Finite Group Databasebeta

← #360147

Subgroups — #360147

Order 512.

2128 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#440C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3C2×#1877C212yesmaximal
4C2×#1803C211yesmaximal, characteristic, complemented
5C2×#1804C211yesmaximal, characteristic
6#13942C211yesmaximal, characteristic, complemented
7#11219C212yesmaximal, complemented
8#11067C212yesmaximal, complemented
9#10843C211yesmaximal, characteristic, complemented
10#10835C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2212yes—
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4—21no—
17C2×#440—24no—
18C2×#440—24no—
19C2×#440C418yescomplemented
20C22×#107C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C22×#102C2212yes—
25C22×#102C2211yescharacteristic
26C22×#97C2211yescharacteristic
27D4×C4⋊C4—24no—
28D4×C4⋊C4—24no—
29C2×#391C2212yes—
30C2×#389C2212yes—
31C2×#385—22no—
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#380—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1877—28no—
37#1804—24no—
38#1803—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1441—21no—
46#1441—21no—
47C4⋊#95C2212yes—
48#1434—22no—
49C23×Q8C2311yescharacteristic
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C24×C4—22no—
53C22×SD16—22no—
54C22×SD16—22no—
55C22×SD16—22no—
56C22×SD16—22no—
57C22×SD16C2×C414yes—
58C22×SD16D414yes—
59C22×SD16Q814yes—
60C23×C8C2312yes—
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8C2×C414yescomplemented
69C2×C4×D4—24no—
70C2×C4×D4—24no—
71C2×C4×D4—24no—
72C2×C4×D4—24no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—42no—
77C2×C4×D4—42no—
78C2×C4×D4C2×C414yes—
79C2×C4×D4D414yes—
80C2×C4×D4D414yes—
81C22×C4⋊C4—28no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—48no—
87C22×C4⋊C4D411yescharacteristic
88C22×C4⋊C4D411yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4—21no—
94C22×C22⋊C4—22no—
95C22×C42—21no—
96C22×C42C2312yes—
97C22×C42C2311yescharacteristic
98#440—232no—
99C2×#107—22no—
100C2×#107—22no—
101C2×#107C2×C414yescomplemented
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—42no—
110C2×#103C2×C414yescomplemented
111C2×#102—22no—
112C2×#102—22no—
113C2×#102—24no—
114C2×#102—24no—
115C2×#102—24no—
116C2×#102C2×C414yes—
117C2×#97—22no—
118C2×#97—22no—
119C2×#97C2×C414yescomplemented
120#391—24no—
121#391—42no—
122#389—24no—
123#389—24no—
124#385—216no—
125#384—22no—
126#384D414yes—
127#382—22no—
128#382—22no—
129#380—28no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—21no—
139C2×#95D412yes—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—42no—
144#336—24no—
145#336—42no—
146C25—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8C22×C412yes—
159C22×Q8C2×D412yes—
160C22×Q8C2×Q812yes—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4C22×C412yes—
169C22×D4C2×D412yes—
170C22×D4C2×D412yes—
171C22×D4C2×D412yes—
172C22×D4C2×Q812yes—
173C22×D4#4112yes—
174C22×D4#4112yes—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—21no—
182C23×C4—28no—
183C23×C4—22no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—82no—
191C23×C4—82no—
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2411yescharacteristic
197C2×SD16—28no—
198C2×SD16—28no—
199C2×SD16—28no—
200C2×SD16—28no—
201C2×SD16—28no—
202C2×SD16—28no—
203C2×SD16C4⋊C4116yescomplemented
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8C22×C412yes—
222C22×C8C22×C412yes—
223C22×C8C2×D412yes—
224C22×C8C2×Q812yes—
225C4×Q8—28no—
226C4×Q8—28no—
227C4×Q8—44no—
228C4×Q8—44no—
229C4×D4—28no—
230C4×D4—28no—
231C4×D4—28no—
232C4×D4—48no—
233C4×D4—48no—
234C4×D4—48no—
235C4×D4—48no—
236C4×D4C22⋊C4116yes—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—432no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—82no—
271C2×C4⋊C4C22×C412yes—
272C2×C4⋊C4C22×C412yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42C22×C412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42#4112yes—
296C2×C42#4112yes—
297#107—28no—
298C4⋊C8—44no—
299#103—28no—
300#103—48no—
301#102—28no—
302#102—48no—
303#97—28no—
304#95—28no—
305#95—24no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—42no—
311#95—44no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24C22×D411yescharacteristic
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8C2×C4⋊C414yescomplemented
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4C2×C22⋊C414yes—
374C2×D4C2×C4⋊C414yes—
375C2×D4#11714yes—
376C2×D4#12114yes—
377C2×D4#12114yes—
378C2×D4C22⋊Q814yes—
379C2×D4C22⋊Q814yes—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—216no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—48no—
407C22×C4—416no—
408C22×C4—48no—
409C22×C4—416no—
410C22×C4—416no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—416no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4C4×D412yes—
441C22×C4C4×D412yes—
442C22×C4#12012yes—
443C22×C4#12012yes—
444C22×C4#12112yes—
445C22×C4#12112yes—
446C22×C4AGL(1,ℤ/8)14yes—
447C22×C4C23×C411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×Q811yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4C2×#4111yescharacteristic
456C22×C4C2×#4111yescharacteristic
457C22×C4H(5,𝔽2)11yescharacteristic
458C22×C4H(5,𝔽2)11yescharacteristic
459C22×C42−1+411yescharacteristic
460C22×C42−1+411yescharacteristic
461SD16—232no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8C2×C4⋊C414yescomplemented
483C4⋊C4—28no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—416no—
494C4⋊C4—42no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—42no—
498C4⋊C4—48no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—84no—
504C4⋊C4C2×C22⋊C414yes—
505C22⋊C4—216no—
506C22⋊C4—416no—
507C42—42no—
508C42—42no—
509C42—42no—
510C42—42no—
511C42—48no—
512C42C2×C22⋊C414yes—
513C42C4×D414yes—
514C42#12114yes—
515C42#12314yes—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23C2×C4×D411yescharacteristic
540C23C2×#12011yescharacteristic
541C23C2×#12111yescharacteristic
542C23D4211yescharacteristic
543C23#54611yescharacteristic
544C23#54811yescharacteristic
545C23D4×Q811yescharacteristic
546C23C2×AGL(1,ℤ/8)12yes—
547C23#57512yes—
548C23#57712yes—
549C23#57812yes—
550Q8—24no—
551Q8—24no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—42no—
557Q8—42no—
558D4—24no—
559D4—24no—
560D4—24no—
561D4—24no—
562D4—24no—
563D4—24no—
564D4—24no—
565D4#38018yes—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—432no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—832no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4#38614yes—
622C2×C4#38614yes—
623C2×C4C22×C22⋊C411yescharacteristic
624C2×C4C22×C4⋊C411yescharacteristic
625C2×C4C2×#11711yescharacteristic
626C2×C4C2×C4×D411yescharacteristic
627C2×C4#52011yescharacteristic
628C2×C4#52011yescharacteristic
629C2×C4C2×#12111yescharacteristic
630C2×C4C2×#12111yescharacteristic
631C2×C4C2×#12111yescharacteristic
632C2×C4C2×C22⋊Q811yescharacteristic
633C2×C4C2×C22⋊Q811yescharacteristic
634C2×C4C2×#12311yescharacteristic
635C2×C4#53511yescharacteristic
636C2×C4#53611yescharacteristic
637C2×C4#53711yescharacteristic
638C2×C4#53711yescharacteristic
639C2×C4D4211yescharacteristic
640C2×C4#54611yescharacteristic
641C2×C4#54711yescharacteristic
642C2×C4#54711yescharacteristic
643C2×C4#54811yescharacteristic
644C2×C4D4×Q811yescharacteristic
645C2×C4#55211yescharacteristic
646C2×C4#55211yescharacteristic
647C2×C4#55311yescharacteristic
648C2×C4#55311yescharacteristic
649C2×C4#55411yescharacteristic
650C2×C4#55411yescharacteristic
651C8—24no—
652C8—42no—
653C8—82no—
654C22—28no—
655C22—44no—
656C22—44no—
657C22—48no—
658C22—48no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—48no—
664C22—48no—
665C22C2×#38611yescharacteristic
666C22D4×C4⋊C411yescharacteristic
667C22#235211yescharacteristic
668C22#240811yescharacteristic
669C22#240911yescharacteristic
670C22#241611yescharacteristic
671C22#241711yescharacteristic
672C22C4×AGL(1,ℤ/8)12yes—
673C22#294712yes—
674C22#300311yescharacteristic
675C22#300511yescharacteristic
676C22#300911yescharacteristic
677C22#301111yescharacteristic
678C22#303612yes—
679C22#304612yes—
680C22#326512yes—
681C22#326912yes—
682C22#331012yes—
683C22#331512yes—
684C22#336712yes—
685C22#337612yes—
686C22#338612yes—
687C22#338812yes—
688C4—24no—
689C4—24no—
690C4—44no—
691C4—44no—
692C4—44no—
693C4—84no—
694C4—82no—
695C4—82no—
696C4—162no—
697C4C2×#38011yescharacteristic
698C4#231111yescharacteristic
699C4D4×C4⋊C411yescharacteristic
700C4#235211yescharacteristic
701C4#241611yescharacteristic
702C4#241711yescharacteristic
703C4#248611yescharacteristic
704C4#248711yescharacteristic
705C2—48no—
706C2#1387211yescharacteristic
707C2#2092511yescharacteristic
708C2#2093111yescharacteristic
709C2#2267712yes—
710C2#2277812yes—
711C2#2425211yescharacteristic
712C2#2425311yescharacteristic
713C2#2434211yescharacteristic
714C2#2434311yescharacteristic
715C2#2474012yes—
716C2#2482512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1803C2semidirect
6. #13942C2semidirect
7. #11219C2semidirect
8. #11067C2semidirect
9. #10843C2semidirect
10. #10835C2semidirect
19. C2×#440C4semidirect
68. C2×C4×Q8C2×C4semidirect
101. C2×#107C2×C4semidirect
110. C2×#103C2×C4semidirect
119. C2×#97C2×C4semidirect
203. C2×SD16C4⋊C4semidirect
351. C2×Q8C2×C4⋊C4semidirect
482. C2×C8C2×C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4402
C2×D4×C4⋊C42
C2×#18772
C2×#18032
C2×#18042
#139422
#112192
#110672
#108432
#108352
#107882