Finite Group Databasebeta

← #405912

Subgroups — #405912

Order 512.

1420 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1873C211yesmaximal, characteristic
2C2×#1872C211yesmaximal, characteristic
3C2×#1853C211yesmaximal, characteristic
4C2×#1798C211yesmaximal, characteristic
5#14258C211yesmaximal, characteristic
6#13795C211yesmaximal, characteristic
7#11316C212yesmaximal, complemented
8#11231C212yesmaximal, complemented
9#11122C212yesmaximal, complemented
10#11077C212yesmaximal, complemented
11#10731C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#106C2212yes—
17C22×C4⋊C8C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#380C2211yescharacteristic
25#1873—24no—
26#1872—24no—
27#1853—24no—
28#1798—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1441—22no—
36#1440—22no—
37C4⋊#95—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C22⋊#95—22no—
43C23×Q8C2311yescharacteristic
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8Q814yes—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—24no—
57C2×#117—24no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—42no—
61C2×#117—42no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4D412yes—
68C22×C4⋊C4D414yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4D412yes—
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#106—24no—
82C2×#106—42no—
83C2×#106—42no—
84C2×C4⋊C8—24no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—42no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103—42no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8D414yes—
99#391—24no—
100#391—42no—
101#380—22no—
102#380—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—42no—
117#336—24no—
118C25C2×D411yescharacteristic
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8C2×Q812yes—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—82no—
143C23×C4—82no—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×Q812yes—
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8C2×D414yes—
168C22⋊Q8—28no—
169C22⋊Q8—28no—
170C22⋊Q8—44no—
171C22⋊Q8—44no—
172#117—48no—
173#117—48no—
174#117—48no—
175#117—48no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—48no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×Q812yes—
212C2×C4⋊C4#4114yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—82no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42#4114yes—
235#106—48no—
236C4⋊C8—48no—
237#103—48no—
238#103—48no—
239C22⋊C8—28no—
240C22⋊C8—44no—
241#95—44no—
242#95—42no—
243#95—44no—
244#95—82no—
245#95—82no—
246#95—82no—
247#95—82no—
248#95—82no—
249#95—82no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—21no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24#12112yes—
259C24#12112yes—
260C24C22⋊Q812yes—
261C24C22×D411yescharacteristic
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—41no—
267C2×Q8—41no—
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—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—48no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
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—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—84no—
345C22×C4—84no—
346C22×C4—88no—
347C22×C4—88no—
348C22×C4#12114yes—
349C22×C4#12112yes—
350C22×C4#12112yes—
351C22×C4C22⋊Q814yes—
352C22×C4C22⋊Q814yes—
353C22×C4C22⋊Q812yes—
354C22×C4#12314yes—
355C22×C4#12614yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×Q811yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4112yes—
363C22×C4C2×#4112yes—
364C22×C4H(5,𝔽2)12yes—
365C22×C4H(5,𝔽2)12yes—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—48no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C22⋊C4—48no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C22⋊C4—84no—
403C42—24no—
404C42—24no—
405C42—48no—
406C42—42no—
407C42—42no—
408C23—24no—
409C23—24no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—24no—
421C23—24no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23#39614yes—
431C23C2×#12111yescharacteristic
432C23C2×#12111yescharacteristic
433C23C2×C22⋊Q811yescharacteristic
434C23C2×#12311yescharacteristic
435C23#52511yescharacteristic
436C23#53411yescharacteristic
437C23#53511yescharacteristic
438C23#57712yes—
439C23#57712yes—
440C23#57712yes—
441C23#57812yes—
442Q8—44no—
443Q8—44no—
444Q8—82no—
445Q8—82no—
446C2×C4—22no—
447C2×C4—22no—
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—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—816no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4#39614yes—
508C2×C4C2×#12111yescharacteristic
509C2×C4C2×C22⋊Q811yescharacteristic
510C2×C4C2×C22⋊Q811yescharacteristic
511C2×C4C2×#12611yescharacteristic
512C2×C4#52912yes—
513C2×C4#53212yes—
514C2×C4#53412yes—
515C2×C4#53812yes—
516C2×C4#54612yes—
517C2×C4#54612yes—
518C2×C4#55112yes—
519C2×C4#55112yes—
520C2×C4#55312yes—
521C2×C4#55312yes—
522C2×C4#55412yes—
523C2×C4#55412yes—
524C8—44no—
525C8—82no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—42no—
530C22—42no—
531C22—44no—
532C22—44no—
533C22—42no—
534C22—42no—
535C22—44no—
536C22—44no—
537C22C2×#39611yescharacteristic
538C22#241111yescharacteristic
539C22#241211yescharacteristic
540C22#248711yescharacteristic
541C22#257911yescharacteristic
542C22#261711yescharacteristic
543C22#285411yescharacteristic
544C22#304211yescharacteristic
545C22#304211yescharacteristic
546C22#304311yescharacteristic
547C22#304411yescharacteristic
548C22#306012yes—
549C22#306312yes—
550C22#308212yes—
551C22#308412yes—
552C22#309712yes—
553C22#309912yes—
554C22#312012yes—
555C22#312512yes—
556C22#320012yes—
557C22#320112yes—
558C22#322212yes—
559C22#322512yes—
560C4—24no—
561C4—44no—
562C4—84no—
563C4—82no—
564C4—82no—
565C4—84no—
566C4—82no—
567C4—161no—
568C4—161no—
569C4#245012yes—
570C4#251012yes—
571C4#254312yes—
572C4#256412yes—
573C2—44no—
574C2#1492111yescharacteristic
575C2#2361011yescharacteristic
576C2#2361511yescharacteristic
577C2#2451712yes—
578C2#2456912yes—
579C2#2768912yes—
580C2#2925112yes—
581C2#2972311yescharacteristic
582C2#2972811yescharacteristic
583C2#3311911yescharacteristic
584C2#3312111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11316C2semidirect
8. #11231C2semidirect
9. #11122C2semidirect
10. #11077C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18732
C2×#18722
C2×#18532
C2×#17982
#142582
#137952
#113162
#112312
#111222
#110772
#107312