Finite Group Databasebeta

← #471535

Subgroups — #471535

Order 512.

1382 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1957C211yesmaximal, characteristic
2C2×#1916C211yesmaximal, characteristic, complemented
3#15151C211yesmaximal, characteristic, complemented
4#15127C211yesmaximal, characteristic
5#11373C211yesmaximal, characteristic, complemented
6#11189C212yesmaximal, complemented
7#11076C212yesmaximal, complemented
8#11066C212yesmaximal, complemented
9#11066C212yesmaximal, complemented
10#11008C211yesmaximal, characteristic
11#10820C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#398—21no—
20C2×#398—21no—
21C2×#394—21no—
22C2×#394—21no—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#389C2212yes—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1957—24no—
29#1916—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1445—21no—
39#1445—21no—
40#1438—21no—
41#1438—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×D4—21no—
46C23×D4—21no—
47C23×D4C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×#127—24no—
53C2×#127—22no—
54C2×#127—22no—
55C2×#127—42no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C2×C4⋊C8—22no—
72C2×C4⋊C8—42no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—42no—
79C2×#102—42no—
80C2×#102—42no—
81C2×#102—42no—
82C2×#102—42no—
83C2×#102—42no—
84C2×#102—42no—
85C2×C4×C8—42no—
86C2×C4×C8D414yes—
87#398—44no—
88#398—44no—
89#394—44no—
90#394—44no—
91#390—24no—
92#390—42no—
93#390—42no—
94#390—42no—
95#389—42no—
96#389—42no—
97#384—22no—
98#384—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95D411yescharacteristic
109C2×#95D411yescharacteristic
110#340—24no—
111#340—42no—
112#340—42no—
113#340—42no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118C25—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—44no—
133C22×D4—44no—
134C22×D4—42no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—82no—
141C22×D4—82no—
142C22×D4#4114yes—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173#127—28no—
174#127—48no—
175#127—44no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
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—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4D814yes—
208C2×C4⋊C4#4114yes—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42C2×D412yes—
234C2×C42#4112yes—
235C4⋊C8—44no—
236#102—48no—
237#102—48no—
238#102—48no—
239#102—48no—
240C4×C8—44no—
241#95—22no—
242#95—22no—
243#95—44no—
244#95—44no—
245#95—42no—
246#95—42no—
247#95—44no—
248#95—42no—
249#95—42no—
250C24—24no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24C22×D411yescharacteristic
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—48no—
273C2×D4—48no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—88no—
297C2×D4—88no—
298C2×D4—88no—
299C2×D4—88no—
300C2×D4—84no—
301C2×D4—84no—
302C2×D4—84no—
303C2×D4—84no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
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—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4#12112yes—
350C22×C4#12112yes—
351C22×C4C2×D812yes—
352C22×C4C2×D812yes—
353C22×C4C2×D812yes—
354C22×C4#13512yes—
355C22×C4AGL(1,ℤ/8)12yes—
356C22×C4AGL(1,ℤ/8)12yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4112yes—
359C22×C4C2×#4112yes—
360C22×C4C2×#4111yescharacteristic
361C22×C4H(5,𝔽2)12yes—
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C4H(5,𝔽2)11yescharacteristic
364C22×C4H(5,𝔽2)11yescharacteristic
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C42−1+412yes—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C4⋊C4—28no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—88no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C22⋊C4—88no—
405C22⋊C4—88no—
406C22⋊C4—88no—
407C22⋊C4—88no—
408C42—22no—
409C42—22no—
410C42—24no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—42no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—82no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23C2×#12111yescharacteristic
454C23#53511yescharacteristic
455C23#53511yescharacteristic
456C23#54612yes—
457C23#54812yes—
458C23C22×D811yescharacteristic
459C23C2×#13511yescharacteristic
460C23C2×AGL(1,ℤ/8)11yescharacteristic
461C23C2×AGL(1,ℤ/8)11yescharacteristic
462C23#57712yes—
463C23#57712yes—
464D4—44no—
465D4—44no—
466D4—44no—
467D4—44no—
468D4—816no—
469D4—816no—
470D4—82no—
471D4—82no—
472D4—82no—
473D4—82no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
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—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#46614yes—
526C2×C4#46914yes—
527C2×C4#52511yescharacteristic
528C2×C4#53211yescharacteristic
529C2×C4#53411yescharacteristic
530C2×C4#53511yescharacteristic
531C2×C4#53511yescharacteristic
532C2×C4#53611yescharacteristic
533C2×C4#53812yes—
534C2×C4#53912yes—
535C2×C4#53912yes—
536C2×C4#54012yes—
537C2×C4#54011yescharacteristic
538C2×C4#54011yescharacteristic
539C2×C4#55112yes—
540C2×C4#55112yes—
541C2×C4#55312yes—
542C2×C4#55612yes—
543C2×C4#56212yes—
544C2×C4#56212yes—
545C8—42no—
546C8—42no—
547C8—81no—
548C8—81no—
549C22—84no—
550C22—84no—
551C22—84no—
552C22—84no—
553C22—84no—
554C22—84no—
555C22—84no—
556C22—162no—
557C22—162no—
558C22—162no—
559C22—162no—
560C22—162no—
561C22—162no—
562C22—162no—
563C22—162no—
564C22#253411yescharacteristic
565C22#254511yescharacteristic
566C22#261911yescharacteristic
567C22#269612yes—
568C22#271112yes—
569C22C2×#46611yescharacteristic
570C22C2×#46911yescharacteristic
571C22#305711yescharacteristic
572C22#305711yescharacteristic
573C22#320511yescharacteristic
574C22#320511yescharacteristic
575C22#320711yescharacteristic
576C22#320711yescharacteristic
577C22#322112yes—
578C22#322212yes—
579C22#329412yes—
580C22#329612yes—
581C22#330612yes—
582C22#331012yes—
583C22#335812yes—
584C22#336512yes—
585C22#336812yes—
586C22#337312yes—
587C4—24no—
588C4—24no—
589C4—42no—
590C4—42no—
591C4—81no—
592C4—84no—
593C4—81no—
594C4—162no—
595C4—162no—
596C4#260311yescharacteristic
597C4#263311yescharacteristic
598C4#266111yescharacteristic
599C4#274012yes—
600C4#278612yes—
601C4#281611yescharacteristic
602C2—164no—
603C2#1680011yescharacteristic
604C2#2809411yescharacteristic
605C2#2810711yescharacteristic
606C2#2863711yescharacteristic
607C2#2864811yescharacteristic
608C2#2980611yescharacteristic
609C2#2981111yescharacteristic
610C2#3163412yes—
611C2#3163712yes—
612C2#3207812yes—
613C2#3208112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
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
2. C2×#1916C2semidirect
3. #15151C2semidirect
5. #11373C2semidirect
6. #11189C2semidirect
7. #11076C2semidirect
8. #11066C2semidirect
9. #11066C2semidirect
11. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19572
C2×#19162
#151512
#151272
#113732
#111892
#110762
#110662
#110082
#108202