Finite Group Databasebeta

← #472268

Subgroups — #472268

Order 512.

1264 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2034C212yesmaximal
2#14340C211yesmaximal, characteristic, complemented
3#14257C211yesmaximal, characteristic
4#11323C211yesmaximal, characteristic, complemented
5#11268C211yesmaximal, characteristic, complemented
6#11257C212yesmaximal, complemented
7#11199C212yesmaximal, complemented
8#11189C212yesmaximal, complemented
9#11051C211yesmaximal, characteristic, complemented
10#11049C211yesmaximal, characteristic, complemented
11#10713C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#106C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×C22⋊C8C2212yes—
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#380C2212yes—
23C2×#380C2211yescharacteristic
24#2034—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33#1441—22no—
34#1439—22no—
35#1439—22no—
36#1438—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C23×D4—21no—
41C23×D4C2311yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#121—22no—
46C2×#121—42no—
47C2×#121—42no—
48C2×#121D414yes—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—42no—
52C2×#117—42no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4D412yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4D412yes—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#106—24no—
69C2×#106—42no—
70C2×#106—42no—
71C2×#102—24no—
72C2×#102—24no—
73C2×#102—24no—
74C2×#102—42no—
75C2×#102—42no—
76C2×#102—42no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—42no—
80#390—22no—
81#390—22no—
82#390—42no—
83#384—42no—
84#382—42no—
85#380—24no—
86#380—24no—
87#380—42no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95#340—24no—
96#340—24no—
97#340—42no—
98#340—42no—
99#340—42no—
100#336—22no—
101#336—22no—
102#336—42no—
103C25—21no—
104C25C2×D411yescharacteristic
105C22×D4—24no—
106C22×D4—21no—
107C22×D4—21no—
108C22×D4—22no—
109C22×D4—21no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4—42no—
116C22×D4—42no—
117C22×D4—82no—
118C22×D4—82no—
119C22×D4C2×D412yes—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4#4112yes—
138C23×C4C2411yescharacteristic
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8#4114yes—
153#121—48no—
154#121—48no—
155#117—48no—
156#117—48no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4C2×D412yes—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4—82no—
201C2×C22⋊C4—82no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42#4112yes—
211C2×C42#4112yes—
212#106—48no—
213#102—48no—
214#102—48no—
215C22⋊C8—48no—
216#95—42no—
217#95—44no—
218#95—44no—
219#95—82no—
220#95—82no—
221#95—82no—
222#95—82no—
223C24—21no—
224C24—21no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—22no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24—41no—
235C24—41no—
236C24—41no—
237C24—41no—
238C24—41no—
239C24—41no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24#12112yes—
245C24C22×D411yescharacteristic
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—48no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—88no—
264C2×D4—88no—
265C2×D4—84no—
266C2×D4—84no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4#12112yes—
332C22×C4#12314yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4112yes—
338C22×C4H(5,𝔽2)12yes—
339C22×C4H(5,𝔽2)12yes—
340C22×C4H(5,𝔽2)12yes—
341C22×C42−1+411yescharacteristic
342C22×C42−1+411yescharacteristic
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C4⋊C4—24no—
358C4⋊C4—48no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—88no—
373C22⋊C4—48no—
374C22⋊C4—48no—
375C22⋊C4—88no—
376C22⋊C4—88no—
377C22⋊C4—88no—
378C22⋊C4—84no—
379C42—24no—
380C42—24no—
381C42—42no—
382C42—42no—
383C23—22no—
384C23—22no—
385C23—24no—
386C23—24no—
387C23—22no—
388C23—41no—
389C23—42no—
390C23—42no—
391C23—41no—
392C23—42no—
393C23—41no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—42no—
404C23—42no—
405C23—41no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—44no—
410C23—42no—
411C23—81no—
412C23—82no—
413C23—82no—
414C23—81no—
415C23—82no—
416C23—81no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23—82no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23—82no—
427C23—82no—
428C23—81no—
429C23C2×#12111yescharacteristic
430C23C2×#12311yescharacteristic
431C23#53211yescharacteristic
432C23#53411yescharacteristic
433C23#53411yescharacteristic
434C23#53511yescharacteristic
435C23#53611yescharacteristic
436C23#57612yes—
437C23#57712yes—
438C23#57712yes—
439C23#57712yes—
440D4—44no—
441D4—44no—
442D4—816no—
443D4—82no—
444D4—82no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—82no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4#53412yes—
500C2×C4#53812yes—
501C2×C4#53812yes—
502C2×C4#53812yes—
503C2×C4#53912yes—
504C2×C4#54212yes—
505C2×C4#54612yes—
506C2×C4#54711yescharacteristic
507C2×C4#54711yescharacteristic
508C2×C4#55211yescharacteristic
509C2×C4#55211yescharacteristic
510C2×C4#55312yes—
511C2×C4#55312yes—
512C2×C4#55512yes—
513C2×C4#56212yes—
514C2×C4#56212yes—
515C8—44no—
516C8—82no—
517C22—44no—
518C22—42no—
519C22—42no—
520C22—44no—
521C22—42no—
522C22—81no—
523C22—81no—
524C22—82no—
525C22—82no—
526C22—81no—
527C22—81no—
528C22—84no—
529C22—82no—
530C22—82no—
531C22—84no—
532C22—82no—
533C22—161no—
534C22—161no—
535C22—162no—
536C22—162no—
537C22—161no—
538C22—161no—
539C22#248611yescharacteristic
540C22#250011yescharacteristic
541C22#254311yescharacteristic
542C22#261711yescharacteristic
543C22#262211yescharacteristic
544C22#266311yescharacteristic
545C22#281611yescharacteristic
546C22#304212yes—
547C22#304512yes—
548C22#309612yes—
549C22#309912yes—
550C22#317312yes—
551C22#317912yes—
552C22#319812yes—
553C22#319812yes—
554C22#320012yes—
555C22#320412yes—
556C22#322011yescharacteristic
557C22#322111yescharacteristic
558C22#322211yescharacteristic
559C22#322211yescharacteristic
560C22#324112yes—
561C22#324412yes—
562C4—42no—
563C4—44no—
564C4—82no—
565C4—82no—
566C4—161no—
567C4—162no—
568C4—162no—
569C4—161no—
570C4#268612yes—
571C4#270812yes—
572C4#277912yes—
573C4#278112yes—
574C2—82no—
575C2—162no—
576C2#1681611yescharacteristic
577C2#2765412yes—
578C2#2789811yescharacteristic
579C2#2790311yescharacteristic
580C2#2858512yes—
581C2#2972311yescharacteristic
582C2#2972511yescharacteristic
583C2#2989912yes—
584C2#3037212yes—
585C2#3278211yescharacteristic
586C2#3278511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14340C2semidirect
4. #11323C2semidirect
5. #11268C2semidirect
6. #11257C2semidirect
7. #11199C2semidirect
8. #11189C2semidirect
9. #11051C2semidirect
10. #11049C2semidirect
11. #10713C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20342
#143402
#142572
#113232
#112682
#112572
#111992
#111892
#110512
#110492
#107132