Finite Group Databasebeta

← #360165

Subgroups — #360165

Order 512.

2448 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#1896C212yesmaximal
4C2×#1806C212yesmaximal, complemented
5#20626C211yesmaximal, characteristic, complemented
6#11226C212yesmaximal, complemented
7#11075C212yesmaximal, complemented
8#10852C212yesmaximal, complemented
9#10719C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C23×C4⋊C4—21no—
13C23×C42C2212yes—
14C23×C42C2211yescharacteristic
15C2×#423C2212yes—
16C22×#104C2214yes—
17C2×#407C2212yes—
18C4×C22⋊Q8C412yescomplemented
19C4×C22⋊Q8C412yescomplemented
20C4×#123—21no—
21C4×#123—21no—
22C4×#121C412yescomplemented
23C4×#121C412yescomplemented
24C4×#120—21no—
25C4×#120—21no—
26#2297C2211yescharacteristic
27#2296C2211yescharacteristic
28#2290—22no—
29#2280C2211yescharacteristic
30C42×D4—22no—
31C4×#117C2211yescharacteristic
32C2×C4×C4⋊C4—21no—
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4—28no—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C22×#95—21no—
38#1896—28no—
39#1806—28no—
40#1797—22no—
41#1797—22no—
42C2×#339C2214yes—
43C2×#339C2214yes—
44C2×#339C2214yes—
45#1728C2212yes—
46C22×#41C2311yescharacteristic
47C24×C4—22no—
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C22×M16C2314yes—
52#525—22no—
53#525—22no—
54#525D414yes—
55#525Q814yes—
56C2×#123—21no—
57C2×C22⋊Q8C2×C412yes—
58C2×#121C2×C412yes—
59C2×#120—21no—
60C4×#41—24no—
61C4×#41—24no—
62C2×C4×Q8C2×C411yescharacteristic, complemented
63C2×C4×Q8C2×C411yescharacteristic, complemented
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4C2×C411yescharacteristic, complemented
70C2×C4×D4C2×C411yescharacteristic, complemented
71C2×#117—21no—
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C4⋊C4—216no—
76C22×C4⋊C4—28no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C42—24no—
80C22×C42—24no—
81C22×C42—22no—
82C22×C42—28no—
83C22×C42—21no—
84C22×C42—24no—
85C22×C42—24no—
86C22×C42—44no—
87C22×C42C2×C412yes—
88C22×C42C2×C412yes—
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D412yes—
92C22×C42D412yes—
93C22×C42C2314yes—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#423—24no—
97#423—24no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×#104—28no—
101C2×#104—28no—
102C2×#104—44no—
103#407—24no—
104#407—24no—
105C2×C22⋊C8—24no—
106#391C2×C412yescomplemented
107#390C2×C412yescomplemented
108#389C2×C412yescomplemented
109#389C2×C412yescomplemented
110#388—22no—
111#388—22no—
112#387—22no—
113#387—22no—
114#386—22no—
115#386—22no—
116#386—21no—
117#386—21no—
118#385—22no—
119#385—22no—
120#385—21no—
121#385—21no—
122#381—21no—
123#380C2×C412yes—
124#380C2×C412yes—
125#379—21no—
126C4×C4⋊C4—28no—
127C4×C4⋊C4—28no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—28no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4C2×C412yescomplemented
134C4×C4⋊C4C2×C412yescomplemented
135C4×C4⋊C4D418yescomplemented
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144#376—21no—
145C2×#95—28no—
146C2×#95—22no—
147C2×#95—22no—
148C43—21no—
149#339—28no—
150#339—28no—
151#339—44no—
152#336—24no—
153C25C2×D411yescharacteristic
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—22no—
160C2×#41C2×D412yes—
161C2×#41C2×Q812yes—
162C22×Q8C22×C411yescharacteristic
163C22×D4—22no—
164C22×D4—21no—
165C22×D4C22×C411yescharacteristic
166C23×C4—28no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—216no—
174C23×C4—28no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—44no—
182C23×C4—48no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4C22⋊C412yes—
186C23×C4C22⋊C412yes—
187C23×C4C22⋊C412yes—
188C23×C4C22⋊C412yes—
189C23×C4C22×C412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×Q812yes—
198C23×C4#4112yes—
199C23×C4#4112yes—
200C23×C4C2411yescharacteristic
201C2×M16—24no—
202C2×M16—24no—
203C2×M16—24no—
204C2×M16—24no—
205C2×M16—24no—
206C2×M16—24no—
207C22×C8—24no—
208#123—24no—
209#123—24no—
210C22⋊Q8—24no—
211C22⋊Q8C4⋊C418yescomplemented
212#121—24no—
213#121C4⋊C418yescomplemented
214#120—24no—
215#120—24no—
216C4×Q8—24no—
217C4×Q8—24no—
218C4×D4—28no—
219C4×D4—28no—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—48no—
223C4×D4—48no—
224C4×D4—42no—
225C4×D4—42no—
226C4×D4—42no—
227C4×D4—42no—
228#117—21no—
229#117—21no—
230#117—22no—
231#117—22no—
232#117—22no—
233#117—22no—
234#117—44no—
235#117C2×D412yes—
236#117C2×Q812yes—
237C2×C4⋊C4—264no—
238C2×C4⋊C4—216no—
239C2×C4⋊C4—216no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C22×C412yes—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—28no—
270C2×C42—28no—
271C2×C42—28no—
272C2×C42—28no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—28no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—416no—
286C2×C42—416no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42C22⋊C412yes—
291C2×C42C22⋊C412yes—
292C2×C42C22⋊C412yes—
293C2×C42C22⋊C412yes—
294C2×C42C22×C412yes—
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C2×D418yes—
298C2×C42C2×D412yes—
299C2×C42C2×D414yes—
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42#4114yes—
303C4⋊C8—44no—
304C4⋊C8—44no—
305#104—416no—
306C22⋊C8—48no—
307#95—28no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—28no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—24no—
322C24—24no—
323C24—21no—
324C24C2×C22⋊C412yes—
325C24#12112yes—
326C24C22⋊Q812yes—
327C24C22×D411yescharacteristic
328#41—24no—
329#41—24no—
330#41—24no—
331#41—24no—
332#41—42no—
333#41—42no—
334#41—42no—
335#41—42no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8C2×C4⋊C412yescomplemented
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—28no—
343C2×D4—21no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4C2×C4⋊C412yescomplemented
353C22×C4—216no—
354C22×C4—216no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—232no—
381C22×C4—216no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—28no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—416no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—416no—
405C22×C4—432no—
406C22×C4—432no—
407C22×C4—48no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4#10418yescomplemented
416C22×C4C2×C22⋊C412yes—
417C22×C4C2×C22⋊C412yes—
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C412yes—
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C4⋊C414yes—
424C22×C4#11714yes—
425C22×C4#12012yes—
426C22×C4#12012yes—
427C22×C4#12114yes—
428C22×C4#12112yes—
429C22×C4C22⋊Q814yes—
430C22×C4C22⋊Q812yes—
431C22×C4#12312yes—
432C22×C4#12312yes—
433C22×C4C23×C411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×Q811yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4112yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C42−1+412yes—
443M16—44no—
444M16—44no—
445M16—44no—
446M16—44no—
447C2×C8—44no—
448C2×C8—44no—
449C2×C8—44no—
450C4⋊C4—232no—
451C4⋊C4—232no—
452C4⋊C4—22no—
453C4⋊C4—28no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4C2×C4⋊C414yescomplemented
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—28no—
465C22⋊C4—28no—
466C22⋊C4—44no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—44no—
476C42—24no—
477C42—24no—
478C42—22no—
479C42—22no—
480C42—24no—
481C42—24no—
482C42—24no—
483C42—24no—
484C42—28no—
485C42—28no—
486C42—416no—
487C42—44no—
488C42—42no—
489C42—42no—
490C42—44no—
491C42—44no—
492C42—42no—
493C42—42no—
494C42—44no—
495C42C4×D414yes—
496C42C4×D414yes—
497C42#12114yes—
498C42#12114yes—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—24no—
504C23—28no—
505C23—22no—
506C23—28no—
507C23—28no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23#38014yes—
523C23C2×#10414yes—
524C23#42114yes—
525C23C22×C22⋊C411yescharacteristic
526C23C2×#12011yescharacteristic
527C23C2×#12111yescharacteristic
528C23C2×C22⋊Q811yescharacteristic
529C23C2×#12311yescharacteristic
530C23#53511yescharacteristic
531C23#53611yescharacteristic
532Q8—24no—
533Q8—42no—
534D4—24no—
535D4—416no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—42no—
541C2×C4—24no—
542C2×C4—216no—
543C2×C4—24no—
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—22no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—216no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—216no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—464no—
573C2×C4—416no—
574C2×C4—416no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—48no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4#38014yes—
595C2×C4#38612yes—
596C2×C4#38612yes—
597C2×C4#38612yes—
598C2×C4#38612yes—
599C2×C4C2×#10414yes—
600C2×C4C2×#10414yes—
601C2×C4C22×C4⋊C411yescharacteristic
602C2×C4C2×#11711yescharacteristic
603C2×C4C2×C4×D412yes—
604C2×C4#52012yes—
605C2×C4C2×#12111yescharacteristic
606C2×C4C2×#12112yes—
607C2×C4C2×C22⋊Q811yescharacteristic
608C2×C4#53712yes—
609C2×C4D4212yes—
610C2×C4#54612yes—
611C2×C4#54712yes—
612C2×C4#54812yes—
613C2×C4D4×Q812yes—
614C2×C4#55212yes—
615C2×C4#55312yes—
616C2×C4#55412yes—
617C8—84no—
618C22—28no—
619C22—28no—
620C22—28no—
621C22—28no—
622C22—24no—
623C22—22no—
624C22—22no—
625C22—24no—
626C22—24no—
627C22—22no—
628C22—22no—
629C22—24no—
630C22—21no—
631C22—21no—
632C22—24no—
633C22—21no—
634C22—21no—
635C22—24no—
636C22—42no—
637C22—44no—
638C22—82no—
639C22—81no—
640C22—81no—
641C22#180618yescomplemented
642C22C2×#38011yescharacteristic
643C22C2×#38611yescharacteristic
644C22#231111yescharacteristic
645C22#240811yescharacteristic
646C22#240911yescharacteristic
647C22#248611yescharacteristic
648C22#248711yescharacteristic
649C22C22×#10412yes—
650C22C2×#42112yes—
651C4—24no—
652C4—22no—
653C4—24no—
654C4—22no—
655C4—22no—
656C4—416no—
657C4—42no—
658C4—44no—
659C4—42no—
660C4—82no—
661C4—81no—
662C4—81no—
663C4#189614yescomplemented
664C4D4×C4⋊C412yes—
665C4#235212yes—
666C4#241612yes—
667C4#241712yes—
668C2—21no—
669C2—28no—
670C2—21no—
671C2—21no—
672C2—21no—
673C2—82no—
674C2#1387211yescharacteristic
675C2C2×#180612yes—
676C2C2×#189611yescharacteristic
677C2#2097911yescharacteristic
678C2#2189912yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×C4⋊C4C2semidirect
4. C2×#1806C2semidirect
5. #20626C2semidirect
6. #11226C2semidirect
7. #11075C2semidirect
8. #10852C2semidirect
9. #10719C2semidirect
18. C4×C22⋊Q8C4semidirect
19. C4×C22⋊Q8C4semidirect
22. C4×#121C4semidirect
23. C4×#121C4semidirect
62. C2×C4×Q8C2×C4semidirect
63. C2×C4×Q8C2×C4semidirect
69. C2×C4×D4C2×C4semidirect
70. C2×C4×D4C2×C4semidirect
106. #391C2×C4semidirect
107. #390C2×C4semidirect
108. #389C2×C4semidirect
109. #389C2×C4semidirect
133. C4×C4⋊C4C2×C4semidirect
134. C4×C4⋊C4C2×C4semidirect
135. C4×C4⋊C4D4semidirect
211. C22⋊Q8C4⋊C4semidirect
213. #121C4⋊C4semidirect
339. C2×Q8C2×C4⋊C4semidirect
352. C2×D4C2×C4⋊C4semidirect
415. C22×C4#104semidirect
461. C4⋊C4C2×C4⋊C4semidirect
641. C22#1806semidirect
663. C4#1896semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
C22×C4×C4⋊C42
C2×#18962
C2×#18062
#206262
#112262
#110752
#108522
#107192