Finite Group Databasebeta

← #355607

Subgroups — #355607

Order 512.

1766 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×#1996C212yesmaximal
3C4×#380C211yesmaximal, characteristic
4#13753C211yesmaximal, characteristic
5#10912C214yesmaximal
6#10854C212yesmaximal
7#10801C214yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic, complemented
10C23×C42C2211yescharacteristic
11C4×C22⋊Q8—21no—
12C4×C22⋊Q8—21no—
13C4×C22⋊Q8—21no—
14C4×C22⋊Q8—21no—
15C4×C22⋊Q8—21no—
16C4×C22⋊Q8—21no—
17C4×C22⋊Q8—21no—
18C4×C22⋊Q8—21no—
19C2×#391C411yescharacteristic, complemented
20C2×#389C411yescharacteristic, complemented
21C2×#387C412yescomplemented
22C2×#385C2212yes—
23C2×#380C2212yes—
24C2×#380C2211yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C412yescomplemented
27C2×C4×C4⋊C4C411yescharacteristic, complemented
28C2×C4×C22⋊C4C2212yes—
29C2×C4×C22⋊C4C2212yes—
30C2×C4×C22⋊C4C2212yes—
31C2×#376—21no—
32#1996C4116yescomplemented
33C2×#342C2218yes—
34C2×#342C2218yes—
35C4⋊#95—21no—
36#1434—21no—
37#1434—21no—
38#1434—21no—
39C22⋊#95C2212yes—
40C22⋊#95C2212yes—
41C22⋊#95C2212yes—
42C4×#95—21no—
43C4×#95—21no—
44C23×Q8C2×C411yescharacteristic
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8C2×C418yes—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—44no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C412yes—
66C22×C22⋊C4C2312yes—
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2314yes—
69C22×C22⋊C4C2312yes—
70C22×C42—22no—
71C22×C42—22no—
72C22×C42—44no—
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C412yes—
75C22×C42C2×C412yes—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87#391—21no—
88#391—21no—
89#391—21no—
90#391—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#385—22no—
100#385—22no—
101#385—22no—
102#385—22no—
103#380—21no—
104#380—21no—
105#380—21no—
106#380—21no—
107#380—24no—
108#380C2×C418yes—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—44no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—24no—
123C4×C22⋊C4—44no—
124C4×C22⋊C4C2×C418yes—
125#376—44no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95C2×C412yes—
136C2×#95C2×C412yes—
137C2×#95C2×C412yes—
138C2×#95C2×C412yes—
139C2×#95C2×C412yes—
140#342—48no—
141#342C2×C4132yes—
142C25C2411yescharacteristic
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8C4214yescomplemented
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C412yes—
165C23×C4C22×C412yes—
166C23×C4C22×C412yes—
167C23×C4C22×C412yes—
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C414yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4#4114yes—
184C22⋊Q8—28no—
185C22⋊Q8—28no—
186C22⋊Q8—28no—
187C22⋊Q8—28no—
188C4×Q8—44no—
189C4×Q8—44no—
190C4×Q8—44no—
191C4×Q8—44no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4C4214yescomplemented
214C2×C4⋊C4C4218yescomplemented
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4C22×C418yes—
231C2×C22⋊C4C22×C418yes—
232C2×C22⋊C4C22×C418yes—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—28no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—44no—
251C2×C42—44no—
252C2×C42—44no—
253C2×C42—44no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—28no—
284#95—28no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24C23×C411yescharacteristic
297C24C23×C412yes—
298C24C22×D411yescharacteristic
299C24C22×D411yescharacteristic
300C24C2×#4112yes—
301C24H(5,𝔽2)12yes—
302C242−1+414yes—
303C242−1+412yes—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—48no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—48no—
310C2×Q8—42no—
311C2×Q8—42no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—48no—
375C22×C4—416no—
376C22×C4—48no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4C2×C4212yes—
398C22×C4C2×C4214yes—
399C22×C4C2×C4214yes—
400C22×C4C2×C4212yes—
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4#11712yes—
414C22×C4#11712yes—
415C22×C4C4×D414yes—
416C22×C4C4×D414yes—
417C22×C4C4×D414yes—
418C22×C4C4×D414yes—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—28no—
422C4⋊C4—28no—
423C4⋊C4—42no—
424C4⋊C4—416no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C22⋊C4—28no—
433C22⋊C4—28no—
434C22⋊C4—28no—
435C22⋊C4—28no—
436C22⋊C4—216no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—48no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—44no—
459C42—416no—
460C42—44no—
461C42—44no—
462C42—44no—
463C42—88no—
464C42#9914yes—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23—42no—
487C23C22×C4211yescharacteristic
488C23C22×C22⋊C411yescharacteristic
489C23C2×#11711yescharacteristic
490C23C2×C4×D412yes—
491C23C2×C4×D412yes—
492C23#51912yes—
493C23#51914yes—
494C23#52012yes—
495C23#52014yes—
496C23#53511yescharacteristic
497C23#53612yes—
498C23#53612yes—
499C23#53611yescharacteristic
500C23#53712yes—
501C23#54114yes—
502C23#54214yes—
503Q8—48no—
504Q8—48no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4C4×C22⋊C412yes—
586C2×C4C4×C22⋊C412yes—
587C2×C4C4×C22⋊C412yes—
588C2×C4C4×C22⋊C412yes—
589C2×C4C2×#9912yes—
590C2×C4C2×#9912yes—
591C2×C4C2×#9912yes—
592C2×C4#41012yes—
593C2×C4#41012yes—
594C2×C4#41012yes—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—21no—
602C22—21no—
603C22—22no—
604C22—22no—
605C22—21no—
606C22—22no—
607C22—22no—
608C22—21no—
609C22—21no—
610C22—22no—
611C22—22no—
612C22—21no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—21no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22C2×C4×C22⋊C411yescharacteristic
622C22#227612yes—
623C22#231011yescharacteristic
624C22#231111yescharacteristic
625C22#232012yes—
626C22#232314yes—
627C22#232612yes—
628C22#232712yes—
629C22C22×#9911yescharacteristic
630C22C2×#41011yescharacteristic
631C22C2×#41011yescharacteristic
632C22C2×#41011yescharacteristic
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—44no—
645C4—44no—
646C4—88no—
647C4—88no—
648C4—82no—
649C4—82no—
650C4C4×#9914yes—
651C2—22no—
652C2—24no—
653C2—22no—
654C2—22no—
655C2—22no—
656C2#1378611yescharacteristic
657C2C2×C4×#9911yescharacteristic
658C2C4×#41011yescharacteristic
659C2#2182511yescharacteristic
660C2#2182611yescharacteristic
661C2#2187411yescharacteristic
662C2#2187511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, prime radical, cosocleC25C24
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C22×C4×Q8C4semidirect
19. C2×#391C4semidirect
20. C2×#389C4semidirect
21. C2×#387C4semidirect
26. C2×C4×C4⋊C4C4semidirect
27. C2×C4×C4⋊C4C4semidirect
32. #1996C4semidirect
148. C22×Q8C42semidirect
213. C2×C4⋊C4C42semidirect
214. C2×C4⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C2×#19962
C4×#3802
#137532
#109122
#108542
#108012