Finite Group Databasebeta

← #354259

Subgroups — #354259

Order 512.

2232 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#440C211yesmaximal, characteristic
2C2×C42×Q8C211yesmaximal, characteristic, complemented
3C2×#1880C212yesmaximal
4C2×#1763C211yesmaximal, characteristic, complemented
5C2×#1762C211yesmaximal, characteristic
6C4×#390C211yesmaximal, characteristic
7#10931C212yesmaximal
8#10907C212yesmaximal, complemented
9#10789C212yesmaximal, complemented
10#10703C211yesmaximal, characteristic, complemented
11#10693C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#440—24no—
17C2×#440—24no—
18C2×#440C418yescomplemented
19C22×#107C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2212yes—
22C22×#102C2211yescharacteristic
23C22×#102C2212yes—
24C22×#97C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C2×#390C2212yes—
27C42×Q8—24no—
28C42×Q8—24no—
29C2×C4×C4⋊C4—22no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4—22no—
35C2×C43—21no—
36C2×C43C2211yescharacteristic
37#1880C4116yescomplemented
38#1763—24no—
39#1762—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1437—21no—
47#1437—21no—
48C4⋊#95C2212yes—
49C23×Q8C2311yescharacteristic
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C22×SD16—22no—
53C22×SD16—22no—
54C22×SD16—24no—
55C22×SD16C2×C418yes—
56C22×SD16C2×C414yes—
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C2×C4×Q8—24no—
61C2×C4×Q8—24no—
62C2×C4×Q8—24no—
63C2×C4×Q8—24no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—24no—
67C2×C4×Q8—22no—
68C2×C4×Q8C2×C414yes—
69C2×C4×Q8C2×C418yescomplemented
70C2×C4×Q8D414yes—
71C2×C4×Q8D414yes—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4C2×C414yes—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C42—22no—
88C22×C42—21no—
89C22×C42—22no—
90C22×C42—21no—
91C22×C42—21no—
92C22×C42D411yescharacteristic
93C22×C42D412yes—
94C22×C42D411yescharacteristic
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97#440—232no—
98C2×#107—22no—
99C2×#107—22no—
100C2×#107C2×C414yes—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103C2×C414yes—
106C2×#103C2×C418yescomplemented
107C2×#102—22no—
108C2×#102—22no—
109C2×#102—24no—
110C2×#102—24no—
111C2×#102C2×C418yes—
112C2×#102C2×C414yes—
113C2×#97—22no—
114C2×#97—22no—
115C2×#97—22no—
116C2×#97—22no—
117C2×#97—22no—
118C2×#97C2×C414yes—
119#390—24no—
120#390C2×C418yes—
121C4×C4⋊C4—28no—
122C4×C4⋊C4—216no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4D414yes—
129C4×C22⋊C4—48no—
130C2×#95—22no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—21no—
135C43—28no—
136C43—22no—
137C43—22no—
138#340—24no—
139#340—24no—
140#340—24no—
141#340C2×C418yescomplemented
142#336—24no—
143#336C2×C418yescomplemented
144C25—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8C22×C412yes—
151C22×Q8C22×C414yes—
152C22×Q8C2×D412yes—
153C22×Q8C2×D412yes—
154C22×Q8#4114yes—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—21no—
162C22×D4—22no—
163C22×D4C22×C412yes—
164C22×D4C22×C414yes—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—24no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2411yescharacteristic
180C2×SD16—28no—
181C2×SD16—28no—
182C2×SD16—216no—
183C2×SD16—216no—
184C2×SD16C42116yescomplemented
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8C22×C414yes—
199C22×C8C22×C412yes—
200C22×C8C22×C414yes—
201C22×C8C22×C412yes—
202C4×Q8—216no—
203C4×Q8—216no—
204C4×Q8—216no—
205C4×Q8—216no—
206C4×Q8—28no—
207C4×Q8—28no—
208C4×Q8—28no—
209C4×Q8C22⋊C4116yescomplemented
210C4×D4—28no—
211C4×D4—28no—
212C4×D4—44no—
213C4×D4—44no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
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—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C414yes—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C414yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4#4114yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C42—28no—
254C2×C42—28no—
255C2×C42—24no—
256C2×C42—28no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—28no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42—42no—
277C2×C42C22×C414yes—
278C2×C42C22×C412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42#4114yes—
283C2×C42#4112yes—
284C2×C42#4112yes—
285#107—28no—
286#103—28no—
287#103—216no—
288#102—28no—
289#102—216no—
290#97—28no—
291#97—44no—
292#95—28no—
293#95—24no—
294#95—44no—
295#95—42no—
296#95—44no—
297C24—24no—
298C24—22no—
299C24—41no—
300C24—41no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24C22×D411yescharacteristic
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—22no—
310C2×Q8—24no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8C2×C4214yes—
321C2×Q8C2×C22⋊C414yes—
322C2×Q8#11714yes—
323C2×Q8C4×D418yes—
324C2×Q8C4×D418yes—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4C2×C4214yes—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—416no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—48no—
388C22×C4—88no—
389C22×C4—84no—
390C22×C4—84no—
391C22×C4C4×D414yes—
392C22×C4C4×D412yes—
393C22×C4C4×D412yes—
394C22×C4C4×D414yes—
395C22×C4#12012yes—
396C22×C4#12012yes—
397C22×C4#12112yes—
398C22×C4#12112yes—
399C22×C4AGL(1,ℤ/8)14yes—
400C22×C4#13714yes—
401C22×C4C23×C411yescharacteristic
402C22×C4C23×C412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C2×#4112yes—
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4112yes—
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4112yes—
411SD16—232no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8C2×C4214yes—
429C4⋊C4—232no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—216no—
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⋊C4C2×C22⋊C414yes—
446C4⋊C4C4×D418yes—
447C22⋊C4—416no—
448C22⋊C4—48no—
449C22⋊C4—416no—
450C42—28no—
451C42—232no—
452C42—24no—
453C42—24no—
454C42—24no—
455C42—24no—
456C42—216no—
457C42—48no—
458C42—44no—
459C42—48no—
460C42—416no—
461C42—84no—
462C42C2×C22⋊C414yes—
463C42C4×D414yes—
464C42#12114yes—
465C42#12314yes—
466C23—24no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—42no—
474C23—42no—
475C23—44no—
476C23—44no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—81no—
482C23—81no—
483C23—82no—
484C23—82no—
485C23—81no—
486C23—81no—
487C23C2×C4×D412yes—
488C23C2×C4×D411yescharacteristic
489C23C2×#12011yescharacteristic
490C23C2×#12111yescharacteristic
491C23#52512yes—
492C23C2×AGL(1,ℤ/8)12yes—
493C23C2×#13712yes—
494C23#57512yes—
495C23#57512yes—
496Q8—24no—
497Q8—24no—
498Q8—24no—
499Q8—28no—
500Q8—28no—
501Q8C4×C22⋊C418yescomplemented
502D4—24no—
503D4—24no—
504D4—42no—
505D4—42no—
506D4—44no—
507D4—44no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—816no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4#38614yes—
560C2×C4#38614yes—
561C2×C4#44018yescomplemented
562C2×C4C22×C4211yescharacteristic
563C2×C4C22×C22⋊C411yescharacteristic
564C2×C4C2×#11711yescharacteristic
565C2×C4C2×C4×D412yes—
566C2×C4C2×C4×D412yes—
567C2×C4C2×C4×D411yescharacteristic
568C2×C4C2×C4×D412yes—
569C2×C4C4×#4111yescharacteristic
570C2×C4C4×#4112yes—
571C2×C4C4×#4111yescharacteristic
572C2×C4C4×#4112yes—
573C2×C4C2×#12111yescharacteristic
574C2×C4C2×#12311yescharacteristic
575C2×C4#52512yes—
576C2×C4#52512yes—
577C2×C4#52912yes—
578C2×C4#52912yes—
579C2×C4#53212yes—
580C8—24no—
581C8—42no—
582C8—44no—
583C22—48no—
584C22—44no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22—82no—
589C22—82no—
590C22—84no—
591C22—84no—
592C22C42×D411yescharacteristic
593C22C2×#38611yescharacteristic
594C22#229011yescharacteristic
595C22C4×#12012yes—
596C22C4×#12112yes—
597C22C2×#44012yes—
598C22#294312yes—
599C22C4×AGL(1,ℤ/8)12yes—
600C22C4×AGL(1,ℤ/8)12yes—
601C22C4×#13712yes—
602C22C4×#13712yes—
603C22#300411yescharacteristic
604C22#300511yescharacteristic
605C22#300612yes—
606C22#303712yes—
607C22#303811yescharacteristic
608C22#304011yescharacteristic
609C22#311112yes—
610C22#311212yes—
611C22#311512yes—
612C22#311712yes—
613C4—24no—
614C4—24no—
615C4—28no—
616C4—28no—
617C4—44no—
618C4—44no—
619C4—48no—
620C4—84no—
621C4—82no—
622C4C2×C4×C22⋊C411yescharacteristic
623C4C42×D411yescharacteristic
624C4#229011yescharacteristic
625C4#229711yescharacteristic
626C4C4×#12112yes—
627C4C4×#12312yes—
628C2—84no—
629C2C4×#38611yescharacteristic
630C2C4×#44012yes—
631C2#2092611yescharacteristic
632C2#2092811yescharacteristic
633C2#2112412yes—
634C2#2145412yes—
635C2#2145612yes—
636C2#2150512yes—
637C2#2150812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C42×Q8C2semidirect
4. C2×#1763C2semidirect
8. #10907C2semidirect
9. #10789C2semidirect
10. #10703C2semidirect
11. #10693C2semidirect
18. C2×#440C4semidirect
37. #1880C4semidirect
69. C2×C4×Q8C2×C4semidirect
106. C2×#103C2×C4semidirect
141. #340C2×C4semidirect
143. #336C2×C4semidirect
184. C2×SD16C42semidirect
209. C4×Q8C22⋊C4semidirect
501. Q8C4×C22⋊C4semidirect
561. C2×C4#440semidirect

Maximal subgroups

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

subgroupindex
C22×#4402
C2×C42×Q82
C2×#18802
C2×#17632
C2×#17622
C4×#3902
#109312
#109072
#107892
#107032
#106932