Finite Group Databasebeta

← #368208

Subgroups — #368208

Order 512.

2110 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×D4×C4⋊C4C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3C2×#1852C211yesmaximal, characteristic, complemented
4C22×#340C211yesmaximal, characteristic, complemented
5#13872C211yesmaximal, characteristic
6#11267C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic, complemented
10#10730C212yesmaximal, complemented
11#10721C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×C4×D4—21no—
14C22×C4×D4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C22×#107C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21D4×C4⋊C4—24no—
22D4×C4⋊C4—24no—
23C2×#389—21no—
24C2×#386—21no—
25C2×#385—21no—
26C2×#385—21no—
27C2×#384C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1869C418yescomplemented
34#1852—24no—
35C2×#340—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1441—21no—
45#1439—21no—
46C4⋊#95C2212yes—
47C22⋊#95—21no—
48C23×D4D411yescharacteristic
49C24×C4—21no—
50C24×C4—21no—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—42no—
57C23×C8—42no—
58C23×C8C2×C414yescomplemented
59C23×C8C2312yes—
60C23×C8C2312yes—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—28no—
66C2×C4×D4—44no—
67C2×C4×D4—44no—
68C2×C4×D4C2×C414yes—
69C2×C4×D4Q814yes—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—42no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4D411yescharacteristic
87C22×C4⋊C4D414yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—22no—
98C22×C22⋊C4D411yescharacteristic
99C22×C22⋊C4D411yescharacteristic
100C22×C42—21no—
101C22×C42C2311yescharacteristic
102C2×#107—24no—
103C2×#107—24no—
104C2×#107—42no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—42no—
108C2×C22⋊C8—42no—
109C2×C22⋊C8C2×C414yescomplemented
110C2×C22⋊C8C2×C418yescomplemented
111#389—28no—
112#386—44no—
113#385—28no—
114#385—28no—
115#384—22no—
116#384—22no—
117#380—22no—
118#380—22no—
119#380—22no—
120#380—22no—
121#380—22no—
122#380—24no—
123#380—42no—
124#380D414yes—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—22no—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—24no—
138#340—42no—
139#340—42no—
140#340—42no—
141#336C2×C418yescomplemented
142C25C2×D411yescharacteristic
143C25C2×D411yescharacteristic
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C4⋊C414yes—
149C22×D4D814yes—
150C22×D4Q1614yes—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—82no—
182C23×C4—82no—
183C23×C4—82no—
184C23×C4C22×C412yes—
185C23×C4C22×C412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4C2×Q812yes—
193C23×C4C2×Q812yes—
194C23×C4#4112yes—
195C23×C4#4112yes—
196C23×C4C2411yescharacteristic
197C22×C8—28no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
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—48no—
211C22×C8—44no—
212C22×C8—44no—
213C22×C8—48no—
214C22×C8C22×C414yes—
215C22×C8C22×C414yes—
216C4×D4—28no—
217C4×D4—28no—
218C4×D4—28no—
219C4×D4—28no—
220C4×D4—432no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
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—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—48no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4—48no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—82no—
265C2×C4⋊C4C4⋊C414yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4#4114yes—
271C2×C4⋊C4#4114yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—82no—
288C2×C22⋊C4C4⋊C414yes—
289C2×C22⋊C4C4⋊C414yes—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—24no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42C22×C412yes—
298C2×C42C2×Q812yes—
299#107—48no—
300C22⋊C8—28no—
301C22⋊C8—48no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—44no—
306#95—42no—
307#95—44no—
308#95—82no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—21no—
314C24—21no—
315C24—22no—
316C24—22no—
317C24—21no—
318C24—21no—
319C24—22no—
320C24—22no—
321C24C2×C4⋊C412yes—
322C24C2×C4⋊C412yes—
323C24#12012yes—
324C24#12312yes—
325C24C2×D812yes—
326C24C2×D812yes—
327C24C2×Q1612yes—
328C24C2×Q1612yes—
329C24AGL(1,ℤ/8)12yes—
330C24#13712yes—
331C24C22×D411yescharacteristic
332C2×D4—28no—
333C2×D4—28no—
334C2×D4—28no—
335C2×D4—28no—
336C2×D4—28no—
337C2×D4—28no—
338C2×D4#107116yescomplemented
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—48no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4—88no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4C2×C4⋊C412yes—
435C22×C4C2×C4⋊C412yes—
436C22×C4C2×C4⋊C412yes—
437C22×C4C2×C4⋊C412yes—
438C22×C4C4×D414yes—
439C22×C4C4×D414yes—
440C22×C4#12012yes—
441C22×C4C22⋊Q814yes—
442C22×C4C22⋊Q814yes—
443C22×C4#12312yes—
444C22×C4C2×D812yes—
445C22×C4C2×Q1612yes—
446C22×C4C23×C411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×Q811yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C4H(5,𝔽2)12yes—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—416no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—28no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—84no—
496C4⋊C4#12314yes—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—88no—
502C22⋊C4—84no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—48no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—22no—
536C23—42no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23#38514yes—
547C23C2×#10714yes—
548C23C2×#10714yes—
549C23#42814yes—
550C23#44714yes—
551C23#45114yes—
552C23#48014yes—
553C23#48414yes—
554C23C22×C4⋊C411yescharacteristic
555C23C2×#12011yescharacteristic
556C23C2×#12311yescharacteristic
557C23#54612yes—
558C23#54612yes—
559C23C22×D811yescharacteristic
560C23C22×Q1611yescharacteristic
561C23C2×AGL(1,ℤ/8)11yescharacteristic
562C23C2×#13711yescharacteristic
563C23#57712yes—
564C23#57712yes—
565D4—232no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—816no—
616C2×C4—816no—
617C2×C4—816no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4#38514yes—
636C2×C4C2×#10714yes—
637C2×C4C2×C4×D411yescharacteristic
638C2×C4C2×C4×D411yescharacteristic
639C2×C4C4×#4112yes—
640C2×C4#51812yes—
641C2×C4C2×C22⋊Q811yescharacteristic
642C2×C4C2×C22⋊Q811yescharacteristic
643C2×C4C2×#12311yescharacteristic
644C2×C4#52511yescharacteristic
645C2×C4#52912yes—
646C2×C4#53411yescharacteristic
647C2×C4#53511yescharacteristic
648C2×C4#53812yes—
649C2×C4#54612yes—
650C2×C4#54612yes—
651C2×C4#55312yes—
652C2×C4#55312yes—
653C2×C4#55412yes—
654C2×C4#55412yes—
655C8—44no—
656C8—44no—
657C22—28no—
658C22—28no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—28no—
663C22—28no—
664C22—24no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—28no—
669C22—44no—
670C22—44no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—44no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22#185218yes—
679C22C2×#38511yescharacteristic
680C22#234912yes—
681C22#240411yescharacteristic
682C22#240511yescharacteristic
683C22#244412yes—
684C22C22×#10711yescharacteristic
685C22C2×#42811yescharacteristic
686C22#291212yes—
687C22C2×#44711yescharacteristic
688C22C2×#45111yescharacteristic
689C22#300911yescharacteristic
690C22#301011yescharacteristic
691C22C2×#48011yescharacteristic
692C22C2×#48411yescharacteristic
693C22#309912yes—
694C22#329412yes—
695C22#329412yes—
696C22#329912yes—
697C22#329912yes—
698C22#330612yes—
699C22#331012yes—
700C22#331612yes—
701C22#332012yes—
702C4—24no—
703C4—24no—
704C4—42no—
705C4—42no—
706C4—44no—
707C4—82no—
708C4—84no—
709C4—82no—
710C4—82no—
711C4—162no—
712C4C4×#12311yescharacteristic
713C4#232111yescharacteristic
714C4#236412yes—
715C4#245012yes—
716C4#248711yescharacteristic
717C4#257911yescharacteristic
718C2—28no—
719C2—44no—
720C2#1401611yescharacteristic
721C2C2×#185211yescharacteristic
722C2#2084811yescharacteristic
723C2#2270512yes—
724C2#2271112yes—
725C2#2396011yescharacteristic
726C2#2397011yescharacteristic
727C2#2403211yescharacteristic
728C2#2404211yescharacteristic
729C2#2619812yes—
730C2#2620412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1852C2semidirect
4. C22×#340C2semidirect
6. #11267C2semidirect
7. #11096C2semidirect
8. #10907C2semidirect
9. #10730C2semidirect
10. #10730C2semidirect
11. #10721C2semidirect
33. #1869C4semidirect
58. C23×C8C2×C4semidirect
109. C2×C22⋊C8C2×C4semidirect
110. C2×C22⋊C8C2×C4semidirect
141. #336C2×C4semidirect
338. C2×D4#107semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
C2×#18692
C2×#18522
C22×#3402
#138722
#112672
#110962
#109072
#107302
#107212