Finite Group Databasebeta

← #188342

Subgroups — #188342

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2374C211yesmaximal, characteristic
2#21932C212yesmaximal, complemented
3C2×#1468C212yesmaximal
4#8562C212yesmaximal, complemented
5#8561C212yesmaximal, complemented
6#8560C211yesmaximal, characteristic, complemented
7#8559C211yesmaximal, characteristic, complemented
8#8558C211yesmaximal, characteristic, complemented
9#8557C211yesmaximal, characteristic, complemented
10#7911C211yesmaximal, characteristic, complemented
11#7907C211yesmaximal, characteristic, complemented
12C22×#123C2211yescharacteristic
13C2×#407C2212yes—
14C22×C22⋊C8C2212yes—
15#2374—22no—
16#2374—22no—
17#2374—22no—
18#2374—22no—
19C2×#390C411yescharacteristic
20C2×#388C411yescharacteristic
21C2×#388C411yescharacteristic
22C2×#388C411yescharacteristic
23C2×#388C2211yescharacteristic
24C2×#387C411yescharacteristic
25C2×#387C2211yescharacteristic
26C2×#386C411yescharacteristic
27C2×C4×C4⋊C4C411yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C411yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1851—22no—
34#1851—22no—
35#1850—22no—
36#1850—22no—
37#1797C2212yes—
38#1797C2212yes—
39#1797C2212yes—
40#1797C2212yes—
41#1468—28no—
42C2×#324C2212yes—
43C2×#324C2212yes—
44C2×#323C2212yes—
45C2×#323C2212yes—
46#1280C2211yescharacteristic
47#1280C2211yescharacteristic
48#1280C2211yescharacteristic
49#1280C2211yescharacteristic
50#1280C2212yes—
51#1280C2212yes—
52C23×D4C2×C411yescharacteristic
53C24×C4C2×C411yescharacteristic
54C24×C4C2311yescharacteristic
55C22×M16—22no—
56C23×C8—22no—
57C2×#123—22no—
58C2×#123—22no—
59C2×#123—24no—
60C2×#123D414yes—
61C2×#123Q814yes—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C22⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C22⋊C4C2311yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88#407—24no—
89#407—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8C2312yes—
95C2×C22⋊C8C2312yes—
96C2×C22⋊C8C2312yes—
97C2×C22⋊C8C2312yes—
98#390—22no—
99#390—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#386—22no—
113#386—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4D412yes—
129C4×C22⋊C4D412yes—
130C4×C22⋊C4D412yes—
131C4×C22⋊C4D412yes—
132C2×#95C2×C411yescharacteristic
133C2×#95C2×C411yescharacteristic
134C2×#95C2×C411yescharacteristic
135C2×#95C2×C411yescharacteristic
136C2×#95C2×C411yescharacteristic
137C2×#95C2×C411yescharacteristic
138C2×#95C2×C411yescharacteristic
139C2×#95C2×C411yescharacteristic
140C2×#95C2311yescharacteristic
141C2×#95C2311yescharacteristic
142#336—22no—
143#336—22no—
144#336—22no—
145#336—22no—
146#324—24no—
147#324—24no—
148#323—24no—
149#323—24no—
150C25C22×C411yescharacteristic
151C25C22×C411yescharacteristic
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—81no—
155C22×D4—81no—
156C22×D4C4⋊C414yes—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—41no—
161C23×C4—42no—
162C23×C4—81no—
163C23×C4—81no—
164C23×C4—81no—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×Q812yes—
181C23×C4#4114yes—
182C23×C4C2411yescharacteristic
183C2×M16—44no—
184C2×M16—82no—
185C2×M16—82no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—44no—
191C22×C8—82no—
192C22×C8—82no—
193#123—28no—
194#123—28no—
195#123—28no—
196#123—28no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—81no—
210C2×C4⋊C4C4⋊C414yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×Q812yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—81no—
236C2×C22⋊C4C22⋊C412yes—
237C2×C22⋊C4C22⋊C412yes—
238C2×C22⋊C4C22⋊C412yes—
239C2×C22⋊C4C22⋊C412yes—
240C2×C22⋊C4C4⋊C414yes—
241C2×C22⋊C4C4⋊C414yes—
242C2×C22⋊C4C2×D412yes—
243C2×C22⋊C4C2×Q812yes—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—41no—
256C2×C42—42no—
257C2×C42—81no—
258C2×C42—81no—
259C2×C42—81no—
260C2×C42—81no—
261C2×C42—81no—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C4⋊C8—82no—
267C4⋊C8—82no—
268C4⋊C8—82no—
269C4⋊C8—82no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—22no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278C22⋊C8—44no—
279C22⋊C8—44no—
280#95—22no—
281#95—21no—
282#95—21no—
283#95—22no—
284#95—21no—
285#95—21no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95C22⋊C411yescharacteristic
291#95C22⋊C411yescharacteristic
292#95C22⋊C411yescharacteristic
293#95C22⋊C411yescharacteristic
294#95C22⋊C412yes—
295#95C22⋊C412yes—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24—41no—
310C24—41no—
311C24—41no—
312C24—41no—
313C24—41no—
314C24C2×C22⋊C412yes—
315C24C2×C4⋊C412yes—
316C24C2×C4⋊C412yes—
317C24#11712yes—
318C24C23×C411yescharacteristic
319C2×D4—28no—
320C2×D4—82no—
321C2×D4—82no—
322C2×D4—84no—
323C2×D4—84no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
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—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—84no—
389C22×C4—84no—
390C22×C4—84no—
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C412yes—
393C22×C4C2×C22⋊C412yes—
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C4⋊C412yes—
400C22×C4C2×C4⋊C412yes—
401C22×C4C2×C4⋊C412yes—
402C22×C4C2×C4⋊C412yes—
403C22×C4#11712yes—
404C22×C4#12114yes—
405C22×C4#12114yes—
406C22×C4C22⋊Q814yes—
407C22×C4C22⋊Q814yes—
408C22×C4#13112yes—
409C22×C4#13112yes—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×Q811yescharacteristic
413C22×C4C2×#4112yes—
414M16—88no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—88no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—82no—
436C22⋊C4—28no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—82no—
451C42—42no—
452C42—42no—
453C42—42no—
454C42—42no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42—42no—
459C42—81no—
460C42—81no—
461C42—81no—
462C42—81no—
463C42—81no—
464C42—81no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—81no—
499C23—81no—
500C23—81no—
501C23—81no—
502C23—81no—
503C23—81no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23—81no—
508C23—81no—
509C23—81no—
510C23—81no—
511C23—81no—
512C23—81no—
513C23—81no—
514C23#38014yes—
515C23C22×C22⋊C411yescharacteristic
516C23C22×C4⋊C411yescharacteristic
517C23C2×#11711yescharacteristic
518C23C2×#13111yescharacteristic
519C23C2×#13111yescharacteristic
520C23#56811yescharacteristic
521C23#56811yescharacteristic
522D4—88no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—84no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—84no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4#38012yes—
602C2×C4#38012yes—
603C2×C4#41312yes—
604C2×C4#41312yes—
605C2×C4C2×#12112yes—
606C2×C4C2×C22⋊Q812yes—
607C8—82no—
608C8—82no—
609C8—82no—
610C8—82no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22C2×#38011yescharacteristic
646C22#288011yescharacteristic
647C22#288011yescharacteristic
648C22#288311yescharacteristic
649C22#288311yescharacteristic
650C22C2×#41311yescharacteristic
651C22C2×#41311yescharacteristic
652C22#290611yescharacteristic
653C22#290611yescharacteristic
654C22#292211yescharacteristic
655C22#292311yescharacteristic
656C4—22no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—41no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—41no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—81no—
672C4—81no—
673C2—21no—
674C2—21no—
675C2—21no—
676C2—21no—
677C2—44no—
678C2—82no—
679C2#1742211yescharacteristic
680C2#1742411yescharacteristic
681C2#1742711yescharacteristic
682C2#1742911yescharacteristic
683C2#2063011yescharacteristic
684C2#2063311yescharacteristic
685C2#2063711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21932C2semidirect
4. #8562C2semidirect
5. #8561C2semidirect
6. #8560C2semidirect
7. #8559C2semidirect
8. #8558C2semidirect
9. #8557C2semidirect
10. #7911C2semidirect
11. #7907C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23742
#219322
C2×#14682
#85622
#85612
#85602
#85592
#85582
#85572
#79112
#79072