Finite Group Databasebeta

← #6514136

Subgroups — #6514136

Order 512.

1944 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3249C211yesmaximal, characteristic, complemented
2C2×#3110C211yesmaximal, characteristic
3#35240C211yesmaximal, characteristic
4#34399C211yesmaximal, characteristic, complemented
5#28288C212yesmaximal, complemented
6#28258C212yesmaximal, complemented
7#28211C212yesmaximal, complemented
8#28168C212yesmaximal, complemented
9#25206C214yesmaximal, complemented
10#25084C214yesmaximal
11#24827C212yesmaximal, complemented
12#24743C212yesmaximal, complemented
13#24694C212yesmaximal, complemented
14#24645C212yesmaximal, complemented
15#21977C211yesmaximal, characteristic, complemented
16#21866C211yesmaximal, characteristic, complemented
17C2×#1804C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#3249—28no—
22#3110—28no—
23C2×#484C2212yes—
24C2×#477C2212yescomplemented
25C2×#477C2212yescomplemented
26C2×#474C2211yescharacteristic
27C2×#474C2211yescharacteristic
28C2×#467C2214yes—
29C2×#464C2214yes—
30C2×#462C2212yescomplemented
31C2×#462C2212yescomplemented
32C2×#461C2211yescharacteristic
33C2×#461C2211yescharacteristic
34C2×#450C2212yes—
35C2×#440C2212yes—
36C2×#440C2212yes—
37C2×#432C2211yescharacteristic
38C22×C4⋊C8C2211yescharacteristic
39C22×#102C2212yes—
40#2848—22no—
41#2633—22no—
42#2547C2211yescharacteristic
43#2547C2211yescharacteristic
44#2542C2211yescharacteristic
45#2542C2211yescharacteristic
46#2536—22no—
47#2502C2212yes—
48#2482—22no—
49#2435C2212yes—
50#2382—22no—
51#2360C2214yes—
52#2356C2212yescomplemented
53#2351C2212yescomplemented
54#2343—22no—
55#2331C2212yescomplemented
56#2329—22no—
57#2327C2212yescomplemented
58#2326C2212yes—
59#2326C2212yes—
60#2320—22no—
61#2283C2211yescharacteristic
62C2×#382C2211yescharacteristic
63#2083C2214yescomplemented
64#2072C2214yescomplemented
65#2068C2214yescomplemented
66#2067C2214yes—
67#2056C2214yescomplemented
68#2055C2214yes—
69#2042C2214yes—
70#2015C2214yes—
71#1993C2212yescomplemented
72#1983C2214yes—
73#1960C2212yescomplemented
74#1947C2214yescomplemented
75#1940C2212yescomplemented
76#1940C2212yescomplemented
77#1928C2214yescomplemented
78#1925C2214yes—
79#1880C2214yes—
80#1878C2214yescomplemented
81#1872C2214yes—
82#1850C2214yes—
83#1843C2212yescomplemented
84#1832C2212yescomplemented
85#1805C2212yescomplemented
86#1805C2212yescomplemented
87#1804C2212yes—
88#1804C2212yes—
89#1804C2212yes—
90#1804C2212yes—
91#1803C2214yescomplemented
92C2×#340C2212yes—
93C23×D4C2311yescharacteristic
94C24×C4—21no—
95C22×Q16C2314yes—
96C22×SD16C2312yes—
97C22×SD16C2312yes—
98C23×C8C2312yes—
99#533—24no—
100#533—24no—
101#533—24no—
102#533D418yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×#125—21no—
108C2×#125—21no—
109C2×#124C2312yes—
110C2×#124C2312yes—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2312yes—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2314yes—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—24no—
125C2×C4×D4—24no—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×#117—21no—
131C2×#117—21no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C22⋊C4—21no—
135C22×C42C2311yescharacteristic
136#484—28no—
137#477—28no—
138#477—28no—
139#474—24no—
140#474—24no—
141#467—216no—
142#464—48no—
143#462—28no—
144#462—28no—
145#461—24no—
146#461—24no—
147#450—28no—
148#440—28no—
149#440—28no—
150#432—24no—
151C2×#107C2314yes—
152C2×#107C2312yes—
153C2×#107C2312yes—
154C2×C4⋊C8—24no—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2311yescharacteristic
158C2×C4⋊C8C2311yescharacteristic
159C2×C4⋊C8C2314yes—
160C2×C4⋊C8C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#102—44no—
168C2×#102C2314yes—
169C2×#102C2312yes—
170C2×#102C2312yes—
171C2×#102C2314yes—
172C2×C22⋊C8C2312yes—
173C2×#97C2312yes—
174#399—22no—
175#399—22no—
176#398—24no—
177#397—22no—
178#397—22no—
179#394—24no—
180#391—24no—
181#391C2314yes—
182#391C2312yes—
183#391C2312yes—
184#390C2312yes—
185#390C2312yes—
186#389—24no—
187#389—24no—
188#389C2314yes—
189#388—24no—
190#388—24no—
191#387—24no—
192#387—24no—
193#387—24no—
194#387—24no—
195#387—24no—
196#385—24no—
197#385—24no—
198#385—24no—
199#385—24no—
200#384C2312yes—
201#384C2312yes—
202#383—22no—
203#383D412yes—
204#382—21no—
205#382—21no—
206#382—22no—
207#382D412yes—
208#382C2312yes—
209#382C2312yes—
210#381D412yes—
211#380—22no—
212#380—22no—
213#380D412yes—
214#379—24no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—24no—
220C4×C22⋊C4C2312yes—
221#376—22no—
222#376C2312yes—
223C2×#95—21no—
224C2×#95—21no—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#336C2314yes—
230#336C2314yes—
231C25—21no—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8C2412yes—
235C22×Q8C2412yes—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—41no—
242C22×D4—41no—
243C22×D4C2411yescharacteristic
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—44no—
252C23×C4—84no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×Q16—28no—
257C2×Q16—44no—
258C2×Q16—44no—
259C2×SD16—24no—
260C2×SD16—24no—
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—42no—
264C2×SD16—42no—
265C22×C8—24no—
266C22×C8—42no—
267C22×C8—42no—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277C4⋊Q8C2×D414yes—
278C4⋊Q8C2×D414yes—
279#125—28no—
280#125—44no—
281#124—24no—
282#124—24no—
283#124—42no—
284#124—42no—
285C22⋊Q8—28no—
286C22⋊Q8—24no—
287C22⋊Q8—24no—
288C22⋊Q8—24no—
289C22⋊Q8—44no—
290C22⋊Q8—84no—
291C22⋊Q8—84no—
292C22⋊Q8C2×D418yes—
293C4×Q8—28no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—44no—
298C4×Q8—44no—
299C4×Q8—44no—
300C4×Q8C2×D418yes—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—44no—
310C4×D4—44no—
311#117—28no—
312#117—44no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4C2×D414yes—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D414yes—
359C2×C42#4114yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—24no—
365#107—24no—
366#107—28no—
367C4⋊C8—24no—
368C4⋊C8—24no—
369C4⋊C8—22no—
370C4⋊C8—22no—
371C4⋊C8—44no—
372C4⋊C8—44no—
373#103—24no—
374#103—24no—
375#103—28no—
376#103—28no—
377#103—44no—
378#103—44no—
379#102—28no—
380#102—24no—
381#102—24no—
382#102—44no—
383C22⋊C8—24no—
384#97—24no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95C2×D412yes—
396#95C2×D414yes—
397#95C2×D414yes—
398#95C2×D414yes—
399#95C2×D412yes—
400C24—21no—
401C24—21no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—44no—
406C24—82no—
407C24C22×D411yescharacteristic
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—44no—
421C2×Q8—42no—
422C2×Q8—82no—
423C2×Q8C22×D414yes—
424C2×D4—28no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—44no—
440C2×D4—42no—
441C2×D4H(5,𝔽2)12yes—
442C2×D42−1+412yes—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—816no—
470C22×C4#12114yes—
471C22×C4#12114yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C2×#4112yes—
481C22×C4C2511yescharacteristic
482Q16—88no—
483SD16—48no—
484SD16—48no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—84no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—84no—
518C4⋊C4—82no—
519C4⋊C4—84no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4C22×D414yes—
523C22⋊C4—28no—
524C22⋊C4—28no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C42—28no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—44no—
540C42—84no—
541C42—84no—
542C42#12114yes—
543C42#12114yes—
544C42C22×D412yes—
545C42C22×D412yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C23—22no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—48no—
554C23—42no—
555C23—42no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—88no—
563C23—88no—
564C23—84no—
565C23—84no—
566C23C2×#12112yes—
567C23#57814yes—
568C23C23×D411yescharacteristic
569Q8—24no—
570Q8—24no—
571Q8—42no—
572Q8—42no—
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—84no—
578Q8—84no—
579D4—22no—
580D4—22no—
581D4—22no—
582D4—22no—
583D4—48no—
584D4—41no—
585D4—41no—
586D4—41no—
587D4—41no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—164no—
629C2×C4C2×#12112yes—
630C2×C4C2×#12112yes—
631C2×C4C2×#12112yes—
632C2×C4C2×#12112yes—
633C2×C4C2×#12112yes—
634C2×C4C23×D411yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C2×C4#58511yescharacteristic
639C2×C4#58511yescharacteristic
640C8—44no—
641C8—42no—
642C8—42no—
643C22—22no—
644C22—24no—
645C22—22no—
646C22—24no—
647C22—44no—
648C22—42no—
649C22—42no—
650C22—44no—
651C22—816no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—82no—
657C22—82no—
658C22#304414yes—
659C22C22×#12111yescharacteristic
660C22#348811yescharacteristic
661C22#348911yescharacteristic
662C22C2×#57812yes—
663C22#358512yes—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—44no—
670C4—44no—
671C4—44no—
672C4—44no—
673C4—81no—
674C4—81no—
675C4—82no—
676C4—84no—
677C4—82no—
678C4#348811yescharacteristic
679C4#348911yescharacteristic
680C4#349211yescharacteristic
681C4#349311yescharacteristic
682C2—24no—
683C2—82no—
684C2—82no—
685C2#3551811yescharacteristic
686C2C2×#304411yescharacteristic
687C2#6042611yescharacteristic
688C2#6201712yes—
689C2#6216512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3249C2semidirect
4. #34399C2semidirect
5. #28288C2semidirect
6. #28258C2semidirect
7. #28211C2semidirect
8. #28168C2semidirect
9. #25206C2semidirect
11. #24827C2semidirect
12. #24743C2semidirect
13. #24694C2semidirect
14. #24645C2semidirect
15. #21977C2semidirect
16. #21866C2semidirect
24. C2×#477C22semidirect
25. C2×#477C22semidirect
30. C2×#462C22semidirect
31. C2×#462C22semidirect
52. #2356C22semidirect
53. #2351C22semidirect
55. #2331C22semidirect
57. #2327C22semidirect
63. #2083C22semidirect
64. #2072C22semidirect
65. #2068C22semidirect
67. #2056C22semidirect
71. #1993C22semidirect
73. #1960C22semidirect
74. #1947C22semidirect
75. #1940C22semidirect
76. #1940C22semidirect
77. #1928C22semidirect
80. #1878C22semidirect
83. #1843C22semidirect
84. #1832C22semidirect
85. #1805C22semidirect
86. #1805C22semidirect
91. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32492
C2×#31102
#352402
#343992
#282882
#282582
#282112
#281682
#252062
#250842
#248272
#247432
#246942
#246452
#219772
#218662
C2×#18042