Finite Group Databasebeta

← #368316

Subgroups — #368316

Order 512.

2052 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2366C211yesmaximal, characteristic
2C2×#1982C211yesmaximal, characteristic
3C2×#1957C211yesmaximal, characteristic
4C2×#1864C212yesmaximal, complemented
5C2×C4×#106C211yesmaximal, characteristic, complemented
6#14151C211yesmaximal, characteristic, complemented
7#11121C212yesmaximal, complemented
8#11066C212yesmaximal, complemented
9#10849C211yesmaximal, characteristic, complemented
10#10834C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal, complemented
12C22×#127C2211yescharacteristic
13C22×C4×D4—21no—
14C22×C4×D4C2212yes—
15C22×#106C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×C4×C8C2211yescharacteristic
23#2366—24no—
24#2366—24no—
25C2×#390—21no—
26C2×#390C2212yes—
27C2×#389—21no—
28C2×#389C2212yes—
29C2×#386—22no—
30C2×#384C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1982—24no—
35#1957C418yescomplemented
36#1864—28no—
37C4×#106—24no—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1441—22no—
45C4⋊#95—21no—
46C4⋊#95C2212yes—
47C23×D4—21no—
48C23×D4—21no—
49C23×D4C2312yes—
50C24×C4—22no—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×#127—22no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127C2×C414yes—
58C2×#127D414yes—
59C2×#127Q814yes—
60C2×C4×D4—28no—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—24no—
65C2×C4×D4—42no—
66C2×C4×D4—42no—
67C2×C4×D4—44no—
68C2×C4×D4—44no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86C2×#106—22no—
87C2×#106—22no—
88C2×#106—22no—
89C2×#106—22no—
90C2×#106—22no—
91C2×#106D414yescomplemented
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—42no—
102C2×#102—42no—
103C2×#102—42no—
104C2×#102—42no—
105C2×#102C2×C418yescomplemented
106C2×#102C2×C418yescomplemented
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8C2×C414yescomplemented
110#390—28no—
111#390—24no—
112#390—24no—
113#389—24no—
114#389—42no—
115#389—44no—
116#386—216no—
117#384—22no—
118#384C2×C414yescomplemented
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4D414yes—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95D412yes—
131#340—24no—
132#340—22no—
133#340—22no—
134#340—42no—
135#336—24no—
136#336—42no—
137C25—22no—
138C22×D4—24no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—24no—
142C22×D4—24no—
143C22×D4—24no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4—81no—
160C22×D4—81no—
161C22×D4—81no—
162C22×D4—81no—
163C22×D4C22×C414yes—
164C22×D4C2×D414yes—
165C22×D4C2×Q814yes—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—28no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4—82no—
177C23×C4—82no—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2411yescharacteristic
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8C22×C412yes—
201C22×C8C22×C412yes—
202C22×C8C2×D412yes—
203C22×C8#4112yes—
204#127—28no—
205#127—28no—
206#127—28no—
207#127C4⋊C4116yescomplemented
208C4×D4—48no—
209C4×D4—48no—
210C4×D4—48no—
211C4×D4—432no—
212C4×D4—48no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—81no—
248C2×C4⋊C4—81no—
249C2×C4⋊C4—82no—
250C2×C4⋊C4SD1614yes—
251C2×C4⋊C4C22×C414yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4#4112yes—
257C2×C4⋊C4#4112yes—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—82no—
266C2×C22⋊C4—82no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42C22×C412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×Q812yes—
283C2×C42#4112yes—
284C2×C42#4112yes—
285#106—28no—
286#106—28no—
287C4⋊C8—44no—
288#102—216no—
289#102—48no—
290#102—48no—
291#102—48no—
292C4×C8—28no—
293#95—24no—
294#95—24no—
295#95—44no—
296#95—44no—
297#95—42no—
298#95—82no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24C22×D411yescharacteristic
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—216no—
331C2×D4—216no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—84no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4—84no—
354C2×D4—84no—
355C2×D4—84no—
356C2×D4C2×C4⋊C418yes—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
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—24no—
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—416no—
382C22×C4—416no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—88no—
412C22×C4—88no—
413C22×C4—88no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4C4×D412yes—
417C22×C4C4×D412yes—
418C22×C4#12012yes—
419C22×C4#12012yes—
420C22×C4C22⋊Q812yes—
421C22×C4C22⋊Q812yes—
422C22×C4C2×SD1614yes—
423C22×C4C2×SD1612yes—
424C22×C4#13512yes—
425C22×C4AGL(1,ℤ/8)12yes—
426C22×C4AGL(1,ℤ/8)12yes—
427C22×C4C23×C411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×Q811yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4H(5,𝔽2)11yescharacteristic
438C22×C4H(5,𝔽2)11yescharacteristic
439C22×C4H(5,𝔽2)11yescharacteristic
440C22×C42−1+411yescharacteristic
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8C4×D414yescomplemented
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—84no—
485C4⋊C4—88no—
486C4⋊C4#12114yes—
487C22⋊C4—416no—
488C22⋊C4—416no—
489C22⋊C4—88no—
490C42—42no—
491C42—42no—
492C42—42no—
493C42—42no—
494C42—48no—
495C42—84no—
496C42C2×C4⋊C414yes—
497C42C4×D414yes—
498C42C22⋊Q814yes—
499C42#12314yes—
500C23—28no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23—82no—
525C23—82no—
526C23—82no—
527C23—82no—
528C23—82no—
529C23—82no—
530C23—82no—
531C23—82no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23C2×C4×D411yescharacteristic
536C23C2×#12011yescharacteristic
537C23C2×C22⋊Q811yescharacteristic
538C23D4211yescharacteristic
539C23#54611yescharacteristic
540C23#54811yescharacteristic
541C23#55011yescharacteristic
542C23C22×SD1611yescharacteristic
543C23C2×#13511yescharacteristic
544C23C2×AGL(1,ℤ/8)11yescharacteristic
545C23C2×AGL(1,ℤ/8)11yescharacteristic
546C23#57611yescharacteristic
547C23#57611yescharacteristic
548C23#57711yescharacteristic
549C23#57711yescharacteristic
550D4—28no—
551D4—28no—
552D4—44no—
553D4—44no—
554D4—432no—
555D4—44no—
556D4—44no—
557D4—44no—
558D4—44no—
559D4—816no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—832no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—81no—
616C2×C4#38514yes—
617C2×C4#38514yes—
618C2×C4C4×SD1614yes—
619C2×C4#44214yes—
620C2×C4#44914yes—
621C2×C4#45014yes—
622C2×C4#47614yes—
623C2×C4#47814yes—
624C2×C4C22×C4⋊C411yescharacteristic
625C2×C4C2×C4×D411yescharacteristic
626C2×C4C2×C4×D411yescharacteristic
627C2×C4C4×#4111yescharacteristic
628C2×C4#51811yescharacteristic
629C2×C4#52011yescharacteristic
630C2×C4C2×#12111yescharacteristic
631C2×C4C2×C22⋊Q811yescharacteristic
632C2×C4C2×#12311yescharacteristic
633C2×C4#52511yescharacteristic
634C2×C4#52911yescharacteristic
635C2×C4#53211yescharacteristic
636C2×C4#53511yescharacteristic
637C2×C4#53711yescharacteristic
638C2×C4#54011yescharacteristic
639C2×C4#54211yescharacteristic
640C2×C4D4211yescharacteristic
641C2×C4#54611yescharacteristic
642C2×C4#54611yescharacteristic
643C2×C4#54711yescharacteristic
644C2×C4D4×Q811yescharacteristic
645C2×C4#55011yescharacteristic
646C2×C4#55411yescharacteristic
647C2×C4#55411yescharacteristic
648C2×C4#55411yescharacteristic
649C2×C4#55611yescharacteristic
650C2×C4#55911yescharacteristic
651C2×C4#55911yescharacteristic
652C8—24no—
653C8—42no—
654C8—82no—
655C22—48no—
656C22—48no—
657C22—48no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—84no—
664C22—84no—
665C22—84no—
666C22—84no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22C2×#38511yescharacteristic
671C22#236611yescharacteristic
672C22#236711yescharacteristic
673C22#240711yescharacteristic
674C22#240811yescharacteristic
675C22#245211yescharacteristic
676C22#245411yescharacteristic
677C22C2×C4×SD1611yescharacteristic
678C22C2×#44211yescharacteristic
679C22#294811yescharacteristic
680C22#294911yescharacteristic
681C22C2×#45011yescharacteristic
682C22C2×#44911yescharacteristic
683C22#301211yescharacteristic
684C22#301211yescharacteristic
685C22C2×#47611yescharacteristic
686C22C2×#47811yescharacteristic
687C22#308111yescharacteristic
688C22#308111yescharacteristic
689C22#327411yescharacteristic
690C22#327411yescharacteristic
691C22D4×SD1611yescharacteristic
692C22#328411yescharacteristic
693C22#329511yescharacteristic
694C22#330511yescharacteristic
695C22#332411yescharacteristic
696C22#333211yescharacteristic
697C22#336111yescharacteristic
698C22#336211yescharacteristic
699C22#336811yescharacteristic
700C22#337311yescharacteristic
701C22#339211yescharacteristic
702C22#339411yescharacteristic
703C22#339911yescharacteristic
704C22#339911yescharacteristic
705C4—24no—
706C4—44no—
707C4—44no—
708C4—82no—
709C4—82no—
710C4—84no—
711C4—82no—
712C4—82no—
713C4#186418yescomplemented
714C4C4×#12111yescharacteristic
715C4#232411yescharacteristic
716C4D4×C4⋊C411yescharacteristic
717C4#234911yescharacteristic
718C4#244511yescharacteristic
719C4#244711yescharacteristic
720C4#253811yescharacteristic
721C4#258011yescharacteristic
722C2—88no—
723C2#1401811yescharacteristic
724C2C2×#186411yescharacteristic
725C2#2087911yescharacteristic
726C2#2318211yescharacteristic
727C2#2318811yescharacteristic
728C2#2321511yescharacteristic
729C2#2322011yescharacteristic
730C2#2418611yescharacteristic
731C2#2418811yescharacteristic
732C2#2423811yescharacteristic
733C2#2424011yescharacteristic
734C2#2646311yescharacteristic
735C2#2646411yescharacteristic
736C2#2652911yescharacteristic
737C2#2653011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
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×#1864C2semidirect
5. C2×C4×#106C2semidirect
6. #14151C2semidirect
7. #11121C2semidirect
8. #11066C2semidirect
9. #10849C2semidirect
10. #10834C2semidirect
11. #10788C2semidirect
35. #1957C4semidirect
91. C2×#106D4semidirect
105. C2×#102C2×C4semidirect
106. C2×#102C2×C4semidirect
109. C2×C4×C8C2×C4semidirect
118. #384C2×C4semidirect
207. #127C4⋊C4semidirect
461. C2×C8C4×D4semidirect
713. C4#1864semidirect

Maximal subgroups

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

subgroupindex
C2×#23662
C2×#19822
C2×#19572
C2×#18642
C2×C4×#1062
#141512
#111212
#110662
#108492
#108342
#107882