Finite Group Databasebeta

← #360884

Subgroups — #360884

Order 512.

1690 conjugacy classes of subgroups.

The subgroup lattice has 769 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1797C211yesmaximal, characteristic
2C2×#1749C211yesmaximal, characteristic
3C2×C4×#95C211yesmaximal, characteristic
4#11298C212yesmaximal, complemented
5#11255C212yesmaximal, complemented
6#10877C212yesmaximal, complemented
7#10798C212yesmaximal, complemented
8#10712C211yesmaximal, characteristic
9#10712C211yesmaximal, characteristic
10#10712C211yesmaximal, characteristic
11#10712C211yesmaximal, characteristic
12C23×C42C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×#97C2212yes—
21C22×#95C411yescharacteristic
22C22×#95C411yescharacteristic
23C22×#95C411yescharacteristic
24C22×#95C411yescharacteristic
25#1797—24no—
26#1749—24no—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C4×#95C411yescharacteristic
38C4×#95C411yescharacteristic
39C4×#95C411yescharacteristic
40C4×#95C411yescharacteristic
41C4×#95C2211yescharacteristic
42C4×#95C2211yescharacteristic
43C4×#95C2211yescharacteristic
44C4×#95C2211yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C22×C42—22no—
57C22×C42—22no—
58C22×C42—22no—
59C22×C42—22no—
60C22×C42—22no—
61C22×C42—22no—
62C22×C42—24no—
63C22×C42—24no—
64C22×C42—42no—
65C22×C42—42no—
66C22×C42—42no—
67C22×C42—42no—
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×C4⋊C8—24no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—42no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8D414yes—
94C2×C22⋊C8Q814yes—
95C2×#97—24no—
96C2×#97—24no—
97C2×#97—42no—
98C2×#95—24no—
99C2×#95—24no—
100C2×#95—24no—
101C2×#95—24no—
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
114C2×#95C2×C411yescharacteristic
115C2×#95C2×C411yescharacteristic
116C2×#95C2×C411yescharacteristic
117C2×#95C2×C411yescharacteristic
118#336—24no—
119#336—24no—
120#336—24no—
121#336—24no—
122#336—42no—
123#336—42no—
124#336—42no—
125#336—42no—
126#336—42no—
127#336—42no—
128C25C22×C411yescharacteristic
129C23×C4—22no—
130C23×C4—22no—
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—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—41no—
160C23×C4—41no—
161C23×C4—41no—
162C23×C4—41no—
163C23×C4—41no—
164C23×C4—41no—
165C23×C4—41no—
166C23×C4—41no—
167C23×C4—41no—
168C23×C4—41no—
169C23×C4—41no—
170C23×C4—41no—
171C23×C4—41no—
172C23×C4—41no—
173C23×C4M1614yes—
174C23×C4M1614yes—
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4C2×Q812yes—
190C23×C4#4112yes—
191C23×C4#4112yes—
192C23×C4#4112yes—
193C23×C4#4112yes—
194C23×C4C2411yescharacteristic
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
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—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8C2×D414yes—
222C22×C8C2×Q814yes—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
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—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—48no—
256C2×C42—48no—
257C2×C42—48no—
258C2×C42—48no—
259C2×C42—41no—
260C2×C42—41no—
261C2×C42—41no—
262C2×C42—41no—
263C2×C42—41no—
264C2×C42—41no—
265C2×C42—41no—
266C2×C42—41no—
267C2×C42—48no—
268C2×C42—48no—
269C2×C42#4114yes—
270C2×C42#4114yes—
271C4⋊C8—48no—
272C22⋊C8—28no—
273C22⋊C8—28no—
274C22⋊C8—44no—
275C22⋊C8—44no—
276#97—48no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95M1614yes—
292#95M1614yes—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24#11712yes—
302C24C4×D412yes—
303C24C4×Q812yes—
304C24C2×M1612yes—
305C24C2×M1612yes—
306C24#13112yes—
307C24#13112yes—
308C24C23×C411yescharacteristic
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—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—28no—
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—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4#11712yes—
438C22×C4#11712yes—
439C22×C4#11712yes—
440C22×C4#11712yes—
441C22×C4#11712yes—
442C22×C4C4×D412yes—
443C22×C4C4×Q812yes—
444C22×C4C22⋊Q814yes—
445C22×C4#12314yes—
446C22×C4#12514yes—
447C22×C4#12614yes—
448C22×C4C2×M1612yes—
449C22×C4C2×M1612yes—
450C22×C4C2×M1612yes—
451C22×C4C2×M1612yes—
452C22×C4C2×M1612yes—
453C22×C4C2×M1612yes—
454C22×C4C2×M1612yes—
455C22×C4C2×M1612yes—
456C22×C4C2×M1612yes—
457C22×C4C2×M1612yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×Q811yescharacteristic
460C22×C4C2×#4111yescharacteristic
461C22×C4C2×#4111yescharacteristic
462C22×C4C2×#4111yescharacteristic
463C22×C4C2×#4111yescharacteristic
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C4H(5,𝔽2)12yes—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—42no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—42no—
490C2×C8—42no—
491C2×C8—42no—
492C2×C8—42no—
493C2×C8—42no—
494C2×C8—42no—
495C2×C8—42no—
496C42—24no—
497C42—24no—
498C42—24no—
499C42—24no—
500C42—44no—
501C42—44no—
502C42—44no—
503C42—44no—
504C42—44no—
505C42—44no—
506C42—42no—
507C42—42no—
508C42—42no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—42no—
513C42—42no—
514C23—22no—
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—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23#38714yes—
543C23#43214yes—
544C23#43314yes—
545C23#43514yes—
546C23#43614yes—
547C23#44614yes—
548C23#44614yes—
549C23C2×#11711yescharacteristic
550C23C2×#11711yescharacteristic
551C23C2×#11711yescharacteristic
552C23C2×C4×D411yescharacteristic
553C23C2×C4×Q811yescharacteristic
554C23#51811yescharacteristic
555C23#51811yescharacteristic
556C23C22×M1611yescharacteristic
557C23C22×M1611yescharacteristic
558C23C22×M1611yescharacteristic
559C23C22×M1611yescharacteristic
560C23C2×#13111yescharacteristic
561C23C2×#13111yescharacteristic
562C23#56811yescharacteristic
563C23#56811yescharacteristic
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—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
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—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—416no—
614C2×C4—416no—
615C2×C4—416no—
616C2×C4—416no—
617C2×C4—416no—
618C2×C4—416no—
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—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4#38714yes—
664C2×C4#43214yes—
665C2×C4C2×C22⋊Q811yescharacteristic
666C2×C4C2×#12311yescharacteristic
667C2×C4C2×#12511yescharacteristic
668C2×C4C2×#12611yescharacteristic
669C2×C4#52912yes—
670C2×C4#52912yes—
671C2×C4#53812yes—
672C2×C4#53812yes—
673C2×C4#54612yes—
674C2×C4#54612yes—
675C2×C4#55112yes—
676C2×C4#55112yes—
677C2×C4#55312yes—
678C2×C4#55312yes—
679C2×C4#55412yes—
680C2×C4#55412yes—
681C8—44no—
682C8—44no—
683C8—82no—
684C8—82no—
685C22—24no—
686C22—24no—
687C22—24no—
688C22—24no—
689C22—24no—
690C22—24no—
691C22—24no—
692C22—24no—
693C22—24no—
694C22—24no—
695C22—24no—
696C22—24no—
697C22—24no—
698C22—24no—
699C22#191718yes—
700C22#228311yescharacteristic
701C22C2×#38711yescharacteristic
702C22#231211yescharacteristic
703C22#234211yescharacteristic
704C22#234211yescharacteristic
705C22#234511yescharacteristic
706C22#234511yescharacteristic
707C22C2×#43211yescharacteristic
708C22C2×#43211yescharacteristic
709C22C2×#43311yescharacteristic
710C22#292011yescharacteristic
711C22#292011yescharacteristic
712C22#292011yescharacteristic
713C22#292011yescharacteristic
714C22#292111yescharacteristic
715C22#292411yescharacteristic
716C22#292411yescharacteristic
717C22#292411yescharacteristic
718C22#292411yescharacteristic
719C22C2×#43511yescharacteristic
720C22C2×#43611yescharacteristic
721C22D4×M1611yescharacteristic
722C22D4×M1611yescharacteristic
723C22C2×#44611yescharacteristic
724C22C2×#44611yescharacteristic
725C22Q8×M1611yescharacteristic
726C22Q8×M1611yescharacteristic
727C22#297311yescharacteristic
728C22#297311yescharacteristic
729C22#297311yescharacteristic
730C22#297411yescharacteristic
731C22#297411yescharacteristic
732C22#297411yescharacteristic
733C22#297411yescharacteristic
734C22#297511yescharacteristic
735C4—24no—
736C4—24no—
737C4—42no—
738C4—42no—
739C4—42no—
740C4—42no—
741C4—42no—
742C4—42no—
743C4—42no—
744C4—42no—
745C4—42no—
746C4—42no—
747C4—42no—
748C4#244412yes—
749C4#245012yes—
750C4#250512yes—
751C4#251012yes—
752C2—28no—
753C2#1388311yescharacteristic
754C2#2077811yescharacteristic
755C2#2077911yescharacteristic
756C2C2×#191711yescharacteristic
757C2#2102311yescharacteristic
758C2#2190411yescharacteristic
759C2#2190611yescharacteristic
760C2#2255311yescharacteristic
761C2#2255411yescharacteristic
762C2#2255511yescharacteristic
763C2#2255511yescharacteristic
764C2#2262711yescharacteristic
765C2#2262711yescharacteristic
766C2#2262711yescharacteristic
767C2#2262911yescharacteristic
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
4. #11298C2semidirect
5. #11255C2semidirect
6. #10877C2semidirect
7. #10798C2semidirect

Maximal subgroups

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

subgroupindex
C2×#17972
C2×#17492
C2×C4×#952
#112982
#112552
#108772
#107982
#107122