Finite Group Databasebeta

← #354264

Subgroups — #354264

Order 512.

2220 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#440C211yesmaximal, characteristic
2C2×#2275C211yesmaximal, characteristic
3C2×#1877C212yesmaximal
4C2×C4×#103C211yesmaximal, characteristic, complemented
5C2×C4×#102C211yesmaximal, characteristic
6#13786C211yesmaximal, characteristic, complemented
7#10932C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10788C212yesmaximal
10#10703C211yesmaximal, characteristic, complemented
11#10694C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2212yes—
15C22×C4×D4C2211yescharacteristic
16C2×#440—24no—
17C2×#440—24no—
18C2×#440C418yescomplemented
19C22×#107C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2212yes—
22C22×#102C2211yescharacteristic
23C22×#102C2212yes—
24C22×#97C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×#391C2212yes—
27#2275—24no—
28#2275—24no—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4—22no—
33C2×C4×C22⋊C4—21no—
34C2×#376C2211yescharacteristic
35C22×#95—21no—
36#1877C4116yescomplemented
37C4×#103—24no—
38C4×#102—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1436—21no—
46#1436—21no—
47C4⋊#95C2212yes—
48#1434—22no—
49C23×Q8C2311yescharacteristic
50C23×D4C2311yescharacteristic
51C24×C4—22no—
52C24×C4—21no—
53C22×SD16—22no—
54C22×SD16—22no—
55C22×SD16—24no—
56C22×SD16C2×C418yes—
57C22×SD16C2×C414yes—
58C23×C8C2312yes—
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8C2×C414yescomplemented
67C2×C4×D4—24no—
68C2×C4×D4—24no—
69C2×C4×D4—24no—
70C2×C4×D4—24no—
71C2×C4×D4—22no—
72C2×C4×D4—24no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4C2×C418yes—
76C2×C4×D4C2×C414yes—
77C2×C4×D4D414yes—
78C2×C4×D4D414yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—21no—
86C22×C42—21no—
87C22×C42—22no—
88C22×C42—21no—
89C22×C42—21no—
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92#440—232no—
93C2×#107—22no—
94C2×#107—22no—
95C2×#107C2×C414yescomplemented
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103C2×C418yescomplemented
101C2×#103C2×C414yescomplemented
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102C2×C418yes—
107C2×#102C2×C414yes—
108C2×#97—22no—
109C2×#97—22no—
110C2×#97C2×C414yescomplemented
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114#391—24no—
115#391C2×C418yescomplemented
116C4×C4⋊C4—24no—
117C4×C4⋊C4—24no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4D414yes—
122C4×C22⋊C4—216no—
123C4×C22⋊C4—28no—
124#376—22no—
125#376—22no—
126C2×#95—28no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95D412yes—
135C2×#95D411yescharacteristic
136C2×#95D411yescharacteristic
137#340—24no—
138#340—24no—
139#340—24no—
140#340C2×C418yescomplemented
141#336—24no—
142#336C2×C418yescomplemented
143C25—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8C22×C412yes—
153C22×Q8C22×C414yes—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4C22×C412yes—
160C22×D4C22×C414yes—
161C22×D4C2×D412yes—
162C22×D4C2×D412yes—
163C22×D4#4114yes—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—28no—
168C23×C4—28no—
169C23×C4—28no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—82no—
176C23×C4—82no—
177C23×C4—82no—
178C23×C4—82no—
179C23×C4—82no—
180C23×C4—82no—
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2411yescharacteristic
186C2×SD16—28no—
187C2×SD16—28no—
188C2×SD16—216no—
189C2×SD16—216no—
190C2×SD16C42116yescomplemented
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8C22×C414yes—
205C22×C8C22×C412yes—
206C22×C8C22×C414yes—
207C22×C8C22×C412yes—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—44no—
211C4×Q8—44no—
212C4×D4—216no—
213C4×D4—216no—
214C4×D4—216no—
215C4×D4—216no—
216C4×D4—28no—
217C4×D4—28no—
218C4×D4—28no—
219C4×D4C22⋊C4116yes—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C414yes—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C22×C414yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4#4114yes—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—28no—
259C2×C42—24no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—82no—
264C2×C42—82no—
265C2×C42—82no—
266C2×C42—82no—
267C2×C42C22×C414yes—
268C2×C42C22×C412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42#4114yes—
273C2×C42#4112yes—
274C2×C42#4112yes—
275#107—28no—
276#103—28no—
277#103—216no—
278#102—28no—
279#102—216no—
280#97—28no—
281C4×C8—44no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—22no—
286#95—24no—
287#95—24no—
288#95—22no—
289#95—42no—
290#95—82no—
291#95—82no—
292C24—22no—
293C24—24no—
294C24—22no—
295C24—22no—
296C24—24no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24C22×D411yescharacteristic
302C2×Q8—22no—
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8C2×C4214yes—
320C2×D4—22no—
321C2×D4—24no—
322C2×D4—22no—
323C2×D4—24no—
324C2×D4—22no—
325C2×D4—24no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4C2×C4214yes—
334C2×D4C2×C22⋊C414yes—
335C2×D4#11714yes—
336C2×D4C4×D418yes—
337C2×D4C4×D418yes—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—232no—
363C22×C4—216no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4—88no—
390C22×C4—88no—
391C22×C4—88no—
392C22×C4—88no—
393C22×C4—88no—
394C22×C4—88no—
395C22×C4C4×D414yes—
396C22×C4C4×D414yes—
397C22×C4C4×D412yes—
398C22×C4C4×D412yes—
399C22×C4#12012yes—
400C22×C4#12012yes—
401C22×C4#12112yes—
402C22×C4#12112yes—
403C22×C4C23×C411yescharacteristic
404C22×C4C23×C412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4112yes—
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4112yes—
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4112yes—
413SD16—232no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8C2×C4214yes—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—48no—
446C4⋊C4C2×C22⋊C414yes—
447C4⋊C4C4×D418yes—
448C22⋊C4—232no—
449C22⋊C4—216no—
450C42—24no—
451C42—24no—
452C42—24no—
453C42—24no—
454C42—48no—
455C42—84no—
456C42—84no—
457C42—88no—
458C42C2×C22⋊C414yes—
459C42C4×D414yes—
460C42#12114yes—
461C42#12314yes—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—28no—
466C23—28no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23C2×C4×D411yescharacteristic
484C23C2×C4×D412yes—
485C23C2×#12011yescharacteristic
486C23C2×#12111yescharacteristic
487C23#52512yes—
488C23#57612yes—
489C23#57712yes—
490C23#57712yes—
491C23#57812yes—
492Q8—24no—
493Q8—24no—
494Q8—42no—
495Q8—42no—
496Q8—44no—
497Q8—44no—
498D4—24no—
499D4—24no—
500D4—24no—
501D4—28no—
502D4—28no—
503D4C4×C22⋊C418yes—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—816no—
549C2×C4—832no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4#38614yes—
558C2×C4#38614yes—
559C2×C4C22×C4211yescharacteristic
560C2×C4C22×C22⋊C411yescharacteristic
561C2×C4C2×#11711yescharacteristic
562C2×C4C2×C4×D412yes—
563C2×C4C2×C4×D412yes—
564C2×C4C2×C4×D411yescharacteristic
565C2×C4C2×C4×D412yes—
566C2×C4C4×#4111yescharacteristic
567C2×C4C4×#4112yes—
568C2×C4C4×#4111yescharacteristic
569C2×C4C4×#4112yes—
570C2×C4C2×#12111yescharacteristic
571C2×C4C2×#12311yescharacteristic
572C2×C4#52512yes—
573C2×C4#52512yes—
574C2×C4#52912yes—
575C2×C4#52912yes—
576C2×C4#53212yes—
577C8—24no—
578C8—42no—
579C8—44no—
580C22—28no—
581C22—48no—
582C22—48no—
583C22—48no—
584C22—48no—
585C22—44no—
586C22—44no—
587C22—48no—
588C22—48no—
589C22C42×D411yescharacteristic
590C22C2×#38611yescharacteristic
591C22#229011yescharacteristic
592C22C4×#12012yes—
593C22C4×#12112yes—
594C22#294612yes—
595C22#294612yes—
596C22#294712yes—
597C22#294712yes—
598C22#294912yes—
599C22#295112yes—
600C22#300911yescharacteristic
601C22#301111yescharacteristic
602C22#301212yes—
603C22#304111yescharacteristic
604C22#304211yescharacteristic
605C22#304612yes—
606C22#311812yes—
607C22#312012yes—
608C22#312312yes—
609C22#312412yes—
610C4—24no—
611C4—24no—
612C4—44no—
613C4—44no—
614C4—44no—
615C4—48no—
616C4—44no—
617C4—84no—
618C4—82no—
619C4C2×C4×C22⋊C411yescharacteristic
620C4C42×D411yescharacteristic
621C4#229011yescharacteristic
622C4#229711yescharacteristic
623C4C4×#12112yes—
624C4C4×#12312yes—
625C2—48no—
626C2C4×#38611yescharacteristic
627C2#2050412yes—
628C2#2092911yescharacteristic
629C2#2093111yescharacteristic
630C2#2112512yes—
631C2#2145912yes—
632C2#2146512yes—
633C2#2150912yes—
634C2#2151412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C2×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
4. C2×C4×#103C2semidirect
6. #13786C2semidirect
7. #10932C2semidirect
8. #10907C2semidirect
10. #10703C2semidirect
11. #10694C2semidirect
18. C2×#440C4semidirect
36. #1877C4semidirect
66. C2×C4×Q8C2×C4semidirect
95. C2×#107C2×C4semidirect
100. C2×#103C2×C4semidirect
101. C2×#103C2×C4semidirect
110. C2×#97C2×C4semidirect
115. #391C2×C4semidirect
140. #340C2×C4semidirect
142. #336C2×C4semidirect
190. C2×SD16C42semidirect

Maximal subgroups

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

subgroupindex
C22×#4402
C2×#22752
C2×#18772
C2×C4×#1032
C2×C4×#1022
#137862
#109322
#109072
#107882
#107032
#106942