characteristic, complemented
3C2×#3230C211yesmaximal, characteristic, complemented
4#34947C211yesmaximal, characteristic
5#25936C212yesmaximal, complemented
6#25841C212yesmaximal, complemented
7#25706C212yesmaximal, complemented
8#25573C212yesmaximal, complemented
9#25545C218yesmaximal, complemented
10#24302C214yesmaximal, complemented
11#24155C214yesmaximal, complemented
12#21847C211yesmaximal, characteristic
13#21385C211yesmaximal, characteristic, complemented
14C2×#1987C211yesmaximal, characteristic, complemented
15#3503—28no—
16#3503—28no—
17C2×Q82C2212yes—
18C2×#557—21no—
19C2×#554C2212yes—
20C2×#554C2214yes—
21C2×D4×Q8—22no—
22C2×D4×Q8—21no—
23C2×#544—22no—
24C2×#544—21no—
25C2×#533C2212yes—
26C2×#533C2211yescharacteristic
27C2×#533C2211yescharacteristic
28C22×C4⋊Q8C2212yes—
29C22×C4⋊Q8C2211yescharacteristic
30C2×#520—21no—
31C2×C4×#41C2211yescharacteristic
32#3249—28no—
33#3230—28no—
34C2×#500C2212yes—
35C2×#487C2212yes—
36C2×#484C2214yes—
37C2×#477C2214yes—
38C2×#477C2212yes—
39C2×#477C2212yes—
40C2×#467C2214yes—
41C2×#462C2212yes—
42C2×#462C2212yes—
43C2×#431C2211yescharacteristic
44C22×C4⋊C8C2211yescharacteristic
45#2568C2211yescharacteristic
46#2567C2211yescharacteristic
47#2566—21no—
48#2565—21no—
49#2564—22no—
50#2563C2211yescharacteristic
51#2542C2211yescharacteristic
52#2417C2214yes—
53#2396—22no—
54C4×C22⋊Q8C2214yes—
55#2297—21no—
56#2285—21no—
57C2×C4×C4⋊C4C2211yescharacteristic
58#2057C2218yes—
59#2036C2218yes—
60#2024C2218yes—
61#2001C2218yes—
62#1987C2212yes—
63#1987C2214yes—
64#1987C2212yes—
65#1987C2212yes—
66#1984C2212yes—
67#1981C2212yes—
68#1954C2212yes—
69#1878C2218yes—
70#1872C2214yes—
71#1872C2218yes—
72#1869C2214yes—
73#1863C2218yes—
74#1852C2218yes—
75#1797C2214yes—
76C4×#103C2212yes—
77C4×#103C2212yes—
78C2×#340C2214yes—
79C22×#41D411yescharacteristic
80C23×Q8—22no—
81C23×Q8—21no—
82C22×Q16C2314yes—
83C23×C8C2312yes—
84Q82—28no—
85Q82—28no—
86#557—216no—
87#554—28no—
88#554—216no—
89#554—28no—
90#554—216no—
91D4×Q8—232no—
92D4×Q8—216no—
93#544—232no—
94#544—216no—
95#533—28no—
96#533—28no—
97#533—24no—
98#533—24no—
99#533—24no—
100#533D418yes—
101C2×C4⋊Q8—28no—
102C2×C4⋊Q8—28no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—22no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—24no—
110C2×C4⋊Q8D418yes—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2314yes—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2312yes—
118C2×#125—22no—
119C2×#125—24no—
120C2×#125—22no—
121C2×#125—21no—
122C2×#125—21no—
123C2×C22⋊Q8—22no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—24no—
127C2×C22⋊Q8—24no—
128C2×C22⋊Q8—24no—
129C2×C22⋊Q8C2312yes—
130C2×C22⋊Q8C2314yes—
131C2×C22⋊Q8C2314yes—
132C2×C22⋊Q8C2312yes—
133#520—216no—
134C4×#41—24no—
135C4×#41—24no—
136C2×C4×Q8—22no—
137C2×C4×Q8—22no—
138C2×C4×Q8—21no—
139C2×C4×Q8—21no—
140C2×C4×Q8D411yescharacteristic
141C2×C4×Q8C2312yes—
142C2×C4×Q8C2314yes—
143C2×C4×Q8C2312yes—
144C2×C4×Q8C2311yescharacteristic
145C2×C4×D4—24no—
146C2×C4×D4—22no—
147C2×C4×D4D412yes—
148C2×C4×D4D411yescharacteristic
149C2×C4×D4C2311yescharacteristic
150C2×C4×D4C2312yes—
151C2×#117—22no—
152C2×#117—21no—
153C2×#117—21no—
154C2×#117D412yes—
155C2×#117D411yescharacteristic
156C22×C4⋊C4—22no—
157C22×C4⋊C4—21no—
158C22×C4⋊C4C2314yes—
159C22×C4⋊C4C2312yes—
160C22×C4⋊C4C2312yes—
161C22×C42—21no—
162C22×C42C2312yes—
163C22×C42C2311yescharacteristic
164#500—28no—
165#487—28no—
166#484—216no—
167#477—216no—
168#477—28no—
169#477—28no—
170#467—216no—
171#462—28no—
172#462—28no—
173#431—24no—
174C2×#107C2314yes—
175C2×#107C2318yes—
176C2×C4⋊C8—24no—
177C2×C4⋊C8C2312yes—
178C2×C4⋊C8C2312yes—
179C2×C4⋊C8C2312yes—
180C2×#103C2318yes—
181C2×#103C2314yes—
182C2×#103C2314yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×C22⋊C8C2312yes—
186C2×C22⋊C8C2318yes—
187C2×C4×C8C2312yes—
188#400—22no—
189#399—22no—
190#398—24no—
191#396—24no—
192#395—22no—
193#393—22no—
194#391—21no—
195#391—21no—
196#391C2314yes—
197#390—21no—
198#390—21no—
199#389C2314yes—
200#387—24no—
201#386—24no—
202#385—24no—
203#384—21no—
204#384—21no—
205#382—21no—
206#382—21no—
207#380—24no—
208#380C2314yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—24no—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—24no—
216C2×#95—21no—
217#340C2314yes—
218#340C2318yes—
219#340C2314yes—
220#340C2318yes—
221#336C2318yes—
222C2×#41—24no—
223C2×#41—24no—
224C2×#41Q1618yescomplemented
225C22×Q8—216no—
226C22×Q8—28no—
227C22×Q8—22no—
228C22×Q8—21no—
229C22×Q8—24no—
230C22×Q8—22no—
231C22×Q8—21no—
232C22×Q8—84no—
233C22×Q8—88no—
234C22×Q8C2×D411yescharacteristic
235C22×Q8C2412yes—
236C22×Q8C2412yes—
237C22×D4C2×D412yes—
238C22×D4C2×D411yescharacteristic
239C23×C4—24no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C2×Q16—28no—
249C2×Q16—48no—
250C22×C8—24no—
251C22×C8—44no—
252C22×C8C2414yes—
253C22×C8C2414yes—
254C4⋊Q8—216no—
255C4⋊Q8—216no—
256C4⋊Q8—232no—
257C4⋊Q8—216no—
258C4⋊Q8—28no—
259C4⋊Q8—28no—
260C4⋊Q8—28no—
261C4⋊Q8—28no—
262C4⋊Q8—22no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8C2×D414yes—
271C4⋊Q8C2×D414yes—
272C4⋊Q8C2×D418yes—
273C4⋊Q8C2×D418yes—
274#125—216no—
275#125—232no—
276#125—216no—
277#125—28no—
278#125—44no—
279C22⋊Q8—216no—
280C22⋊Q8—232no—
281C22⋊Q8—232no—
282C22⋊Q8—232no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C22⋊Q8—24no—
286C22⋊Q8—28no—
287C22⋊Q8—24no—
288C22⋊Q8—24no—
289C22⋊Q8—28no—
290C22⋊Q8—44no—
291C22⋊Q8—44no—
292C22⋊Q8C2×D418yes—
293C4×Q8—216no—
294C4×Q8—216no—
295C4×Q8—28no—
296C4×Q8—28no—
297C4×Q8—28no—
298C4×Q8—24no—
299C4×Q8—22no—
300C4×Q8—22no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—44no—
305C4×Q8C2×D418yes—
306C4×D4—232no—
307C4×D4—216no—
308C4×D4—28no—
309C4×D4—24no—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—22no—
313C4×D4—22no—
314#117—216no—
315#117—28no—
316#117—24no—
317#117—28no—
318#117—44no—
319C2×C4⋊C4—216no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—28no—
335C2×C4⋊C4—28no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—42no—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D418yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2414yes—
355C2×C4⋊C4C2412yes—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C42—24no—
365C2×C42—22no—
366C2×C42—24no—
367C2×C42—24no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—44no—
373C2×C42C2×D412yes—
374C2×C42C2×D412yes—
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D414yes—
378C2×C42C2×D414yes—
379C2×C42C2412yes—
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382#107—28no—
383#107—216no—
384C4⋊C8—24no—
385C4⋊C8—24no—
386C4⋊C8—24no—
387#103—28no—
388#103—28no—
389#103—216no—
390#103—28no—
391#103—44no—
392C22⋊C8—24no—
393C22⋊C8—48no—
394C4×C8—24no—
395#95—22no—
396#95—21no—
397#95—21no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402C24C22×D412yes—
403C24C22×D411yescharacteristic
404#41—232no—
405C2×Q8—216no—
406C2×Q8—22no—
407C2×Q8—28no—
408C2×Q8—24no—
409C2×Q8—28no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—48no—
419C2×Q8—44no—
420C2×Q8—42no—
421C2×Q8—82no—
422C2×Q8—84no—
423C2×Q8—88no—
424C2×Q8—816no—
425C2×Q8—832no—
426C2×Q8C2×Q1614yes—
427C2×Q8C22×D414yes—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4C2×Q1618yes—
433C2×D4C2×Q1614yes—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—216no—
443C22×C4—28no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—82no—
457C22×C4—84no—
458C22×C4—88no—
459C22×C4#12014yes—
460C22×C4#12014yes—
461C22×C4C2×Q1618yes—
462C22×C4C2×Q1614yes—
463C22×C4#13714yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C2511yescharacteristic
475Q16—88no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—28no—
481C2×C8—84no—
482C4⋊C4—216no—
483C4⋊C4—216no—
484C4⋊C4—216no—
485C4⋊C4—216no—
486C4⋊C4—24no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—84no—
513C4⋊C4C22×D414yes—
514C4⋊C4C22×D418yes—
515C4⋊C42−1+414yes—
516C22⋊C4—216no—
517C22⋊C4—28no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—84no—
524C42—216no—
525C42—24no—
526C42—24no—
527C42—28no—
528C42—22no—
529C42—22no—
530C42—44no—
531C42—42no—
532C42—44no—
533C42—42no—
534C42#12014yes—
535C42#12014yes—
536C42C22×D412yes—
537C42C22×D412yes—
538C42C22×D412yes—
539C42C22×D412yes—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—22no—
544C23—24no—
545C23—22no—
546C23C2×#12012yes—
547C23C22×Q1614yes—
548C23C22×Q1612yes—
549C23C2×#13712yes—
550C23C23×D411yescharacteristic
551Q8—28no—
552Q8—24no—
553Q8—24no—
554Q8—416no—
555Q8—48no—
556Q8—44no—
557Q8—44no—
558Q8—42no—
559Q8—42no—
560Q8—88no—
561Q8—84no—
562Q8—816no—
563D4—216no—
564D4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—216no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—48no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—88no—
600C2×C4—816no—
601C2×C4—88no—
602C2×C4#451116yes—
603C2×C4C2×#12012yes—
604C2×C4C2×#12012yes—
605C2×C4C2×#12012yes—
606C2×C4C2×#12012yes—
607C2×C4C2×#12012yes—
608C2×C4C22×Q1614yes—
609C2×C4C22×Q1612yes—
610C2×C4C22×Q1612yes—
611C2×C4C2×#13712yes—
612C2×C4C2×#13712yes—
613C2×C4C23×D411yescharacteristic
614C2×C4C23×D411yescharacteristic
615C2×C4C2×2−1+412yes—
616C2×C4#58512yes—
617C8—44no—
618C8—44no—
619C22—22no—
620C22—24no—
621C22—24no—
622C22—22no—
623C22—28no—
624C22—24no—
625C22—42no—
626C22—44no—
627C22—42no—
628C22C2×#45114yes—
629C22C22×#12011yescharacteristic
630C22#348912yes—
631C22C23×Q1611yescharacteristic
632C22C22×#13711yescharacteristic
633C22#358512yes—
634C4—22no—
635C4—24no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—44no—
640C4—44no—
641C4—42no—
642C4—48no—
643C4—44no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4C2×#45114yes—
648C4C2×#45114yes—
649C4#349712yes—
650C4#349911yescharacteristic
651C4#350011yescharacteristic
652C2—24no—
653C2—44no—
654C2#3566011yescharacteristic
655C2C22×#45111yescharacteristic
656C2#6036711yescharacteristic
657C2#6210012yes—
658C2#6210812yes—
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×#3249C2semidirect
3. C2×#3230C2semidirect
5. #25936C2semidirect
6. #25841C2semidirect
7. #25706C2semidirect
8. #25573C2semidirect
9. #25545C2semidirect
10. #24302C2semidirect
11. #24155C2semidirect
13. #21385C2semidirect
14. C2×#1987C2semidirect
224. C2×#41Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#35032
C2×#32492
C2×#32302
#349472
#259362
#258412
#257062
#255732
#255452
#243022
#241552
#218472
#213852
C2×#19872