Finite Group Databasebeta

← #368002

Subgroups — #368002

Order 512.

1958 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#435C211yesmaximal, characteristic
2C22×#424C211yesmaximal, characteristic, complemented
3C2×#1869C211yesmaximal, characteristic
4C2×#1850C212yesmaximal, complemented
5C2×#1850C211yesmaximal, characteristic, complemented
6#13839C211yesmaximal, characteristic
7#11322C212yesmaximal, complemented
8#11042C212yesmaximal, complemented
9#10877C212yesmaximal, complemented
10#10715C211yesmaximal, characteristic, complemented
11#10713C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C2×#435—24no—
18C2×#435—24no—
19C2×#435—24no—
20C2×#424—28no—
21C2×#424—24no—
22C22×C4⋊C8C2212yes—
23C22×C4⋊C8C2211yescharacteristic
24C22×C4⋊C8C2212yes—
25C22×C4⋊C8C2212yes—
26C22×C22⋊C8C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28C22×C22⋊C8C2212yes—
29C22×C22⋊C8C2211yescharacteristic
30C22×#97C2211yescharacteristic
31C2×C4×C22⋊C4C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33#1869—24no—
34#1850—28no—
35#1850—24no—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1441C412yes—
43#1440C411yescharacteristic
44#1440C411yescharacteristic
45#1439C412yes—
46#1438C411yescharacteristic
47#1438C411yescharacteristic
48C4×#95C2212yes—
49C23×D4C2×C411yescharacteristic
50C24×C4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—24no—
55C22×M16—42no—
56C22×M16D414yes—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—24no—
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—22no—
63C23×C8—22no—
64C23×C8—24no—
65C23×C8—42no—
66C23×C8—42no—
67C23×C8C2312yes—
68C23×C8C2311yescharacteristic
69C23×C8C2312yes—
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4D414yes—
78C2×C4×D4Q814yes—
79C2×#117—22no—
80C2×#117—22no—
81C2×#117—24no—
82C2×#117—24no—
83C22×C4⋊C4C2×C412yes—
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C22⋊C4C2×C411yescharacteristic
86C22×C22⋊C4C2×C411yescharacteristic
87C22×C22⋊C4C2×C412yes—
88C22×C22⋊C4C2×C411yescharacteristic
89C22×C22⋊C4C2×C412yes—
90C22×C22⋊C4C2×C411yescharacteristic
91C22×C22⋊C4C2×C411yescharacteristic
92C22×C22⋊C4C2×C411yescharacteristic
93C22×C42C2312yes—
94C22×C42C2311yescharacteristic
95C22×C42C2312yes—
96#435—232no—
97#424—232no—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—24no—
100C2×C4⋊C8—24no—
101C2×C4⋊C8—24no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—24no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—24no—
107C2×C4⋊C8—24no—
108C2×C4⋊C8—42no—
109C2×C4⋊C8D418yescomplemented
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—24no—
116C2×C22⋊C8—22no—
117C2×C22⋊C8—22no—
118C2×C22⋊C8—22no—
119C2×C22⋊C8—24no—
120C2×C22⋊C8—42no—
121C2×C22⋊C8D414yes—
122C2×#97—22no—
123C2×#97—22no—
124C2×#97—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—24no—
128C4×C22⋊C4—24no—
129C2×#95C2×C412yes—
130C2×#95C2×C412yes—
131C2×#95C2×C412yes—
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134C2×#95C2×C412yes—
135C2×#95C2×C412yes—
136#336—24no—
137#336—24no—
138#336—24no—
139#336—24no—
140#336—24no—
141#336—42no—
142C25C22×C411yescharacteristic
143C25C22×C411yescharacteristic
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—24no—
147C22×D4—42no—
148C22×D4C4⋊C414yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C412yes—
166C23×C4C22×C412yes—
167C23×C4C22×C412yes—
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×Q812yes—
174C23×C4C2×Q812yes—
175C23×C4#4112yes—
176C23×C4#4112yes—
177C23×C4C2411yescharacteristic
178C2×M16—28no—
179C2×M16—28no—
180C2×M16—44no—
181C2×M16—48no—
182C2×M16—48no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—28no—
187C22×C8—28no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—22no—
198C22×C8—21no—
199C22×C8—21no—
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—22no—
213C22×C8—48no—
214C22×C8—48no—
215C22×C8—48no—
216C22×C8—48no—
217C22×C8—44no—
218C22×C8—44no—
219C22×C8—44no—
220C22×C8—44no—
221C22×C8C2×D414yes—
222C22×C8C2×D412yes—
223C22×C8C2×D412yes—
224C22×C8C2×D412yes—
225C22×C8C2×D414yes—
226C22×C8C2×D412yes—
227C4×D4—28no—
228C4×D4—28no—
229C4×D4—28no—
230C4×D4—28no—
231#117—216no—
232#117—28no—
233#117—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4C4⋊C414yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4C4⋊C414yes—
259C2×C22⋊C4C4⋊C414yes—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42C2×D412yes—
275C2×C42C2×D414yes—
276C2×C42C2×Q812yes—
277C2×C42#4114yes—
278C4⋊C8—216no—
279C4⋊C8—28no—
280C4⋊C8—216no—
281C4⋊C8—48no—
282C22⋊C8—28no—
283C22⋊C8—28no—
284C22⋊C8—48no—
285C22⋊C8—44no—
286#97—28no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—42no—
292#95—42no—
293#95—42no—
294C24—22no—
295C24—22no—
296C24—21no—
297C24—21no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24C2×C4⋊C412yes—
308C24C2×C4⋊C412yes—
309C24C4×D412yes—
310C24C4×D412yes—
311C24#13114yes—
312C24C23×C411yescharacteristic
313C2×D4—28no—
314C2×D4—28no—
315C2×D4—44no—
316C2×D4—48no—
317C2×D4—48no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
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—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4C2×C4⋊C412yes—
390C22×C4C2×C4⋊C412yes—
391C22×C4C2×C4⋊C412yes—
392C22×C4C2×C4⋊C412yes—
393C22×C4C4×D414yes—
394C22×C4C4×D414yes—
395C22×C4C4×D412yes—
396C22×C4C4×D412yes—
397C22×C4#12014yes—
398C22×C4C22⋊Q814yes—
399C22×C4C22⋊Q814yes—
400C22×C4#12314yes—
401C22×C4#13114yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D412yes—
406C22×C4C22×Q811yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4112yes—
410C22×C4H(5,𝔽2)12yes—
411C22×C42−1+412yes—
412M16—416no—
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—416no—
428C2×C8—416no—
429C2×C8—42no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—42no—
434C2×C8—41no—
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—42no—
447C2×C8—42no—
448C2×C8#12014yes—
449C4⋊C4—216no—
450C4⋊C4—28no—
451C22⋊C4—216no—
452C22⋊C4—28no—
453C22⋊C4—28no—
454C22⋊C4—28no—
455C22⋊C4—28no—
456C22⋊C4—28no—
457C22⋊C4—48no—
458C22⋊C4—44no—
459C42—22no—
460C42—24no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—24no—
465C42—24no—
466C42—24no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—42no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—24no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—41no—
494C23—42no—
495C23—41no—
496C23—42no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23#38514yes—
512C23#42414yescomplemented
513C23C22×C4⋊C411yescharacteristic
514C23C2×C4×D411yescharacteristic
515C23C2×C4×D412yes—
516C23C2×C4×D411yescharacteristic
517C23#52012yes—
518C23C2×#13112yes—
519C23C2×#13112yes—
520C23#56812yes—
521C23#56812yes—
522D4—416no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
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—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
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—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4#38514yes—
583C2×C4C2×#12011yescharacteristic
584C2×C4C2×#12011yescharacteristic
585C2×C4C2×C22⋊Q811yescharacteristic
586C2×C4C2×C22⋊Q811yescharacteristic
587C2×C4C2×#12311yescharacteristic
588C2×C4#52511yescharacteristic
589C2×C4#52512yes—
590C2×C4#53411yescharacteristic
591C2×C4#53611yescharacteristic
592C2×C4#53912yes—
593C2×C4D4212yes—
594C2×C4D4212yes—
595C2×C4#54712yes—
596C2×C4#54712yes—
597C2×C4#54812yes—
598C2×C4D4×Q812yes—
599C2×C4#55312yes—
600C2×C4#55412yes—
601C8—24no—
602C8—42no—
603C8—44no—
604C8—42no—
605C8—42no—
606C8—82no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—28no—
611C22—24no—
612C22—44no—
613C22—44no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—42no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22C2×#38511yescharacteristic
626C22C4×#12011yescharacteristic
627C22#232211yescharacteristic
628C22D4×C4⋊C412yes—
629C22#236112yes—
630C22C2×#42411yescharacteristic
631C22#290511yescharacteristic
632C22#290612yes—
633C22#292911yescharacteristic
634C22#292912yes—
635C22#293011yescharacteristic
636C22#293212yes—
637C22#293312yes—
638C22#293312yes—
639C22#293312yes—
640C22#293512yes—
641C22#293512yes—
642C22#299212yes—
643C22#299212yes—
644C22#299412yes—
645C22#299512yes—
646C4—24no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—82no—
654C4#240311yescharacteristic
655C4#240611yescharacteristic
656C4#240911yescharacteristic
657C4#244312yes—
658C4#244912yes—
659C4#257911yescharacteristic
660C2—44no—
661C2—44no—
662C2#1401511yescharacteristic
663C2#2083611yescharacteristic
664C2#2083811yescharacteristic
665C2#2144312yes—
666C2#2207212yes—
667C2#2264712yes—
668C2#2265212yes—
669C2#2299112yes—
670C2#2299312yes—
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
2. C22×#424C2semidirect
4. C2×#1850C2semidirect
5. C2×#1850C2semidirect
7. #11322C2semidirect
8. #11042C2semidirect
9. #10877C2semidirect
10. #10715C2semidirect
11. #10713C2semidirect
109. C2×C4⋊C8D4semidirect
512. C23#424semidirect

Maximal subgroups

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

subgroupindex
C22×#4352
C22×#4242
C2×#18692
C2×#18502
#138392
#113222
#110422
#108772
#107152
#107132