Finite Group Databasebeta

← #397169

Subgroups — #397169

Order 512.

1824 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#467C211yesmaximal, characteristic
2C2×#2056C211yesmaximal, characteristic, complemented
3C2×#1872C212yesmaximal
4C2×#1850C211yesmaximal, characteristic
5C22×#340C211yesmaximal, characteristic
6#14139C212yesmaximal
7#11256C211yesmaximal, characteristic, complemented
8#11232C211yesmaximal, characteristic, complemented
9#11190C212yesmaximal, complemented
10#11077C212yesmaximal, complemented
11#11024C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C23×C4⋊C4C2212yes—
16C2×#467—24no—
17C2×#467—28no—
18C22×#107C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#2056—24no—
29#1872—28no—
30#1850—24no—
31C2×#340—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1441—22no—
40#1441—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C22⋊#95—22no—
44C22⋊#95—22no—
45C23×Q8C2312yes—
46C24×C4—22no—
47C24×C4C2311yescharacteristic
48C22×Q16—24no—
49C22×Q16—22no—
50C22×Q16—22no—
51C22×Q16—22no—
52C22×Q16—42no—
53C23×C8—24no—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—42no—
57C23×C8D414yes—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—24no—
65C2×C22⋊Q8—24no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8D418yes—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4—42no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D411yescharacteristic
81C22×C22⋊C4D411yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#467—232no—
85C2×#107—22no—
86C2×#107—24no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—42no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—42no—
100C2×#103—42no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—24no—
103#391—24no—
104#391—24no—
105#391—42no—
106#391—42no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D411yescharacteristic
118C2×#95D411yescharacteristic
119#340—22no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340D414yes—
124#336—22no—
125#336—22no—
126C25C2×D411yescharacteristic
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8C2×D414yes—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—82no—
157C23×C4—82no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D414yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4#4112yes—
166C23×C4#4114yes—
167C23×C4C2411yescharacteristic
168C2×Q16—28no—
169C2×Q16—48no—
170C2×Q16—48no—
171C2×Q16—44no—
172C2×Q16—44no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—28no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—28no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—22no—
190C22×C8—21no—
191C22×C8—44no—
192C22×C8—48no—
193C22×C8—48no—
194C22×C8C2×D412yes—
195C22×C8C2×D412yes—
196C22×C8C2×D412yes—
197C22×C8#4112yes—
198C22⋊Q8—216no—
199C22⋊Q8—216no—
200C22⋊Q8—48no—
201C22⋊Q8—48no—
202C2×C4⋊C4—216no—
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—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—48no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4#4114yes—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4Q1614yescomplemented
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—21no—
248#107—28no—
249C4⋊C8—44no—
250C4⋊C8—44no—
251#103—216no—
252#103—48no—
253#103—48no—
254C22⋊C8—44no—
255#95—22no—
256#95—22no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—82no—
262#95—82no—
263#95—82no—
264#95—82no—
265C24—21no—
266C24—22no—
267C24—21no—
268C24—21no—
269C24—22no—
270C24—21no—
271C24#12114yes—
272C24#12112yes—
273C24C2×Q1612yes—
274C24#13512yes—
275C24#13714yes—
276C24C22×D411yescharacteristic
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—416no—
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—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—88no—
373C22×C4—88no—
374C22×C4—88no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4#12014yes—
379C22×C4#12112yes—
380C22×C4#12114yes—
381C22×C4#12318yes—
382C22×C4#12414yes—
383C22×C4C2×Q1612yes—
384C22×C4C2×Q1612yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D412yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4112yes—
391C22×C4C2×#4112yes—
392C22×C4H(5,𝔽2)12yes—
393C22×C4H(5,𝔽2)11yescharacteristic
394C22×C42−1+411yescharacteristic
395Q16—416no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—416no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8#12114yes—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—416no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—416no—
431C22⋊C4—28no—
432C22⋊C4—416no—
433C22⋊C4—44no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—42no—
441C23—22no—
442C23—22no—
443C23—24no—
444C23—22no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—22no—
449C23—22no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—22no—
458C23—22no—
459C23#39414yes—
460C23#46218yes—
461C23#46418yes—
462C23#46714yes—
463C23#47014yes—
464C23C2×#12111yescharacteristic
465C23C2×#12112yes—
466C23D4211yescharacteristic
467C23#54612yes—
468C23#54711yescharacteristic
469C23C22×Q1611yescharacteristic
470C23C2×#13511yescharacteristic
471C23C2×#13712yes—
472C23#57611yescharacteristic
473C23#57712yes—
474C23#57811yescharacteristic
475Q8—44no—
476Q8—44no—
477Q8—44no—
478Q8—44no—
479Q8—82no—
480Q8—82no—
481Q8—82no—
482Q8—82no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—832no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4#39414yes—
545C2×C4C2×#12011yescharacteristic
546C2×C4C2×#12111yescharacteristic
547C2×C4C2×#12312yes—
548C2×C4#52512yes—
549C2×C4C2×#12411yescharacteristic
550C2×C4#52912yes—
551C2×C4#53211yescharacteristic
552C2×C4#53411yescharacteristic
553C2×C4#53711yescharacteristic
554C2×C4#53812yes—
555C2×C4#53912yes—
556C2×C4D4212yes—
557C2×C4#54612yes—
558C2×C4#54612yes—
559C2×C4#54712yes—
560C2×C4#54811yescharacteristic
561C2×C4#55312yes—
562C2×C4#55311yescharacteristic
563C8—24no—
564C8—42no—
565C8—81no—
566C8—81no—
567C22—28no—
568C22—28no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—28no—
576C22—28no—
577C22#205618yescomplemented
578C22C2×#39411yescharacteristic
579C22#241611yescharacteristic
580C22#242412yes—
581C22#242512yes—
582C22#254311yescharacteristic
583C22C2×#46212yes—
584C22C2×#46412yes—
585C22#304512yes—
586C22#304612yes—
587C22C2×#46711yescharacteristic
588C22C2×#47011yescharacteristic
589C22#306011yescharacteristic
590C22#306111yescharacteristic
591C22#326612yes—
592C22#328011yescharacteristic
593C22D4×Q1611yescharacteristic
594C22#329612yes—
595C22#329912yes—
596C22#331212yes—
597C22#331412yes—
598C22#333611yescharacteristic
599C22#333811yescharacteristic
600C22#334412yes—
601C4—24no—
602C4—42no—
603C4—42no—
604C4—81no—
605C4—84no—
606C4—81no—
607C4—84no—
608C4—84no—
609C4—84no—
610C4#241011yescharacteristic
611C4#243011yescharacteristic
612C4#244312yes—
613C4#244412yes—
614C4#263911yescharacteristic
615C4#285011yescharacteristic
616C2—28no—
617C2#1474711yescharacteristic
618C2C2×#205611yescharacteristic
619C2#2354311yescharacteristic
620C2#2478111yescharacteristic
621C2#2478311yescharacteristic
622C2#2508212yes—
623C2#2508412yes—
624C2#2520612yes—
625C2#2520812yes—
626C2#2855311yescharacteristic
627C2#2856311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2056C2semidirect
7. #11256C2semidirect
8. #11232C2semidirect
9. #11190C2semidirect
10. #11077C2semidirect
11. #11024C2semidirect
237. C2×C22⋊C4Q16semidirect
577. C22#2056semidirect

Maximal subgroups

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

subgroupindex
C22×#4672
C2×#20562
C2×#18722
C2×#18502
C22×#3402
#141392
#112562
#112322
#111902
#110772
#110242