Finite Group Databasebeta

← #369232

Subgroups — #369232

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×Q8C211yesmaximal, characteristic, complemented
2C2×#2592C211yesmaximal, characteristic
3C2×#1983C211yesmaximal, characteristic
4C2×#1958C211yesmaximal, characteristic
5#13925C211yesmaximal, characteristic, complemented
6#11077C212yesmaximal, complemented
7#11076C212yesmaximal
8#10926C212yesmaximal, complemented
9#10925C212yesmaximal
10#10850C211yesmaximal, characteristic, complemented
11C4×#340C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×C8×Q8—24no—
15C2×C8×Q8—24no—
16C2×C8×Q8—24no—
17C22×C4⋊C8C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2212yes—
24C22×#102C2211yescharacteristic
25C22×C4×C8C2212yes—
26C22×C4×C8C2211yescharacteristic
27#2592—24no—
28#2592C418yescomplemented
29C2×#397—21no—
30C2×#393—21no—
31C2×#391C2212yes—
32C2×#390C2212yes—
33C2×#388—22no—
34C2×#382C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1983—24no—
38#1958—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1446—21no—
44#1445—21no—
45#1436—22no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×Q8D411yescharacteristic
49C23×Q8C2311yescharacteristic
50C23×D4C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C2×#124—22no—
55C2×#124—22no—
56C2×#124—22no—
57C2×#124—22no—
58C2×#124—22no—
59C2×#124C2×C414yes—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8C2×C414yes—
65C2×C4×Q8D414yes—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D412yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80C8×Q8—232no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—24no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8D414yes—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—42no—
93C2×#103—42no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—42no—
100C2×#102—42no—
101C2×C4×C8—24no—
102C2×C4×C8—24no—
103C2×C4×C8—24no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8D414yes—
107#397—28no—
108#393—28no—
109#391—24no—
110#391C2×C418yes—
111#390—24no—
112#390C2×C418yes—
113#388—216no—
114#382—22no—
115#382—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C414yes—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130#340—24no—
131#340—24no—
132#340—42no—
133#340—42no—
134C25—21no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—24no—
138C22×Q8—24no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8C22⋊C414yes—
149C22×Q8C22×C412yes—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4C22×C412yes—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2411yescharacteristic
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8C2×D414yes—
184C22×C8C2×D412yes—
185C22×C8C2×D412yes—
186C22×C8#4114yes—
187#124—28no—
188#124—28no—
189#124—44no—
190#124—44no—
191C4×Q8—28no—
192C4×Q8—28no—
193C4×Q8—28no—
194C4×Q8—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
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—22no—
206C2×C4⋊C4—22no—
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⋊C4C22⋊C418yes—
218C2×C4⋊C4C22⋊C414yes—
219C2×C4⋊C4C22×C414yes—
220C2×C4⋊C4#4114yes—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—82no—
232C2×C22⋊C4—82no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—42no—
239C2×C42—81no—
240C2×C42—81no—
241C2×C42—81no—
242C2×C42—81no—
243C2×C42C22×C414yes—
244C2×C42C22×C412yes—
245C2×C42C2×D414yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251C4⋊C8—216no—
252C4⋊C8—28no—
253#103—48no—
254#103—44no—
255#103—44no—
256#102—48no—
257#102—44no—
258#102—44no—
259C4×C8—216no—
260C4×C8—28no—
261#95—24no—
262#95—28no—
263#95—28no—
264#95—42no—
265#95—42no—
266#95—42no—
267#95—44no—
268#95—44no—
269#95—82no—
270#95—82no—
271C24—22no—
272C24—41no—
273C24—41no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24C22×D411yescharacteristic
283C2×Q8—28no—
284C2×Q8—28no—
285C2×Q8—216no—
286C2×Q8—28no—
287C2×Q8—216no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—41no—
297C2×Q8—42no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—42no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4C2×C22⋊C414yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C414yes—
373C22×C4#12112yes—
374C22×C4#12112yes—
375C22×C4#12112yes—
376C22×C4#12112yes—
377C22×C4#13514yes—
378C22×C4#13514yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C42−1+412yes—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8#12414yes—
406C4⋊C4—216no—
407C4⋊C4—28no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—416no—
425C22⋊C4—88no—
426C42—24no—
427C42—24no—
428C42—24no—
429C42—24no—
430C42—84no—
431C42—84no—
432C42C4×D414yes—
433C42C4×D414yes—
434C42#12014yes—
435C42#12314yes—
436C23—44no—
437C23—44no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—81no—
451C23—81no—
452C23—82no—
453C23—81no—
454C23—81no—
455C23—81no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23C22×C22⋊C411yescharacteristic
460C23C2×#12111yescharacteristic
461C23C2×#12111yescharacteristic
462C23#54812yes—
463C23#55012yes—
464C23C2×#13512yes—
465C23C2×#13512yes—
466C23#57611yescharacteristic
467C23#57711yescharacteristic
468C23#57711yescharacteristic
469C23#57811yescharacteristic
470Q8—232no—
471Q8—42no—
472Q8—44no—
473Q8—42no—
474Q8—81no—
475Q8—81no—
476Q8—82no—
477D4—42no—
478D4—44no—
479D4—42no—
480D4—81no—
481D4—81no—
482D4—82no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4#38614yes—
536C2×C4#38614yes—
537C2×C4#41618yes—
538C2×C4C2×C4×D411yescharacteristic
539C2×C4C2×C4×D411yescharacteristic
540C2×C4C4×#4112yes—
541C2×C4#52012yes—
542C2×C4C2×#12011yescharacteristic
543C2×C4C2×#12311yescharacteristic
544C2×C4C2×#12411yescharacteristic
545C2×C4#52912yes—
546C2×C4#53211yescharacteristic
547C2×C4#53511yescharacteristic
548C2×C4#53611yescharacteristic
549C2×C4#54212yes—
550C2×C4#54612yes—
551C2×C4#54812yes—
552C2×C4#54812yes—
553C2×C4#55012yes—
554C2×C4#55612yes—
555C2×C4#55912yes—
556C8—24no—
557C8—42no—
558C8—44no—
559C22—84no—
560C22—84no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—161no—
565C22—161no—
566C22—162no—
567C22—162no—
568C22—161no—
569C22—161no—
570C22C2×#38611yescharacteristic
571C22#234112yes—
572C22#242612yes—
573C22#254511yescharacteristic
574C22#254811yescharacteristic
575C22C2×#41612yes—
576C22#289711yescharacteristic
577C22#289811yescharacteristic
578C22#305011yescharacteristic
579C22#305011yescharacteristic
580C22#305012yes—
581C22#306011yescharacteristic
582C22#306111yescharacteristic
583C22#306211yescharacteristic
584C22#306311yescharacteristic
585C22#336012yes—
586C22#336112yes—
587C22#336412yes—
588C22#336512yes—
589C22#339312yes—
590C22#339412yes—
591C22#339412yes—
592C22#339612yes—
593C4—24no—
594C4—28no—
595C4—24no—
596C4—44no—
597C4—84no—
598C4—82no—
599C4—84no—
600C4—161no—
601C4—161no—
602C4C4×#12411yescharacteristic
603C4#232611yescharacteristic
604C4#236712yes—
605C4#243111yescharacteristic
606C4#243611yescharacteristic
607C4#244812yes—
608C2—162no—
609C2#1402611yescharacteristic
610C2#2096711yescharacteristic
611C2#2097611yescharacteristic
612C2#2254112yes—
613C2#2254312yes—
614C2#2529612yes—
615C2#2529712yes—
616C2#2864111yescharacteristic
617C2#2864211yescharacteristic
618C2#2873911yescharacteristic
619C2#2874011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×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
1. C22×C8×Q8C2semidirect
5. #13925C2semidirect
6. #11077C2semidirect
8. #10926C2semidirect
10. #10850C2semidirect
11. C4×#340C2semidirect
28. #2592C4semidirect

Maximal subgroups

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

subgroupindex
C22×C8×Q82
C2×#25922
C2×#19832
C2×#19582
#139252
#110772
#110762
#109262
#109252
#108502
C4×#3402