Finite Group Databasebeta

← #374176

Subgroups — #374176

Order 512.

1940 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic
3C2×#1892C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic
5#14139C211yesmaximal, characteristic
6#11220C212yesmaximal, complemented
7#11067C212yesmaximal, complemented
8#10932C212yesmaximal, complemented
9#10908C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#117—21no—
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#483—24no—
18C2×#483—24no—
19C2×#483C418yescomplemented
20C22×#106C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26#2344—24no—
27#2344—24no—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#387—22no—
31C2×#381—21no—
32C2×#380C2211yescharacteristic
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C22⋊C4C2211yescharacteristic
35#1892—24no—
36C2×#340—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1441—22no—
45C4⋊#95C2212yes—
46C22⋊#95—21no—
47C22⋊#95—21no—
48C23×Q8C2311yescharacteristic
49C24×C4—21no—
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C23×C8—22no—
53C23×C8—24no—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—42no—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C23×C8C2312yes—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8C2×C414yes—
66C2×#117—28no—
67C2×#117—44no—
68C2×#117—44no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4C2×C414yes—
81C22×C4⋊C4D414yes—
82C22×C4⋊C4D414yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4D411yescharacteristic
91C22×C22⋊C4D411yescharacteristic
92C22×C42—22no—
93C22×C42C2312yes—
94#483—232no—
95C2×#106—22no—
96C2×#106—22no—
97C2×#106—22no—
98C2×#106—22no—
99C2×#106C2×C414yescomplemented
100C2×#106C2×C414yescomplemented
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—42no—
108C2×#103C2×C414yescomplemented
109C2×#103C2×C414yescomplemented
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8C2×C414yes—
113#391—24no—
114#391—42no—
115#389—24no—
116#389—24no—
117#387—216no—
118#381—28no—
119#380—22no—
120#380D414yes—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—24no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95D412yes—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—42no—
138#336—24no—
139#336—42no—
140C25C2×D411yescharacteristic
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8C22×C412yes—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—44no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—82no—
173C23×C4C22×C412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D412yes—
180C23×C4#4112yes—
181C23×C4#4112yes—
182C23×C4C2411yescharacteristic
183C22×C8—28no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
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—48no—
199C22×C8—48no—
200C22×C8—44no—
201C22×C8—44no—
202C22×C8C22×C412yes—
203C22×C8C22×C412yes—
204C22×C8#4114yes—
205C22⋊Q8—28no—
206C22⋊Q8—28no—
207C22⋊Q8—44no—
208C22⋊Q8—44no—
209#117—432no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4C22⋊C4116yes—
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C22×C412yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4#4114yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4SD1614yes—
270C2×C42—28no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—44no—
278C2×C42—44no—
279#106—28no—
280#106—28no—
281#103—28no—
282#103—28no—
283#103—48no—
284C22⋊C8—28no—
285#95—28no—
286#95—28no—
287#95—24no—
288#95—24no—
289#95—44no—
290#95—44no—
291#95—82no—
292#95—82no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24C4×D412yes—
298C24#12012yes—
299C24#12112yes—
300C24C2×SD1612yes—
301C24#13512yes—
302C24AGL(1,ℤ/8)12yes—
303C24#13712yes—
304C24C22×D411yescharacteristic
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—42no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—216no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—88no—
399C22×C4—88no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4C2×C22⋊C414yes—
403C22×C4C4×D414yes—
404C22×C4C4×D412yes—
405C22×C4#12012yes—
406C22×C4#12114yes—
407C22×C4#12112yes—
408C22×C4#12314yes—
409C22×C4C2×SD1614yes—
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—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—416no—
428C2×C8—42no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—42no—
432C2×C8—41no—
433C2×C8—42no—
434C2×C8—41no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—216no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—48no—
452C4⋊C4—416no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—48no—
460C4⋊C4C2×C22⋊C414yes—
461C4⋊C4C2×C22⋊C414yes—
462C22⋊C4—28no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—44no—
466C42—416no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—42no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23#38614yes—
488C23C4×SD1614yes—
489C23#44014yes—
490C23#45014yes—
491C23#45214yes—
492C23#46114yes—
493C23#46314yes—
494C23C2×C4×D411yescharacteristic
495C23C2×#12011yescharacteristic
496C23C2×#12111yescharacteristic
497C23#54612yes—
498C23#54712yes—
499C23C22×SD1611yescharacteristic
500C23C2×#13511yescharacteristic
501C23C2×AGL(1,ℤ/8)11yescharacteristic
502C23C2×#13711yescharacteristic
503C23#57712yes—
504C23#57812yes—
505Q8—42no—
506Q8—44no—
507Q8—42no—
508Q8—81no—
509Q8—81no—
510Q8—82no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—42no—
527C2×C4—432no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—816no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4#38614yes—
576C2×C4C22×C22⋊C411yescharacteristic
577C2×C4C2×C4×D411yescharacteristic
578C2×C4C4×#4112yes—
579C2×C4#52012yes—
580C2×C4C2×#12111yescharacteristic
581C2×C4C2×#12311yescharacteristic
582C2×C4#52511yescharacteristic
583C2×C4#52511yescharacteristic
584C2×C4#53212yes—
585C2×C4#53511yescharacteristic
586C2×C4#53611yescharacteristic
587C2×C4#53712yes—
588C2×C4#54612yes—
589C2×C4#54712yes—
590C2×C4#55112yes—
591C2×C4#55112yes—
592C2×C4#55212yes—
593C2×C4#55212yes—
594C8—42no—
595C8—44no—
596C8—42no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—28no—
601C22—28no—
602C22—24no—
603C22—24no—
604C22—28no—
605C22—24no—
606C22—28no—
607C22#187718yes—
608C22C2×#38611yescharacteristic
609C22#235912yes—
610C22#241111yescharacteristic
611C22#241311yescharacteristic
612C22#243012yes—
613C22C2×C4×SD1611yescharacteristic
614C22C2×#44011yescharacteristic
615C22#294712yes—
616C22C2×#45011yescharacteristic
617C22C2×#45211yescharacteristic
618C22#301312yes—
619C22C2×#46111yescharacteristic
620C22C2×#46311yescharacteristic
621C22#304311yescharacteristic
622C22#304411yescharacteristic
623C22#329612yes—
624C22#329812yes—
625C22#330612yes—
626C22#331212yes—
627C22#333512yes—
628C22#333612yes—
629C22#334312yes—
630C22#334412yes—
631C4—24no—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—82no—
636C4—84no—
637C4—84no—
638C4—82no—
639C4—82no—
640C4#229711yescharacteristic
641C4#231111yescharacteristic
642C4#235912yes—
643C4#241111yescharacteristic
644C4#241311yescharacteristic
645C4#249812yes—
646C2—28no—
647C2#1413511yescharacteristic
648C2C2×#187711yescharacteristic
649C2#2093011yescharacteristic
650C2#2292012yes—
651C2#2293412yes—
652C2#2447711yescharacteristic
653C2#2448011yescharacteristic
654C2#2457711yescharacteristic
655C2#2458011yescharacteristic
656C2#2553112yes—
657C2#2554112yes—
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. #11067C2semidirect
8. #10932C2semidirect
9. #10908C2semidirect
19. C2×#483C4semidirect
99. C2×#106C2×C4semidirect
100. C2×#106C2×C4semidirect
108. C2×#103C2×C4semidirect
109. C2×#103C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
C2×#23442
C2×#18922
C22×#3402
#141392
#112202
#110672
#109322
#109082
#107302
#107142