Finite Group Databasebeta

← #416882

Subgroups — #416882

Order 512.

1360 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C2×#1989C211yesmaximal, characteristic, complemented
3C2×#1832C211yesmaximal, characteristic, complemented
4C2×#1802C211yesmaximal, characteristic
5#14283C211yesmaximal, characteristic, complemented
6#13785C211yesmaximal, characteristic
7#11257C212yesmaximal, complemented
8#11256C212yesmaximal, complemented
9#11189C212yesmaximal
10#11189C212yesmaximal
11#10849C211yesmaximal, 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×#384C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×#376C2211yescharacteristic
28#1989—24no—
29#1832—24no—
30#1802—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1441—21no—
39#1441—21no—
40#1441—22no—
41#1437—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—21no—
46C22⋊#95—21no—
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⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79#478—232no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#106—22no—
84C2×#106—22no—
85C2×#106—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8D414yes—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101#390—24no—
102#390—24no—
103#384—22no—
104#384—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107#376—22no—
108#376—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95D411yescharacteristic
117C2×#95D411yescharacteristic
118C2×#95D412yes—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#336—24no—
125#336—24no—
126C25—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4C2×D412yes—
136C22×D4#4114yes—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—82no—
146C23×C4—82no—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
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×D412yes—
168C22×C8C2×D412yes—
169#125—28no—
170#125—28no—
171#125—44no—
172#125—44no—
173C4×D4—28no—
174C4×D4—28no—
175C4×D4—44no—
176C4×D4—44no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
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—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—81no—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4#4114yes—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—82no—
224C2×C22⋊C4—82no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4—82no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—42no—
234C2×C42—81no—
235C2×C42—81no—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42#4112yes—
242#107—28no—
243#106—28no—
244C4⋊C8—28no—
245C4⋊C8—44no—
246#102—28no—
247#102—28no—
248#102—48no—
249#95—24no—
250#95—22no—
251#95—22no—
252#95—42no—
253#95—82no—
254#95—82no—
255#95—82no—
256#95—82no—
257#95—82no—
258#95—82no—
259C24—22no—
260C24—24no—
261C24—41no—
262C24—41no—
263C24—42no—
264C24—41no—
265C24—42no—
266C24—41no—
267C24—42no—
268C24—41no—
269C24—41no—
270C24C22×D411yescharacteristic
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4#12414yes—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—42no—
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—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
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—84no—
342C22×C4—84no—
343C22×C4—88no—
344C22×C4—88no—
345C22×C4#12112yes—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4#12112yes—
349C22×C4#12112yes—
350C22×C4#12112yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4112yes—
358C22×C4C2×#4112yes—
359C22×C42−1+412yes—
360C22×C42−1+412yes—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
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—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C22⋊C4—48no—
403C22⋊C4—88no—
404C22⋊C4—88no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42—84no—
410C42#12014yes—
411C42#12314yes—
412C42#12314yes—
413C42#12414yes—
414C23—24no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—42no—
420C23—42no—
421C23—44no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—44no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—82no—
437C23C2×#12111yescharacteristic
438C23C2×#12111yescharacteristic
439C23C2×#12111yescharacteristic
440C23#52512yes—
441C23#53612yes—
442C23#57612yes—
443C23#57611yescharacteristic
444C23#57711yescharacteristic
445C23#57712yes—
446C23#57711yescharacteristic
447C23#57811yescharacteristic
448D4—24no—
449D4—24no—
450D4—44no—
451D4—42no—
452D4—44no—
453D4—42no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—816no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4#39414yes—
511C2×C4#39414yes—
512C2×C4C2×#12011yescharacteristic
513C2×C4C2×#12311yescharacteristic
514C2×C4C2×#12311yescharacteristic
515C2×C4C2×#12411yescharacteristic
516C2×C4C2×#12411yescharacteristic
517C2×C4#52912yes—
518C2×C4#53211yescharacteristic
519C2×C4#53611yescharacteristic
520C2×C4#53611yescharacteristic
521C2×C4#54112yes—
522C2×C4#54712yes—
523C2×C4#54812yes—
524C2×C4#54812yes—
525C2×C4#54812yes—
526C2×C4#55212yes—
527C2×C4#55212yes—
528C2×C4#55212yes—
529C2×C4#55612yes—
530C8—42no—
531C8—42no—
532C8—82no—
533C22—44no—
534C22—44no—
535C22—44no—
536C22—84no—
537C22—84no—
538C22—84no—
539C22—84no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22C2×#39411yescharacteristic
545C22#241512yes—
546C22#253612yes—
547C22#262111yescharacteristic
548C22#285011yescharacteristic
549C22#304111yescharacteristic
550C22#304211yescharacteristic
551C22#304511yescharacteristic
552C22#304511yescharacteristic
553C22#304512yes—
554C22#304611yescharacteristic
555C22#304611yescharacteristic
556C22#306011yescharacteristic
557C22#306111yescharacteristic
558C22#306212yes—
559C22#311912yes—
560C22#312012yes—
561C22#312212yes—
562C22#312312yes—
563C22#324112yes—
564C22#324112yes—
565C22#324312yes—
566C22#324412yes—
567C4—24no—
568C4—24no—
569C4—84no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—162no—
576C4#242112yes—
577C4#243211yescharacteristic
578C4#243611yescharacteristic
579C4#248912yes—
580C4#249911yescharacteristic
581C4#250311yescharacteristic
582C2—84no—
583C2#1523611yescharacteristic
584C2#2354311yescharacteristic
585C2#2354411yescharacteristic
586C2#2471312yes—
587C2#2471712yes—
588C2#2821412yes—
589C2#2821512yes—
590C2#2987211yescharacteristic
591C2#2987411yescharacteristic
592C2#3303811yescharacteristic
593C2#3304011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ 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×#1832C2semidirect
5. #14283C2semidirect
7. #11257C2semidirect
8. #11256C2semidirect
11. #10849C2semidirect

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×#18322
C2×#18022
#142832
#137852
#112572
#112562
#111892
#108492