Finite Group Databasebeta

← #360148

Subgroups — #360148

Order 512.

1760 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#441C211yesmaximal, characteristic
2C2×#2351C211yesmaximal, characteristic
3C2×#1878C212yesmaximal
4C2×#1805C211yesmaximal, characteristic
5C2×#1803C211yesmaximal, characteristic
6#13926C211yesmaximal, characteristic
7#11220C212yesmaximal
8#11067C212yesmaximal
9#10843C211yesmaximal, characteristic
10#10835C211yesmaximal, characteristic
11#10789C212yesmaximal
12C23×Q16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C22×C4×Q8C2211yescharacteristic
16C2×#441—24no—
17C2×#441—24no—
18C2×#441C418yescomplemented
19C22×#106C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#97C2211yescharacteristic
26#2351—24no—
27#2351—24no—
28C2×#391C2212yes—
29C2×#389—22no—
30C2×#389C2212yes—
31C2×#384C2211yescharacteristic
32C2×#382—21no—
33C2×#382C2211yescharacteristic
34C2×C4×C4⋊C4—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1878—28no—
37#1805—24no—
38#1803—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1441—21no—
46#1441—21no—
47C4⋊#95—22no—
48C4⋊#95C2212yes—
49C23×Q8C2311yescharacteristic
50C23×Q8C2311yescharacteristic
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—22no—
54C22×Q16—22no—
55C22×Q16C2×C414yes—
56C22×Q16D414yes—
57C22×Q16Q814yes—
58C23×C8C2312yes—
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4×Q8—24no—
62C2×C4×Q8—24no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—24no—
68C2×C4×Q8—24no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—42no—
74C2×C4×Q8—42no—
75C2×C4×Q8C2×C414yes—
76C2×C4×Q8C2×C414yes—
77C2×C4×Q8D414yes—
78C2×C4×Q8D414yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D412yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C42—21no—
91C22×C42—21no—
92C22×C42—22no—
93C22×C42C2312yes—
94C22×C42C2311yescharacteristic
95#441—232no—
96C2×#106—22no—
97C2×#106—22no—
98C2×#106C2×C414yes—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103—42no—
112C2×#103C2×C414yes—
113C2×#103C2×C414yes—
114C2×#97—22no—
115C2×#97—22no—
116C2×#97C2×C414yes—
117#391—24no—
118#391—42no—
119#389—216no—
120#389—24no—
121#389—24no—
122#384—22no—
123#384—22no—
124#382—28no—
125#382—22no—
126#382D414yes—
127C4×C4⋊C4—28no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95D411yescharacteristic
138C2×#95D411yescharacteristic
139#340—24no—
140#340—24no—
141#340—24no—
142#340—42no—
143#336—24no—
144#336—42no—
145C22×Q8—21no—
146C22×Q8—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×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8C22×C412yes—
164C22×Q8C22×C412yes—
165C22×Q8C2×D412yes—
166C22×Q8C2×D412yes—
167C22×Q8C2×D412yes—
168C22×Q8C2×D412yes—
169C22×Q8C2×Q812yes—
170C22×Q8C2×Q812yes—
171C22×Q8#4112yes—
172C22×Q8#4112yes—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C2×Q16—28no—
185C2×Q16—28no—
186C2×Q16—28no—
187C2×Q16—28no—
188C2×Q16—28no—
189C2×Q16—28no—
190C2×Q16C4⋊C4116yescomplemented
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8C22×C412yes—
209C22×C8C22×C412yes—
210C22×C8C2×D412yes—
211C22×C8C2×Q812yes—
212C4×Q8—28no—
213C4×Q8—28no—
214C4×Q8—28no—
215C4×Q8—28no—
216C4×Q8—28no—
217C4×Q8—48no—
218C4×Q8—48no—
219C4×Q8—48no—
220C4×Q8—48no—
221C4×Q8—44no—
222C4×Q8—44no—
223C4×Q8C22⋊C4116yes—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—82no—
262C2×C4⋊C4C22×C412yes—
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—28no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—44no—
279C2×C42—44no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42C22×C412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42#4112yes—
287C2×C42#4112yes—
288#106—28no—
289C4⋊C8—44no—
290#103—28no—
291#103—28no—
292#103—48no—
293#103—48no—
294#97—28no—
295#95—28no—
296#95—24no—
297#95—24no—
298#95—22no—
299#95—22no—
300#95—44no—
301#95—44no—
302#95—82no—
303#95—82no—
304C24C22×D411yescharacteristic
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
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—22no—
340C2×Q8—22no—
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×Q8—41no—
352C2×Q8—41no—
353C2×Q8C2×C22⋊C414yes—
354C2×Q8C2×C4⋊C414yes—
355C2×Q8C2×C4⋊C414yes—
356C2×Q8#11714yes—
357C2×Q8#12114yes—
358C2×Q8#12114yes—
359C2×Q8C22⋊Q814yes—
360C2×Q8C22⋊Q814yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4C4×D412yes—
417C22×C4C4×D412yes—
418C22×C4#12012yes—
419C22×C4#12012yes—
420C22×C4#12112yes—
421C22×C4#12112yes—
422C22×C4#13714yes—
423C22×C4C23×C411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×Q811yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4H(5,𝔽2)11yescharacteristic
434C22×C4H(5,𝔽2)11yescharacteristic
435C22×C42−1+411yescharacteristic
436C22×C42−1+411yescharacteristic
437Q16—232no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8C2×C4⋊C414yes—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—216no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—48no—
468C4⋊C4—416no—
469C4⋊C4—416no—
470C4⋊C4—42no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—42no—
474C4⋊C4—48no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C4⋊C4C2×C22⋊C414yes—
482C42—216no—
483C42—416no—
484C42—42no—
485C42—42no—
486C42—42no—
487C42—42no—
488C42—48no—
489C42C2×C22⋊C414yes—
490C42C4×D414yes—
491C42#12114yes—
492C42#12314yes—
493C23C2×C4×D411yescharacteristic
494C23C2×#12011yescharacteristic
495C23C2×#12111yescharacteristic
496C23D4211yescharacteristic
497C23#54611yescharacteristic
498C23#54811yescharacteristic
499C23D4×Q811yescharacteristic
500C23C2×#13712yes—
501C23#57512yes—
502C23#57712yes—
503C23#57812yes—
504Q8—24no—
505Q8—24no—
506Q8—24no—
507Q8—24no—
508Q8—24no—
509Q8—24no—
510Q8—24no—
511Q8—24no—
512Q8—24no—
513Q8—42no—
514Q8—42no—
515Q8—42no—
516Q8—42no—
517Q8—42no—
518Q8—42no—
519Q8#38018yes—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4#38614yes—
581C2×C4#38614yes—
582C2×C4C22×C22⋊C411yescharacteristic
583C2×C4C22×C4⋊C411yescharacteristic
584C2×C4C2×#11711yescharacteristic
585C2×C4C2×C4×D411yescharacteristic
586C2×C4#52011yescharacteristic
587C2×C4#52011yescharacteristic
588C2×C4C2×#12111yescharacteristic
589C2×C4C2×#12111yescharacteristic
590C2×C4C2×#12111yescharacteristic
591C2×C4C2×C22⋊Q811yescharacteristic
592C2×C4C2×C22⋊Q811yescharacteristic
593C2×C4C2×#12311yescharacteristic
594C2×C4#53511yescharacteristic
595C2×C4#53611yescharacteristic
596C2×C4#53711yescharacteristic
597C2×C4#53711yescharacteristic
598C2×C4D4211yescharacteristic
599C2×C4#54611yescharacteristic
600C2×C4#54711yescharacteristic
601C2×C4#54711yescharacteristic
602C2×C4#54811yescharacteristic
603C2×C4D4×Q811yescharacteristic
604C2×C4#55211yescharacteristic
605C2×C4#55211yescharacteristic
606C2×C4#55311yescharacteristic
607C2×C4#55311yescharacteristic
608C2×C4#55411yescharacteristic
609C2×C4#55411yescharacteristic
610C8—24no—
611C8—42no—
612C8—82no—
613C22C2×#38611yescharacteristic
614C22D4×C4⋊C411yescharacteristic
615C22#235211yescharacteristic
616C22#240811yescharacteristic
617C22#240911yescharacteristic
618C22#241611yescharacteristic
619C22#241711yescharacteristic
620C22C4×#13712yes—
621C22#294712yes—
622C22#300712yes—
623C22#301312yes—
624C22#303911yescharacteristic
625C22#304011yescharacteristic
626C22#304311yescharacteristic
627C22#304411yescharacteristic
628C22#327012yes—
629C22#327112yes—
630C22#331312yes—
631C22#331612yes—
632C22#337112yes—
633C22#337412yes—
634C22#338712yes—
635C22#339012yes—
636C4—28no—
637C4—24no—
638C4—24no—
639C4—44no—
640C4—44no—
641C4—44no—
642C4—82no—
643C4—82no—
644C4—84no—
645C4—162no—
646C4C2×#38011yescharacteristic
647C4#231111yescharacteristic
648C4D4×C4⋊C411yescharacteristic
649C4#235211yescharacteristic
650C4#241611yescharacteristic
651C4#241711yescharacteristic
652C4#248611yescharacteristic
653C4#248711yescharacteristic
654C2#1387211yescharacteristic
655C2#2092711yescharacteristic
656C2#2093011yescharacteristic
657C2#2267912yes—
658C2#2277712yes—
659C2#2425812yes—
660C2#2434812yes—
661C2#2474611yescharacteristic
662C2#2474711yescharacteristic
663C2#2483111yescharacteristic
664C2#2483211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
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
18. C2×#441C4semidirect
190. C2×Q16C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4412
C2×#23512
C2×#18782
C2×#18052
C2×#18032
#139262
#112202
#110672
#108432
#108352
#107892