Finite Group Databasebeta

← #416079

Subgroups — #416079

Order 512.

1358 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#488C211yesmaximal, characteristic
2C2×#1992C211yesmaximal, characteristic, complemented
3C2×#1959C211yesmaximal, characteristic
4C2×#1768C211yesmaximal, characteristic, complemented
5#14283C211yesmaximal, characteristic, complemented
6#14273C211yesmaximal, characteristic
7#11190C212yesmaximal, complemented
8#11189C212yesmaximal
9#11077C212yesmaximal, complemented
10#11076C212yesmaximal
11#10841C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C2×#488—24no—
15C2×#488—24no—
16C2×#488—24no—
17C22×C4⋊C8C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×#97C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#1992—24no—
32#1959—24no—
33#1768—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39#1441—22no—
40#1441—22no—
41#1439—22no—
42#1439—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×Q8C2311yescharacteristic
46C23×D4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125D414yes—
55C2×#124—22no—
56C2×#124—22no—
57C2×#124—22no—
58C2×#124—22no—
59C2×#124—22no—
60C2×#124D414yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75#488—232no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—42no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—42no—
91C2×#97—22no—
92C2×#97—22no—
93C2×#97—22no—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96C2×C4×C8D414yes—
97#391—24no—
98#391—42no—
99#390—24no—
100#390—42no—
101#384—22no—
102#384—22no—
103#384—22no—
104#384—22no—
105#376—22no—
106#376—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D412yes—
115C2×#95D412yes—
116#340—24no—
117#340—24no—
118#340—42no—
119#340—42no—
120#336—24no—
121C25—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8C2×D412yes—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4C2×D412yes—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2411yescharacteristic
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
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×C8C2×D414yes—
165C22×C8#4114yes—
166#125—28no—
167#125—28no—
168#125—44no—
169#125—44no—
170#124—28no—
171#124—28no—
172#124—44no—
173#124—44no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—28no—
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—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—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C2×D414yes—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—82no—
210C2×C22⋊C4—82no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42#4114yes—
224C2×C42#4112yes—
225C2×C42#4112yes—
226C2×C42#4112yes—
227C4⋊C8—216no—
228#103—216no—
229#103—48no—
230#102—216no—
231#102—48no—
232#97—44no—
233C4×C8—28no—
234#95—24no—
235#95—24no—
236#95—42no—
237#95—42no—
238#95—44no—
239#95—82no—
240#95—82no—
241#95—82no—
242#95—82no—
243C24—22no—
244C24—41no—
245C24—41no—
246C24—41no—
247C24—41no—
248C24—41no—
249C24—41no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24C22×D411yescharacteristic
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—41no—
276C2×D4—42no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C22×C4—22no—
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—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
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#12114yes—
339C22×C4#12114yes—
340C22×C4#12114yes—
341C22×C4#12414yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4112yes—
349C22×C4C2×#4112yes—
350C22×C4H(5,𝔽2)12yes—
351C22×C42−1+412yes—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—416no—
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—
380C4⋊C4—88no—
381C22⋊C4—416no—
382C22⋊C4—88no—
383C22⋊C4—84no—
384C22⋊C4—84no—
385C42—24no—
386C42—24no—
387C42—42no—
388C42—42no—
389C42#12014yes—
390C42#12314yes—
391C42#12314yes—
392C42#12414yes—
393C23—42no—
394C23—44no—
395C23—44no—
396C23—42no—
397C23—42no—
398C23—81no—
399C23—81no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—82no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—81no—
413C23—81no—
414C23—82no—
415C23—82no—
416C23C2×#12111yescharacteristic
417C23C2×#12111yescharacteristic
418C23C2×#12111yescharacteristic
419C23C2×#12411yescharacteristic
420C23#53211yescharacteristic
421C23#53511yescharacteristic
422C23#53611yescharacteristic
423C23#57612yes—
424C23#57712yes—
425C23#57712yes—
426C23#57812yes—
427Q8—44no—
428Q8—44no—
429Q8—82no—
430Q8—82no—
431D4—44no—
432D4—44no—
433D4—82no—
434D4—82no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4#39418yes—
490C2×C4C2×#12011yescharacteristic
491C2×C4C2×#12311yescharacteristic
492C2×C4C2×#12311yescharacteristic
493C2×C4#52512yes—
494C2×C4C2×#12411yescharacteristic
495C2×C4#52912yes—
496C2×C4#53712yes—
497C2×C4#54212yes—
498C2×C4#54612yes—
499C2×C4#54712yes—
500C2×C4#54812yes—
501C2×C4#55012yes—
502C2×C4#55612yes—
503C2×C4#55612yes—
504C2×C4#55912yes—
505C2×C4#55912yes—
506C8—44no—
507C8—44no—
508C22—84no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—84no—
513C22—162no—
514C22—161no—
515C22—161no—
516C22—161no—
517C22—161no—
518C22—162no—
519C22C2×#39411yescharacteristic
520C22#254411yescharacteristic
521C22#254511yescharacteristic
522C22#254711yescharacteristic
523C22#254811yescharacteristic
524C22#262011yescharacteristic
525C22#285011yescharacteristic
526C22#304512yes—
527C22#304512yes—
528C22#304612yes—
529C22#304612yes—
530C22#306012yes—
531C22#306112yes—
532C22#313612yes—
533C22#313712yes—
534C22#317812yes—
535C22#317912yes—
536C22#322011yescharacteristic
537C22#322111yescharacteristic
538C22#322211yescharacteristic
539C22#322311yescharacteristic
540C22#324312yes—
541C22#324412yes—
542C4—28no—
543C4—44no—
544C4—82no—
545C4—81no—
546C4—81no—
547C4—84no—
548C4—161no—
549C4—161no—
550C4—162no—
551C4#240712yes—
552C4#244712yes—
553C4#244812yes—
554C4#259212yes—
555C2—162no—
556C2#1521711yescharacteristic
557C2#2354312yes—
558C2#2862911yescharacteristic
559C2#2863011yescharacteristic
560C2#2866812yes—
561C2#2873012yes—
562C2#2876011yescharacteristic
563C2#2876111yescharacteristic
564C2#2984911yescharacteristic
565C2#2985011yescharacteristic
566C2#3304712yes—
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×#1992C2semidirect
4. C2×#1768C2semidirect
5. #14283C2semidirect
7. #11190C2semidirect
9. #11077C2semidirect
11. #10841C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4882
C2×#19922
C2×#19592
C2×#17682
#142832
#142732
#111902
#111892
#110772
#110762
#108412