Finite Group Databasebeta

← #380552

Subgroups — #380552

Order 512.

1906 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C2×#1871C214yesmaximal, complemented
3C2×#1728C211yesmaximal, characteristic, complemented
4C4×#380C211yesmaximal, characteristic, complemented
5#11359C212yesmaximal, complemented
6#11075C214yesmaximal, complemented
7#11011C212yesmaximal, complemented
8C23×M16C2212yes—
9C2×#532C2211yescharacteristic
10C2×#525C2212yes—
11C23×C42C2211yescharacteristic
12C22×#104C2218yes—
13#2857C411yescharacteristic, complemented
14#2857C411yescharacteristic, complemented
15#2856C411yescharacteristic, complemented
16#2856C411yescharacteristic, complemented
17#2854—21no—
18#2854—21no—
19#2850C412yescomplemented
20#2850C412yescomplemented
21#2847—21no—
22#2847—21no—
23C4×#121—21no—
24C4×#121—21no—
25C4×#121—21no—
26C4×#121—21no—
27#2297C2212yes—
28#2285C2211yescharacteristic
29C2×#380—21no—
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C22⋊C4C2212yes—
32#1871—216no—
33C2×#339C2218yes—
34#1728—21no—
35#1728—21no—
36#1728C2212yes—
37#1728C2212yes—
38C2×#336—21no—
39C2×#336—21no—
40C4⋊#95—21no—
41#1434—21no—
42C22⋊#95—21no—
43C4×#95C2212yes—
44C22×#41C2312yes—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—24no—
50C22×M16—44no—
51C22×M16—44no—
52C22×M16D418yescomplemented
53C22×M16C2314yes—
54C23×C8—22no—
55#532—44no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#127C2×C411yescharacteristic
58C2×#124C2×C412yes—
59#525—28no—
60#525—28no—
61C2×#123—22no—
62C2×C22⋊Q8C2×C412yes—
63C2×C22⋊Q8C2×C412yes—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121C2×C412yes—
71C2×#121C2×C412yes—
72C2×#120—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×#117C2312yes—
78C22×C4⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C42—44no—
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42C2314yes—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C2×#104—216no—
95C2×#104—48no—
96C2×#104—48no—
97#391C2×C412yes—
98#391C2×C412yes—
99#390—21no—
100#390—21no—
101#390—21no—
102#390—21no—
103#390C2×C412yes—
104#390C2×C412yes—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#386—22no—
119#384C2×C411yescharacteristic
120#384C2×C411yescharacteristic
121#382C2×C412yes—
122#380—28no—
123#380—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127#380—21no—
128#380—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—44no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—28no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139#339—216no—
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144C25C2×D411yescharacteristic
145C2×#41—24no—
146C2×#41—24no—
147C2×#41—28no—
148C2×#41—24no—
149C2×#41—24no—
150C22×Q8C22×C412yes—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4C22×C412yes—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D414yes—
185C23×C4C2×D411yescharacteristic
186C23×C4#4114yes—
187C23×C4C2411yescharacteristic
188C2×M16—28no—
189C2×M16—24no—
190C2×M16—28no—
191C2×M16—28no—
192C2×M16—416no—
193C2×M16—416no—
194C2×M16—44no—
195C2×M16—44no—
196C2×M16C2×D418yes—
197C22×C8—28no—
198C22×C8—24no—
199C22×C8—44no—
200C22×C8—44no—
201C4⋊Q8—42no—
202#127—42no—
203#124—44no—
204#123—44no—
205#123—44no—
206C22⋊Q8—28no—
207C22⋊Q8—28no—
208#121—28no—
209#121—28no—
210#121—44no—
211#121—44no—
212#121—44no—
213#121—44no—
214#121—44no—
215#121—44no—
216#120—44no—
217#120—44no—
218C4×D4—44no—
219C4×D4—44no—
220C4×D4—48no—
221C4×D4—48no—
222#117—24no—
223#117—24no—
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—28no—
255C2×C42—28no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284#104—432no—
285#95—28no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—44no—
293#95—44no—
294#95—44no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—41no—
301C24—41no—
302C24C2×C22⋊C411yescharacteristic
303C24C2×C22⋊C411yescharacteristic
304C24#12412yes—
305C24#12712yes—
306C24C22×D411yescharacteristic
307#41—48no—
308#41—48no—
309#41—48no—
310#41—48no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—42no—
332C2×D4—42no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—28no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
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—28no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—416no—
389C22×C4—416no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4#10414yescomplemented
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C4×D414yes—
412C22×C4C4×D414yes—
413C22×C4#12114yes—
414C22×C4#12114yes—
415C22×C4#12412yes—
416C22×C4#12712yes—
417C22×C4C23×C411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C2×#4112yes—
421C22×C4H(5,𝔽2)14yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C42−1+412yes—
424M16—216no—
425M16—416no—
426M16—48no—
427C2×C8—28no—
428C2×C8—416no—
429C2×C8—44no—
430C2×C8—44no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—416no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—416no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42—44no—
472C42—48no—
473C42—48no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—24no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23#39012yes—
506C23#39012yes—
507C23C2×#10412yes—
508C23#42112yes—
509C23C22×C22⋊C411yescharacteristic
510C23C2×#12411yescharacteristic
511C23C2×#12711yescharacteristic
512C23#53412yes—
513C23#53712yes—
514Q8—48no—
515D4—48no—
516D4—84no—
517D4—84no—
518D4—88no—
519D4—84no—
520D4—84no—
521D4—88no—
522C2×C4—28no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4#39012yes—
585C2×C4#39012yes—
586C2×C4C2×#10412yes—
587C2×C4C2×#10412yes—
588C2×C4#42112yes—
589C2×C4#42112yes—
590C2×C4C2×C4×D412yes—
591C2×C4#51812yes—
592C2×C4#52012yes—
593C2×C4C2×#12112yes—
594C2×C4#53414yes—
595C2×C4#53412yes—
596C2×C4#53511yescharacteristic
597C2×C4#53611yescharacteristic
598C2×C4#53714yes—
599C2×C4#53814yes—
600C2×C4#53914yes—
601C8—48no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—48no—
620C22—48no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22#198014yes—
633C22C2×#39011yescharacteristic
634C22#230811yescharacteristic
635C22#231111yescharacteristic
636C22#263912yes—
637C22#265612yes—
638C22C22×#10411yescharacteristic
639C22C2×#42111yescharacteristic
640C22C2×#42111yescharacteristic
641C22C2×#42111yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—88no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4#232514yes—
657C4#261712yes—
658C4#261812yes—
659C2—22no—
660C2—22no—
661C2—24no—
662C2—42no—
663C2—162no—
664C2#1427111yescharacteristic
665C2C2×#198011yescharacteristic
666C2#2125211yescharacteristic
667C2#2177311yescharacteristic
668C2#2177411yescharacteristic
669C2#2185611yescharacteristic
670C2#2185711yescharacteristic
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
2. C2×#1871C2semidirect
3. C2×#1728C2semidirect
4. C4×#380C2semidirect
5. #11359C2semidirect
6. #11075C2semidirect
7. #11011C2semidirect
13. #2857C4semidirect
14. #2857C4semidirect
15. #2856C4semidirect
16. #2856C4semidirect
19. #2850C4semidirect
20. #2850C4semidirect
52. C22×M16D4semidirect
400. C22×C4#104semidirect

Maximal subgroups

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

subgroupindex
#366242
C2×#18712
C2×#17282
C4×#3802
#113592
#110752
#110112