Finite Group Databasebeta

← #190491

Subgroups — #190491

Order 512.

1274 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2#20716C212yesmaximal, complemented
3#8704C212yesmaximal, complemented
4#8561C212yesmaximal, complemented
5#8560C211yesmaximal, characteristic, complemented
6#8558C211yesmaximal, characteristic, complemented
7#8301C211yesmaximal, characteristic
8#8301C211yesmaximal, characteristic
9#8155C212yesmaximal, complemented
10#8153C211yesmaximal, characteristic, complemented
11C4×#324C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C411yescharacteristic
14C22×#117C2211yescharacteristic
15C2×#433—22no—
16#2354—22no—
17#2354—22no—
18#2354—22no—
19#2354—22no—
20C2×#390C411yescharacteristic
21C2×#390C411yescharacteristic
22C2×#388C411yescharacteristic
23C2×#388C411yescharacteristic
24C2×#388C2211yescharacteristic
25C2×#388C2211yescharacteristic
26C2×#384C411yescharacteristic
27C2×#380C411yescharacteristic
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4C411yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1818—22no—
33#1798—22no—
34#1797C2212yes—
35#1797C2212yes—
36#1779C2212yes—
37#1779C2212yes—
38#1750—22no—
39#1749C2212yes—
40C4×C22⋊C8C2212yes—
41C2×#324C2211yescharacteristic
42C2×#324C2211yescharacteristic
43C2×#324C2211yescharacteristic
44C2×#324C2211yescharacteristic
45C2×#323C2212yes—
46C2×#323C2212yes—
47#1280C2212yes—
48#1280C2212yes—
49#1280C2212yes—
50#1280C2212yes—
51C23×D4C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4C2311yescharacteristic
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—24no—
59C2×#117—21no—
60C2×#117—21no—
61C2×#117—21no—
62C2×#117—21no—
63C2×#117—44no—
64C2×#117—44no—
65C2×#117D412yes—
66C2×#117D412yes—
67C2×#117D412yes—
68C2×#117D412yes—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C22⋊C4C2×C411yescharacteristic
83C22×C22⋊C4C2×C411yescharacteristic
84C22×C22⋊C4C2×C411yescharacteristic
85C22×C22⋊C4C2×C411yescharacteristic
86C22×C22⋊C4C2×C411yescharacteristic
87C22×C22⋊C4C2311yescharacteristic
88C22×C42C2×C411yescharacteristic
89C22×C42C2×C411yescharacteristic
90C22×C42C2×C411yescharacteristic
91C22×C42C2×C411yescharacteristic
92C22×C42C2×C411yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#433—48no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8C2312yes—
103C2×C22⋊C8C2312yes—
104C2×C22⋊C8C2312yes—
105C2×C22⋊C8C2312yes—
106C2×#97—22no—
107C2×C4×C8—22no—
108#390—22no—
109#390—22no—
110#390—22no—
111#390—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#388—22no—
117#388—22no—
118#388—22no—
119#388—22no—
120#384—22no—
121#384—22no—
122#380—22no—
123#380—22no—
124#380—22no—
125#380—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C2×#95C2×C411yescharacteristic
135C2×#95C2×C411yescharacteristic
136C2×#95C2×C411yescharacteristic
137C2×#95C2×C411yescharacteristic
138C2×#95C2311yescharacteristic
139C2×#95C2311yescharacteristic
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144#324—22no—
145#324—22no—
146#324—22no—
147#324—22no—
148#323—24no—
149#323—24no—
150C25C22×C411yescharacteristic
151C25C22×C411yescharacteristic
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—41no—
158C22×D4—41no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2411yescharacteristic
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C4×D4—28no—
190C4×D4—28no—
191C4×D4—28no—
192C4×D4—28no—
193#117—28no—
194#117—44no—
195#117—44no—
196#117—44no—
197#117—44no—
198#117—44no—
199#117—44no—
200#117—44no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—81no—
220C2×C4⋊C4—81no—
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C22⋊C411yescharacteristic
224C2×C4⋊C4C22⋊C412yes—
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4#4112yes—
228C2×C4⋊C4#4112yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—81no—
248C2×C22⋊C4—81no—
249C2×C22⋊C4—81no—
250C2×C22⋊C4—81no—
251C2×C22⋊C4C22⋊C412yes—
252C2×C22⋊C4C22⋊C412yes—
253C2×C22⋊C4C22⋊C412yes—
254C2×C22⋊C4C22⋊C412yes—
255C2×C22⋊C4#4114yes—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—41no—
273C2×C42—41no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C4⋊C8—44no—
281C4⋊C8—44no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8—44no—
291C22⋊C8—44no—
292C22⋊C8—44no—
293C22⋊C8—44no—
294#97—44no—
295C4×C8—44no—
296#95—22no—
297#95—22no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—22no—
303#95—22no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—41no—
317C24—41no—
318C24—41no—
319C24—41no—
320C24—41no—
321C24—41no—
322C24—41no—
323C24—41no—
324C24C2×C22⋊C412yes—
325C24#11712yes—
326C24C23×C411yescharacteristic
327C2×D4—28no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
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—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
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—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C412yes—
410C22×C4C2×C22⋊C412yes—
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4#11712yes—
416C22×C4#11712yes—
417C22×C4#11712yes—
418C22×C4#13112yes—
419C22×C4#13112yes—
420C22×C4C22×D412yes—
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4112yes—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4#9914yes—
452C22⋊C4—28no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—82no—
464C22⋊C4—82no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C42—24no—
468C42—24no—
469C42—24no—
470C42—24no—
471C42—42no—
472C42—42no—
473C42—42no—
474C42—42no—
475C42—42no—
476C42—42no—
477C42—42no—
478C42—42no—
479C42—42no—
480C42—42no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
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—81no—
516C23—81no—
517C23—81no—
518C23—81no—
519C23—81no—
520C23—81no—
521C23—81no—
522C23—81no—
523C23—81no—
524C23—81no—
525C23—81no—
526C23—81no—
527C23—81no—
528C23—81no—
529C23—81no—
530C23—81no—
531C23C22×C22⋊C411yescharacteristic
532C23C2×#11711yescharacteristic
533C23C2×#11711yescharacteristic
534C23C2×#13111yescharacteristic
535C23C2×#13111yescharacteristic
536C23#56811yescharacteristic
537C23#56811yescharacteristic
538D4—416no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4C2×#9912yes—
617C2×C4C2×#9912yes—
618C2×C4C2×#9912yes—
619C2×C4#41012yes—
620C2×C4#43314yescomplemented
621C2×C4#52512yes—
622C2×C4#52512yes—
623C8—82no—
624C8—82no—
625C8—82no—
626C8—82no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22#228011yescharacteristic
662C22#288011yescharacteristic
663C22#288011yescharacteristic
664C22C22×#9911yescharacteristic
665C22C2×#41011yescharacteristic
666C22#288811yescharacteristic
667C22#288911yescharacteristic
668C22C2×#43311yescharacteristic
669C22#292111yescharacteristic
670C22#292411yescharacteristic
671C22#292411yescharacteristic
672C4—22no—
673C4—22no—
674C4—24no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—41no—
680C4—41no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—81no—
686C4—81no—
687C4—81no—
688C4—81no—
689C2—21no—
690C2—21no—
691C2—21no—
692C2—21no—
693C2—82no—
694C2—82no—
695C2#1741611yescharacteristic
696C2#1741811yescharacteristic
697C2#1741911yescharacteristic
698C2#1742011yescharacteristic
699C2#2071611yescharacteristic
700C2#2072211yescharacteristic
701C2#2073111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20716C2semidirect
3. #8704C2semidirect
4. #8561C2semidirect
5. #8560C2semidirect
6. #8558C2semidirect
9. #8155C2semidirect
10. #8153C2semidirect
11. C4×#324C2semidirect
620. C2×C4#433semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
#207162
#87042
#85612
#85602
#85582
#83012
#81552
#81532
C4×#3242