Finite Group Databasebeta

← #400371

Subgroups — #400371

Order 512.

1832 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic, complemented
3C2×#1921C211yesmaximal, characteristic, complemented
4C2×#1852C212yesmaximal, complemented
5#15181C211yesmaximal, characteristic
6#11371C211yesmaximal, characteristic, complemented
7#11200C212yesmaximal, complemented
8#11096C212yesmaximal, complemented
9#10964C211yesmaximal, characteristic
10#10807C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal, complemented
12C22×C4×D4—21no—
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4C2212yes—
15C22×#107C2212yes—
16C22×#107C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19D4×C4⋊C4—24no—
20D4×C4⋊C4—24no—
21C2×#389—21no—
22C2×#389C2211yescharacteristic
23C2×#385—22no—
24C2×#384C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2083—24no—
30#1921—24no—
31#1852—28no—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1442—21no—
43#1439—22no—
44#1438—21no—
45C4⋊#95C2211yescharacteristic
46C23×D4D411yescharacteristic
47C24×C4—22no—
48C24×C4C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C2×C4×D4—24no—
53C2×C4×D4—24no—
54C2×C4×D4—28no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—48no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4—42no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4D412yes—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C2×#107—22no—
82C2×#107—22no—
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—42no—
86C2×#107D414yescomplemented
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—42no—
93#389—28no—
94#389—22no—
95#389—22no—
96#385—216no—
97#384—22no—
98#384D414yes—
99#380—24no—
100#380—24no—
101#380—24no—
102#380—24no—
103#380—24no—
104#380—42no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4D414yes—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—42no—
122#336—22no—
123#336—22no—
124C25C2×D412yes—
125C22×D4—24no—
126C22×D4—24no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4Q1614yes—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—28no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—82no—
154C23×C4—82no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4#4114yes—
159C23×C4#4114yes—
160C23×C4C2411yescharacteristic
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8C2×D412yes—
177C22×C8#4114yes—
178C4×D4—28no—
179C4×D4—28no—
180C4×D4—216no—
181C4×D4—432no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
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—24no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—48no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4#4112yes—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—82no—
240C2×C22⋊C4—82no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—44no—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247#107—28no—
248#107—48no—
249C22⋊C8—216no—
250C22⋊C8—48no—
251#95—28no—
252#95—24no—
253#95—44no—
254#95—42no—
255#95—44no—
256#95—42no—
257#95—81no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24#12314yes—
271C24C2×Q1614yes—
272C24AGL(1,ℤ/8)14yes—
273C24C22×D411yescharacteristic
274C2×D4—28no—
275C2×D4—28no—
276C2×D4—216no—
277C2×D4—216no—
278C2×D4—28no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—24no—
295C22×C4—24no—
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—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—416no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—416no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—88no—
356C22×C4—88no—
357C22×C4—88no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4#12012yes—
361C22×C4#12012yes—
362C22×C4#12314yes—
363C22×C4C2×Q1612yes—
364C22×C4#13712yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4112yes—
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4H(5,𝔽2)11yescharacteristic
373C22×C4H(5,𝔽2)11yescharacteristic
374C22×C42−1+412yes—
375C2×C8—22no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—42no—
391C2×C8—42no—
392C4⋊C4—28no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—416no—
403C4⋊C4—48no—
404C4⋊C4—42no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—84no—
417C4⋊C4—84no—
418C4⋊C4#12114yes—
419C22⋊C4—216no—
420C22⋊C4—416no—
421C22⋊C4—88no—
422C22⋊C4—88no—
423C42—22no—
424C42—22no—
425C42—24no—
426C42—48no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23#48418yes—
462C23C2×#12011yescharacteristic
463C23C2×#12312yes—
464C23D4211yescharacteristic
465C23#54611yescharacteristic
466C23#54712yes—
467C23C22×Q1611yescharacteristic
468C23C2×AGL(1,ℤ/8)12yes—
469C23C2×#13711yescharacteristic
470C23#57612yes—
471C23#57711yescharacteristic
472C23#57811yescharacteristic
473D4—232no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—832no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—82no—
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—82no—
527C2×C4—81no—
528C2×C4—82no—
529C2×C4—81no—
530C2×C4—82no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4#45114yes—
534C2×C4C2×#12111yescharacteristic
535C2×C4#52511yescharacteristic
536C2×C4#52511yescharacteristic
537C2×C4#52512yes—
538C2×C4#52512yes—
539C2×C4#52912yes—
540C2×C4#53211yescharacteristic
541C2×C4#53511yescharacteristic
542C2×C4#53611yescharacteristic
543C2×C4#53912yes—
544C2×C4#53912yes—
545C2×C4D4211yescharacteristic
546C2×C4#54611yescharacteristic
547C2×C4#54812yes—
548C2×C4#54812yes—
549C2×C4#55112yes—
550C2×C4#55212yes—
551C2×C4#55312yes—
552C8—42no—
553C8—44no—
554C8—81no—
555C22—28no—
556C22—28no—
557C22—28no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22#240811yescharacteristic
571C22#241912yes—
572C22#243211yescharacteristic
573C22#244312yes—
574C22#284911yescharacteristic
575C22C2×#45111yescharacteristic
576C22#300311yescharacteristic
577C22#300811yescharacteristic
578C22#301311yescharacteristic
579C22C2×#48412yes—
580C22#309212yes—
581C22#309612yes—
582C22#309812yes—
583C22#326412yes—
584C22#327811yescharacteristic
585C22D4×Q1611yescharacteristic
586C22#329911yescharacteristic
587C22#331812yes—
588C22#332811yescharacteristic
589C22#333812yes—
590C22#334312yes—
591C22#334512yes—
592C22#335712yes—
593C4—24no—
594C4—44no—
595C4—44no—
596C4—42no—
597C4—82no—
598C4—82no—
599C4—84no—
600C4—162no—
601C4—161no—
602C4#239211yescharacteristic
603C4#246012yes—
604C4#246312yes—
605C4#253611yescharacteristic
606C4#260311yescharacteristic
607C4#263311yescharacteristic
608C2—48no—
609C2#1480111yescharacteristic
610C2#2424311yescharacteristic
611C2#2424811yescharacteristic
612C2#2493912yes—
613C2#2495112yes—
614C2#2570611yescharacteristic
615C2#2570911yescharacteristic
616C2#2613212yes—
617C2#2614212yes—
618C2#3298311yescharacteristic
619C2#3298611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2083C2semidirect
3. C2×#1921C2semidirect
4. C2×#1852C2semidirect
6. #11371C2semidirect
7. #11200C2semidirect
8. #11096C2semidirect
10. #10807C2semidirect
11. #10730C2semidirect
86. C2×#107D4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
C2×#20832
C2×#19212
C2×#18522
#151812
#113712
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