Finite Group Databasebeta

← #403382

Subgroups — #403382

Order 512.

2748 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#2059C212yesmaximal, complemented
3C2×#2058C211yesmaximal, characteristic, complemented
4C2×#1864C212yesmaximal
5C2×#1862C211yesmaximal, characteristic
6#15158C211yesmaximal, characteristic, complemented
7#11385C212yesmaximal, complemented
8#11384C211yesmaximal, characteristic, complemented
9#11380C211yesmaximal, characteristic, complemented
10#10788C212yesmaximal
11#10788C211yesmaximal, characteristic
12C2×#554—24no—
13C2×#554—28no—
14C2×#554—24no—
15C2×#554—28no—
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125C2212yes—
18C22×C22⋊Q8—21no—
19C22×C22⋊Q8—22no—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2212yes—
23C23×C4⋊C4—21no—
24C22×#107C2211yescharacteristic
25C22×#106C2212yes—
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2212yes—
29C2×#395—21no—
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32#2059—28no—
33#2058—24no—
34#1864—28no—
35#1862—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44C2×#336C2212yes—
45#1441—21no—
46#1441—22no—
47C4⋊#95C2211yescharacteristic
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C23×Q8D411yescharacteristic
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C24×C4—22no—
54C23×C8C2311yescharacteristic
55C23×C8C2312yes—
56C23×C8C2311yescharacteristic
57#554—264no—
58#554—232no—
59#554—232no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8D414yes—
64C2×#125—24no—
65C2×#125—24no—
66C2×#125—24no—
67C2×#125—24no—
68C2×#125—24no—
69C2×#125D418yes—
70C2×C22⋊Q8—28no—
71C2×C22⋊Q8—216no—
72C2×C22⋊Q8—216no—
73C2×C22⋊Q8—216no—
74C2×C22⋊Q8—216no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—24no—
78C2×C4×Q8—24no—
79C2×C4×D4—24no—
80C2×C4×D4—24no—
81C2×C4×D4—24no—
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C2×C4×D4—24no—
85C2×C4×D4—24no—
86C2×C4×D4—22no—
87C2×C4×D4D414yes—
88C2×C4×D4D418yes—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—216no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4D411yescharacteristic
97C22×C4⋊C4D412yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—21no—
104C22×C22⋊C4—22no—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C2×#107—24no—
108C2×#107—22no—
109C2×#106—24no—
110C2×#106—24no—
111C2×#106—24no—
112C2×#102—22no—
113C2×#102—24no—
114C2×#102—22no—
115C2×#102—24no—
116C2×#102—22no—
117C2×#102—24no—
118C2×#102Q814yescomplemented
119C2×#102Q818yescomplemented
120#395—44no—
121#389—22no—
122#389—24no—
123#389—22no—
124#389—24no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—21no—
130C2×#95—22no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—22no—
135#340—24no—
136#336—22no—
137#336—24no—
138#336—22no—
139#336—24no—
140C25—21no—
141C22×Q8—28no—
142C22×Q8—24no—
143C22×Q8—22no—
144C22×Q8—24no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—42no—
148C22×Q8—81no—
149C22×Q8—82no—
150C22×Q8—81no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4C2×D412yes—
157C22×D4C2×D414yes—
158C22×D4C2×Q812yes—
159C22×D4C2×Q814yes—
160C22×D4#4112yes—
161C23×C4—28no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—28no—
165C23×C4—28no—
166C23×C4—24no—
167C23×C4—28no—
168C23×C4—21no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—44no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—82no—
175C23×C4—82no—
176C23×C4—82no—
177C23×C4—82no—
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2411yescharacteristic
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—21no—
194C22×C8—22no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—21no—
198C22×C8C2×Q814yes—
199C22×C8C2×Q812yes—
200C4⋊Q8—28no—
201C4⋊Q8—28no—
202C4⋊Q8—216no—
203#125—216no—
204#125—216no—
205#125—216no—
206#125—216no—
207C22⋊Q8—2128no—
208C22⋊Q8—264no—
209C4×Q8—28no—
210C4×Q8—28no—
211C4×Q8—216no—
212C4×D4—216no—
213C4×D4—216no—
214C4×D4—216no—
215C4×D4—216no—
216C4×D4—28no—
217C4×D4—216no—
218C4×D4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—264no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
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—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×Q812yes—
264C2×C4⋊C4C2×Q814yes—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42C2×D414yes—
277C2×C42C2×D412yes—
278C2×C42C2×D414yes—
279C2×C42C2×D412yes—
280#107—44no—
281#106—48no—
282#102—28no—
283#102—216no—
284#102—44no—
285#95—44no—
286#95—42no—
287#95—44no—
288#95—42no—
289#95—82no—
290#95—81no—
291C24—22no—
292C24—24no—
293C24—22no—
294C24—22no—
295C24—24no—
296C24—41no—
297C24—41no—
298C24—42no—
299C24—41no—
300C24—41no—
301C24—42no—
302C24C22×D411yescharacteristic
303C2×Q8—28no—
304C2×Q8—216no—
305C2×Q8—48no—
306C2×Q8—48no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—84no—
310C2×Q8—84no—
311C2×Q8—88no—
312C2×Q8—88no—
313C2×D4—24no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—24no—
317C2×D4—22no—
318C2×D4—24no—
319C2×D4—22no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—22no—
325C2×D4—24no—
326C2×D4#12014yes—
327C2×D4C22⋊Q814yes—
328C2×D4C22⋊Q818yes—
329C2×D4C4⋊Q814yes—
330C2×D4C4⋊Q818yes—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—216no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—232no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—416no—
359C22×C4—48no—
360C22×C4—416no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—48no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4—88no—
389C22×C4—88no—
390C22×C4—88no—
391C22×C4—88no—
392C22×C4#12012yes—
393C22×C4#12014yes—
394C22×C4#12014yes—
395C22×C4#12012yes—
396C22×C4#13512yes—
397C22×C4AGL(1,ℤ/8)12yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D412yes—
402C22×C4C22×Q811yescharacteristic
403C22×C4C22×Q812yes—
404C22×C4C2×#4111yescharacteristic
405C22×C42−1+411yescharacteristic
406C22×C42−1+412yes—
407C22×C42−1+411yescharacteristic
408C2×C8—24no—
409C2×C8—22no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—42no—
414C2×C8—41no—
415C2×C8—42no—
416C2×C8—41no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—42no—
423C2×C8—41no—
424C2×C8—42no—
425C4⋊C4—28no—
426C4⋊C4—232no—
427C4⋊C4—216no—
428C4⋊C4—216no—
429C4⋊C4—232no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—41no—
440C4⋊C4—42no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—41no—
444C4⋊C4—42no—
445C4⋊C4—41no—
446C4⋊C4—88no—
447C4⋊C4—84no—
448C4⋊C4#12018yes—
449C4⋊C4#12014yes—
450C22⋊C4—232no—
451C22⋊C4—216no—
452C42—24no—
453C42—24no—
454C42—24no—
455C42—24no—
456C42—22no—
457C42—24no—
458C42—22no—
459C23—24no—
460C23—24no—
461C23—28no—
462C23—24no—
463C23—28no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—42no—
468C23—44no—
469C23—42no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—42no—
475C23—44no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—42no—
480C23—42no—
481C23—44no—
482C23C2×#12011yescharacteristic
483C23C2×#12012yes—
484C23#54811yescharacteristic
485C23D4×Q811yescharacteristic
486C23D4×Q812yes—
487C23C2×#13511yescharacteristic
488C23C2×AGL(1,ℤ/8)11yescharacteristic
489C23#57512yes—
490C23#57611yescharacteristic
491C23#57711yescharacteristic
492C23#57712yes—
493Q8—416no—
494Q8—816no—
495D4—24no—
496D4—28no—
497D4—24no—
498D4—28no—
499D4—24no—
500D4#39518yescomplemented
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—432no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—832no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—81no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4#44914yes—
558C2×C4C2×#12011yescharacteristic
559C2×C4C2×#12012yes—
560C2×C4C2×#12011yescharacteristic
561C2×C4C2×C22⋊Q811yescharacteristic
562C2×C4C2×C22⋊Q812yes—
563C2×C4C2×C4⋊Q811yescharacteristic
564C2×C4C2×C4⋊Q812yes—
565C2×C4#53611yescharacteristic
566C2×C4#53612yes—
567C2×C4#53611yescharacteristic
568C2×C4#53612yes—
569C2×C4#54811yescharacteristic
570C2×C4#54812yes—
571C2×C4D4×Q812yes—
572C2×C4D4×Q812yes—
573C2×C4D4×Q812yes—
574C2×C4D4×Q811yescharacteristic
575C2×C4D4×Q812yes—
576C8—44no—
577C8—42no—
578C8—81no—
579C22—28no—
580C22—44no—
581C22—48no—
582C22—44no—
583C22—48no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—48no—
588C22—44no—
589C22—48no—
590C22#240911yescharacteristic
591C22#240912yes—
592C22#243411yescharacteristic
593C22#243412yes—
594C22#284611yescharacteristic
595C22C2×#44911yescharacteristic
596C22#300212yes—
597C22#300511yescharacteristic
598C22#300612yes—
599C22#300912yes—
600C22#300911yescharacteristic
601C22#301211yescharacteristic
602C22#301212yes—
603C22#336011yescharacteristic
604C22#336611yescharacteristic
605C22#337612yes—
606C22#338011yescharacteristic
607C22#338112yes—
608C22#338411yescharacteristic
609C22#338412yes—
610C22#338812yes—
611C22#338812yes—
612C22#338912yes—
613C4—24no—
614C4—44no—
615C4—44no—
616C4—44no—
617C4—42no—
618C4—48no—
619C4—82no—
620C4—84no—
621C4—161no—
622C4C2×#39511yescharacteristic
623C4#240912yes—
624C4#240911yescharacteristic
625C4#243412yes—
626C4#243411yescharacteristic
627C4#261411yescharacteristic
628C2—48no—
629C2#1487411yescharacteristic
630C2#2432711yescharacteristic
631C2#2433712yes—
632C2#2434311yescharacteristic
633C2#2434712yes—
634C2#2583511yescharacteristic
635C2#2584312yes—
636C2#2584511yescharacteristic
637C2#2585112yes—
638C2#3288011yescharacteristic
639C2#3288611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2059C2semidirect
3. C2×#2058C2semidirect
6. #15158C2semidirect
7. #11385C2semidirect
8. #11384C2semidirect
9. #11380C2semidirect
118. C2×#102Q8semidirect
119. C2×#102Q8semidirect
500. D4#395semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#20592
C2×#20582
C2×#18642
C2×#18622
#151582
#113852
#113842
#113802
#107882