Finite Group Databasebeta

← #481840

Subgroups — #481840

Order 512.

1117 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36009C211yesmaximal, characteristic
2#20689C211yesmaximal, characteristic, complemented
3#17367C211yesmaximal, characteristic, complemented
4#11801C212yesmaximal, complemented
5#11799C212yesmaximal, complemented
6#11798C212yesmaximal, complemented
7#11797C212yesmaximal, complemented
8#11692C212yesmaximal, complemented
9C2×#1344C212yesmaximal, complemented
10C2×#547C411yescharacteristic, complemented
11C2×#546—21no—
12C2×#518C2211yescharacteristic
13#2973—21no—
14#2621C2211yescharacteristic
15#2617C2211yescharacteristic
16#2617C2211yescharacteristic
17#2547C411yescharacteristic, complemented
18#2543C411yescharacteristic, complemented
19#2543C411yescharacteristic, complemented
20#2535C411yescharacteristic, complemented
21#2486C411yescharacteristic, complemented
22#2486C411yescharacteristic, complemented
23#2427—21no—
24#2425—21no—
25#2425—21no—
26#2415—21no—
27#2412—21no—
28#2412—21no—
29#2392C2211yescharacteristic
30C2×#397C2211yescharacteristic
31C2×#397C2211yescharacteristic
32#2349C411yescharacteristic, complemented
33#2339—21no—
34#2308—21no—
35#1820—21no—
36#1820C2212yes—
37#1819—21no—
38#1819C2212yes—
39C2×#342—21no—
40C2×#342C2212yes—
41#1466—21no—
42#1465—21no—
43#1464C2212yes—
44#1462C2212yes—
45#1459—21no—
46C2×#323C2212yes—
47#1344C2214yes—
48#1344C2214yes—
49#1344C2214yes—
50#1343C2214yes—
51C22×#41—21no—
52C23×D4D411yescharacteristic
53#547—44no—
54#546—48no—
55C2×C4⋊Q8C2×C411yescharacteristic, complemented
56C2×#124—21no—
57C2×#123—21no—
58C2×#123—21no—
59C2×#123—21no—
60C2×#123—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121C2×C411yescharacteristic
67C2×#121C2×C411yescharacteristic, complemented
68C2×#120—21no—
69C2×#120—21no—
70#518—44no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C411yescharacteristic, complemented
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×C4×D4D411yescharacteristic
81C2×C4×D4D411yescharacteristic
82C2×#117D411yescharacteristic
83C2×#117C2311yescharacteristic
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C4⋊C4C2×C411yescharacteristic, complemented
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—22no—
89C22×C22⋊C4D411yescharacteristic
90C22×C22⋊C4D411yescharacteristic
91C22×C22⋊C4C2311yescharacteristic
92C22×C22⋊C4C2311yescharacteristic
93#435—44no—
94#432—22no—
95#407—22no—
96C2×C22⋊C8—22no—
97#398—21no—
98#398C2×C411yescharacteristic, complemented
99#398C2×C411yescharacteristic, complemented
100#398C2×C411yescharacteristic, complemented
101#397—21no—
102#397—21no—
103#397—21no—
104#397—21no—
105#397—21no—
106#397C2×C411yescharacteristic
107#397C2×C411yescharacteristic
108#397C2×C411yescharacteristic
109#397C2×C411yescharacteristic, complemented
110#397C2×C411yescharacteristic, complemented
111#397C2×C411yescharacteristic, complemented
112#397C2311yescharacteristic
113#397C2311yescharacteristic
114#397C2311yescharacteristic
115#396—21no—
116#394—21no—
117#394—21no—
118#394—21no—
119#394—21no—
120#394—21no—
121#394C2×C411yescharacteristic, complemented
122#392C2×C411yescharacteristic
123#392C2×C411yescharacteristic, complemented
124#392C2311yescharacteristic
125#391—21no—
126#391—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#388—21no—
131#388—21no—
132#386—21no—
133#386—21no—
134#386—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#386—44no—
141#385—21no—
142#385—21no—
143#382C2×C411yescharacteristic, complemented
144#381—21no—
145#381—21no—
146#381—22no—
147#380—21no—
148#380—21no—
149#380—21no—
150#380—21no—
151#380—21no—
152#380C2×C411yescharacteristic
153#380C2×C411yescharacteristic, complemented
154#379—22no—
155C4×C4⋊C4C2×C411yescharacteristic, complemented
156C4×C22⋊C4—21no—
157C2×#95—21no—
158C2×#95—21no—
159C2×#95C2×C411yescharacteristic
160C2×#95C2×C411yescharacteristic, complemented
161#342—22no—
162#342—22no—
163#342—22no—
164#342C2314yes—
165#324—22no—
166#323—22no—
167#323—22no—
168#323C2314yes—
169C25C2×D411yescharacteristic
170C25C2×D411yescharacteristic
171C2×#41—44no—
172C2×#41—44no—
173C2×#41—44no—
174C22×Q8—21no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—42no—
183C22×D4C22⋊C411yescharacteristic
184C22×D4C22⋊C411yescharacteristic
185C22×D4C22⋊C411yescharacteristic
186C22×D4C22⋊C411yescharacteristic
187C22×D4C22×C411yescharacteristic
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—42no—
194C23×C4C22⋊C411yescharacteristic
195C23×C4C22⋊C411yescharacteristic
196C23×C4C22⋊C411yescharacteristic
197C23×C4C22⋊C411yescharacteristic
198C23×C4C22×C411yescharacteristic
199C23×C4C22×C411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C2×M16—42no—
204C22×C8—42no—
205C4⋊Q8—42no—
206#124—44no—
207#123—44no—
208#123—44no—
209#123—44no—
210#123—44no—
211C22⋊Q8—44no—
212C22⋊Q8—44no—
213#121—42no—
214#121—42no—
215#121—44no—
216#121—44no—
217#121—44no—
218#120—44no—
219#120—44no—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—42no—
223C4×D4—42no—
224C4×D4—42no—
225C4×D4—42no—
226C4×D4—42no—
227C4×D4—42no—
228C4×D4—42no—
229C4×D4—44no—
230C4×D4—44no—
231#117—42no—
232#117—42no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4C22⋊C411yescharacteristic
284C2×C22⋊C4C22⋊C411yescharacteristic
285C2×C22⋊C4C22⋊C411yescharacteristic
286C2×C22⋊C4C22⋊C411yescharacteristic
287C2×C22⋊C4C22×C411yescharacteristic
288C2×C22⋊C4C22×C411yescharacteristic
289C2×C22⋊C4C2×D411yescharacteristic
290C2×C22⋊C4C2×D411yescharacteristic
291C2×C22⋊C4C2×D411yescharacteristic
292C2×C22⋊C4C2×D411yescharacteristic
293C2×C22⋊C4C2×D411yescharacteristic
294C2×C22⋊C4C2411yescharacteristic
295C2×C42—21no—
296C2×C42C22⋊C411yescharacteristic
297C2×C42C22⋊C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C4⋊C8—42no—
301C22⋊C8—24no—
302C22⋊C8—42no—
303C22⋊C8—42no—
304#97—42no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—44no—
321#95C22×C411yescharacteristic
322#95C22×C411yescharacteristic
323#95C22×C411yescharacteristic
324#95C22×C411yescharacteristic
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—21no—
330C24—41no—
331C24—81no—
332C24—81no—
333C24—81no—
334C24—81no—
335C24C2×C22⋊C411yescharacteristic
336C24C2×C22⋊C411yescharacteristic
337C24C2×C22⋊C411yescharacteristic
338C24C2×C22⋊C411yescharacteristic
339C24AGL(1,ℤ/8)14yes—
340C24C22×D411yescharacteristic
341#41—816no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—81no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4—81no—
372C2×D4—81no—
373C2×D4—81no—
374C2×D4—81no—
375C2×D4#9914yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4—84no—
419C22×C4—162no—
420C22×C4—162no—
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C23×C411yescharacteristic
430C22×C4C22×D411yescharacteristic
431M16—162no—
432C2×C8—44no—
433C2×C8—162no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—81no—
481C22⋊C4—81no—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C22⋊C4—84no—
485C22⋊C4—162no—
486C22⋊C4—162no—
487C42—22no—
488C42—22no—
489C42—41no—
490C42—41no—
491C42—42no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—24no—
499C23—24no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—44no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23—82no—
525C23—84no—
526C23—84no—
527C23—162no—
528C23—162no—
529C23C2×#9912yes—
530C23C2×#9912yes—
531C23#41812yes—
532C23#41812yes—
533C23#42112yes—
534C23C22×C22⋊C411yescharacteristic
535C23C2×AGL(1,ℤ/8)12yes—
536Q8—84no—
537D4—28no—
538D4—44no—
539D4—44no—
540D4—44no—
541D4—82no—
542D4—84no—
543D4—84no—
544D4—84no—
545D4—162no—
546D4—162no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
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—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—84no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—161no—
601C2×C4—161no—
602C2×C4—161no—
603C2×C4—161no—
604C2×C4—162no—
605C2×C4—162no—
606C2×C4—168no—
607C2×C4C2×#9912yes—
608C8—84no—
609C22—21no—
610C22—22no—
611C22—21no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—84no—
633C22—81no—
634C22—81no—
635C22—84no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—164no—
645C22#151014yes—
646C22C22×#9911yescharacteristic
647C22C2×#41811yescharacteristic
648C22C2×#42111yescharacteristic
649C4—22no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—164no—
667C2—21no—
668C2—21no—
669C2—24no—
670C2—42no—
671C2—42no—
672C2—42no—
673C2—81no—
674C2—161no—
675C2C2×#151011yescharacteristic
676C2#3335011yescharacteristic
677C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#36009C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20689C2semidirect
3. #17367C2semidirect
4. #11801C2semidirect
5. #11799C2semidirect
6. #11798C2semidirect
7. #11797C2semidirect
8. #11692C2semidirect
9. C2×#1344C2semidirect
10. C2×#547C4semidirect
17. #2547C4semidirect
18. #2543C4semidirect
19. #2543C4semidirect
20. #2535C4semidirect
21. #2486C4semidirect
22. #2486C4semidirect
32. #2349C4semidirect
55. C2×C4⋊Q8C2×C4semidirect
67. C2×#121C2×C4semidirect
77. C2×C4×D4C2×C4semidirect
85. C22×C4⋊C4C2×C4semidirect
98. #398C2×C4semidirect
99. #398C2×C4semidirect
100. #398C2×C4semidirect
109. #397C2×C4semidirect
110. #397C2×C4semidirect
111. #397C2×C4semidirect
121. #394C2×C4semidirect
123. #392C2×C4semidirect
143. #382C2×C4semidirect
153. #380C2×C4semidirect
155. C4×C4⋊C4C2×C4semidirect
160. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#360092
#206892
#173672
#118012
#117992
#117982
#117972
#116922
C2×#13442