Finite Group Databasebeta

← #406709

Subgroups — #406709

Order 512.

1840 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#1823C212yesmaximal
3C2×C4⋊#95C211yesmaximal, characteristic
4#11384C212yesmaximal, complemented
5#11292C214yesmaximal, complemented
6#11267C212yesmaximal, complemented
7#10730C212yesmaximal
8#10712C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×C22⋊Q8—21no—
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15#2857—28no—
16C2×#391—21no—
17C2×#389—21no—
18C2×#389C2212yes—
19C2×#384C2211yescharacteristic
20C2×#380—21no—
21C2×#380C2212yes—
22C22×#95—21no—
23C22×#95—21no—
24#1823—28no—
25C2×#340C2214yes—
26C2×#340C2214yes—
27C2×#340C2214yes—
28C2×#340C2214yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C4⋊#95—21no—
32C4⋊#95—21no—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C23×Q8—21no—
36C24×C4—21no—
37C24×C4—21no—
38C24×C4D411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4C2311yescharacteristic
41C23×C8C2314yes—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—42no—
45C2×C22⋊Q8—28no—
46C2×C22⋊Q8—44no—
47C2×C22⋊Q8—44no—
48C22×C4⋊C4—24no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—42no—
60C22×C4⋊C4Q814yescomplemented
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—22no—
66C22×C42—24no—
67C22×C42—24no—
68C22×C42—42no—
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76#391—28no—
77#389—28no—
78#389—24no—
79#389—24no—
80#384—24no—
81#380—28no—
82#380—24no—
83#380D418yes—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—44no—
91C2×#95—44no—
92#340—28no—
93#340—28no—
94#340—28no—
95#340—44no—
96#336—24no—
97#336—42no—
98C25C2×D411yescharacteristic
99C22×Q8—24no—
100C22×Q8—42no—
101C22×Q8—42no—
102C22×Q8—42no—
103C22×Q8—42no—
104C22×Q8—42no—
105C22×Q8—42no—
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—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—41no—
125C23×C4—41no—
126C23×C4—41no—
127C23×C4—41no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4D814yes—
141C23×C4SD1614yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D414yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×Q812yes—
149C23×C4#4114yes—
150C23×C4#4112yes—
151C23×C4C2411yescharacteristic
152C22×C8—24no—
153C22×C8—24no—
154C22×C8—24no—
155C22×C8—24no—
156C22×C8—24no—
157C22×C8—24no—
158C22×C8—24no—
159C4⋊Q8—416no—
160C22⋊Q8—432no—
161C2×C4⋊C4—216no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
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—24no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—416no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—48no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4C2×D414yes—
196C2×C4⋊C4C2×Q814yes—
197C2×C4⋊C4#4114yes—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—416no—
211C2×C42—41no—
212C2×C42—41no—
213C2×C42#4114yes—
214C22⋊C8—48no—
215#95—28no—
216#95—28no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—44no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—21no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24#12012yes—
230C24C22⋊Q812yes—
231C24#12412yes—
232C24C2×D812yes—
233C24C2×SD1612yes—
234C24AGL(1,ℤ/8)12yes—
235C24#13712yes—
236C24C22×D411yescharacteristic
237C2×Q8—48no—
238C2×Q8—48no—
239C2×Q8—48no—
240C2×Q8—84no—
241C2×Q8—84no—
242C2×Q8—84no—
243C2×Q8—84no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—28no—
261C22×C4—28no—
262C22×C4—28no—
263C22×C4—28no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—28no—
279C22×C4—28no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
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—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—84no—
325C22×C4—84no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4—84no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4#12012yes—
333C22×C4C22⋊Q818yes—
334C22×C4C22⋊Q812yes—
335C22×C4#12312yes—
336C22×C4#12312yes—
337C22×C4#12418yes—
338C22×C4#12412yes—
339C22×C4C2×D812yes—
340C22×C4C2×D812yes—
341C22×C4C2×SD1612yes—
342C22×C4C2×SD1612yes—
343C22×C4#13712yes—
344C22×C4#13712yes—
345C22×C4C22×D412yes—
346C22×C4C22×Q811yescharacteristic
347C22×C4C2×#4112yes—
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4112yes—
350C22×C4C2×#4112yes—
351C22×C4H(5,𝔽2)12yes—
352C22×C42−1+412yes—
353C2×C8—44no—
354C2×C8—44no—
355C2×C8—44no—
356C2×C8—44no—
357C2×C8—44no—
358C2×C8—44no—
359C2×C8—44no—
360C4⋊C4—28no—
361C4⋊C4—28no—
362C4⋊C4—28no—
363C4⋊C4—28no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C4⋊C4—416no—
368C4⋊C4—48no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—48no—
374C22⋊C4—416no—
375C42—28no—
376C42—44no—
377C42—44no—
378C42—44no—
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—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23#39314yes—
408C23#44714yes—
409C23#44814yes—
410C23#47414yes—
411C23#47614yes—
412C23#48614yes—
413C23#48914yes—
414C23C2×#12011yescharacteristic
415C23C2×C22⋊Q811yescharacteristic
416C23C2×#12311yescharacteristic
417C23#52511yescharacteristic
418C23C2×#12411yescharacteristic
419C23#53411yescharacteristic
420C23#53611yescharacteristic
421C23C22×D811yescharacteristic
422C23C22×SD1611yescharacteristic
423C23C2×AGL(1,ℤ/8)11yescharacteristic
424C23C2×#13711yescharacteristic
425C23C2×#13711yescharacteristic
426C23C2×#13711yescharacteristic
427C23#57511yescharacteristic
428C23#57511yescharacteristic
429Q8—816no—
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—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—216no—
449C2×C4—216no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—416no—
457C2×C4—416no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—816no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—816no—
496C2×C4#39314yes—
497C2×C4#48014yes—
498C2×C4#48314yes—
499C2×C4C2×C22⋊Q812yes—
500C2×C4C2×#12412yes—
501C2×C4#52912yes—
502C2×C4#53312yes—
503C2×C4#53912yes—
504C2×C4#54412yes—
505C2×C4#54614yes—
506C2×C4#54714yes—
507C2×C4#55114yes—
508C2×C4#55214yes—
509C8—84no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—24no—
514C22—24no—
515C22—24no—
516C22—24no—
517C22—24no—
518C22—24no—
519C22—24no—
520C22—24no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22#202318yes—
525C22C2×#39311yescharacteristic
526C22#240511yescharacteristic
527C22#240611yescharacteristic
528C22#250311yescharacteristic
529C22#256111yescharacteristic
530C22#262711yescharacteristic
531C22#285511yescharacteristic
532C22C2×#44711yescharacteristic
533C22C2×#44811yescharacteristic
534C22#300711yescharacteristic
535C22#300711yescharacteristic
536C22C2×#47411yescharacteristic
537C22C2×#47611yescharacteristic
538C22#307811yescharacteristic
539C22#307811yescharacteristic
540C22C2×#48011yescharacteristic
541C22C2×#48311yescharacteristic
542C22#309411yescharacteristic
543C22#309511yescharacteristic
544C22#310011yescharacteristic
545C22#310111yescharacteristic
546C22#311211yescharacteristic
547C22#311311yescharacteristic
548C22C2×#48611yescharacteristic
549C22C2×#48911yescharacteristic
550C22#313911yescharacteristic
551C22#314011yescharacteristic
552C22#318611yescharacteristic
553C22#318711yescharacteristic
554C22#319311yescharacteristic
555C22#319511yescharacteristic
556C22#322611yescharacteristic
557C22#322911yescharacteristic
558C22#323411yescharacteristic
559C22#323611yescharacteristic
560C4—24no—
561C4—24no—
562C4—24no—
563C4—42no—
564C4—42no—
565C4—84no—
566C4—84no—
567C4—84no—
568C4—84no—
569C4#243814yes—
570C4#248814yes—
571C2—28no—
572C2#1493411yescharacteristic
573C2C2×#202311yescharacteristic
574C2#2345511yescharacteristic
575C2#2400211yescharacteristic
576C2#2400311yescharacteristic
577C2#2406011yescharacteristic
578C2#2406311yescharacteristic
579C2#2796011yescharacteristic
580C2#2796911yescharacteristic
581C2#2910311yescharacteristic
582C2#2911111yescharacteristic
583C2#2996611yescharacteristic
584C2#2996811yescharacteristic
585C2#3314611yescharacteristic
586C2#3314811yescharacteristic
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
4. #11384C2semidirect
5. #11292C2semidirect
6. #11267C2semidirect
60. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#18232
C2×C4⋊#952
#113842
#112922
#112672
#107302
#107122