Finite Group Databasebeta

← #362649

Subgroups — #362649

Order 512.

2018 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×#2366C211yesmaximal, characteristic
2C2×#1982C211yesmaximal, characteristic
3C2×#1957C211yesmaximal, characteristic
4C2×#1939C211yesmaximal, characteristic, complemented
5C2×#1804C212yesmaximal, complemented
6#13911C211yesmaximal, characteristic, complemented
7#11076C212yesmaximal, complemented
8#11051C212yesmaximal, complemented
9#10849C211yesmaximal, characteristic, complemented
10#10788C212yesmaximal, complemented
11C4×#340C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×C4×D4—21no—
14C22×C4×D4C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×C4×C8C2211yescharacteristic
22#2366—24no—
23#2366C418yescomplemented
24C2×#390—21no—
25C2×#390C2212yes—
26C2×#386—22no—
27C2×#384C2211yescharacteristic
28C2×#382C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1982—24no—
33#1957—24no—
34#1939—24no—
35#1804—28no—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#336C2212yes—
43#1442—22no—
44C4⋊#95—21no—
45C4⋊#95—21no—
46C4⋊#95C2212yes—
47C23×D4—21no—
48C23×D4—21no—
49C23×D4C2312yes—
50C24×C4—22no—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×#127—22no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127C2×C414yes—
58C2×#127D414yes—
59C2×#127Q814yes—
60C2×C4×D4—28no—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—42no—
65C2×C4×D4—42no—
66C2×C4×D4—44no—
67C2×C4×D4—44no—
68C2×C4×D4C2×C418yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42C2312yes—
84C22×C42C2311yescharacteristic
85C2×C4⋊C8—24no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—42no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—42no—
100C2×#102—42no—
101C2×#102—42no—
102C2×#102—42no—
103C2×C4×C8—22no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106#390—28no—
107#390—24no—
108#390C2×C418yes—
109#386—216no—
110#384—22no—
111#384—22no—
112#382—24no—
113#382—42no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4C2×C414yes—
118C4×C4⋊C4C2×C414yes—
119C4×C4⋊C4D414yes—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95D412yes—
127#340—22no—
128#340—24no—
129#340—22no—
130#340—22no—
131#340—42no—
132#340D414yescomplemented
133#336—24no—
134C25—22no—
135C22×D4—24no—
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—24no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—81no—
157C22×D4—81no—
158C22×D4—81no—
159C22×D4—81no—
160C22×D4C22×C414yes—
161C22×D4C2×D414yes—
162C22×D4C2×Q814yes—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—28no—
168C23×C4—22no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—82no—
174C23×C4—82no—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2411yescharacteristic
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8C2×D412yes—
200C22×C8#4112yes—
201#127—28no—
202#127—28no—
203#127—28no—
204#127C4⋊C4116yes—
205C4×D4—216no—
206C4×D4—48no—
207C4×D4—48no—
208C4×D4—48no—
209C4×D4—432no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—81no—
243C2×C4⋊C4—81no—
244C2×C4⋊C4C22⋊C414yes—
245C2×C4⋊C4C22⋊C414yes—
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C22×C412yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4#4112yes—
251C2×C4⋊C4#4112yes—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—82no—
260C2×C22⋊C4—82no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42C22×C414yes—
272C2×C42C22×C412yes—
273C2×C42C2×D414yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×Q812yes—
277C2×C42#4112yes—
278C2×C42#4112yes—
279C4⋊C8—48no—
280C4⋊C8—44no—
281#102—48no—
282#102—48no—
283#102—48no—
284#102—48no—
285C4×C8—44no—
286#95—24no—
287#95—44no—
288#95—44no—
289#95—42no—
290#95—82no—
291#95—82no—
292#95C22⋊C418yes—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24C22×D411yescharacteristic
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—216no—
325C2×D4—216no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4C2×C4⋊C418yes—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
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—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—416no—
374C22×C4—416no—
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—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—88no—
405C22×C4—88no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4—88no—
409C22×C4C2×C22⋊C414yes—
410C22×C4C2×C22⋊C414yes—
411C22×C4C2×C22⋊C412yes—
412C22×C4C2×C22⋊C412yes—
413C22×C4#12112yes—
414C22×C4#12112yes—
415C22×C4C22⋊Q812yes—
416C22×C4C22⋊Q812yes—
417C22×C4#13512yes—
418C22×C4#13512yes—
419C22×C4AGL(1,ℤ/8)12yes—
420C22×C4AGL(1,ℤ/8)12yes—
421C22×C4C23×C411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×Q811yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4H(5,𝔽2)11yescharacteristic
432C22×C4H(5,𝔽2)11yescharacteristic
433C22×C4H(5,𝔽2)11yescharacteristic
434C22×C42−1+411yescharacteristic
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C4⋊C4C4×D414yes—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—88no—
483C42—24no—
484C42—24no—
485C42—24no—
486C42—24no—
487C42—48no—
488C42—84no—
489C42C2×C4⋊C414yes—
490C42#11714yes—
491C42#12114yes—
492C42C22⋊Q814yes—
493C23—28no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—82no—
513C23—82no—
514C23—82no—
515C23—82no—
516C23—82no—
517C23—82no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23—82no—
525C23—82no—
526C23—82no—
527C23—82no—
528C23C22×C22⋊C411yescharacteristic
529C23C2×#12111yescharacteristic
530C23C2×C22⋊Q811yescharacteristic
531C23D4211yescharacteristic
532C23#54611yescharacteristic
533C23#54811yescharacteristic
534C23#55011yescharacteristic
535C23C2×#13511yescharacteristic
536C23C2×#13511yescharacteristic
537C23C2×AGL(1,ℤ/8)11yescharacteristic
538C23C2×AGL(1,ℤ/8)11yescharacteristic
539C23#57611yescharacteristic
540C23#57611yescharacteristic
541C23#57711yescharacteristic
542C23#57711yescharacteristic
543D4—28no—
544D4—28no—
545D4—432no—
546D4—44no—
547D4—44no—
548D4—44no—
549D4—44no—
550D4—44no—
551D4—44no—
552D4—816no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
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—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—832no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4#38014yes—
610C2×C4#38014yes—
611C2×C4#41614yes—
612C2×C4#41814yes—
613C2×C4#46314yes—
614C2×C4#46314yes—
615C2×C4#47814yes—
616C2×C4#47814yes—
617C2×C4C22×C4⋊C411yescharacteristic
618C2×C4C2×#11711yescharacteristic
619C2×C4C2×C4×D411yescharacteristic
620C2×C4C4×#4111yescharacteristic
621C2×C4#51811yescharacteristic
622C2×C4#52011yescharacteristic
623C2×C4C2×#12111yescharacteristic
624C2×C4C2×C22⋊Q811yescharacteristic
625C2×C4#52911yescharacteristic
626C2×C4#52911yescharacteristic
627C2×C4#53211yescharacteristic
628C2×C4#53211yescharacteristic
629C2×C4#54011yescharacteristic
630C2×C4#54011yescharacteristic
631C2×C4#54211yescharacteristic
632C2×C4#54211yescharacteristic
633C2×C4D4211yescharacteristic
634C2×C4D4211yescharacteristic
635C2×C4#54611yescharacteristic
636C2×C4#54611yescharacteristic
637C2×C4#54611yescharacteristic
638C2×C4#54711yescharacteristic
639C2×C4#54811yescharacteristic
640C2×C4D4×Q811yescharacteristic
641C2×C4#55011yescharacteristic
642C2×C4#55411yescharacteristic
643C2×C4#55411yescharacteristic
644C2×C4#55411yescharacteristic
645C8—42no—
646C8—42no—
647C8—82no—
648C22—48no—
649C22—48no—
650C22—48no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22C2×#38011yescharacteristic
664C22D4×C22⋊C411yescharacteristic
665C22#234111yescharacteristic
666C22#242711yescharacteristic
667C22#242811yescharacteristic
668C22#245211yescharacteristic
669C22#245411yescharacteristic
670C22C2×#41611yescharacteristic
671C22C2×#41811yescharacteristic
672C22#289711yescharacteristic
673C22#289711yescharacteristic
674C22C2×#46311yescharacteristic
675C22C2×#46311yescharacteristic
676C22#304111yescharacteristic
677C22#304211yescharacteristic
678C22C2×#47811yescharacteristic
679C22C2×#47811yescharacteristic
680C22#308111yescharacteristic
681C22#308111yescharacteristic
682C22#327211yescharacteristic
683C22#327411yescharacteristic
684C22#328411yescharacteristic
685C22#328711yescharacteristic
686C22#330411yescharacteristic
687C22#330511yescharacteristic
688C22#332411yescharacteristic
689C22#332511yescharacteristic
690C22#336111yescharacteristic
691C22#336511yescharacteristic
692C22#336611yescharacteristic
693C22#336811yescharacteristic
694C22#339411yescharacteristic
695C22#339611yescharacteristic
696C22#339711yescharacteristic
697C22#339911yescharacteristic
698C4—24no—
699C4—44no—
700C4—44no—
701C4—44no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—162no—
706C4#180418yescomplemented
707C4#229311yescharacteristic
708C4#231911yescharacteristic
709C4D4×C4⋊C411yescharacteristic
710C4#234911yescharacteristic
711C4#242711yescharacteristic
712C4#242811yescharacteristic
713C4#245211yescharacteristic
714C4#245411yescharacteristic
715C2—88no—
716C2#1392011yescharacteristic
717C2C2×#180411yescharacteristic
718C2#2065011yescharacteristic
719C2#2241811yescharacteristic
720C2#2242811yescharacteristic
721C2#2254311yescharacteristic
722C2#2254511yescharacteristic
723C2#2532111yescharacteristic
724C2#2532511yescharacteristic
725C2#2536711yescharacteristic
726C2#2537111yescharacteristic
727C2#2646311yescharacteristic
728C2#2646711yescharacteristic
729C2#2652911yescharacteristic
730C2#2653311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1939C2semidirect
5. C2×#1804C2semidirect
6. #13911C2semidirect
7. #11076C2semidirect
8. #11051C2semidirect
9. #10849C2semidirect
10. #10788C2semidirect
11. C4×#340C2semidirect
23. #2366C4semidirect
132. #340D4semidirect
706. C4#1804semidirect

Maximal subgroups

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

subgroupindex
C2×#23662
C2×#19822
C2×#19572
C2×#19392
C2×#18042
#139112
#110762
#110512
#108492
#107882
C4×#3402