Finite Group Databasebeta

← #368626

Subgroups — #368626

Order 512.

1682 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1982C211yesmaximal, characteristic
2C2×#1982C211yesmaximal, characteristic
3C2×#1934C212yesmaximal, complemented
4C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
5#14713C211yesmaximal, characteristic
6C4×#389C211yesmaximal, characteristic, complemented
7#11189C212yesmaximal, complemented
8#11066C212yesmaximal, complemented
9#10925C212yesmaximal, complemented
10#10849C211yesmaximal, characteristic, complemented
11#10849C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×C4×C8C2212yes—
20C2×#390—21no—
21C2×#390—21no—
22C2×#390—21no—
23C2×#390—21no—
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C43C2211yescharacteristic
33#1982—24no—
34#1982C418yescomplemented
35#1934—28no—
36C4×C4⋊C8—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42#1436—21no—
43#1436—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C4×#95—21no—
47C23×D4—21no—
48C23×D4—21no—
49C23×D4C2312yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#127—22no—
53C2×#127—22no—
54C2×#127—22no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127C2×C414yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8C2×C414yes—
85C2×C4⋊C8D414yes—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—42no—
93C2×#102—42no—
94C2×#102—42no—
95C2×#102—42no—
96C2×#102C2×C418yes—
97C2×#102C2×C418yes—
98C2×C4×C8—24no—
99C2×C4×C8—24no—
100C2×C4×C8—42no—
101#390—24no—
102#390—24no—
103#390—42no—
104#390—42no—
105#390—44no—
106#390—44no—
107#390—44no—
108#390—44no—
109#389—22no—
110#389—22no—
111#389—22no—
112#389D414yescomplemented
113#384—22no—
114#384—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—44no—
119C4×C4⋊C4C2×C414yes—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C43—22no—
129C43—22no—
130#340—24no—
131#340—24no—
132#340—42no—
133#340—42no—
134#336—24no—
135C25—22no—
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
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—42no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4C22×C414yes—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2411yescharacteristic
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8C22×C414yes—
189C22×C8C2×D414yes—
190#127—28no—
191#127—28no—
192#127—44no—
193#127—44no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4#4112yes—
235C2×C4⋊C4#4112yes—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—41no—
268C2×C42—41no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42D814yes—
275C2×C42SD1614yes—
276C2×C42C22×C412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42#4112yes—
280C2×C42#4112yes—
281C2×C42#4112yes—
282C2×C42#4112yes—
283C2×C42#4112yes—
284C4⋊C8—28no—
285C4⋊C8—28no—
286#102—216no—
287#102—48no—
288#102—48no—
289#102—48no—
290C4×C8—48no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—42no—
295#95—42no—
296#95—42no—
297#95—42no—
298#95—42no—
299C24—24no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24C22×D411yescharacteristic
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—48no—
322C2×D4—48no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
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—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—84no—
350C2×D4—84no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4—84no—
354C2×D4—84no—
355C2×D4—84no—
356C2×D4—84no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
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—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4C4×D412yes—
417C22×C4C4×D412yes—
418C22×C4#12112yes—
419C22×C4#12112yes—
420C22×C4#12112yes—
421C22×C4#12112yes—
422C22×C4#12412yes—
423C22×C4#12412yes—
424C22×C4C2×D812yes—
425C22×C4C2×D812yes—
426C22×C4C2×D812yes—
427C22×C4C2×SD1612yes—
428C22×C4C2×SD1612yes—
429C22×C4C2×SD1612yes—
430C22×C4#13512yes—
431C22×C4#13512yes—
432C22×C4AGL(1,ℤ/8)12yes—
433C22×C4AGL(1,ℤ/8)12yes—
434C22×C4C23×C411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4H(5,𝔽2)11yescharacteristic
445C22×C4H(5,𝔽2)11yescharacteristic
446C22×C4H(5,𝔽2)11yescharacteristic
447C22×C42−1+411yescharacteristic
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—48no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4C4×D414yes—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—28no—
495C42—28no—
496C42—44no—
497C42—44no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—48no—
503C42—84no—
504C42#11714yes—
505C42C4×D414yes—
506C42#12314yes—
507C42#12614yes—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
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—
528C23—82no—
529C23—82no—
530C23—82no—
531C23—82no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23—82no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23C2×C4×D411yescharacteristic
544C23C2×#12111yescharacteristic
545C23C2×#12111yescharacteristic
546C23#52511yescharacteristic
547C23C2×#12411yescharacteristic
548C23#53511yescharacteristic
549C23#53711yescharacteristic
550C23C22×D811yescharacteristic
551C23C22×SD1611yescharacteristic
552C23C2×#13511yescharacteristic
553C23C2×#13511yescharacteristic
554C23C2×AGL(1,ℤ/8)11yescharacteristic
555C23C2×AGL(1,ℤ/8)11yescharacteristic
556C23#57511yescharacteristic
557C23#57511yescharacteristic
558D4—44no—
559D4—44no—
560D4—44no—
561D4—44no—
562D4—816no—
563D4—82no—
564D4—816no—
565D4—82no—
566D4—82no—
567D4—82no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4#38814yes—
636C2×C4#38814yes—
637C2×C4C4×D814yes—
638C2×C4C4×SD1614yes—
639C2×C4#45914yes—
640C2×C4#46014yes—
641C2×C4#46514yes—
642C2×C4#46614yes—
643C2×C4#48614yes—
644C2×C4#48814yes—
645C2×C4C2×#11711yescharacteristic
646C2×C4C2×C4×D411yescharacteristic
647C2×C4C2×C4×D411yescharacteristic
648C2×C4C4×#4111yescharacteristic
649C2×C4#51811yescharacteristic
650C2×C4#52011yescharacteristic
651C2×C4C2×#12311yescharacteristic
652C2×C4C2×#12611yescharacteristic
653C2×C4#52911yescharacteristic
654C2×C4#53211yescharacteristic
655C2×C4#54011yescharacteristic
656C2×C4#54211yescharacteristic
657C2×C4D4211yescharacteristic
658C2×C4#54611yescharacteristic
659C2×C4#54711yescharacteristic
660C2×C4#54811yescharacteristic
661C2×C4#55011yescharacteristic
662C2×C4#55011yescharacteristic
663C2×C4#55011yescharacteristic
664C2×C4#55311yescharacteristic
665C2×C4#55311yescharacteristic
666C2×C4#55511yescharacteristic
667C2×C4#55611yescharacteristic
668C2×C4#55611yescharacteristic
669C2×C4#55911yescharacteristic
670C2×C4#55911yescharacteristic
671C2×C4#55911yescharacteristic
672C2×C4#55911yescharacteristic
673C8—44no—
674C8—44no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—84no—
682C22—162no—
683C22—162no—
684C22—162no—
685C22—162no—
686C22—162no—
687C22—162no—
688C22—162no—
689C22—162no—
690C22C2×#38811yescharacteristic
691C22C4×#12111yescharacteristic
692C22#232411yescharacteristic
693C22#241411yescharacteristic
694C22#241611yescharacteristic
695C22#254811yescharacteristic
696C22#259211yescharacteristic
697C22C2×C4×D811yescharacteristic
698C22C2×C4×SD1611yescharacteristic
699C22C4×AGL(1,ℤ/8)11yescharacteristic
700C22C4×AGL(1,ℤ/8)11yescharacteristic
701C22C2×#45911yescharacteristic
702C22C2×#46011yescharacteristic
703C22#303511yescharacteristic
704C22#303511yescharacteristic
705C22C2×#46511yescharacteristic
706C22C2×#46611yescharacteristic
707C22#305511yescharacteristic
708C22#305511yescharacteristic
709C22#310011yescharacteristic
710C22#310311yescharacteristic
711C22#310611yescharacteristic
712C22#310611yescharacteristic
713C22C2×#48611yescharacteristic
714C22C2×#48811yescharacteristic
715C22#313811yescharacteristic
716C22#314011yescharacteristic
717C22#320511yescharacteristic
718C22#320811yescharacteristic
719C22#321211yescharacteristic
720C22#321211yescharacteristic
721C22#324511yescharacteristic
722C22#324611yescharacteristic
723C22#324711yescharacteristic
724C22#324711yescharacteristic
725C4—24no—
726C4—24no—
727C4—24no—
728C4—42no—
729C4—42no—
730C4—44no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—162no—
735C4#193418yescomplemented
736C4#235411yescharacteristic
737C4#235711yescharacteristic
738C4#236611yescharacteristic
739C4#236711yescharacteristic
740C4#244511yescharacteristic
741C4#244811yescharacteristic
742C4#251211yescharacteristic
743C4#251311yescharacteristic
744C2—164no—
745C2#1402211yescharacteristic
746C2C2×#193411yescharacteristic
747C2#2109511yescharacteristic
748C2C4×#45911yescharacteristic
749C2C4×#46011yescharacteristic
750C2#2214511yescharacteristic
751C2#2214711yescharacteristic
752C2#2465011yescharacteristic
753C2#2465311yescharacteristic
754C2#2475011yescharacteristic
755C2#2475311yescharacteristic
756C2#2874911yescharacteristic
757C2#2875211yescharacteristic
758C2#2934611yescharacteristic
759C2#2934811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1934C2semidirect
4. C2×C4×C4⋊C8C2semidirect
6. C4×#389C2semidirect
7. #11189C2semidirect
8. #11066C2semidirect
9. #10925C2semidirect
10. #10849C2semidirect
11. #10849C2semidirect
34. #1982C4semidirect
112. #389D4semidirect
735. C4#1934semidirect

Maximal subgroups

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

subgroupindex
C2×#19822
C2×#19342
C2×C4×C4⋊C82
#147132
C4×#3892
#111892
#110662
#109252
#108492