Finite Group Databasebeta

← #401689

Subgroups — #401689

Order 512.

1824 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#389C211yesmaximal, characteristic
2C2×#2100C212yesmaximal, complemented
3#14686C211yesmaximal, characteristic
4#11293C214yesmaximal, complemented
5#11207C212yesmaximal, complemented
6#10730C212yesmaximal
7#10730C212yesmaximal
8#10712C211yesmaximal, characteristic
9C23×C4⋊C4—21no—
10C23×C4⋊C4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C2×#389—21no—
15C2×#389—21no—
16C2×#389—28no—
17C2×#389C2212yes—
18C2×#389C2212yes—
19C2×#384C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#380—21no—
22C2×#380—21no—
23C2×#380C2212yes—
24C22×#95—21no—
25#2100—28no—
26C2×#340C2214yes—
27C2×#340C2214yes—
28C2×#340C2214yes—
29C2×#340C2214yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C4⋊#95—21no—
33C4⋊#95—21no—
34C4⋊#95C2212yes—
35C24×C4—21no—
36C24×C4—21no—
37C24×C4D411yescharacteristic
38C24×C4D411yescharacteristic
39C24×C4C2311yescharacteristic
40C23×C8C2314yes—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—24no—
44C22×C4⋊C4—24no—
45C22×C4⋊C4—28no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—42no—
56C22×C4⋊C4—44no—
57C22×C4⋊C4—44no—
58C22×C4⋊C4Q814yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—22no—
64C22×C42—24no—
65C22×C42—24no—
66C22×C42—42no—
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C2×C22⋊C8—24no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74#389—28no—
75#389—28no—
76#389—24no—
77#389—24no—
78#389—24no—
79#389D418yes—
80#384—24no—
81#384—42no—
82#380—28no—
83#380—44no—
84#380—44no—
85C2×#95—28no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91#340—28no—
92#340—28no—
93#340—28no—
94#340—44no—
95#336—24no—
96#336—42no—
97C25C2×D411yescharacteristic
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4—24no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—24no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—41no—
119C23×C4—41no—
120C23×C4—41no—
121C23×C4—41no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4SD1614yes—
133C23×C4Q1614yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×Q812yes—
140C23×C4#4112yes—
141C23×C4#4114yes—
142C23×C4#4114yes—
143C23×C4C2411yescharacteristic
144C22×C8—24no—
145C22×C8—24no—
146C22×C8—24no—
147C22×C8—24no—
148C22×C8—24no—
149C22×C8—24no—
150C22×C8—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—216no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—216no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—416no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—432no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4C2×D414yes—
192C2×C4⋊C4C2×D414yes—
193C2×C4⋊C4C2×Q814yes—
194C2×C4⋊C4#4114yes—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—416no—
208C2×C42—41no—
209C2×C42—41no—
210C2×C42C2×D414yes—
211C22⋊C8—48no—
212#95—28no—
213#95—28no—
214#95—44no—
215#95—44no—
216#95—44no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—21no—
225C24C22⋊Q812yes—
226C24#12312yes—
227C24#12412yes—
228C24C2×SD1612yes—
229C24C2×Q1612yes—
230C24AGL(1,ℤ/8)12yes—
231C24AGL(1,ℤ/8)12yes—
232C24C22×D411yescharacteristic
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—28no—
252C22×C4—28no—
253C22×C4—216no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—28no—
270C22×C4—28no—
271C22×C4—28no—
272C22×C4—28no—
273C22×C4—24no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—48no—
287C22×C4—48no—
288C22×C4—48no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—48no—
294C22×C4—48no—
295C22×C4—48no—
296C22×C4—48no—
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—84no—
318C22×C4—84no—
319C22×C4—84no—
320C22×C4—84no—
321C22×C4#12012yes—
322C22×C4#12012yes—
323C22×C4C22⋊Q812yes—
324C22×C4#12312yes—
325C22×C4#12412yes—
326C22×C4#12518yes—
327C22×C4#12618yes—
328C22×C4C2×SD1612yes—
329C22×C4C2×SD1612yes—
330C22×C4C2×Q1612yes—
331C22×C4C2×Q1612yes—
332C22×C4AGL(1,ℤ/8)12yes—
333C22×C4#13712yes—
334C22×C4C22×D412yes—
335C22×C4C22×Q811yescharacteristic
336C22×C4C2×#4112yes—
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4112yes—
339C22×C4C2×#4112yes—
340C22×C4H(5,𝔽2)12yes—
341C22×C42−1+412yes—
342C2×C8—44no—
343C2×C8—44no—
344C2×C8—44no—
345C2×C8—44no—
346C2×C8—44no—
347C2×C8—44no—
348C2×C8—44no—
349C4⋊C4—216no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—28no—
354C4⋊C4—48no—
355C4⋊C4—416no—
356C4⋊C4—44no—
357C4⋊C4—416no—
358C4⋊C4—48no—
359C4⋊C4—48no—
360C4⋊C4—84no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C4⋊C4—84no—
364C4⋊C4C22⋊Q818yes—
365C22⋊C4—88no—
366C42—28no—
367C42—44no—
368C42—44no—
369C42—44no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23#40014yes—
399C23#47614yes—
400C23#47714yes—
401C23#48114yes—
402C23#48414yes—
403C23#48714yes—
404C23#49014yes—
405C23C2×#12011yescharacteristic
406C23C2×C22⋊Q811yescharacteristic
407C23C2×#12311yescharacteristic
408C23#52511yescharacteristic
409C23C2×#12411yescharacteristic
410C23#53411yescharacteristic
411C23#53611yescharacteristic
412C23C22×SD1611yescharacteristic
413C23C22×Q1611yescharacteristic
414C23C2×AGL(1,ℤ/8)11yescharacteristic
415C23C2×AGL(1,ℤ/8)11yescharacteristic
416C23C2×AGL(1,ℤ/8)11yescharacteristic
417C23C2×#13711yescharacteristic
418C23#57511yescharacteristic
419C23#57511yescharacteristic
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—216no—
439C2×C4—216no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—28no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—416no—
448C2×C4—416no—
449C2×C4—432no—
450C2×C4—44no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—44no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—816no—
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—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4#40014yes—
487C2×C4#45014yes—
488C2×C4#45114yes—
489C2×C4C2×C22⋊Q812yes—
490C2×C4#52512yes—
491C2×C4C2×#12512yes—
492C2×C4C2×#12612yes—
493C2×C4#53912yes—
494C2×C4#54412yes—
495C2×C4#54614yes—
496C2×C4#54814yes—
497C2×C4#55214yes—
498C2×C4#55314yes—
499C8—84no—
500C22—24no—
501C22—24no—
502C22—24no—
503C22—24no—
504C22—24no—
505C22—24no—
506C22—24no—
507C22—24no—
508C22—24no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—24no—
514C22#210018yescomplemented
515C22C2×#40011yescharacteristic
516C22#240511yescharacteristic
517C22#243611yescharacteristic
518C22#248411yescharacteristic
519C22#259111yescharacteristic
520C22#262711yescharacteristic
521C22#284811yescharacteristic
522C22C2×#45011yescharacteristic
523C22C2×#45111yescharacteristic
524C22#300611yescharacteristic
525C22#300611yescharacteristic
526C22C2×#47611yescharacteristic
527C22C2×#47711yescharacteristic
528C22#307711yescharacteristic
529C22#307811yescharacteristic
530C22C2×#48111yescharacteristic
531C22C2×#48411yescharacteristic
532C22#309411yescharacteristic
533C22#309511yescharacteristic
534C22#310111yescharacteristic
535C22#310411yescharacteristic
536C22#310911yescharacteristic
537C22#311211yescharacteristic
538C22C2×#48711yescharacteristic
539C22C2×#49011yescharacteristic
540C22#314111yescharacteristic
541C22#314211yescharacteristic
542C22#318711yescharacteristic
543C22#318911yescharacteristic
544C22#319211yescharacteristic
545C22#319311yescharacteristic
546C22#322911yescharacteristic
547C22#323011yescharacteristic
548C22#323411yescharacteristic
549C22#323411yescharacteristic
550C4—24no—
551C4—24no—
552C4—24no—
553C4—48no—
554C4—42no—
555C4—42no—
556C4—84no—
557C4—162no—
558C4—162no—
559C4#244214yes—
560C4#248114yes—
561C2—28no—
562C2#1482211yescharacteristic
563C2C2×#210011yescharacteristic
564C2#2374711yescharacteristic
565C2#2400511yescharacteristic
566C2#2400611yescharacteristic
567C2#2590211yescharacteristic
568C2#2591311yescharacteristic
569C2#2761311yescharacteristic
570C2#2761411yescharacteristic
571C2#2932911yescharacteristic
572C2#2933411yescharacteristic
573C2#2996811yescharacteristic
574C2#2996911yescharacteristic
575C2#3295611yescharacteristic
576C2#3295711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2100C2semidirect
4. #11293C2semidirect
5. #11207C2semidirect
514. C22#2100semidirect

Maximal subgroups

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

subgroupindex
C22×#3892
C2×#21002
#146862
#112932
#112072
#107302
#107122