Finite Group Databasebeta

← #482049

Subgroups — #482049

Order 512.

1317 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35000C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic, complemented
3C2×#1460C211yesmaximal, characteristic, complemented
4#11876C214yesmaximal, complemented
5#11874C212yesmaximal, complemented
6#11873C212yesmaximal, complemented
7#11692C214yesmaximal, complemented
8C2×#534—21no—
9C22×#121C411yescharacteristic, complemented
10C22×C4×D4C2211yescharacteristic
11C2×#431—21no—
12C22×#99—21no—
13#2854—21no—
14#2847—21no—
15#2618C412yescomplemented
16#2617C2212yes—
17#2543C412yescomplemented
18#2543C2212yes—
19#2534—21no—
20#2534—21no—
21#2486C411yescharacteristic, complemented
22#2486C2211yescharacteristic
23#2416—21no—
24#2416—21no—
25#2408C411yescharacteristic, complemented
26#2408C2211yescharacteristic
27#2308—21no—
28C2×#380—21no—
29C2×#380C411yescharacteristic, complemented
30#1799—21no—
31#1799—21no—
32#1799C2212yes—
33#1799C2212yescomplemented
34C2×#342C2212yescomplemented
35#1460—21no—
36#1460—21no—
37#1460C2212yes—
38#1460C2212yescomplemented
39C2×#323—21no—
40C2×#323C2212yescomplemented
41#1344C2218yescomplemented
42#1343C2218yescomplemented
43C23×D4—21no—
44C23×D4C2×C411yescharacteristic, complemented
45C23×D4D411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48#534—44no—
49#534—44no—
50C2×#123—21no—
51C2×#123—21no—
52C2×#123—22no—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121—21no—
56C2×#121—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—41no—
60C2×#121—41no—
61C2×#121—41no—
62C2×#121—41no—
63C2×#121C2×C412yes—
64C2×#121C2×C412yescomplemented
65C2×#120—22no—
66C2×#120—21no—
67C2×#120—21no—
68C2×C4×D4—41no—
69C2×C4×D4—41no—
70C2×C4×D4—41no—
71C2×C4×D4—41no—
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C2×C4×D4D411yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C22×C4⋊C4—21no—
81C22×C4⋊C4—41no—
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C4⋊C4C2×C411yescharacteristic, complemented
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4D411yescharacteristic
88C22×C22⋊C4C2311yescharacteristic
89C22×C42D411yescharacteristic
90#431—22no—
91#431—22no—
92#431—41no—
93#431—41no—
94C2×C4⋊C8—41no—
95C2×#99—22no—
96C2×#99—22no—
97C2×#99—44no—
98C2×C22⋊C8—22no—
99C2×C4×C8—41no—
100#398C2×C412yescomplemented
101#398C2×C412yescomplemented
102#397C2×C412yes—
103#397C2312yes—
104#396—22no—
105#396—22no—
106#394—22no—
107#394—22no—
108#394C2×C412yescomplemented
109#394C2×C412yescomplemented
110#392C2×C412yes—
111#392C2312yes—
112#390—21no—
113#390—21no—
114#389—21no—
115#389—21no—
116#388—22no—
117#388—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#381—21no—
125#381—21no—
126#380—21no—
127#380—21no—
128#380—21no—
129#380—21no—
130#380—22no—
131#380—22no—
132#380—41no—
133#380—41no—
134#380C2×C412yes—
135#380C2×C411yescharacteristic, complemented
136#380C2×C411yescharacteristic, complemented
137#379—21no—
138#379—21no—
139C2×#95—41no—
140C2×#95C2×C412yescomplemented
141#342—22no—
142#342C2314yes—
143#323—22no—
144#323—22no—
145#323—22no—
146#323C2314yes—
147C25—21no—
148C25C22⋊C411yescharacteristic
149C25C2×D411yescharacteristic
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—41no—
156C22×D4—41no—
157C22×D4—42no—
158C22×D4—41no—
159C22×D4—41no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4—44no—
163C22×D4—44no—
164C22×D4—82no—
165C22×D4C22⋊C411yescharacteristic
166C22×D4C22⋊C411yescharacteristic
167C22×D4D814yes—
168C22×D4SD1614yes—
169C22×D4C22×C412yes—
170C22×D4C2×D411yescharacteristic
171C22×D4C2×D411yescharacteristic
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—22no—
182C23×C4—42no—
183C23×C4—82no—
184C23×C4—82no—
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C22×C8—42no—
199#123—44no—
200#123—42no—
201#123—42no—
202#123—48no—
203#121—42no—
204#121—42no—
205#121—42no—
206#121—42no—
207#121—48no—
208#121—44no—
209#121—44no—
210#121—41no—
211#121—41no—
212#121—41no—
213#121—41no—
214#121—42no—
215#121—42no—
216#121—82no—
217#121—82no—
218#120—44no—
219#120—44no—
220#120—44no—
221#120—42no—
222#120—42no—
223C4×D4—41no—
224C4×D4—41no—
225C4×D4—41no—
226C4×D4—41no—
227C4×D4—41no—
228C4×D4—41no—
229C4×D4—41no—
230C4×D4—41no—
231C4×D4—41no—
232C4×D4—41no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—41no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—82no—
252C2×C4⋊C4—84no—
253C2×C4⋊C4C22⋊C411yescharacteristic
254C2×C4⋊C4C22⋊C411yescharacteristic
255C2×C4⋊C4C22×C412yes—
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C22⋊C4C22⋊C411yescharacteristic
284C2×C22⋊C4C22×C411yescharacteristic
285C2×C22⋊C4C2×D411yescharacteristic
286C2×C22⋊C4C2×D411yescharacteristic
287C2×C22⋊C4C2×D411yescharacteristic
288C2×C22⋊C4C2×D411yescharacteristic
289C2×C22⋊C4C2411yescharacteristic
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42C22⋊C411yescharacteristic
295C2×C42C22⋊C411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C4⋊C8—42no—
299C4⋊C8—42no—
300#99—44no—
301#99—44no—
302C22⋊C8—24no—
303C22⋊C8—42no—
304C4×C8—42no—
305C4×C8—42no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—42no—
312#95—42no—
313#95C22×C412yes—
314#95C22×C412yes—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—42no—
323C24—42no—
324C24—42no—
325C24—81no—
326C24—81no—
327C24—81no—
328C24—81no—
329C24—82no—
330C24—82no—
331C24#10214yes—
332C24C2×C22⋊C411yescharacteristic
333C24C2×C22⋊C411yescharacteristic
334C24C2×C22⋊C411yescharacteristic
335C24C2×C22⋊C411yescharacteristic
336C24C2×D812yes—
337C24C2×SD1612yes—
338C24C22×D411yescharacteristic
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—44no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—84no—
367C2×D4—84no—
368C2×D4—82no—
369C2×D4—82no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4—82no—
373C2×D4—88no—
374C2×D4—88no—
375C2×D4#10214yes—
376C2×D4#10214yes—
377C2×D4C2×D812yes—
378C2×D4C2×D812yes—
379C2×D4C2×SD1612yes—
380C2×D4C2×SD1612yes—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4#9911yescharacteristic
440C22×C4#9911yescharacteristic
441C22×C4#9911yescharacteristic
442C22×C4#9911yescharacteristic
443C22×C4#10214yes—
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C23×C411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C2×C8—44no—
455C2×C8—84no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—82no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—44no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—84no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—81no—
538C23—81no—
539C23—81no—
540C23—81no—
541C23—81no—
542C23—81no—
543C23—84no—
544C23—82no—
545C23—82no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—164no—
552C23—164no—
553C23C2×#9911yescharacteristic
554C23C2×#9911yescharacteristic
555C23C2×#10212yes—
556C23C2×#10212yes—
557C23C2×#10212yes—
558C23C2×#10212yes—
559C23#42112yes—
560C23C22×C22⋊C411yescharacteristic
561C23C22×D811yescharacteristic
562C23C22×SD1611yescharacteristic
563D4—44no—
564D4—44no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—42no—
569D4—82no—
570D4—84no—
571D4—88no—
572D4—84no—
573D4—84no—
574D4—82no—
575D4—162no—
576D4—162no—
577D4—164no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—164no—
638C2×C4—168no—
639C2×C4C2×#9911yescharacteristic
640C2×C4C2×#9911yescharacteristic
641C2×C4C2×#9911yescharacteristic
642C2×C4C2×#9911yescharacteristic
643C2×C4C2×#10212yes—
644C2×C4C2×#10212yes—
645C8—84no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—22no—
654C22—22no—
655C22—21no—
656C22—22no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—41no—
661C22—41no—
662C22—42no—
663C22—42no—
664C22—44no—
665C22—42no—
666C22—44no—
667C22—42no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—81no—
677C22—81no—
678C22—84no—
679C22—82no—
680C22—82no—
681C22—84no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—168no—
690C22#151912yes—
691C22#151912yes—
692C22C22×#9911yescharacteristic
693C22C22×#10211yescharacteristic
694C22C2×#42111yescharacteristic
695C4—21no—
696C4—21no—
697C4—21no—
698C4—22no—
699C4—22no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—164no—
710C4#212112yes—
711C2—21no—
712C2—21no—
713C2—22no—
714C2—22no—
715C2—41no—
716C2—41no—
717C2—42no—
718C2—42no—
719C2—42no—
720C2—162no—
721C2C2×#151911yescharacteristic
722C2C2×#212111yescharacteristic
723C2#3335811yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35000C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1799C2semidirect
3. C2×#1460C2semidirect
4. #11876C2semidirect
5. #11874C2semidirect
6. #11873C2semidirect
7. #11692C2semidirect
9. C22×#121C4semidirect
15. #2618C4semidirect
17. #2543C4semidirect
21. #2486C4semidirect
25. #2408C4semidirect
29. C2×#380C4semidirect
33. #1799C22semidirect
34. C2×#342C22semidirect
38. #1460C22semidirect
40. C2×#323C22semidirect
41. #1344C22semidirect
42. #1343C22semidirect
44. C23×D4C2×C4semidirect
64. C2×#121C2×C4semidirect
83. C22×C4⋊C4C2×C4semidirect
100. #398C2×C4semidirect
101. #398C2×C4semidirect
108. #394C2×C4semidirect
109. #394C2×C4semidirect
135. #380C2×C4semidirect
136. #380C2×C4semidirect
140. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350002
C2×#17992
C2×#14602
#118762
#118742
#118732
#116922