Finite Group Databasebeta

← #456925

Subgroups — #456925

Order 512.

1494 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2068C212yesmaximal, complemented
2C2×#2036C212yesmaximal
3C2×#1872C212yesmaximal
4#15178C211yesmaximal, characteristic
5#14130C211yesmaximal, characteristic
6#11256C212yesmaximal, complemented
7#11067C212yesmaximal, complemented
8#11042C211yesmaximal, characteristic
9#11010C212yesmaximal, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C23×C4⋊C4C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2212yes—
20C2×#391C2212yes—
21C2×#389C2212yes—
22C2×#385—21no—
23C2×#380C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25#2068—28no—
26#2036—28no—
27#1872—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33#1444—21no—
34#1444—21no—
35#1444—21no—
36#1442—21no—
37#1439—21no—
38#1439—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C22⋊#95—21no—
42C23×Q8C2312yes—
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—42no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8—41no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—42no—
56C2×C22⋊Q8—42no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—41no—
64C22×C4⋊C4—41no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4D411yescharacteristic
72C22×C22⋊C4D411yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#107—24no—
76C2×#107—24no—
77C2×#107—42no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—41no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—42no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103—42no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—42no—
95#391—24no—
96#391—42no—
97#389—24no—
98#389—42no—
99#385—44no—
100#380—22no—
101#380D414yes—
102C4×C22⋊C4—41no—
103C4×C22⋊C4—41no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D411yescharacteristic
113C2×#95D411yescharacteristic
114#340—24no—
115#340—24no—
116#340D418yes—
117#336—42no—
118#336—41no—
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×Q8#4114yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—44no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—82no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4#4112yes—
158C23×C4C2411yescharacteristic
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×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8C2×D414yes—
173C4⋊Q8—44no—
174C4⋊Q8—48no—
175C4⋊Q8—44no—
176C22⋊Q8—216no—
177C22⋊Q8—48no—
178C22⋊Q8—48no—
179C22⋊Q8—48no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
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—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—44no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4C2×D414yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4Q1614yes—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233#107—48no—
234C4⋊C8—44no—
235#103—48no—
236#103—48no—
237#103—48no—
238#103—48no—
239C22⋊C8—48no—
240#95—24no—
241#95—41no—
242#95—41no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—42no—
247#95—42no—
248#95—44no—
249#95—82no—
250C24—21no—
251C24—21no—
252C24—22no—
253C24—22no—
254C24—21no—
255C24—21no—
256C24—22no—
257C24—22no—
258C24#12112yes—
259C24C2×Q1612yes—
260C24#13712yes—
261C24C22×D411yescharacteristic
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
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—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—48no—
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—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4#12112yes—
363C22×C4C2×Q1612yes—
364C22×C4C2×Q1612yes—
365C22×C4AGL(1,ℤ/8)12yes—
366C22×C4C22×D412yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4112yes—
370C22×C4H(5,𝔽2)11yescharacteristic
371C22×C4H(5,𝔽2)12yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)11yescharacteristic
374C22×C42−1+411yescharacteristic
375C22×C42−1+411yescharacteristic
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—416no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—48no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—88no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—48no—
410C22⋊C4—416no—
411C42—42no—
412C42—42no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C23—22no—
418C23—22no—
419C23—24no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—24no—
424C23—24no—
425C23—22no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—41no—
430C23—42no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23#46214yes—
440C23C2×#12111yescharacteristic
441C23#53411yescharacteristic
442C23#53611yescharacteristic
443C23D4212yes—
444C23#54612yes—
445C23C22×Q1611yescharacteristic
446C23C2×AGL(1,ℤ/8)11yescharacteristic
447C23C2×#13711yescharacteristic
448C23#57511yescharacteristic
449C23#57712yes—
450C23#57812yes—
451Q8—44no—
452Q8—44no—
453Q8—44no—
454Q8—44no—
455Q8—82no—
456Q8—82no—
457Q8—82no—
458Q8—82no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—816no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4#52512yes—
520C2×C4#53512yes—
521C2×C4#53712yes—
522C2×C4#53711yescharacteristic
523C2×C4#53711yescharacteristic
524C2×C4#53812yes—
525C2×C4#53812yes—
526C2×C4#53911yescharacteristic
527C2×C4#53911yescharacteristic
528C2×C4#53912yes—
529C2×C4D4212yes—
530C2×C4D4211yescharacteristic
531C2×C4#54612yes—
532C2×C4#54812yes—
533C2×C4#55111yescharacteristic
534C2×C4#56011yescharacteristic
535C2×C4#56011yescharacteristic
536C2×C4#56212yes—
537C8—44no—
538C8—82no—
539C22—24no—
540C22—28no—
541C22—44no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—44no—
546C22—42no—
547C22—44no—
548C22—42no—
549C22—42no—
550C22#242412yes—
551C22#261811yescharacteristic
552C22#266911yescharacteristic
553C22#268112yes—
554C22#269711yescharacteristic
555C22C2×#46211yescharacteristic
556C22#303611yescharacteristic
557C22#304612yes—
558C22#318911yescharacteristic
559C22#319311yescharacteristic
560C22#320011yescharacteristic
561C22#320211yescharacteristic
562C22#323011yescharacteristic
563C22#323611yescharacteristic
564C22#324212yes—
565C22#326612yes—
566C22#326912yes—
567C22#327212yes—
568C22D4×Q1612yes—
569C22#329912yes—
570C22#330812yes—
571C22#331212yes—
572C22#332612yes—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—84no—
578C4—84no—
579C4—81no—
580C4—84no—
581C4—84no—
582C4#246311yescharacteristic
583C4#263612yes—
584C4#267511yescharacteristic
585C4#267612yes—
586C4#269311yescharacteristic
587C4#282611yescharacteristic
588C2—44no—
589C2#1628811yescharacteristic
590C2#2508212yes—
591C2#2510412yes—
592C2#2975711yescharacteristic
593C2#2976611yescharacteristic
594C2#3050711yescharacteristic
595C2#3051011yescharacteristic
596C2#3112212yes—
597C2#3114412yes—
598C2#3166911yescharacteristic
599C2#3168311yescharacteristic
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. C2×#2068C2semidirect
6. #11256C2semidirect
7. #11067C2semidirect
9. #11010C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20682
C2×#20362
C2×#18722
#151782
#141302
#112562
#110672
#110422
#110102