Finite Group Databasebeta

← #362412

Subgroups — #362412

Order 512.

1534 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1923C212yesmaximal
2C2×#1831C211yesmaximal, characteristic, complemented
3C2×#1797C211yesmaximal, characteristic
4C42×C22⋊C4C211yesmaximal, characteristic
5#11322C212yesmaximal, complemented
6#11042C212yesmaximal
7#10876C212yesmaximal, complemented
8#10832C211yesmaximal, characteristic, complemented
9#10795C211yesmaximal, characteristic, complemented
10#10712C211yesmaximal, characteristic
11C4×#336C211yesmaximal, characteristic, complemented
12C23×C42C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C4×C8C2212yes—
19C2×C4×C22⋊C4C411yescharacteristic
20C2×C4×C22⋊C4C411yescharacteristic
21C2×C4×C22⋊C4C411yescharacteristic
22C2×C4×C22⋊C4C411yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25C2×C43C2211yescharacteristic
26C2×C43C2211yescharacteristic
27#1923—28no—
28#1831—24no—
29#1797—24no—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
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
49C23×C8C2312yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C42—22no—
58C22×C42—24no—
59C22×C42—22no—
60C22×C42—42no—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×C4⋊C8—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—42no—
83C2×C4⋊C8—42no—
84C2×C4⋊C8—42no—
85C2×C4⋊C8—42no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—42no—
91C2×C22⋊C8—42no—
92C2×C4×C8—24no—
93C2×C4×C8—24no—
94C2×C4×C8—42no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4C818yescomplemented
100C4×C22⋊C4D414yes—
101C4×C22⋊C4Q814yes—
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
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114C43—22no—
115C43—22no—
116C43—42no—
117#336—24no—
118#336—24no—
119#336—22no—
120#336—22no—
121#336—24no—
122#336—24no—
123#336—24no—
124#336—24no—
125#336—22no—
126#336D414yes—
127C25C22×C411yescharacteristic
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—41no—
146C23×C4—41no—
147C23×C4—41no—
148C23×C4—41no—
149C23×C4C2×C814yes—
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×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×Q812yes—
165C23×C4C2411yescharacteristic
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
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×C8C2×D414yes—
193C22×C8#4114yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4C4⋊C414yes—
202C2×C22⋊C4C4⋊C414yes—
203C2×C22⋊C4C2×C814yes—
204C2×C22⋊C4C2×C814yes—
205C2×C22⋊C4M1614yes—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—22no—
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—21no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—48no—
242C2×C42—48no—
243C2×C42—41no—
244C2×C42—41no—
245C2×C42—41no—
246C2×C42—41no—
247C2×C42—41no—
248C2×C42—41no—
249C2×C42—41no—
250C2×C42—41no—
251C2×C42—41no—
252C2×C42—41no—
253C2×C42C2×C814yes—
254C2×C42C2×C814yes—
255C2×C42M1614yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×Q812yes—
260C2×C42C2×Q812yes—
261C2×C42#4114yes—
262C2×C42#4114yes—
263C2×C42#4112yes—
264C4⋊C8—48no—
265C4⋊C8—48no—
266C4⋊C8—48no—
267C22⋊C8—48no—
268C22⋊C8—48no—
269C4×C8—48no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95C4⋊C414yes—
278#95C4⋊C414yes—
279#95C2×C814yes—
280#95C2×C814yes—
281#95M1614yes—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24C2×C4⋊C412yes—
295C24C22×C812yes—
296C24C2×M1612yes—
297C24C23×C411yescharacteristic
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
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—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—28no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
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—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4C2×C4⋊C412yes—
393C22×C4C2×C4⋊C412yes—
394C22×C4C2×C4⋊C412yes—
395C22×C4C2×C4⋊C412yes—
396C22×C4C2×C4⋊C412yes—
397C22×C4#11712yes—
398C22×C4#11712yes—
399C22×C4#11712yes—
400C22×C4#11712yes—
401C22×C4C4×D412yes—
402C22×C4C4×D412yes—
403C22×C4C22×C812yes—
404C22×C4C22×C812yes—
405C22×C4C22×C812yes—
406C22×C4C22×C812yes—
407C22×C4C22×C812yes—
408C22×C4C22×C812yes—
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×C4#13112yes—
418C22×C4#13112yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×Q811yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4H(5,𝔽2)11yescharacteristic
427C22×C4H(5,𝔽2)12yes—
428C22×C42−1+411yescharacteristic
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
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—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—
457C22⋊C4—28no—
458C22⋊C4—28no—
459C22⋊C4—28no—
460C22⋊C4C4⋊C8116yescomplemented
461C42—24no—
462C42—24no—
463C42—22no—
464C42—24no—
465C42—24no—
466C42—24no—
467C42—22no—
468C42—22no—
469C42—24no—
470C42—24no—
471C42—22no—
472C42—24no—
473C42—28no—
474C42—44no—
475C42—44no—
476C42—44no—
477C42—44no—
478C42—44no—
479C42—44no—
480C42—44no—
481C42—44no—
482C42—42no—
483C42—42no—
484C42—44no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
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—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23C2×C4⋊C814yes—
520C23C22×C4⋊C411yescharacteristic
521C23C2×#11711yescharacteristic
522C23C2×#11711yescharacteristic
523C23C2×C4×D411yescharacteristic
524C23C4×#4111yescharacteristic
525C23#51811yescharacteristic
526C23#52011yescharacteristic
527C23C23×C811yescharacteristic
528C23C22×M1611yescharacteristic
529C23C22×M1611yescharacteristic
530C23C22×M1611yescharacteristic
531C23C2×#13111yescharacteristic
532C23C2×#13111yescharacteristic
533C23#56811yescharacteristic
534C23#56811yescharacteristic
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
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—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—416no—
596C2×C4—416no—
597C2×C4—41no—
598C2×C4—41no—
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—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4C2×C4⋊C814yes—
626C2×C4C2×C4⋊C814yes—
627C2×C4#42314yes—
628C2×C4#43114yes—
629C2×C4#43114yes—
630C2×C4C8×D414yes—
631C2×C4#43614yes—
632C2×C4#52912yes—
633C2×C4#53212yes—
634C2×C4#53212yes—
635C2×C4#53312yes—
636C2×C4#53812yes—
637C2×C4#53912yes—
638C2×C4D4211yescharacteristic
639C2×C4#54612yes—
640C2×C4#54612yes—
641C2×C4#54711yescharacteristic
642C2×C4D4×Q811yescharacteristic
643C2×C4#55112yes—
644C2×C4#55212yes—
645C2×C4#55411yescharacteristic
646C2×C4#55412yes—
647C2×C4#55512yes—
648C8—44no—
649C8—44no—
650C8—82no—
651C8—82no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22#228511yescharacteristic
668C22#229311yescharacteristic
669C22#231811yescharacteristic
670C22#234211yescharacteristic
671C22#234611yescharacteristic
672C22D4×C4⋊C411yescharacteristic
673C22#234911yescharacteristic
674C22C22×C4⋊C811yescharacteristic
675C22C2×#42311yescharacteristic
676C22#290611yescharacteristic
677C22#290611yescharacteristic
678C22C2×#43111yescharacteristic
679C22C2×#43111yescharacteristic
680C22#292011yescharacteristic
681C22#292011yescharacteristic
682C22#292211yescharacteristic
683C22#292311yescharacteristic
684C22#292411yescharacteristic
685C22#292411yescharacteristic
686C22C2×C8×D411yescharacteristic
687C22C2×#43611yescharacteristic
688C22D4×M1611yescharacteristic
689C22D4×M1611yescharacteristic
690C22C8×#4111yescharacteristic
691C22#296511yescharacteristic
692C22#297011yescharacteristic
693C22#297011yescharacteristic
694C22#297211yescharacteristic
695C22#297311yescharacteristic
696C22#297411yescharacteristic
697C22#297411yescharacteristic
698C22#298811yescharacteristic
699C22#298911yescharacteristic
700C22#299011yescharacteristic
701C22#299111yescharacteristic
702C4—24no—
703C4—24no—
704C4—24no—
705C4—24no—
706C4—24no—
707C4—24no—
708C4—24no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4#255312yes—
716C4#255512yes—
717C4#255612yes—
718C4#257412yes—
719C2—44no—
720C2#1391511yescharacteristic
721C2#2080811yescharacteristic
722C2#2081311yescharacteristic
723C2#2126411yescharacteristic
724C2#2127111yescharacteristic
725C2#2199611yescharacteristic
726C2#2199811yescharacteristic
727C2#2255211yescharacteristic
728C2#2255511yescharacteristic
729C2#2263211yescharacteristic
730C2#2263411yescharacteristic
731C2D4×C4⋊C811yescharacteristic
732C2#2265011yescharacteristic
733C2#2269611yescharacteristic
734C2#2269811yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C23 ▷ 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. #11322C2semidirect
7. #10876C2semidirect
8. #10832C2semidirect
9. #10795C2semidirect
11. C4×#336C2semidirect
99. C4×C22⋊C4C8semidirect
460. C22⋊C4C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×#19232
C2×#18312
C2×#17972
C42×C22⋊C42
#113222
#110422
#108762
#108322
#107952
#107122
C4×#3362