Finite Group Databasebeta

← #366773

Subgroups — #366773

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic
2C2×#1872C211yesmaximal, characteristic
3C2×#1852C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5C2×#1749C211yesmaximal, characteristic
6#14702C211yesmaximal, characteristic
7C2×C4⋊#95C211yesmaximal, characteristic
8#11378C212yesmaximal, complemented
9#11124C212yesmaximal, complemented
10#10928C212yesmaximal, complemented
11#10863C212yesmaximal, complemented
12C22×C22⋊Q8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C2×#483—24no—
17C2×#483—24no—
18C2×#483C418yescomplemented
19C22×#107C2212yes—
20C22×#106C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C22×#97C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C22×#95—22no—
30#1872—24no—
31#1852—24no—
32#1823—24no—
33#1749—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39#1436—22no—
40C4⋊#95—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—22no—
46C23×Q8C2311yescharacteristic
47C24×C4—21no—
48C24×C4—21no—
49C24×C4D412yes—
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8C2×C414yes—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4C2×C414yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C42—22no—
80C22×C42—22no—
81C22×C42—24no—
82C22×C42—42no—
83C22×C42D412yes—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#483—232no—
87C2×#107—24no—
88C2×#107—24no—
89C2×#107—42no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#106C2×C418yescomplemented
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—42no—
98C2×#103C2×C418yescomplemented
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8C2×C414yes—
102C2×C22⋊C8D414yes—
103C2×C22⋊C8D414yes—
104C2×C22⋊C8D414yes—
105C2×#97—24no—
106C2×#97—24no—
107C2×#97—42no—
108#391—24no—
109#391—42no—
110#380—22no—
111#380—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—24no—
117C2×#95—24no—
118C2×#95—21no—
119C2×#95—48no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#336—24no—
125C25C2×D411yescharacteristic
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8C22×C412yes—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—44no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4C22×C412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4#4112yes—
171C23×C4#4112yes—
172C23×C4C2411yescharacteristic
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8C22×C414yes—
184C22×C8C2×D414yes—
185C22×C8C2×D414yes—
186C22×C8C2×D414yes—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C22⋊Q8—44no—
190C22⋊Q8—44no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—48no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4C22×C414yes—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4#4114yes—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—48no—
247C2×C42—48no—
248C2×C42—42no—
249C2×C42#4114yes—
250#107—48no—
251#106—216no—
252#103—216no—
253#103—48no—
254C22⋊C8—28no—
255C22⋊C8C22⋊C4116yes—
256#97—48no—
257#95—44no—
258#95—48no—
259#95—48no—
260#95—44no—
261#95—44no—
262#95—42no—
263#95—44no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24C4×D412yes—
271C24#12112yes—
272C24#12712yes—
273C24AGL(1,ℤ/8)14yes—
274C24#13714yes—
275C24C22×D411yescharacteristic
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—42no—
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—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—216no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—48no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—416no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—48no—
352C22×C4—416no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—84no—
364C22×C4—88no—
365C22×C4—88no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4C2×C22⋊C414yes—
370C22×C4C4×D414yes—
371C22×C4C4×D412yes—
372C22×C4#12114yes—
373C22×C4#12112yes—
374C22×C4#12314yes—
375C22×C4#12414yes—
376C22×C4#12712yes—
377C22×C4#13714yes—
378C22×C4C23×C411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4112yes—
384C22×C4C2×#4112yes—
385C22×C4C2×#4111yescharacteristic
386C22×C4H(5,𝔽2)12yes—
387C22×C42−1+412yes—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8C2×C22⋊C418yes—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—48no—
409C4⋊C4—42no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C22⋊C4—48no—
422C22⋊C4—84no—
423C42—24no—
424C42—24no—
425C42—48no—
426C42—42no—
427C42—42no—
428C23—22no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—22no—
443C23—22no—
444C23—24no—
445C23—22no—
446C23#39014yes—
447C23#44018yes—
448C23#46818yes—
449C23#49614yes—
450C23#49714yes—
451C23C2×C4×D411yescharacteristic
452C23C2×#12111yescharacteristic
453C23C2×#12311yescharacteristic
454C23#52511yescharacteristic
455C23C2×#12711yescharacteristic
456C23#53411yescharacteristic
457C23#53611yescharacteristic
458C23C2×AGL(1,ℤ/8)12yes—
459C23C2×#13712yes—
460C23C2×#13712yes—
461C23#57512yes—
462Q8—44no—
463Q8—44no—
464Q8—82no—
465Q8—82no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—432no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—48no—
493C2×C4—48no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—816no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—84no—
510C2×C4—816no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—84no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4#39014yes—
526C2×C4C22×C22⋊C411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C4×#4112yes—
529C2×C4#52012yes—
530C2×C4C2×#12111yescharacteristic
531C2×C4C2×#12411yescharacteristic
532C2×C4#52912yes—
533C2×C4#53912yes—
534C2×C4#54612yes—
535C2×C4#54612yes—
536C2×C4#54612yes—
537C2×C4#54712yes—
538C2×C4#54712yes—
539C2×C4#54712yes—
540C2×C4#55112yes—
541C2×C4#55212yes—
542C8—28no—
543C8—44no—
544C22—24no—
545C22—24no—
546C22—28no—
547C22—28no—
548C22—28no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—28no—
553C22—24no—
554C22#197318yes—
555C22C2×#39011yescharacteristic
556C22C4×#12311yescharacteristic
557C22#232211yescharacteristic
558C22#241211yescharacteristic
559C22#241311yescharacteristic
560C22#250111yescharacteristic
561C22#253911yescharacteristic
562C22C2×#44012yes—
563C22C4×#13712yes—
564C22C2×#46812yes—
565C22#305612yes—
566C22#309212yes—
567C22#309512yes—
568C22#311312yes—
569C22#311412yes—
570C22C2×#49611yescharacteristic
571C22C2×#49711yescharacteristic
572C22#315211yescharacteristic
573C22#315311yescharacteristic
574C22#319412yes—
575C22#319512yes—
576C22#323612yes—
577C22#323812yes—
578C4—24no—
579C4—44no—
580C4—44no—
581C4—84no—
582C4—82no—
583C4—84no—
584C4—82no—
585C4—161no—
586C4—161no—
587C4#233912yes—
588C4#235912yes—
589C4#242512yes—
590C4#248812yes—
591C2—28no—
592C2#1399611yescharacteristic
593C2C2×#197311yescharacteristic
594C2#2123711yescharacteristic
595C2#2157912yes—
596C2#2208412yes—
597C2#2455912yes—
598C2#2460912yes—
599C2#2791411yescharacteristic
600C2#2791911yescharacteristic
601C2#2831211yescharacteristic
602C2#2831311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11378C2semidirect
9. #11124C2semidirect
10. #10928C2semidirect
11. #10863C2semidirect
18. C2×#483C4semidirect
92. C2×#106C2×C4semidirect
98. C2×#103C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
C2×#18722
C2×#18522
C2×#18232
C2×#17492
#147022
C2×C4⋊#952
#113782
#111242
#109282
#108632