Finite Group Databasebeta

← #414035

Subgroups — #414035

Order 512.

1316 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1990C211yesmaximal, characteristic, complemented
2C2×C8×C22⋊C4C211yesmaximal, characteristic
3#15135C211yesmaximal, characteristic
4#15125C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11357C211yesmaximal, characteristic, complemented
7#11220C212yesmaximal, complemented
8#11095C212yesmaximal
9#10880C212yesmaximal, complemented
10#10861C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal
12C24×C8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#106C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#399—21no—
19C2×#393—21no—
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1990—24no—
26C8×C22⋊C4—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1446—21no—
37#1445—21no—
38#1444—21no—
39#1442—21no—
40#1439—21no—
41#1439—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C24×C4C2311yescharacteristic
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—42no—
50C23×C8—42no—
51C23×C8Q814yescomplemented
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—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4D411yescharacteristic
71C22×C22⋊C4D411yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#106—24no—
75C2×#106—24no—
76C2×#106—42no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8Q814yescomplemented
80C2×C4×C8—22no—
81C2×C4×C8—42no—
82C2×C4×C8—42no—
83C2×C4×C8D414yes—
84#399—44no—
85#393—44no—
86#389—24no—
87#389—42no—
88#384—22no—
89#384—22no—
90#380—24no—
91#380—24no—
92#380—42no—
93#380—42no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95D411yescharacteristic
106C2×#95D411yescharacteristic
107#340—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—42no—
112#340—42no—
113#340—42no—
114#336—22no—
115#336—22no—
116C25C2×D411yescharacteristic
117C22×Q8—44no—
118C22×Q8—42no—
119C22×Q8—42no—
120C22×Q8—42no—
121C22×Q8—82no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×Q812yes—
140C23×C4C2×Q812yes—
141C23×C4C2411yescharacteristic
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—28no—
155C22×C8—48no—
156C22×C8—44no—
157C22×C8—44no—
158C22×C8—48no—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C22×C8C2×Q812yes—
162C22×C8C2×Q812yes—
163C22×C8C2×Q812yes—
164C22×C8C2×Q812yes—
165C22×C8#4112yes—
166C22×C8#4112yes—
167C4⋊Q8—28no—
168C4⋊Q8—48no—
169C4⋊Q8—44no—
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—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4#4114yes—
203C2×C4⋊C4#4114yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4SD1614yes—
215C2×C22⋊C4SD1614yes—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42C2×D412yes—
223C2×C42#4112yes—
224#106—48no—
225C22⋊C8—28no—
226C4×C8—44no—
227C4×C8—44no—
228#95—22no—
229#95—22no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—42no—
235#95—42no—
236#95—42no—
237#95—42no—
238#95—82no—
239C24—21no—
240C24—21no—
241C24—22no—
242C24—22no—
243C24—21no—
244C24—21no—
245C24—22no—
246C24—22no—
247C24C4⋊Q812yes—
248C24C2×SD1612yes—
249C24C2×SD1612yes—
250C24C22×D411yescharacteristic
251C2×Q8—48no—
252C2×Q8—88no—
253C2×Q8—88no—
254C2×Q8—84no—
255C2×Q8—84no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
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—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4#12112yes—
319C22×C4#12112yes—
320C22×C4C4⋊Q812yes—
321C22×C4C2×SD1612yes—
322C22×C4C2×SD1612yes—
323C22×C4C2×SD1612yes—
324C22×C4C2×SD1612yes—
325C22×C4#13512yes—
326C22×C4#13512yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×Q811yescharacteristic
329C22×C4C22×Q811yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4112yes—
333C22×C4C2×#4112yes—
334C22×C4C2×#4111yescharacteristic
335C22×C4H(5,𝔽2)12yes—
336C22×C4H(5,𝔽2)12yes—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—416no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8C22⋊Q814yes—
357C2×C8C22⋊Q814yes—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—48no—
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—88no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C22⋊C4—28no—
379C22⋊C4—28no—
380C22⋊C4—88no—
381C22⋊C4—88no—
382C42—22no—
383C42—22no—
384C42—24no—
385C42—41no—
386C42—41no—
387C42—42no—
388C23—22no—
389C23—22no—
390C23—24no—
391C23—22no—
392C23—22no—
393C23—24no—
394C23—24no—
395C23—24no—
396C23—24no—
397C23—22no—
398C23—22no—
399C23—24no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—41no—
404C23—42no—
405C23—42no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23#49814yes—
411C23C2×#12111yescharacteristic
412C23C2×C4⋊Q811yescharacteristic
413C23#53211yescharacteristic
414C23#54612yes—
415C23#54612yes—
416C23C22×SD1611yescharacteristic
417C23C22×SD1611yescharacteristic
418C23C2×#13511yescharacteristic
419C23C2×#13511yescharacteristic
420C23#57612yes—
421C23#57812yes—
422Q8—816no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
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—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4#46514yes—
481C2×C4#46514yes—
482C2×C4C2×C22⋊Q811yescharacteristic
483C2×C4C2×C22⋊Q811yescharacteristic
484C2×C4#52511yescharacteristic
485C2×C4#52912yes—
486C2×C4#53211yescharacteristic
487C2×C4#53311yescharacteristic
488C2×C4#53311yescharacteristic
489C2×C4#53312yes—
490C2×C4#53411yescharacteristic
491C2×C4#53511yescharacteristic
492C2×C4#53812yes—
493C2×C4#54312yes—
494C2×C4#55112yes—
495C2×C4#55112yes—
496C2×C4#55412yes—
497C2×C4#55412yes—
498C2×C4#55912yes—
499C2×C4#55912yes—
500C8—24no—
501C8—24no—
502C8—42no—
503C8—42no—
504C22—28no—
505C22—24no—
506C22—44no—
507C22—44no—
508C22—42no—
509C22—44no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22#253411yescharacteristic
516C22#254411yescharacteristic
517C22#254912yes—
518C22#255712yes—
519C22#285711yescharacteristic
520C22C2×#46511yescharacteristic
521C22C2×#46511yescharacteristic
522C22#305811yescharacteristic
523C22#305911yescharacteristic
524C22C2×#49811yescharacteristic
525C22#316011yescharacteristic
526C22#316412yes—
527C22#316711yescharacteristic
528C22#316711yescharacteristic
529C22#317512yes—
530C22#329512yes—
531C22#329512yes—
532C22#329712yes—
533C22#329812yes—
534C22#329812yes—
535C22#330012yes—
536C22#330212yes—
537C22#330512yes—
538C4—24no—
539C4—24no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—84no—
544C4—84no—
545C4—162no—
546C4—162no—
547C4#239611yescharacteristic
548C4#256411yescharacteristic
549C4#258011yescharacteristic
550C4#258912yes—
551C4#260712yes—
552C4#262611yescharacteristic
553C2—44no—
554C2#1516111yescharacteristic
555C2#2809711yescharacteristic
556C2#2809811yescharacteristic
557C2#2859911yescharacteristic
558C2#2860011yescharacteristic
559C2#2877112yes—
560C2#2877312yes—
561C2#2897912yes—
562C2#2898112yes—
563C2#3320711yescharacteristic
564C2#3321511yescharacteristic
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. #11357C2semidirect
7. #11220C2semidirect
9. #10880C2semidirect
10. #10861C2semidirect
51. C23×C8Q8semidirect
79. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19902
C2×C8×C22⋊C42
#151352
#151252
#113842
#113572
#112202
#110952
#108802
#108612
#107212