Finite Group Databasebeta

← #408593

Subgroups — #408593

Order 512.

1496 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#470C211yesmaximal, characteristic
2C2×#1933C212yesmaximal, complemented
3C2×#1753C211yesmaximal, characteristic
4#15179C212yesmaximal
5#11327C211yesmaximal, characteristic, complemented
6#11257C211yesmaximal, characteristic, complemented
7#11024C211yesmaximal, characteristic, complemented
8#11024C211yesmaximal, characteristic, complemented
9#10970C212yesmaximal
10#10928C212yesmaximal, complemented
11#10722C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C2×#470—28no—
16C2×#470—24no—
17C22×#106C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×#380C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29#1933—28no—
30#1753—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1442—22no—
39#1441—22no—
40#1440—22no—
41#1439—22no—
42C4⋊#95C2212yes—
43C23×Q8C2312yes—
44C24×C4C2311yescharacteristic
45C22×Q16—24no—
46C22×Q16—22no—
47C22×Q16—22no—
48C22×Q16—22no—
49C22×Q16—42no—
50C22×M16—24no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—42no—
54C22×M16D414yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—42no—
64C2×C22⋊Q8—42no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4D412yes—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75#470—232no—
76C2×#106—22no—
77C2×#106—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103D418yes—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×#97—22no—
91C2×#97—42no—
92C2×C4×C8—24no—
93C2×C4×C8—22no—
94#391—24no—
95#391—24no—
96#391—42no—
97#391—42no—
98#389—24no—
99#389—42no—
100#380—24no—
101#380—42no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—22no—
104C2×#95—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—22no—
115#340—22no—
116#340—24no—
117#340—42no—
118#336—22no—
119#336—22no—
120C25C2×D411yescharacteristic
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×Q8—22no—
134C22×Q8C2×D414yes—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C2×Q16—28no—
154C2×Q16—48no—
155C2×Q16—48no—
156C2×Q16—44no—
157C2×Q16—44no—
158C2×M16—28no—
159C2×M16—28no—
160C2×M16—44no—
161C2×M16—48no—
162C2×M16—48no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8C2×D414yes—
177C22×C8C2×D412yes—
178C22×C8C2×D412yes—
179C22×C8#4112yes—
180C22×C8#4112yes—
181C22×C8#4112yes—
182C22×C8#4112yes—
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—82no—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4#4114yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—82no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228#106—28no—
229#103—216no—
230#103—216no—
231#103—48no—
232C22⋊C8—28no—
233#97—44no—
234C4×C8—44no—
235#95—24no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—82no—
244C24—21no—
245C24—21no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24#12112yes—
253C24C22×D411yescharacteristic
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—42no—
261C2×Q8—42no—
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—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
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—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
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—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#12112yes—
346C22×C4#13714yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D412yes—
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4112yes—
354C22×C4C2×#4111yescharacteristic
355C22×C4H(5,𝔽2)12yes—
356C22×C4H(5,𝔽2)12yes—
357Q16—416no—
358M16—416no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—42no—
377C2×C8#12014yes—
378C2×C8#12314yes—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—416no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—88no—
393C22⋊C4—48no—
394C22⋊C4—416no—
395C22⋊C4—88no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—42no—
401C42—42no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—24no—
406C23—24no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—41no—
414C23—41no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23C2×#12111yescharacteristic
424C23#53211yescharacteristic
425C23#53211yescharacteristic
426C23D4212yes—
427C23#54612yes—
428C23C2×#13712yes—
429C23#57512yes—
430C23#57712yes—
431C23#57812yes—
432Q8—44no—
433Q8—44no—
434Q8—44no—
435Q8—44no—
436Q8—82no—
437Q8—82no—
438Q8—82no—
439Q8—82no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—82no—
475C2×C4—82no—
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×C4C2×#12011yescharacteristic
494C2×C4C2×#12311yescharacteristic
495C2×C4#52511yescharacteristic
496C2×C4#52512yes—
497C2×C4#52911yescharacteristic
498C2×C4#52911yescharacteristic
499C2×C4#53211yescharacteristic
500C2×C4#53212yes—
501C2×C4#53411yescharacteristic
502C2×C4#53511yescharacteristic
503C2×C4#53812yes—
504C2×C4#54012yes—
505C2×C4#54612yes—
506C2×C4#54612yes—
507C2×C4#54612yes—
508C2×C4#55312yes—
509C2×C4#55312yes—
510C2×C4#55512yes—
511C8—24no—
512C8—24no—
513C8—42no—
514C8—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—44no—
519C22—44no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22#240811yescharacteristic
525C22#246412yes—
526C22#254311yescharacteristic
527C22#255512yes—
528C22#285411yescharacteristic
529C22#305612yes—
530C22#306312yes—
531C22#317212yes—
532C22#317311yescharacteristic
533C22#317711yescharacteristic
534C22#317812yes—
535C22#318311yescharacteristic
536C22#318511yescharacteristic
537C22#326612yes—
538C22#326912yes—
539C22#327012yes—
540C22#327112yes—
541C22#332312yes—
542C22#332712yes—
543C22#332812yes—
544C22#333112yes—
545C4—42no—
546C4—44no—
547C4—84no—
548C4—84no—
549C4—82no—
550C4—84no—
551C4—162no—
552C4#238811yescharacteristic
553C4#240511yescharacteristic
554C4#247612yes—
555C4#250011yescharacteristic
556C4#255912yes—
557C4#263511yescharacteristic
558C2—82no—
559C2#1499611yescharacteristic
560C2#2427412yes—
561C2#2697012yes—
562C2#2697212yes—
563C2#2856812yes—
564C2#2892512yes—
565C2#2892812yes—
566C2#3314111yescharacteristic
567C2#3314411yescharacteristic
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×#1933C2semidirect
5. #11327C2semidirect
6. #11257C2semidirect
7. #11024C2semidirect
8. #11024C2semidirect
10. #10928C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4702
C2×#19332
C2×#17532
#151792
#113272
#112572
#110242
#109702
#109282
#107222