Finite Group Databasebeta

← #405565

Subgroups — #405565

Order 512.

1548 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#491C211yesmaximal, characteristic, complemented
2C2×#1892C212yesmaximal
3C2×#1752C211yesmaximal, characteristic
4#15216C211yesmaximal, characteristic
5#15178C211yesmaximal, characteristic
6#11385C211yesmaximal, characteristic, complemented
7#11190C212yesmaximal, complemented
8#11010C211yesmaximal, characteristic, complemented
9#10970C212yesmaximal
10#10928C212yesmaximal, complemented
11#10821C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#125C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C2×#491—28no—
17C2×#491—24no—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C2×#391C2212yes—
26C2×#380C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1892—28no—
29#1752—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1444—22no—
37#1442—22no—
38#1439—22no—
39#1439—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C23×Q8C2312yes—
44C24×C4C2311yescharacteristic
45C23×C8—22no—
46C23×C8—22no—
47C23×C8—22no—
48C23×C8—24no—
49C23×C8—42no—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×#125—24no—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125—42no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—24no—
65C2×C22⋊Q8—24no—
66C2×C22⋊Q8—42no—
67C2×C22⋊Q8—42no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4D412yes—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#491—232no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103D418yes—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×#97—24no—
96C2×#97—22no—
97C2×#97—42no—
98C2×#97D414yes—
99#391—24no—
100#391—42no—
101#380—24no—
102#380—42no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D412yes—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—42no—
117#336—22no—
118#336—22no—
119C25C2×D411yescharacteristic
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8C2×D414yes—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4#4112yes—
150C23×C4#4112yes—
151C23×C4C2411yescharacteristic
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—28no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—48no—
169C22×C8—48no—
170C22×C8—44no—
171C22×C8—44no—
172C22×C8C2×D412yes—
173C22×C8C2×D414yes—
174C22×C8C2×D412yes—
175C22×C8#4112yes—
176C22×C8#4112yes—
177C4⋊Q8—28no—
178C4⋊Q8—48no—
179C4⋊Q8—44no—
180#125—28no—
181#125—48no—
182#125—44no—
183C22⋊Q8—216no—
184C22⋊Q8—48no—
185C22⋊Q8—48no—
186C22⋊Q8—48no—
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—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4#4114yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—82no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42C2×D412yes—
230C2×C42#4112yes—
231#103—216no—
232#103—216no—
233#103—48no—
234#103—48no—
235C22⋊C8—28no—
236#97—28no—
237#97—44no—
238#95—24no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—82no—
245#95—82no—
246C24—21no—
247C24—21no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24#12412yes—
254C24#13714yes—
255C24C22×D411yescharacteristic
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4#12114yes—
348C22×C4#12412yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D412yes—
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2×#4111yescharacteristic
357C22×C4H(5,𝔽2)12yes—
358C22×C4H(5,𝔽2)12yes—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—416no—
371C2×C8—41no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8#12014yes—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—416no—
385C4⋊C4—416no—
386C4⋊C4—416no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C22⋊C4—48no—
396C22⋊C4—416no—
397C22⋊C4—88no—
398C42—22no—
399C42—22no—
400C42—24no—
401C42—41no—
402C42—41no—
403C42—42no—
404C23—22no—
405C23—22no—
406C23—24no—
407C23—24no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—41no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23#49114yescomplemented
425C23C2×#12111yescharacteristic
426C23C2×#12411yescharacteristic
427C23#53211yescharacteristic
428C23D4212yes—
429C23#54612yes—
430C23C2×#13712yes—
431C23#57512yes—
432C23#57712yes—
433C23#57812yes—
434Q8—44no—
435Q8—44no—
436Q8—44no—
437Q8—44no—
438Q8—82no—
439Q8—82no—
440Q8—82no—
441Q8—82no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
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—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×C4C2×#12011yescharacteristic
496C2×C4#52511yescharacteristic
497C2×C4#52511yescharacteristic
498C2×C4#52512yes—
499C2×C4#52911yescharacteristic
500C2×C4#52911yescharacteristic
501C2×C4#53211yescharacteristic
502C2×C4#53212yes—
503C2×C4#53511yescharacteristic
504C2×C4#53511yescharacteristic
505C2×C4#53812yes—
506C2×C4#53812yes—
507C2×C4#54612yes—
508C2×C4#54612yes—
509C2×C4#54612yes—
510C2×C4#55112yes—
511C2×C4#55512yes—
512C2×C4#55912yes—
513C8—24no—
514C8—42no—
515C8—42no—
516C8—42no—
517C22—24no—
518C22—28no—
519C22—44no—
520C22—44no—
521C22—42no—
522C22—42no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22#240811yescharacteristic
527C22#247212yes—
528C22#254411yescharacteristic
529C22#255512yes—
530C22#285011yescharacteristic
531C22#305612yes—
532C22#306312yes—
533C22C2×#49111yescharacteristic
534C22#313511yescharacteristic
535C22#313712yes—
536C22#317411yescharacteristic
537C22#317711yescharacteristic
538C22#318112yes—
539C22#326612yes—
540C22#326912yes—
541C22#327012yes—
542C22#327112yes—
543C22#330712yes—
544C22#331212yes—
545C22#331512yes—
546C22#332012yes—
547C4—42no—
548C4—44no—
549C4—84no—
550C4—84no—
551C4—82no—
552C4—84no—
553C4—162no—
554C4#238811yescharacteristic
555C4#240411yescharacteristic
556C4#245912yes—
557C4#255712yes—
558C4#261011yescharacteristic
559C4#263111yescharacteristic
560C2—44no—
561C2#1491511yescharacteristic
562C2#2427412yes—
563C2#2726712yes—
564C2#2726812yes—
565C2#2861112yes—
566C2#2893412yes—
567C2#2893612yes—
568C2#3305311yescharacteristic
569C2#3305611yescharacteristic
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. C22×#491C2semidirect
6. #11385C2semidirect
7. #11190C2semidirect
8. #11010C2semidirect
10. #10928C2semidirect
11. #10821C2semidirect
424. C23#491semidirect

Maximal subgroups

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

subgroupindex
C22×#4912
C2×#18922
C2×#17522
#152162
#151782
#113852
#111902
#110102
#109702
#109282
#108212