Finite Group Databasebeta

← #196375

Subgroups — #196375

Order 512.

1701 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2#36626C211yesmaximal, characteristic
3C2×#1504C212yesmaximal
4C2×#1434C211yesmaximal, characteristic, complemented
5#9118C212yesmaximal, complemented
6#9116C212yesmaximal, complemented
7#9114C212yesmaximal, complemented
8#9112C212yesmaximal, complemented
9#8020C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C23×C42C2211yescharacteristic
12#2920—28no—
13C2×#432—21no—
14C2×#432C2212yes—
15C2×#431—21no—
16C2×#431C2212yes—
17C2×#423—21no—
18C2×#407C2212yes—
19C2×C4×M16—21no—
20#2855C411yescharacteristic, complemented
21#2855C411yescharacteristic, complemented
22#2855C2211yescharacteristic
23#2855C2211yescharacteristic
24#2854C411yescharacteristic, complemented
25#2854C411yescharacteristic, complemented
26#2854C2211yescharacteristic
27#2854C2211yescharacteristic
28#2853—21no—
29#2853—21no—
30#2849—21no—
31#2849—21no—
32C4×#123—21no—
33C4×#123—21no—
34C4×#123—21no—
35C4×#123—21no—
36#2297C411yescharacteristic, complemented
37#2297C411yescharacteristic, complemented
38#2296C411yescharacteristic, complemented
39#2296C411yescharacteristic, complemented
40#2280C2211yescharacteristic
41C22×#95—21no—
42C22×#95—21no—
43C22×#95C2212yes—
44#1504—28no—
45C2×#328C2214yes—
46C2×#327C2214yes—
47#1434—21no—
48#1434—21no—
49#1434C2212yes—
50#1434C2212yes—
51#1294C2214yes—
52#1294C2214yes—
53C22×#41C2×C411yescharacteristic
54C24×C4—22no—
55C24×C4D411yescharacteristic
56C24×C4D411yescharacteristic
57C24×C4C2311yescharacteristic
58C22×M16—22no—
59#525—24no—
60#525—24no—
61C2×#123—21no—
62C2×#123—21no—
63C2×#123—21no—
64C2×#123—21no—
65C2×#123—21no—
66C2×C22⋊Q8C2×C411yescharacteristic
67C2×C22⋊Q8C2311yescharacteristic
68C2×#121C2×C411yescharacteristic
69C2×#121C2311yescharacteristic
70C2×#120—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C411yescharacteristic
75C2×#117C2×C411yescharacteristic
76C22×C42—24no—
77C22×C42—24no—
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#432—28no—
91#432—28no—
92#431—28no—
93#431—28no—
94#423—28no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97#407—24no—
98#407D418yes—
99C2×C22⋊C8—22no—
100C2×C22⋊C8C2314yes—
101C4×M16—28no—
102C2×#97—22no—
103C2×C4×C8—22no—
104#391C2×C411yescharacteristic, complemented
105#391C2×C411yescharacteristic, complemented
106#391C2×C411yescharacteristic, complemented
107#391C2×C411yescharacteristic, complemented
108#390C2×C411yescharacteristic, complemented
109#390C2×C411yescharacteristic, complemented
110#390C2×C411yescharacteristic, complemented
111#390C2×C411yescharacteristic, complemented
112#389C2×C411yescharacteristic, complemented
113#389C2×C411yescharacteristic, complemented
114#389C2×C411yescharacteristic, complemented
115#389C2×C411yescharacteristic, complemented
116#389C2×C411yescharacteristic, complemented
117#389C2×C411yescharacteristic, complemented
118#389C2×C411yescharacteristic, complemented
119#389C2×C411yescharacteristic, complemented
120#388—21no—
121#388—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#387—21no—
129#387—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#386—21no—
141#386—21no—
142#386—21no—
143#386—21no—
144#385—21no—
145#385—21no—
146#385—21no—
147#385—21no—
148#385—21no—
149#385—21no—
150#385—21no—
151#385—21no—
152#381—21no—
153#381—21no—
154#381—21no—
155#381—21no—
156#381—21no—
157#380C2×C411yescharacteristic
158#380C2×C411yescharacteristic
159#380C2311yescharacteristic
160#380C2311yescharacteristic
161#379—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C4×C22⋊C4—21no—
167C4×C22⋊C4—21no—
168C4×C22⋊C4—21no—
169C4×C22⋊C4—21no—
170C4×C22⋊C4—21no—
171C4×C22⋊C4—21no—
172C2×#95—28no—
173C2×#95—22no—
174C2×#95—22no—
175C2×#95—24no—
176C2×#95—24no—
177C2×#95—22no—
178C2×#95—22no—
179C2×#95—22no—
180C2×#95—44no—
181C2×#95C2314yes—
182#328—28no—
183#327—28no—
184C25C2×D411yescharacteristic
185C2×#41—24no—
186C2×#41—42no—
187C2×#41—42no—
188C22×Q8C22×C411yescharacteristic
189C22×D4—21no—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4C22×C411yescharacteristic
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—21no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4C22⋊C411yescharacteristic
212C23×C4C22⋊C411yescharacteristic
213C23×C4C22⋊C411yescharacteristic
214C23×C4C22⋊C411yescharacteristic
215C23×C4C22⋊C411yescharacteristic
216C23×C4C22⋊C411yescharacteristic
217C23×C4C22⋊C411yescharacteristic
218C23×C4C22⋊C411yescharacteristic
219C23×C4C22×C411yescharacteristic
220C23×C4C22×C411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D414yes—
229C23×C4#4114yes—
230C23×C4C2411yescharacteristic
231C2×M16—28no—
232C2×M16—44no—
233C2×M16—44no—
234C22×C8—24no—
235#123—44no—
236#123—44no—
237#123—44no—
238#123—42no—
239#123—42no—
240#123—42no—
241#123—42no—
242C22⋊Q8—24no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245#121—24no—
246#121—22no—
247#121—22no—
248#120—44no—
249C4×D4—42no—
250C4×D4—44no—
251C4×D4—42no—
252#117—24no—
253#117—42no—
254#117—42no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42C22⋊C411yescharacteristic
312C2×C42C22⋊C411yescharacteristic
313C2×C42C22⋊C411yescharacteristic
314C2×C42C22⋊C411yescharacteristic
315C2×C42C22⋊C411yescharacteristic
316C2×C42C22⋊C411yescharacteristic
317C2×C42C22⋊C411yescharacteristic
318C2×C42C22⋊C411yescharacteristic
319C2×C42C22×C411yescharacteristic
320C2×C42C22×C411yescharacteristic
321C2×C42C22×C411yescharacteristic
322C2×C42C22×C411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C4⋊C8—28no—
328C4⋊C8—28no—
329C22⋊C8—28no—
330C22⋊C8—24no—
331C22⋊C8C2×D418yes—
332#97—28no—
333C4×C8—28no—
334#95—28no—
335#95—28no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—24no—
357#95—24no—
358#95—44no—
359#95—44no—
360#95—44no—
361C24—21no—
362C24—21no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—21no—
369C24C2×C22⋊C411yescharacteristic
370C24C2×C22⋊C411yescharacteristic
371C24#12114yes—
372C24C22×D411yescharacteristic
373#41—88no—
374C2×Q8—22no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×D4—22no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—42no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—416no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—416no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4#10414yes—
467C22×C4C2×C22⋊C411yescharacteristic
468C22×C4C2×C22⋊C411yescharacteristic
469C22×C4C2×C22⋊C411yescharacteristic
470C22×C4C2×C22⋊C411yescharacteristic
471C22×C4C2×C22⋊C411yescharacteristic
472C22×C4C2×C22⋊C411yescharacteristic
473C22×C4C2×C22⋊C411yescharacteristic
474C22×C4C2×C22⋊C411yescharacteristic
475C22×C4C2×C22⋊C411yescharacteristic
476C22×C4C2×C22⋊C411yescharacteristic
477C22×C4C4×D414yes—
478C22×C4C4×D414yes—
479C22×C4#12012yes—
480C22×C4#12012yes—
481C22×C4#12114yes—
482C22×C4#12312yes—
483C22×C4#12312yes—
484C22×C4AGL(1,ℤ/8)12yes—
485C22×C4#13712yes—
486C22×C4C23×C411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C2×#4112yes—
490M16—416no—
491C2×C8—28no—
492C2×C8—44no—
493C2×C8—44no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—41no—
526C22⋊C4—41no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—42no—
553C42—42no—
554C42—42no—
555C23—24no—
556C23—28no—
557C23—24no—
558C23—28no—
559C23—24no—
560C23—24no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—28no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—28no—
575C23—24no—
576C23—24no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23#38612yes—
585C23#38612yes—
586C23#41012yes—
587C23#41012yes—
588C23C2×#10412yes—
589C23#42112yes—
590C23C22×C22⋊C411yescharacteristic
591C23C2×#12112yes—
592C23C2×AGL(1,ℤ/8)11yescharacteristic
593C23C2×#13711yescharacteristic
594C23#57512yes—
595Q8—82no—
596D4—82no—
597D4—84no—
598D4—84no—
599D4—82no—
600D4—82no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—28no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—24no—
615C2×C4—28no—
616C2×C4—24no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—48no—
653C2×C4—48no—
654C2×C4—48no—
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—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—84no—
677C2×C4—832no—
678C2×C4—84no—
679C2×C4—84no—
680C2×C4—84no—
681C2×C4—84no—
682C2×C4#38612yes—
683C2×C4#38612yes—
684C2×C4#41712yes—
685C2×C4#41712yes—
686C2×C4C2×#10412yes—
687C2×C4C2×#10412yes—
688C2×C4C2×C4×D412yes—
689C2×C4C2×#12011yescharacteristic
690C2×C4C2×#12311yescharacteristic
691C8—48no—
692C22—216no—
693C22—216no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—21no—
698C22—21no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—21no—
704C22—21no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—22no—
711C22—22no—
712C22—22no—
713C22—22no—
714C22—81no—
715C22—81no—
716C22—81no—
717C22—81no—
718C22—81no—
719C22—81no—
720C22—81no—
721C22#187114yes—
722C22#189914yes—
723C22C2×#38611yescharacteristic
724C22C2×#41011yescharacteristic
725C22C2×#41011yescharacteristic
726C22C2×#41711yescharacteristic
727C22#289611yescharacteristic
728C22C22×#10411yescharacteristic
729C22C2×#42111yescharacteristic
730C22#305512yes—
731C22#305612yes—
732C4—22no—
733C4—22no—
734C4—22no—
735C4—22no—
736C4—21no—
737C4—22no—
738C4—22no—
739C4—21no—
740C4—22no—
741C4—41no—
742C4—41no—
743C4—41no—
744C4—41no—
745C4—88no—
746C4—81no—
747C4—81no—
748C4—81no—
749C4—81no—
750C4—81no—
751C4—81no—
752C4—81no—
753C2—22no—
754C2—24no—
755C2—24no—
756C2—22no—
757C2—161no—
758C2#1754111yescharacteristic
759C2#1754611yescharacteristic
760C2#1758811yescharacteristic
761C2#1759311yescharacteristic
762C2C2×#187111yescharacteristic
763C2#2097111yescharacteristic
764C2C2×#189911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2920C2semidirect
4. C2×#1434C2semidirect
5. #9118C2semidirect
6. #9116C2semidirect
7. #9114C2semidirect
8. #9112C2semidirect
9. #8020C2semidirect
20. #2855C4semidirect
21. #2855C4semidirect
24. #2854C4semidirect
25. #2854C4semidirect
36. #2297C4semidirect
37. #2297C4semidirect
38. #2296C4semidirect
39. #2296C4semidirect
104. #391C2×C4semidirect
105. #391C2×C4semidirect
106. #391C2×C4semidirect
107. #391C2×C4semidirect
108. #390C2×C4semidirect
109. #390C2×C4semidirect
110. #390C2×C4semidirect
111. #390C2×C4semidirect
112. #389C2×C4semidirect
113. #389C2×C4semidirect
114. #389C2×C4semidirect
115. #389C2×C4semidirect
116. #389C2×C4semidirect
117. #389C2×C4semidirect
118. #389C2×C4semidirect
119. #389C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
#366262
C2×#15042
C2×#14342
#91182
#91162
#91142
#91122
#80202