Finite Group Databasebeta

← #456911

Subgroups — #456911

Order 512.

1434 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2066C212yesmaximal, complemented
2C2×#2001C211yesmaximal, characteristic
3C2×#1892C212yesmaximal
4C2×#1850C211yesmaximal, characteristic
5#15179C212yesmaximal
6#11292C212yesmaximal, complemented
7#11095C211yesmaximal, characteristic
8#11077C212yesmaximal, complemented
9#11024C212yesmaximal, complemented
10C23×Q16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×C22⋊Q8C2212yes—
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C2×#391C2212yes—
20C2×#391C2212yes—
21C2×#389C2212yes—
22C2×#380C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#2066—28no—
25#2001—24no—
26#1892—28no—
27#1850—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#336C2212yes—
34#1442—22no—
35#1441—22no—
36#1440—22no—
37#1439—22no—
38C4⋊#95C2212yes—
39C23×Q8C2312yes—
40C24×C4C2311yescharacteristic
41C22×Q16—24no—
42C22×Q16—22no—
43C22×Q16—42no—
44C22×Q16—41no—
45C22×Q16—41no—
46C23×C8—22no—
47C23×C8—24no—
48C23×C8—41no—
49C23×C8—41no—
50C23×C8—42no—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—42no—
59C2×C22⋊Q8—42no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4D411yescharacteristic
68C22×C22⋊C4D411yescharacteristic
69C22×C42C2312yes—
70C2×#106—24no—
71C2×#106—41no—
72C2×C4⋊C8—24no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—42no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×C22⋊C8—41no—
85C2×C22⋊C8—41no—
86C2×C22⋊C8D414yes—
87#391—24no—
88#391—24no—
89#391—42no—
90#391—42no—
91#389—24no—
92#389—42no—
93#380—24no—
94#380—42no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95D412yes—
106#340—22no—
107#340—24no—
108#340—42no—
109#340—41no—
110#340D418yes—
111#336—24no—
112C25C2×D411yescharacteristic
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8#4114yes—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4#4112yes—
144C23×C4C2411yescharacteristic
145C2×Q16—48no—
146C2×Q16—44no—
147C2×Q16—44no—
148C2×Q16—48no—
149C2×Q16—44no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—44no—
165C22×C8—48no—
166C22×C8—48no—
167C22×C8—44no—
168C22×C8—44no—
169C22×C8C2×D414yes—
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C22⋊Q8—216no—
173C22⋊Q8—48no—
174C22⋊Q8—48no—
175C22⋊Q8—48no—
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—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×D414yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—82no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216#106—44no—
217C4⋊C8—48no—
218#103—48no—
219#103—48no—
220#103—48no—
221C22⋊C8—44no—
222#95—24no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—82no—
231C24—21no—
232C24—21no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24#12112yes—
239C24#13714yes—
240C24C22×D411yescharacteristic
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
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—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4#12112yes—
333C22×C4#13514yes—
334C22×C4C22×D412yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4112yes—
338C22×C4H(5,𝔽2)11yescharacteristic
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C4H(5,𝔽2)12yes—
341C22×C4H(5,𝔽2)12yes—
342C22×C42−1+411yescharacteristic
343C22×C42−1+411yescharacteristic
344Q16—416no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—416no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—41no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—416no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—88no—
377C22⋊C4—28no—
378C22⋊C4—416no—
379C22⋊C4—44no—
380C22⋊C4—88no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—42no—
385C23—22no—
386C23—24no—
387C23—24no—
388C23—24no—
389C23—24no—
390C23—24no—
391C23—22no—
392C23—24no—
393C23—24no—
394C23—24no—
395C23—41no—
396C23—42no—
397C23—41no—
398C23—42no—
399C23—41no—
400C23—41no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23#47014yes—
406C23C2×#12111yescharacteristic
407C23#53711yescharacteristic
408C23#53711yescharacteristic
409C23D4212yes—
410C23#54612yes—
411C23C2×#13512yes—
412C23C2×#13712yes—
413C23#57712yes—
414C23#57812yes—
415Q8—44no—
416Q8—44no—
417Q8—44no—
418Q8—44no—
419Q8—82no—
420Q8—82no—
421Q8—82no—
422Q8—82no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
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—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4#52512yes—
477C2×C4#53411yescharacteristic
478C2×C4#53512yes—
479C2×C4#53611yescharacteristic
480C2×C4#53712yes—
481C2×C4#53812yes—
482C2×C4#53812yes—
483C2×C4#53911yescharacteristic
484C2×C4#53911yescharacteristic
485C2×C4#53912yes—
486C2×C4D4212yes—
487C2×C4D4211yescharacteristic
488C2×C4#54612yes—
489C2×C4#54812yes—
490C2×C4#55111yescharacteristic
491C2×C4#56011yescharacteristic
492C2×C4#56011yescharacteristic
493C2×C4#56212yes—
494C8—42no—
495C8—44no—
496C8—42no—
497C22—28no—
498C22—24no—
499C22—44no—
500C22—44no—
501C22—42no—
502C22—44no—
503C22—42no—
504C22—44no—
505C22—44no—
506C22#242412yes—
507C22#261811yescharacteristic
508C22#267511yescharacteristic
509C22#267612yes—
510C22#269311yescharacteristic
511C22#305011yescharacteristic
512C22C2×#47011yescharacteristic
513C22#306312yes—
514C22#324812yes—
515C22#324812yes—
516C22#325812yes—
517C22#326111yescharacteristic
518C22#326311yescharacteristic
519C22#326612yes—
520C22#327012yes—
521C22#328212yes—
522C22#328312yes—
523C22#330012yes—
524C22#330412yes—
525C22#331212yes—
526C22#331612yes—
527C4—44no—
528C4—42no—
529C4—84no—
530C4—82no—
531C4—84no—
532C4—84no—
533C4—162no—
534C4#246311yescharacteristic
535C4#263612yes—
536C4#266911yescharacteristic
537C4#268112yes—
538C4#269711yescharacteristic
539C4#282611yescharacteristic
540C2—44no—
541C2#1628811yescharacteristic
542C2#2511312yes—
543C2#2512612yes—
544C2#2977311yescharacteristic
545C2#2978611yescharacteristic
546C2#3085012yes—
547C2#3091212yes—
548C2#3091312yes—
549C2#3155812yes—
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×#2066C2semidirect
6. #11292C2semidirect
8. #11077C2semidirect
9. #11024C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20662
C2×#20012
C2×#18922
C2×#18502
#151792
#112922
#110952
#110772
#110242