Finite Group Databasebeta

← #6508806

Subgroups — #6508806

Order 512.

2174 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#482C211yesmaximal, characteristic, complemented
2C2×#2972C211yesmaximal, characteristic, complemented
3#35076C211yesmaximal, characteristic
4#34809C211yesmaximal, characteristic, complemented
5#28097C212yesmaximal, complemented
6#28094C212yesmaximal, complemented
7#27662C212yesmaximal, complemented
8#27658C212yesmaximal, complemented
9#25237C214yesmaximal, complemented
10#25235C214yesmaximal, complemented
11#24605C214yesmaximal, complemented
12#24555C214yesmaximal, complemented
13#21939C211yesmaximal, characteristic, complemented
14#21761C211yesmaximal, characteristic, complemented
15#20699C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#482—28no—
20C2×#482C2214yes—
21C2×#482C2214yescomplemented
22#2972—28no—
23C2×C8×D4C2212yes—
24C2×C8×D4C2212yes—
25C2×C8×D4C2212yescomplemented
26C2×C8×D4C2212yescomplemented
27C2×#431C2212yescomplemented
28C22×#107C2211yescharacteristic, complemented
29C22×#106C2211yescharacteristic, complemented
30C22×#102C2211yescharacteristic, complemented
31C22×#102C2211yescharacteristic, complemented
32C22×C22⋊C8C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34C22×C22⋊C8C2212yescomplemented
35#2854C2212yes—
36#2847C2212yes—
37#2635—22no—
38#2591—21no—
39#2539—21no—
40#2487C2212yes—
41#2487C2212yes—
42#2486C2212yes—
43#2485C2214yescomplemented
44#2484—21no—
45#2483—21no—
46#2482—21no—
47#2414C2214yes—
48#2408C2212yes—
49#2405—21no—
50#2405—21no—
51#2403—21no—
52#2395—22no—
53#2321—22no—
54C4×#123—22no—
55#2296C2212yescomplemented
56#2280C2211yescharacteristic
57#2280C2211yescharacteristic
58#2280C2212yescomplemented
59#2088C2214yescomplemented
60#2087C2214yescomplemented
61#2077C2214yescomplemented
62#2076C2214yescomplemented
63#2067C2214yescomplemented
64#2065C2214yescomplemented
65#2034C2214yescomplemented
66#2033C2214yescomplemented
67#1942C2214yescomplemented
68#1941C2214yescomplemented
69#1934C2214yescomplemented
70#1934C2214yescomplemented
71#1893C2212yescomplemented
72#1893C2212yescomplemented
73#1893C2214yescomplemented
74#1893C2214yescomplemented
75#1875C2214yes—
76#1854C2214yescomplemented
77#1854C2214yescomplemented
78#1825C2214yescomplemented
79#1824C2212yescomplemented
80#1824C2212yescomplemented
81#1823C2212yescomplemented
82#1808C2214yescomplemented
83#1808C2212yescomplemented
84#1807C2214yescomplemented
85#1807C2212yescomplemented
86#1786C2212yescomplemented
87#1735C2212yescomplemented
88#1735C2212yescomplemented
89C23×D4—21no—
90C23×D4D411yescharacteristic
91C23×D4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C23×C8C2312yes—
95C23×C8C2312yes—
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98C23×C8C2311yescharacteristic
99C23×C8C2311yescharacteristic
100C2×#123—22no—
101C2×#123—22no—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121—24no—
105C2×#121—24no—
106C2×#121C2312yes—
107C2×#121C2312yes—
108C2×#121C2314yes—
109C2×#120—22no—
110C2×#120—22no—
111C2×C4×D4—24no—
112C2×C4×D4—24no—
113C2×C4×D4D412yes—
114C2×C4×D4D412yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×#117—21no—
118C2×#117—21no—
119C2×#117—24no—
120C2×#117—24no—
121C2×#117C2312yes—
122C2×#117C2312yes—
123C2×#117C2312yes—
124C2×#117C2312yes—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4D411yescharacteristic
132C22×C22⋊C4D411yescharacteristic
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135#482—216no—
136#482—216no—
137C8×D4—28no—
138C8×D4—28no—
139C8×D4—28no—
140C8×D4—28no—
141#431—28no—
142C2×#107—24no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#106—24no—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×#102—24no—
151C2×#102—24no—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×C22⋊C8—24no—
157C2×C22⋊C8—28no—
158C2×C22⋊C8D418yescomplemented
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2312yes—
165C2×C4×C8C2312yes—
166C2×C4×C8C2312yes—
167#400—22no—
168#400—22no—
169#399—22no—
170#399—22no—
171#398—22no—
172#397—22no—
173#397—24no—
174#394—24no—
175#394—22no—
176#394—24no—
177#392—24no—
178#392—22no—
179#390—24no—
180#390C2314yes—
181#389C2314yes—
182#389C2314yes—
183#389C2314yes—
184#388—24no—
185#388—24no—
186#388—24no—
187#387—24no—
188#386—22no—
189#386—22no—
190#386—24no—
191#386—22no—
192#386—22no—
193#385—22no—
194#385—22no—
195#385—22no—
196#385—22no—
197#381—21no—
198#381—21no—
199#381—22no—
200#381—22no—
201#380C2314yes—
202#380C2312yes—
203#380C2312yes—
204#380C2314yes—
205#380C2314yes—
206#380C2312yes—
207#380C2312yes—
208#379—21no—
209#379—22no—
210#379—21no—
211#379—22no—
212C4×C4⋊C4—24no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—24no—
217C4×C22⋊C4—21no—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226C25—21no—
227C25C2×D411yescharacteristic
228C25C2×D411yescharacteristic
229C22×D4—24no—
230C22×D4—28no—
231C22×D4—22no—
232C22×D4—24no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—21no—
236C22×D4—22no—
237C22×D4—42no—
238C22×D4—42no—
239C22×D4—44no—
240C22×D4—84no—
241C22×D4C2×D412yes—
242C22×D4C2×D412yes—
243C22×D4C2412yes—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—42no—
250C23×C4—42no—
251C23×C4—42no—
252C23×C4—42no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D414yes—
261C23×C4#4114yes—
262C23×C4C2412yes—
263C23×C4C2411yescharacteristic
264C23×C4C2411yescharacteristic
265C23×C4C2411yescharacteristic
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—24no—
269C22×C8—24no—
270C22×C8—42no—
271C22×C8—42no—
272C22×C8—44no—
273C22×C8—44no—
274C22×C8—42no—
275C22×C8—42no—
276C22×C8C2×D414yes—
277C22×C8C2×D414yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282#123—48no—
283#123—84no—
284#121—24no—
285#121—24no—
286#121—24no—
287#121—24no—
288#121—44no—
289#121—44no—
290#121—44no—
291#121—44no—
292#120—48no—
293#120—84no—
294C4×D4—28no—
295C4×D4—28no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—24no—
300#117—24no—
301#117—24no—
302#117—24no—
303#117—24no—
304#117—42no—
305#117—42no—
306#117—44no—
307#117—44no—
308#117—42no—
309#117—42no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2414yes—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—44no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—44no—
361C2×C42C2×D412yes—
362C2×C42C2412yes—
363#107—28no—
364#107—44no—
365#106—28no—
366#106—44no—
367C4⋊C8—24no—
368C4⋊C8—24no—
369#102—28no—
370#102—28no—
371#102—44no—
372#102—44no—
373C22⋊C8—24no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376C22⋊C8—24no—
377C22⋊C8C2×D418yes—
378C22⋊C8C2×D418yes—
379C4×C8—24no—
380C4×C8—24no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391C24—22no—
392C24—24no—
393C24—24no—
394C24—22no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—44no—
401C24—82no—
402C24—82no—
403C24#12114yes—
404C24C22×D411yescharacteristic
405C24C22×D411yescharacteristic
406C24C22×D412yes—
407C24C22×D411yescharacteristic
408C2×D4—232no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—48no—
425C2×D4—42no—
426C2×D4—84no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4—82no—
430C2×D4—84no—
431C2×D4—84no—
432C2×D4—816no—
433C2×D4#13518yes—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—84no—
470C22×C4—82no—
471C22×C4—82no—
472C22×C4#12114yes—
473C22×C4#12114yes—
474C22×C4#12114yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2511yescharacteristic
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—44no—
495C2×C8—44no—
496C2×C8—84no—
497C2×C8—84no—
498C2×C8—82no—
499C2×C8—82no—
500C2×C8C22×D414yes—
501C2×C8C22×D414yes—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—48no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—48no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C22⋊C4—82no—
537C22⋊C4—84no—
538C22⋊C4—82no—
539C22⋊C4—84no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—44no—
544C42—44no—
545C42—44no—
546C42—82no—
547C23—22no—
548C23—22no—
549C23—28no—
550C23—22no—
551C23—24no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—28no—
561C23—44no—
562C23—44no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—44no—
567C23—44no—
568C23—82no—
569C23—88no—
570C23—88no—
571C23—82no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23—168no—
576C23C2×#12112yes—
577C23C2×#12112yes—
578C23C2×#12112yes—
579C23C2×#13514yes—
580C23C2×#13514yes—
581C23C23×D411yescharacteristic
582D4—216no—
583D4—216no—
584D4—42no—
585D4—42no—
586D4—44no—
587D4—84no—
588D4—88no—
589D4—88no—
590D4—168no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—84no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4C2×#12112yes—
636C2×C4C2×#12112yes—
637C2×C4C2×#12112yes—
638C2×C4C2×#13514yes—
639C2×C4C23×D411yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4#58512yes—
642C2×C4#58512yes—
643C8—24no—
644C8—24no—
645C8—42no—
646C8—42no—
647C22—28no—
648C22—22no—
649C22—22no—
650C22—28no—
651C22—22no—
652C22—22no—
653C22—24no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—42no—
662C22—42no—
663C22—88no—
664C22—88no—
665C22—84no—
666C22—84no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—168no—
671C22#305014yes—
672C22C22×#12111yescharacteristic
673C22#348512yes—
674C22C22×#13512yes—
675C22#358512yes—
676C4—24no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—42no—
681C4—42no—
682C4—84no—
683C4—84no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4#348511yescharacteristic
688C4#348511yescharacteristic
689C4#349112yes—
690C2—24no—
691C2—44no—
692C2—42no—
693C2—42no—
694C2—162no—
695C2#3544711yescharacteristic
696C2C2×#305011yescharacteristic
697C2#6045411yescharacteristic
698C2#6170214yes—
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×#482C2semidirect
2. C2×#2972C2semidirect
4. #34809C2semidirect
5. #28097C2semidirect
6. #28094C2semidirect
7. #27662C2semidirect
8. #27658C2semidirect
9. #25237C2semidirect
10. #25235C2semidirect
11. #24605C2semidirect
12. #24555C2semidirect
13. #21939C2semidirect
14. #21761C2semidirect
15. #20699C2semidirect
21. C2×#482C22semidirect
25. C2×C8×D4C22semidirect
26. C2×C8×D4C22semidirect
27. C2×#431C22semidirect
28. C22×#107C22semidirect
29. C22×#106C22semidirect
30. C22×#102C22semidirect
31. C22×#102C22semidirect
34. C22×C22⋊C8C22semidirect
43. #2485C22semidirect
55. #2296C22semidirect
58. #2280C22semidirect
59. #2088C22semidirect
60. #2087C22semidirect
61. #2077C22semidirect
62. #2076C22semidirect
63. #2067C22semidirect
64. #2065C22semidirect
65. #2034C22semidirect
66. #2033C22semidirect
67. #1942C22semidirect
68. #1941C22semidirect
69. #1934C22semidirect
70. #1934C22semidirect
71. #1893C22semidirect
72. #1893C22semidirect
73. #1893C22semidirect
74. #1893C22semidirect
76. #1854C22semidirect
77. #1854C22semidirect
78. #1825C22semidirect
79. #1824C22semidirect
80. #1824C22semidirect
81. #1823C22semidirect
82. #1808C22semidirect
83. #1808C22semidirect
84. #1807C22semidirect
85. #1807C22semidirect
86. #1786C22semidirect
87. #1735C22semidirect
88. #1735C22semidirect
158. C2×C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4822
C2×#29722
#350762
#348092
#280972
#280942
#276622
#276582
#252372
#252352
#246052
#245552
#219392
#217612
#206992