Finite Group Databasebeta

← #196842

Subgroups — #196842

Order 512.

1748 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2C2×#2349C211yesmaximal, characteristic
3#21738C211yesmaximal, characteristic
4C2×#1520C211yesmaximal, characteristic, complemented
5C22×#328C211yesmaximal, characteristic, complemented
6#9152C212yesmaximal, complemented
7#9151C212yesmaximal, complemented
8#9144C212yesmaximal, complemented
9#9143C212yesmaximal, complemented
10#8023C211yesmaximal, characteristic, complemented
11C2×#1294C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×D4—21no—
14C22×C4×D4C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17#2855C2212yes—
18#2848C2212yes—
19#2627—21no—
20#2626—21no—
21#2579—22no—
22#2564—22no—
23#2487C412yescomplemented
24#2417C414yescomplemented
25#2409C412yescomplemented
26#2349—24no—
27#2349—24no—
28#2315—21no—
29#2308—21no—
30C2×#389—21no—
31C2×#385—21no—
32C2×#385—21no—
33C2×#382C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C22×#95C2211yescharacteristic
39C22×#95C2211yescharacteristic
40#1869—22no—
41#1869—22no—
42#1817C2212yes—
43#1779C2212yes—
44#1520—24no—
45C2×#328—24no—
46C2×#328C2212yes—
47C2×#328C2212yes—
48C2×#328C2212yes—
49C2×#328C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54C23×Q8C2×C411yescharacteristic, complemented
55C23×D4D411yescharacteristic
56C24×C4—21no—
57C24×C4—21no—
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C23×C8—21no—
61C23×C8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8C2×C412yescomplemented
69C2×C22⋊Q8C2×C414yescomplemented
70C2×C22⋊Q8C2312yes—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—28no—
76C2×C4×D4—44no—
77C2×C4×D4—44no—
78C2×C4×D4C2×C412yes—
79C2×C4×D4C2×C412yes—
80C2×C4×D4D412yes—
81C2×C4×D4D412yes—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4C2×C411yescharacteristic, complemented
88C22×C4⋊C4C2×C412yescomplemented
89C22×C4⋊C4D411yescharacteristic
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4D411yescharacteristic
95C22×C22⋊C4D411yescharacteristic
96C22×C42—21no—
97C22×C42C2311yescharacteristic
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8C2312yes—
106C2×C22⋊C8C2312yes—
107#400—22no—
108#399—22no—
109#398—24no—
110#396—24no—
111#395—22no—
112#393—22no—
113#391C2×C414yescomplemented
114#389—28no—
115#389C2×C414yescomplemented
116#387—24no—
117#386—24no—
118#386—22no—
119#386—22no—
120#385—28no—
121#385—28no—
122#385—24no—
123#385—22no—
124#385—22no—
125#385—22no—
126#385—22no—
127#382—22no—
128#382—22no—
129#381—22no—
130#380—21no—
131#380—22no—
132#380—21no—
133#380—24no—
134#380—22no—
135#380—22no—
136#380C2×C414yescomplemented
137#380C2×C412yescomplemented
138#380D414yes—
139#380C2312yes—
140#379—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C2×#95—21no—
144C2×#95—22no—
145C2×#95—21no—
146C2×#95—21no—
147C2×#95—21no—
148C2×#95—22no—
149C2×#95—22no—
150C2×#95D414yes—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155#336—24no—
156#328—24no—
157#328—24no—
158#328—24no—
159#328—24no—
160C25C2×D411yescharacteristic
161C25C2×D411yescharacteristic
162C22×Q8—21no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—82no—
166C22×Q8C22×C412yes—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4C22⋊C412yes—
172C22×D4C22⋊C412yes—
173C22×D4SD1614yes—
174C22×D4Q1614yes—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—22no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—82no—
195C23×C4—82no—
196C23×C4—82no—
197C23×C4—82no—
198C23×C4C22×C412yes—
199C23×C4C22×C411yescharacteristic
200C23×C4C22×C411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D412yes—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D412yes—
209C23×C4C2×D412yes—
210C23×C4#4112yes—
211C23×C4#4112yes—
212C23×C4C2411yescharacteristic
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—42no—
216C22×C8—42no—
217C22×C8—82no—
218C22×C8—82no—
219C22⋊Q8—24no—
220C22⋊Q8—24no—
221C22⋊Q8—22no—
222C22⋊Q8—22no—
223C22⋊Q8—22no—
224C22⋊Q8—22no—
225C22⋊Q8—42no—
226C22⋊Q8—42no—
227C22⋊Q8—42no—
228C22⋊Q8—42no—
229C22⋊Q8—84no—
230C4×D4—24no—
231C4×D4—24no—
232C4×D4—24no—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—432no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—82no—
259C2×C4⋊C4—82no—
260C2×C4⋊C4—82no—
261C2×C4⋊C4C22⋊C412yes—
262C2×C4⋊C4C22⋊C412yes—
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C22×C414yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4C22⋊C412yes—
286C2×C22⋊C4C22⋊C412yes—
287C2×C22⋊C4C22⋊C412yes—
288C2×C22⋊C4C22⋊C412yes—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—24no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42C22×C412yes—
297C2×C42C2×D412yes—
298C22⋊C8—22no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—22no—
302C22⋊C8—42no—
303C22⋊C8—42no—
304C22⋊C8—84no—
305#95—24no—
306#95—24no—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—22no—
317#95—21no—
318#95C2×D412yes—
319#95C2×D412yes—
320#95C2×D412yes—
321#95C2×D412yes—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24C2×C22⋊C412yes—
331C24C2×C22⋊C411yescharacteristic
332C24C2×C22⋊C411yescharacteristic
333C24#12012yes—
334C24#12312yes—
335C24C2×SD1612yes—
336C24C2×SD1612yes—
337C24C2×Q1612yes—
338C24C2×Q1612yes—
339C24AGL(1,ℤ/8)12yes—
340C24#13712yes—
341C24C22×D411yescharacteristic
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—44no—
345C2×Q8—42no—
346C2×Q8—82no—
347C2×Q8—84no—
348C2×Q8—84no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—28no—
354C2×D4—28no—
355C2×D4—28no—
356C2×D4—28no—
357C2×D4#10318yes—
358C2×D4#10318yes—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—48no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—48no—
410C22×C4—44no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4C2×C22⋊C412yes—
430C22×C4C2×C22⋊C412yes—
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C412yes—
434C22×C4C4×D412yes—
435C22×C4C4×D412yes—
436C22×C4C4×D412yes—
437C22×C4C4×D412yes—
438C22×C4#12012yes—
439C22×C4#12112yes—
440C22×C4#12112yes—
441C22×C4#12112yes—
442C22×C4#12112yes—
443C22×C4#12312yes—
444C22×C4C2×SD1612yes—
445C22×C4C2×Q1612yes—
446C22×C4C23×C411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C2×C8—44no—
453C2×C8—44no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—84no—
457C2×C8—84no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—84no—
470C4⋊C4—82no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—48no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—48no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—48no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—22no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—42no—
521C23—44no—
522C23—42no—
523C23#38612yes—
524C23#38612yes—
525C23C2×#10314yes—
526C23C2×#10312yes—
527C23C2×#10312yes—
528C23#41712yes—
529C23#41712yes—
530C23#45014yes—
531C23#45114yes—
532C23#48314yes—
533C23#48414yes—
534C23C22×C22⋊C411yescharacteristic
535C23C2×#12011yescharacteristic
536C23C2×#12311yescharacteristic
537C23C22×SD1611yescharacteristic
538C23C22×Q1611yescharacteristic
539C23C2×AGL(1,ℤ/8)11yescharacteristic
540C23C2×#13711yescharacteristic
541Q8—82no—
542Q8—82no—
543Q8—84no—
544D4—216no—
545D4—216no—
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—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—816no—
604C2×C4—816no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—164no—
614C2×C4#38612yes—
615C2×C4#38612yes—
616C2×C4C2×#10314yes—
617C2×C4C2×C4×D411yescharacteristic
618C2×C4C2×C4×D411yescharacteristic
619C2×C4C2×#12111yescharacteristic
620C2×C4C2×#12111yescharacteristic
621C8—82no—
622C8—82no—
623C22—22no—
624C22—22no—
625C22—24no—
626C22—24no—
627C22—22no—
628C22—24no—
629C22—22no—
630C22—24no—
631C22—21no—
632C22—21no—
633C22—22no—
634C22—22no—
635C22—21no—
636C22—21no—
637C22—22no—
638C22—22no—
639C22—24no—
640C22—28no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—42no—
651C22—42no—
652C22#187214yes—
653C22#187214yes—
654C22C2×#38611yescharacteristic
655C22#288312yes—
656C22C22×#10311yescharacteristic
657C22C2×#41711yescharacteristic
658C22#290112yes—
659C22C2×#45011yescharacteristic
660C22C2×#45111yescharacteristic
661C22C2×#48311yescharacteristic
662C22C2×#48411yescharacteristic
663C4—24no—
664C4—21no—
665C4—21no—
666C4—21no—
667C4—21no—
668C4—22no—
669C4—22no—
670C4—44no—
671C4—42no—
672C4—42no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—84no—
677C4—82no—
678C4—82no—
679C2—21no—
680C2—21no—
681C2—21no—
682C2—21no—
683C2—24no—
684C2—24no—
685C2—44no—
686C2#1761412yes—
687C2#1762212yes—
688C2C2×#187211yescharacteristic
689C2#2098011yescharacteristic
690C2#2098411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1520C2semidirect
5. C22×#328C2semidirect
6. #9152C2semidirect
7. #9151C2semidirect
8. #9144C2semidirect
9. #9143C2semidirect
10. #8023C2semidirect
11. C2×#1294C2semidirect
23. #2487C4semidirect
24. #2417C4semidirect
25. #2409C4semidirect
54. C23×Q8C2×C4semidirect
68. C2×C22⋊Q8C2×C4semidirect
69. C2×C22⋊Q8C2×C4semidirect
87. C22×C4⋊C4C2×C4semidirect
88. C22×C4⋊C4C2×C4semidirect
113. #391C2×C4semidirect
115. #389C2×C4semidirect
136. #380C2×C4semidirect
137. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350102
C2×#23492
#217382
C2×#15202
C22×#3282
#91522
#91512
#91442
#91432
#80232
C2×#12942