Finite Group Databasebeta

← #6477461

Subgroups — #6477461

Order 512.

3284 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic, complemented
2C2×#3193C211yesmaximal, characteristic, complemented
3C22×#481C211yesmaximal, characteristic, complemented
4#35076C211yesmaximal, characteristic
5#33005C212yesmaximal, complemented
6#32980C212yesmaximal, complemented
7#29640C212yesmaximal, complemented
8#29593C212yesmaximal, complemented
9#25555C218yesmaximal, complemented
10#24095C214yesmaximal, complemented
11#23966C214yesmaximal, complemented
12#21762C211yesmaximal, characteristic, complemented
13#21740C211yesmaximal, characteristic, complemented
14#20700C211yesmaximal, characteristic, complemented
15C2×#552—28no—
16C2×#552—22no—
17C2×#552—22no—
18C2×#552—28no—
19C2×#552C2214yes—
20C2×#552C2214yescomplemented
21C22×#126—21no—
22C22×#125—21no—
23C22×#125C2211yescharacteristic, complemented
24C22×#123—21no—
25C22×C22⋊Q8C2211yescharacteristic
26C22×C22⋊Q8C2211yescharacteristic, complemented
27C22×#121C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic, complemented
29C22×C4×D4C2211yescharacteristic
30C22×#117—21no—
31C22×#117C2211yescharacteristic, complemented
32C23×C4⋊C4C2211yescharacteristic
33#3193—28no—
34C2×#484C2212yescomplemented
35C2×#481—28no—
36C2×#481C2212yes—
37C2×#481C2212yes—
38C2×#481C2214yescomplemented
39C2×#480C2212yes—
40C2×#468C2214yescomplemented
41C2×#448C2212yes—
42C2×#450C2212yescomplemented
43C22×#106C2212yescomplemented
44C22×#102C2212yescomplemented
45C2×#407C2211yescharacteristic
46C22×C22⋊C8C2211yescharacteristic
47#2854C2212yes—
48#2847C2212yes—
49#2635—22no—
50#2591—21no—
51#2539—21no—
52#2487C2212yes—
53#2414C2214yes—
54#2408C2212yes—
55#2405—21no—
56#2403—21no—
57#2321—22no—
58#2280C2211yescharacteristic
59#2059C2218yescomplemented
60#2035C2218yescomplemented
61#2025C2218yescomplemented
62#2000C2218yescomplemented
63#1892C2214yescomplemented
64#1880C2218yescomplemented
65#1875C2218yescomplemented
66#1873C2214yescomplemented
67#1870C2214yescomplemented
68#1869C2214yes—
69#1865C2218yescomplemented
70#1854C2218yescomplemented
71#1824C2212yescomplemented
72#1824C2212yescomplemented
73#1823C2212yescomplemented
74#1823C2214yescomplemented
75#1817C2212yescomplemented
76#1787C2212yescomplemented
77#1779C2212yescomplemented
78#1736C2214yescomplemented
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4D411yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C22×SD16C2314yes—
87C22×M16C2312yes—
88C23×C8C2312yes—
89#552—232no—
90#552—232no—
91#552—216no—
92#552—216no—
93#552—216no—
94#552—216no—
95C2×#126—216no—
96C2×#126—24no—
97C2×#126—24no—
98C2×#125—216no—
99C2×#125—24no—
100C2×#125—24no—
101C2×#125—24no—
102C2×#125—24no—
103C2×#125—22no—
104C2×#125C2314yes—
105C2×#123—216no—
106C2×#123—24no—
107C2×#123—24no—
108C2×#123—22no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8D418yescomplemented
116C2×C22⋊Q8C2314yes—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119C2×#121—21no—
120C2×#121—21no—
121C2×#121—24no—
122C2×#121—24no—
123C2×#121C2312yes—
124C2×#121C2312yes—
125C2×#121C2314yes—
126C2×#120—22no—
127C2×#120—22no—
128C2×C4×Q8—24no—
129C2×C4×Q8—24no—
130C2×C4×Q8—22no—
131C2×C4×Q8C2314yes—
132C2×C4×D4—24no—
133C2×C4×D4—24no—
134C2×C4×D4D412yes—
135C2×C4×D4D412yes—
136C2×C4×D4C2312yes—
137C2×C4×D4C2312yes—
138C2×#117—216no—
139C2×#117—22no—
140C2×#117—24no—
141C2×#117—24no—
142C2×#117—24no—
143C2×#117—24no—
144C2×#117C2314yes—
145C2×#117C2312yes—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—24no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4D411yescharacteristic
156C22×C4⋊C4D418yescomplemented
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2311yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C4⋊C4C2311yescharacteristic
161C22×C4⋊C4C2312yes—
162C22×C4⋊C4C2312yes—
163C22×C22⋊C4—21no—
164C22×C22⋊C4—21no—
165C22×C22⋊C4—21no—
166C22×C22⋊C4—21no—
167C22×C22⋊C4D411yescharacteristic
168C22×C22⋊C4D411yescharacteristic
169C22×C42—21no—
170C22×C42—21no—
171C22×C42C2311yescharacteristic
172#484—28no—
173#481—28no—
174#481—28no—
175#481—216no—
176#480—28no—
177#468—216no—
178#450—28no—
179#448—28no—
180C2×#107C2314yes—
181C2×#106—28no—
182C2×#106C2314yes—
183C2×#106C2314yes—
184C2×#103C2314yes—
185C2×#103C2318yes—
186C2×#102—28no—
187C2×#102C2314yes—
188C2×#102C2314yes—
189C2×#102C2314yes—
190#407—24no—
191C2×C22⋊C8—24no—
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2312yes—
194C2×C22⋊C8C2312yes—
195C2×C22⋊C8C2312yes—
196C2×C22⋊C8C2318yes—
197#400—22no—
198#399—22no—
199#394—22no—
200#394—24no—
201#392—24no—
202#392—22no—
203#390—24no—
204#390C2314yes—
205#389C2314yes—
206#388—24no—
207#388—24no—
208#386—22no—
209#386—24no—
210#386—22no—
211#386—22no—
212#385—22no—
213#381—21no—
214#380C2312yes—
215#380C2312yes—
216#380C2314yes—
217#379—21no—
218#379—22no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221#340C2318yes—
222#340C2318yes—
223#340C2318yes—
224#336C2318yes—
225C25—21no—
226C25C2×D411yescharacteristic
227C25C2×D411yescharacteristic
228C22×Q8—22no—
229C22×Q8—21no—
230C22×Q8—44no—
231C22×Q8C2×D414yes—
232C22×Q8C2412yes—
233C22×D4—24no—
234C22×D4—22no—
235C22×D4—24no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—44no—
241C22×D4—44no—
242C22×D4—44no—
243C22×D4—84no—
244C22×D4C2×D412yes—
245C22×D4C2×D412yes—
246C22×D4C2412yes—
247C23×C4—28no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—44no—
258C23×C4—44no—
259C23×C4—48no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D414yes—
268C23×C4C2×D414yes—
269C23×C4C2412yes—
270C23×C4C2411yescharacteristic
271C23×C4C2411yescharacteristic
272C23×C4C2411yescharacteristic
273C2×SD16—28no—
274C2×SD16—48no—
275C2×M16—24no—
276C2×M16—44no—
277C22×C8—24no—
278C22×C8—44no—
279C22×C8C2414yes—
280C22×C8C2414yes—
281#126—232no—
282#126—232no—
283#125—232no—
284#125—232no—
285#125—216no—
286#125—28no—
287#125—28no—
288#123—232no—
289#123—232no—
290#123—84no—
291C22⋊Q8—216no—
292C22⋊Q8—28no—
293C22⋊Q8—28no—
294C22⋊Q8—28no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C22⋊Q8—28no—
298C22⋊Q8C2×D418yes—
299C22⋊Q8C2×D418yes—
300#121—24no—
301#121—24no—
302#121—24no—
303#121—24no—
304#121—44no—
305#121—44no—
306#121—48no—
307#120—48no—
308#120—84no—
309C4×Q8—216no—
310C4×Q8—28no—
311C4×Q8—28no—
312C4×D4—28no—
313C4×D4—28no—
314C4×D4—24no—
315C4×D4—24no—
316C4×D4—24no—
317C4×D4—24no—
318#117—216no—
319#117—232no—
320#117—232no—
321#117—28no—
322#117—28no—
323#117—44no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—28no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—28no—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—28no—
330C2×C4⋊C4—28no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2×D418yes—
358C2×C4⋊C4C2×D418yes—
359C2×C4⋊C4C2×D414yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2414yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—28no—
366C2×C22⋊C4—28no—
367C2×C22⋊C4—28no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—44no—
384C2×C22⋊C4C2×D412yes—
385C2×C22⋊C4C2×D412yes—
386C2×C22⋊C4C2×D412yes—
387C2×C22⋊C4C2×D412yes—
388C2×C42—28no—
389C2×C42—28no—
390C2×C42—24no—
391C2×C42—24no—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42—22no—
395C2×C42C2×D412yes—
396C2×C42C2412yes—
397#107—28no—
398#106—28no—
399#106—28no—
400#103—28no—
401#103—48no—
402#102—28no—
403#102—28no—
404#102—28no—
405C22⋊C8—24no—
406C22⋊C8—24no—
407C22⋊C8—24no—
408C22⋊C8—24no—
409C22⋊C8—48no—
410#95—22no—
411#95—22no—
412#95—24no—
413#95—24no—
414C24—24no—
415C24—24no—
416C24—22no—
417C24—24no—
418C24—22no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—44no—
424C24—82no—
425C24—82no—
426C24#12014yes—
427C24C22×D412yes—
428C24C22×D411yescharacteristic
429C24C22×D411yescharacteristic
430C24C22×D411yescharacteristic
431C2×Q8—24no—
432C2×Q8—22no—
433C2×Q8—28no—
434C2×Q8—48no—
435C2×Q8—44no—
436C2×Q8—48no—
437C2×Q8C22×D414yes—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—24no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—28no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—48no—
448C2×D4—416no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—48no—
454C2×D4—42no—
455C2×D4—84no—
456C2×D4—82no—
457C2×D4—82no—
458C2×D4—82no—
459C2×D4—84no—
460C2×D4—84no—
461C2×D4—816no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—28no—
473C22×C4—28no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—28no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—432no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—48no—
499C22×C4—44no—
500C22×C4—48no—
501C22×C4—44no—
502C22×C4—42no—
503C22×C4—82no—
504C22×C4#12014yes—
505C22×C4#12014yes—
506C22×C4#12014yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C2511yescharacteristic
520SD16—88no—
521M16—84no—
522C2×C8—24no—
523C2×C8—24no—
524C2×C8—28no—
525C2×C8—28no—
526C2×C8—48no—
527C4⋊C4—216no—
528C4⋊C4—216no—
529C4⋊C4—216no—
530C4⋊C4—216no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—28no—
534C4⋊C4—28no—
535C4⋊C4—28no—
536C4⋊C4—28no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—28no—
546C4⋊C4—42no—
547C4⋊C4—84no—
548C4⋊C4C22×D414yes—
549C4⋊C4C22×D414yes—
550C4⋊C4C22×D414yes—
551C22⋊C4—216no—
552C22⋊C4—216no—
553C22⋊C4—28no—
554C22⋊C4—28no—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C22⋊C4—48no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—84no—
565C22⋊C4—84no—
566C22⋊C4—82no—
567C22⋊C4—84no—
568C42—216no—
569C42—216no—
570C42—24no—
571C42—22no—
572C42—22no—
573C42—82no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—28no—
580C23—28no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—24no—
585C23—22no—
586C23—22no—
587C23—24no—
588C23—44no—
589C23—416no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—42no—
594C23—44no—
595C23—44no—
596C23—42no—
597C23—42no—
598C23—88no—
599C23—84no—
600C23—84no—
601C23—84no—
602C23—168no—
603C23C2×#12012yes—
604C23C2×#12012yes—
605C23C2×#12012yes—
606C23#57514yes—
607C23#57714yes—
608C23C23×D411yescharacteristic
609Q8—28no—
610Q8—48no—
611Q8—44no—
612D4—48no—
613D4—48no—
614D4—44no—
615D4—44no—
616D4—84no—
617D4—88no—
618D4—88no—
619D4—168no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—24no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—48no—
644C2×C4—44no—
645C2×C4—48no—
646C2×C4—48no—
647C2×C4—416no—
648C2×C4—44no—
649C2×C4—48no—
650C2×C4—44no—
651C2×C4—416no—
652C2×C4—44no—
653C2×C4—48no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—84no—
666C2×C4C2×#12012yes—
667C2×C4C2×#12012yes—
668C2×C4C2×#12012yes—
669C2×C4C23×D411yescharacteristic
670C2×C4C23×D411yescharacteristic
671C2×C4#58514yes—
672C8—44no—
673C8—44no—
674C22—22no—
675C22—22no—
676C22—28no—
677C22—28no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—416no—
684C22—44no—
685C22—42no—
686C22—44no—
687C22—44no—
688C22—42no—
689C22—42no—
690C22—88no—
691C22—82no—
692C22—84no—
693C22—84no—
694C22—82no—
695C22—82no—
696C22—168no—
697C22#300514yes—
698C22#301014yes—
699C22C22×#12011yescharacteristic
700C22#348512yes—
701C22C2×#57512yes—
702C22C2×#57712yes—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—42no—
707C4—44no—
708C4—42no—
709C4—44no—
710C4—44no—
711C4—42no—
712C4—44no—
713C4—84no—
714C4—82no—
715C4#344912yes—
716C4#348512yes—
717C2—24no—
718C2—24no—
719C2—42no—
720C2—82no—
721C2—162no—
722C2#3508811yescharacteristic
723C2C2×#300511yescharacteristic
724C2C2×#301011yescharacteristic
725C2#6172914yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35076C2
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
1. C22×#552C2semidirect
2. C2×#3193C2semidirect
3. C22×#481C2semidirect
5. #33005C2semidirect
6. #32980C2semidirect
7. #29640C2semidirect
8. #29593C2semidirect
9. #25555C2semidirect
10. #24095C2semidirect
11. #23966C2semidirect
12. #21762C2semidirect
13. #21740C2semidirect
14. #20700C2semidirect
20. C2×#552C22semidirect
23. C22×#125C22semidirect
26. C22×C22⋊Q8C22semidirect
28. C22×C4×Q8C22semidirect
31. C22×#117C22semidirect
34. C2×#484C22semidirect
38. C2×#481C22semidirect
40. C2×#468C22semidirect
42. C2×#450C22semidirect
43. C22×#106C22semidirect
44. C22×#102C22semidirect
59. #2059C22semidirect
60. #2035C22semidirect
61. #2025C22semidirect
62. #2000C22semidirect
63. #1892C22semidirect
64. #1880C22semidirect
65. #1875C22semidirect
66. #1873C22semidirect
67. #1870C22semidirect
69. #1865C22semidirect
70. #1854C22semidirect
71. #1824C22semidirect
72. #1824C22semidirect
73. #1823C22semidirect
74. #1823C22semidirect
75. #1817C22semidirect
76. #1787C22semidirect
77. #1779C22semidirect
78. #1736C22semidirect
115. C2×C22⋊Q8D4semidirect
156. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#31932
C22×#4812
#350762
#330052
#329802
#296402
#295932
#255552
#240952
#239662
#217622
#217402
#207002