Finite Group Databasebeta

← #361682

Subgroups — #361682

Order 512.

3910 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Q8C211yesmaximal, characteristic
2C2×#1981C211yesmaximal, characteristic, complemented
3C2×#1956C211yesmaximal, characteristic, complemented
4C2×#1954C211yesmaximal, characteristic
5C2×#1803C211yesmaximal, characteristic, complemented
6C2×#1787C214yesmaximal, complemented
7C2×C4×#103C211yesmaximal, characteristic, complemented
8#13906C211yesmaximal, characteristic
9#10970C214yesmaximal, complemented
10C24×Q8C2212yes—
11C2×D4×Q8—24no—
12C2×D4×Q8—24no—
13C2×D4×Q8—24no—
14C2×D4×Q8—24no—
15C2×D4×Q8—24no—
16C2×D4×Q8C418yescomplemented
17C22×C4⋊Q8—21no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—22no—
20C22×C22⋊Q8C2212yes—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4—21no—
23C22×C4×D4C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2212yes—
26C22×#103C2214yes—
27C22×#103C2214yes—
28C22×#103C2212yes—
29C22×C22⋊C8C2214yes—
30C22×C4×C8C2211yescharacteristic
31C2×#390—21no—
32C2×#384C2211yescharacteristic
33C2×#380C2214yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35#1981—24no—
36#1956—24no—
37#1954—24no—
38#1803—24no—
39#1787—216no—
40C4×#103—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1439—22no—
44C4⋊#95—21no—
45C23×Q8—24no—
46C23×Q8—24no—
47C23×Q8—24no—
48C23×Q8—24no—
49C23×Q8—22no—
50C23×Q8—24no—
51C23×Q8—28no—
52C23×Q8—28no—
53C23×Q8—44no—
54C23×Q8—44no—
55C23×Q8C2×C418yes—
56C23×Q8C2312yes—
57C23×Q8C2311yescharacteristic
58C23×D4D411yescharacteristic
59C24×C4—22no—
60C24×C4C2312yes—
61C23×C8C2312yes—
62C23×C8C2312yes—
63D4×Q8—232no—
64D4×Q8—232no—
65D4×Q8—232no—
66D4×Q8—232no—
67C2×C4⋊Q8—28no—
68C2×C4⋊Q8—28no—
69C2×C4⋊Q8—28no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8C2×C414yes—
74C2×C4⋊Q8D414yes—
75C2×C4⋊Q8D414yes—
76C2×C22⋊Q8—216no—
77C2×C22⋊Q8—216no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—216no—
84C2×C22⋊Q8C2×C418yes—
85C2×C4×Q8—22no—
86C2×C4×Q8—22no—
87C2×C4×Q8—22no—
88C2×C4×Q8C2×C414yes—
89C2×C4×Q8D414yes—
90C2×C4×Q8D414yes—
91C2×C4×D4—28no—
92C2×C4×D4—28no—
93C2×C4×D4—28no—
94C2×C4×D4—22no—
95C2×C4×D4—22no—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4C2×C414yes—
99C2×C4×D4D414yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—22no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4D412yes—
113C22×C42—21no—
114C22×C42—21no—
115C22×C42C2311yescharacteristic
116C2×C4⋊C8—22no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—22no—
119C2×#103—24no—
120C2×#103—24no—
121C2×#103—24no—
122C2×#103—28no—
123C2×#103—28no—
124C2×#103—28no—
125C2×#103—28no—
126C2×#103—28no—
127C2×#103—44no—
128C2×#103—44no—
129C2×#103D418yescomplemented
130C2×C22⋊C8—28no—
131C2×C22⋊C8—28no—
132C2×C22⋊C8—44no—
133C2×C4×C8—22no—
134C2×C4×C8—22no—
135C2×C4×C8—22no—
136#390—44no—
137#384—22no—
138#384—22no—
139#380—28no—
140#380—44no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C2×#95—24no—
144C2×#95—21no—
145#340—24no—
146#340—24no—
147C25C2×D412yes—
148C22×Q8—28no—
149C22×Q8—28no—
150C22×Q8—216no—
151C22×Q8—216no—
152C22×Q8—216no—
153C22×Q8—216no—
154C22×Q8—28no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—232no—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—48no—
169C22×Q8—48no—
170C22×Q8—416no—
171C22×Q8—416no—
172C22×Q8—416no—
173C22×Q8—416no—
174C22×Q8—416no—
175C22×Q8—416no—
176C22×Q8—416no—
177C22×Q8—416no—
178C22×Q8—48no—
179C22×Q8—48no—
180C22×Q8—416no—
181C22×Q8—48no—
182C22×Q8—48no—
183C22×Q8C22×C414yes—
184C22×Q8C22×C412yes—
185C22×Q8C2×D414yes—
186C22×Q8C2×D414yes—
187C22×Q8C2×D412yes—
188C22×Q8C2×D412yes—
189C22×Q8C2×D414yes—
190C22×Q8#4114yes—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4C22⋊C414yes—
196C22×D4SD1614yes—
197C22×D4Q1614yes—
198C23×C4—21no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—21no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—48no—
215C23×C4—48no—
216C23×C4C22×C414yes—
217C23×C4C2×D412yes—
218C23×C4C2×D414yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2411yescharacteristic
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—24no—
231C22×C8—22no—
232C22×C8—24no—
233C22×C8C2×D414yes—
234C4⋊Q8—264no—
235C4⋊Q8—28no—
236C4⋊Q8—28no—
237C4⋊Q8—28no—
238C4⋊Q8C22⋊C4116yescomplemented
239C22⋊Q8—216no—
240C22⋊Q8—216no—
241C22⋊Q8—2128no—
242C22⋊Q8—216no—
243C22⋊Q8—216no—
244C4×Q8—28no—
245C4×Q8—28no—
246C4×Q8—28no—
247C4×Q8C22⋊C4116yescomplemented
248C4×D4—264no—
249C4×D4—28no—
250C4×D4—28no—
251C4×D4—28no—
252C4×D4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—82no—
284C2×C4⋊C4C22⋊C414yes—
285C2×C4⋊C4C22×C412yes—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—84no—
297C2×C22⋊C4C22⋊C418yes—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42C22×C412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C4⋊C8—44no—
311#103—216no—
312#103—416no—
313#103—416no—
314#103—48no—
315C22⋊C8—416no—
316C4×C8—44no—
317#95—48no—
318#95—42no—
319C24—22no—
320C24—22no—
321C24—24no—
322C24—24no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24C2×C22⋊C414yes—
328C24C2×SD1614yes—
329C24C2×Q1614yes—
330C24C22×D411yescharacteristic
331C2×Q8—232no—
332C2×Q8—22no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—28no—
349C2×Q8—28no—
350C2×Q8—28no—
351C2×Q8—28no—
352C2×Q8—28no—
353C2×Q8—28no—
354C2×Q8—432no—
355C2×Q8—432no—
356C2×Q8—416no—
357C2×Q8—416no—
358C2×Q8—464no—
359C2×Q8—432no—
360C2×Q8—464no—
361C2×Q8—432no—
362C2×Q8—432no—
363C2×Q8—432no—
364C2×Q8—44no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—44no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8C2×C22⋊C418yes—
373C2×Q8C2×C22⋊C414yes—
374C2×Q8C4×D418yes—
375C2×Q8#12118yes—
376C2×Q8#12414yes—
377C2×Q8#12714yes—
378C2×D4—28no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4#103116yescomplemented
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—232no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—432no—
413C22×C4—432no—
414C22×C4—416no—
415C22×C4—416no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4C2×C22⋊C414yes—
442C22×C4C2×C22⋊C412yes—
443C22×C4C2×C22⋊C412yes—
444C22×C4#12012yes—
445C22×C4#12012yes—
446C22×C4#12012yes—
447C22×C4#12012yes—
448C22×C4C2×SD1612yes—
449C22×C4C2×Q1612yes—
450C22×C4#13712yes—
451C22×C4#13712yes—
452C22×C4C23×C411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4112yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—44no—
473C2×C8—44no—
474C4⋊C4—232no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—216no—
479C4⋊C4—216no—
480C4⋊C4—216no—
481C4⋊C4—28no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4C2×C22⋊C414yes—
492C22⋊C4—232no—
493C22⋊C4—216no—
494C22⋊C4—816no—
495C42—216no—
496C42—84no—
497C42C2×C22⋊C414yes—
498C42C2×C22⋊C414yes—
499C42#12014yes—
500C42#12014yes—
501C23—28no—
502C23—24no—
503C23—28no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—24no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23C2×#10318yes—
524C23C22×C22⋊C411yescharacteristic
525C23C2×#12011yescharacteristic
526C23C2×#12011yescharacteristic
527C23D4212yes—
528C23#54612yes—
529C23C22×SD1611yescharacteristic
530C23C22×Q1611yescharacteristic
531C23C2×#13711yescharacteristic
532C23C2×#13711yescharacteristic
533C23#57712yes—
534C23#57812yes—
535Q8—24no—
536Q8—28no—
537Q8—28no—
538Q8—24no—
539Q8—24no—
540Q8—216no—
541Q8—464no—
542Q8—44no—
543Q8—48no—
544Q8—44no—
545Q8—44no—
546Q8—44no—
547Q8#39018yescomplemented
548D4—232no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—464no—
563C2×C4—48no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4#37914yes—
593C2×C4#37914yes—
594C2×C4C2×#10314yes—
595C2×C4#41914yes—
596C2×C4#44814yes—
597C2×C4#44814yes—
598C2×C4#45114yes—
599C2×C4#45114yes—
600C2×C4C22×C22⋊C411yescharacteristic
601C2×C4C22×C22⋊C411yescharacteristic
602C2×C4C2×C4×D412yes—
603C2×C4#51812yes—
604C2×C4C2×#12011yescharacteristic
605C2×C4C2×#12011yescharacteristic
606C2×C4C2×#12112yes—
607C2×C4C2×#12411yescharacteristic
608C2×C4C2×#12711yescharacteristic
609C2×C4#53512yes—
610C2×C4#53511yescharacteristic
611C2×C4#53511yescharacteristic
612C2×C4D4212yes—
613C2×C4D4212yes—
614C2×C4D4212yes—
615C2×C4#54612yes—
616C2×C4#54612yes—
617C2×C4#54612yes—
618C8—44no—
619C8—82no—
620C22—28no—
621C22—216no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—48no—
628C22—44no—
629C22—48no—
630C22—48no—
631C22C2×#37911yescharacteristic
632C22D4×C22⋊C412yes—
633C22#240812yes—
634C22#243311yescharacteristic
635C22#243511yescharacteristic
636C22C22×#10311yescharacteristic
637C22C2×#41911yescharacteristic
638C22#289812yes—
639C22C2×#44811yescharacteristic
640C22C2×#44811yescharacteristic
641C22C2×#45111yescharacteristic
642C22C2×#45111yescharacteristic
643C22#301011yescharacteristic
644C22#301111yescharacteristic
645C22#301312yes—
646C22#327612yes—
647C22#327812yes—
648C22D4×SD1612yes—
649C22D4×Q1612yes—
650C22#329812yes—
651C22#329912yes—
652C22#332812yes—
653C22#332912yes—
654C4—28no—
655C4—24no—
656C4—44no—
657C4—48no—
658C4—44no—
659C4—84no—
660C4—162no—
661C4#178718yescomplemented
662C4C2×#39011yescharacteristic
663C4#230911yescharacteristic
664C4D4×C22⋊C412yes—
665C4#240812yes—
666C4#243311yescharacteristic
667C4#243511yescharacteristic
668C2—48no—
669C2#1390211yescharacteristic
670C2C2×#178711yescharacteristic
671C2#2060311yescharacteristic
672C2D4×#10312yes—
673C2#2243012yes—
674C2#2423912yes—
675C2#2424312yes—
676C2#2575611yescharacteristic
677C2#2576011yescharacteristic
678C2#2588411yescharacteristic
679C2#2588811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1981C2semidirect
3. C2×#1956C2semidirect
5. C2×#1803C2semidirect
6. C2×#1787C2semidirect
7. C2×C4×#103C2semidirect
9. #10970C2semidirect
16. C2×D4×Q8C4semidirect
129. C2×#103D4semidirect
238. C4⋊Q8C22⋊C4semidirect
247. C4×Q8C22⋊C4semidirect
384. C2×D4#103semidirect
547. Q8#390semidirect
661. C4#1787semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Q82
C2×#19812
C2×#19562
C2×#19542
C2×#18032
C2×#17872
C2×C4×#1032
#139062
#109702