Finite Group Databasebeta

← #368186

Subgroups — #368186

Order 512.

1948 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×#1872C211yesmaximal, characteristic
3C2×#1872C211yesmaximal, characteristic
4C2×#1852C211yesmaximal, characteristic
5C2×C4⋊#95C211yesmaximal, characteristic
6#11293C212yesmaximal, complemented
7#11124C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10789C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10712C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C22×#107C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C4×C22⋊Q8—24no—
23C4×C22⋊Q8—24no—
24C2×#391C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#387—21no—
27C2×#385—21no—
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4—21no—
32C22×#95—21no—
33C22×#95—21no—
34#1872—24no—
35#1872C418yescomplemented
36#1852—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C4⋊#95—21no—
45C4⋊#95—21no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C24×C4—21no—
50C24×C4—21no—
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4C2311yescharacteristic
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8C2×C414yes—
60C2×C22⋊Q8D414yes—
61C2×C22⋊Q8Q814yes—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—24no—
65C2×C4×Q8—42no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—42no—
76C22×C4⋊C4—42no—
77C22×C4⋊C4—42no—
78C22×C4⋊C4C2×C414yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C42—22no—
87C22×C42—22no—
88C22×C42—21no—
89C22×C42—21no—
90C22×C42—24no—
91C22×C42—42no—
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C2×#107—24no—
97C2×#107—24no—
98C2×#107—42no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—42no—
102C2×#103—42no—
103C2×#103—42no—
104C2×#103C2×C418yescomplemented
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8C2×C414yes—
111#391—22no—
112#391—22no—
113#389—22no—
114#389—22no—
115#387—28no—
116#385—28no—
117#380—22no—
118#380D414yes—
119C4×C4⋊C4—28no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C22⋊C4—28no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—44no—
133C2×#95—44no—
134#340—24no—
135#340—24no—
136#340—42no—
137#340—42no—
138#340C2×C418yescomplemented
139#336—24no—
140#336—42no—
141C25C2×D411yescharacteristic
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8C22×C412yes—
150C22×Q8C2×D412yes—
151C22×Q8C2×Q812yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—41no—
174C23×C4—41no—
175C23×C4—41no—
176C23×C4—41no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4Q1614yes—
190C23×C4C22×C412yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D412yes—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D412yes—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×Q812yes—
199C23×C4#4112yes—
200C23×C4#4112yes—
201C23×C4C2411yescharacteristic
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8C22×C414yes—
216C22⋊Q8—28no—
217C22⋊Q8—28no—
218C22⋊Q8—28no—
219C22⋊Q8C4⋊C4116yes—
220C4×Q8—48no—
221C4×Q8—48no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
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—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—48no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—48no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4C22×C414yes—
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×Q812yes—
266C2×C4⋊C4#4114yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—48no—
287C2×C42—48no—
288C2×C42—41no—
289C2×C42—41no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42#4114yes—
295#107—48no—
296#103—48no—
297#103—48no—
298C22⋊C8—28no—
299C22⋊C8—44no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24C4×D412yes—
318C24#12012yes—
319C24C22⋊Q812yes—
320C24C2×Q1612yes—
321C24#13512yes—
322C24AGL(1,ℤ/8)12yes—
323C24#13712yes—
324C24C22×D411yescharacteristic
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8C2×C4⋊C414yes—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—28no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—44no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—42no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—48no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—84no—
449C22×C4C2×C4⋊C414yes—
450C22×C4C4×D414yes—
451C22×C4C4×D412yes—
452C22×C4#12012yes—
453C22×C4C22⋊Q814yes—
454C22×C4C22⋊Q812yes—
455C22×C4#12312yes—
456C22×C4#12314yes—
457C22×C4#12312yes—
458C22×C4C2×Q1612yes—
459C22×C4C2×Q1612yes—
460C22×C4AGL(1,ℤ/8)12yes—
461C22×C4#13712yes—
462C22×C4C23×C411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×Q811yescharacteristic
466C22×C4C2×#4111yescharacteristic
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4111yescharacteristic
470C22×C4H(5,𝔽2)12yes—
471C22×C4H(5,𝔽2)12yes—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8—42no—
485C2×C8—42no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—216no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4C2×C4⋊C414yes—
507C22⋊C4—216no—
508C22⋊C4—48no—
509C42—24no—
510C42—24no—
511C42—44no—
512C42—44no—
513C42—48no—
514C42—48no—
515C42—42no—
516C42—42no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23#38514yes—
546C23C4×Q1614yes—
547C23#44014yes—
548C23#44914yes—
549C23#45114yes—
550C23#47714yes—
551C23#47914yes—
552C23C2×C4×D411yescharacteristic
553C23C2×#12011yescharacteristic
554C23C2×C22⋊Q811yescharacteristic
555C23C2×#12311yescharacteristic
556C23#52511yescharacteristic
557C23#53411yescharacteristic
558C23#53511yescharacteristic
559C23C22×Q1611yescharacteristic
560C23C2×#13511yescharacteristic
561C23C2×AGL(1,ℤ/8)11yescharacteristic
562C23C2×AGL(1,ℤ/8)11yescharacteristic
563C23C2×#13711yescharacteristic
564C23C2×#13711yescharacteristic
565C23#57511yescharacteristic
566C23#57511yescharacteristic
567Q8—28no—
568Q8—44no—
569Q8—44no—
570Q8—44no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—416no—
599C2×C4—416no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—816no—
637C2×C4—816no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—82no—
644C2×C4—81no—
645C2×C4—82no—
646C2×C4—81no—
647C2×C4—82no—
648C2×C4—82no—
649C2×C4—81no—
650C2×C4—82no—
651C2×C4—82no—
652C2×C4—82no—
653C2×C4—82no—
654C2×C4#38514yes—
655C2×C4#48414yes—
656C2×C4C22×C4⋊C411yescharacteristic
657C2×C4C2×C4×D411yescharacteristic
658C2×C4C4×#4112yes—
659C2×C4#51812yes—
660C2×C4C2×C22⋊Q811yescharacteristic
661C2×C4C2×#12311yescharacteristic
662C2×C4#52912yes—
663C2×C4#53812yes—
664C2×C4#54612yes—
665C2×C4#54612yes—
666C2×C4#54612yes—
667C2×C4#54612yes—
668C2×C4#55312yes—
669C2×C4#55312yes—
670C2×C4#55412yes—
671C2×C4#55412yes—
672C8—44no—
673C8—82no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—24no—
681C22—24no—
682C22—24no—
683C22—24no—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—24no—
688C22#186318yes—
689C22C2×#38511yescharacteristic
690C22C4×#12311yescharacteristic
691C22#232111yescharacteristic
692C22#240411yescharacteristic
693C22#240511yescharacteristic
694C22#248711yescharacteristic
695C22#257911yescharacteristic
696C22C2×C4×Q1611yescharacteristic
697C22C2×#44011yescharacteristic
698C22C4×AGL(1,ℤ/8)11yescharacteristic
699C22C4×#13711yescharacteristic
700C22C2×#45111yescharacteristic
701C22C2×#44911yescharacteristic
702C22#300611yescharacteristic
703C22#300711yescharacteristic
704C22C2×#47711yescharacteristic
705C22C2×#47911yescharacteristic
706C22#307911yescharacteristic
707C22#308011yescharacteristic
708C22C2×#48411yescharacteristic
709C22#309111yescharacteristic
710C22#309211yescharacteristic
711C22#309511yescharacteristic
712C22#310411yescharacteristic
713C22#310511yescharacteristic
714C22#311311yescharacteristic
715C22#311411yescharacteristic
716C22#318911yescharacteristic
717C22#319011yescharacteristic
718C22#319311yescharacteristic
719C22#319411yescharacteristic
720C22#320911yescharacteristic
721C22#321111yescharacteristic
722C22#321511yescharacteristic
723C22#321711yescharacteristic
724C4—24no—
725C4—24no—
726C4—44no—
727C4—44no—
728C4—42no—
729C4—42no—
730C4—42no—
731C4—84no—
732C4—82no—
733C4—84no—
734C4—82no—
735C4#234912yes—
736C4#236412yes—
737C4#244412yes—
738C4#245012yes—
739C2—28no—
740C2#1401611yescharacteristic
741C2C2×#186311yescharacteristic
742C2#2087511yescharacteristic
743C2C4×#48411yescharacteristic
744C2#2157911yescharacteristic
745C2#2204411yescharacteristic
746C2#2204911yescharacteristic
747C2#2393411yescharacteristic
748C2#2393611yescharacteristic
749C2#2400411yescharacteristic
750C2#2400611yescharacteristic
751C2#2768111yescharacteristic
752C2#2768611yescharacteristic
753C2#2924111yescharacteristic
754C2#2924311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11293C2semidirect
7. #11124C2semidirect
8. #10907C2semidirect
9. #10789C2semidirect
35. #1872C4semidirect
104. C2×#103C2×C4semidirect
138. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C2×#18722
C2×#18522
C2×C4⋊#952
#112932
#111242
#109072
#107892
#107302
#107122