Finite Group Databasebeta

← #361458

Subgroups — #361458

Order 512.

1852 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×C8×D4C211yesmaximal, characteristic
2C2×#1854C212yesmaximal, complemented
3C2×#1807C211yesmaximal, characteristic, complemented
4C2×#1735C211yesmaximal, characteristic, complemented
5#13871C212yesmaximal
6#11231C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#10862C211yesmaximal, characteristic, complemented
9#10820C211yesmaximal, characteristic, complemented
10#10721C211yesmaximal, characteristic, complemented
11#10721C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#117C2212yes—
16C2×C8×D4—28no—
17C2×C8×D4C418yescomplemented
18C22×#107C2212yes—
19C22×#106C2212yes—
20C22×C4⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27#1854—28no—
28#1807—24no—
29#1735—24no—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38#1440—22no—
39#1438—22no—
40#1437—22no—
41#1436—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C23×D4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8—24no—
48C23×C8—22no—
49C23×C8—24no—
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—42no—
54C23×C8—42no—
55C23×C8C2×C414yes—
56C23×C8C2×C414yes—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4C2×C414yes—
64C2×C4×D4Q814yes—
65C2×#117—24no—
66C2×#117—24no—
67C2×#117—24no—
68C2×#117—24no—
69C2×#117—42no—
70C2×#117—42no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D411yescharacteristic
81C22×C22⋊C4D411yescharacteristic
82C22×C42—22no—
83C22×C42C2311yescharacteristic
84C8×D4—232no—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—42no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—42no—
91C2×C4⋊C8—24no—
92C2×C4⋊C8C2×C414yes—
93C2×C22⋊C8—24no—
94C2×C22⋊C8C2×C414yes—
95C2×C22⋊C8C2×C414yes—
96C2×C22⋊C8D418yes—
97C2×C4×C8—24no—
98C2×C4×C8C2×C414yes—
99#380—24no—
100#380—24no—
101#380—24no—
102#380—24no—
103#380—42no—
104#380—42no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112#340—22no—
113#340—24no—
114#340—24no—
115#340—22no—
116#340—22no—
117#340—24no—
118#340—42no—
119#340D414yescomplemented
120C25C2×D411yescharacteristic
121C25C2×D411yescharacteristic
122C22×D4—24no—
123C22×D4—24no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4C4⋊C414yes—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4C22×C412yes—
146C23×C4C22×C412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D414yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×Q812yes—
153C23×C4C2×Q812yes—
154C23×C4#4114yes—
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—28no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—28no—
170C22×C8—28no—
171C22×C8—28no—
172C22×C8—44no—
173C22×C8—44no—
174C22×C8—48no—
175C22×C8—48no—
176C22×C8—48no—
177C22×C8—48no—
178C22×C8C22×C412yes—
179C22×C8C22×C412yes—
180C22×C8C22×C412yes—
181C22×C8C22×C412yes—
182C22×C8C2×D414yes—
183C22×C8C2×D414yes—
184C22×C8C2×D412yes—
185C22×C8#4112yes—
186C4×D4—28no—
187C4×D4—28no—
188C4×D4—216no—
189#117—216no—
190#117—48no—
191#117—48no—
192#117—48no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
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—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4C4⋊C414yes—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4#4114yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—82no—
230C2×C22⋊C4—82no—
231C2×C22⋊C4—82no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4C4⋊C414yes—
234C2×C22⋊C4C4⋊C414yes—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—44no—
238C2×C42—44no—
239C2×C42—44no—
240C2×C42C22×C412yes—
241C2×C42C2×Q812yes—
242#107—48no—
243#106—48no—
244C4⋊C8—28no—
245C22⋊C8—28no—
246C22⋊C8C22⋊C4116yescomplemented
247C4×C8—28no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—82no—
254#95—82no—
255C24—22no—
256C24—21no—
257C24—21no—
258C24—22no—
259C24—22no—
260C24—21no—
261C24—21no—
262C24—22no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—22no—
270C24—22no—
271C24—21no—
272C24—21no—
273C24C2×C4⋊C412yes—
274C24C2×C4⋊C412yes—
275C24#12114yes—
276C24#13514yes—
277C24C22×D411yescharacteristic
278C2×D4—216no—
279C2×D4—216no—
280C2×D4—28no—
281C2×D4—28no—
282C2×D4—28no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
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—42no—
342C22×C4—42no—
343C22×C4—42no—
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—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4C2×C22⋊C414yes—
358C22×C4C2×C4⋊C412yes—
359C22×C4C2×C4⋊C412yes—
360C22×C4C2×C4⋊C412yes—
361C22×C4C2×C4⋊C412yes—
362C22×C4#11714yes—
363C22×C4#12114yes—
364C22×C4C22⋊Q818yes—
365C22×C4#13512yes—
366C22×C4#13512yes—
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D412yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×Q811yescharacteristic
371C22×C4C2×#4112yes—
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4112yes—
374C22×C4H(5,𝔽2)12yes—
375C22×C4H(5,𝔽2)11yescharacteristic
376C22×C42−1+411yescharacteristic
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—416no—
393C2×C8—416no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8C2×C22⋊C414yes—
397C2×C8C2×C22⋊C414yes—
398C2×C8C4×D414yes—
399C4⋊C4—28no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C22⋊C4—28no—
417C22⋊C4—28no—
418C22⋊C4—416no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C22⋊C4—88no—
422C42—22no—
423C42—22no—
424C42—24no—
425C42—416no—
426C23—24no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—22no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—42no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23#38014yes—
471C23#42714yes—
472C23C22×C4⋊C411yescharacteristic
473C23C2×#12112yes—
474C23D4211yescharacteristic
475C23#54612yes—
476C23#54711yescharacteristic
477C23C2×#13512yes—
478C23C2×#13511yescharacteristic
479C23C2×#13511yescharacteristic
480C23#57611yescharacteristic
481C23#57712yes—
482C23#57811yescharacteristic
483D4—232no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4#38014yes—
543C2×C4#42714yes—
544C2×C4C22×C22⋊C411yescharacteristic
545C2×C4C2×#11711yescharacteristic
546C2×C4C2×C4×D411yescharacteristic
547C2×C4C4×#4111yescharacteristic
548C2×C4#51811yescharacteristic
549C2×C4#52011yescharacteristic
550C2×C4C2×C22⋊Q812yes—
551C2×C4#52512yes—
552C2×C4#52912yes—
553C2×C4#53812yes—
554C2×C4#53912yes—
555C2×C4D4212yes—
556C2×C4#54612yes—
557C2×C4#54612yes—
558C2×C4#54712yes—
559C2×C4D4×Q811yescharacteristic
560C2×C4#55412yes—
561C2×C4#55411yescharacteristic
562C8—24no—
563C8—24no—
564C8—24no—
565C8—42no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22C2×#38011yescharacteristic
589C22D4×C4⋊C411yescharacteristic
590C22#234911yescharacteristic
591C22#242412yes—
592C22#242512yes—
593C22C2×#42711yescharacteristic
594C22C2×#42711yescharacteristic
595C22#291211yescharacteristic
596C22#291311yescharacteristic
597C22#305012yes—
598C22#305012yes—
599C22#306212yes—
600C22#306312yes—
601C22#328212yes—
602C22#328311yescharacteristic
603C22#328711yescharacteristic
604C22#329712yes—
605C22#330012yes—
606C22#330312yes—
607C22#330412yes—
608C22#333712yes—
609C22#334111yescharacteristic
610C22#334211yescharacteristic
611C4—24no—
612C4—24no—
613C4—42no—
614C4—42no—
615C4—84no—
616C4—84no—
617C4—162no—
618C4—162no—
619C4#229011yescharacteristic
620C4#231811yescharacteristic
621C4D4×C22⋊C411yescharacteristic
622C4#233911yescharacteristic
623C4#244912yes—
624C4#245012yes—
625C2—44no—
626C2—44no—
627C2#1389911yescharacteristic
628C2#2065811yescharacteristic
629C2#2066411yescharacteristic
630C2#2265411yescharacteristic
631C2#2265811yescharacteristic
632C2#2270411yescharacteristic
633C2#2270811yescharacteristic
634C2#2511312yes—
635C2#2511512yes—
636C2#2523712yes—
637C2#2523912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×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×#1854C2semidirect
3. C2×#1807C2semidirect
4. C2×#1735C2semidirect
6. #11231C2semidirect
7. #11096C2semidirect
8. #10862C2semidirect
9. #10820C2semidirect
10. #10721C2semidirect
11. #10721C2semidirect
17. C2×C8×D4C4semidirect
119. #340D4semidirect
246. C22⋊C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D42
C2×#18542
C2×#18072
C2×#17352
#138712
#112312
#110962
#108622
#108202
#107212