Finite Group Databasebeta

← #365639

Subgroups — #365639

Order 512.

1832 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#435C211yesmaximal, characteristic
2C2×#1955C211yesmaximal, characteristic
3C2×#1940C211yesmaximal, characteristic, complemented
4C2×#1851C212yesmaximal, complemented
5C2×#1850C212yesmaximal, complemented
6C2×#1842C211yesmaximal, characteristic, complemented
7C4×#390C211yesmaximal, characteristic
8#11042C212yesmaximal, complemented
9#11042C212yesmaximal, complemented
10#10835C211yesmaximal, characteristic, complemented
11#10693C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#435—24no—
16C2×#435—24no—
17C2×#435—24no—
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2212yes—
20C22×C4⋊C8C2212yes—
21C22×C4⋊C8C2212yes—
22C22×C4⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C22×C22⋊C8C2212yes—
26C22×C22⋊C8C2212yes—
27C22×#97C2211yescharacteristic
28C22×#97C2211yescharacteristic
29C2×#390C412yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2212yes—
32C2×C4×C22⋊C4C2212yes—
33C2×C43C2211yescharacteristic
34#1955—24no—
35#1940—24no—
36#1851—28no—
37#1850—28no—
38#1842—24no—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1437C411yescharacteristic
45#1437C411yescharacteristic
46#1437C411yescharacteristic
47#1437C411yescharacteristic
48C4⋊#95C412yes—
49C23×D4C2×C411yescharacteristic
50C24×C4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—24no—
56C22×M16—42no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—24no—
61C23×C8—42no—
62C23×C8C2312yes—
63C23×C8C2311yescharacteristic
64C23×C8C2312yes—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4D414yes—
73C2×C4×D4Q814yes—
74C22×C4⋊C4C2×C412yes—
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4C2×C412yes—
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C412yes—
84C22×C42C2×C411yescharacteristic
85C22×C42C2312yes—
86C22×C42C2312yes—
87C22×C42C2311yescharacteristic
88#435—232no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—24no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8—24no—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—24no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—24no—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—42no—
102C2×C4⋊C8—42no—
103C2×C4⋊C8D414yes—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—24no—
111C2×C22⋊C8—24no—
112C2×C22⋊C8—24no—
113C2×C22⋊C8—24no—
114C2×C22⋊C8—42no—
115C2×C22⋊C8—42no—
116C2×#97—22no—
117C2×#97—22no—
118C2×#97—22no—
119C2×#97—22no—
120C2×#97—22no—
121C2×#97D414yes—
122#390—28no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C22⋊C4—24no—
126C4×C22⋊C4—24no—
127C4×C22⋊C4—24no—
128C4×C22⋊C4—24no—
129C2×#95C2×C412yes—
130C2×#95C2×C412yes—
131C2×#95C2×C412yes—
132C2×#95C2×C412yes—
133C2×#95C2×C411yescharacteristic
134C2×#95C2×C411yescharacteristic
135C43—22no—
136C43—22no—
137#336—24no—
138#336—24no—
139#336—24no—
140#336—24no—
141#336D418yescomplemented
142C25C22×C411yescharacteristic
143C25C22×C411yescharacteristic
144C22×D4—24no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—42no—
148C22×D4C4⋊C414yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4C22×C412yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C412yes—
169C23×C4C22×C412yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×Q812yes—
176C23×C4C2×Q812yes—
177C23×C4C2411yescharacteristic
178C2×M16—28no—
179C2×M16—48no—
180C2×M16—48no—
181C2×M16—44no—
182C2×M16—44no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—28no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
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×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—48no—
213C22×C8—48no—
214C22×C8—44no—
215C22×C8—44no—
216C22×C8C2×D412yes—
217C22×C8C2×D412yes—
218C22×C8C2×D414yes—
219C22×C8C2×D412yes—
220C22×C8C2×D414yes—
221C22×C8C2×D412yes—
222C4×D4—28no—
223C4×D4—28no—
224C4×D4—28no—
225C4×D4—28no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4C4⋊C414yes—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
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—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4C4⋊C414yes—
248C2×C22⋊C4C4⋊C414yes—
249C2×C22⋊C4M1618yescomplemented
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42C2×D414yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×Q812yes—
272C2×C42#4114yes—
273C2×C42#4112yes—
274C2×C42#4112yes—
275C4⋊C8—28no—
276C4⋊C8—48no—
277C4⋊C8—48no—
278C4⋊C8—48no—
279C4⋊C8—44no—
280C22⋊C8—28no—
281C22⋊C8—28no—
282C22⋊C8—48no—
283C22⋊C8—48no—
284#97—28no—
285#97—44no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—22no—
291#95—22no—
292C24—22no—
293C24—21no—
294C24—21no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—21no—
301C24—21no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24C2×C4⋊C412yes—
308C24C2×C4⋊C412yes—
309C24C2×M1614yes—
310C24C23×C411yescharacteristic
311C2×D4—28no—
312C2×D4—28no—
313C2×D4—44no—
314C2×D4—48no—
315C2×D4—48no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
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—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
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×C4C2×C4⋊C412yes—
388C22×C4C2×C4⋊C412yes—
389C22×C4C2×C4⋊C412yes—
390C22×C4C2×C4⋊C412yes—
391C22×C4C4×D414yes—
392C22×C4C4×D412yes—
393C22×C4C4×D412yes—
394C22×C4C4×D412yes—
395C22×C4C4×D412yes—
396C22×C4C4×D414yes—
397C22×C4C2×M1614yes—
398C22×C4C2×M1614yes—
399C22×C4#13114yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D412yes—
404C22×C4C22×Q811yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4112yes—
408C22×C4H(5,𝔽2)12yes—
409C22×C42−1+412yes—
410M16—416no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—24no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—416no—
426C2×C8—42no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—42no—
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#12714yes—
446C4⋊C4—28no—
447C4⋊C4—48no—
448C22⋊C4—28no—
449C22⋊C4—28no—
450C22⋊C4—216no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C42—28no—
455C42—24no—
456C42—24no—
457C42—24no—
458C42—24no—
459C42—48no—
460C42—44no—
461C42—42no—
462C42—42no—
463C42—42no—
464C42—42no—
465C42#12014yes—
466C42C22⋊Q814yes—
467C42C22⋊Q814yes—
468C42#12314yes—
469C23—24no—
470C23—22no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23#42314yes—
511C23C22×C4⋊C411yescharacteristic
512C23C2×C4×D411yescharacteristic
513C23C2×C4×D412yes—
514C23C2×C4×D411yescharacteristic
515C23#52012yes—
516C23C22×M1612yes—
517C23C2×#13112yes—
518C23#56812yes—
519C23#56812yes—
520D4—416no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
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—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
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—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
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#38514yes—
579C2×C4#38514yes—
580C2×C4#43518yescomplemented
581C2×C4C2×#12011yescharacteristic
582C2×C4C2×C22⋊Q811yescharacteristic
583C2×C4C2×C22⋊Q811yescharacteristic
584C2×C4C2×#12311yescharacteristic
585C2×C4C2×#12711yescharacteristic
586C2×C4#53211yescharacteristic
587C2×C4#53212yes—
588C2×C4#53511yescharacteristic
589C2×C4#53611yescharacteristic
590C2×C4#54212yes—
591C2×C4D4212yes—
592C2×C4D4212yes—
593C2×C4#54712yes—
594C2×C4#54712yes—
595C2×C4#54812yes—
596C2×C4D4×Q812yes—
597C2×C4#55312yes—
598C2×C4#55412yes—
599C8—24no—
600C8—44no—
601C8—42no—
602C8—42no—
603C8—42no—
604C8—82no—
605C22—24no—
606C22—28no—
607C22—42no—
608C22—44no—
609C22—42no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—44no—
617C22—44no—
618C22—42no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22C2×#38511yescharacteristic
624C22C4×#12711yescharacteristic
625C22#232611yescharacteristic
626C22D4×C4⋊C412yes—
627C22#236112yes—
628C22C2×#42311yescharacteristic
629C22#290511yescharacteristic
630C22#290612yes—
631C22C2×#43512yes—
632C22#292911yescharacteristic
633C22#293011yescharacteristic
634C22#293212yes—
635C22D4×M1612yes—
636C22D4×M1612yes—
637C22D4×M1612yes—
638C22#293512yes—
639C22#293512yes—
640C22#298912yes—
641C22#299012yes—
642C22#299212yes—
643C22#299512yes—
644C4—24no—
645C4—24no—
646C4—28no—
647C4—42no—
648C4—42no—
649C4—44no—
650C4—44no—
651C4—82no—
652C4#242212yes—
653C4#243211yescharacteristic
654C4#243311yescharacteristic
655C4#245212yes—
656C4#254211yescharacteristic
657C4#256611yescharacteristic
658C2—44no—
659C2—82no—
660C2#1397711yescharacteristic
661C2#2083711yescharacteristic
662C2#2083811yescharacteristic
663C2#2168212yes—
664C2#2221612yes—
665C2#2264812yes—
666C2#2265212yes—
667C2#2298712yes—
668C2#2298812yes—
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
3. C2×#1940C2semidirect
4. C2×#1851C2semidirect
5. C2×#1850C2semidirect
6. C2×#1842C2semidirect
8. #11042C2semidirect
9. #11042C2semidirect
10. #10835C2semidirect
11. #10693C2semidirect
141. #336D4semidirect
249. C2×C22⋊C4M16semidirect
580. C2×C4#435semidirect

Maximal subgroups

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

subgroupindex
C22×#4352
C2×#19552
C2×#19402
C2×#18512
C2×#18502
C2×#18422
C4×#3902
#110422
#108352
#106932