Finite Group Databasebeta

← #415021

Subgroups — #415021

Order 512.

1416 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C2×#1989C211yesmaximal, characteristic, complemented
3C2×#1843C211yesmaximal, characteristic, complemented
4C2×#1802C211yesmaximal, characteristic
5#14283C211yesmaximal, characteristic, complemented
6#13785C211yesmaximal, characteristic
7#11232C212yesmaximal, complemented
8#11231C212yesmaximal, complemented
9#11076C212yesmaximal
10#11076C212yesmaximal
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⋊C8C2212yes—
21C22×C4⋊C8C2212yes—
22C22×C4⋊C8C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2212yes—
26C2×#390C2212yes—
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#1989—24no—
31#1843—24no—
32#1802—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
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—24no—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—24no—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—42no—
96C2×C4⋊C8—42no—
97C2×C4⋊C8D414yes—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—24no—
105C2×#102—42no—
106C2×#102—42no—
107#390—24no—
108#390Q818yes—
109#384—22no—
110#384—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113#376—22no—
114#376—22no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95D411yescharacteristic
123C2×#95D411yescharacteristic
124C2×#95D412yes—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—42no—
129#340—42no—
130C25—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4C2×D412yes—
140C22×D4C2×Q814yes—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—82no—
150C23×C4—82no—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
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×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173#125—28no—
174#125—28no—
175#125—44no—
176#125—44no—
177C4×D4—28no—
178C4×D4—28no—
179C4×D4—44no—
180C4×D4—44no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—81no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—81no—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×Q814yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4—82no—
230C2×C22⋊C4—82no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—42no—
236C2×C42—81no—
237C2×C42—81no—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×Q814yes—
242C2×C42#4114yes—
243C2×C42#4112yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246#107—28no—
247#106—28no—
248C4⋊C8—28no—
249C4⋊C8—48no—
250C4⋊C8—48no—
251C4⋊C8—44no—
252#102—28no—
253#102—28no—
254#102—48no—
255#95—24no—
256#95—22no—
257#95—22no—
258#95—42no—
259#95—82no—
260#95—82no—
261#95—82no—
262#95—82no—
263#95—82no—
264#95—82no—
265C24—22no—
266C24—24no—
267C24—41no—
268C24—41no—
269C24—42no—
270C24—41no—
271C24—42no—
272C24—41no—
273C24—42no—
274C24—41no—
275C24—41no—
276C24C22×D411yescharacteristic
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—24no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4C4⋊Q814yes—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
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—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—48no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4—88no—
348C22×C4—88no—
349C22×C4#12112yes—
350C22×C4#12112yes—
351C22×C4#12112yes—
352C22×C4#12112yes—
353C22×C4#12112yes—
354C22×C4#12112yes—
355C22×C4C22⋊Q814yes—
356C22×C4C22⋊Q814yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×Q812yes—
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4112yes—
365C22×C4H(5,𝔽2)12yes—
366C22×C42−1+412yes—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—84no—
407C4⋊C4—84no—
408C22⋊C4—48no—
409C22⋊C4—88no—
410C22⋊C4—88no—
411C42—24no—
412C42—24no—
413C42—24no—
414C42—24no—
415C42—84no—
416C42#12014yes—
417C42#12314yes—
418C42#12314yes—
419C42#12414yes—
420C23—24no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—42no—
426C23—42no—
427C23—44no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—44no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—82no—
443C23C2×#12111yescharacteristic
444C23C2×#12111yescharacteristic
445C23C2×#12111yescharacteristic
446C23C2×C22⋊Q812yes—
447C23#53712yes—
448C23#57611yescharacteristic
449C23#57612yes—
450C23#57711yescharacteristic
451C23#57712yes—
452C23#57711yescharacteristic
453C23#57811yescharacteristic
454D4—24no—
455D4—24no—
456D4—44no—
457D4—42no—
458D4—44no—
459D4—42no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
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—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—816no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4#39414yes—
517C2×C4#39414yes—
518C2×C4C2×#12011yescharacteristic
519C2×C4C2×#12311yescharacteristic
520C2×C4C2×#12311yescharacteristic
521C2×C4C2×#12411yescharacteristic
522C2×C4C2×C4⋊Q811yescharacteristic
523C2×C4#53211yescharacteristic
524C2×C4#53312yes—
525C2×C4#53511yescharacteristic
526C2×C4#53611yescharacteristic
527C2×C4#54212yes—
528C2×C4#54612yes—
529C2×C4D4×Q812yes—
530C2×C4D4×Q812yes—
531C2×C4#55012yes—
532C2×C4#55212yes—
533C2×C4#55412yes—
534C2×C4#55412yes—
535C2×C4#55612yes—
536C8—42no—
537C8—42no—
538C8—82no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22—84no—
543C22—84no—
544C22—84no—
545C22—84no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22C2×#39411yescharacteristic
551C22#241712yes—
552C22#253812yes—
553C22#262011yescharacteristic
554C22#285711yescharacteristic
555C22#304111yescharacteristic
556C22#304211yescharacteristic
557C22#304511yescharacteristic
558C22#304512yes—
559C22#304511yescharacteristic
560C22#304611yescharacteristic
561C22#304611yescharacteristic
562C22#306011yescharacteristic
563C22#306111yescharacteristic
564C22#306212yes—
565C22#308112yes—
566C22#308112yes—
567C22#308312yes—
568C22#308412yes—
569C22#325712yes—
570C22#325812yes—
571C22#325912yes—
572C22#326012yes—
573C4—24no—
574C4—24no—
575C4—44no—
576C4—44no—
577C4—84no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—162no—
582C4#242312yes—
583C4#243111yescharacteristic
584C4#243411yescharacteristic
585C4#249212yes—
586C4#249911yescharacteristic
587C4#250211yescharacteristic
588C2—84no—
589C2#1518211yescharacteristic
590C2#2354311yescharacteristic
591C2#2354411yescharacteristic
592C2#2484812yes—
593C2#2485212yes—
594C2#2829212yes—
595C2#2829312yes—
596C2#2983411yescharacteristic
597C2#2983611yescharacteristic
598C2#3320311yescharacteristic
599C2#3320511yescharacteristic
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×#1843C2semidirect
5. #14283C2semidirect
7. #11232C2semidirect
8. #11231C2semidirect
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×#18432
C2×#18022
#142832
#137852
#112322
#112312
#110762
#108492