Finite Group Databasebeta

← #431008

Subgroups — #431008

Order 512.

1614 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#2058C212yesmaximal, complemented
4C2×#1853C212yesmaximal
5C2×#1844C211yesmaximal, characteristic, complemented
6#15135C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic, complemented
8#11327C212yesmaximal, complemented
9#11095C212yesmaximal
10#10921C211yesmaximal, characteristic
11#10880C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C23×C4⋊C4C2211yescharacteristic
15C22×#106C2212yes—
16C22×#106C2212yes—
17C22×#106C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C4×C8C2211yescharacteristic
20#2440—24no—
21#2440—24no—
22C2×#399—21no—
23C2×#398—21no—
24C2×#395—21no—
25C2×#389C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2211yescharacteristic
28C2×#385—21no—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#2084—24no—
35#2058—28no—
36#1853—28no—
37#1844—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2211yescharacteristic
45#1445—21no—
46#1444—21no—
47#1439—21no—
48C4⋊#95C2211yescharacteristic
49C23×Q8—21no—
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—42no—
58C2×C22⋊Q8—28no—
59C2×C22⋊Q8—44no—
60C2×C22⋊Q8—44no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—42no—
94C2×#106—42no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8Q818yescomplemented
98C2×C4×C8—24no—
99C2×C4×C8—22no—
100#399—44no—
101#398—28no—
102#395—28no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—24no—
107#389—42no—
108#389D414yes—
109#385—28no—
110#384—22no—
111#384—22no—
112#384—22no—
113#384D414yes—
114#380—22no—
115#380—22no—
116#380—24no—
117#380—42no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95D411yescharacteristic
128C2×#95D411yescharacteristic
129#340—24no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340D418yes—
134#336—24no—
135#336—22no—
136C25C2×D411yescharacteristic
137C22×Q8—24no—
138C22×Q8—44no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—82no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—44no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—82no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×Q814yes—
166C23×C4C2411yescharacteristic
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8C2×D414yes—
178C22×C8C2×Q814yes—
179C22×C8C2×Q814yes—
180C22×C8#4114yes—
181C4⋊Q8—48no—
182C4⋊Q8—44no—
183C4⋊Q8—44no—
184C22⋊Q8—432no—
185C2×C4⋊C4—21no—
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—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D412yes—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—82no—
249C2×C22⋊C4SD1614yes—
250C2×C22⋊C4SD1614yes—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259#106—48no—
260#106—48no—
261#106—48no—
262C22⋊C8—216no—
263C4×C8—28no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—22no—
268#95—22no—
269#95—44no—
270#95—44no—
271#95—42no—
272#95—42no—
273C24—21no—
274C24—21no—
275C24—22no—
276C24—22no—
277C24—21no—
278C24—21no—
279C24—22no—
280C24—22no—
281C24C4⋊Q812yes—
282C24C2×SD1612yes—
283C24C2×SD1612yes—
284C24C22×D411yescharacteristic
285C2×Q8—48no—
286C2×Q8—48no—
287C2×Q8—88no—
288C2×Q8—84no—
289C2×Q8—88no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—48no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—48no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—84no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4#12012yes—
362C22×C4#12012yes—
363C22×C4C4⋊Q812yes—
364C22×C4C2×SD1612yes—
365C22×C4C2×SD1612yes—
366C22×C4C2×SD1612yes—
367C22×C4C2×SD1612yes—
368C22×C4AGL(1,ℤ/8)12yes—
369C22×C4#13712yes—
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×Q812yes—
374C22×C4C2×#4112yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C42−1+412yes—
377C22×C42−1+412yes—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8C22⋊Q818yescomplemented
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—48no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—84no—
417C4⋊C4—88no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4#12714yes—
421C22⋊C4—28no—
422C22⋊C4—28no—
423C22⋊C4—48no—
424C22⋊C4—88no—
425C42—22no—
426C42—22no—
427C42—24no—
428C42—41no—
429C42—41no—
430C42—42no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—24no—
442C23—24no—
443C23—42no—
444C23—42no—
445C23—41no—
446C23—42no—
447C23—41no—
448C23—42no—
449C23—41no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23#49814yes—
454C23C2×#12011yescharacteristic
455C23C2×C4⋊Q811yescharacteristic
456C23#53611yescharacteristic
457C23D4212yes—
458C23#54712yes—
459C23C22×SD1611yescharacteristic
460C23C22×SD1611yescharacteristic
461C23C2×AGL(1,ℤ/8)11yescharacteristic
462C23C2×#13711yescharacteristic
463C23#57712yes—
464C23#57712yes—
465Q8—816no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—816no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4#44814yes—
529C2×C4#45014yes—
530C2×C4C2×C22⋊Q812yes—
531C2×C4#52511yescharacteristic
532C2×C4C2×#12711yescharacteristic
533C2×C4#53511yescharacteristic
534C2×C4#53611yescharacteristic
535C2×C4#53612yes—
536C2×C4#53912yes—
537C2×C4#54412yes—
538C2×C4D4212yes—
539C2×C4#54712yes—
540C2×C4#54712yes—
541C2×C4D4×Q812yes—
542C2×C4D4×Q812yes—
543C2×C4D4×Q812yes—
544C2×C4#55412yes—
545C2×C4#56312yes—
546C8—28no—
547C8—44no—
548C22—28no—
549C22—24no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—44no—
559C22#240611yescharacteristic
560C22#243311yescharacteristic
561C22#243411yescharacteristic
562C22#244012yes—
563C22#268012yes—
564C22C2×#45011yescharacteristic
565C22C2×#44811yescharacteristic
566C22#300911yescharacteristic
567C22#301011yescharacteristic
568C22C2×#49811yescharacteristic
569C22#316312yes—
570C22#316511yescharacteristic
571C22#322711yescharacteristic
572C22#322911yescharacteristic
573C22#324412yes—
574C22#327212yes—
575C22#327812yes—
576C22D4×SD1612yes—
577C22D4×SD1612yes—
578C22#333512yes—
579C22#333512yes—
580C22#334412yes—
581C22#334512yes—
582C4—24no—
583C4—24no—
584C4—42no—
585C4—42no—
586C4—44no—
587C4—42no—
588C4—82no—
589C4—84no—
590C4—82no—
591C4—82no—
592C4—162no—
593C4#244912yes—
594C4#254212yes—
595C4#267712yes—
596C4#272012yes—
597C2—44no—
598C2#1543011yescharacteristic
599C2#2408911yescharacteristic
600C2#2409711yescharacteristic
601C2#2575311yescharacteristic
602C2#2575611yescharacteristic
603C2#2581111yescharacteristic
604C2#2582011yescharacteristic
605C2#2606012yes—
606C2#2607812yes—
607C2#3108612yes—
608C2#3108912yes—
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
2. C2×#2084C2semidirect
3. C2×#2058C2semidirect
5. C2×#1844C2semidirect
7. #11357C2semidirect
8. #11327C2semidirect
11. #10880C2semidirect
97. C2×C22⋊C8Q8semidirect
391. C2×C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
C2×#20842
C2×#20582
C2×#18532
C2×#18442
#151352
#113572
#113272
#110952
#109212
#108802