Finite Group Databasebeta

← #394881

Subgroups — #394881

Order 512.

1998 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3C2×#2057C211yesmaximal, characteristic, complemented
4C2×#1862C212yesmaximal
5#15128C211yesmaximal, characteristic, complemented
6#11328C212yesmaximal, complemented
7#11324C212yesmaximal, complemented
8#11227C211yesmaximal, characteristic, complemented
9#11202C211yesmaximal, characteristic, complemented
10#10788C212yesmaximal
11#10788C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4—21no—
17C2×#474—28no—
18C2×#474—24no—
19C22×#107C2212yes—
20C22×#107C2212yes—
21C22×C4⋊C8C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2212yes—
24C2×#395—21no—
25D4×C4⋊C4—24no—
26D4×C4⋊C4—24no—
27C2×#389C2211yescharacteristic
28C2×#389C2212yes—
29C2×#389C2211yescharacteristic
30C2×#385—21no—
31C2×#385—21no—
32C2×#384C2211yescharacteristic
33C2×#380—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35#2057—24no—
36#1862—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1446—21no—
45#1444—21no—
46#1441—21no—
47C4⋊#95C2212yes—
48C4⋊#95C2211yescharacteristic
49C23×Q8—21no—
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C24×C4—21no—
53C24×C4—21no—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—42no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—24no—
67C2×C4×D4—24no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—42no—
71C2×C4×D4—42no—
72C2×C4×D4D414yes—
73C2×C4×D4Q814yescomplemented
74C22×C4⋊C4—28no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—44no—
82C22×C4⋊C4—44no—
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D411yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#474—232no—
97C2×#107—24no—
98C2×#107—24no—
99C2×#107—24no—
100C2×#107—24no—
101C2×#107—24no—
102C2×#107—42no—
103C2×C4⋊C8—24no—
104C2×C4⋊C8—22no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102—24no—
109C2×#102D418yes—
110C2×#102Q818yescomplemented
111#395—44no—
112#389—22no—
113#389—22no—
114#389—22no—
115#389—24no—
116#389—42no—
117#389D414yes—
118#385—28no—
119#385—28no—
120#384—22no—
121#384—22no—
122#380—28no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95D411yescharacteristic
133C2×#95D411yescharacteristic
134#340—24no—
135#340—24no—
136#340—24no—
137#340—42no—
138#336—24no—
139#336—22no—
140#336—24no—
141C25—21no—
142C22×Q8—44no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—82no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4C2×D412yes—
153C22×D4C2×D414yes—
154C22×D4C2×Q812yes—
155C22×D4C2×Q814yes—
156C22×D4#4112yes—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—21no—
160C23×C4—24no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—82no—
178C23×C4—82no—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2411yescharacteristic
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
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×C8C2×D414yes—
199C22×C8C2×Q814yes—
200C4⋊Q8—28no—
201C4⋊Q8—48no—
202C4⋊Q8—44no—
203C4×D4—28no—
204C4×D4—28no—
205C4×D4—28no—
206C4×D4—28no—
207C4×D4—28no—
208C4×D4—44no—
209C4×D4—48no—
210C4×D4—48no—
211C4×D4—44no—
212C4×D4—44no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—432no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—82no—
253C2×C4⋊C4—82no—
254C2×C4⋊C4D814yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×Q814yes—
260C2×C4⋊C4C2×Q812yes—
261C2×C4⋊C4#4112yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C42—22no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—22no—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×Q812yes—
282C2×C42#4112yes—
283#107—216no—
284#107—48no—
285C4⋊C8—28no—
286#102—216no—
287#102—216no—
288#95—24no—
289#95—24no—
290#95—22no—
291#95—22no—
292#95—44no—
293#95—42no—
294#95—42no—
295#95—44no—
296#95—82no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—24no—
301C24—24no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—42no—
307C24—42no—
308C24C22×D411yescharacteristic
309C2×Q8—48no—
310C2×Q8—88no—
311C2×Q8—88no—
312C2×Q8—84no—
313C2×Q8—84no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
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#12014yes—
328C2×D4C22⋊Q818yes—
329C2×D4C22⋊Q814yes—
330C2×D4C4⋊Q814yes—
331C2×D4C4⋊Q818yes—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—216no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
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—48no—
397C22×C4—84no—
398C22×C4—88no—
399C22×C4—88no—
400C22×C4—88no—
401C22×C4—84no—
402C22×C4—88no—
403C22×C4#12012yes—
404C22×C4#12012yes—
405C22×C4C22⋊Q812yes—
406C22×C4C22⋊Q812yes—
407C22×C4C2×D812yes—
408C22×C4C2×D812yes—
409C22×C4C2×D812yes—
410C22×C4AGL(1,ℤ/8)12yes—
411C22×C4#13712yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×Q811yescharacteristic
416C22×C4C22×Q812yes—
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4H(5,𝔽2)12yes—
421C22×C42−1+412yes—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—28no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—48no—
459C4⋊C4—88no—
460C4⋊C4—84no—
461C4⋊C4C22⋊Q814yes—
462C22⋊C4—216no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—24no—
472C42—41no—
473C42—41no—
474C42—42no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—28no—
479C23—28no—
480C23—42no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23C2×#12011yescharacteristic
499C23C2×C22⋊Q811yescharacteristic
500C23#52511yescharacteristic
501C23D4212yes—
502C23D4×Q812yes—
503C23C22×D811yescharacteristic
504C23C2×AGL(1,ℤ/8)11yescharacteristic
505C23C2×#13711yescharacteristic
506C23#57511yescharacteristic
507C23#57612yes—
508C23#57812yes—
509Q8—816no—
510D4—24no—
511D4—24no—
512D4—24no—
513D4—28no—
514D4—28no—
515D4#39518yescomplemented
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—432no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—816no—
567C2×C4—816no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4#44714yes—
581C2×C4#47414yescomplemented
582C2×C4C2×#12011yescharacteristic
583C2×C4C2×C22⋊Q811yescharacteristic
584C2×C4C2×C22⋊Q812yes—
585C2×C4C2×C22⋊Q811yescharacteristic
586C2×C4#52511yescharacteristic
587C2×C4#52511yescharacteristic
588C2×C4C2×C4⋊Q811yescharacteristic
589C2×C4C2×C4⋊Q812yes—
590C2×C4#53411yescharacteristic
591C2×C4#53611yescharacteristic
592C2×C4#53912yes—
593C2×C4#54412yes—
594C2×C4D4212yes—
595C2×C4#54712yes—
596C2×C4#54712yes—
597C2×C4D4×Q812yes—
598C2×C4#55412yes—
599C2×C4#55412yes—
600C8—44no—
601C8—44no—
602C22—28no—
603C22—48no—
604C22—48no—
605C22—44no—
606C22—44no—
607C22—48no—
608C22—44no—
609C22—44no—
610C22—48no—
611C22—44no—
612C22—44no—
613C22#240311yescharacteristic
614C22#243411yescharacteristic
615C22#244012yes—
616C22#244912yes—
617C22#284811yescharacteristic
618C22C2×#44711yescharacteristic
619C22#300711yescharacteristic
620C22#300811yescharacteristic
621C22#301111yescharacteristic
622C22C2×#47411yescharacteristic
623C22#307711yescharacteristic
624C22#308312yes—
625C22#310011yescharacteristic
626C22#311011yescharacteristic
627C22#312412yes—
628C22#326412yes—
629C22#326912yes—
630C22#327612yes—
631C22D4×D812yes—
632C22Q8×D812yes—
633C22#338512yes—
634C22#338612yes—
635C22#338912yes—
636C4—24no—
637C4—24no—
638C4—42no—
639C4—42no—
640C4—44no—
641C4—42no—
642C4—84no—
643C4—82no—
644C4—82no—
645C4—84no—
646C4—81no—
647C4C2×#39511yescharacteristic
648C4#240611yescharacteristic
649C4#244012yes—
650C4#244912yes—
651C4#258311yescharacteristic
652C4#262711yescharacteristic
653C2—48no—
654C2#1472011yescharacteristic
655C2#2384811yescharacteristic
656C2#2386511yescharacteristic
657C2#2583811yescharacteristic
658C2#2585011yescharacteristic
659C2#2604012yes—
660C2#2605212yes—
661C2#2636812yes—
662C2#2638412yes—
663C2#3295311yescharacteristic
664C2#3296311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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
3. C2×#2057C2semidirect
5. #15128C2semidirect
6. #11328C2semidirect
7. #11324C2semidirect
8. #11227C2semidirect
9. #11202C2semidirect
73. C2×C4×D4Q8semidirect
110. C2×#102Q8semidirect
515. D4#395semidirect
581. C2×C4#474semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
C2×D4×C4⋊C42
C2×#20572
C2×#18622
#151282
#113282
#113242
#112272
#112022
#107882