Finite Group Databasebeta

← #415065

Subgroups — #415065

Order 512.

1558 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#1985C211yesmaximal, characteristic, complemented
3C2×#1847C211yesmaximal, characteristic, complemented
4C2×#1804C211yesmaximal, characteristic
5C2×#1762C211yesmaximal, characteristic
6#13925C211yesmaximal, characteristic, complemented
7#11220C212yesmaximal, complemented
8#11219C212yesmaximal, complemented
9#11123C212yesmaximal
10#11121C212yesmaximal
11#10849C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8—21no—
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C22×#102C2211yescharacteristic
21C22×#97C2211yescharacteristic
22#2538—24no—
23#2538—24no—
24C2×#396—22no—
25C2×#390C2212yes—
26C2×#389C2212yes—
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#380—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31#1985—24no—
32#1847—24no—
33#1804—24no—
34#1762—24no—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1446—21no—
43#1445—21no—
44#1436—22no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8D411yescharacteristic
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2312yes—
53C2×C22⋊Q8—28no—
54C2×C22⋊Q8—48no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4D414yes—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C42—21no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—42no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—42no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—42no—
97C2×#102—42no—
98C2×#102Q818yescomplemented
99C2×#97—22no—
100C2×#97—22no—
101C2×#97D414yes—
102#396—216no—
103#390—24no—
104#390—24no—
105#389—24no—
106#389—24no—
107#384—22no—
108#384D414yes—
109#382—22no—
110#382—22no—
111#380—28no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122#340—24no—
123#340—22no—
124#340—22no—
125#340—42no—
126#340—42no—
127#336—24no—
128#336—24no—
129C25—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—24no—
134C22×Q8—42no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4C2×D412yes—
144C22×D4C2×Q814yes—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8C2×D414yes—
175C22×C8C2×Q814yes—
176C22×C8#4114yes—
177C22⋊Q8—432no—
178C4×Q8—44no—
179C4×Q8—44no—
180C4×Q8—44no—
181C4×Q8—44no—
182C4×D4—28no—
183C4×D4—28no—
184C4×D4—44no—
185C4×D4—44no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
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—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—81no—
222C2×C4⋊C4—81no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×Q814yes—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—42no—
242C2×C42—81no—
243C2×C42—81no—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250#107—48no—
251#106—48no—
252C4⋊C8—44no—
253#102—216no—
254#102—44no—
255#102—44no—
256#97—28no—
257#95—28no—
258#95—44no—
259#95—42no—
260#95—42no—
261#95—42no—
262#95—44no—
263#95—82no—
264#95—82no—
265C24—22no—
266C24—24no—
267C24—41no—
268C24—41no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24C22×D411yescharacteristic
277C2×Q8—28no—
278C2×Q8—48no—
279C2×Q8—48no—
280C2×Q8—44no—
281C2×Q8—44no—
282C2×D4—22no—
283C2×D4—24no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—24no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4C4⋊Q814yes—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
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—48no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—88no—
352C22×C4—88no—
353C22×C4#12012yes—
354C22×C4#12012yes—
355C22×C4#12112yes—
356C22×C4#12112yes—
357C22×C4#12312yes—
358C22×C4#12312yes—
359C22×C4AGL(1,ℤ/8)14yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×Q812yes—
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4112yes—
367C22×C4C2×#4111yescharacteristic
368C22×C4H(5,𝔽2)12yes—
369C22×C42−1+412yes—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8C22⋊Q818yescomplemented
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—48no—
394C4⋊C4—416no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—48no—
402C4⋊C4—44no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—84no—
410C22⋊C4—416no—
411C22⋊C4—48no—
412C22⋊C4—88no—
413C42—42no—
414C42—42no—
415C42—42no—
416C42—42no—
417C42—84no—
418C42#12114yes—
419C42#12114yes—
420C42#12314yes—
421C42#12414yes—
422C23—24no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—42no—
428C23—44no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—44no—
435C23—82no—
436C23—81no—
437C23—81no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—81no—
444C23—81no—
445C23C2×#12011yescharacteristic
446C23C2×#12111yescharacteristic
447C23C2×#12311yescharacteristic
448C23D4×Q812yes—
449C23#55012yes—
450C23C2×AGL(1,ℤ/8)12yes—
451C23#57512yes—
452C23#57611yescharacteristic
453C23#57711yescharacteristic
454C23#57711yescharacteristic
455C23#57811yescharacteristic
456Q8—416no—
457D4—24no—
458D4—24no—
459D4—42no—
460D4—44no—
461D4—44no—
462D4—42no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—816no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
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—82no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4#39414yes—
518C2×C4#39414yes—
519C2×C4C2×#12111yescharacteristic
520C2×C4C2×#12111yescharacteristic
521C2×C4C2×C22⋊Q812yes—
522C2×C4C2×#12311yescharacteristic
523C2×C4C2×#12411yescharacteristic
524C2×C4C2×C4⋊Q811yescharacteristic
525C2×C4#53211yescharacteristic
526C2×C4#53312yes—
527C2×C4#53511yescharacteristic
528C2×C4#53611yescharacteristic
529C2×C4#53712yes—
530C2×C4#54212yes—
531C2×C4#54612yes—
532C2×C4D4×Q812yes—
533C2×C4#55212yes—
534C2×C4#55412yes—
535C2×C4#55412yes—
536C2×C4#55612yes—
537C8—28no—
538C8—81no—
539C8—81no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—82no—
544C22—82no—
545C22—84no—
546C22—84no—
547C22—82no—
548C22—82no—
549C22—84no—
550C22—84no—
551C22C2×#39411yescharacteristic
552C22#242312yes—
553C22#243111yescharacteristic
554C22#243411yescharacteristic
555C22#253812yes—
556C22#300311yescharacteristic
557C22#300511yescharacteristic
558C22#301211yescharacteristic
559C22#301311yescharacteristic
560C22#305512yes—
561C22#306211yescharacteristic
562C22#306311yescharacteristic
563C22#309412yes—
564C22#309811yescharacteristic
565C22#309911yescharacteristic
566C22#338412yes—
567C22#338612yes—
568C22#338812yes—
569C22#338912yes—
570C22#339712yes—
571C22#339912yes—
572C22#340112yes—
573C22#340212yes—
574C4—24no—
575C4—24no—
576C4—44no—
577C4—84no—
578C4—84no—
579C4—82no—
580C4—82no—
581C4—161no—
582C4—161no—
583C4#241712yes—
584C4#249212yes—
585C4#249911yescharacteristic
586C4#250211yescharacteristic
587C4#262011yescharacteristic
588C4#285711yescharacteristic
589C2—84no—
590C2#1518211yescharacteristic
591C2#2348011yescharacteristic
592C2#2348511yescharacteristic
593C2#2507212yes—
594C2#2507312yes—
595C2#2563711yescharacteristic
596C2#2563811yescharacteristic
597C2#2584411yescharacteristic
598C2#2584511yescharacteristic
599C2#2827512yes—
600C2#2827612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1985C2semidirect
3. C2×#1847C2semidirect
6. #13925C2semidirect
7. #11220C2semidirect
8. #11219C2semidirect
11. #10849C2semidirect
98. C2×#102Q8semidirect
386. C2×C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#19852
C2×#18472
C2×#18042
C2×#17622
#139252
#112202
#112192
#111232
#111212
#108492