Finite Group Databasebeta

← #402370

Subgroups — #402370

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1864C211yesmaximal, characteristic
2C2×#1802C211yesmaximal, characteristic
3#15145C211yesmaximal, characteristic, complemented
4#13820C211yesmaximal, characteristic
5#11371C211yesmaximal, characteristic, complemented
6#11322C211yesmaximal, characteristic, complemented
7#11228C211yesmaximal, characteristic, complemented
8#11076C212yesmaximal
9#11049C212yesmaximal
10#10908C212yesmaximal, complemented
11#10849C212yesmaximal, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C2×#398—22no—
20C2×#390C2212yes—
21C2×#389C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#384C2212yes—
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C22⋊C4—21no—
26#1864—24no—
27#1802—24no—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36#1444—21no—
37#1444—21no—
38#1437—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C22⋊#95—21no—
42C4×#95C2211yescharacteristic
43C23×D4C2311yescharacteristic
44C24×C4—21no—
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C2×C4×D4—24no—
49C2×C4×D4—22no—
50C2×C4×D4—42no—
51C2×C4×D4D414yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C42—22no—
66C22×C42C2311yescharacteristic
67C22×C42C2312yes—
68C2×#106—22no—
69C2×#106—24no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8—42no—
73C2×C4⋊C8—42no—
74C2×C4⋊C8—42no—
75C2×#102—22no—
76C2×#102—22no—
77C2×#102—22no—
78C2×#102—24no—
79C2×#102—22no—
80C2×#102—42no—
81C2×#102—42no—
82C2×#102—42no—
83#398—48no—
84#390—24no—
85#390Q818yes—
86#389—22no—
87#389—22no—
88#384—24no—
89#384—22no—
90#384—22no—
91#384—42no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—42no—
94C4×C22⋊C4—44no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95D412yes—
102C2×#95D411yescharacteristic
103C2×#95D411yescharacteristic
104#340—24no—
105#340—24no—
106#340—22no—
107#340—42no—
108#340—42no—
109#340—42no—
110#336—22no—
111#336—24no—
112C25—21no—
113C22×D4—22no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4C2×D412yes—
122C22×D4C2×Q814yes—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—24no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—44no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—82no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2411yescharacteristic
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8#4112yes—
155C4×D4—28no—
156C4×D4—44no—
157C4×D4—48no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×Q814yes—
194C2×C4⋊C4#4112yes—
195C2×C4⋊C4#4114yes—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—44no—
211C2×C42—44no—
212C2×C42—82no—
213C2×C42C2×D412yes—
214C2×C42C2×Q814yes—
215C2×C42#4112yes—
216C2×C42#4114yes—
217#106—44no—
218C4⋊C8—44no—
219C4⋊C8—48no—
220#102—28no—
221#102—44no—
222#102—48no—
223#95—24no—
224#95—22no—
225#95—44no—
226#95—44no—
227#95—42no—
228#95—42no—
229#95—44no—
230#95—81no—
231#95SD1614yes—
232C24—22no—
233C24—24no—
234C24—42no—
235C24—41no—
236C24—41no—
237C24—41no—
238C24—42no—
239C24—41no—
240C24—42no—
241C24—41no—
242C24—41no—
243C24C22×D411yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—24no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4C4⋊Q814yes—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—48no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—48no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—84no—
310C22×C4—88no—
311C22×C4—88no—
312C22×C4#12112yes—
313C22×C4#12112yes—
314C22×C4C22⋊Q814yes—
315C22×C4C22⋊Q814yes—
316C22×C4C2×SD1612yes—
317C22×C4C2×SD1614yes—
318C22×C4AGL(1,ℤ/8)14yes—
319C22×C4#13712yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C22×Q812yes—
322C22×C4C2×#4112yes—
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4112yes—
326C22×C4C2×#4111yescharacteristic
327C22×C4H(5,𝔽2)12yes—
328C22×C4H(5,𝔽2)11yescharacteristic
329C22×C4H(5,𝔽2)11yescharacteristic
330C2×C8—22no—
331C2×C8—24no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—88no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C4⋊C4—84no—
368C4⋊C4—88no—
369C4⋊C4#12314yes—
370C22⋊C4—48no—
371C22⋊C4—416no—
372C22⋊C4—88no—
373C42—22no—
374C42—24no—
375C42—22no—
376C42—24no—
377C42—24no—
378C42—24no—
379C42—24no—
380C42—88no—
381C23—24no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—42no—
386C23—42no—
387C23—44no—
388C23—44no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—44no—
394C23—81no—
395C23—82no—
396C23—81no—
397C23—81no—
398C23—82no—
399C23—82no—
400C23—81no—
401C23—82no—
402C23—82no—
403C23—82no—
404C23C2×#12111yescharacteristic
405C23C2×C22⋊Q812yes—
406C23#54611yescharacteristic
407C23#54612yes—
408C23#55011yescharacteristic
409C23C22×SD1611yescharacteristic
410C23C2×AGL(1,ℤ/8)12yes—
411C23C2×#13711yescharacteristic
412C23#57612yes—
413C23#57711yescharacteristic
414C23#57811yescharacteristic
415D4—24no—
416D4—24no—
417D4—44no—
418D4—42no—
419D4—42no—
420D4—44no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—816no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4#46114yes—
474C2×C4#47618yes—
475C2×C4C2×#12311yescharacteristic
476C2×C4#52511yescharacteristic
477C2×C4#52912yes—
478C2×C4#52912yes—
479C2×C4C2×C4⋊Q811yescharacteristic
480C2×C4#53211yescharacteristic
481C2×C4#53312yes—
482C2×C4#53411yescharacteristic
483C2×C4#53511yescharacteristic
484C2×C4#53812yes—
485C2×C4#54012yes—
486C2×C4#55112yes—
487C2×C4#55311yescharacteristic
488C2×C4#55412yes—
489C2×C4#55412yes—
490C2×C4#55412yes—
491C2×C4#55412yes—
492C2×C4#55511yescharacteristic
493C8—42no—
494C8—81no—
495C8—82no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—84no—
500C22—82no—
501C22—84no—
502C22—82no—
503C22—84no—
504C22—84no—
505C22—82no—
506C22—82no—
507C22#241211yescharacteristic
508C22#241411yescharacteristic
509C22#245012yes—
510C22#245412yes—
511C22#285711yescharacteristic
512C22C2×#46111yescharacteristic
513C22#303811yescharacteristic
514C22#304111yescharacteristic
515C22#304611yescharacteristic
516C22C2×#47612yes—
517C22#307912yes—
518C22#308112yes—
519C22#308312yes—
520C22#329512yes—
521C22#329811yescharacteristic
522C22#330812yes—
523C22#331412yes—
524C22#331812yes—
525C22#332911yescharacteristic
526C22#333212yes—
527C22#339211yescharacteristic
528C22#339811yescharacteristic
529C22#339912yes—
530C4—28no—
531C4—24no—
532C4—44no—
533C4—44no—
534C4—42no—
535C4—82no—
536C4—84no—
537C4—162no—
538C4—161no—
539C4#239511yescharacteristic
540C4#250211yescharacteristic
541C4#252212yes—
542C4#257412yes—
543C4#258311yescharacteristic
544C4#266111yescharacteristic
545C2—84no—
546C2#1484011yescharacteristic
547C2#2453011yescharacteristic
548C2#2454211yescharacteristic
549C2#2463011yescharacteristic
550C2#2464211yescharacteristic
551C2#2641012yes—
552C2#2642512yes—
553C2#2653012yes—
554C2#2654212yes—
555C2#3319711yescharacteristic
556C2#3320311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
3. #15145C2semidirect
5. #11371C2semidirect
6. #11322C2semidirect
7. #11228C2semidirect
10. #10908C2semidirect
11. #10849C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18642
C2×#18022
#151452
#138202
#113712
#113222
#112282
#110762
#110492
#109082
#108492