Finite Group Databasebeta

← #356603

Subgroups — #356603

Order 512.

2164 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D8C211yesmaximal, characteristic
2C2×#1879C212yesmaximal
3C2×#1762C212yesmaximal, complemented
4#13820C211yesmaximal, characteristic, complemented
5D4×#95C211yesmaximal, characteristic, complemented
6#10927C212yesmaximal, complemented
7#10907C212yesmaximal, complemented
8#10820C211yesmaximal, characteristic, complemented
9#10788C211yesmaximal, characteristic, complemented
10#10788C211yesmaximal, characteristic, complemented
11#10696C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×C4×D4C2212yes—
16C2×C4×D8—28no—
17C2×C4×D8C418yescomplemented
18C22×#107C2211yescharacteristic
19C22×#102C2212yes—
20C22×#102C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2211yescharacteristic
23C22×#97C2212yes—
24C22×C4×C8C2211yescharacteristic
25C2×#390C2212yes—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4—21no—
28C22×#95—21no—
29#1879C4116yescomplemented
30#1762—28no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37#1437—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C22⋊#95—21no—
42C22⋊#95—22no—
43C22⋊#95—21no—
44C4×#95C2211yescharacteristic
45C23×D4C2311yescharacteristic
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C24×C4—22no—
49C24×C4—21no—
50C22×D8—24no—
51C22×D8—22no—
52C22×D8—22no—
53C22×D8C2×C418yes—
54C22×D8C2×C414yes—
55C23×C8C2312yes—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C2×C4×D4—24no—
59C2×C4×D4—24no—
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—24no—
65C2×C4×D4—42no—
66C2×C4×D4—42no—
67C2×C4×D4—42no—
68C2×C4×D4—42no—
69C2×C4×D4C2×C414yes—
70C2×C4×D4C2×C418yescomplemented
71C2×C4×D4C2×C414yescomplemented
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C42—22no—
82C22×C42C2311yescharacteristic
83C22×C42C2312yes—
84C4×D8—232no—
85C2×#107—24no—
86C2×#107C2×C414yescomplemented
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—42no—
91C2×#102—42no—
92C2×#102C2×C414yes—
93C2×#102C2×C418yescomplemented
94C2×#102C2×C414yescomplemented
95C2×#102C2×C418yescomplemented
96C2×#102D418yes—
97C2×#97—24no—
98C2×#97—24no—
99C2×#97—42no—
100C2×C4×C8—24no—
101C2×C4×C8C2×C414yescomplemented
102#390—24no—
103#390C2×C418yescomplemented
104C4×C4⋊C4—24no—
105C4×C4⋊C4—42no—
106C4×C22⋊C4—44no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—44no—
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117C2×#95D412yes—
118#340—24no—
119#340—24no—
120#340—42no—
121#340C2×C418yescomplemented
122#336—22no—
123#336C2×C418yescomplemented
124C25—21no—
125C25—21no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4C22×C412yes—
140C22×D4C22×C414yes—
141C22×D4C22×C412yes—
142C22×D4C22×C414yes—
143C22×D4C2×D414yes—
144C22×D4#4112yes—
145C22×D4#4112yes—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—28no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—82no—
164C23×C4—82no—
165C23×C4—82no—
166C23×C4—82no—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C2×D8—28no—
173C2×D8—28no—
174C2×D8—216no—
175C2×D8—216no—
176C2×D8C42116yescomplemented
177C22×C8—22no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8C22×C414yes—
190C22×C8C22×C414yes—
191C22×C8C22×C412yes—
192C22×C8C22×C412yes—
193C22×C8C2×D414yes—
194C4×D4—28no—
195C4×D4—216no—
196C4×D4—28no—
197C4×D4—44no—
198C4×D4—48no—
199C4×D4—44no—
200C4×D4—48no—
201C4×D4—48no—
202C4×D4—48no—
203C4×D4—48no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C414yes—
221C2×C4⋊C4C22×C414yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C2×D414yes—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4—82no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42—82no—
246C2×C42C22×C412yes—
247C2×C42C22×C414yes—
248C2×C42#4112yes—
249C2×C42#4114yes—
250#107—28no—
251#102—216no—
252#102—28no—
253#102—48no—
254#102C22⋊C4116yescomplemented
255#97—48no—
256C4×C8—28no—
257#95—22no—
258#95—22no—
259#95—42no—
260#95—44no—
261#95—44no—
262#95—42no—
263#95—42no—
264#95—81no—
265#95—81no—
266#95—82no—
267C24—22no—
268C24—24no—
269C24—24no—
270C24—22no—
271C24—22no—
272C24—24no—
273C24—22no—
274C24—41no—
275C24—41no—
276C24—42no—
277C24—41no—
278C24—42no—
279C24—41no—
280C24—42no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—42no—
286C24—41no—
287C24—42no—
288C24—41no—
289C24C22×D411yescharacteristic
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—22no—
295C2×D4—24no—
296C2×D4—22no—
297C2×D4—24no—
298C2×D4—22no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—24no—
304C2×D4—22no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—22no—
311C2×D4—24no—
312C2×D4—22no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—41no—
318C2×D4—42no—
319C2×D4—41no—
320C2×D4C2×C4214yes—
321C2×D4C2×C4214yes—
322C2×D4C2×C22⋊C414yes—
323C2×D4#11714yes—
324C2×D4C4×D418yes—
325C2×D4C4×D418yes—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—416no—
352C22×C4—416no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—84no—
375C22×C4—84no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4—88no—
379C22×C4—88no—
380C22×C4—88no—
381C22×C4—88no—
382C22×C4—88no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4C4×D412yes—
386C22×C4C4×D414yes—
387C22×C4C4×D414yes—
388C22×C4C4×D412yes—
389C22×C4AGL(1,ℤ/8)12yes—
390C22×C4AGL(1,ℤ/8)12yes—
391C22×C4C23×C411yescharacteristic
392C22×C4C23×C412yes—
393C22×C4C22×D412yes—
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4112yes—
398C22×C4H(5,𝔽2)11yescharacteristic
399C22×C4H(5,𝔽2)12yes—
400C22×C42−1+411yescharacteristic
401D8—232no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8C2×C4214yes—
418C2×C8C2×C22⋊C414yes—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4C2×C22⋊C414yes—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—416no—
436C22⋊C4—416no—
437C22⋊C4—88no—
438C42—22no—
439C42—24no—
440C42—24no—
441C42—22no—
442C42—24no—
443C42—24no—
444C42—24no—
445C42—88no—
446C23—24no—
447C23—24no—
448C23—28no—
449C23—28no—
450C23—24no—
451C23—24no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—44no—
456C23—42no—
457C23—42no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—42no—
462C23—44no—
463C23—42no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—44no—
473C23—44no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—42no—
480C23—44no—
481C23—42no—
482C23—81no—
483C23—81no—
484C23—81no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—81no—
492C23C2×C4×D411yescharacteristic
493C23C2×C4×D412yes—
494C23D4212yes—
495C23#54611yescharacteristic
496C23#54811yescharacteristic
497C23C2×AGL(1,ℤ/8)11yescharacteristic
498C23C2×AGL(1,ℤ/8)11yescharacteristic
499C23#57512yes—
500C23#57611yescharacteristic
501C23#57612yes—
502C23#57811yescharacteristic
503D4—24no—
504D4—24no—
505D4—28no—
506D4—28no—
507D4—24no—
508D4—24no—
509D4—24no—
510D4—42no—
511D4—42no—
512D4—44no—
513D4—44no—
514D4C4×C22⋊C418yescomplemented
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—816no—
553C2×C4—816no—
554C2×C4—832no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4#44214yes—
566C2×C4C22×C4211yescharacteristic
567C2×C4C22×C22⋊C411yescharacteristic
568C2×C4C2×#11711yescharacteristic
569C2×C4C2×C4×D412yes—
570C2×C4C2×C4×D412yes—
571C2×C4C4×#4111yescharacteristic
572C2×C4C4×#4112yes—
573C2×C4#51811yescharacteristic
574C2×C4#51812yes—
575C2×C4#52011yescharacteristic
576C2×C4#52012yes—
577C2×C4#54612yes—
578C2×C4#54612yes—
579C2×C4#54712yes—
580C2×C4#55112yes—
581C2×C4#55211yescharacteristic
582C2×C4#55312yes—
583C2×C4#55311yescharacteristic
584C8—24no—
585C8—24no—
586C8—44no—
587C22—28no—
588C22—44no—
589C22—44no—
590C22—48no—
591C22—48no—
592C22—44no—
593C22—44no—
594C22—48no—
595C22—48no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—84no—
602C22—84no—
603C22—82no—
604C22—82no—
605C22—84no—
606C22—84no—
607C22—82no—
608C22—82no—
609C22C42×D411yescharacteristic
610C22D4×C22⋊C411yescharacteristic
611C22#235211yescharacteristic
612C22#235812yes—
613C22#236112yes—
614C22C2×#44211yescharacteristic
615C22#294311yescharacteristic
616C22C4×AGL(1,ℤ/8)12yes—
617C22C4×AGL(1,ℤ/8)12yes—
618C22#294812yes—
619C22#294811yescharacteristic
620C22#295112yes—
621C22#295111yescharacteristic
622C22#326412yes—
623C22#326412yes—
624C22#326812yes—
625C22#326912yes—
626C22#332212yes—
627C22#332511yescharacteristic
628C22#333011yescharacteristic
629C22#336611yescharacteristic
630C22#336711yescharacteristic
631C22#337012yes—
632C4—24no—
633C4—44no—
634C4—44no—
635C4—44no—
636C4—42no—
637C4—84no—
638C4—82no—
639C4—84no—
640C4C2×C4×C22⋊C411yescharacteristic
641C4#227511yescharacteristic
642C4#233911yescharacteristic
643C4#235311yescharacteristic
644C4#235912yes—
645C4#236412yes—
646C2—48no—
647C2—84no—
648C2#1381311yescharacteristic
649C2C4×#44211yescharacteristic
650C2#2050811yescharacteristic
651C2#2243611yescharacteristic
652C2#2243911yescharacteristic
653C2#2275811yescharacteristic
654C2#2276411yescharacteristic
655C2#2287812yes—
656C2#2288012yes—
657C2#2300812yes—
658C2#2301112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1762C2semidirect
4. #13820C2semidirect
5. D4×#95C2semidirect
6. #10927C2semidirect
7. #10907C2semidirect
8. #10820C2semidirect
9. #10788C2semidirect
10. #10788C2semidirect
11. #10696C2semidirect
17. C2×C4×D8C4semidirect
29. #1879C4semidirect
70. C2×C4×D4C2×C4semidirect
71. C2×C4×D4C2×C4semidirect
86. C2×#107C2×C4semidirect
93. C2×#102C2×C4semidirect
94. C2×#102C2×C4semidirect
95. C2×#102C2×C4semidirect
101. C2×C4×C8C2×C4semidirect
103. #390C2×C4semidirect
121. #340C2×C4semidirect
123. #336C2×C4semidirect
176. C2×D8C42semidirect
254. #102C22⋊C4semidirect
514. D4C4×C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D82
C2×#18792
C2×#17622
#138202
D4×#952
#109272
#109072
#108202
#107882
#106962