Finite Group Databasebeta

← #6477473

Subgroups — #6477473

Order 512.

3154 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C2×#3189C211yesmaximal, characteristic, complemented
3C2×#3095C211yesmaximal, characteristic, complemented
4#35084C211yesmaximal, characteristic
5#33010C212yesmaximal, complemented
6#32983C212yesmaximal, complemented
7#29643C212yesmaximal, complemented
8#29599C212yesmaximal, complemented
9#25546C218yesmaximal, complemented
10#24110C214yesmaximal, complemented
11#23983C214yesmaximal, complemented
12#21768C211yesmaximal, characteristic, complemented
13#21740C211yesmaximal, characteristic
14#20699C211yesmaximal, characteristic, complemented
15C2×#552—28no—
16C2×#552—22no—
17C2×#552—22no—
18C2×#552—28no—
19C2×#552C2214yes—
20C2×#552C2214yes—
21C22×#126—21no—
22C22×#125—21no—
23C22×#125C2211yescharacteristic
24C22×#123—21no—
25C22×C22⋊Q8C2211yescharacteristic
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30C22×#117—21no—
31C22×#117C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#3189—28no—
34#3095—28no—
35C2×#484C2212yes—
36C2×#484C2212yes—
37C2×#484C2212yescomplemented
38C2×#485C2214yescomplemented
39C2×#483C2212yes—
40C2×#467C2214yescomplemented
41C2×#451C2212yes—
42C2×#451C2212yescomplemented
43C2×#428C2212yescomplemented
44C2×#419C2212yescomplemented
45C2×#407C2211yescharacteristic
46C22×C22⋊C8C2211yescharacteristic
47#2855C2212yes—
48#2848C2212yes—
49#2638—22no—
50#2583—21no—
51#2561—21no—
52#2486C2212yes—
53#2483—21no—
54#2415C2214yes—
55#2409C2212yes—
56#2405—21no—
57#2322—22no—
58#2280C2211yescharacteristic
59#2059C2218yescomplemented
60#2035C2218yescomplemented
61#2026C2218yescomplemented
62#2001C2218yescomplemented
63#1894C2214yescomplemented
64#1878C2218yescomplemented
65#1873C2214yescomplemented
66#1872C2218yescomplemented
67#1870C2214yes—
68#1869C2214yes—
69#1863C2218yescomplemented
70#1852C2218yescomplemented
71#1825C2214yescomplemented
72#1824C2212yes—
73#1824C2212yes—
74#1823C2212yes—
75#1817C2212yes—
76#1787C2212yes—
77#1779C2212yes—
78#1735C2214yescomplemented
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×D4D411yescharacteristic
82C24×C4—21no—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C22×Q16C2314yes—
86C22×M16C2312yes—
87C23×C8C2312yes—
88#552—232no—
89#552—232no—
90#552—216no—
91#552—216no—
92#552—216no—
93#552—216no—
94C2×#126—216no—
95C2×#126—24no—
96C2×#126—24no—
97C2×#125—216no—
98C2×#125—24no—
99C2×#125—24no—
100C2×#125—24no—
101C2×#125—24no—
102C2×#125—22no—
103C2×#125C2314yes—
104C2×#123—216no—
105C2×#123—24no—
106C2×#123—24no—
107C2×#123—22no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8D418yes—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2312yes—
121C2×C22⋊Q8C2314yes—
122C2×C22⋊Q8C2312yes—
123C2×C22⋊Q8C2312yes—
124C2×C22⋊Q8C2314yes—
125C2×C4×Q8—24no—
126C2×C4×Q8—24no—
127C2×C4×Q8—22no—
128C2×C4×Q8C2314yes—
129C2×C4×D4—24no—
130C2×C4×D4—24no—
131C2×C4×D4D412yes—
132C2×C4×D4D412yes—
133C2×C4×D4C2312yes—
134C2×C4×D4C2312yes—
135C2×#117—216no—
136C2×#117—24no—
137C2×#117—21no—
138C2×#117—21no—
139C2×#117—24no—
140C2×#117—24no—
141C2×#117D418yes—
142C2×#117C2312yes—
143C2×#117C2312yes—
144C2×#117C2312yes—
145C22×C4⋊C4—21no—
146C22×C4⋊C4—22no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—24no—
153C22×C4⋊C4—24no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4D411yescharacteristic
156C22×C4⋊C4C2311yescharacteristic
157C22×C4⋊C4C2314yes—
158C22×C4⋊C4C2311yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C4⋊C4C2311yescharacteristic
161C22×C22⋊C4—21no—
162C22×C22⋊C4—21no—
163C22×C22⋊C4—21no—
164C22×C22⋊C4—21no—
165C22×C22⋊C4D411yescharacteristic
166C22×C22⋊C4D411yescharacteristic
167C22×C42—21no—
168C22×C42—21no—
169C22×C42C2311yescharacteristic
170#485—216no—
171#484—28no—
172#484—28no—
173#484—28no—
174#483—28no—
175#467—216no—
176#451—28no—
177#451—28no—
178#428—28no—
179C2×#107C2314yes—
180C2×#107C2314yes—
181C2×#106C2314yes—
182#419—28no—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2314yes—
187C2×#103C2318yes—
188#407—24no—
189C2×C22⋊C8—24no—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2312yes—
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2312yes—
194C2×C22⋊C8C2318yes—
195#398—22no—
196#397—22no—
197#397—24no—
198#395—22no—
199#394—24no—
200#393—22no—
201#391C2314yes—
202#389—24no—
203#389C2314yes—
204#388—24no—
205#387—24no—
206#386—22no—
207#385—24no—
208#385—22no—
209#385—22no—
210#385—22no—
211#381—22no—
212#381—21no—
213#381—22no—
214#380C2312yes—
215#380C2314yes—
216#380C2312yes—
217#379—22no—
218#379—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221#340C2318yes—
222#340C2318yes—
223#340C2318yes—
224#336C2318yes—
225C25C2×D411yescharacteristic
226C25C2×D411yescharacteristic
227C22×Q8—22no—
228C22×Q8—21no—
229C22×Q8—21no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—44no—
233C22×Q8—44no—
234C22×Q8C2×D414yes—
235C22×Q8C2412yes—
236C22×Q8C2412yes—
237C22×D4—24no—
238C22×D4—44no—
239C22×D4C2×D412yes—
240C22×D4C2×D412yes—
241C23×C4—28no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—24no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4—48no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D414yes—
262C23×C4C2×D414yes—
263C23×C4C2412yes—
264C23×C4C2411yescharacteristic
265C23×C4C2411yescharacteristic
266C23×C4C2411yescharacteristic
267C2×Q16—28no—
268C2×Q16—48no—
269C2×M16—24no—
270C2×M16—44no—
271C22×C8—24no—
272C22×C8—44no—
273C22×C8C2414yes—
274C22×C8C2414yes—
275#126—232no—
276#126—232no—
277#125—232no—
278#125—232no—
279#125—216no—
280#125—28no—
281#125—28no—
282#123—232no—
283#123—232no—
284#123—48no—
285C22⋊Q8—216no—
286C22⋊Q8—28no—
287C22⋊Q8—28no—
288C22⋊Q8—28no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291C22⋊Q8—28no—
292C22⋊Q8—24no—
293C22⋊Q8—24no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—44no—
297C22⋊Q8—44no—
298C22⋊Q8—48no—
299C22⋊Q8C2×D418yes—
300C22⋊Q8C2×D418yes—
301C4×Q8—216no—
302C4×Q8—28no—
303C4×Q8—28no—
304C4×D4—28no—
305C4×D4—28no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—24no—
309C4×D4—24no—
310#117—232no—
311#117—232no—
312#117—28no—
313#117—28no—
314#117—24no—
315#117—24no—
316#117—44no—
317#117C2×D418yes—
318#117C2×D418yes—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—216no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—28no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—28no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—28no—
341C2×C4⋊C4—28no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2414yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—28no—
361C2×C22⋊C4—28no—
362C2×C22⋊C4—28no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—44no—
378C2×C22⋊C4C2×D412yes—
379C2×C22⋊C4C2×D412yes—
380C2×C22⋊C4C2×D412yes—
381C2×C22⋊C4C2×D412yes—
382C2×C42—28no—
383C2×C42—28no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—24no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42C2×D412yes—
391C2×C42C2412yes—
392#107—28no—
393#107—28no—
394#106—28no—
395#103—28no—
396#103—28no—
397#103—28no—
398#103—28no—
399#103—48no—
400C22⋊C8—24no—
401C22⋊C8—24no—
402C22⋊C8—24no—
403C22⋊C8—24no—
404C22⋊C8—48no—
405#95—22no—
406#95—22no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95—24no—
411C24—22no—
412C24—24no—
413C24—24no—
414C24—24no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24#12014yes—
419C24C22×D411yescharacteristic
420C24C22×D411yescharacteristic
421C24C22×D412yes—
422C24C22×D411yescharacteristic
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—28no—
428C2×Q8—28no—
429C2×Q8—48no—
430C2×Q8—44no—
431C2×Q8—48no—
432C2×Q8—48no—
433C2×Q8—42no—
434C2×Q8—82no—
435C2×Q8—84no—
436C2×Q8C22×D414yes—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—416no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—216no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—28no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—432no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—48no—
481C22×C4—44no—
482C22×C4—48no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—82no—
488C22×C4#12014yes—
489C22×C4#12014yes—
490C22×C4#12014yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C2511yescharacteristic
504Q16—88no—
505M16—48no—
506C2×C8—24no—
507C2×C8—24no—
508C2×C8—28no—
509C2×C8—28no—
510C2×C8—84no—
511C4⋊C4—216no—
512C4⋊C4—216no—
513C4⋊C4—216no—
514C4⋊C4—216no—
515C4⋊C4—216no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—28no—
519C4⋊C4—28no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—28no—
529C4⋊C4—48no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—84no—
534C4⋊C4C22×D414yes—
535C4⋊C4C22×D414yes—
536C4⋊C4C22×D414yes—
537C22⋊C4—216no—
538C22⋊C4—28no—
539C22⋊C4—28no—
540C22⋊C4—28no—
541C22⋊C4—28no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—48no—
550C22⋊C4—44no—
551C22⋊C4—82no—
552C22⋊C4—84no—
553C42—216no—
554C42—28no—
555C42—28no—
556C42—24no—
557C42—22no—
558C42—22no—
559C42—82no—
560C23—22no—
561C23—22no—
562C23—28no—
563C23—22no—
564C23—22no—
565C23—24no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—24no—
571C23—22no—
572C23—22no—
573C23—28no—
574C23—44no—
575C23—44no—
576C23—416no—
577C23—44no—
578C23—44no—
579C23—88no—
580C23C2×#12012yes—
581C23C2×#12012yes—
582C23C2×#12012yes—
583C23#57514yes—
584C23#57814yes—
585C23C23×D411yescharacteristic
586Q8—28no—
587Q8—48no—
588Q8—44no—
589Q8—44no—
590Q8—44no—
591Q8—84no—
592D4—48no—
593D4—48no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—48no—
618C2×C4—44no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—432no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—48no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4C2×#12012yes—
642C2×C4C2×#12012yes—
643C2×C4C2×#12012yes—
644C2×C4C23×D411yescharacteristic
645C2×C4C23×D411yescharacteristic
646C2×C4#58514yes—
647C8—44no—
648C8—44no—
649C22—22no—
650C22—28no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—28no—
655C22—24no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—416no—
661C22—42no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—42no—
666C22—88no—
667C22#301114yes—
668C22C22×#12011yescharacteristic
669C22#348512yes—
670C22C2×#57512yes—
671C22C2×#57812yes—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—42no—
676C4—44no—
677C4—42no—
678C4—44no—
679C4—44no—
680C4—42no—
681C4—44no—
682C4—82no—
683C4—84no—
684C4—82no—
685C4#344912yes—
686C4#348512yes—
687C2—24no—
688C2—44no—
689C2—42no—
690C2—42no—
691C2#3508811yescharacteristic
692C2#6035911yescharacteristic
693C2C2×#301111yescharacteristic
694C2#6173814yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3189C2semidirect
3. C2×#3095C2semidirect
5. #33010C2semidirect
6. #32983C2semidirect
7. #29643C2semidirect
8. #29599C2semidirect
9. #25546C2semidirect
10. #24110C2semidirect
11. #23983C2semidirect
12. #21768C2semidirect
14. #20699C2semidirect
37. C2×#484C22semidirect
38. C2×#485C22semidirect
40. C2×#467C22semidirect
42. C2×#451C22semidirect
43. C2×#428C22semidirect
44. C2×#419C22semidirect
59. #2059C22semidirect
60. #2035C22semidirect
61. #2026C22semidirect
62. #2001C22semidirect
63. #1894C22semidirect
64. #1878C22semidirect
65. #1873C22semidirect
66. #1872C22semidirect
69. #1863C22semidirect
70. #1852C22semidirect
71. #1825C22semidirect
78. #1735C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#31892
C2×#30952
#350842
#330102
#329832
#296432
#295992
#255462
#241102
#239832
#217682
#217402
#206992