Finite Group Databasebeta

← #411300

Subgroups — #411300

Order 512.

1468 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic, complemented
2C2×#1926C212yesmaximal, complemented
3#14257C211yesmaximal, characteristic
4#11323C211yesmaximal, characteristic, complemented
5#11268C211yesmaximal, characteristic, complemented
6#11095C212yesmaximal, complemented
7#11051C211yesmaximal, characteristic, complemented
8#11049C211yesmaximal, characteristic, complemented
9#10969C212yesmaximal
10#10806C212yesmaximal, complemented
11#10713C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#106C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×C22⋊C8C2212yes—
18#2314—28no—
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#388—22no—
22C2×#387—22no—
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4C2212yes—
28#1926—28no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37#1439—22no—
38#1438—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C23×D4—21no—
42C23×D4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#121—22no—
47C2×#121—22no—
48C2×#121—24no—
49C2×#121—42no—
50C2×#117—24no—
51C2×#117—24no—
52C2×#117—22no—
53C2×#117—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4D412yes—
67C22×C42—22no—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#106—24no—
71C2×#106—42no—
72C2×#106—42no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—42no—
78C2×#102—42no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—42no—
82#390—24no—
83#390—22no—
84#390—22no—
85#388—216no—
86#387—216no—
87#384—24no—
88#382—24no—
89#380—24no—
90#380—24no—
91#380—42no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4D418yescomplemented
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—42no—
104#340—42no—
105#336—24no—
106#336—22no—
107#336—22no—
108C25—21no—
109C25C2×D411yescharacteristic
110C22×D4—24no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—21no—
117C22×D4—22no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4—42no—
121C22×D4—42no—
122C22×D4—82no—
123C22×D4—82no—
124C22×D4#4112yes—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4#4114yes—
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8#4114yes—
158#121—48no—
159#121—48no—
160#117—216no—
161#117—216no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C2×D414yes—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4#4112yes—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4—82no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4—82no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—28no—
215C2×C42—82no—
216C2×C42—82no—
217C2×C42C2×D414yes—
218C2×C42#4112yes—
219C2×C42#4112yes—
220#106—48no—
221#102—216no—
222#102—48no—
223C22⋊C8—48no—
224#95—28no—
225#95—28no—
226#95—28no—
227#95—42no—
228#95—82no—
229#95—82no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—41no—
242C24—41no—
243C24—41no—
244C24—41no—
245C24—41no—
246C24—41no—
247C24—42no—
248C24—42no—
249C24—42no—
250C24—42no—
251C24#12312yes—
252C24C22×D411yescharacteristic
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—48no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—88no—
271C2×D4—88no—
272C2×D4—84no—
273C2×D4—84no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
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—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4#12014yes—
338C22×C4#12314yes—
339C22×C4#12312yes—
340C22×C4C22×D412yes—
341C22×C4C2×#4112yes—
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4112yes—
346C22×C4C2×#4111yescharacteristic
347C22×C4H(5,𝔽2)11yescharacteristic
348C22×C4H(5,𝔽2)12yes—
349C22×C42−1+411yescharacteristic
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C4⋊C4—216no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—88no—
380C22⋊C4—216no—
381C22⋊C4—48no—
382C22⋊C4—88no—
383C22⋊C4—88no—
384C22⋊C4—88no—
385C22⋊C4—84no—
386C42—24no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—88no—
391C23—24no—
392C23—24no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—41no—
401C23—41no—
402C23—42no—
403C23—42no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—41no—
413C23—42no—
414C23—44no—
415C23—44no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—82no—
420C23—82no—
421C23—82no—
422C23—82no—
423C23—81no—
424C23—81no—
425C23—82no—
426C23—82no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23—82no—
432C23—82no—
433C23—81no—
434C23—82no—
435C23—81no—
436C23—82no—
437C23C2×#12011yescharacteristic
438C23C2×#12311yescharacteristic
439C23C2×#12311yescharacteristic
440C23#52511yescharacteristic
441C23#53211yescharacteristic
442C23#53511yescharacteristic
443C23#53711yescharacteristic
444C23#57612yes—
445C23#57712yes—
446C23#57712yes—
447C23#57712yes—
448D4—44no—
449D4—44no—
450D4—816no—
451D4—82no—
452D4—82no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4#52512yes—
508C2×C4#52512yes—
509C2×C4#52512yes—
510C2×C4#53212yes—
511C2×C4#53512yes—
512C2×C4#53712yes—
513C2×C4#55112yes—
514C2×C4#55112yes—
515C2×C4#55112yes—
516C2×C4#55212yes—
517C2×C4#55211yescharacteristic
518C2×C4#55311yescharacteristic
519C2×C4#55312yes—
520C2×C4#55512yes—
521C2×C4#55511yescharacteristic
522C2×C4#55611yescharacteristic
523C8—44no—
524C8—82no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—44no—
529C22—44no—
530C22—82no—
531C22—81no—
532C22—81no—
533C22—82no—
534C22—81no—
535C22—81no—
536C22—82no—
537C22—84no—
538C22—82no—
539C22—82no—
540C22—84no—
541C22—162no—
542C22—162no—
543C22—161no—
544C22—161no—
545C22—161no—
546C22—161no—
547C22#238811yescharacteristic
548C22#248211yescharacteristic
549C22#248511yescharacteristic
550C22#249811yescharacteristic
551C22#250011yescharacteristic
552C22#261511yescharacteristic
553C22#284911yescharacteristic
554C22#301012yes—
555C22#301212yes—
556C22#309612yes—
557C22#309612yes—
558C22#309912yes—
559C22#309912yes—
560C22#311812yes—
561C22#312512yes—
562C22#317312yes—
563C22#317912yes—
564C22#322011yescharacteristic
565C22#322111yescharacteristic
566C22#322211yescharacteristic
567C22#322211yescharacteristic
568C22#325712yes—
569C22#326012yes—
570C4—42no—
571C4—44no—
572C4—44no—
573C4—84no—
574C4—82no—
575C4—84no—
576C4—161no—
577C4—161no—
578C4#259812yes—
579C4#260112yes—
580C4#260912yes—
581C4#261012yes—
582C2—82no—
583C2—162no—
584C2#1511311yescharacteristic
585C2#2358312yes—
586C2#2759411yescharacteristic
587C2#2759711yescharacteristic
588C2#2764112yes—
589C2#2786912yes—
590C2#2789811yescharacteristic
591C2#2790311yescharacteristic
592C2#2967111yescharacteristic
593C2#2967511yescharacteristic
594C2#3298712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
1. C2×#2314C2semidirect
2. C2×#1926C2semidirect
4. #11323C2semidirect
5. #11268C2semidirect
6. #11095C2semidirect
7. #11051C2semidirect
8. #11049C2semidirect
10. #10806C2semidirect
11. #10713C2semidirect
93. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#19262
#142572
#113232
#112682
#110952
#110512
#110492
#109692
#108062
#107132