Finite Group Databasebeta

← #405900

Subgroups — #405900

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1894C211yesmaximal, characteristic
2C2×#1873C211yesmaximal, characteristic
3C2×#1851C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5#14139C211yesmaximal, characteristic
6#13951C211yesmaximal, characteristic
7#11316C212yesmaximal, complemented
8#11227C212yesmaximal, complemented
9#11077C212yesmaximal, complemented
10#11067C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C2×#391C2212yes—
22C2×#389C2212yes—
23C2×#380C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25#1894—24no—
26#1873—24no—
27#1851—24no—
28#1736—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#336C2212yes—
37#1441—21no—
38#1441—21no—
39#1439—22no—
40C4⋊#95—21no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43#1434—21no—
44C22⋊#95—22no—
45C23×Q8C2311yescharacteristic
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—42no—
51C22×M16—42no—
52C22×M16D414yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2312yes—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8Q814yes—
62C2×#117—22no—
63C2×#117—24no—
64C2×#117—42no—
65C2×#117D414yes—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4D412yes—
82C22×C42—21no—
83C22×C42C2312yes—
84C2×C4⋊C8—24no—
85C2×C4⋊C8—42no—
86C2×C4⋊C8—42no—
87C2×#103—24no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—24no—
91C2×#103—42no—
92C2×#103—42no—
93C2×#103—42no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97#391—24no—
98#391—42no—
99#389—24no—
100#389—42no—
101#380—22no—
102#380—22no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D411yescharacteristic
114C2×#95D411yescharacteristic
115#340—24no—
116#340—22no—
117#340—22no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—42no—
122#336—24no—
123C25C2×D411yescharacteristic
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8C2×Q812yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—44no—
145C23×C4—42no—
146C23×C4—44no—
147C23×C4—81no—
148C23×C4—81no—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×Q812yes—
155C23×C4#4112yes—
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4C2411yescharacteristic
159C2×M16—28no—
160C2×M16—48no—
161C2×M16—44no—
162C2×M16—44no—
163C2×M16—48no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8C2×D414yes—
181C22⋊Q8—28no—
182C22⋊Q8—28no—
183C22⋊Q8—44no—
184C22⋊Q8—44no—
185#117—28no—
186#117—44no—
187#117—48no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
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—48no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×Q812yes—
223C2×C4⋊C4#4114yes—
224C2×C4⋊C4#4114yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—81no—
235C2×C22⋊C4—81no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—44no—
244C4⋊C8—48no—
245#103—44no—
246#103—48no—
247#103—44no—
248C22⋊C8—44no—
249#95—22no—
250#95—22no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—42no—
256#95—82no—
257#95—82no—
258#95—82no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24#12112yes—
268C24#12112yes—
269C24C22⋊Q812yes—
270C24C22×D411yescharacteristic
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—42no—
282C2×Q8—41no—
283C2×Q8—42no—
284C2×Q8—41no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—48no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
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—41no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—88no—
356C22×C4—88no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4#12114yes—
360C22×C4#12112yes—
361C22×C4#12112yes—
362C22×C4C22⋊Q814yes—
363C22×C4C22⋊Q814yes—
364C22×C4C22⋊Q812yes—
365C22×C4#12614yes—
366C22×C4#13714yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×Q811yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4112yes—
374C22×C4C2×#4112yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C4H(5,𝔽2)12yes—
377M16—416no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—41no—
394C2×C8—42no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—48no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—48no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—88no—
415C4⋊C4#12314yes—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C22⋊C4—48no—
419C22⋊C4—84no—
420C42—48no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—42no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—24no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23#39614yes—
448C23C2×#12111yescharacteristic
449C23C2×#12111yescharacteristic
450C23C2×C22⋊Q811yescharacteristic
451C23#54612yes—
452C23#54612yes—
453C23C2×#13712yes—
454C23#57512yes—
455C23#57611yescharacteristic
456C23#57711yescharacteristic
457C23#57711yescharacteristic
458C23#57811yescharacteristic
459Q8—44no—
460Q8—42no—
461Q8—42no—
462Q8—81no—
463Q8—81no—
464Q8—82no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—816no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4#39614yes—
527C2×C4C2×#12111yescharacteristic
528C2×C4C2×C22⋊Q811yescharacteristic
529C2×C4C2×C22⋊Q811yescharacteristic
530C2×C4C2×#12311yescharacteristic
531C2×C4#52511yescharacteristic
532C2×C4C2×#12611yescharacteristic
533C2×C4#52912yes—
534C2×C4#53212yes—
535C2×C4#53412yes—
536C2×C4#53411yescharacteristic
537C2×C4#53511yescharacteristic
538C2×C4#53812yes—
539C2×C4#55112yes—
540C2×C4#55112yes—
541C2×C4#55312yes—
542C2×C4#55312yes—
543C2×C4#55412yes—
544C2×C4#55412yes—
545C8—44no—
546C8—81no—
547C8—81no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22C2×#39611yescharacteristic
560C22#241111yescharacteristic
561C22#241211yescharacteristic
562C22#245012yes—
563C22#254312yes—
564C22#303911yescharacteristic
565C22#304011yescharacteristic
566C22#304511yescharacteristic
567C22#304611yescharacteristic
568C22#305612yes—
569C22#306211yescharacteristic
570C22#306311yescharacteristic
571C22#307812yes—
572C22#308311yescharacteristic
573C22#308411yescharacteristic
574C22#331412yes—
575C22#331712yes—
576C22#331912yes—
577C22#332012yes—
578C22#332312yes—
579C22#332712yes—
580C22#332812yes—
581C22#333112yes—
582C4—24no—
583C4—44no—
584C4—44no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4—84no—
589C4—82no—
590C4—162no—
591C4#248711yescharacteristic
592C4#251012yes—
593C4#256412yes—
594C4#257911yescharacteristic
595C4#261711yescharacteristic
596C4#285411yescharacteristic
597C2—44no—
598C2#1492111yescharacteristic
599C2#2361611yescharacteristic
600C2#2361911yescharacteristic
601C2#2449511yescharacteristic
602C2#2449611yescharacteristic
603C2#2454311yescharacteristic
604C2#2454811yescharacteristic
605C2#2642812yes—
606C2#2643412yes—
607C2#2856012yes—
608C2#2856812yes—
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
7. #11316C2semidirect
8. #11227C2semidirect
9. #11077C2semidirect
10. #11067C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18942
C2×#18732
C2×#18512
C2×#17362
#141392
#139512
#113162
#112272
#110772
#110672
#107302