Finite Group Databasebeta

← #416880

Subgroups — #416880

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C2×#1989C211yesmaximal, characteristic, complemented
3C2×#1831C211yesmaximal, characteristic, complemented
4C2×#1804C211yesmaximal, characteristic
5#14709C211yesmaximal, characteristic, complemented
6C4×#390C211yesmaximal, characteristic
7#11257C212yesmaximal, complemented
8#11256C212yesmaximal, complemented
9#11189C212yesmaximal
10#11189C212yesmaximal
11#10850C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#478—24no—
15C2×#478—24no—
16C2×#478—24no—
17C22×#107C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#102C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2212yes—
24C2×#390C2212yes—
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4—22no—
28C2×C43C2211yescharacteristic
29#1989—24no—
30#1831—24no—
31#1804—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1437—21no—
40#1437—21no—
41#1436—21no—
42#1436—22no—
43#1436—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125D414yes—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C42—21no—
75C22×C42D411yescharacteristic
76C22×C42D412yes—
77C22×C42D411yescharacteristic
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80#478—232no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#106—22no—
85C2×#106—22no—
86C2×#106—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8D414yes—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102#390—24no—
103#390—24no—
104#382—22no—
105#382—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C22⋊C4—48no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C43—22no—
119C43—22no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#336—24no—
126#336—24no—
127C25—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4C2×D412yes—
137C22×D4#4114yes—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
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×C8—22no—
168C22×C8C2×D412yes—
169C22×C8C2×D412yes—
170#125—28no—
171#125—28no—
172#125—44no—
173#125—44no—
174C4×D4—28no—
175C4×D4—28no—
176C4×D4—44no—
177C4×D4—44no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4#4114yes—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—44no—
237C2×C42—42no—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243C2×C42#4112yes—
244#107—28no—
245#106—28no—
246C4⋊C8—28no—
247C4⋊C8—44no—
248#102—28no—
249#102—28no—
250#102—48no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—42no—
255#95—42no—
256#95—42no—
257#95—44no—
258#95—44no—
259#95—44no—
260C24—22no—
261C24—24no—
262C24—41no—
263C24—41no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24C22×D411yescharacteristic
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—24no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4#12414yes—
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—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—416no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
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—48no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—84no—
336C22×C4—84no—
337C22×C4—88no—
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4#12112yes—
341C22×C4#12112yes—
342C22×C4#12112yes—
343C22×C4#12112yes—
344C22×C4#13514yes—
345C22×C4AGL(1,ℤ/8)14yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4112yes—
353C22×C4C2×#4112yes—
354C22×C42−1+412yes—
355C22×C42−1+412yes—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C22⋊C4—416no—
396C22⋊C4—416no—
397C22⋊C4—48no—
398C42—28no—
399C42—48no—
400C42—44no—
401C42—42no—
402C42—42no—
403C42—42no—
404C42—42no—
405C42—84no—
406C42#12014yes—
407C42#12314yes—
408C42#12314yes—
409C42#12414yes—
410C23—24no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—42no—
416C23—42no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—82no—
431C23C2×#12111yescharacteristic
432C23C2×#12111yescharacteristic
433C23C2×#12111yescharacteristic
434C23#52512yes—
435C23#53612yes—
436C23C2×#13512yes—
437C23C2×AGL(1,ℤ/8)12yes—
438C23#57512yes—
439C23#57512yes—
440D4—24no—
441D4—24no—
442D4—44no—
443D4—42no—
444D4—44no—
445D4—42no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—816no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4#39414yes—
499C2×C4#39414yes—
500C2×C4C2×#12011yescharacteristic
501C2×C4C2×#12311yescharacteristic
502C2×C4C2×#12311yescharacteristic
503C2×C4C2×#12411yescharacteristic
504C2×C4C2×#12411yescharacteristic
505C2×C4#52912yes—
506C2×C4#53211yescharacteristic
507C2×C4#53611yescharacteristic
508C2×C4#53611yescharacteristic
509C2×C4#54112yes—
510C2×C4#54712yes—
511C2×C4#54812yes—
512C2×C4#54812yes—
513C2×C4#54812yes—
514C2×C4#55212yes—
515C2×C4#55212yes—
516C2×C4#55212yes—
517C2×C4#55612yes—
518C8—42no—
519C8—42no—
520C8—82no—
521C22—48no—
522C22—44no—
523C22—84no—
524C22—84no—
525C22—84no—
526C22—84no—
527C22—84no—
528C22—82no—
529C22—82no—
530C22C2×#39411yescharacteristic
531C22#241512yes—
532C22#253612yes—
533C22#262111yescharacteristic
534C22#285011yescharacteristic
535C22#303512yes—
536C22#303512yes—
537C22#303612yes—
538C22#303811yescharacteristic
539C22#304011yescharacteristic
540C22#305011yescharacteristic
541C22#305411yescharacteristic
542C22#305512yes—
543C22#310212yes—
544C22#310512yes—
545C22#311112yes—
546C22#311412yes—
547C22#322812yes—
548C22#323212yes—
549C22#323412yes—
550C22#324012yes—
551C4—24no—
552C4—24no—
553C4—44no—
554C4—44no—
555C4—84no—
556C4—82no—
557C4—82no—
558C4—82no—
559C4—162no—
560C4#242112yes—
561C4#243211yescharacteristic
562C4#243611yescharacteristic
563C4#248912yes—
564C4#249911yescharacteristic
565C4#250311yescharacteristic
566C2—84no—
567C2#1523611yescharacteristic
568C2#2353911yescharacteristic
569C2#2354611yescharacteristic
570C2#2470412yes—
571C2#2470612yes—
572C2#2820712yes—
573C2#2820812yes—
574C2#2987012yes—
575C2#3302812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1989C2semidirect
3. C2×#1831C2semidirect
5. #14709C2semidirect
7. #11257C2semidirect
8. #11256C2semidirect
11. #10850C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4782
C2×#19892
C2×#18312
C2×#18042
#147092
C4×#3902
#112572
#112562
#111892
#108502