Finite Group Databasebeta

← #188462

Subgroups — #188462

Order 512.

1424 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34246C211yesmaximal, characteristic
2#21932C211yesmaximal, characteristic, complemented
3C2×#1507C211yesmaximal, characteristic
4C22×#327C211yesmaximal, characteristic
5#13782C211yesmaximal, characteristic, complemented
6#8577C212yesmaximal, complemented
7#8575C212yesmaximal, complemented
8#8573C212yesmaximal, complemented
9#8571C212yesmaximal, complemented
10#8020C211yesmaximal, characteristic, complemented
11#8012C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C2×#407C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15#2364—22no—
16#2363C412yescomplemented
17#2363C412yescomplemented
18#2345C411yescharacteristic, complemented
19#2345C411yescharacteristic, complemented
20D4×C22⋊C4—21no—
21D4×C22⋊C4—21no—
22D4×C22⋊C4C411yescharacteristic, complemented
23D4×C22⋊C4C411yescharacteristic, complemented
24#2333—22no—
25#2312—21no—
26C4×#121C2212yes—
27C4×#121C2212yes—
28C2×#385—21no—
29#2280C2211yescharacteristic
30#2273—21no—
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C22×#95C2211yescharacteristic
34#1851—22no—
35#1850—21no—
36#1850—21no—
37#1797C2212yescomplemented
38#1797C2212yescomplemented
39#1507—24no—
40C2×#327—24no—
41C2×#327C2212yes—
42C2×#327C2212yes—
43C2×#327C2211yescharacteristic
44C2×#327C2211yescharacteristic
45C2×#327C2211yescharacteristic
46C2×#327C2211yescharacteristic
47#1436—22no—
48C4⋊#95C2212yescomplemented
49#1434C2212yescomplemented
50C22⋊#95—21no—
51#1294C2212yescomplemented
52#1294C2212yescomplemented
53#1294C2212yescomplemented
54#1294C2212yescomplemented
55C23×D4—21no—
56C23×D4C2×C411yescharacteristic
57C24×C4—21no—
58C24×C4C2311yescharacteristic
59C22×M16—21no—
60C23×C8—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121C2×C411yescharacteristic
66C2×#121C2×C411yescharacteristic
67C2×#121D414yes—
68C2×#121Q814yes—
69C2×#121C2311yescharacteristic
70C2×#121C2311yescharacteristic
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4C2×C412yescomplemented
74C2×C4×D4C2×C412yescomplemented
75C2×#117C2×C412yes—
76C22×C4⋊C4—22no—
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C42—21no—
85C22×C42C2312yes—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88#407—22no—
89#407—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8C2312yes—
95C2×C22⋊C8C2311yescharacteristic
96C2×C22⋊C8C2311yescharacteristic
97#390—22no—
98#390—22no—
99#390C2×C412yescomplemented
100#390C2×C412yescomplemented
101#389C2×C412yescomplemented
102#389C2×C412yescomplemented
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#387—22no—
108#387—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#385—28no—
116#381—21no—
117#380—21no—
118#380—21no—
119#380C2×C411yescharacteristic
120#380C2×C411yescharacteristic
121#380C2312yes—
122#379—21no—
123#379—21no—
124#379—21no—
125C4×C4⋊C4C2×C412yescomplemented
126C4×C4⋊C4C2×C412yescomplemented
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4C2×C411yescharacteristic
138C4×C22⋊C4C2×C411yescharacteristic
139C4×C22⋊C4C2×C412yes—
140C4×C22⋊C4D411yescharacteristic
141C4×C22⋊C4D412yes—
142C4×C22⋊C4D411yescharacteristic
143C2×#95—22no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95—21no—
147C2×#95—22no—
148C2×#95—22no—
149C2×#95D412yes—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2311yescharacteristic
153#336—22no—
154#336—22no—
155#327—24no—
156#327—24no—
157#327—22no—
158#327—22no—
159#327—22no—
160#327—22no—
161C25—21no—
162C25C2×D411yescharacteristic
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—21no—
170C22×D4—22no—
171C22×D4—82no—
172C22×D4—82no—
173C22×D4—82no—
174C22×D4C4⋊C414yescomplemented
175C22×D4C22×C411yescharacteristic
176C22×D4C22×C411yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—82no—
187C23×C4—82no—
188C23×C4—82no—
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D412yes—
197C23×C4C2×Q812yes—
198C23×C4#4112yes—
199C23×C4#4112yes—
200C23×C4C2411yescharacteristic
201C2×M16—42no—
202C2×M16—82no—
203C22×C8—22no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—42no—
207C22×C8—82no—
208#121—22no—
209#121—22no—
210#121—24no—
211#121—24no—
212#121—24no—
213#121—24no—
214#121—21no—
215#121—21no—
216#121—21no—
217#121—21no—
218#121C4⋊C414yescomplemented
219#121C4⋊C414yescomplemented
220#121C2×D412yes—
221#121C2×D412yes—
222#121C2×Q812yes—
223#121C2×Q812yes—
224C4×D4—82no—
225C4×D4—82no—
226C4×D4—82no—
227C4×D4—82no—
228C4×D4—82no—
229C4×D4—82no—
230#117—82no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—82no—
237C2×C4⋊C4—82no—
238C2×C4⋊C4C4⋊C414yescomplemented
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C22×C412yes—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4C22⋊C411yescharacteristic
267C2×C22⋊C4C22⋊C411yescharacteristic
268C2×C22⋊C4C22⋊C411yescharacteristic
269C2×C22⋊C4C22⋊C412yes—
270C2×C22⋊C4C22⋊C412yes—
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—42no—
276C2×C42—82no—
277C2×C42—82no—
278C2×C42C22×C412yes—
279C2×C42C22×C412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C4⋊C8—82no—
283C4⋊C8—82no—
284C22⋊C8—22no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—22no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—42no—
291C22⋊C8—42no—
292#95—28no—
293#95—22no—
294#95—22no—
295#95—21no—
296#95—21no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—21no—
302#95—21no—
303#95—22no—
304#95—21no—
305#95—21no—
306#95—42no—
307#95—82no—
308#95—82no—
309#95C22⋊C412yes—
310#95C22⋊C412yes—
311#95C22⋊C412yes—
312#95C22⋊C412yes—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—21no—
318C24—21no—
319C24—24no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24—82no—
324C24—82no—
325C24C2×C22⋊C411yescharacteristic
326C24C2×C22⋊C411yescharacteristic
327C24#12112yes—
328C24C22⋊Q812yes—
329C24C22×D411yescharacteristic
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—24no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—82no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—82no—
349C2×D4—82no—
350C2×D4—88no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4C2×C4⋊C412yes—
354C2×D4C2×C4⋊C412yes—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—48no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4—82no—
403C22×C4—88no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—88no—
407C22×C4C2×C22⋊C412yes—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C412yes—
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C412yes—
413C22×C4C2×C4⋊C412yes—
414C22×C4C2×C4⋊C412yes—
415C22×C4#11712yes—
416C22×C4#11712yes—
417C22×C4#12112yes—
418C22×C4#12112yes—
419C22×C4#12112yes—
420C22×C4C22⋊Q812yes—
421C22×C4C22⋊Q812yes—
422C22×C4C22⋊Q812yes—
423C22×C4C23×C411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×Q811yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429M16—84no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—84no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—24no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4—88no—
446C4⋊C4C2×C4⋊C412yes—
447C4⋊C4C2×C4⋊C412yes—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—42no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—44no—
466C22⋊C4—82no—
467C22⋊C4—82no—
468C22⋊C4—82no—
469C22⋊C4—82no—
470C22⋊C4—82no—
471C22⋊C4#10414yes—
472C42—42no—
473C42—42no—
474C42—42no—
475C42—42no—
476C42—82no—
477C42—82no—
478C42—82no—
479C42—84no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—21no—
484C23—21no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—21no—
491C23—21no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—48no—
511C23—84no—
512C23—84no—
513C23—164no—
514C23—164no—
515C23#38012yes—
516C23#38012yes—
517C23C2×#10412yes—
518C23#42112yes—
519C23C22×C22⋊C411yescharacteristic
520C23C2×#12111yescharacteristic
521C23C2×C22⋊Q811yescharacteristic
522C23#57612yes—
523C23#57712yes—
524D4—42no—
525D4—42no—
526D4—42no—
527D4—42no—
528D4—44no—
529D4—44no—
530D4—42no—
531D4—42no—
532D4—42no—
533D4—42no—
534D4—84no—
535D4—84no—
536D4—84no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—84no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—816no—
596C2×C4—81no—
597C2×C4—82no—
598C2×C4—81no—
599C2×C4—164no—
600C2×C4—164no—
601C2×C4#38012yes—
602C2×C4#38012yes—
603C2×C4C2×#10414yes—
604C2×C4C22×C4⋊C411yescharacteristic
605C2×C4C2×#11711yescharacteristic
606C2×C4C2×#12111yescharacteristic
607C2×C4C2×C22⋊Q811yescharacteristic
608C8—82no—
609C8—81no—
610C8—81no—
611C22—22no—
612C22—22no—
613C22—24no—
614C22—21no—
615C22—22no—
616C22—21no—
617C22—22no—
618C22—21no—
619C22—21no—
620C22—22no—
621C22—22no—
622C22—24no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—44no—
627C22—44no—
628C22—48no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—168no—
642C22#180614yes—
643C22C2×#38011yescharacteristic
644C22#288312yes—
645C22#289712yes—
646C22C22×#10411yescharacteristic
647C22C2×#42111yescharacteristic
648C22#304111yescharacteristic
649C22#304211yescharacteristic
650C22#308112yes—
651C4—24no—
652C4—21no—
653C4—21no—
654C4—21no—
655C4—21no—
656C4—22no—
657C4—22no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—82no—
664C4—82no—
665C4—84no—
666C4—82no—
667C4—82no—
668C2—21no—
669C2—21no—
670C2—24no—
671C2—22no—
672C2—42no—
673C2—82no—
674C2—82no—
675C2#1754712yes—
676C2#1759712yes—
677C2#2063311yescharacteristic
678C2#2065011yescharacteristic
679C2C2×#180611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ 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
5. #13782C2semidirect
6. #8577C2semidirect
7. #8575C2semidirect
8. #8573C2semidirect
9. #8571C2semidirect
10. #8020C2semidirect
11. #8012C2semidirect
16. #2363C4semidirect
17. #2363C4semidirect
18. #2345C4semidirect
19. #2345C4semidirect
22. D4×C22⋊C4C4semidirect
23. D4×C22⋊C4C4semidirect
37. #1797C22semidirect
38. #1797C22semidirect
48. C4⋊#95C22semidirect
49. #1434C22semidirect
51. #1294C22semidirect
52. #1294C22semidirect
53. #1294C22semidirect
54. #1294C22semidirect
73. C2×C4×D4C2×C4semidirect
74. C2×C4×D4C2×C4semidirect
99. #390C2×C4semidirect
100. #390C2×C4semidirect
101. #389C2×C4semidirect
102. #389C2×C4semidirect
125. C4×C4⋊C4C2×C4semidirect
126. C4×C4⋊C4C2×C4semidirect
174. C22×D4C4⋊C4semidirect
218. #121C4⋊C4semidirect
219. #121C4⋊C4semidirect
238. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342462
#219322
C2×#15072
C22×#3272
#137822
#85772
#85752
#85732
#85712
#80202
#80122