Finite Group Databasebeta

← #460397

Subgroups — #460397

Order 512.

1552 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2429C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic, complemented
3C2×#2043C212yesmaximal, complemented
4C2×#2035C212yesmaximal
5C2×#1803C211yesmaximal, characteristic, complemented
6#15135C211yesmaximal, characteristic
7#11327C212yesmaximal, complemented
8#11077C211yesmaximal, characteristic, complemented
9#11042C211yesmaximal, characteristic
10#10970C212yesmaximal
11#10789C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C22×#106C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×C22⋊C8C2212yes—
23#2429—24no—
24#2429—24no—
25C2×#399—21no—
26C2×#398—21no—
27C2×#396—21no—
28C2×#391C2211yescharacteristic
29C2×#389—21no—
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#388—21no—
33C2×#384C2211yescharacteristic
34C2×#380C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36#2096—24no—
37#2043—28no—
38#2035—28no—
39#1803—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2211yescharacteristic
44C2×#336C2212yes—
45#1445—21no—
46#1444—21no—
47#1439—21no—
48C4⋊#95C2211yescharacteristic
49C23×Q8C2311yescharacteristic
50C23×Q8C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—42no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—42no—
65C2×C22⋊Q8—42no—
66C2×C4×Q8—22no—
67C2×C4×Q8—24no—
68C2×C4×Q8—42no—
69C2×C4×Q8D414yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4D411yescharacteristic
87C22×C22⋊C4D411yescharacteristic
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—42no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—42no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—42no—
103C2×#103—42no—
104C2×#103—42no—
105C2×#103—42no—
106C2×#103—42no—
107C2×#103D418yes—
108C2×C22⋊C8—24no—
109C2×C22⋊C8—42no—
110C2×C22⋊C8—42no—
111#399—44no—
112#398—28no—
113#396—28no—
114#391—22no—
115#391D414yes—
116#389—28no—
117#389—22no—
118#389—22no—
119#389—24no—
120#389—42no—
121#388—28no—
122#384—22no—
123#384—22no—
124#380—24no—
125#380—42no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95D411yescharacteristic
136C2×#95D411yescharacteristic
137#340—24no—
138#340—24no—
139#340—24no—
140#336—24no—
141#336—22no—
142C25C2×D411yescharacteristic
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—21no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8C2×D412yes—
161C22×Q8C2×D412yes—
162C22×Q8C2×D414yes—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8C2×D414yes—
199C4⋊Q8—48no—
200C4⋊Q8—44no—
201C4⋊Q8—44no—
202C22⋊Q8—48no—
203C22⋊Q8—48no—
204C22⋊Q8—44no—
205C22⋊Q8—44no—
206C22⋊Q8—44no—
207C22⋊Q8—44no—
208C4×Q8—28no—
209C4×Q8—44no—
210C4×Q8—48no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—82no—
247C2×C4⋊C4—82no—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4#4114yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—82no—
264C2×C22⋊C4SD1614yes—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—22no—
272C2×C42—44no—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275#106—48no—
276C4⋊C8—44no—
277#103—48no—
278#103—48no—
279#103—48no—
280#103—48no—
281C22⋊C8—48no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—22no—
286#95—22no—
287#95—44no—
288#95—44no—
289#95—42no—
290#95—42no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—22no—
295C24—22no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—22no—
300C24—22no—
301C24C2×SD1612yes—
302C24C22×D411yescharacteristic
303C2×Q8—22no—
304C2×Q8—24no—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—22no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8#12714yes—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
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#12012yes—
411C22×C4#12012yes—
412C22×C4C2×SD1612yes—
413C22×C4C2×SD1612yes—
414C22×C4#13712yes—
415C22×C4#13712yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C2×#4112yes—
420C22×C4H(5,𝔽2)11yescharacteristic
421C22×C4H(5,𝔽2)12yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C42−1+411yescharacteristic
424C22×C42−1+411yescharacteristic
425C22×C42−1+411yescharacteristic
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—88no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C4⋊C4#12314yes—
463C22⋊C4—28no—
464C22⋊C4—44no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—88no—
468C42—22no—
469C42—22no—
470C42—24no—
471C42—48no—
472C42—41no—
473C42—41no—
474C42—42no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—42no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—41no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23C2×#12011yescharacteristic
499C23#53611yescharacteristic
500C23#53711yescharacteristic
501C23D4212yes—
502C23#54612yes—
503C23C22×SD1611yescharacteristic
504C23C2×#13711yescharacteristic
505C23C2×#13711yescharacteristic
506C23#57511yescharacteristic
507C23#57712yes—
508C23#57812yes—
509Q8—24no—
510Q8—24no—
511Q8—42no—
512Q8—42no—
513Q8—44no—
514Q8—44no—
515Q8—44no—
516Q8—44no—
517Q8—81no—
518Q8—81no—
519Q8—81no—
520Q8—81no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—81no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4#44814yes—
586C2×C4C2×#12311yescharacteristic
587C2×C4C2×#12711yescharacteristic
588C2×C4#53411yescharacteristic
589C2×C4#53412yes—
590C2×C4#53511yescharacteristic
591C2×C4#53611yescharacteristic
592C2×C4#53611yescharacteristic
593C2×C4#53711yescharacteristic
594C2×C4#53712yes—
595C2×C4#53711yescharacteristic
596C2×C4#54012yes—
597C2×C4#54212yes—
598C2×C4D4212yes—
599C2×C4#54612yes—
600C2×C4#54712yes—
601C2×C4#54712yes—
602C2×C4#56212yes—
603C2×C4#56212yes—
604C8—44no—
605C8—82no—
606C22—44no—
607C22—42no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—81no—
612C22—82no—
613C22—81no—
614C22—82no—
615C22—81no—
616C22—81no—
617C22#240511yescharacteristic
618C22#243311yescharacteristic
619C22#261611yescharacteristic
620C22#267112yes—
621C22#267212yes—
622C22C2×#44811yescharacteristic
623C22#300711yescharacteristic
624C22#301011yescharacteristic
625C22#301111yescharacteristic
626C22#322911yescharacteristic
627C22#323711yescharacteristic
628C22#324212yes—
629C22#324611yescharacteristic
630C22#325511yescharacteristic
631C22#325812yes—
632C22#326912yes—
633C22#327012yes—
634C22#327812yes—
635C22D4×SD1612yes—
636C22#329812yes—
637C22#330712yes—
638C22#331212yes—
639C22#333112yes—
640C4—24no—
641C4—42no—
642C4—42no—
643C4—44no—
644C4—42no—
645C4—44no—
646C4—84no—
647C4—82no—
648C4—81no—
649C4—82no—
650C4—162no—
651C4#250111yescharacteristic
652C4#262411yescharacteristic
653C4#265611yescharacteristic
654C4#267112yes—
655C4#267212yes—
656C4#281611yescharacteristic
657C2—82no—
658C2#1636311yescharacteristic
659C2#2404111yescharacteristic
660C2#2405511yescharacteristic
661C2#2575611yescharacteristic
662C2#2576711yescharacteristic
663C2#2968811yescharacteristic
664C2#2969911yescharacteristic
665C2#3059712yes—
666C2#3062012yes—
667C2#3063312yes—
668C2#3065212yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2096C2semidirect
3. C2×#2043C2semidirect
5. C2×#1803C2semidirect
7. #11327C2semidirect
8. #11077C2semidirect
11. #10789C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24292
C2×#20962
C2×#20432
C2×#20352
C2×#18032
#151352
#113272
#110772
#110422
#109702
#107892