Finite Group Databasebeta

← #360998

Subgroups — #360998

Order 512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2360C211yesmaximal, characteristic
2#21049C212yesmaximal
3C2×#1821C211yesmaximal, characteristic
4C2×C4×#101C211yesmaximal, characteristic, complemented
5#11045C212yesmaximal, complemented
6#10968C212yesmaximal
7#10913C212yesmaximal, complemented
8#10847C211yesmaximal, characteristic, complemented
9#10802C211yesmaximal, characteristic, complemented
10#10763C211yesmaximal, characteristic, complemented
11#10727C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#423C2212yes—
16C22×#101C2212yes—
17C22×#101C2212yes—
18C2×#407C2212yes—
19C2×#407C2212yes—
20C2×C4×M16C2212yes—
21#2360—22no—
22#2360—22no—
23#2360—22no—
24#2360—22no—
25C2×#391C2211yescharacteristic
26C2×#389C412yes—
27C2×#389C2211yescharacteristic
28C2×#387C412yes—
29C2×#385C412yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C412yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33#1924—22no—
34#1924—22no—
35#1923—22no—
36#1923—22no—
37#1821—24no—
38C4×#101—24no—
39C2×#343C2211yescharacteristic
40C2×#343C2211yescharacteristic
41C2×#343C2212yes—
42C2×#343C2212yes—
43C2×#343C2212yes—
44C2×#343C2211yescharacteristic
45C2×#343C2212yes—
46C2×#343C2211yescharacteristic
47#1728C2212yes—
48#1728C2212yes—
49C23×Q8C2311yescharacteristic
50C24×C4C2×C412yes—
51C24×C4C2311yescharacteristic
52C22×M16C2312yes—
53C22×M16C2312yes—
54C22×M16C2312yes—
55C22×M16C2312yes—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—42no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4C2×C412yes—
74C22×C4⋊C4C2×C412yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C42C2×C412yes—
82C22×C42C2×C412yes—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#423—24no—
86#423—42no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×#101—24no—
90C2×#101—24no—
91C2×#101—24no—
92C2×#101—24no—
93C2×#101—42no—
94C2×#101—42no—
95#407—24no—
96#407—24no—
97#407—42no—
98#407—42no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C4×M16—24no—
104C4×M16—42no—
105C2×#97—22no—
106C2×C4×C8—22no—
107#391—22no—
108#391—22no—
109#389—22no—
110#389—22no—
111#389—24no—
112#389—24no—
113#387—24no—
114#387—24no—
115#385—24no—
116#385—24no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4D412yes—
124C4×C22⋊C4D412yes—
125C4×C22⋊C4D412yes—
126C4×C22⋊C4D412yes—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129C2×#95C2×C412yes—
130C2×#95C2×C412yes—
131C2×#95C2×C411yescharacteristic
132C2×#95C2×C411yescharacteristic
133#343—24no—
134#343—22no—
135#343—22no—
136#343—24no—
137#343—24no—
138#343—22no—
139#343—42no—
140#343D414yes—
141#336—22no—
142#336—22no—
143#336—22no—
144#336—22no—
145C25C22×C411yescharacteristic
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8#4114yes—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—42no—
169C23×C4—44no—
170C23×C4—82no—
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C412yes—
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C412yes—
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C412yes—
179C23×C4C22×C412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2411yescharacteristic
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16C2×D414yes—
206C2×M16#4114yes—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22⋊Q8—28no—
212C22⋊Q8—28no—
213C22⋊Q8—28no—
214C22⋊Q8—28no—
215C4×Q8—44no—
216C4×Q8—44no—
217C4×Q8—44no—
218C4×Q8—44no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—82no—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4#4112yes—
244C2×C4⋊C4#4112yes—
245C2×C4⋊C4#4112yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4C22⋊C411yescharacteristic
253C2×C22⋊C4C22⋊C411yescharacteristic
254C2×C22⋊C4C22⋊C411yescharacteristic
255C2×C22⋊C4C22⋊C412yes—
256C2×C22⋊C4C22⋊C412yes—
257C2×C22⋊C4C22⋊C411yescharacteristic
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—42no—
270C2×C42—82no—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C4⋊C8—44no—
276C4⋊C8—44no—
277#101—216no—
278#101—48no—
279C22⋊C8—44no—
280C22⋊C8—44no—
281C22⋊C8—44no—
282C22⋊C8—44no—
283#97—44no—
284C4×C8—44no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—24no—
290#95—24no—
291#95C22⋊C412yes—
292#95C22⋊C412yes—
293#95C22⋊C412yes—
294#95C22⋊C412yes—
295C24—21no—
296C24—21no—
297C24—22no—
298C24—22no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—41no—
304C24—41no—
305C24C2×C22⋊C411yescharacteristic
306C24C2×C22⋊C411yescharacteristic
307C24C23×C411yescharacteristic
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—42no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—42no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
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—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—88no—
394C22×C4—88no—
395C22×C4C2×C22⋊C412yes—
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4#11714yes—
403C22×C4#11714yes—
404C22×C4C4×D414yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D412yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4112yes—
411C22×C4C2×#4111yescharacteristic
412C22×C4H(5,𝔽2)11yescharacteristic
413C22×C4H(5,𝔽2)12yes—
414C22×C42−1+411yescharacteristic
415M16—24no—
416M16—24no—
417M16—24no—
418M16—24no—
419M16—42no—
420M16—42no—
421M16—42no—
422M16—42no—
423M16—42no—
424M16—42no—
425M16—42no—
426M16—42no—
427M16—42no—
428M16—42no—
429M16—42no—
430M16—42no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C4⋊C4—216no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—84no—
453C22⋊C4—28no—
454C22⋊C4—28no—
455C22⋊C4—22no—
456C22⋊C4—24no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—42no—
460C22⋊C4#10114yescomplemented
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—44no—
470C42—44no—
471C42—82no—
472C42—82no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—24no—
478C23—24no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23C2×#10112yes—
506C23#41312yes—
507C23C22×C22⋊C411yescharacteristic
508C23C2×#11711yescharacteristic
509C23C2×#11711yescharacteristic
510C23C2×C4×D411yescharacteristic
511C23C4×#4111yescharacteristic
512C23#51811yescharacteristic
513C23#52011yescharacteristic
514Q8—44no—
515Q8—44no—
516Q8—44no—
517Q8—44no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—24no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—88no—
578C2×C4—88no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4C2×#10112yes—
582C2×C4C2×#10112yes—
583C2×C4#52512yes—
584C2×C4#52512yes—
585C2×C4#52512yes—
586C2×C4#52912yes—
587C2×C4#53812yes—
588C2×C4#53912yes—
589C2×C4D4212yes—
590C2×C4#54612yes—
591C2×C4#54612yes—
592C2×C4#54711yescharacteristic
593C2×C4#54712yes—
594C2×C4#55011yescharacteristic
595C2×C4#55312yes—
596C2×C4#55511yescharacteristic
597C2×C4#55611yescharacteristic
598C2×C4#55912yes—
599C8—44no—
600C8—44no—
601C8—82no—
602C8—82no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22#228011yescharacteristic
634C22#229011yescharacteristic
635C22#231811yescharacteristic
636C22D4×C22⋊C411yescharacteristic
637C22#233911yescharacteristic
638C22#235411yescharacteristic
639C22#235711yescharacteristic
640C22C22×#10111yescharacteristic
641C22C2×#41311yescharacteristic
642C22#288811yescharacteristic
643C22#288911yescharacteristic
644C4—22no—
645C4—22no—
646C4—42no—
647C4—44no—
648C4—42no—
649C4—44no—
650C4—41no—
651C4—42no—
652C4—41no—
653C4—44no—
654C4—82no—
655C4#245512yes—
656C4#245612yes—
657C4#245912yes—
658C4#246612yes—
659C2—21no—
660C2—21no—
661C2—21no—
662C2—21no—
663C2—42no—
664C2—42no—
665C2#1388611yescharacteristic
666C2#2072711yescharacteristic
667C2#2073111yescharacteristic
668C2#2240811yescharacteristic
669C2D4×#10111yescharacteristic
670C2#2248011yescharacteristic
671C2#2248111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×C4×#101C2semidirect
5. #11045C2semidirect
7. #10913C2semidirect
8. #10847C2semidirect
9. #10802C2semidirect
10. #10763C2semidirect
460. C22⋊C4#101semidirect

Maximal subgroups

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

subgroupindex
C2×#23602
#210492
C2×#18212
C2×C4×#1012
#110452
#109682
#109132
#108472
#108022
#107632
#107272