Finite Group Databasebeta

← #364770

Subgroups — #364770

Order 512.

1708 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3C2×#1990C211yesmaximal, characteristic, complemented
4C2×C4×#102C211yesmaximal, characteristic
5#13906C211yesmaximal, characteristic
6#11220C212yesmaximal, complemented
7#11066C212yesmaximal
8#10931C212yesmaximal
9#10908C212yesmaximal, complemented
10#10849C211yesmaximal, characteristic, complemented
11#10834C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#476—24no—
15C2×#476—24no—
16C2×#476C418yescomplemented
17C22×#106C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×#102C2211yescharacteristic
20C22×#102C2212yes—
21C22×#102C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2369—24no—
24#2369—24no—
25C2×#391—22no—
26C2×#390C2212yes—
27C2×#389—21no—
28C2×#389C2212yes—
29C2×#384C2211yescharacteristic
30C2×#380—21no—
31C2×#380—21no—
32C2×C4×C4⋊C4—21no—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1990—24no—
37C4×#102—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1439—22no—
47C4⋊#95C2212yes—
48C23×Q8—21no—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8C2×C414yes—
58C2×C4⋊Q8D414yes—
59C2×C4⋊Q8D414yes—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4C2×C414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C42—21no—
81C22×C42—22no—
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84#476—232no—
85C2×#106—24no—
86C2×#106—24no—
87C2×#106C2×C418yes—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8C2×C414yes—
91C2×#102—22no—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—22no—
98C2×#102C2×C418yes—
99C2×#102Q814yescomplemented
100C2×C4×C8—22no—
101C2×C4×C8—22no—
102C2×C4×C8—22no—
103#391—216no—
104#390—24no—
105#390—42no—
106#389—28no—
107#389—24no—
108#389—24no—
109#384—22no—
110#384—22no—
111#380—44no—
112#380—44no—
113C4×C4⋊C4—28no—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4D414yes—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95D412yes—
126#340—24no—
127#340—24no—
128#340—22no—
129#340—22no—
130#340—24no—
131#340—42no—
132#336—24no—
133#336—42no—
134C25—21no—
135C22×Q8—24no—
136C22×Q8—24no—
137C22×Q8—24no—
138C22×Q8—81no—
139C22×Q8—81no—
140C22×Q8—81no—
141C22×Q8—81no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4C22×C412yes—
154C22×D4C2×Q812yes—
155C22×D4#4112yes—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8C22×C414yes—
192C22×C8C2×Q812yes—
193C22×C8#4112yes—
194C4⋊Q8—28no—
195C4⋊Q8—28no—
196C4⋊Q8—28no—
197C4⋊Q8C22⋊C4116yescomplemented
198C4×D4—28no—
199C4×D4—28no—
200C4×D4—44no—
201C4×D4—44no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
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—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—81no—
238C2×C4⋊C4—81no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4—82no—
241C2×C4⋊C4SD1614yes—
242C2×C4⋊C4C22×C412yes—
243C2×C4⋊C4C22×C414yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2×D414yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×Q812yes—
249C2×C4⋊C4#4112yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—82no—
258C2×C22⋊C4—82no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—28no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—82no—
270C2×C42—82no—
271C2×C42C22×C412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42#4112yes—
276C2×C42#4112yes—
277#106—216no—
278C4⋊C8—28no—
279#102—28no—
280#102—216no—
281#102—28no—
282C4×C8—44no—
283#95—28no—
284#95—28no—
285#95—24no—
286#95—24no—
287#95—44no—
288#95—44no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24C22×D411yescharacteristic
305C2×Q8—216no—
306C2×Q8—84no—
307C2×Q8—84no—
308C2×Q8—84no—
309C2×Q8—84no—
310C2×Q8—84no—
311C2×Q8—84no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4C4×Q814yes—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
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—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
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—48no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4C4×D412yes—
400C22×C4C4×D412yes—
401C22×C4#12012yes—
402C22×C4#12012yes—
403C22×C4#12112yes—
404C22×C4#12112yes—
405C22×C4C2×SD1614yes—
406C22×C4C2×SD1612yes—
407C22×C4#13512yes—
408C22×C4AGL(1,ℤ/8)12yes—
409C22×C4#13712yes—
410C22×C4C23×C411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×Q811yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4H(5,𝔽2)11yescharacteristic
421C22×C42−1+411yescharacteristic
422C22×C42−1+411yescharacteristic
423C22×C42−1+411yescharacteristic
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—41no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—216no—
452C4⋊C4—28no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—84no—
467C4⋊C4—88no—
468C4⋊C4—84no—
469C4⋊C4C2×C22⋊C418yes—
470C4⋊C4C22⋊Q814yes—
471C22⋊C4—48no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C42—42no—
475C42—42no—
476C42—42no—
477C42—42no—
478C42—48no—
479C42—88no—
480C42C2×C22⋊C414yes—
481C42C4×D414yes—
482C42#12114yes—
483C42#12314yes—
484C23—24no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—81no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23—81no—
508C23—81no—
509C23—81no—
510C23—81no—
511C23—81no—
512C23—81no—
513C23—81no—
514C23—81no—
515C23—81no—
516C23—81no—
517C23—81no—
518C23—81no—
519C23C2×C4×D411yescharacteristic
520C23C2×#12011yescharacteristic
521C23C2×#12111yescharacteristic
522C23#54611yescharacteristic
523C23#54711yescharacteristic
524C23#54811yescharacteristic
525C23D4×Q811yescharacteristic
526C23C22×SD1611yescharacteristic
527C23C2×#13511yescharacteristic
528C23C2×AGL(1,ℤ/8)11yescharacteristic
529C23C2×#13711yescharacteristic
530C23#57611yescharacteristic
531C23#57611yescharacteristic
532C23#57711yescharacteristic
533C23#57711yescharacteristic
534Q8—816no—
535D4—24no—
536D4—24no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—42no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
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—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—816no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4#38614yes—
600C2×C4#38614yes—
601C2×C4C4×SD1614yes—
602C2×C4#44014yes—
603C2×C4#44814yes—
604C2×C4#44914yes—
605C2×C4#46014yes—
606C2×C4#46414yes—
607C2×C4C22×C22⋊C411yescharacteristic
608C2×C4C2×C4×D411yescharacteristic
609C2×C4C2×C4×Q811yescharacteristic
610C2×C4C4×#4111yescharacteristic
611C2×C4#51911yescharacteristic
612C2×C4#52011yescharacteristic
613C2×C4C2×#12111yescharacteristic
614C2×C4C2×C22⋊Q811yescharacteristic
615C2×C4C2×#12311yescharacteristic
616C2×C4#52511yescharacteristic
617C2×C4#52911yescharacteristic
618C2×C4#53311yescharacteristic
619C2×C4#53611yescharacteristic
620C2×C4#53711yescharacteristic
621C2×C4#54111yescharacteristic
622C2×C4#54211yescharacteristic
623C2×C4#54811yescharacteristic
624C2×C4#54811yescharacteristic
625C2×C4#54811yescharacteristic
626C2×C4#54811yescharacteristic
627C2×C4D4×Q811yescharacteristic
628C2×C4D4×Q811yescharacteristic
629C2×C4#55011yescharacteristic
630C2×C4#55011yescharacteristic
631C2×C4#55211yescharacteristic
632C2×C4#55211yescharacteristic
633C2×C4#55411yescharacteristic
634C2×C4#55611yescharacteristic
635C8—44no—
636C8—42no—
637C8—42no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22C2×#38611yescharacteristic
654C22#236011yescharacteristic
655C22#236211yescharacteristic
656C22#240711yescharacteristic
657C22#240911yescharacteristic
658C22#242611yescharacteristic
659C22#242911yescharacteristic
660C22C2×C4×SD1611yescharacteristic
661C22C2×#44011yescharacteristic
662C22#294611yescharacteristic
663C22#294611yescharacteristic
664C22C2×#44811yescharacteristic
665C22C2×#44911yescharacteristic
666C22#301211yescharacteristic
667C22#301211yescharacteristic
668C22C2×#46011yescharacteristic
669C22C2×#46411yescharacteristic
670C22#304111yescharacteristic
671C22#304211yescharacteristic
672C22#329511yescharacteristic
673C22#330311yescharacteristic
674C22#332511yescharacteristic
675C22#333111yescharacteristic
676C22#333511yescharacteristic
677C22#334111yescharacteristic
678C22#335411yescharacteristic
679C22#335711yescharacteristic
680C22#336011yescharacteristic
681C22#336211yescharacteristic
682C22#336611yescharacteristic
683C22#336911yescharacteristic
684C22Q8×SD1611yescharacteristic
685C22#338011yescharacteristic
686C22#338411yescharacteristic
687C22#338511yescharacteristic
688C4—24no—
689C4—44no—
690C4—44no—
691C4—48no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—162no—
696C4#188018yes—
697C4C4×C22⋊Q811yescharacteristic
698C4#232311yescharacteristic
699C4Q8×C22⋊C411yescharacteristic
700C4#234111yescharacteristic
701C4#242111yescharacteristic
702C4#242311yescharacteristic
703C4#253611yescharacteristic
704C4#253811yescharacteristic
705C2—84no—
706C2#1396911yescharacteristic
707C2C2×#188011yescharacteristic
708C2#2092911yescharacteristic
709C2#2296311yescharacteristic
710C2#2296911yescharacteristic
711C2#2306111yescharacteristic
712C2#2306711yescharacteristic
713C2#2411711yescharacteristic
714C2#2411811yescharacteristic
715C2#2432711yescharacteristic
716C2#2432911yescharacteristic
717C2#2527311yescharacteristic
718C2#2527411yescharacteristic
719C2#2545011yescharacteristic
720C2#2545111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
2. C2×#2369C2semidirect
3. C2×#1990C2semidirect
6. #11220C2semidirect
9. #10908C2semidirect
10. #10849C2semidirect
11. #10834C2semidirect
16. C2×#476C4semidirect
99. C2×#102Q8semidirect
197. C4⋊Q8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
C2×#23692
C2×#19902
C2×C4×#1022
#139062
#112202
#110662
#109312
#109082
#108492
#108342