Finite Group Databasebeta

← #367453

Subgroups — #367453

Order 512.

1382 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1945C211yesmaximal, characteristic, complemented
2C2×#1855C211yesmaximal, characteristic
3#20843C211yesmaximal, characteristic
4#20836C211yesmaximal, characteristic
5#20628C211yesmaximal, characteristic
6C2×C4×C22⋊C8C211yesmaximal, characteristic
7C22×#339C211yesmaximal, characteristic
8#11329C211yesmaximal, characteristic, complemented
9#11258C211yesmaximal, characteristic, complemented
10#11221C211yesmaximal, characteristic, complemented
11#11221C211yesmaximal, characteristic, complemented
12#10882C211yesmaximal, characteristic, complemented
13#10882C211yesmaximal, characteristic, complemented
14#10859C211yesmaximal, characteristic, complemented
15#10719C211yesmaximal, characteristic
16C23×M16C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C42C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C22×#108C2211yescharacteristic
21C2×#427C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C2×#424C2211yescharacteristic
24C2×#424C2211yescharacteristic
25C2×#424C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#407C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32#2280C2211yescharacteristic
33#1945—24no—
34#1855—24no—
35#1854C412yescomplemented
36#1854C412yescomplemented
37#1853C412yescomplemented
38#1852C412yescomplemented
39#1851—22no—
40#1850—22no—
41#1798—22no—
42#1797—22no—
43C4×C22⋊C8—24no—
44C2×#341C2211yescharacteristic
45C2×#341C2211yescharacteristic
46#1735C2211yescharacteristic
47#1735C2211yescharacteristic
48C2×#339—24no—
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C24×C4—21no—
64C24×C4C2311yescharacteristic
65C22×M16—21no—
66C22×M16—22no—
67C22×M16—22no—
68C22×M16—22no—
69C22×M16—22no—
70C22×M16—41no—
71C22×M16—41no—
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C23×C8—21no—
77C23×C8C2311yescharacteristic
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C2×#117—21no—
82C2×#117—21no—
83C2×#117—22no—
84C2×#117—22no—
85C2×#117—41no—
86C2×#117—41no—
87C2×#117C2×C412yes—
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C2×#117C2311yescharacteristic
92C22×C4⋊C4C2×C412yes—
93C22×C22⋊C4—21no—
94C22×C42—22no—
95C22×C42—22no—
96C22×C42—21no—
97C22×C42—22no—
98C22×C42—21no—
99C22×C42—41no—
100C22×C42—41no—
101C22×C42Q814yes—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C2×#108—22no—
105C2×#108—22no—
106C2×#108—22no—
107C2×#108—22no—
108C2×#108—41no—
109C2×#108—41no—
110#427—22no—
111#427—22no—
112#427—41no—
113#427—41no—
114C2×#107C2×C412yescomplemented
115C2×#107C2×C412yescomplemented
116C2×#106C2×C412yescomplemented
117C2×#106C2×C412yescomplemented
118#424—22no—
119#424—22no—
120#424—22no—
121#424—22no—
122#424—41no—
123#424—41no—
124#424—41no—
125#424—41no—
126C2×C4⋊C8—22no—
127C2×C4⋊C8—22no—
128C2×C4⋊C8—22no—
129C2×C4⋊C8—22no—
130#407—22no—
131#407—22no—
132C2×C22⋊C8—22no—
133C2×C22⋊C8—22no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8C2×C411yescharacteristic
137C2×C22⋊C8C2×C411yescharacteristic
138C2×C22⋊C8C2×C411yescharacteristic
139C2×C22⋊C8C2×C411yescharacteristic
140C2×C22⋊C8D412yes—
141C2×C22⋊C8D412yes—
142C2×C22⋊C8D414yes—
143C2×C22⋊C8Q814yes—
144C2×C4×C8—22no—
145C2×C4×C8—22no—
146C2×C4×C8—22no—
147C2×C4×C8—22no—
148C2×C4×C8—41no—
149C2×C4×C8—41no—
150#381—21no—
151#380C2×C412yes—
152#380C2×C412yes—
153#379—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156#341—22no—
157#341—22no—
158#340C2×C412yescomplemented
159#340C2×C412yescomplemented
160#340C2×C412yescomplemented
161#340C2×C412yescomplemented
162#339—22no—
163#339—22no—
164#339—22no—
165#339—22no—
166#339—22no—
167#339—41no—
168#339—41no—
169#339D414yes—
170#336—22no—
171#336—22no—
172#336—22no—
173#336—22no—
174#336—22no—
175#336—22no—
176C25#4111yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—28no—
187C23×C4—24no—
188C23×C4—21no—
189C23×C4—22no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—44no—
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C2×D412yes—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×Q812yes—
199C23×C4C2×Q812yes—
200C23×C4#4111yescharacteristic
201C23×C4#4112yes—
202C23×C4C2411yescharacteristic
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—28no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—42no—
227C2×M16—42no—
228C2×M16—44no—
229C2×M16—44no—
230C2×M16—44no—
231C2×M16—42no—
232C2×M16C2×D412yes—
233C2×M16#4112yes—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—42no—
259C22×C8—42no—
260C22×C8—42no—
261C22×C8C22×C411yescharacteristic
262C22×C8C22×C411yescharacteristic
263C22×C8C22×C411yescharacteristic
264C22×C8C22×C411yescharacteristic
265C22×C8C2×D412yes—
266C22×C8C2×D412yes—
267C22×C8C2×D412yes—
268C22×C8C2×D412yes—
269C22×C8C2×Q812yes—
270C22×C8C2×Q812yes—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117—21no—
284#117—21no—
285#117—24no—
286#117—42no—
287#117—42no—
288#117—42no—
289#117—42no—
290#117—42no—
291#117C2×D412yes—
292#117#4112yes—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4C22×C412yes—
297C2×C4⋊C4C22×C412yes—
298C2×C4⋊C4C22×C412yes—
299C2×C4⋊C4C22×C412yes—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—28no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—44no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42—44no—
328C2×C42—44no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42C2×D412yes—
332C2×C42C2×Q812yes—
333C2×C42C2×Q812yes—
334C2×C42#4112yes—
335#108—44no—
336#108—44no—
337#107—24no—
338#107—42no—
339#106—24no—
340#106—42no—
341C4⋊C8—44no—
342C4⋊C8—44no—
343C4⋊C8—44no—
344C4⋊C8—44no—
345C22⋊C8—24no—
346C22⋊C8—24no—
347C22⋊C8—42no—
348C22⋊C8—42no—
349C22⋊C8C4⋊C414yes—
350C22⋊C8C4⋊C414yes—
351C22⋊C8C4⋊C414yes—
352C22⋊C8C4⋊C414yes—
353C4×C8—44no—
354C4×C8—44no—
355#95—22no—
356#95—22no—
357C24—21no—
358C24—21no—
359C24—21no—
360C24—21no—
361C24—24no—
362C24—42no—
363C24C4×D411yescharacteristic
364C24C4×D411yescharacteristic
365C24C22⋊Q812yes—
366C24#12512yes—
367C24C2×#4111yescharacteristic
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
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—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—216no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—416no—
419C22×C4—416no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—44no—
425C22×C4—82no—
426C22×C4C2×C4⋊C412yes—
427C22×C4C2×C4⋊C412yes—
428C22×C4C4×D411yescharacteristic
429C22×C4C4×D411yescharacteristic
430C22×C4C4×Q812yes—
431C22×C4C4×Q812yes—
432C22×C4#12112yes—
433C22×C4#12112yes—
434C22×C4C22⋊Q812yes—
435C22×C4#12512yes—
436C22×C4C4⋊Q812yes—
437C22×C4C4⋊Q812yes—
438C22×C4C23×C411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×Q811yescharacteristic
443C22×C4C22×Q811yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4H(5,𝔽2)11yescharacteristic
449C22×C4H(5,𝔽2)11yescharacteristic
450C22×C4H(5,𝔽2)11yescharacteristic
451C22×C42−1+411yescharacteristic
452M16—22no—
453M16—22no—
454M16—22no—
455M16—22no—
456M16—48no—
457M16—41no—
458M16—41no—
459M16—41no—
460M16—41no—
461M16—41no—
462M16—41no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—48no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8C2×C4⋊C412yes—
513C2×C8C2×C4⋊C412yes—
514C2×C8C2×C4⋊C412yes—
515C2×C8C2×C4⋊C412yes—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—44no—
524C4⋊C4C4×D414yes—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—44no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—44no—
535C22⋊C4—82no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—48no—
544C42—44no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—41no—
552C42—41no—
553C42C22⋊Q814yes—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—21no—
560C23—21no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—24no—
566C23—24no—
567C23—22no—
568C23—22no—
569C23—42no—
570C23—42no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23#38912yes—
576C23#38912yes—
577C23#44314yes—
578C23C2×C4×D411yescharacteristic
579C23C2×C22⋊Q811yescharacteristic
580C23C2×#12511yescharacteristic
581C23#54611yescharacteristic
582C23#55111yescharacteristic
583C23#55111yescharacteristic
584C23#55211yescharacteristic
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—432no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—88no—
633C2×C4#38912yes—
634C2×C4#38912yes—
635C2×C4C22×C4⋊C411yescharacteristic
636C2×C4C2×C4×D411yescharacteristic
637C2×C4C2×C4×Q811yescharacteristic
638C2×C4C4×#4111yescharacteristic
639C2×C4#51811yescharacteristic
640C2×C4#52011yescharacteristic
641C2×C4C2×#12111yescharacteristic
642C2×C4C2×C22⋊Q811yescharacteristic
643C2×C4#52511yescharacteristic
644C2×C4#52911yescharacteristic
645C2×C4C2×C4⋊Q811yescharacteristic
646C2×C4#53311yescharacteristic
647C2×C4#53811yescharacteristic
648C2×C4#53911yescharacteristic
649C2×C4#54311yescharacteristic
650C2×C4#54411yescharacteristic
651C2×C4D4211yescharacteristic
652C2×C4D4211yescharacteristic
653C2×C4#54611yescharacteristic
654C2×C4#54611yescharacteristic
655C2×C4#54611yescharacteristic
656C2×C4#54611yescharacteristic
657C2×C4#54711yescharacteristic
658C2×C4#54711yescharacteristic
659C2×C4D4×Q811yescharacteristic
660C2×C4#55411yescharacteristic
661C2×C4#55411yescharacteristic
662C2×C4#55411yescharacteristic
663C8—22no—
664C8—22no—
665C8—22no—
666C8—22no—
667C8—42no—
668C8—42no—
669C8—42no—
670C8—42no—
671C8—81no—
672C8—81no—
673C22—22no—
674C22—24no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—24no—
679C22—21no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—42no—
693C22—42no—
694C22#194514yescomplemented
695C22C2×#38911yescharacteristic
696C22#235811yescharacteristic
697C22#235911yescharacteristic
698C22#243711yescharacteristic
699C22#243811yescharacteristic
700C22#247311yescharacteristic
701C22#250511yescharacteristic
702C22C2×#44312yes—
703C22#295612yes—
704C4—28no—
705C4—22no—
706C4—22no—
707C4—44no—
708C4—44no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4C4×C22⋊Q811yescharacteristic
713C4#232911yescharacteristic
714C4D4×C4⋊C411yescharacteristic
715C4#234911yescharacteristic
716C4#242411yescharacteristic
717C4#242511yescharacteristic
718C4#244011yescharacteristic
719C4#254911yescharacteristic
720C2—21no—
721C2—21no—
722C2—24no—
723C2—21no—
724C2—21no—
725C2—42no—
726C2#1400411yescharacteristic
727C2C2×#194511yescharacteristic
728C2#2116511yescharacteristic
729C2#2288812yes—
730C2#2294812yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1945C2semidirect
8. #11329C2semidirect
9. #11258C2semidirect
10. #11221C2semidirect
11. #11221C2semidirect
12. #10882C2semidirect
13. #10882C2semidirect
14. #10859C2semidirect
35. #1854C4semidirect
36. #1854C4semidirect
37. #1853C4semidirect
38. #1852C4semidirect
114. C2×#107C2×C4semidirect
115. C2×#107C2×C4semidirect
116. C2×#106C2×C4semidirect
117. C2×#106C2×C4semidirect
158. #340C2×C4semidirect
159. #340C2×C4semidirect
160. #340C2×C4semidirect
161. #340C2×C4semidirect
694. C22#1945semidirect

Maximal subgroups

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

subgroupindex
C2×#19452
C2×#18552
#208432
#208362
#206282
C2×C4×C22⋊C82
C22×#3392
#113292
#112582
#112212
#108822
#108592
#107192