Finite Group Databasebeta

← #6453149

Subgroups — #6453149

Order 512.

2170 conjugacy classes of subgroups.

The subgroup lattice has 763 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3094C211yesmaximal, characteristic
2C2×#2921C211yesmaximal, characteristic, complemented
3C2×#2624C211yesmaximal, characteristic
4#34052C211yesmaximal, characteristic, complemented
5#24618C214yesmaximal, complemented
6#24617C214yesmaximal, complemented
7#24615C214yesmaximal, complemented
8#24612C214yesmaximal, complemented
9#23670C212yesmaximal, complemented
10#23618C212yesmaximal, complemented
11#21159C212yesmaximal, complemented
12#21095C212yesmaximal, complemented
13#20912C211yesmaximal, characteristic
14#20846C211yesmaximal, characteristic, complemented
15#20702C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#3094—28no—
21C2×#481C2212yes—
22C2×#482C2214yes—
23C2×#480C2212yes—
24#2921—28no—
25C2×#433C2214yescomplemented
26C2×#433C2214yescomplemented
27C2×#428C2212yes—
28C2×#428C2212yes—
29C2×#423C2212yes—
30C2×#418C2212yes—
31C2×#418C2212yes—
32C2×#407C2211yescharacteristic
33C2×#407C2211yescharacteristic
34C2×C4×M16C2212yes—
35#2624—22no—
36#2624—22no—
37#2624C2214yes—
38#2624C2214yes—
39C2×#399—21no—
40C2×#397—21no—
41C2×#394—21no—
42C2×#394—21no—
43#2317—21no—
44#2317—21no—
45#2317C2212yescomplemented
46#2317C2212yescomplemented
47#2316C2212yescomplemented
48#2316C2212yescomplemented
49#2315—21no—
50#2315—21no—
51#2296—22no—
52#2296C2212yes—
53#2296C2212yes—
54C2×#390C2211yescharacteristic
55C2×#390C2211yescharacteristic
56C2×#389C2211yescharacteristic
57C2×#389C2211yescharacteristic
58#2285—22no—
59C2×#380C2211yescharacteristic
60#2088C2214yescomplemented
61#2088C2214yescomplemented
62#2087C2214yescomplemented
63#2086C2214yescomplemented
64#2067C2214yescomplemented
65#2067C2214yescomplemented
66#2065C2214yescomplemented
67#2065C2214yescomplemented
68#1944C2214yescomplemented
69#1943C2214yescomplemented
70#1942C2214yescomplemented
71#1941C2214yescomplemented
72#1936C2214yescomplemented
73#1936C2214yescomplemented
74#1934C2214yescomplemented
75#1934C2214yescomplemented
76#1875C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1854C2214yescomplemented
80#1853C2212yescomplemented
81#1852C2212yescomplemented
82#1825C2214yescomplemented
83#1824C2212yescomplemented
84#1823C2212yescomplemented
85#1736C2212yes—
86#1736C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89C23×D4—21no—
90C23×D4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×M16C2312yes—
93C22×M16C2312yes—
94C2×#121—21no—
95C2×#121—21no—
96C2×#121—24no—
97C2×#121—24no—
98C2×#121C2312yes—
99C2×#121C2312yes—
100C2×#117—21no—
101C2×#117—21no—
102C2×#117—24no—
103C2×#117—24no—
104C2×#117—28no—
105C2×#117—21no—
106C2×#117—21no—
107C2×#117—24no—
108C2×#117—24no—
109C2×#117D412yes—
110C2×#117D412yes—
111C2×#117C2312yes—
112C2×#117C2312yes—
113C2×#117C2312yes—
114C2×#117C2312yes—
115C2×#117C2312yes—
116C2×#117C2312yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4D411yescharacteristic
137C22×C22⋊C4D411yescharacteristic
138C22×C42—21no—
139C22×C42—21no—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#482—216no—
143#481—28no—
144#480—28no—
145#433—216no—
146#433—216no—
147#428—28no—
148#428—28no—
149C2×#107C2314yes—
150C2×#107C2314yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153#423—28no—
154C2×C4⋊C8C2314yes—
155C2×C4⋊C8C2314yes—
156#418—28no—
157#418—28no—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162#407—24no—
163#407D418yes—
164C2×C22⋊C8C2312yes—
165C2×C22⋊C8C2312yes—
166C2×C22⋊C8C2312yes—
167C2×C22⋊C8C2312yes—
168C4×M16—28no—
169C2×#97C2314yes—
170C2×C4×C8C2314yes—
171#399—24no—
172#399—24no—
173#397—24no—
174#397—24no—
175#394—24no—
176#394—24no—
177#394—24no—
178#394—24no—
179#390—22no—
180#390—22no—
181#390C2314yes—
182#390C2314yes—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2314yes—
198#389C2314yes—
199#385—22no—
200#385—22no—
201#385—22no—
202#385—22no—
203#385—22no—
204#385—22no—
205#385—22no—
206#385—22no—
207#384—22no—
208#384—22no—
209#382—22no—
210#382—22no—
211#380—21no—
212#380—21no—
213#380—22no—
214#380—22no—
215#380—22no—
216#380—22no—
217#380C2312yes—
218#380C2312yes—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C2×#95—21no—
224C2×#95—21no—
225C2×#95—21no—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234C25—21no—
235C25C2×D411yescharacteristic
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—24no—
242C22×D4—24no—
243C22×D4—44no—
244C22×D4—82no—
245C22×D4—82no—
246C22×D4C2412yes—
247C23×C4—24no—
248C23×C4—24no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D414yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4#4114yes—
265C23×C4C2411yescharacteristic
266C23×C4C2411yescharacteristic
267C23×C4C2411yescharacteristic
268C2×M16—24no—
269C2×M16—24no—
270C2×M16—24no—
271C2×M16—24no—
272C2×M16—24no—
273C2×M16C2×D418yes—
274C22×C8C2414yes—
275C22×C8C2414yes—
276#121—24no—
277#121—24no—
278#121—24no—
279#121—24no—
280#121—44no—
281#121—44no—
282#117—28no—
283#117—28no—
284#117—28no—
285#117—28no—
286#117—24no—
287#117—24no—
288#117—24no—
289#117—24no—
290#117—24no—
291#117—24no—
292#117—24no—
293#117—24no—
294#117—24no—
295#117—24no—
296#117—24no—
297#117—44no—
298#117—44no—
299#117#4118yes—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—82no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—44no—
358C2×C22⋊C4—44no—
359C2×C22⋊C4—82no—
360C2×C22⋊C4—82no—
361C2×C22⋊C4—82no—
362C2×C22⋊C4—82no—
363C2×C22⋊C4—82no—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D412yes—
367C2×C22⋊C4C2×D412yes—
368C2×C42—28no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—44no—
378C2×C42C2×D412yes—
379C2×C42C2×D412yes—
380C2×C42C2412yes—
381C2×C42C2412yes—
382#107—28no—
383#107—28no—
384#106—28no—
385#106—28no—
386C4⋊C8—28no—
387C4⋊C8—28no—
388#102—28no—
389#102—28no—
390#102—28no—
391#102—28no—
392C22⋊C8—24no—
393C22⋊C8—24no—
394C22⋊C8C2×D418yes—
395C22⋊C8C2×D418yes—
396#97—28no—
397C4×C8—28no—
398#95—22no—
399#95—22no—
400#95—24no—
401#95—24no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—41no—
414C24—41no—
415C24—41no—
416C24—41no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—42no—
424C24#12114yes—
425C24C22×D411yescharacteristic
426C24C22×D411yescharacteristic
427C24C22×D411yescharacteristic
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—24no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—48no—
438C2×D4—48no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—816no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—216no—
451C22×C4—216no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—81no—
489C22×C4—81no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—81no—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—81no—
496C22×C4#12114yes—
497C22×C4#12114yes—
498C22×C4#12114yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D412yes—
509C22×C4C2×#4112yes—
510C22×C4C2×#4112yes—
511C22×C4C2×#4112yes—
512C22×C4C2×#4114yes—
513C22×C42−1+414yes—
514C22×C4C2511yescharacteristic
515M16—216no—
516M16—48no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8C22×D418yes—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—24no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—84no—
553C4⋊C4—84no—
554C4⋊C4C2×#4114yes—
555C4⋊C4C2×#4114yes—
556C22⋊C4—28no—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—24no—
561C22⋊C4—28no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—84no—
567C22⋊C4—84no—
568C22⋊C4—84no—
569C22⋊C4—84no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C42—28no—
573C42—24no—
574C42—24no—
575C42—28no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—44no—
581C42—44no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—28no—
599C23—44no—
600C23—44no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—82no—
609C23—82no—
610C23—82no—
611C23—82no—
612C23—84no—
613C23—84no—
614C23—84no—
615C23—84no—
616C23—84no—
617C23—84no—
618C23—84no—
619C23C2×#12112yes—
620C23C2×#12112yes—
621C23C2×#12112yes—
622C23#52512yes—
623C23#53612yes—
624C23C23×D411yescharacteristic
625D4—44no—
626D4—44no—
627D4—88no—
628D4—88no—
629D4—84no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—28no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—28no—
636C2×C4—28no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—82no—
675C2×C4—82no—
676C2×C4—82no—
677C2×C4—82no—
678C2×C4—84no—
679C2×C4—84no—
680C2×C4—84no—
681C2×C4—84no—
682C2×C4—84no—
683C2×C4—84no—
684C2×C4—84no—
685C2×C4C2×#12112yes—
686C2×C4C2×#12112yes—
687C2×C4C2×#12112yes—
688C2×C4#52512yes—
689C2×C4#53612yes—
690C2×C4#54718yes—
691C2×C4#55218yes—
692C2×C4C23×D411yescharacteristic
693C2×C4C22×#4112yes—
694C2×C4C22×#4111yescharacteristic
695C2×C4C2×2−1+412yes—
696C8—28no—
697C8—44no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22—41no—
715C22—41no—
716C22—41no—
717C22—41no—
718C22—88no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—168no—
727C22#241314yes—
728C22C22×#12111yescharacteristic
729C22C2×#52511yescharacteristic
730C22C2×#53611yescharacteristic
731C22#358511yescharacteristic
732C22#358511yescharacteristic
733C22#358511yescharacteristic
734C22#358611yescharacteristic
735C4—22no—
736C4—22no—
737C4—42no—
738C4—42no—
739C4—42no—
740C4—42no—
741C4—44no—
742C4—44no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4#241314yes—
749C4C2×#54712yes—
750C4C2×#55212yes—
751C2—24no—
752C2—42no—
753C2—42no—
754C2—162no—
755C2C2×#241311yescharacteristic
756C2#6045411yescharacteristic
757C2#6045611yescharacteristic
758C2#6056711yescharacteristic
759C2#6056811yescharacteristic
760C2#6086811yescharacteristic
761C2#6087011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2921C2semidirect
4. #34052C2semidirect
5. #24618C2semidirect
6. #24617C2semidirect
7. #24615C2semidirect
8. #24612C2semidirect
9. #23670C2semidirect
10. #23618C2semidirect
11. #21159C2semidirect
12. #21095C2semidirect
14. #20846C2semidirect
15. #20702C2semidirect
25. C2×#433C22semidirect
26. C2×#433C22semidirect
45. #2317C22semidirect
46. #2317C22semidirect
47. #2316C22semidirect
48. #2316C22semidirect
60. #2088C22semidirect
61. #2088C22semidirect
62. #2087C22semidirect
63. #2086C22semidirect
64. #2067C22semidirect
65. #2067C22semidirect
66. #2065C22semidirect
67. #2065C22semidirect
68. #1944C22semidirect
69. #1943C22semidirect
70. #1942C22semidirect
71. #1941C22semidirect
72. #1936C22semidirect
73. #1936C22semidirect
74. #1934C22semidirect
75. #1934C22semidirect
79. #1854C22semidirect
80. #1853C22semidirect
81. #1852C22semidirect
82. #1825C22semidirect
83. #1824C22semidirect
84. #1823C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30942
C2×#29212
C2×#26242
#340522
#246182
#246172
#246152
#246122
#236702
#236182
#211592
#210952
#209122
#208462
#207022