Finite Group Databasebeta

← #409713

Subgroups — #409713

Order 512.

1194 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1990C211yesmaximal, characteristic, complemented
2C2×#1753C211yesmaximal, characteristic
3#15195C211yesmaximal, characteristic
4#15125C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11371C211yesmaximal, characteristic, complemented
7#11315C212yesmaximal, complemented
8#11199C212yesmaximal
9#10821C212yesmaximal, complemented
10#10820C211yesmaximal, characteristic, complemented
11#10722C212yesmaximal
12C23×M16C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C22×#97C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C2×#393—21no—
18C2×#384C2211yescharacteristic
19C2×#380C2212yes—
20C2×#380C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22#1990—24no—
23#1753—24no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34#1446—21no—
35#1442—21no—
36#1442—21no—
37#1442—21no—
38#1439—21no—
39#1439—21no—
40#1439—21no—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—24no—
51C22×M16—42no—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4D411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×#97—22no—
77C2×#97—24no—
78C2×C4×C8—42no—
79C2×C4×C8D414yes—
80#393—44no—
81#384—22no—
82#384—22no—
83#380—24no—
84#380—24no—
85#380—24no—
86#380—42no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95D411yescharacteristic
100C2×#95D411yescharacteristic
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340—42no—
109#336—22no—
110#336—22no—
111C25C2×D411yescharacteristic
112C22×Q8—44no—
113C22×Q8—42no—
114C22×Q8—42no—
115C22×Q8—42no—
116C22×Q8—82no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4#4112yes—
135C23×C4#4112yes—
136C23×C4C2411yescharacteristic
137C2×M16—28no—
138C2×M16—48no—
139C2×M16—48no—
140C2×M16—44no—
141C2×M16—44no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8C2×D412yes—
158C22×C8C2×D412yes—
159C22×C8#4112yes—
160C22×C8#4112yes—
161C22×C8#4114yes—
162C4⋊Q8—28no—
163C4⋊Q8—48no—
164C4⋊Q8—44no—
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—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4#4114yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4—82no—
206C2×C22⋊C4—82no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42C2×D412yes—
214C2×C42#4112yes—
215C22⋊C8—28no—
216#97—44no—
217C4×C8—44no—
218#95—22no—
219#95—22no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—42no—
224#95—42no—
225#95—42no—
226#95—42no—
227#95—82no—
228#95—82no—
229#95—82no—
230C24—21no—
231C24—21no—
232C24—22no—
233C24—22no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—22no—
239C24—22no—
240C24#12412yes—
241C24C22×D411yescharacteristic
242C2×Q8—48no—
243C2×Q8—88no—
244C2×Q8—88no—
245C2×Q8—84no—
246C2×Q8—84no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
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—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4#12112yes—
312C22×C4#12112yes—
313C22×C4#12412yes—
314C22×C4AGL(1,ℤ/8)12yes—
315C22×C4#13712yes—
316C22×C4#13712yes—
317C22×C4#13712yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4112yes—
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4112yes—
325C22×C4C2×#4111yescharacteristic
326C22×C4H(5,𝔽2)12yes—
327C22×C42−1+412yes—
328M16—416no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—24no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—24no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8#12414yes—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—48no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—84no—
367C4⋊C4—84no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—88no—
371C22⋊C4—88no—
372C42—22no—
373C42—22no—
374C42—24no—
375C42—41no—
376C42—41no—
377C42—42no—
378C23—22no—
379C23—24no—
380C23—22no—
381C23—22no—
382C23—24no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—42no—
397C23—41no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23C2×#12111yescharacteristic
402C23C2×#12411yescharacteristic
403C23#53211yescharacteristic
404C23#54612yes—
405C23#54712yes—
406C23C2×AGL(1,ℤ/8)11yescharacteristic
407C23C2×#13711yescharacteristic
408C23C2×#13711yescharacteristic
409C23C2×#13711yescharacteristic
410C23#57612yes—
411C23#57712yes—
412Q8—816no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—44no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4#46814yes—
471C2×C4#47014yes—
472C2×C4#52511yescharacteristic
473C2×C4#52511yescharacteristic
474C2×C4#52511yescharacteristic
475C2×C4C2×#12411yescharacteristic
476C2×C4#52912yes—
477C2×C4#52912yes—
478C2×C4#52911yescharacteristic
479C2×C4#52911yescharacteristic
480C2×C4#53411yescharacteristic
481C2×C4#53611yescharacteristic
482C2×C4#53912yes—
483C2×C4#54212yes—
484C2×C4#55112yes—
485C2×C4#55112yes—
486C2×C4#55112yes—
487C2×C4#55212yes—
488C2×C4#55612yes—
489C2×C4#55612yes—
490C8—24no—
491C8—42no—
492C8—42no—
493C8—42no—
494C22—42no—
495C22—42no—
496C22—44no—
497C22—44no—
498C22—42no—
499C22—81no—
500C22—81no—
501C22—82no—
502C22—82no—
503C22—81no—
504C22—81no—
505C22#248812yes—
506C22#252112yes—
507C22#253411yescharacteristic
508C22#254711yescharacteristic
509C22#285011yescharacteristic
510C22C2×#46811yescharacteristic
511C22C2×#47011yescharacteristic
512C22#305711yescharacteristic
513C22#305811yescharacteristic
514C22#313612yes—
515C22#313911yescharacteristic
516C22#314111yescharacteristic
517C22#317912yes—
518C22#318211yescharacteristic
519C22#318511yescharacteristic
520C22#331412yes—
521C22#331612yes—
522C22#331812yes—
523C22#332012yes—
524C22#335412yes—
525C22#335612yes—
526C22#335712yes—
527C22#335712yes—
528C4—24no—
529C4—24no—
530C4—42no—
531C4—81no—
532C4—81no—
533C4—84no—
534C4—162no—
535C4—162no—
536C4—162no—
537C4#239811yescharacteristic
538C4#252312yes—
539C4#252712yes—
540C4#256111yescharacteristic
541C4#259911yescharacteristic
542C4#263811yescharacteristic
543C2—82no—
544C2#1505011yescharacteristic
545C2#2772012yes—
546C2#2772112yes—
547C2#2808812yes—
548C2#2809112yes—
549C2#2810411yescharacteristic
550C2#2810811yescharacteristic
551C2#2871111yescharacteristic
552C2#2871311yescharacteristic
553C2#3304511yescharacteristic
554C2#3304911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1990C2semidirect
5. #11384C2semidirect
6. #11371C2semidirect
7. #11315C2semidirect
9. #10821C2semidirect
10. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19902
C2×#17532
#151952
#151252
#113842
#113712
#113152
#111992
#108212
#108202
#107222