Finite Group Databasebeta

← #364769

Subgroups — #364769

Order 512.

1816 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3C2×#1990C211yesmaximal, characteristic, complemented
4C2×C4×#102C211yesmaximal, characteristic
5#13906C211yesmaximal, characteristic
6#11219C212yesmaximal, complemented
7#11066C212yesmaximal
8#10931C212yesmaximal
9#10907C212yesmaximal, complemented
10#10849C211yesmaximal, characteristic, complemented
11#10834C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×C4×D4C2211yescharacteristic
15C2×#474—24no—
16C2×#474—24no—
17C2×#474C418yescomplemented
18C22×#107C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#102C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24#2365—24no—
25#2365—24no—
26C2×#391—21no—
27C2×#390C2212yes—
28C2×#389C2212yes—
29C2×#387—22no—
30C2×#384C2211yescharacteristic
31C2×#380—21no—
32C2×#380—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×Q8—28no—
61C2×C4×Q8—44no—
62C2×C4×Q8—44no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4C2×C414yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C42—22no—
84C22×C42—21no—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87#474—232no—
88C2×#107—24no—
89C2×#107—24no—
90C2×#107C2×C418yes—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8C2×C414yes—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—22no—
101C2×#102C2×C418yes—
102C2×#102Q814yescomplemented
103C2×C4×C8—22no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106#391—28no—
107#390—24no—
108#390—42no—
109#389—24no—
110#389—24no—
111#387—216no—
112#384—22no—
113#384—22no—
114#380—44no—
115#380—44no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—24no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4D414yes—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95D412yes—
128#340—24no—
129#340—24no—
130#340—22no—
131#340—22no—
132#340—24no—
133#340—42no—
134#336—24no—
135#336—42no—
136C25—21no—
137C22×Q8—24no—
138C22×Q8—24no—
139C22×Q8—24no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
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×D4—21no—
154C22×D4—21no—
155C22×D4C22×C412yes—
156C22×D4C2×Q812yes—
157C22×D4#4112yes—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2411yescharacteristic
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8C22×C414yes—
194C22×C8C2×Q812yes—
195C22×C8#4112yes—
196C4⋊Q8—28no—
197C4⋊Q8—28no—
198C4⋊Q8—28no—
199C4⋊Q8C22⋊C4116yescomplemented
200C4×Q8—432no—
201C4×D4—28no—
202C4×D4—28no—
203C4×D4—44no—
204C4×D4—44no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
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—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
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⋊C4D814yes—
245C2×C4⋊C4C22×C412yes—
246C2×C4⋊C4C22×C414yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×Q812yes—
252C2×C4⋊C4#4112yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—82no—
261C2×C22⋊C4—82no—
262C2×C42—28no—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—44no—
271C2×C42—44no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42C22×C412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42#4112yes—
279C2×C42#4112yes—
280#107—216no—
281C4⋊C8—28no—
282#102—28no—
283#102—216no—
284#102—28no—
285C4×C8—44no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—44no—
291#95—44no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24C22×D411yescharacteristic
308C2×Q8—216no—
309C2×Q8—48no—
310C2×Q8—48no—
311C2×Q8—48no—
312C2×Q8—48no—
313C2×Q8—48no—
314C2×Q8—48no—
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—22no—
328C2×D4—22no—
329C2×D4—22no—
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×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4C4×Q814yescomplemented
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
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—22no—
367C22×C4—22no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
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—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4C4×D412yes—
403C22×C4C4×D412yes—
404C22×C4#12012yes—
405C22×C4#12012yes—
406C22×C4#12112yes—
407C22×C4#12112yes—
408C22×C4C2×D814yes—
409C22×C4C2×D812yes—
410C22×C4#13512yes—
411C22×C4AGL(1,ℤ/8)12yes—
412C22×C4AGL(1,ℤ/8)12yes—
413C22×C4C23×C411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×Q811yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4H(5,𝔽2)11yescharacteristic
424C22×C42−1+411yescharacteristic
425C22×C42−1+411yescharacteristic
426C22×C42−1+411yescharacteristic
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—41no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—216no—
455C4⋊C4—28no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—416no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—84no—
471C4⋊C4—84no—
472C4⋊C4C2×C22⋊C418yes—
473C4⋊C4C22⋊Q814yes—
474C22⋊C4—48no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C42—416no—
478C42—42no—
479C42—42no—
480C42—42no—
481C42—42no—
482C42—48no—
483C42C2×C22⋊C414yes—
484C42C4×D414yes—
485C42#12114yes—
486C42#12314yes—
487C23—24no—
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—42no—
504C23—42no—
505C23—42no—
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—
519C23—81no—
520C23—81no—
521C23—81no—
522C23C2×C4×D411yescharacteristic
523C23C2×#12011yescharacteristic
524C23C2×#12111yescharacteristic
525C23#54611yescharacteristic
526C23#54711yescharacteristic
527C23#54811yescharacteristic
528C23D4×Q811yescharacteristic
529C23C22×D811yescharacteristic
530C23C2×#13511yescharacteristic
531C23C2×AGL(1,ℤ/8)11yescharacteristic
532C23C2×AGL(1,ℤ/8)11yescharacteristic
533C23#57711yescharacteristic
534C23#57711yescharacteristic
535C23#57811yescharacteristic
536C23#57811yescharacteristic
537Q8—432no—
538D4—24no—
539D4—24no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—42no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
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—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—816no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4#38614yes—
603C2×C4#38614yes—
604C2×C4C4×D814yes—
605C2×C4#44214yes—
606C2×C4#44714yes—
607C2×C4#44914yes—
608C2×C4#45914yes—
609C2×C4#46314yes—
610C2×C4C22×C22⋊C411yescharacteristic
611C2×C4C2×C4×D411yescharacteristic
612C2×C4C2×C4×Q811yescharacteristic
613C2×C4C4×#4111yescharacteristic
614C2×C4#51911yescharacteristic
615C2×C4#52011yescharacteristic
616C2×C4C2×#12111yescharacteristic
617C2×C4C2×C22⋊Q811yescharacteristic
618C2×C4C2×#12311yescharacteristic
619C2×C4#52511yescharacteristic
620C2×C4#52911yescharacteristic
621C2×C4#53311yescharacteristic
622C2×C4#53611yescharacteristic
623C2×C4#53711yescharacteristic
624C2×C4#54111yescharacteristic
625C2×C4#54211yescharacteristic
626C2×C4#54811yescharacteristic
627C2×C4#54811yescharacteristic
628C2×C4#54811yescharacteristic
629C2×C4#54811yescharacteristic
630C2×C4D4×Q811yescharacteristic
631C2×C4D4×Q811yescharacteristic
632C2×C4#55011yescharacteristic
633C2×C4#55011yescharacteristic
634C2×C4#55211yescharacteristic
635C2×C4#55211yescharacteristic
636C2×C4#55411yescharacteristic
637C2×C4#55611yescharacteristic
638C8—44no—
639C8—42no—
640C8—42no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22C2×#38611yescharacteristic
657C22#236011yescharacteristic
658C22#236211yescharacteristic
659C22#240711yescharacteristic
660C22#240911yescharacteristic
661C22#242611yescharacteristic
662C22#242911yescharacteristic
663C22C2×C4×D811yescharacteristic
664C22C2×#44211yescharacteristic
665C22#294711yescharacteristic
666C22#294711yescharacteristic
667C22C2×#44711yescharacteristic
668C22C2×#44911yescharacteristic
669C22#301311yescharacteristic
670C22#301311yescharacteristic
671C22C2×#45911yescharacteristic
672C22C2×#46311yescharacteristic
673C22#304311yescharacteristic
674C22#304411yescharacteristic
675C22#329411yescharacteristic
676C22#330211yescharacteristic
677C22#332411yescharacteristic
678C22#333011yescharacteristic
679C22#333411yescharacteristic
680C22#334011yescharacteristic
681C22#335311yescharacteristic
682C22#335411yescharacteristic
683C22#335811yescharacteristic
684C22#336411yescharacteristic
685C22#336711yescharacteristic
686C22#336811yescharacteristic
687C22Q8×D811yescharacteristic
688C22#338111yescharacteristic
689C22#338411yescharacteristic
690C22#338611yescharacteristic
691C4—24no—
692C4—44no—
693C4—44no—
694C4—48no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—162no—
699C4#187918yes—
700C4C4×C22⋊Q811yescharacteristic
701C4#232311yescharacteristic
702C4Q8×C22⋊C411yescharacteristic
703C4#234111yescharacteristic
704C4#242111yescharacteristic
705C4#242311yescharacteristic
706C4#253611yescharacteristic
707C4#253811yescharacteristic
708C2—84no—
709C2#1396911yescharacteristic
710C2C2×#187911yescharacteristic
711C2#2093011yescharacteristic
712C2#2296211yescharacteristic
713C2#2296811yescharacteristic
714C2#2306011yescharacteristic
715C2#2306611yescharacteristic
716C2#2411611yescharacteristic
717C2#2411911yescharacteristic
718C2#2432611yescharacteristic
719C2#2432811yescharacteristic
720C2#2527211yescharacteristic
721C2#2527511yescharacteristic
722C2#2544911yescharacteristic
723C2#2545211yescharacteristic
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×#2365C2semidirect
3. C2×#1990C2semidirect
6. #11219C2semidirect
9. #10907C2semidirect
10. #10849C2semidirect
11. #10834C2semidirect
17. C2×#474C4semidirect
102. C2×#102Q8semidirect
199. C4⋊Q8C22⋊C4semidirect
342. C2×D4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
C2×#23652
C2×#19902
C2×C4×#1022
#139062
#112192
#110662
#109312
#109072
#108492
#108342