Finite Group Databasebeta

← #412588

Subgroups — #412588

Order 512.

1304 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2071C212yesmaximal
2C2×#1802C211yesmaximal, characteristic
3C2×#1762C211yesmaximal, characteristic
4#13941C211yesmaximal, characteristic
5#13910C211yesmaximal, characteristic, complemented
6#11316C212yesmaximal, complemented
7#11267C212yesmaximal, complemented
8#11189C212yesmaximal
9#10850C211yesmaximal, characteristic, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10835C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#97C2211yescharacteristic
18C2×#390C2212yes—
19C2×#389—22no—
20C2×#384C2211yescharacteristic
21C2×#384C2212yes—
22C2×#382C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#2071—28no—
26#1802—24no—
27#1762—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1444—21no—
38#1444—21no—
39#1439—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C22⋊#95—21no—
43C22⋊#95—21no—
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4D414yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×#102—22no—
75C2×#102—22no—
76C2×#102—22no—
77C2×#102—22no—
78C2×#102—22no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102D414yes—
83C2×#97—22no—
84C2×#97—22no—
85C2×#97—22no—
86#390—24no—
87#390—42no—
88#389—48no—
89#384—24no—
90#384—22no—
91#384—22no—
92#384—42no—
93#382—22no—
94#382—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D411yescharacteristic
107C2×#95D411yescharacteristic
108#340—24no—
109#340—24no—
110#340—24no—
111#340—22no—
112#340—22no—
113#340—24no—
114#340—42no—
115#336—24no—
116#336—24no—
117C25—21no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4C2×D412yes—
130C22×D4C2×D412yes—
131C22×D4#4112yes—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—24no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4—82no—
141C23×C4—82no—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C2×D412yes—
167C22×C8#4112yes—
168C4×D4—28no—
169C4×D4—28no—
170C4×D4—44no—
171C4×D4—44no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4#4112yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4—82no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—44no—
231C2×C42—42no—
232C2×C42C2×D412yes—
233C2×C42#4112yes—
234C2×C42#4112yes—
235C2×C42#4112yes—
236C2×C42#4112yes—
237C2×C42#4112yes—
238C4⋊C8—44no—
239#102—28no—
240#102—28no—
241#102—48no—
242#97—44no—
243#95—22no—
244#95—22no—
245#95—44no—
246#95—42no—
247#95—42no—
248#95—42no—
249#95—44no—
250#95—82no—
251#95—82no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—41no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24C22×D411yescharacteristic
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4#12114yes—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—48no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4—88no—
341C22×C4—88no—
342C22×C4#12112yes—
343C22×C4#12112yes—
344C22×C4#12312yes—
345C22×C4#12312yes—
346C22×C4#12412yes—
347C22×C4#12412yes—
348C22×C4AGL(1,ℤ/8)14yes—
349C22×C4#13714yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C42−1+411yescharacteristic
362C22×C42−1+411yescharacteristic
363C22×C42−1+411yescharacteristic
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C4⋊C4—216no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—88no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C4⋊C4—88no—
410C22⋊C4—48no—
411C22⋊C4—88no—
412C22⋊C4—88no—
413C42—42no—
414C42—42no—
415C42—42no—
416C42—42no—
417C42—84no—
418C42#12314yes—
419C42#12314yes—
420C42#12614yes—
421C42#12614yes—
422C23—24no—
423C23—44no—
424C23—44no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—44no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23C2×#12111yescharacteristic
446C23C2×#12311yescharacteristic
447C23C2×#12411yescharacteristic
448C23D4211yescharacteristic
449C23#54711yescharacteristic
450C23#54811yescharacteristic
451C23#54811yescharacteristic
452C23C2×AGL(1,ℤ/8)12yes—
453C23C2×#13712yes—
454C23#57612yes—
455C23#57712yes—
456D4—24no—
457D4—24no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—42no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—816no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4#39714yes—
517C2×C4#39714yes—
518C2×C4C2×#12111yescharacteristic
519C2×C4C2×#12311yescharacteristic
520C2×C4C2×#12311yescharacteristic
521C2×C4#52511yescharacteristic
522C2×C4C2×#12611yescharacteristic
523C2×C4C2×#12611yescharacteristic
524C2×C4#52911yescharacteristic
525C2×C4#52911yescharacteristic
526C2×C4#53211yescharacteristic
527C2×C4#53211yescharacteristic
528C2×C4#53611yescharacteristic
529C2×C4#53711yescharacteristic
530C2×C4#54111yescharacteristic
531C2×C4#54111yescharacteristic
532C2×C4#54211yescharacteristic
533C2×C4#54211yescharacteristic
534C2×C4#54711yescharacteristic
535C2×C4#54711yescharacteristic
536C2×C4#54711yescharacteristic
537C2×C4#55211yescharacteristic
538C2×C4#55211yescharacteristic
539C2×C4#55211yescharacteristic
540C2×C4#55211yescharacteristic
541C2×C4#55311yescharacteristic
542C2×C4#55311yescharacteristic
543C2×C4#55511yescharacteristic
544C2×C4#55611yescharacteristic
545C2×C4#55611yescharacteristic
546C8—42no—
547C8—42no—
548C8—82no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—82no—
553C22—82no—
554C22—84no—
555C22—82no—
556C22—82no—
557C22—84no—
558C22—84no—
559C22—84no—
560C22C2×#39711yescharacteristic
561C22#241511yescharacteristic
562C22#241611yescharacteristic
563C22#242111yescharacteristic
564C22#242211yescharacteristic
565C22#248911yescharacteristic
566C22#249111yescharacteristic
567C22#303811yescharacteristic
568C22#303911yescharacteristic
569C22#304111yescharacteristic
570C22#304211yescharacteristic
571C22#309212yes—
572C22#309612yes—
573C22#313612yes—
574C22#314112yes—
575C22#326412yes—
576C22#326612yes—
577C22#334512yes—
578C22#334612yes—
579C22#336612yes—
580C22#337212yes—
581C22#337312yes—
582C22#337512yes—
583C4—24no—
584C4—24no—
585C4—44no—
586C4—44no—
587C4—82no—
588C4—82no—
589C4—82no—
590C4—162no—
591C4—162no—
592C4#248511yescharacteristic
593C4#248611yescharacteristic
594C4#249411yescharacteristic
595C4#249511yescharacteristic
596C4#249611yescharacteristic
597C4#249611yescharacteristic
598C4#262511yescharacteristic
599C4#285411yescharacteristic
600C2—84no—
601C2#1513211yescharacteristic
602C2#2363011yescharacteristic
603C2#2363211yescharacteristic
604C2#2469211yescharacteristic
605C2#2469611yescharacteristic
606C2#2474411yescharacteristic
607C2#2474811yescharacteristic
608C2#2500212yes—
609C2#2502212yes—
610C2#2773912yes—
611C2#2778312yes—
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
5. #13910C2semidirect
6. #11316C2semidirect
7. #11267C2semidirect
9. #10850C2semidirect
10. #10843C2semidirect
11. #10835C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20712
C2×#18022
C2×#17622
#139412
#139102
#113162
#112672
#111892
#108502
#108432
#108352