Finite Group Databasebeta

← #6514214

Subgroups — #6514214

Order 512.

1934 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×#3255C211yesmaximal, characteristic, complemented
2C2×#3101C211yesmaximal, characteristic
3#35240C211yesmaximal, characteristic
4#34373C211yesmaximal, characteristic, complemented
5#28289C212yesmaximal, complemented
6#28258C212yesmaximal, complemented
7#28212C212yesmaximal, complemented
8#28168C212yesmaximal, complemented
9#25219C214yesmaximal, complemented
10#25105C214yesmaximal
11#24813C212yesmaximal, complemented
12#24729C212yesmaximal, complemented
13#24678C212yesmaximal, complemented
14#24629C212yesmaximal, complemented
15#21975C211yesmaximal, characteristic, complemented
16#21864C211yesmaximal, characteristic, complemented
17#20646C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#3255—28no—
22#3101—28no—
23C2×#483C2212yes—
24C2×#475C2212yescomplemented
25C2×#475C2212yescomplemented
26C2×#476C2211yescharacteristic
27C2×#476C2211yescharacteristic
28C2×#470C2214yes—
29C2×#464C2214yes—
30C2×#462C2212yescomplemented
31C2×#462C2212yescomplemented
32C2×#461C2211yescharacteristic
33C2×#461C2211yescharacteristic
34C2×#450C2212yes—
35C2×C4×SD16C2212yes—
36C2×C4×SD16C2212yes—
37C2×#431C2211yescharacteristic
38C2×#423C2211yescharacteristic
39C2×#418C2212yes—
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#2378—22no—
51#2362C2214yes—
52#2355—22no—
53#2355C2212yescomplemented
54Q8×C4⋊C4C2212yescomplemented
55#2336—22no—
56#2331C2212yescomplemented
57#2327C2212yescomplemented
58#2326C2212yes—
59#2326C2212yes—
60#2314—22no—
61#2285C2211yescharacteristic
62#2283C2211yescharacteristic
63#2084C2214yescomplemented
64#2072C2214yescomplemented
65#2066C2214yescomplemented
66#2065C2214yes—
67#2056C2214yescomplemented
68#2055C2214yes—
69#2042C2214yes—
70#2015C2214yes—
71#1990C2212yescomplemented
72#1983C2214yes—
73#1956C2212yescomplemented
74#1948C2214yescomplemented
75#1939C2212yescomplemented
76#1939C2212yescomplemented
77#1927C2214yescomplemented
78#1926C2214yes—
79#1880C2214yes—
80#1878C2214yescomplemented
81#1873C2214yes—
82#1851C2214yes—
83#1843C2212yescomplemented
84#1832C2212yescomplemented
85#1805C2212yescomplemented
86#1805C2212yescomplemented
87#1804C2212yes—
88#1804C2212yes—
89#1803C2214yescomplemented
90#1802C2212yes—
91#1802C2212yes—
92#1736C2212yes—
93C23×D4C2311yescharacteristic
94C24×C4—21no—
95C22×Q16C2314yes—
96C22×SD16C2312yes—
97C22×SD16C2312yes—
98C22×M16C2312yes—
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—
132C2×#117C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C22⋊C4—21no—
135C22×C42C2311yescharacteristic
136#483—28no—
137#476—24no—
138#476—24no—
139#475—28no—
140#475—28no—
141#470—216no—
142#464—48no—
143#462—28no—
144#462—28no—
145#461—24no—
146#461—24no—
147#450—28no—
148C4×SD16—28no—
149C4×SD16—28no—
150#431—24no—
151C2×#106C2314yes—
152C2×#106C2312yes—
153C2×#106C2312yes—
154#423—24no—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2311yescharacteristic
158C2×C4⋊C8C2311yescharacteristic
159C2×C4⋊C8C2314yes—
160C2×C4⋊C8C2314yes—
161#418—44no—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2312yes—
165C2×#103C2312yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#102C2314yes—
169C2×#102C2312yes—
170C2×#102C2312yes—
171C2×#102C2314yes—
172C2×C22⋊C8C2312yes—
173C2×C4×C8C2312yes—
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#389C2314yes—
188#388—24no—
189#388—24no—
190#388—24no—
191#388—24no—
192#387—24no—
193#387—24no—
194#387—24no—
195#387—24no—
196#387—24no—
197#385—24no—
198#385—24no—
199#385—24no—
200#384—22no—
201#384C2312yes—
202#384C2312yes—
203#384C2312yes—
204#383D412yes—
205#382—22no—
206#382D412yes—
207#382C2312yes—
208#381D412yes—
209#380—21no—
210#380—21no—
211#380—22no—
212#380—22no—
213#380D412yes—
214#379—24no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—24no—
220C4×C4⋊C4C2314yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4C2312yes—
224#376C2312yes—
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—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—44no—
251C23×C4—84no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×Q16—28no—
256C2×Q16—44no—
257C2×Q16—44no—
258C2×SD16—24no—
259C2×SD16—24no—
260C2×SD16—24no—
261C2×SD16—24no—
262C2×SD16—42no—
263C2×SD16—42no—
264C2×M16—24no—
265C2×M16—42no—
266C2×M16—42no—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C4⋊Q8—22no—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276C4⋊Q8C2×D414yes—
277C4⋊Q8C2×D414yes—
278#125—28no—
279#125—44no—
280#124—24no—
281#124—24no—
282#124—42no—
283#124—42no—
284C22⋊Q8—28no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—24no—
288C22⋊Q8—44no—
289C22⋊Q8—84no—
290C22⋊Q8—84no—
291C22⋊Q8C2×D418yes—
292C4×Q8—28no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—44no—
297C4×Q8—44no—
298C4×Q8—44no—
299C4×Q8C2×D418yes—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—44no—
309C4×D4—44no—
310#117—28no—
311#117—44no—
312#117—42no—
313#117—42no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
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—22no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4C2×D414yes—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D414yes—
361C2×C42#4114yes—
362C2×C42C2412yes—
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#106—24no—
367#106—24no—
368#106—28no—
369C4⋊C8—24no—
370C4⋊C8—24no—
371C4⋊C8—22no—
372C4⋊C8—22no—
373C4⋊C8—44no—
374C4⋊C8—44no—
375#103—24no—
376#103—24no—
377#103—28no—
378#103—28no—
379#103—44no—
380#103—44no—
381#102—28no—
382#102—24no—
383#102—24no—
384#102—44no—
385C22⋊C8—24no—
386C4×C8—24no—
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×D414yes—
396#95C2×D414yes—
397#95C2×D414yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—21no—
401C24—21no—
402C24—42no—
403C24—42no—
404C24—44no—
405C24—42no—
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—42no—
421C2×Q8—44no—
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—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—816no—
466C22×C4—82no—
467C22×C4#12114yes—
468C22×C4#12114yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C2×#4112yes—
478C22×C4C2511yescharacteristic
479Q16—88no—
480SD16—48no—
481SD16—48no—
482M16—84no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—82no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4C22×D414yes—
521C22⋊C4—28no—
522C22⋊C4—44no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—44no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—82no—
533C42—28no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—44no—
539C42—84no—
540C42—82no—
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—22no—
554C23—48no—
555C23—42no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—42no—
562C23—42no—
563C23—84no—
564C23—84no—
565C23—88no—
566C23—88no—
567C23C2×#12112yes—
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—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4C2×#12112yes—
629C2×C4C2×#12112yes—
630C2×C4C2×#12112yes—
631C2×C4C2×#12112yes—
632C2×C4C2×#12112yes—
633C2×C4#57714yes—
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—22no—
645C22—22no—
646C22—42no—
647C22—42no—
648C22—44no—
649C22—42no—
650C22—42no—
651C22—44no—
652C22—816no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—82no—
658C22—82no—
659C22C22×#12111yescharacteristic
660C22#348811yescharacteristic
661C22#348911yescharacteristic
662C22C2×#57712yes—
663C22#358612yes—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—44no—
669C4—44no—
670C4—44no—
671C4—44no—
672C4—81no—
673C4—81no—
674C4—82no—
675C4—84no—
676C4—82no—
677C4#304314yes—
678C4#348811yescharacteristic
679C4#348911yescharacteristic
680C4#349211yescharacteristic
681C4#349311yescharacteristic
682C2—42no—
683C2—82no—
684C2—82no—
685C2#3551811yescharacteristic
686C2C2×#304311yescharacteristic
687C2#6042711yescharacteristic
688C2#6204212yes—
689C2#6218912yes—
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×#3255C2semidirect
4. #34373C2semidirect
5. #28289C2semidirect
6. #28258C2semidirect
7. #28212C2semidirect
8. #28168C2semidirect
9. #25219C2semidirect
11. #24813C2semidirect
12. #24729C2semidirect
13. #24678C2semidirect
14. #24629C2semidirect
15. #21975C2semidirect
16. #21864C2semidirect
24. C2×#475C22semidirect
25. C2×#475C22semidirect
30. C2×#462C22semidirect
31. C2×#462C22semidirect
53. #2355C22semidirect
54. Q8×C4⋊C4C22semidirect
56. #2331C22semidirect
57. #2327C22semidirect
63. #2084C22semidirect
64. #2072C22semidirect
65. #2066C22semidirect
67. #2056C22semidirect
71. #1990C22semidirect
73. #1956C22semidirect
74. #1948C22semidirect
75. #1939C22semidirect
76. #1939C22semidirect
77. #1927C22semidirect
80. #1878C22semidirect
83. #1843C22semidirect
84. #1832C22semidirect
85. #1805C22semidirect
86. #1805C22semidirect
89. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32552
C2×#31012
#352402
#343732
#282892
#282582
#282122
#281682
#252192
#251052
#248132
#247292
#246782
#246292
#219752
#218642
#206462