Finite Group Databasebeta

← #404650

Subgroups — #404650

Order 512.

1418 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1982C211yesmaximal, characteristic
2C2×#1918C211yesmaximal, characteristic, complemented
3#15140C211yesmaximal, characteristic
4#13810C211yesmaximal, characteristic, complemented
5#11255C211yesmaximal, characteristic, complemented
6#11227C211yesmaximal, characteristic, complemented
7#11189C212yesmaximal, complemented
8#11123C212yesmaximal, complemented
9#11008C211yesmaximal, characteristic
10#10931C212yesmaximal, complemented
11#10931C212yesmaximal, complemented
12C22×#127C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C2×#394—21no—
20C2×#394—21no—
21C2×#390C2212yes—
22C2×#390C2212yes—
23C2×#389C2211yescharacteristic
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#376—21no—
27#1982—24no—
28#1918—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1438—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40#1434—21no—
41#1434—22no—
42C4×#95C2211yescharacteristic
43C23×D4—21no—
44C23×D4—21no—
45C23×D4C2312yes—
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#127—24no—
50C2×#127—22no—
51C2×#127—22no—
52C2×#127—42no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C42—22no—
64C22×C42—21no—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C2×#107—22no—
68C2×#107—22no—
69C2×#107—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—42no—
72C2×#102—24no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—42no—
81C2×#102—42no—
82C2×#102—42no—
83C2×#102—42no—
84#394—44no—
85#394—44no—
86#390—24no—
87#390—24no—
88#390—42no—
89#390—42no—
90#389—22no—
91#389—22no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4D414yes—
96#376—44no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95D411yescharacteristic
105C2×#95D412yes—
106C2×#95D411yescharacteristic
107#340—22no—
108#340—22no—
109#340—22no—
110#340—42no—
111#340—42no—
112#336—22no—
113#336—42no—
114#336—42no—
115C25—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—22no—
129C22×D4—44no—
130C22×D4—44no—
131C22×D4—44no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4—42no—
136C22×D4—82no—
137C22×D4—81no—
138C22×D4—81no—
139C22×D4#4114yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8#4112yes—
167#127—28no—
168#127—48no—
169#127—44no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4#4114yes—
197C2×C4⋊C4#4112yes—
198C2×C4⋊C4#4114yes—
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—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4—82no—
209C2×C22⋊C4—82no—
210C2×C22⋊C4—82no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—28no—
214C2×C42—24no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—44no—
222C2×C42—44no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42C2×D412yes—
226C2×C42#4112yes—
227#107—28no—
228C4⋊C8—44no—
229#102—48no—
230#102—48no—
231#102—48no—
232#102—48no—
233#95—28no—
234#95—24no—
235#95—22no—
236#95—42no—
237#95—42no—
238#95—44no—
239#95—42no—
240#95—81no—
241#95—82no—
242#95D814yes—
243C24—24no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24C22×D411yescharacteristic
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—48no—
266C2×D4—48no—
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—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—88no—
290C2×D4—88no—
291C2×D4—84no—
292C2×D4—84no—
293C2×D4—84no—
294C2×D4—88no—
295C2×D4—88no—
296C2×D4—84no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4#12112yes—
348C22×C4#12112yes—
349C22×C4C2×D814yes—
350C22×C4C2×D812yes—
351C22×C4AGL(1,ℤ/8)12yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C2×#4112yes—
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4112yes—
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4112yes—
359C22×C4H(5,𝔽2)11yescharacteristic
360C22×C4H(5,𝔽2)12yes—
361C22×C4H(5,𝔽2)11yescharacteristic
362C2×C8—22no—
363C2×C8—24no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—42no—
396C4⋊C4—84no—
397C4⋊C4#12414yes—
398C22⋊C4—88no—
399C22⋊C4—88no—
400C22⋊C4—88no—
401C22⋊C4—88no—
402C42—22no—
403C42—22no—
404C42—24no—
405C42—416no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—48no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
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—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23C2×#12111yescharacteristic
447C23#52512yes—
448C23#54612yes—
449C23#54611yescharacteristic
450C23#55011yescharacteristic
451C23C22×D811yescharacteristic
452C23C2×AGL(1,ℤ/8)11yescharacteristic
453C23#57512yes—
454C23#57611yescharacteristic
455C23#57712yes—
456C23#57711yescharacteristic
457D4—44no—
458D4—44no—
459D4—44no—
460D4—44no—
461D4—816no—
462D4—816no—
463D4—82no—
464D4—82no—
465D4—82no—
466D4—82no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—44no—
479C2×C4—44no—
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—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—82no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4#45914yes—
522C2×C4#52511yescharacteristic
523C2×C4#52511yescharacteristic
524C2×C4C2×#12411yescharacteristic
525C2×C4#52912yes—
526C2×C4#52912yes—
527C2×C4#52912yes—
528C2×C4#53211yescharacteristic
529C2×C4#53511yescharacteristic
530C2×C4#53511yescharacteristic
531C2×C4#54012yes—
532C2×C4#54012yes—
533C2×C4#55112yes—
534C2×C4#55112yes—
535C2×C4#55112yes—
536C2×C4#55312yes—
537C2×C4#55311yescharacteristic
538C2×C4#55512yes—
539C2×C4#55511yescharacteristic
540C8—42no—
541C8—81no—
542C8—82no—
543C22—84no—
544C22—84no—
545C22—84no—
546C22—84no—
547C22—84no—
548C22—84no—
549C22—84no—
550C22—162no—
551C22—162no—
552C22—162no—
553C22—162no—
554C22—162no—
555C22—162no—
556C22—162no—
557C22—162no—
558C22#241111yescharacteristic
559C22#241411yescharacteristic
560C22#247112yes—
561C22#247612yes—
562C22#285011yescharacteristic
563C22C2×#45911yescharacteristic
564C22#304011yescharacteristic
565C22#304311yescharacteristic
566C22#304511yescharacteristic
567C22#310012yes—
568C22#311512yes—
569C22#311912yes—
570C22#312612yes—
571C22#329411yescharacteristic
572C22#329412yes—
573C22#331012yes—
574C22#331712yes—
575C22#331912yes—
576C22#332312yes—
577C22#332511yescharacteristic
578C22#339111yescharacteristic
579C22#339711yescharacteristic
580C22#340112yes—
581C4—24no—
582C4—44no—
583C4—42no—
584C4—44no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4—84no—
589C4—161no—
590C4#239811yescharacteristic
591C4#252612yes—
592C4#253112yes—
593C4#260311yescharacteristic
594C4#261011yescharacteristic
595C4#264111yescharacteristic
596C2—164no—
597C2#1489811yescharacteristic
598C2#2447611yescharacteristic
599C2#2449511yescharacteristic
600C2#2462611yescharacteristic
601C2#2464911yescharacteristic
602C2#2719012yes—
603C2#2721412yes—
604C2#2739012yes—
605C2#2741412yes—
606C2#3302411yescharacteristic
607C2#3303911yescharacteristic
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×#1918C2semidirect
4. #13810C2semidirect
5. #11255C2semidirect
6. #11227C2semidirect
7. #11189C2semidirect
8. #11123C2semidirect
10. #10931C2semidirect
11. #10931C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19822
C2×#19182
#151402
#138102
#112552
#112272
#111892
#111232
#110082
#109312