Finite Group Databasebeta

← #364659

Subgroups — #364659

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic
2C2×#2362C211yesmaximal, characteristic
3C2×#1894C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5#13924C211yesmaximal, characteristic
6#11220C212yesmaximal, complemented
7#11190C212yesmaximal, complemented
8#10908C212yesmaximal, complemented
9#10789C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#483—24no—
18C2×#483—24no—
19C2×#483C418yescomplemented
20C22×#106C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24#2362—24no—
25#2362—24no—
26C2×#391C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#388—21no—
29C2×#388—21no—
30C2×#387—21no—
31C2×#380C2211yescharacteristic
32C2×C4×C4⋊C4—21no—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35#1894—24no—
36#1736—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1445—21no—
45#1445—21no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C22⋊#95—21no—
49C22⋊#95—21no—
50C23×Q8C2311yescharacteristic
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—24no—
57C22×M16—42no—
58C23×C8C2312yes—
59C23×C8C2312yes—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8C2×C414yes—
64C2×C22⋊Q8D414yes—
65C2×C22⋊Q8D414yes—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—24no—
69C2×C4×Q8—42no—
70C2×#117—22no—
71C2×#117—24no—
72C2×#117—42no—
73C2×#117D414yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4C2×C414yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4D411yescharacteristic
94C22×C42—21no—
95C22×C42—21no—
96C22×C42—21no—
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99#483—232no—
100C2×#106—24no—
101C2×#106—24no—
102C2×#106C2×C418yes—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—42no—
108C2×#103C2×C418yes—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8C2×C414yes—
112#391—22no—
113#391—22no—
114#389—22no—
115#389—22no—
116#388—28no—
117#388—28no—
118#387—28no—
119#380—22no—
120#380—22no—
121C4×C4⋊C4—28no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95D411yescharacteristic
136C2×#95D411yescharacteristic
137#340—24no—
138#340—24no—
139#340—24no—
140#340—24no—
141#340—42no—
142#336—24no—
143#336—42no—
144C25C2×D411yescharacteristic
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8C22×C412yes—
153C22×Q8C2×D412yes—
154C22×Q8C2×D412yes—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—82no—
169C23×C4—82no—
170C23×C4C22×C412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4#4112yes—
180C23×C4#4112yes—
181C23×C4C2411yescharacteristic
182C2×M16—28no—
183C2×M16—48no—
184C2×M16—48no—
185C2×M16—44no—
186C2×M16—44no—
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×C8—22no—
199C22×C8C22×C414yes—
200C22×C8#4114yes—
201C22⋊Q8—28no—
202C22⋊Q8—28no—
203C22⋊Q8—28no—
204C22⋊Q8C22⋊C4116yes—
205C4×Q8—48no—
206C4×Q8—44no—
207C4×Q8—44no—
208#117—28no—
209#117—44no—
210#117—48no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4C22×C414yes—
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4#4114yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—44no—
278C2×C42—44no—
279C2×C42—42no—
280C2×C42—42no—
281#106—216no—
282#103—216no—
283#103—48no—
284C22⋊C8—28no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—22no—
290#95—22no—
291#95—42no—
292#95—82no—
293#95—82no—
294#95—82no—
295#95—82no—
296C24—22no—
297C24—22no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—22no—
303C24—22no—
304C24C4×D412yes—
305C24#12012yes—
306C24#12112yes—
307C24C22×D411yescharacteristic
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—42no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8C2×C22⋊C414yes—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—48no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—88no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4—88no—
401C22×C4C2×C22⋊C414yes—
402C22×C4C4×D414yes—
403C22×C4C4×D412yes—
404C22×C4#12012yes—
405C22×C4#12114yes—
406C22×C4#12112yes—
407C22×C4#12314yes—
408C22×C4#13712yes—
409C22×C4#13712yes—
410C22×C4C23×C411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4111yescharacteristic
418C22×C4H(5,𝔽2)12yes—
419C22×C42−1+412yes—
420M16—416no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—216no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—48no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4C2×C22⋊C414yes—
456C4⋊C4#12314yes—
457C22⋊C4—216no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—84no—
462C42—48no—
463C42—48no—
464C42—48no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—41no—
469C42—42no—
470C42—42no—
471C23—24no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23#38614yes—
494C23C2×C4×D411yescharacteristic
495C23C2×#12011yescharacteristic
496C23C2×#12111yescharacteristic
497C23#54612yes—
498C23#54712yes—
499C23C2×#13711yescharacteristic
500C23C2×#13711yescharacteristic
501C23#57511yescharacteristic
502C23#57511yescharacteristic
503C23#57712yes—
504C23#57712yes—
505Q8—28no—
506Q8—42no—
507Q8—44no—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—816no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4#38614yes—
579C2×C4C22×C22⋊C411yescharacteristic
580C2×C4C2×C4×D411yescharacteristic
581C2×C4C4×#4112yes—
582C2×C4#52012yes—
583C2×C4C2×#12111yescharacteristic
584C2×C4C2×#12311yescharacteristic
585C2×C4C2×#12311yescharacteristic
586C2×C4#52511yescharacteristic
587C2×C4#52912yes—
588C2×C4#53411yescharacteristic
589C2×C4#53611yescharacteristic
590C2×C4#53912yes—
591C2×C4#54612yes—
592C2×C4#54612yes—
593C2×C4#54712yes—
594C2×C4#54712yes—
595C2×C4#55112yes—
596C2×C4#55212yes—
597C8—44no—
598C8—44no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—42no—
609C22—42no—
610C22C2×#38611yescharacteristic
611C22#235912yes—
612C22#240511yescharacteristic
613C22#240611yescharacteristic
614C22#242512yes—
615C22C4×#13711yescharacteristic
616C22C4×#13711yescharacteristic
617C22#294912yes—
618C22#300711yescharacteristic
619C22#300711yescharacteristic
620C22#301011yescharacteristic
621C22#301011yescharacteristic
622C22#303911yescharacteristic
623C22#304011yescharacteristic
624C22#304312yes—
625C22#330712yes—
626C22#331612yes—
627C22#332612yes—
628C22#333112yes—
629C22#334412yes—
630C22#335012yes—
631C22#335112yes—
632C22#335612yes—
633C4—24no—
634C4—42no—
635C4—44no—
636C4—44no—
637C4—42no—
638C4—44no—
639C4—81no—
640C4—84no—
641C4—82no—
642C4—81no—
643C4—162no—
644C4C4×#12311yescharacteristic
645C4#232211yescharacteristic
646C4#233912yes—
647C4#241912yes—
648C4#248611yescharacteristic
649C4#253411yescharacteristic
650C2—44no—
651C2#1396811yescharacteristic
652C2#2092711yescharacteristic
653C2#2093611yescharacteristic
654C2#2292712yes—
655C2#2294512yes—
656C2#2405211yescharacteristic
657C2#2405511yescharacteristic
658C2#2410811yescharacteristic
659C2#2411111yescharacteristic
660C2#2521912yes—
661C2#2523112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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
6. #11220C2semidirect
7. #11190C2semidirect
8. #10908C2semidirect
9. #10789C2semidirect
19. C2×#483C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
C2×#23622
C2×#18942
C2×#17362
#139242
#112202
#111902
#109082
#107892
#107312
#107152