Finite Group Databasebeta

← #474270

Subgroups — #474270

Order 512.

1304 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15152C211yesmaximal, characteristic, complemented
2#15127C211yesmaximal, characteristic
3#11315C212yesmaximal, complemented
4#11189C212yesmaximal, complemented
5#11076C212yesmaximal, complemented
6#11049C212yesmaximal, complemented
7#11008C212yesmaximal
8#10925C212yesmaximal, complemented
9#10834C211yesmaximal, characteristic, complemented
10C22×#127C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C22×#102C2212yes—
13C22×#102C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×C4×C8C2211yescharacteristic
17C2×#400—21no—
18C2×#394—21no—
19C2×#394—21no—
20C2×#390C2212yes—
21C2×#390C2212yes—
22C2×#384C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30#1445—21no—
31#1442—21no—
32#1441—21no—
33#1438—21no—
34#1438—21no—
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C23×D4—21no—
40C23×D4—21no—
41C23×D4C2312yes—
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×#127—24no—
45C2×#127—42no—
46C2×#127—41no—
47C2×#127—41no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×C4⋊C8—22no—
63C2×C4⋊C8—41no—
64C2×C4⋊C8—41no—
65C2×#102—24no—
66C2×#102—24no—
67C2×#102—24no—
68C2×#102—42no—
69C2×#102—42no—
70C2×#102—42no—
71C2×#102—42no—
72C2×#102—42no—
73C2×#102—42no—
74C2×#102—42no—
75C2×#102—42no—
76C2×#102—42no—
77C2×C4×C8—22no—
78C2×C4×C8—41no—
79C2×C4×C8—41no—
80#400—44no—
81#394—44no—
82#394—44no—
83#390—24no—
84#390—24no—
85#390—24no—
86#390—42no—
87#384—41no—
88#384—41no—
89C4×C4⋊C4—41no—
90C4×C4⋊C4—41no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95D411yescharacteristic
100C2×#95D411yescharacteristic
101#340—24no—
102#340—42no—
103#340—42no—
104#340—42no—
105#336—42no—
106#336—42no—
107C25—22no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—44no—
122C22×D4—44no—
123C22×D4—42no—
124C22×D4—42no—
125C22×D4—42no—
126C22×D4—42no—
127C22×D4—42no—
128C22×D4—42no—
129C22×D4—82no—
130C22×D4—82no—
131C22×D4#4114yes—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
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—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8#4114yes—
156#127—44no—
157#127—48no—
158#127—44no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
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—22no—
168C2×C4⋊C4—22no—
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—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4D814yes—
187C2×C4⋊C4SD1614yes—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4—82no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42#4112yes—
213C2×C42#4112yes—
214C4⋊C8—44no—
215#102—48no—
216#102—48no—
217#102—48no—
218#102—48no—
219C4×C8—44no—
220#95—41no—
221#95—41no—
222#95—42no—
223#95—42no—
224#95—42no—
225#95—44no—
226#95—44no—
227#95—42no—
228#95—82no—
229#95—82no—
230C24—24no—
231C24—42no—
232C24—42no—
233C24—42no—
234C24—42no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24C22×D411yescharacteristic
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—48no—
253C2×D4—48no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
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—88no—
277C2×D4—88no—
278C2×D4—84no—
279C2×D4—84no—
280C2×D4—88no—
281C2×D4—88no—
282C2×D4—84no—
283C2×D4—84no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
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#12412yes—
330C22×C4#12412yes—
331C22×C4C2×D812yes—
332C22×C4C2×D812yes—
333C22×C4C2×D812yes—
334C22×C4C2×SD1612yes—
335C22×C4C2×SD1612yes—
336C22×C4C2×SD1612yes—
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4112yes—
339C22×C4C2×#4112yes—
340C22×C4C2×#4111yescharacteristic
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C4H(5,𝔽2)12yes—
344C22×C4H(5,𝔽2)11yescharacteristic
345C22×C4H(5,𝔽2)12yes—
346C22×C42−1+411yescharacteristic
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C4⋊C4—44no—
362C4⋊C4—44no—
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—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C22⋊C4—88no—
378C22⋊C4—88no—
379C22⋊C4—88no—
380C22⋊C4—88no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—42no—
386C42—42no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23—82no—
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—
426C23C2×#12411yescharacteristic
427C23#53411yescharacteristic
428C23#53711yescharacteristic
429C23#54612yes—
430C23#55012yes—
431C23C22×D811yescharacteristic
432C23C22×SD1611yescharacteristic
433C23#57511yescharacteristic
434C23#57511yescharacteristic
435C23#57612yes—
436C23#57712yes—
437D4—44no—
438D4—44no—
439D4—44no—
440D4—44no—
441D4—816no—
442D4—816no—
443D4—82no—
444D4—82no—
445D4—82no—
446D4—82no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
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—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4#48614yes—
500C2×C4#52912yes—
501C2×C4#53811yescharacteristic
502C2×C4#53812yes—
503C2×C4#53812yes—
504C2×C4#53811yescharacteristic
505C2×C4#53911yescharacteristic
506C2×C4#53912yes—
507C2×C4#53911yescharacteristic
508C2×C4#54012yes—
509C2×C4#54212yes—
510C2×C4#55112yes—
511C2×C4#55111yescharacteristic
512C2×C4#55312yes—
513C2×C4#55312yes—
514C2×C4#55511yescharacteristic
515C2×C4#56012yes—
516C2×C4#56011yescharacteristic
517C2×C4H(3,𝔽4)11yescharacteristic
518C8—44no—
519C8—82no—
520C22—84no—
521C22—84no—
522C22—84no—
523C22—84no—
524C22—84no—
525C22—84no—
526C22—84no—
527C22—162no—
528C22—162no—
529C22—162no—
530C22—162no—
531C22—162no—
532C22—162no—
533C22—162no—
534C22—162no—
535C22#249012yes—
536C22#263911yescharacteristic
537C22#269711yescharacteristic
538C22#270512yes—
539C22#273311yescharacteristic
540C22C2×#48611yescharacteristic
541C22#313511yescharacteristic
542C22#313612yes—
543C22#318611yescharacteristic
544C22#318711yescharacteristic
545C22#319911yescharacteristic
546C22#320011yescharacteristic
547C22#324511yescharacteristic
548C22#324611yescharacteristic
549C22#325712yes—
550C22#329412yes—
551C22#329512yes—
552C22#332212yes—
553C22#332312yes—
554C22#339112yes—
555C22#339212yes—
556C22#340112yes—
557C22#340312yes—
558C4—24no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—162no—
565C4—162no—
566C4—162no—
567C4#257211yescharacteristic
568C4#264912yes—
569C4#277812yes—
570C4#278211yescharacteristic
571C4#279511yescharacteristic
572C4#282811yescharacteristic
573C2—164no—
574C2#1693211yescharacteristic
575C2#2774712yes—
576C2#2775312yes—
577C2#3018511yescharacteristic
578C2#3019211yescharacteristic
579C2#3166611yescharacteristic
580C2#3166711yescharacteristic
581C2#3189812yes—
582C2#3190212yes—
583C2#3266811yescharacteristic
584C2#3266911yescharacteristic
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
1. #15152C2semidirect
3. #11315C2semidirect
4. #11189C2semidirect
5. #11076C2semidirect
6. #11049C2semidirect
8. #10925C2semidirect
9. #10834C2semidirect

Maximal subgroups

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

subgroupindex
#151522
#151272
#113152
#111892
#110762
#110492
#110082
#109252
#108342