Finite Group Databasebeta

← #379946

Subgroups — #379946

Order 512.

1884 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2#36624C211yesmaximal, characteristic
3C2×#1980C212yesmaximal, complemented
4C4×#380C211yesmaximal, characteristic, complemented
5#11053C214yesmaximal, complemented
6#10971C214yesmaximal, complemented
7#10851C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525C2212yes—
10C23×C42C2211yescharacteristic
11#2920—28no—
12C2×#432—21no—
13C2×#432—21no—
14C2×#431—21no—
15C2×#431—21no—
16C2×#423C2212yes—
17C22×#104C2218yes—
18C2×#407C2212yes—
19C2×C4×M16C2212yes—
20#2857C411yescharacteristic, complemented
21#2857C411yescharacteristic, complemented
22#2856C411yescharacteristic, complemented
23#2856C411yescharacteristic, complemented
24#2854—21no—
25#2854—21no—
26#2850C412yescomplemented
27#2850C412yescomplemented
28#2847—21no—
29#2847—21no—
30C4×#121—21no—
31C4×#121—21no—
32C4×#121—21no—
33C4×#121—21no—
34#2297C2212yes—
35#2285C2211yescharacteristic
36C2×#380—21no—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39#1980—28no—
40C2×#339C2218yes—
41C4⋊#95—21no—
42#1434—21no—
43C22⋊#95—21no—
44C4×#95C2212yes—
45C22×#41C2312yes—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16C2314yes—
50#532—24no—
51#532—24no—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#127C2×C411yescharacteristic
54C2×#124C2×C412yes—
55#525—48no—
56C2×#123—22no—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2×C412yes—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121C2×C412yes—
66C2×#121C2×C412yes—
67C2×#120—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×#117C2312yes—
73C22×C4⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C42—24no—
76C22×C42—24no—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2314yes—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#432—28no—
91#432—28no—
92#431—28no—
93#431—28no—
94#423—28no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×#104—216no—
98C2×#104—48no—
99C2×#104—48no—
100#407—28no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C4×M16—24no—
104C4×M16D418yescomplemented
105C2×#97—22no—
106C2×C4×C8—22no—
107#391C2×C412yes—
108#391C2×C412yes—
109#390—21no—
110#390—21no—
111#390—21no—
112#390—21no—
113#390C2×C412yes—
114#390C2×C412yes—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#388—22no—
120#388—22no—
121#388—22no—
122#388—22no—
123#386—22no—
124#386—22no—
125#386—22no—
126#386—22no—
127#386—22no—
128#386—22no—
129#384C2×C411yescharacteristic
130#384C2×C411yescharacteristic
131#382C2×C412yes—
132#380—28no—
133#380—21no—
134#380—21no—
135#380—21no—
136#380—21no—
137#380—21no—
138#380—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—24no—
142C4×C4⋊C4—24no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4—44no—
146C2×#95—22no—
147C2×#95—22no—
148C2×#95—22no—
149C2×#95—22no—
150#339—48no—
151C25C2×D411yescharacteristic
152C2×#41—24no—
153C2×#41—24no—
154C2×#41—28no—
155C2×#41—24no—
156C2×#41—24no—
157C22×Q8C22×C412yes—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4C22×C412yes—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2411yescharacteristic
195C2×M16—24no—
196C2×M16—24no—
197C2×M16—24no—
198C2×M16—24no—
199C2×M16—24no—
200C2×M16C2×D418yes—
201C22×C8—24no—
202C4⋊Q8—24no—
203#127—24no—
204#124—28no—
205#123—48no—
206C22⋊Q8—44no—
207C22⋊Q8—44no—
208#121—44no—
209#121—44no—
210#121—44no—
211#121—44no—
212#121—44no—
213#121—44no—
214#121—44no—
215#121—44no—
216#120—48no—
217C4×D4—48no—
218C4×D4—48no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223#117—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C412yes—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—28no—
271C2×C42—28no—
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C22⋊C411yescharacteristic
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22×C411yescharacteristic
281C2×C42C22×C411yescharacteristic
282C2×C42C22×C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D414yes—
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42#4114yes—
290C4⋊C8—28no—
291C4⋊C8—28no—
292#104—432no—
293C22⋊C8—28no—
294C22⋊C8—28no—
295#97—28no—
296C4×C8—28no—
297#95—28no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—44no—
305#95—44no—
306#95—44no—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—41no—
315C24—41no—
316C24C2×C22⋊C411yescharacteristic
317C24C2×C22⋊C411yescharacteristic
318C24C22×D411yescharacteristic
319#41—48no—
320#41—48no—
321#41—48no—
322#41—48no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—28no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—416no—
401C22×C4—416no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4#10414yescomplemented
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4#12114yes—
428C22×C4#12114yes—
429C22×C4C23×C411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C2×#4112yes—
433C22×C4H(5,𝔽2)14yes—
434C22×C4H(5,𝔽2)12yes—
435C22×C42−1+412yes—
436M16—28no—
437M16—28no—
438M16—44no—
439M16—44no—
440C2×C8—28no—
441C2×C8—44no—
442C2×C8—44no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—416no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—416no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C42—22no—
467C42—22no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—44no—
481C42—48no—
482C42—48no—
483C42C4×D414yes—
484C42C4×D414yes—
485C42#12012yes—
486C42#12012yes—
487C42#12312yes—
488C42#12312yes—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—24no—
494C23—24no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23C2×#10412yes—
523C23#42112yes—
524C23C22×C22⋊C411yescharacteristic
525C23C2×#12112yes—
526C23#53512yes—
527C23#53511yescharacteristic
528C23#53611yescharacteristic
529Q8—48no—
530D4—48no—
531D4—84no—
532D4—88no—
533D4—88no—
534D4—84no—
535D4—84no—
536D4—84no—
537C2×C4—28no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4#38612yes—
598C2×C4#38612yes—
599C2×C4#38612yes—
600C2×C4#38612yes—
601C2×C4C2×#10412yes—
602C2×C4C2×#10412yes—
603C2×C4#42112yes—
604C2×C4#42112yes—
605C2×C4C2×C4×D412yes—
606C2×C4#51812yes—
607C2×C4#52012yes—
608C2×C4C2×#12011yescharacteristic
609C2×C4C2×#12311yescharacteristic
610C2×C4#53514yes—
611C2×C4#53512yes—
612C2×C4#53714yes—
613C2×C4#53712yes—
614C2×C4#54014yes—
615C2×C4#54214yes—
616C8—48no—
617C22—22no—
618C22—22no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—48no—
632C22—48no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22C2×#38611yescharacteristic
649C22#230911yescharacteristic
650C22#231111yescharacteristic
651C22#261912yes—
652C22#262012yes—
653C22C22×#10411yescharacteristic
654C22C2×#42111yescharacteristic
655C22C2×#42111yescharacteristic
656C22C2×#42111yescharacteristic
657C4—22no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—41no—
663C4—41no—
664C4—41no—
665C4—41no—
666C4—88no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4#187114yes—
671C4#232414yes—
672C4#261512yes—
673C4#262412yes—
674C2—22no—
675C2—22no—
676C2—42no—
677C2—42no—
678C2—162no—
679C2#1425711yescharacteristic
680C2C2×#187111yescharacteristic
681C2#2090211yescharacteristic
682C2#2181511yescharacteristic
683C2#2181611yescharacteristic
684C2#2189911yescharacteristic
685C2#2190011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2920C2semidirect
3. C2×#1980C2semidirect
4. C4×#380C2semidirect
5. #11053C2semidirect
6. #10971C2semidirect
7. #10851C2semidirect
20. #2857C4semidirect
21. #2857C4semidirect
22. #2856C4semidirect
23. #2856C4semidirect
26. #2850C4semidirect
27. #2850C4semidirect
104. C4×M16D4semidirect
416. C22×C4#104semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
#366242
C2×#19802
C4×#3802
#110532
#109712
#108512