Finite Group Databasebeta

← #469807

Subgroups — #469807

Order 512.

1350 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1982C211yesmaximal, characteristic
2C2×#1920C211yesmaximal, characteristic, complemented
3#15140C211yesmaximal, characteristic
4#13921C211yesmaximal, characteristic, complemented
5#11322C211yesmaximal, characteristic, complemented
6#11228C211yesmaximal, characteristic, complemented
7#11189C212yesmaximal, complemented
8#11121C212yesmaximal, complemented
9#11076C212yesmaximal, complemented
10#11076C212yesmaximal, complemented
11#11008C211yesmaximal, characteristic
12C22×#127C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C2×#394—22no—
21C2×#390C2212yes—
22C2×#390C2212yes—
23C2×#389C2211yescharacteristic
24C2×#383—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26#1982—24no—
27#1920—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35#1443—22no—
36#1438—21no—
37#1438—21no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×D4—21no—
44C23×D4—21no—
45C23×D4C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C2×#127—24no—
50C2×#127—22no—
51C2×#127—22no—
52C2×#127—42no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C42—21no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66C2×#106—22no—
67C2×#106—22no—
68C2×#106—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—42no—
71C2×C4⋊C8—42no—
72C2×C4⋊C8—42no—
73C2×C4⋊C8—42no—
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×#102—42no—
86#394—48no—
87#390—24no—
88#390—24no—
89#390—42no—
90#390—42no—
91#389—22no—
92#389—22no—
93#383—44no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4D414yes—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95D412yes—
105C2×#95D411yescharacteristic
106C2×#95D411yescharacteristic
107#340—22no—
108#340—22no—
109#340—22no—
110#340—42no—
111#340—42no—
112#336—22no—
113#336—42no—
114C25—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—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—44no—
129C22×D4—44no—
130C22×D4—44no—
131C22×D4—42no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4—82no—
136C22×D4—81no—
137C22×D4—81no—
138C22×D4#4114yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8#4112yes—
166#127—28no—
167#127—48no—
168#127—44no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
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—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4#4112yes—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—82no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4—82no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42C2×D412yes—
221C2×C42#4112yes—
222#106—28no—
223C4⋊C8—44no—
224C4⋊C8—48no—
225#102—48no—
226#102—48no—
227#102—48no—
228#102—48no—
229#95—24no—
230#95—22no—
231#95—22no—
232#95—24no—
233#95—44no—
234#95—44no—
235#95—42no—
236#95—44no—
237#95—42no—
238#95—82no—
239#95—81no—
240C24—24no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24C22×D411yescharacteristic
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—48no—
263C2×D4—48no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
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—88no—
287C2×D4—88no—
288C2×D4—84no—
289C2×D4—84no—
290C2×D4—84no—
291C2×D4—88no—
292C2×D4—88no—
293C2×D4—84no—
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—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
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—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4#12112yes—
343C22×C4#12112yes—
344C22×C4#13512yes—
345C22×C4AGL(1,ℤ/8)14yes—
346C22×C4AGL(1,ℤ/8)12yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C2×#4112yes—
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4H(5,𝔽2)11yescharacteristic
353C22×C4H(5,𝔽2)11yescharacteristic
354C22×C4H(5,𝔽2)12yes—
355C22×C4H(5,𝔽2)12yes—
356C22×C42−1+412yes—
357C2×C8—22no—
358C2×C8—24no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—41no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—84no—
393C22⋊C4—88no—
394C22⋊C4—88no—
395C22⋊C4—88no—
396C22⋊C4—88no—
397C42—22no—
398C42—22no—
399C42—24no—
400C42—42no—
401C42—42no—
402C42—42no—
403C42—42no—
404C42—48no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23—82no—
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—
440C23C2×#12111yescharacteristic
441C23#53712yes—
442C23#54611yescharacteristic
443C23#54612yes—
444C23#55011yescharacteristic
445C23C2×#13511yescharacteristic
446C23C2×AGL(1,ℤ/8)11yescharacteristic
447C23C2×AGL(1,ℤ/8)12yes—
448C23#57611yescharacteristic
449C23#57711yescharacteristic
450C23#57712yes—
451D4—44no—
452D4—44no—
453D4—44no—
454D4—44no—
455D4—816no—
456D4—816no—
457D4—82no—
458D4—82no—
459D4—82no—
460D4—82no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
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—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—81no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4#46314yes—
514C2×C4#52511yescharacteristic
515C2×C4#52511yescharacteristic
516C2×C4#53511yescharacteristic
517C2×C4#53511yescharacteristic
518C2×C4#53511yescharacteristic
519C2×C4#53611yescharacteristic
520C2×C4#53912yes—
521C2×C4#53912yes—
522C2×C4#54212yes—
523C2×C4#54212yes—
524C2×C4#54212yes—
525C2×C4#55112yes—
526C2×C4#55112yes—
527C2×C4#55312yes—
528C2×C4#55311yescharacteristic
529C2×C4#55511yescharacteristic
530C2×C4#56012yes—
531C2×C4#56012yes—
532C8—42no—
533C8—82no—
534C8—81no—
535C22—84no—
536C22—84no—
537C22—84no—
538C22—84no—
539C22—84no—
540C22—84no—
541C22—84no—
542C22—162no—
543C22—162no—
544C22—162no—
545C22—162no—
546C22—162no—
547C22—162no—
548C22—162no—
549C22—162no—
550C22#241111yescharacteristic
551C22#241411yescharacteristic
552C22#262011yescharacteristic
553C22#269112yes—
554C22#270512yes—
555C22C2×#46311yescharacteristic
556C22#303811yescharacteristic
557C22#304311yescharacteristic
558C22#304511yescharacteristic
559C22#324712yes—
560C22#325312yes—
561C22#325712yes—
562C22#326012yes—
563C22#329612yes—
564C22#330511yescharacteristic
565C22#330612yes—
566C22#331012yes—
567C22#331812yes—
568C22#332412yes—
569C22#333211yescharacteristic
570C22#339411yescharacteristic
571C22#339712yes—
572C22#339911yescharacteristic
573C4—24no—
574C4—44no—
575C4—42no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—162no—
581C4—161no—
582C4#259811yescharacteristic
583C4#263211yescharacteristic
584C4#263311yescharacteristic
585C4#273712yes—
586C4#275712yes—
587C4#281811yescharacteristic
588C2—164no—
589C2#1666411yescharacteristic
590C2#2447711yescharacteristic
591C2#2449211yescharacteristic
592C2#2463111yescharacteristic
593C2#2464411yescharacteristic
594C2#2982511yescharacteristic
595C2#2983511yescharacteristic
596C2#3149512yes—
597C2#3151412yes—
598C2#3190912yes—
599C2#3191412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C23×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
2. C2×#1920C2semidirect
4. #13921C2semidirect
5. #11322C2semidirect
6. #11228C2semidirect
7. #11189C2semidirect
8. #11121C2semidirect
9. #11076C2semidirect
10. #11076C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19822
C2×#19202
#151402
#139212
#113222
#112282
#111892
#111212
#110762
#110082