Finite Group Databasebeta

← #443486

Subgroups — #443486

Order 512.

1602 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#487C211yesmaximal, characteristic
2C2×#2551C211yesmaximal, characteristic
3C2×#2043C211yesmaximal, characteristic
4C2×#2024C212yesmaximal
5#15180C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic
7#11067C212yesmaximal
8#11050C212yesmaximal
9#10932C212yesmaximal
10#10880C211yesmaximal, characteristic
11#10874C211yesmaximal, characteristic
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C2×#487—28no—
17C2×#487—24no—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×C4×C8C2211yescharacteristic
23#2551—24no—
24#2551—24no—
25C2×#395—21no—
26C2×#395—21no—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#389—21no—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#2043—24no—
36#2024—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44#1442—21no—
45#1442—21no—
46#1441—21no—
47#1441—21no—
48C4⋊#95C2211yescharacteristic
49C23×Q8—21no—
50C23×Q8—21no—
51C23×Q8C2312yes—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—28no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—42no—
64C2×C4⋊Q8—42no—
65C2×C4⋊Q8—42no—
66C2×C4⋊Q8—44no—
67C2×C4⋊Q8—44no—
68C2×C4⋊Q8D414yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#487—232no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—42no—
100C2×#103D418yes—
101C2×C4×C8—24no—
102C2×C4×C8—22no—
103#395—28no—
104#395—28no—
105#391—24no—
106#391—24no—
107#391—24no—
108#391—42no—
109#389—28no—
110#389—22no—
111#389—22no—
112#389—22no—
113#389D414yes—
114#384—22no—
115#384—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95D411yescharacteristic
130C2×#95D411yescharacteristic
131#340—24no—
132#340—24no—
133#340—24no—
134#340—42no—
135#336—24no—
136#336—22no—
137#336—24no—
138C22×Q8—24no—
139C22×Q8—24no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—44no—
154C22×Q8—44no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8—82no—
160C22×Q8—82no—
161C22×Q8C2×D414yes—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2411yescharacteristic
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8C2×D414yes—
189C4⋊Q8—28no—
190C4⋊Q8—28no—
191C4⋊Q8—44no—
192C4⋊Q8—48no—
193C4⋊Q8—48no—
194C4⋊Q8—44no—
195C4⋊Q8—48no—
196C4⋊Q8—432no—
197C4⋊Q8—44no—
198C4⋊Q8—44no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
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—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—82no—
238C2×C4⋊C4SD1614yes—
239C2×C4⋊C4Q1614yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4#4114yes—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—22no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42#4114yes—
261#103—216no—
262#103—216no—
263#103—48no—
264#103—48no—
265C4×C8—28no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—22no—
270#95—22no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—42no—
276#95—42no—
277C24C22×D411yescharacteristic
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—48no—
285C2×Q8—48no—
286C2×Q8—48no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—48no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—84no—
311C2×Q8—88no—
312C2×Q8—88no—
313C2×Q8—88no—
314C2×Q8—84no—
315C2×Q8—88no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
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—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4#12012yes—
371C22×C4#12012yes—
372C22×C4#12412yes—
373C22×C4#12412yes—
374C22×C4C2×SD1612yes—
375C22×C4C2×SD1612yes—
376C22×C4C2×SD1612yes—
377C22×C4C2×Q1612yes—
378C22×C4C2×Q1612yes—
379C22×C4C2×Q1612yes—
380C22×C4AGL(1,ℤ/8)12yes—
381C22×C4#13712yes—
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C42−1+412yes—
389C22×C42−1+412yes—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—28no—
409C4⋊C4—28no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—48no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—84no—
428C4⋊C4#12014yes—
429C4⋊C4#12414yes—
430C42—22no—
431C42—22no—
432C42—24no—
433C42—24no—
434C42—24no—
435C42—48no—
436C42—41no—
437C42—41no—
438C42—42no—
439C23C2×#12011yescharacteristic
440C23C2×#12411yescharacteristic
441C23#53611yescharacteristic
442C23D4212yes—
443C23#54812yes—
444C23C22×SD1611yescharacteristic
445C23C22×Q1611yescharacteristic
446C23C2×AGL(1,ℤ/8)11yescharacteristic
447C23C2×#13711yescharacteristic
448C23#57712yes—
449C23#57812yes—
450Q8—44no—
451Q8—44no—
452Q8—44no—
453Q8—44no—
454Q8—816no—
455Q8—816no—
456Q8—82no—
457Q8—82no—
458Q8—82no—
459Q8—82no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
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#45014yes—
524C2×C4#45114yes—
525C2×C4#48714yescomplemented
526C2×C4C2×#12011yescharacteristic
527C2×C4C2×#12411yescharacteristic
528C2×C4#53212yes—
529C2×C4#53411yescharacteristic
530C2×C4#53611yescharacteristic
531C2×C4#53612yes—
532C2×C4#54212yes—
533C2×C4#54212yes—
534C2×C4D4212yes—
535C2×C4#54712yes—
536C2×C4#54712yes—
537C2×C4#54812yes—
538C2×C4#54812yes—
539C2×C4#54812yes—
540C2×C4#55512yes—
541C2×C4#56312yes—
542C8—44no—
543C8—82no—
544C22#240311yescharacteristic
545C22#243611yescharacteristic
546C22#243611yescharacteristic
547C22#249112yes—
548C22#267412yes—
549C22C2×#45011yescharacteristic
550C22C2×#45111yescharacteristic
551C22#300911yescharacteristic
552C22#301111yescharacteristic
553C22C2×#48711yescharacteristic
554C22#313712yes—
555C22#314111yescharacteristic
556C22#322911yescharacteristic
557C22#323011yescharacteristic
558C22#324212yes—
559C22#326512yes—
560C22#327012yes—
561C22D4×SD1612yes—
562C22D4×Q1612yes—
563C22#335912yes—
564C22#336312yes—
565C22#336712yes—
566C22#336912yes—
567C4—24no—
568C4—24no—
569C4—24no—
570C4—42no—
571C4—44no—
572C4—42no—
573C4—42no—
574C4—44no—
575C4—82no—
576C4—82no—
577C4—84no—
578C4—162no—
579C4#243212yes—
580C4#249112yes—
581C4#267412yes—
582C4#273112yes—
583C2#1576111yescharacteristic
584C2#2384911yescharacteristic
585C2#2385611yescharacteristic
586C2#2590211yescharacteristic
587C2#2591411yescharacteristic
588C2#2592911yescharacteristic
589C2#2593611yescharacteristic
590C2#2776912yes—
591C2#2778512yes—
592C2#3080812yes—
593C2#3081612yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
525. C2×C4#487semidirect

Maximal subgroups

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

subgroupindex
C22×#4872
C2×#25512
C2×#20432
C2×#20242
#151802
#113842
#110672
#110502
#109322
#108802
#108742