Finite Group Databasebeta

← #393359

Subgroups — #393359

Order 512.

2160 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C2×C42×D4C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic, complemented
4C2×#1804C211yesmaximal, characteristic
5#14709C211yesmaximal, characteristic, complemented
6#10908C212yesmaximal, complemented
7#10907C212yesmaximal, complemented
8#10850C211yesmaximal, characteristic, complemented
9#10832C211yesmaximal, characteristic, complemented
10#10788C212yesmaximal
11#10788C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×C4×D4C2212yes—
14C22×C4×D4C2211yescharacteristic
15C23×C42—21no—
16C2×#478—24no—
17C2×#478—24no—
18C2×#478C418yescomplemented
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#102C2212yes—
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C42×D4—24no—
28C42×D4—24no—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4—22no—
32C2×C4×C22⋊C4—21no—
33C2×C43C2211yescharacteristic
34#1989—24no—
35#1804—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1436—21no—
44#1436—22no—
45#1436—21no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×D4C2311yescharacteristic
49C24×C4—22no—
50C24×C4—21no—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125C2×C414yescomplemented
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—22no—
65C2×C4×D4—24no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4C2×C414yes—
69C2×C4×D4D414yes—
70C2×C4×D4D414yes—
71C2×C4×D4Q818yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—21no—
81C22×C42—28no—
82C22×C42—22no—
83C22×C42—21no—
84C22×C42—48no—
85C22×C42D411yescharacteristic
86C22×C42D412yes—
87C22×C42D411yescharacteristic
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90#478—232no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#107C2×C414yescomplemented
94C2×#106—22no—
95C2×#106—22no—
96C2×#106C2×C414yescomplemented
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8C2×C414yescomplemented
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—22no—
106C2×#102—22no—
107C2×#102—24no—
108C2×#102—24no—
109C2×#102—24no—
110C2×#102C2×C414yes—
111C2×#102C2×C414yes—
112#382—22no—
113#382—22no—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4D414yes—
118C4×C22⋊C4—28no—
119C4×C22⋊C4—216no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—21no—
128C43—22no—
129C43—22no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—24no—
135#336—24no—
136#336—24no—
137C25—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4C22×C412yes—
144C22×D4C2×D412yes—
145C22×D4C2×D412yes—
146C22×D4C2×Q814yes—
147C22×D4#4114yes—
148C23×C4—28no—
149C23×C4—28no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—28no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2411yescharacteristic
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
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×C8—22no—
184C22×C8—22no—
185C22×C8C22×C412yes—
186C22×C8C22×C412yes—
187#125—28no—
188#125—28no—
189#125—44no—
190#125—44no—
191C4×D4—216no—
192C4×D4—216no—
193C4×D4—216no—
194C4×D4—216no—
195C4×D4—28no—
196C4×D4—28no—
197C4×D4—28no—
198C4×D4C22⋊C4116yes—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×Q814yes—
234C2×C4⋊C4#4114yes—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—28no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—432no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42—44no—
258C2×C42C22×C412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×Q814yes—
263C2×C42#4114yes—
264C2×C42#4112yes—
265C2×C42#4112yes—
266#107—28no—
267#106—28no—
268C4⋊C8—28no—
269C4⋊C8—44no—
270#102—28no—
271#102—28no—
272#102—48no—
273#95—28no—
274#95—24no—
275#95—44no—
276#95—44no—
277#95—42no—
278#95—42no—
279#95—44no—
280#95—44no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—24no—
285C24—24no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24C22×D411yescharacteristic
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—22no—
294C2×D4—24no—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4C2×C22⋊C414yes—
305C2×D4C4×Q818yes—
306C2×D4C22⋊Q818yes—
307C2×D4#12414yes—
308C2×D4C4⋊Q814yes—
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—232no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—216no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—48no—
334C22×C4—416no—
335C22×C4—416no—
336C22×C4—416no—
337C22×C4—416no—
338C22×C4—416no—
339C22×C4—416no—
340C22×C4—48no—
341C22×C4—416no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—416no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
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×C4C4×D412yes—
365C22×C4C4×D412yes—
366C22×C4#12012yes—
367C22×C4#12012yes—
368C22×C4#12112yes—
369C22×C4#12112yes—
370C22×C4C22⋊Q814yes—
371C22×C4C22⋊Q814yes—
372C22×C4#13514yes—
373C22×C4AGL(1,ℤ/8)14yes—
374C22×C4C23×C411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×Q812yes—
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4112yes—
381C22×C4C2×#4112yes—
382C22×C4C2×#4112yes—
383C22×C4C2×#4111yescharacteristic
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4C2×C22⋊C414yes—
422C4⋊C4C22⋊Q818yes—
423C22⋊C4—232no—
424C22⋊C4—216no—
425C42—28no—
426C42—24no—
427C42—24no—
428C42—24no—
429C42—24no—
430C42—48no—
431C42—44no—
432C42—48no—
433C42—416no—
434C42C2×C22⋊C414yes—
435C42C4×D414yes—
436C42#12114yes—
437C42#12314yes—
438C23—28no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—28no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—42no—
452C23—42no—
453C23—44no—
454C23—44no—
455C23—42no—
456C23—42no—
457C23—44no—
458C23—44no—
459C23C2×C4×D411yescharacteristic
460C23C2×#12011yescharacteristic
461C23C2×#12111yescharacteristic
462C23C2×C22⋊Q812yes—
463C23#52512yes—
464C23C2×#13512yes—
465C23C2×AGL(1,ℤ/8)12yes—
466C23#57512yes—
467C23#57512yes—
468D4—24no—
469D4—28no—
470D4—24no—
471D4—24no—
472D4—28no—
473D4#39118yes—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—24no—
493C2×C4—432no—
494C2×C4—464no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—82no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4#38614yes—
527C2×C4#38614yes—
528C2×C4#47818yescomplemented
529C2×C4C22×C22⋊C411yescharacteristic
530C2×C4C2×C4×D411yescharacteristic
531C2×C4C2×C4×Q812yes—
532C2×C4C4×#4112yes—
533C2×C4C2×#12111yescharacteristic
534C2×C4C2×C22⋊Q812yes—
535C2×C4C2×C22⋊Q812yes—
536C2×C4C2×#12311yescharacteristic
537C2×C4#52512yes—
538C2×C4#52512yes—
539C2×C4C2×#12411yescharacteristic
540C2×C4#52912yes—
541C2×C4#52912yes—
542C2×C4C2×C4⋊Q811yescharacteristic
543C2×C4#53211yescharacteristic
544C2×C4#53211yescharacteristic
545C2×C4#53312yes—
546C2×C4#53312yes—
547C8—42no—
548C8—42no—
549C8—82no—
550C22—28no—
551C22—48no—
552C22—48no—
553C22—48no—
554C22—48no—
555C22—44no—
556C22—44no—
557C22—48no—
558C22—48no—
559C22C2×#38611yescharacteristic
560C22C4×C22⋊Q812yes—
561C22#284812yes—
562C22#285011yescharacteristic
563C22#285711yescharacteristic
564C22C4×#13511yescharacteristic
565C22#294311yescharacteristic
566C22C4×AGL(1,ℤ/8)12yes—
567C22#300212yes—
568C22#300311yescharacteristic
569C22#300511yescharacteristic
570C22#303512yes—
571C22#303612yes—
572C22C2×#47812yes—
573C22#307612yes—
574C22#307712yes—
575C22#308012yes—
576C22#310212yes—
577C22#310512yes—
578C22#311112yes—
579C22#311412yes—
580C4—24no—
581C4—24no—
582C4—28no—
583C4—44no—
584C4—48no—
585C4—44no—
586C4—82no—
587C4—82no—
588C4—162no—
589C4C2×#39111yescharacteristic
590C4#229711yescharacteristic
591C4C4×C22⋊Q812yes—
592C4#285011yescharacteristic
593C4#285512yes—
594C4#285711yescharacteristic
595C2—48no—
596C2#1470211yescharacteristic
597C2#2092411yescharacteristic
598C2#2092511yescharacteristic
599C2C4×#47812yes—
600C2#2161712yes—
601C2#3295912yes—
602C2#3296112yes—
603C2#3302812yes—
604C2#3319912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×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
3. C2×#1989C2semidirect
5. #14709C2semidirect
6. #10908C2semidirect
7. #10907C2semidirect
8. #10850C2semidirect
9. #10832C2semidirect
18. C2×#478C4semidirect
59. C2×#125C2×C4semidirect
93. C2×#107C2×C4semidirect
96. C2×#106C2×C4semidirect
102. C2×C4⋊C8C2×C4semidirect
528. C2×C4#478semidirect

Maximal subgroups

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

subgroupindex
C22×#4782
C2×C42×D42
C2×#19892
C2×#18042
#147092
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