Finite Group Databasebeta

← #6381863

Subgroups — #6381863

Order 512.

2810 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3036C211yesmaximal, characteristic
2C2×#2946C212yesmaximal, complemented
3#34206C211yesmaximal, characteristic, complemented
4#34181C211yesmaximal, characteristic, complemented
5#23012C214yesmaximal, complemented
6#23011C214yesmaximal, complemented
7#23008C214yesmaximal, complemented
8#23007C214yesmaximal, complemented
9#21199C211yesmaximal, characteristic, complemented
10#21170C211yesmaximal, characteristic, complemented
11#21092C212yesmaximal, complemented
12#20929C212yesmaximal, complemented
13#20878C212yesmaximal, complemented
14#20438C211yesmaximal, characteristic, complemented
15C2×#1762C211yesmaximal, characteristic, complemented
16C22×AGL(1,ℤ/8)C2212yes—
17C2×#532C2211yescharacteristic
18C2×#520C2212yes—
19C2×#518C2212yes—
20C22×C4×D4C2211yescharacteristic
21#3036C4116yescomplemented
22C2×#463C2214yes—
23C2×#461C2212yes—
24C2×#459C2212yes—
25#2946—216no—
26C2×#442C2214yes—
27C2×#440C2214yes—
28C2×C4×SD16C2214yes—
29C2×C4×D8C2214yes—
30C2×#428C2212yes—
31C2×#423C2211yescharacteristic
32C2×#417C2212yes—
33C2×#416C2212yes—
34C2×#416C2212yes—
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C2×#405C2212yes—
38C22×#97C2211yescharacteristic
39#2367C2214yescomplemented
40#2366C2212yescomplemented
41#2365C2212yescomplemented
42#2354—21no—
43#2352—22no—
44#2349—22no—
45#2349C2214yes—
46D4×C4⋊C4—21no—
47D4×C4⋊C4—21no—
48D4×C4⋊C4—21no—
49D4×C4⋊C4C2212yes—
50D4×C4⋊C4C2212yes—
51#2325—22no—
52#2316C2212yes—
53#2315—22no—
54#2311C2212yes—
55C4×#121—22no—
56#2285C2211yescharacteristic
57#2285C2211yescharacteristic
58C2×#380—22no—
59C2×C4×C4⋊C4C2211yescharacteristic
60#1961C2212yescomplemented
61#1960C2212yescomplemented
62#1959C2212yescomplemented
63#1959C2212yescomplemented
64#1949C2214yescomplemented
65#1948C2212yescomplemented
66#1947C2212yescomplemented
67#1936C2214yescomplemented
68#1935C2214yescomplemented
69#1934C2214yescomplemented
70#1933C2214yescomplemented
71#1879C2214yes—
72#1879C2214yes—
73#1877C2214yes—
74#1877C2214yes—
75#1865C2214yescomplemented
76#1864C2214yescomplemented
77#1863C2214yescomplemented
78#1862C2214yescomplemented
79#1771C2214yescomplemented
80#1771C2214yescomplemented
81#1762C2214yes—
82#1762C2214yes—
83#1736C2212yes—
84C2×#340C2211yescharacteristic
85C2×#340C2211yescharacteristic
86#1729C2212yes—
87C22×#41C2312yes—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C2×AGL(1,ℤ/8)—28no—
91C2×AGL(1,ℤ/8)—28no—
92C2×AGL(1,ℤ/8)C2×C4116yescomplemented
93C22×SD16C2314yes—
94C22×D8C2314yes—
95C22×M16C2312yes—
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98#532—24no—
99#532C2×C418yescomplemented
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127C2311yescharacteristic
102C2×#124C2312yes—
103C2×#121—21no—
104C2×#121—21no—
105#520—28no—
106#520—44no—
107#518—28no—
108#518—44no—
109C2×C4×Q8C2314yes—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4—24no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—24no—
119C2×C4×D4C2×C418yescomplemented
120C2×C4×D4C2314yes—
121C2×C4×D4C2314yes—
122C2×C4×D4C2314yes—
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×#117—22no—
126C2×#117C2312yes—
127C2×#117C2312yes—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C22⋊C4—21no—
135C22×C22⋊C4—24no—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138#463C2×C4132yescomplemented
139#461C2×C4116yescomplemented
140#459C2×C4116yescomplemented
141#442—216no—
142#440—216no—
143C4×SD16—216no—
144C4×D8—216no—
145#428—28no—
146C2×#107C2314yes—
147C2×#106C2314yes—
148#423C2×C418yescomplemented
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151#417C2×C4116yescomplemented
152#416—28no—
153#416—44no—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#102—24no—
160C2×#102D418yescomplemented
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2312yes—
165C2×#102C2312yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168#405—28no—
169C2×#97—24no—
170C2×#97C2312yes—
171C2×#97C2312yes—
172C2×#97C2314yes—
173C2×C4×C8C2314yes—
174#391C2314yes—
175#390—22no—
176#390C2314yes—
177#389—22no—
178#389—22no—
179#389—22no—
180#389C2314yes—
181#389C2314yes—
182#388—24no—
183#388—24no—
184#387—24no—
185#386—24no—
186#386—24no—
187#385—24no—
188#385—24no—
189#385—24no—
190#385—24no—
191#385—24no—
192#384D411yescharacteristic
193#384D411yescharacteristic
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#382D412yes—
197#382C2312yes—
198#380—21no—
199#380—21no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—24no—
205#380—24no—
206#380D412yes—
207#380D412yes—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—24no—
218C2×#95—21no—
219C2×#95—24no—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#336C2314yes—
225#336C2314yes—
226C25—21no—
227C2×#41—24no—
228C2×#41—24no—
229C2×#41—24no—
230C2×#41—24no—
231C2×#41—24no—
232C2×#41C22×C418yes—
233C22×Q8C2412yes—
234C22×D4—22no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4C22×C414yes—
242C22×D4C2×D414yes—
243C22×D4#4114yes—
244C22×D4C2412yes—
245C22×D4C2412yes—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—28no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—24no—
254C23×C4—24no—
255C23×C4—24no—
256C23×C4—82no—
257C23×C4—82no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2411yescharacteristic
261AGL(1,ℤ/8)—264no—
262C2×SD16—28no—
263C2×SD16—28no—
264C2×SD16C22×C4116yes—
265C2×D8—28no—
266C2×D8—28no—
267C2×D8C22×C4116yes—
268C2×M16—24no—
269C2×M16—24no—
270C2×M16C22×C418yes—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8C2×D414yes—
277C22×C8C2412yes—
278C22×C8C2414yes—
279C22×C8C2412yes—
280C4⋊Q8—22no—
281C4⋊Q8C22×C414yes—
282#127—22no—
283#127C22×C414yes—
284#124—24no—
285#124C22×C418yes—
286#121—28no—
287#121—28no—
288C4×Q8—28no—
289C4×Q8—44no—
290C4×D4—28no—
291C4×D4—28no—
292C4×D4—28no—
293C4×D4—28no—
294C4×D4—28no—
295C4×D4—28no—
296C4×D4—28no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4—416no—
303C4×D4—48no—
304C4×D4—48no—
305C4×D4—84no—
306C4×D4C22×C418yes—
307C4×D4C22×C418yes—
308#117—24no—
309#117—24no—
310#117—24no—
311#117—24no—
312#117—48no—
313C2×C4⋊C4—21no—
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—24no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—48no—
330C2×C4⋊C4—84no—
331C2×C4⋊C4—84no—
332C2×C4⋊C4C22×C414yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D414yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C22⋊C4—28no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—416no—
357C2×C22⋊C4C2×D414yes—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—82no—
363C2×C42C22×C414yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42#4114yes—
367C2×C42C2414yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370#107—28no—
371#106—28no—
372C4⋊C8C22×C418yes—
373C4⋊C8C22×C418yes—
374#103—24no—
375#103—24no—
376#103—42no—
377#103—42no—
378#103C22×C4116yes—
379#102—28no—
380#102—24no—
381#102—24no—
382#102—42no—
383#102—42no—
384#102C22×C4116yes—
385#102C2×D4116yes—
386#97—28no—
387#97—24no—
388#97—24no—
389C4×C8—28no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—24no—
398#95C2×D414yes—
399#95C2×D414yes—
400C24—22no—
401C24—24no—
402C24—24no—
403C24—24no—
404C24—22no—
405C24—81no—
406C24—81no—
407C24—81no—
408C24—81no—
409C24C22×D411yescharacteristic
410#41—216no—
411#41—216no—
412#41—48no—
413#41—48no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8C23×C414yes—
420C2×D4—28no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—28no—
425C2×D4—24no—
426C2×D4—28no—
427C2×D4—24no—
428C2×D4—28no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—81no—
442C2×D4—81no—
443C2×D4C4×D4116yescomplemented
444C2×D4C23×C414yes—
445C2×D4C23×C414yes—
446C2×D4C22×D414yes—
447C2×D4C2×#4114yes—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—28no—
454C22×C4—28no—
455C22×C4—28no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—48no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4—84no—
474C22×C4—84no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4—816no—
480C22×C4—81no—
481C22×C4—81no—
482C22×C4—81no—
483C22×C4—81no—
484C22×C4—81no—
485C22×C4—81no—
486C22×C4C4×D414yes—
487C22×C4C4×D414yes—
488C22×C4C23×C412yes—
489C22×C4C22×D414yes—
490C22×C4C22×D414yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D412yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4H(5,𝔽2)12yes—
500C22×C42−1+412yes—
501C22×C4C2511yescharacteristic
502SD16—232no—
503D8—232no—
504M16—216no—
505C2×C8—28no—
506C2×C8—24no—
507C2×C8—22no—
508C2×C8—24no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—44no—
514C2×C8C23×C418yes—
515C2×C8C22×D414yes—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—48no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—81no—
539C4⋊C4—81no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C4⋊C4C23×C414yes—
543C4⋊C4C22×D414yes—
544C22⋊C4—28no—
545C22⋊C4—28no—
546C22⋊C4—28no—
547C22⋊C4—28no—
548C22⋊C4—28no—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C22⋊C4—44no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C42—24no—
556C42—24no—
557C42—48no—
558C42—44no—
559C42—44no—
560C42—82no—
561C42C4×D414yes—
562C42C4×D414yes—
563C42C23×C412yes—
564C42C23×C412yes—
565C42C2×#4112yes—
566C42C2×#4112yes—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—216no—
575C23—24no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23—84no—
590C23—84no—
591C23—84no—
592C23C2×C4×D412yes—
593C23D4212yes—
594C23#54812yes—
595C23C23×D411yescharacteristic
596Q8—24no—
597Q8—24no—
598Q8—42no—
599Q8—42no—
600D4—216no—
601D4—24no—
602D4—24no—
603D4—24no—
604D4—24no—
605D4—24no—
606D4—48no—
607D4—416no—
608D4—42no—
609D4—42no—
610D4—84no—
611D4C2×C4×D418yes—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—416no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—416no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—88no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—168no—
653C2×C4—168no—
654C2×C4—168no—
655C2×C4C2×C4×D412yes—
656C2×C4C2×C4×D412yes—
657C2×C4C2×C4×D412yes—
658C2×C4C2×C4×D412yes—
659C2×C4C2×C4×D412yes—
660C2×C4C2×C4×D414yes—
661C2×C4#52014yes—
662C2×C4D4212yes—
663C2×C4#54714yes—
664C2×C4#54812yes—
665C2×C4#55314yes—
666C2×C4C24×C411yescharacteristic
667C2×C4C23×D411yescharacteristic
668C2×C4C22×#4111yescharacteristic
669C2×C4C22×#4111yescharacteristic
670C2×C4C2×H(5,𝔽2)11yescharacteristic
671C2×C4C2×2−1+411yescharacteristic
672C8—28no—
673C8—24no—
674C8—42no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—28no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—816no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22#236114yes—
697C22C22×C4×D411yescharacteristic
698C22C2×D4211yescharacteristic
699C22C2×#54811yescharacteristic
700C22#358511yescharacteristic
701C22#358511yescharacteristic
702C22#358511yescharacteristic
703C22#358611yescharacteristic
704C4—22no—
705C4—22no—
706C4—22no—
707C4—22no—
708C4—44no—
709C4—44no—
710C4—44no—
711C4—44no—
712C4—44no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4#236114yes—
717C4C22×C4×D411yescharacteristic
718C4C2×#52011yescharacteristic
719C4C2×#54711yescharacteristic
720C4C2×#55311yescharacteristic
721C2—24no—
722C2—48no—
723C2—84no—
724C2C2×#236111yescharacteristic
725C2#6016811yescharacteristic
726C2#6017011yescharacteristic
727C2#6107911yescharacteristic
728C2#6108011yescharacteristic
729C2#6133211yescharacteristic
730C2#6133311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3036C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2946C2semidirect
3. #34206C2semidirect
4. #34181C2semidirect
5. #23012C2semidirect
6. #23011C2semidirect
7. #23008C2semidirect
8. #23007C2semidirect
9. #21199C2semidirect
10. #21170C2semidirect
11. #21092C2semidirect
12. #20929C2semidirect
13. #20878C2semidirect
14. #20438C2semidirect
15. C2×#1762C2semidirect
21. #3036C4semidirect
39. #2367C22semidirect
40. #2366C22semidirect
41. #2365C22semidirect
60. #1961C22semidirect
61. #1960C22semidirect
62. #1959C22semidirect
63. #1959C22semidirect
64. #1949C22semidirect
65. #1948C22semidirect
66. #1947C22semidirect
67. #1936C22semidirect
68. #1935C22semidirect
69. #1934C22semidirect
70. #1933C22semidirect
75. #1865C22semidirect
76. #1864C22semidirect
77. #1863C22semidirect
78. #1862C22semidirect
79. #1771C22semidirect
80. #1771C22semidirect
92. C2×AGL(1,ℤ/8)C2×C4semidirect
99. #532C2×C4semidirect
119. C2×C4×D4C2×C4semidirect
138. #463C2×C4semidirect
139. #461C2×C4semidirect
140. #459C2×C4semidirect
148. #423C2×C4semidirect
151. #417C2×C4semidirect
160. C2×#102D4semidirect
443. C2×D4C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30362
C2×#29462
#342062
#341812
#230122
#230112
#230082
#230072
#211992
#211702
#210922
#209292
#208782
#204382
C2×#17622