Finite Group Databasebeta

← #361344

Subgroups — #361344

Order 512.

1590 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1851C212yesmaximal
2C2×#1831C211yesmaximal, characteristic, complemented
3C2×#1728C211yesmaximal, characteristic
4C42×C22⋊C4C211yesmaximal, characteristic
5#11255C212yesmaximal, complemented
6#11255C212yesmaximal, complemented
7#10832C211yesmaximal, characteristic, complemented
8#10795C211yesmaximal, characteristic, complemented
9#10795C211yesmaximal, characteristic, complemented
10#10715C212yesmaximal
11#10712C211yesmaximal, characteristic
12C23×M16C2212yes—
13C23×C42C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C2×C4×C22⋊C4C411yescharacteristic
18C2×C4×C22⋊C4C411yescharacteristic
19C2×C4×C22⋊C4C411yescharacteristic
20C2×C4×C22⋊C4C411yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23C2×C43C2211yescharacteristic
24C2×C43C2211yescharacteristic
25#1851—28no—
26#1831—24no—
27#1728—24no—
28C2×#336C2211yescharacteristic
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C4×#95C411yescharacteristic
41C4×#95C411yescharacteristic
42C4×#95C411yescharacteristic
43C4×#95C411yescharacteristic
44C4×#95C2211yescharacteristic
45C4×#95C2211yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—24no—
50C22×M16—24no—
51C22×M16—24no—
52C22×M16—24no—
53C22×M16—24no—
54C22×M16—42no—
55C22×M16—42no—
56C23×C8C2312yes—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C23×C8C2312yes—
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C42—24no—
65C22×C42—22no—
66C22×C42—22no—
67C22×C42—42no—
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×C4⋊C8—24no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—42no—
89C2×C4⋊C8—42no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C4×C22⋊C4—24no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4D414yes—
99C4×C22⋊C4Q814yes—
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C43—22no—
113C43—22no—
114C43—42no—
115#336—24no—
116#336—24no—
117#336—24no—
118#336—22no—
119#336—24no—
120#336—24no—
121#336—22no—
122#336—22no—
123#336—42no—
124#336—42no—
125#336—42no—
126#336D414yes—
127C25C22×C411yescharacteristic
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—41no—
145C23×C4—41no—
146C23×C4—41no—
147C23×C4—41no—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×Q812yes—
163C23×C4#4114yes—
164C23×C4#4114yes—
165C23×C4C2411yescharacteristic
166C2×M16—216no—
167C2×M16—48no—
168C2×M16—48no—
169C2×M16—48no—
170C2×M16—48no—
171C2×M16—48no—
172C2×M16—48no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8C2×D414yes—
200C22×C8#4114yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4C4⋊C414yes—
212C2×C22⋊C4C4⋊C414yes—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—21no—
251C2×C42—48no—
252C2×C42—48no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42—41no—
257C2×C42—41no—
258C2×C42—41no—
259C2×C42—41no—
260C2×C42—41no—
261C2×C42—41no—
262C2×C42—41no—
263C2×C42M1614yescomplemented
264C2×C42M1614yescomplemented
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×Q812yes—
269C2×C42C2×Q812yes—
270C2×C42#4112yes—
271C4⋊C8—48no—
272C4⋊C8—48no—
273C22⋊C8—48no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95C4⋊C414yes—
283#95C4⋊C414yes—
284#95M1614yescomplemented
285#95M1614yescomplemented
286C24—21no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24C2×C4⋊C412yes—
299C24#11712yes—
300C24#11712yes—
301C24C23×C411yescharacteristic
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
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—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
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—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4C2×C4⋊C412yes—
399C22×C4C2×C4⋊C412yes—
400C22×C4C2×C4⋊C412yes—
401C22×C4C2×C4⋊C412yes—
402C22×C4C2×C4⋊C412yes—
403C22×C4#11712yes—
404C22×C4#11712yes—
405C22×C4C4×D412yes—
406C22×C4C4×D412yes—
407C22×C4#12418yes—
408C22×C4#12518yes—
409C22×C4C2×M1612yes—
410C22×C4C2×M1612yes—
411C22×C4C2×M1612yes—
412C22×C4C2×M1612yes—
413C22×C4C2×M1612yes—
414C22×C4C2×M1612yes—
415C22×C4C2×M1612yes—
416C22×C4C2×M1612yes—
417C22×C4C2×M1612yes—
418C22×C4C2×M1612yes—
419C22×C4C2×M1612yes—
420C22×C4C2×M1612yes—
421C22×C4#13112yes—
422C22×C4#13112yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×Q811yescharacteristic
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4111yescharacteristic
430C22×C4H(5,𝔽2)11yescharacteristic
431C22×C4H(5,𝔽2)12yes—
432C22×C42−1+411yescharacteristic
433M16—432no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
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—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C22⋊C4—28no—
463C22⋊C4—28no—
464C22⋊C4—28no—
465C22⋊C4—28no—
466C42—22no—
467C42—24no—
468C42—24no—
469C42—22no—
470C42—22no—
471C42—24no—
472C42—22no—
473C42—24no—
474C42—28no—
475C42—44no—
476C42—44no—
477C42—44no—
478C42—44no—
479C42—44no—
480C42—44no—
481C42—44no—
482C42—44no—
483C42—42no—
484C42—42no—
485C42—42no—
486C42—42no—
487C42—42no—
488C42—42no—
489C42—44no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23#38214yes—
525C23C22×C4⋊C411yescharacteristic
526C23C2×#11711yescharacteristic
527C23C2×#11711yescharacteristic
528C23C2×C4×D411yescharacteristic
529C23C4×#4111yescharacteristic
530C23#51811yescharacteristic
531C23#52011yescharacteristic
532C23C22×M1611yescharacteristic
533C23C22×M1611yescharacteristic
534C23C22×M1611yescharacteristic
535C23C22×M1611yescharacteristic
536C23C2×#13111yescharacteristic
537C23C2×#13111yescharacteristic
538C23#56811yescharacteristic
539C23#56811yescharacteristic
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—28no—
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—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—416no—
599C2×C4—416no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4#38214yes—
633C2×C4#42314yes—
634C2×C4#42314yes—
635C2×C4#43514yes—
636C2×C4#43614yes—
637C2×C4#52512yes—
638C2×C4C2×#12412yes—
639C2×C4C2×#12512yes—
640C2×C4#52912yes—
641C2×C4#53812yes—
642C2×C4#53912yes—
643C2×C4D4211yescharacteristic
644C2×C4#54612yes—
645C2×C4#54612yes—
646C2×C4#54711yescharacteristic
647C2×C4D4×Q811yescharacteristic
648C2×C4#55112yes—
649C2×C4#55112yes—
650C2×C4#55212yes—
651C2×C4#55412yes—
652C2×C4#55411yescharacteristic
653C8—44no—
654C8—44no—
655C8—82no—
656C8—82no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22C2×#38211yescharacteristic
673C22#229011yescharacteristic
674C22#231811yescharacteristic
675C22#234211yescharacteristic
676C22#234311yescharacteristic
677C22D4×C4⋊C411yescharacteristic
678C22#234911yescharacteristic
679C22C2×#42311yescharacteristic
680C22C2×#42311yescharacteristic
681C22#290611yescharacteristic
682C22#290611yescharacteristic
683C22#292011yescharacteristic
684C22#292011yescharacteristic
685C22#292011yescharacteristic
686C22#292011yescharacteristic
687C22#292211yescharacteristic
688C22#292311yescharacteristic
689C22#292411yescharacteristic
690C22#292411yescharacteristic
691C22C2×#43511yescharacteristic
692C22C2×#43611yescharacteristic
693C22D4×M1611yescharacteristic
694C22D4×M1611yescharacteristic
695C22#296511yescharacteristic
696C22#296511yescharacteristic
697C22#297011yescharacteristic
698C22#297111yescharacteristic
699C22#297311yescharacteristic
700C22#297411yescharacteristic
701C22#297411yescharacteristic
702C22#297511yescharacteristic
703C22#298911yescharacteristic
704C22#298911yescharacteristic
705C22#299011yescharacteristic
706C22#299111yescharacteristic
707C4—24no—
708C4—24no—
709C4—24no—
710C4—24no—
711C4—24no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—42no—
719C4—42no—
720C4#247212yes—
721C4#247312yes—
722C4#248812yes—
723C4#250512yes—
724C2—44no—
725C2#1389611yescharacteristic
726C2#2075211yescharacteristic
727C2#2075411yescharacteristic
728C2#2110211yescharacteristic
729C2#2110811yescharacteristic
730C2#2199811yescharacteristic
731C2#2200211yescharacteristic
732C2#2255311yescharacteristic
733C2#2255511yescharacteristic
734C2#2259011yescharacteristic
735C2#2259211yescharacteristic
736C2#2264811yescharacteristic
737C2#2265011yescharacteristic
738C2#2269811yescharacteristic
739C2#2270011yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC25C22×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. C2×#1831C2semidirect
5. #11255C2semidirect
6. #11255C2semidirect
7. #10832C2semidirect
8. #10795C2semidirect
9. #10795C2semidirect
263. C2×C42M16semidirect
264. C2×C42M16semidirect
284. #95M16semidirect
285. #95M16semidirect

Maximal subgroups

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

subgroupindex
C2×#18512
C2×#18312
C2×#17282
C42×C22⋊C42
#112552
#108322
#107952
#107152
#107122