Finite Group Databasebeta

← #403327

Subgroups — #403327

Order 512.

2594 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C2×#2026C212yesmaximal, complemented
3C2×#2024C212yesmaximal, complemented
4C2×#1873C211yesmaximal, characteristic
5C2×#1872C211yesmaximal, characteristic
6#14258C211yesmaximal, characteristic
7#11385C212yesmaximal, complemented
8#11384C212yesmaximal, complemented
9#10731C211yesmaximal, characteristic
10#10730C211yesmaximal, characteristic
11#10713C211yesmaximal, characteristic
12C2×#544—28no—
13C2×#544—28no—
14C2×#544—24no—
15C2×#544—24no—
16C22×C4⋊Q8—21no—
17C22×C4⋊Q8C2212yes—
18C22×#125—21no—
19C22×#125C2212yes—
20C22×C22⋊Q8—22no—
21C22×C22⋊Q8C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2212yes—
27C22×#103C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C2×#391C2212yes—
31C2×#380C2211yescharacteristic
32#2026—28no—
33#2024—28no—
34#1873—24no—
35#1872—24no—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1441—21no—
46#1441—21no—
47#1440—22no—
48C4⋊#95C2212yes—
49C23×Q8—22no—
50C23×Q8C2311yescharacteristic
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56#544—232no—
57#544—232no—
58#544—264no—
59C2×C4⋊Q8—216no—
60C2×C4⋊Q8—28no—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—24no—
63C2×C4⋊Q8—24no—
64C2×C4⋊Q8—24no—
65C2×C4⋊Q8—24no—
66C2×C4⋊Q8D418yes—
67C2×#125—216no—
68C2×#125—28no—
69C2×#125—24no—
70C2×#125—24no—
71C2×#125—24no—
72C2×#125—24no—
73C2×#125—24no—
74C2×#125D418yes—
75C2×C22⋊Q8—216no—
76C2×C22⋊Q8—216no—
77C2×C22⋊Q8—216no—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8Q814yescomplemented
84C2×#117—22no—
85C2×#117—22no—
86C2×#117—22no—
87C2×#117—24no—
88C2×#117—24no—
89C2×#117—24no—
90C2×#117—24no—
91C2×#117D414yes—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4D411yescharacteristic
101C22×C4⋊C4D411yescharacteristic
102C22×C4⋊C4D414yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4D412yes—
112C22×C42—21no—
113C22×C42C2312yes—
114C2×#103—24no—
115C2×#103—22no—
116C2×#103—22no—
117C2×#103—24no—
118C2×#103—24no—
119C2×#103—24no—
120C2×#103—24no—
121C2×C22⋊C8—22no—
122C2×C22⋊C8—22no—
123C2×C22⋊C8—22no—
124C2×C22⋊C8—22no—
125C2×C22⋊C8—22no—
126C2×C22⋊C8—22no—
127#391—24no—
128#391—24no—
129#380—22no—
130#380—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136#340—24no—
137#340—22no—
138#340—22no—
139#340—24no—
140#340—24no—
141#340—24no—
142#340—24no—
143#336—24no—
144#336—24no—
145C25C2×D411yescharacteristic
146C22×Q8—28no—
147C22×Q8—28no—
148C22×Q8—28no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—82no—
159C22×Q8—82no—
160C22×Q8—82no—
161C22×Q8—82no—
162C22×Q8C2×Q812yes—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—22no—
170C23×C4—28no—
171C23×C4—24no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—81no—
176C23×C4—82no—
177C23×C4—81no—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×Q812yes—
184C23×C4#4112yes—
185C23×C4#4112yes—
186C23×C4#4112yes—
187C23×C4C2411yescharacteristic
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C4⋊Q8—264no—
206C4⋊Q8—216no—
207C4⋊Q8—216no—
208C4⋊Q8—216no—
209C4⋊Q8—216no—
210#125—264no—
211#125—216no—
212#125—216no—
213#125—216no—
214#125—216no—
215C22⋊Q8—2128no—
216C22⋊Q8—28no—
217C22⋊Q8—28no—
218C22⋊Q8—44no—
219C22⋊Q8—44no—
220#117—216no—
221#117—28no—
222#117—216no—
223#117—28no—
224#117—28no—
225#117—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×Q812yes—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—81no—
283C2×C22⋊C4—81no—
284C2×C42—28no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42C2×D414yes—
291C2×C42C2×D414yes—
292#103—48no—
293#103—44no—
294#103—44no—
295C22⋊C8—44no—
296C22⋊C8—44no—
297#95—44no—
298#95—44no—
299#95—42no—
300#95—82no—
301#95—82no—
302C24—22no—
303C24—22no—
304C24—21no—
305C24—21no—
306C24—22no—
307C24—21no—
308C24—22no—
309C24—21no—
310C24#12012yes—
311C24C22⋊Q812yes—
312C24#12412yes—
313C24C22×D411yescharacteristic
314C2×Q8—232no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—88no—
334C2×Q8—88no—
335C2×Q8—88no—
336C2×Q8—88no—
337C2×Q8—88no—
338C2×Q8—88no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—216no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
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—84no—
405C22×C4—84no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4#12014yes—
409C22×C4#12014yes—
410C22×C4#12012yes—
411C22×C4C22⋊Q814yes—
412C22×C4C22⋊Q814yes—
413C22×C4C22⋊Q812yes—
414C22×C4#12414yes—
415C22×C4#12414yes—
416C22×C4#12412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×Q811yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C42−1+412yes—
426C22×C42−1+412yes—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—42no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C4⋊C4—216no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—232no—
448C4⋊C4—232no—
449C4⋊C4—232no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—48no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4#12018yes—
467C4⋊C4#12014yes—
468C4⋊C4#12014yes—
469C22⋊C4—216no—
470C22⋊C4—216no—
471C22⋊C4—48no—
472C22⋊C4—84no—
473C42—216no—
474C42—24no—
475C42—24no—
476C42—24no—
477C42—24no—
478C23—24no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23#39314yes—
501C23C2×#12011yescharacteristic
502C23C2×#12012yes—
503C23C2×C22⋊Q811yescharacteristic
504C23C2×#12411yescharacteristic
505C23#53612yes—
506C23#57611yescharacteristic
507C23#57712yes—
508C23#57711yescharacteristic
509C23#57711yescharacteristic
510C23#57812yes—
511C23#57811yescharacteristic
512Q8—42no—
513Q8—42no—
514Q8—44no—
515Q8—832no—
516Q8—81no—
517Q8—81no—
518Q8—82no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—816no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—82no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4#39314yes—
581C2×C4C2×#12011yescharacteristic
582C2×C4C2×#12011yescharacteristic
583C2×C4C2×#12012yes—
584C2×C4C2×C22⋊Q811yescharacteristic
585C2×C4C2×C22⋊Q811yescharacteristic
586C2×C4C2×#12411yescharacteristic
587C2×C4C2×#12411yescharacteristic
588C2×C4#53611yescharacteristic
589C2×C4#53611yescharacteristic
590C2×C4#53612yes—
591C2×C4#54812yes—
592C2×C4#54812yes—
593C2×C4#54812yes—
594C2×C4#54812yes—
595C2×C4#54812yes—
596C2×C4#54812yes—
597C2×C4D4×Q812yes—
598C2×C4D4×Q812yes—
599C8—81no—
600C8—81no—
601C8—82no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—44no—
606C22—44no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22C2×#39311yescharacteristic
614C22#240912yes—
615C22#243612yes—
616C22#261411yescharacteristic
617C22#284611yescharacteristic
618C22#300912yes—
619C22#300911yescharacteristic
620C22#301112yes—
621C22#301111yescharacteristic
622C22#301212yes—
623C22#301211yescharacteristic
624C22#301312yes—
625C22#301311yescharacteristic
626C22#308212yes—
627C22#308311yescharacteristic
628C22#308411yescharacteristic
629C22#313611yescharacteristic
630C22#313711yescharacteristic
631C22#313712yes—
632C22#324212yes—
633C22#324212yes—
634C22#324312yes—
635C22#324412yes—
636C4—24no—
637C4—44no—
638C4—48no—
639C4—44no—
640C4—44no—
641C4—44no—
642C4—82no—
643C4—82no—
644C4—162no—
645C4#240911yescharacteristic
646C4#240911yescharacteristic
647C4#240912yes—
648C4#243611yescharacteristic
649C4#243611yescharacteristic
650C4#243612yes—
651C2—44no—
652C2#1487311yescharacteristic
653C2#2345711yescharacteristic
654C2#2345811yescharacteristic
655C2#2431012yes—
656C2#2431112yes—
657C2#2591412yes—
658C2#2591612yes—
659C2#2964811yescharacteristic
660C2#2964911yescharacteristic
661C2#3288611yescharacteristic
662C2#3288711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2026C2semidirect
3. C2×#2024C2semidirect
7. #11385C2semidirect
8. #11384C2semidirect
83. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#20262
C2×#20242
C2×#18732
C2×#18722
#142582
#113852
#113842
#107312
#107302
#107132