Finite Group Databasebeta

← #406390

Subgroups — #406390

Order 512.

1618 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#487C211yesmaximal, characteristic, complemented
2C2×#2043C211yesmaximal, characteristic, complemented
3C2×#2023C211yesmaximal, characteristic
4C2×#1959C212yesmaximal
5C2×#1934C211yesmaximal, characteristic
6#15243C211yesmaximal, characteristic
7#15143C211yesmaximal, characteristic, complemented
8#11050C212yesmaximal, complemented
9#11009C212yesmaximal
10#10928C212yesmaximal, complemented
11#10696C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#124C2212yes—
15C2×#487—28no—
16C2×#487—24no—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#102C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2212yes—
24C22×#97C2212yes—
25C22×C4×C8C2211yescharacteristic
26C2×#395—22no—
27C2×#391C2212yes—
28C2×#390C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#384C2212yes—
31#2043—24no—
32#2023—24no—
33#1959—28no—
34#1934—24no—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40#1444—22no—
41#1439—22no—
42#1439—22no—
43C4⋊#95C2212yes—
44C4×#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8C2312yes—
47C23×D4C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8—42no—
59C2×#124—24no—
60C2×#124—24no—
61C2×#124—24no—
62C2×#124—24no—
63C2×#124—42no—
64C2×#124—42no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C22⋊C4—24no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2312yes—
79#487—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—24no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103D418yes—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—42no—
100C2×#97—24no—
101C2×#97—24no—
102C2×#97—42no—
103C2×C4×C8—24no—
104C2×C4×C8D414yes—
105#395—48no—
106#391—24no—
107#391—24no—
108#390—22no—
109#390—22no—
110#389—22no—
111#389—22no—
112#384—24no—
113#384—42no—
114C2×#95—24no—
115C2×#95—24no—
116C2×#95—21no—
117C2×#95D412yes—
118C2×#95D412yes—
119#340—22no—
120#340—22no—
121#340—24no—
122#336—24no—
123#336D414yes—
124C25—21no—
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×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—44no—
139C22×Q8—42no—
140C22×Q8—44no—
141C22×Q8—82no—
142C22×Q8C2×D414yes—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—21no—
149C22×D4—22no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4#4112yes—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4C2×D414yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8C2×D414yes—
176C22×C8C2×D412yes—
177C22×C8C2×D412yes—
178C22×C8C2×D414yes—
179C4⋊Q8—28no—
180C4⋊Q8—28no—
181C4⋊Q8—48no—
182C4⋊Q8—48no—
183C4⋊Q8—44no—
184C4⋊Q8—44no—
185#124—216no—
186#124—48no—
187#124—48no—
188#124—48no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—84no—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4#4112yes—
220C2×C22⋊C4—48no—
221C2×C22⋊C4—48no—
222C2×C22⋊C4—48no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42#4114yes—
236C2×C42#4114yes—
237#103—216no—
238#103—216no—
239#103—48no—
240#103—48no—
241#102—28no—
242#102—48no—
243#102—44no—
244#97—48no—
245C4×C8—28no—
246#95—24no—
247#95—24no—
248#95—48no—
249#95—48no—
250#95—42no—
251C24—22no—
252C24—42no—
253C24—41no—
254C24—42no—
255C24—41no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24C22×D411yescharacteristic
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—48no—
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—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—816no—
291C2×Q8—84no—
292C2×Q8—84no—
293C2×D4—22no—
294C2×D4—24no—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—41no—
309C2×D4—42no—
310C2×D4—42no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4#12418yes—
347C22×C4#12712yes—
348C22×C4#12712yes—
349C22×C4AGL(1,ℤ/8)14yes—
350C22×C4#13714yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D412yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4112yes—
358C22×C4H(5,𝔽2)12yes—
359C22×C4H(5,𝔽2)11yescharacteristic
360C22×C42−1+411yescharacteristic
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8#12014yes—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—48no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—42no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—816no—
396C4⋊C4—84no—
397C22⋊C4—432no—
398C22⋊C4—84no—
399C22⋊C4—84no—
400C42—22no—
401C42—24no—
402C42—22no—
403C42—24no—
404C42—24no—
405C42—24no—
406C42—24no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—42no—
411C23—82no—
412C23—82no—
413C23—81no—
414C23—84no—
415C23—84no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—82no—
422C23—84no—
423C23—84no—
424C23C2×#12412yes—
425C23C2×#12711yescharacteristic
426C23D4212yes—
427C23#54611yescharacteristic
428C23#54811yescharacteristic
429C23C2×AGL(1,ℤ/8)12yes—
430C23C2×#13712yes—
431C23#57712yes—
432C23#57812yes—
433Q8—44no—
434Q8—44no—
435Q8—44no—
436Q8—44no—
437Q8—816no—
438Q8—82no—
439Q8—82no—
440Q8—82no—
441Q8—82no—
442D4—42no—
443D4—42no—
444D4—44no—
445D4—81no—
446D4—81no—
447D4—82no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—28no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—44no—
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—48no—
467C2×C4—42no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—84no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—84no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—84no—
486C2×C4#49614yes—
487C2×C4#49714yes—
488C2×C4C2×#12011yescharacteristic
489C2×C4#52511yescharacteristic
490C2×C4#52512yes—
491C2×C4#53211yescharacteristic
492C2×C4#53211yescharacteristic
493C2×C4#53212yes—
494C2×C4#53212yes—
495C2×C4#53511yescharacteristic
496C2×C4#53711yescharacteristic
497C2×C4#53812yes—
498C2×C4#54212yes—
499C2×C4D4212yes—
500C2×C4#54612yes—
501C2×C4#54611yescharacteristic
502C2×C4#54712yes—
503C2×C4#54811yescharacteristic
504C2×C4#55112yes—
505C2×C4#55512yes—
506C8—24no—
507C8—42no—
508C8—44no—
509C22—82no—
510C22—82no—
511C22—88no—
512C22—82no—
513C22—164no—
514C22—161no—
515C22—161no—
516C22—161no—
517C22—161no—
518C22#243311yescharacteristic
519C22#247212yes—
520C22#249112yes—
521C22#254011yescharacteristic
522C22#285811yescharacteristic
523C22#313714yes—
524C22#314114yes—
525C22C2×#49611yescharacteristic
526C22C2×#49711yescharacteristic
527C22#315411yescharacteristic
528C22#315611yescharacteristic
529C22#326514yes—
530C22#327014yes—
531C22#333012yes—
532C22#333112yes—
533C22#336712yes—
534C22#336912yes—
535C4—24no—
536C4—24no—
537C4—48no—
538C4—84no—
539C4—84no—
540C4—82no—
541C4—161no—
542C4—161no—
543C4#238811yescharacteristic
544C4#243011yescharacteristic
545C4#246112yes—
546C4#255512yes—
547C4#261011yescharacteristic
548C4#266211yescharacteristic
549C2—162no—
550C2#1492811yescharacteristic
551C2#2578311yescharacteristic
552C2#2578911yescharacteristic
553C2#2727414yes—
554C2#2778514yes—
555C2#2841411yescharacteristic
556C2#2841911yescharacteristic
557C2#3322911yescharacteristic
558C2#3323811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#487C2semidirect
2. C2×#2043C2semidirect
7. #15143C2semidirect
8. #11050C2semidirect
10. #10928C2semidirect
11. #10696C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4872
C2×#20432
C2×#20232
C2×#19592
C2×#19342
#152432
#151432
#110502
#110092
#109282
#106962