Finite Group Databasebeta

← #395271

Subgroups — #395271

Order 512.

1926 conjugacy classes of subgroups.

The subgroup lattice has 753 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×#2057C211yesmaximal, characteristic, complemented
4C2×#2024C211yesmaximal, characteristic, complemented
5C2×#1863C212yesmaximal, complemented
6#11207C211yesmaximal, characteristic, complemented
7#11096C212yesmaximal
8#11050C211yesmaximal, characteristic, complemented
9#10970C212yesmaximal
10#10885C212yesmaximal, complemented
11#10712C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C22×C22⋊Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C22×#107C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2212yes—
21#2857—28no—
22C4×C22⋊Q8—24no—
23C4×C22⋊Q8—24no—
24C2×#391—21no—
25C2×#391C2211yescharacteristic
26C2×#389—21no—
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#387—21no—
30C2×#385—21no—
31C2×#384C2211yescharacteristic
32C2×#380—21no—
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4—21no—
38#2057—24no—
39#2024—24no—
40#1863—28no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#336C2211yescharacteristic
47C2×#336C2212yes—
48C2×#336C2211yescharacteristic
49C23×Q8—21no—
50C23×Q8C2311yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4C2311yescharacteristic
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—42no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—28no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—44no—
67C2×C22⋊Q8—44no—
68C2×C22⋊Q8Q814yescomplemented
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—24no—
72C2×C4×Q8—42no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C22⋊C4—21no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4—21no—
88C22×C42—21no—
89C22×C42—21no—
90C22×C42—24no—
91C22×C42—22no—
92C22×C42—22no—
93C22×C42—42no—
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C2×#107—24no—
99C2×#107—24no—
100C2×#107—42no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—42no—
107C2×C22⋊C8—24no—
108C2×C22⋊C8—24no—
109C2×C22⋊C8—42no—
110#391—28no—
111#391—22no—
112#391D414yes—
113#389—28no—
114#389—24no—
115#389—22no—
116#389D414yes—
117#389D418yes—
118#387—28no—
119#385—28no—
120#384—24no—
121#380—28no—
122#380—24no—
123#380—24no—
124#380—22no—
125#380—22no—
126C4×C4⋊C4—28no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C22⋊C4—28no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—22no—
137#340—24no—
138#340—24no—
139#340—24no—
140#340—24no—
141#340—24no—
142#336—22no—
143#336—24no—
144#336—22no—
145C25C2×D411yescharacteristic
146C22×Q8—24no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8—42no—
160C22×Q8C2×D412yes—
161C22×Q8C2×Q812yes—
162C22×Q8#4112yes—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—41no—
187C23×C4—41no—
188C23×C4—41no—
189C23×C4—41no—
190C23×C4Q1614yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×Q812yes—
197C23×C4#4112yes—
198C23×C4C2411yescharacteristic
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#4114yes—
213C4⋊Q8—48no—
214C4⋊Q8—48no—
215C22⋊Q8—28no—
216C22⋊Q8—28no—
217C22⋊Q8—28no—
218C22⋊Q8—28no—
219C22⋊Q8—432no—
220C4×Q8—48no—
221C4×Q8—48no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×Q812yes—
268C2×C4⋊C4#4112yes—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—48no—
290C2×C42—48no—
291C2×C42—41no—
292C2×C42—41no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42C2×D414yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300#107—48no—
301#103—216no—
302#103—48no—
303C22⋊C8—48no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311C24—21no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24C22⋊Q812yes—
324C24C2×Q1612yes—
325C24AGL(1,ℤ/8)12yes—
326C24C22×D411yescharacteristic
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—84no—
346C2×Q8—84no—
347C2×Q8—84no—
348C2×Q8—84no—
349C2×Q8C22⋊Q814yescomplemented
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—28no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4#12012yes—
447C22×C4#12012yes—
448C22×C4#12012yes—
449C22×C4#12012yes—
450C22×C4C22⋊Q812yes—
451C22×C4C2×Q1612yes—
452C22×C4C2×Q1612yes—
453C22×C4#13512yes—
454C22×C4AGL(1,ℤ/8)12yes—
455C22×C4#13712yes—
456C22×C4#13712yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×Q811yescharacteristic
461C22×C4C2×#4111yescharacteristic
462C22×C4C2×#4111yescharacteristic
463C22×C4C2×#4111yescharacteristic
464C22×C4C2×#4112yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C42−1+412yes—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C4⋊C4—216no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—416no—
495C4⋊C4—48no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—48no—
503C4⋊C4C22⋊Q814yescomplemented
504C22⋊C4—216no—
505C22⋊C4—48no—
506C22⋊C4—416no—
507C42—24no—
508C42—24no—
509C42—24no—
510C42—24no—
511C42—44no—
512C42—44no—
513C42—48no—
514C42—48no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23#47714yes—
550C23C2×#12011yescharacteristic
551C23C2×#12011yescharacteristic
552C23C2×C22⋊Q811yescharacteristic
553C23#52511yescharacteristic
554C23#53211yescharacteristic
555C23#53411yescharacteristic
556C23#53611yescharacteristic
557C23C22×Q1611yescharacteristic
558C23C2×#13511yescharacteristic
559C23C2×AGL(1,ℤ/8)11yescharacteristic
560C23C2×AGL(1,ℤ/8)11yescharacteristic
561C23C2×#13711yescharacteristic
562C23C2×#13711yescharacteristic
563C23#57511yescharacteristic
564C23#57511yescharacteristic
565Q8—28no—
566Q8—44no—
567Q8—44no—
568Q8—44no—
569Q8—816no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—416no—
597C2×C4—416no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—81no—
637C2×C4—82no—
638C2×C4—81no—
639C2×C4—82no—
640C2×C4—81no—
641C2×C4—82no—
642C2×C4—81no—
643C2×C4—82no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—82no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—82no—
650C2×C4#44914yes—
651C2×C4#45114yes—
652C2×C4#45114yes—
653C2×C4C2×C22⋊Q811yescharacteristic
654C2×C4C2×C22⋊Q811yescharacteristic
655C2×C4#52512yes—
656C2×C4#52511yescharacteristic
657C2×C4#52511yescharacteristic
658C2×C4#53212yes—
659C2×C4#53912yes—
660C2×C4#54212yes—
661C2×C4D4212yes—
662C2×C4D4212yes—
663C2×C4#54712yes—
664C2×C4#54812yes—
665C2×C4#54812yes—
666C2×C4D4×Q812yes—
667C2×C4#55112yes—
668C2×C4#55412yes—
669C8—44no—
670C8—82no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22#238811yescharacteristic
687C22#240311yescharacteristic
688C22#240911yescharacteristic
689C22#243211yescharacteristic
690C22#256411yescharacteristic
691C22#263811yescharacteristic
692C22#284811yescharacteristic
693C22C2×#45111yescharacteristic
694C22C2×#45111yescharacteristic
695C22C2×#44911yescharacteristic
696C22#300211yescharacteristic
697C22#300311yescharacteristic
698C22#300611yescharacteristic
699C22#300711yescharacteristic
700C22#300711yescharacteristic
701C22C2×#47711yescharacteristic
702C22#307611yescharacteristic
703C22#307811yescharacteristic
704C22#307911yescharacteristic
705C22#310411yescharacteristic
706C22#310711yescharacteristic
707C22#311211yescharacteristic
708C22#311511yescharacteristic
709C22#317011yescharacteristic
710C22#317211yescharacteristic
711C22#318011yescharacteristic
712C22#318211yescharacteristic
713C22#318811yescharacteristic
714C22#318911yescharacteristic
715C22#319411yescharacteristic
716C22#319511yescharacteristic
717C22#323011yescharacteristic
718C22#323111yescharacteristic
719C22#323811yescharacteristic
720C22#324011yescharacteristic
721C4—24no—
722C4—24no—
723C4—24no—
724C4—42no—
725C4—42no—
726C4—44no—
727C4—44no—
728C4—84no—
729C4—82no—
730C4—82no—
731C4—84no—
732C4#244912yes—
733C4#245212yes—
734C4#246112yes—
735C4#246312yes—
736C2—44no—
737C2#1472311yescharacteristic
738C2#2357111yescharacteristic
739C2#2357611yescharacteristic
740C2#2392011yescharacteristic
741C2#2392111yescharacteristic
742C2#2430211yescharacteristic
743C2#2430611yescharacteristic
744C2#2567411yescharacteristic
745C2#2567911yescharacteristic
746C2#2916011yescharacteristic
747C2#2916511yescharacteristic
748C2#3016911yescharacteristic
749C2#3017611yescharacteristic
750C2#3295511yescharacteristic
751C2#3295711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2057C2semidirect
4. C2×#2024C2semidirect
5. C2×#1863C2semidirect
6. #11207C2semidirect
8. #11050C2semidirect
10. #10885C2semidirect
68. C2×C22⋊Q8Q8semidirect
349. C2×Q8C22⋊Q8semidirect
503. C4⋊C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×C4×C22⋊Q82
C2×#20572
C2×#20242
C2×#18632
#112072
#110962
#110502
#109702
#108852
#107122