Finite Group Databasebeta

← #6462811

Subgroups — #6462811

Order 512.

2494 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2943C211yesmaximal, characteristic, complemented
2C2×#2932C211yesmaximal, characteristic, complemented
3#35328C211yesmaximal, characteristic, complemented
4#35273C211yesmaximal, characteristic, complemented
5#28155C214yesmaximal, complemented
6#28154C214yesmaximal, complemented
7#28153C214yesmaximal, complemented
8#28152C214yesmaximal, complemented
9#23641C212yesmaximal, complemented
10#23390C212yesmaximal, complemented
11#20976C212yesmaximal, complemented
12#20976C212yesmaximal, complemented
13C2×#1836C211yesmaximal, characteristic, complemented
14#20649C211yesmaximal, characteristic, complemented
15#20648C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C22×#131C2212yes—
18C2×#537C2212yes—
19C2×#537C2212yes—
20C2×#532C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23#2943—28no—
24C2×#442C2212yes—
25C2×#441C2212yes—
26C2×#440C2212yes—
27C2×#440C2212yes—
28#2932—28no—
29C2×#435C2214yes—
30C2×#435C2214yes—
31C2×#427C2211yescharacteristic
32C2×#423C2211yescharacteristic
33C2×#417C2212yes—
34C2×#417C2212yes—
35C2×#416C2211yescharacteristic
36C2×#416C2211yescharacteristic
37C2×#408C2212yes—
38C22×#97C2211yescharacteristic
39#2857—21no—
40#2856—21no—
41#2850—22no—
42#2621—21no—
43#2620—22no—
44#2619—21no—
45#2543—24no—
46#2538C2214yes—
47#2537C2214yes—
48#2536C2214yes—
49#2535C2214yes—
50#2416—24no—
51#2311C2212yes—
52#2311C2212yes—
53C2×#384C2211yescharacteristic
54C2×#382C2211yescharacteristic
55#2077C2214yes—
56#2077C2214yes—
57#2076C2214yes—
58#2075C2214yes—
59#2001C2214yes—
60#2000C2214yes—
61#2000C2214yes—
62#1999C2214yes—
63#1895C2214yes—
64#1895C2214yes—
65#1894C2214yes—
66#1894C2214yes—
67#1893C2214yes—
68#1893C2214yes—
69#1892C2214yes—
70#1892C2214yes—
71#1836C2214yes—
72#1836C2214yes—
73#1805C2214yes—
74#1804C2214yes—
75#1803C2214yes—
76#1802C2214yes—
77#1736C2212yes—
78#1736C2212yes—
79C2×#340C2212yes—
80C2×#340C2212yes—
81C22×#41C2311yescharacteristic
82C22×#41C2311yescharacteristic
83C22×#41C2312yes—
84C2×#135—24no—
85C2×#135—24no—
86C2×#135—24no—
87C22×Q16C2311yescharacteristic
88C22×SD16C2311yescharacteristic
89C22×SD16C2311yescharacteristic
90C22×D8C2311yescharacteristic
91C2×#131—28no—
92C2×#131—44no—
93C2×#131—44no—
94C22×M16C2312yes—
95C22×M16C2314yes—
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98C23×C8C2314yes—
99#537—28no—
100#537—28no—
101#537—44no—
102#537—44no—
103#532—24no—
104#532—24no—
105C2×C4⋊Q8D411yescharacteristic
106C2×#127D411yescharacteristic
107C2×#124D412yes—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2314yes—
111C2×C22⋊Q8C2314yes—
112C2×#121—22no—
113C2×#121—22no—
114C2×#121—24no—
115C2×#121—24no—
116C2×#121—24no—
117C2×#121D414yes—
118C2×#121C2314yes—
119C2×#121C2314yes—
120C4×#41—24no—
121C4×#41—24no—
122C4×#41—24no—
123C4×#41—24no—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×C4×D4—22no—
127C2×C4×D4—22no—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×C4×D4C2314yes—
131C2×#117—21no—
132C2×#117—21no—
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—22no—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C42—21no—
139C22×C42—21no—
140C22×C42C2311yescharacteristic
141#442—28no—
142#441—28no—
143#440—28no—
144#440—28no—
145#435—216no—
146#435—216no—
147#427—24no—
148C2×#107C2312yes—
149C2×#106C2312yes—
150#423—24no—
151C2×C4⋊C8C2314yes—
152#417—44no—
153#417—44no—
154#416—24no—
155#416—24no—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164#408—28no—
165C2×C22⋊C8C2314yes—
166C2×C22⋊C8C2314yes—
167C2×#97D418yescomplemented
168C2×#97C2312yes—
169C2×#97C2312yes—
170#398—24no—
171#397—24no—
172#396—24no—
173#396—24no—
174#394—24no—
175#394—24no—
176#394—24no—
177#392—24no—
178#391—22no—
179#391—22no—
180#390—22no—
181#390—22no—
182#390—24no—
183#389—24no—
184#388—24no—
185#388—24no—
186#386—24no—
187#386—24no—
188#386—24no—
189#386—24no—
190#384—21no—
191#384—21no—
192#384C2314yes—
193#382—21no—
194#382—21no—
195#382C2314yes—
196#380—24no—
197#380C2314yes—
198#380C2314yes—
199#380C2314yes—
200#380C2314yes—
201C2×#95—22no—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210C2×#41—24no—
211C2×#41—24no—
212C2×#41—24no—
213C2×#41—24no—
214C2×#41—22no—
215C2×#41—22no—
216C2×#41—24no—
217C2×#41—24no—
218C2×#41—22no—
219C2×#41—22no—
220C2×#41—22no—
221C2×#41—22no—
222C2×#41—22no—
223C2×#41—22no—
224C2×#41—22no—
225C2×#41—22no—
226C2×#41#4114yes—
227C2×#41#4114yes—
228C22×Q8C2×D412yes—
229C22×Q8C2411yescharacteristic
230C22×Q8C2411yescharacteristic
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—24no—
234C22×D4—24no—
235C22×D4C2×D414yes—
236C22×D4C2×D412yes—
237C22×D4C2×D414yes—
238C22×D4C2411yescharacteristic
239C22×D4C2411yescharacteristic
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4C2×D412yes—
247C23×C4C2414yes—
248C23×C4C2411yescharacteristic
249#135—232no—
250C2×Q16—22no—
251C2×Q16—22no—
252C2×Q16—22no—
253C2×SD16—22no—
254C2×SD16—22no—
255C2×SD16—22no—
256C2×SD16—22no—
257C2×SD16—22no—
258C2×SD16—22no—
259C2×D8—22no—
260C2×D8—22no—
261C2×D8—22no—
262#131—432no—
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—24no—
266C2×M16—28no—
267C2×M16—44no—
268C2×M16—44no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—28no—
274C22×C8—44no—
275C22×C8—44no—
276C22×C8C2×D414yes—
277C22×C8C2×D414yes—
278C22×C8C2414yes—
279C22×C8C2412yes—
280C22×C8C2411yescharacteristic
281C22×C8C2411yescharacteristic
282C4⋊Q8—24no—
283#127—24no—
284#124—28no—
285C22⋊Q8—28no—
286C22⋊Q8—28no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—48no—
290C22⋊Q8—48no—
291#121—216no—
292#121—28no—
293#121—28no—
294#121—44no—
295#121—44no—
296#121—48no—
297#121—48no—
298#121—416no—
299#121—416no—
300#121—88no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×D4—28no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—28no—
309C4×D4—24no—
310C4×D4—24no—
311C4×D4—48no—
312C4×D4—48no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—44no—
318#117—44no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
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—84no—
332C2×C4⋊C4C2×D414yes—
333C2×C4⋊C4C2414yes—
334C2×C4⋊C4C2414yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4C2×D414yes—
350C2×C22⋊C4C2×D414yes—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—44no—
356C2×C42—44no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D414yes—
360C2×C42#4114yes—
361C2×C42C2412yes—
362#107—24no—
363#106—24no—
364C4⋊C8—28no—
365#103—24no—
366#103—24no—
367#103—44no—
368#103—44no—
369#102—24no—
370#102—24no—
371#102—44no—
372#102—44no—
373C22⋊C8—28no—
374C22⋊C8—28no—
375#97C2×D418yes—
376#97C2×D418yes—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383C24—21no—
384C24—21no—
385C24C22×D414yes—
386C24C22×D411yescharacteristic
387#41—28no—
388#41—28no—
389#41—28no—
390#41—28no—
391#41—48no—
392#41—48no—
393#41—44no—
394#41—44no—
395#41—48no—
396#41—48no—
397#41—44no—
398#41—44no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8C2×#4112yes—
413C2×Q8C2×#4112yes—
414C2×D4—28no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—24no—
420C2×D4—28no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—21no—
424C2×D4—21no—
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—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—48no—
440C2×D4—48no—
441C2×D4—84no—
442C2×D4—84no—
443C2×D4—84no—
444C2×D4—84no—
445C2×D4C2×#4112yes—
446C2×D4C2×#4112yes—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4#12118yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4C2511yescharacteristic
491Q16—28no—
492SD16—28no—
493SD16—28no—
494D8—28no—
495M16—416no—
496M16—48no—
497C2×C8—28no—
498C2×C8—28no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—24no—
508C2×C8—416no—
509C2×C8C22×D412yes—
510C2×C8C22×D412yes—
511C2×C8C22×D414yes—
512C4⋊C4—28no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—84no—
529C22⋊C4—28no—
530C22⋊C4—28no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C42—42no—
544C42—42no—
545C42—44no—
546C42—44no—
547C42#12114yes—
548C42#12114yes—
549C42C22×D414yes—
550C42C2×#4114yes—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—22no—
559C23—22no—
560C23—24no—
561C23—24no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23C2×#12112yes—
571C23#54612yes—
572C23#54612yes—
573C23C23×D411yescharacteristic
574Q8—22no—
575Q8—22no—
576Q8—22no—
577Q8—22no—
578Q8—41no—
579Q8—41no—
580Q8—41no—
581Q8—41no—
582Q8—48no—
583D4—22no—
584D4—22no—
585D4—22no—
586D4—22no—
587D4—416no—
588D4—48no—
589D4—48no—
590D4—41no—
591D4—41no—
592D4—41no—
593D4—41no—
594D4—48no—
595D4—88no—
596D4—168no—
597D4—168no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—48no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—48no—
617C2×C4—48no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—88no—
631C2×C4—88no—
632C2×C4—164no—
633C2×C4—164no—
634C2×C4C2×#12114yes—
635C2×C4C2×#12114yes—
636C2×C4C2×#12112yes—
637C2×C4#52514yes—
638C2×C4#53514yes—
639C2×C4#54612yes—
640C2×C4#54612yes—
641C2×C4#55314yes—
642C2×C4#55314yes—
643C2×C4C23×D411yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C8—22no—
650C8—22no—
651C8—24no—
652C8—44no—
653C22—24no—
654C22—22no—
655C22—24no—
656C22—22no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—42no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—44no—
669C22—82no—
670C22—81no—
671C22—82no—
672C22—82no—
673C22—81no—
674C22#253514yes—
675C22C22×#12111yescharacteristic
676C22C2×#54611yescharacteristic
677C22C2×#54611yescharacteristic
678C22#358512yes—
679C22#358612yes—
680C4—24no—
681C4—24no—
682C4—44no—
683C4—42no—
684C4—42no—
685C4—84no—
686C4—84no—
687C4—82no—
688C4—82no—
689C4#253514yes—
690C4C2×#52511yescharacteristic
691C4C2×#53511yescharacteristic
692C4C2×#55311yescharacteristic
693C4C2×#55311yescharacteristic
694C2—24no—
695C2—84no—
696C2—82no—
697C2—82no—
698C2C2×#253511yescharacteristic
699C2#6045612yes—
700C2#6122212yes—
701C2#6122311yescharacteristic
702C2#6122411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2943C2semidirect
2. C2×#2932C2semidirect
3. #35328C2semidirect
4. #35273C2semidirect
5. #28155C2semidirect
6. #28154C2semidirect
7. #28153C2semidirect
8. #28152C2semidirect
9. #23641C2semidirect
10. #23390C2semidirect
11. #20976C2semidirect
12. #20976C2semidirect
13. C2×#1836C2semidirect
14. #20649C2semidirect
15. #20648C2semidirect
167. C2×#97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29432
C2×#29322
#353282
#352732
#281552
#281542
#281532
#281522
#236412
#233902
#209762
C2×#18362
#206492
#206482