Finite Group Databasebeta

← #363598

Subgroups — #363598

Order 512.

1678 conjugacy classes of subgroups.

The subgroup lattice has 747 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#117C211yesmaximal, characteristic
2C2×#1917C212yesmaximal, complemented
3#11346C212yesmaximal, complemented
4#11042C212yesmaximal
5#10922C212yesmaximal
6#10832C211yesmaximal, characteristic, complemented
7#10795C211yesmaximal, characteristic, complemented
8#10795C211yesmaximal, characteristic, complemented
9#10712C211yesmaximal, characteristic
10#10712C211yesmaximal, characteristic
11#10693C211yesmaximal, characteristic, complemented
12C22×#117C411yescharacteristic
13C22×#117C411yescharacteristic
14C23×C42C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×#97C2212yes—
19C4×#117—28no—
20C2×C4×C4⋊C4C411yescharacteristic
21C2×C4×C4⋊C4C411yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C411yescharacteristic
25C2×C4×C22⋊C4C411yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×#376C411yescharacteristic
29C2×#376C411yescharacteristic
30C2×C43C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#1917—28no—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44C2×#336C2212yes—
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#117—24no—
53C2×#117—24no—
54C2×#117—24no—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—24no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C42—24no—
67C22×C42—24no—
68C22×C42—22no—
69C22×C42—22no—
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×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×C4⋊C8—24no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—42no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—42no—
94C2×C22⋊C8—42no—
95C2×#97—24no—
96C2×#97—24no—
97C2×#97—42no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4D418yes—
102C4×C22⋊C4—24no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—24no—
105C4×C22⋊C4—24no—
106#376—24no—
107#376—24no—
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115C2×#95C2×C411yescharacteristic
116C43—24no—
117C43—24no—
118#336—24no—
119#336—24no—
120#336—24no—
121#336—24no—
122#336—24no—
123#336—24no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—42no—
129#336—42no—
130C25C22×C411yescharacteristic
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
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—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—41no—
152C23×C4—41no—
153C23×C4M1614yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4#4114yes—
168C23×C4C2411yescharacteristic
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
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#4114yes—
196C22×C8#4114yes—
197#117—232no—
198#117—232no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4C22⋊C414yes—
209C2×C4⋊C4C22⋊C414yes—
210C2×C4⋊C4M1614yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4M1614yes—
223C2×C42—216no—
224C2×C42—216no—
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—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—41no—
270C2×C42—41no—
271C2×C42—41no—
272C2×C42—41no—
273C2×C42M1614yes—
274C2×C42C2×D414yes—
275C2×C42C2×D414yes—
276C2×C42C2×D414yes—
277C2×C42#4112yes—
278C2×C42#4112yes—
279C4⋊C8—48no—
280C22⋊C8—48no—
281C22⋊C8—48no—
282#97—48no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95C22⋊C414yes—
290#95C22⋊C414yes—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24#11712yes—
304C24C2×M1612yes—
305C24C2×M1612yes—
306C24C23×C411yescharacteristic
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—28no—
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—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—28no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
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—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—44no—
407C22×C4C2×C22⋊C412yes—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C412yes—
410C22×C4C2×C22⋊C412yes—
411C22×C4C2×C22⋊C412yes—
412C22×C4C2×C22⋊C412yes—
413C22×C4#11712yes—
414C22×C4#11712yes—
415C22×C4#11712yes—
416C22×C4C2×M1612yes—
417C22×C4C2×M1612yes—
418C22×C4C2×M1612yes—
419C22×C4C2×M1612yes—
420C22×C4C2×M1612yes—
421C22×C4C2×M1612yes—
422C22×C4C2×M1612yes—
423C22×C4C2×M1612yes—
424C22×C4C2×M1612yes—
425C22×C4C2×M1612yes—
426C22×C4#13112yes—
427C22×C4#13112yes—
428C22×C4C22×D412yes—
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4H(5,𝔽2)11yescharacteristic
435C22×C4H(5,𝔽2)11yescharacteristic
436C22×C42−1+411yescharacteristic
437C22×C42−1+411yescharacteristic
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
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—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C22⋊C4—28no—
471C22⋊C4—28no—
472C22⋊C4—28no—
473C22⋊C4—28no—
474C42—28no—
475C42—28no—
476C42—28no—
477C42—28no—
478C42—28no—
479C42—24no—
480C42—24no—
481C42—24no—
482C42—24no—
483C42—24no—
484C42—24no—
485C42—24no—
486C42—24no—
487C42—28no—
488C42—44no—
489C42—44no—
490C42—44no—
491C42—44no—
492C42—42no—
493C42—42no—
494C42—42no—
495C42—42no—
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—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23#43214yes—
531C23C22×C22⋊C411yescharacteristic
532C23C2×#11711yescharacteristic
533C23C2×#11711yescharacteristic
534C23C4×#4111yescharacteristic
535C23C4×#4111yescharacteristic
536C23#52011yescharacteristic
537C23#52011yescharacteristic
538C23C22×M1611yescharacteristic
539C23C22×M1611yescharacteristic
540C23C22×M1611yescharacteristic
541C23C22×M1611yescharacteristic
542C23C2×#13111yescharacteristic
543C23C2×#13111yescharacteristic
544C23#56811yescharacteristic
545C23#56811yescharacteristic
546C2×C4—232no—
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—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—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—41no—
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—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4#40714yes—
640C2×C4#40714yes—
641C2×C4#43214yes—
642C2×C4#43314yes—
643C2×C4#52512yes—
644C2×C4#52512yes—
645C2×C4#52912yes—
646C2×C4#52912yes—
647C2×C4#54011yescharacteristic
648C2×C4#54111yescharacteristic
649C2×C4#54211yescharacteristic
650C2×C4#54211yescharacteristic
651C2×C4#54612yes—
652C2×C4#54712yes—
653C2×C4#54812yes—
654C2×C4#55012yes—
655C2×C4#55112yes—
656C2×C4#55112yes—
657C2×C4#55212yes—
658C2×C4#55212yes—
659C8—44no—
660C8—44no—
661C8—82no—
662C8—82no—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22#228011yescharacteristic
679C22#229811yescharacteristic
680C22#231411yescharacteristic
681C22#233911yescharacteristic
682C22#234111yescharacteristic
683C22#234311yescharacteristic
684C22#234311yescharacteristic
685C22C2×#40711yescharacteristic
686C22C2×#40711yescharacteristic
687C22#288011yescharacteristic
688C22#288011yescharacteristic
689C22C2×#43211yescharacteristic
690C22C2×#43211yescharacteristic
691C22C2×#43311yescharacteristic
692C22#292011yescharacteristic
693C22#292011yescharacteristic
694C22#292111yescharacteristic
695C22#292411yescharacteristic
696C22#292411yescharacteristic
697C22#296511yescharacteristic
698C22#296511yescharacteristic
699C22#296511yescharacteristic
700C22#296511yescharacteristic
701C22#297011yescharacteristic
702C22#297011yescharacteristic
703C22#297011yescharacteristic
704C22#297111yescharacteristic
705C22#298911yescharacteristic
706C22#298911yescharacteristic
707C22#298911yescharacteristic
708C22#298911yescharacteristic
709C22#299011yescharacteristic
710C22#299011yescharacteristic
711C22#299011yescharacteristic
712C22#299111yescharacteristic
713C4—24no—
714C4—24no—
715C4—24no—
716C4—24no—
717C4—42no—
718C4—42no—
719C4—42no—
720C4—42no—
721C4—42no—
722C4—42no—
723C4—42no—
724C4—42no—
725C4—42no—
726C4#247112yes—
727C4#247512yes—
728C4#247712yes—
729C4#247912yes—
730C2—44no—
731C2#1394411yescharacteristic
732C2#2071511yescharacteristic
733C2#2071611yescharacteristic
734C2#2143111yescharacteristic
735C2#2143511yescharacteristic
736C2#2192411yescharacteristic
737C2#2192611yescharacteristic
738C2#2247111yescharacteristic
739C2#2247311yescharacteristic
740C2#2253211yescharacteristic
741C2#2253211yescharacteristic
742C2#2259011yescharacteristic
743C2#2259111yescharacteristic
744C2#2259111yescharacteristic
745C2#2259211yescharacteristic
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×#1917C2semidirect
3. #11346C2semidirect
6. #10832C2semidirect
7. #10795C2semidirect
8. #10795C2semidirect
11. #10693C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1172
C2×#19172
#113462
#110422
#109222
#108322
#107952
#107122
#106932