Finite Group Databasebeta

← #369324

Subgroups — #369324

Order 512.

1786 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#446C211yesmaximal, characteristic, complemented
2C2×C4×#124C211yesmaximal, characteristic
3C2×#1985C211yesmaximal, characteristic, complemented
4C2×#1955C211yesmaximal, characteristic
5C2×#1955C211yesmaximal, characteristic
6#11042C212yesmaximal, complemented
7#11042C212yesmaximal, complemented
8#10922C212yesmaximal, complemented
9#10921C212yesmaximal, complemented
10#10832C211yesmaximal, characteristic, complemented
11#10693C211yesmaximal, characteristic, complemented
12C22×#124C411yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#446—24no—
16C2×#446—24no—
17C2×#446—24no—
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C22⋊C8C2212yes—
23C22×C22⋊C8C2212yes—
24C22×C22⋊C8C2212yes—
25C22×C22⋊C8C2212yes—
26C22×#97C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C22×#97C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30C4×#124—24no—
31C4×#124—24no—
32C2×#391C411yescharacteristic
33C2×#390C411yescharacteristic
34C2×#388C412yes—
35C2×#388C412yes—
36C2×#382C411yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C43C2211yescharacteristic
40#1985—24no—
41#1955—24no—
42#1955—24no—
43C2×#336C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47C2×#336C2212yes—
48C2×#336C2212yes—
49C23×Q8C2×C411yescharacteristic
50C23×D4C2×C411yescharacteristic
51C24×C4C2312yes—
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C23×C8C2311yescharacteristic
58C2×#124—24no—
59C2×#124—28no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8D414yes—
65C2×C4×Q8D414yes—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C412yes—
80C22×C22⋊C4C2×C412yes—
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C412yes—
83C22×C42C2×C411yescharacteristic
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#446—232no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8D414yes—
101C2×C22⋊C8—24no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—24no—
107C2×C22⋊C8—24no—
108C2×C22⋊C8—24no—
109C2×C22⋊C8—42no—
110C2×C22⋊C8—42no—
111C2×C22⋊C8—42no—
112C2×C22⋊C8—42no—
113C2×#97—22no—
114C2×#97—22no—
115C2×#97—22no—
116C2×#97—22no—
117C2×#97—22no—
118C2×#97—22no—
119C2×#97—22no—
120C2×#97—22no—
121C2×#97D414yes—
122C2×C4×C8—22no—
123C2×C4×C8—22no—
124C2×C4×C8—22no—
125#391—24no—
126#390—24no—
127#388—28no—
128#388—28no—
129#382—24no—
130C4×C22⋊C4—24no—
131C4×C22⋊C4—24no—
132C4×C22⋊C4—24no—
133C4×C22⋊C4—24no—
134C2×#95C2×C412yes—
135C2×#95C2×C412yes—
136C2×#95C2×C412yes—
137C2×#95C2×C412yes—
138C43—22no—
139C43—22no—
140#336—24no—
141#336—24no—
142#336—24no—
143#336—24no—
144#336—42no—
145#336—42no—
146C25C22×C412yes—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—24no—
150C22×Q8—42no—
151C22×Q8C22⋊C414yes—
152C22×D4—24no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—42no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4C22×C412yes—
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C412yes—
176C23×C4C22×C412yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C2411yescharacteristic
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
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×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8C2×D412yes—
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8C2×D412yes—
213C22×C8#4114yes—
214C22×C8#4114yes—
215#124—232no—
216C4×Q8—28no—
217C4×Q8—28no—
218C4×Q8—28no—
219C4×Q8—28no—
220C4×D4—44no—
221C4×D4—44no—
222C4×D4—44no—
223C4×D4—44no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
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—22no—
233C2×C4⋊C4C22⋊C414yes—
234C2×C4⋊C4C22⋊C414yes—
235C2×C4⋊C4C22⋊C414yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4M1618yes—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42#4112yes—
281C2×C42#4112yes—
282C4⋊C8—28no—
283C4⋊C8—28no—
284C4⋊C8—28no—
285C4⋊C8—44no—
286C22⋊C8—48no—
287C22⋊C8—48no—
288C22⋊C8—48no—
289C22⋊C8—48no—
290#97—28no—
291#97—28no—
292#97—44no—
293C4×C8—28no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24C2×M1614yes—
313C24C23×C411yescharacteristic
314C2×Q8—28no—
315C2×Q8—28no—
316C2×Q8—44no—
317C2×Q8—48no—
318C2×Q8—48no—
319C2×D4—28no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—44no—
323C2×D4—44no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
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—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C412yes—
393C22×C4C2×C22⋊C412yes—
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C412yes—
396C22×C4C4×D412yes—
397C22×C4C4×D412yes—
398C22×C4C4×D412yes—
399C22×C4C4×D412yes—
400C22×C4C2×M1614yes—
401C22×C4C2×M1614yes—
402C22×C4#13114yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4H(5,𝔽2)12yes—
412C22×C42−1+412yes—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—24no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—42no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8#12414yescomplemented
447C4⋊C4—28no—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—44no—
451C22⋊C4—216no—
452C22⋊C4—216no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C42—28no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—48no—
466C42—44no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—42no—
471C42#12014yes—
472C42#12114yes—
473C42#12114yes—
474C42#12314yes—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—42no—
483C23—42no—
484C23—42no—
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—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23C22×C22⋊C411yescharacteristic
511C23C2×C4×D411yescharacteristic
512C23C2×C4×D411yescharacteristic
513C23C4×#4112yes—
514C23#52012yes—
515C23C22×M1612yes—
516C23C2×#13112yes—
517C23#56812yes—
518C23#56812yes—
519Q8—416no—
520D4—416no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4#38614yes—
578C2×C4#38614yes—
579C2×C4#40714yes—
580C2×C4C2×#12011yescharacteristic
581C2×C4C2×#12111yescharacteristic
582C2×C4C2×#12111yescharacteristic
583C2×C4C2×#12311yescharacteristic
584C2×C4C2×#12411yescharacteristic
585C2×C4#52912yes—
586C2×C4#53211yescharacteristic
587C2×C4#53511yescharacteristic
588C2×C4#53611yescharacteristic
589C2×C4#54212yes—
590C2×C4#54612yes—
591C2×C4#54812yes—
592C2×C4#54812yes—
593C2×C4#54812yes—
594C2×C4#55012yes—
595C2×C4#55012yes—
596C2×C4#55612yes—
597C2×C4#55912yes—
598C8—24no—
599C8—44no—
600C8—42no—
601C8—42no—
602C8—42no—
603C8—82no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22C2×#38611yescharacteristic
620C22C4×#12411yescharacteristic
621C22#232611yescharacteristic
622C22#234112yes—
623C22#236712yes—
624C22C2×#40711yescharacteristic
625C22#287911yescharacteristic
626C22#288012yes—
627C22#293111yescharacteristic
628C22#293211yescharacteristic
629C22D4×M1612yes—
630C22D4×M1612yes—
631C22#293512yes—
632C22#296512yes—
633C22#296712yes—
634C22#296812yes—
635C22#297012yes—
636C22#298912yes—
637C22#299012yes—
638C22#299212yes—
639C22#299512yes—
640C4—24no—
641C4—24no—
642C4—24no—
643C4—44no—
644C4—44no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—82no—
649C4#242612yes—
650C4#243111yescharacteristic
651C4#243611yescharacteristic
652C4#244812yes—
653C4#254511yescharacteristic
654C4#254811yescharacteristic
655C2—84no—
656C2#1402611yescharacteristic
657C2#2089811yescharacteristic
658C2#2090011yescharacteristic
659C2#2163212yes—
660C2#2221612yes—
661C2#2253212yes—
662C2#2253312yes—
663C2#2320612yes—
664C2#2320712yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#446C2semidirect
3. C2×#1985C2semidirect
6. #11042C2semidirect
7. #11042C2semidirect
8. #10922C2semidirect
9. #10921C2semidirect
10. #10832C2semidirect
11. #10693C2semidirect
446. C2×C8#124semidirect

Maximal subgroups

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

subgroupindex
C22×#4462
C2×C4×#1242
C2×#19852
C2×#19552
#110422
#109222
#109212
#108322
#106932