Finite Group Databasebeta

← #387421

Subgroups — #387421

Order 512.

1670 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3#13872C211yesmaximal, characteristic
4#11384C212yesmaximal, complemented
5#11200C212yesmaximal, complemented
6#11200C212yesmaximal, complemented
7#10807C212yesmaximal, complemented
8#10730C212yesmaximal, complemented
9#10721C212yesmaximal, complemented
10C24×C8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16#2563—24no—
17#2563C418yescomplemented
18C2×#393—21no—
19C2×#393—21no—
20C2×#390—21no—
21C2×#386—21no—
22C2×#382—21no—
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1869—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1441—21no—
39#1439—21no—
40C4⋊#95C2212yes—
41C22⋊#95—21no—
42C23×Q8—21no—
43C23×D4D411yescharacteristic
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8—22no—
48C23×C8—24no—
49C23×C8—41no—
50C23×C8—41no—
51C23×C8—42no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—41no—
56C2×C4⋊Q8—41no—
57C2×C4⋊Q8—41no—
58C2×C4⋊Q8—41no—
59C2×C4⋊Q8C2×C414yescomplemented
60C2×C4×D4—41no—
61C2×C4×D4—41no—
62C2×C4×D4—41no—
63C2×C4×D4—41no—
64C2×C4×D4C2×C414yes—
65C2×C4×D4D414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—41no—
70C22×C4⋊C4—41no—
71C22×C4⋊C4—42no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85C2×C22⋊C8—24no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—41no—
90C2×C22⋊C8—41no—
91#393—28no—
92#393—28no—
93#390—28no—
94#386—44no—
95#382—28no—
96#380—24no—
97#380—24no—
98#380—24no—
99#380—42no—
100#380—41no—
101#380—41no—
102#380C2×C418yes—
103#380C2×C418yescomplemented
104C4×C4⋊C4—22no—
105C4×C4⋊C4C2×C414yescomplemented
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—22no—
113#340—24no—
114#340—24no—
115#340—42no—
116#340—42no—
117#340—42no—
118#340—42no—
119#340—42no—
120#340—42no—
121#336—24no—
122C25C2×D411yescharacteristic
123C25C2×D411yescharacteristic
124C22×Q8—24no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—82no—
128C22×Q8—82no—
129C22×D4—41no—
130C22×D4—41no—
131C22×D4—41no—
132C22×D4—41no—
133C22×D4C22⋊C414yes—
134C22×D4SD1614yes—
135C22×D4Q1614yes—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—44no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—82no—
161C23×C4C22×C412yes—
162C23×C4C22×C412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
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×C8—44no—
184C22×C8—48no—
185C22×C8—48no—
186C22×C8—44no—
187C22×C8—44no—
188C22×C8#4114yes—
189C4⋊Q8—44no—
190C4⋊Q8—44no—
191C4⋊Q8—44no—
192C4⋊Q8—44no—
193C4×D4—44no—
194C4×D4—44no—
195C4×D4—44no—
196C4×D4—44no—
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—24no—
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⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4C22⋊C414yes—
229C2×C4⋊C4C22×C414yes—
230C2×C4⋊C4C22×C414yes—
231C2×C4⋊C4#4114yes—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—82no—
244C2×C22⋊C4—82no—
245C2×C22⋊C4—82no—
246C2×C22⋊C4—82no—
247C2×C22⋊C4—82no—
248C2×C22⋊C4C22⋊C414yes—
249C2×C22⋊C4C22⋊C414yes—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—24no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42C22×C412yes—
258C2×C42C2×D412yes—
259C22⋊C8—48no—
260C22⋊C8—44no—
261#95—24no—
262#95—28no—
263#95—28no—
264#95—44no—
265#95—42no—
266#95—44no—
267#95—82no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24C2×C22⋊C412yes—
285C24C2×C22⋊C412yes—
286C24C2×SD1612yes—
287C24C2×SD1612yes—
288C24C2×Q1612yes—
289C24C2×Q1612yes—
290C24C22×D411yescharacteristic
291C2×Q8—48no—
292C2×Q8—88no—
293C2×Q8—88no—
294C2×Q8—84no—
295C2×Q8—84no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4#103116yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
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—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—48no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4C2×C22⋊C412yes—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×SD1612yes—
393C22×C4C2×Q1612yes—
394C22×C4C23×C411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C2×#4112yes—
397C22×C4C2×#4112yes—
398C22×C4H(5,𝔽2)11yescharacteristic
399C22×C4H(5,𝔽2)11yescharacteristic
400C22×C4H(5,𝔽2)11yescharacteristic
401C22×C4H(5,𝔽2)11yescharacteristic
402C22×C4H(5,𝔽2)12yes—
403C22×C42−1+412yes—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—416no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—44no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C22⋊C4—84no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—48no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—22no—
458C23—22no—
459C23—24no—
460C23—24no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23C2×#10314yes—
494C23C2×#10314yes—
495C23C22×C22⋊C411yescharacteristic
496C23#53411yescharacteristic
497C23#53411yescharacteristic
498C23#54612yes—
499C23#54712yes—
500C23C22×SD1611yescharacteristic
501C23C22×Q1611yescharacteristic
502C23#57511yescharacteristic
503C23#57511yescharacteristic
504C23#57612yes—
505C23#57712yes—
506Q8—816no—
507D4—416no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—816no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4C2×#10314yes—
571C2×C4C4×#4112yes—
572C2×C4#51811yescharacteristic
573C2×C4#51811yescharacteristic
574C2×C4#52012yes—
575C2×C4#53411yescharacteristic
576C2×C4#53411yescharacteristic
577C2×C4#53812yes—
578C2×C4#53812yes—
579C2×C4#54212yes—
580C2×C4#54212yes—
581C2×C4#54612yes—
582C2×C4#54712yes—
583C2×C4#55111yescharacteristic
584C2×C4#55111yescharacteristic
585C2×C4#55512yes—
586C2×C4H(3,𝔽4)11yescharacteristic
587C2×C4#56212yes—
588C2×C4#56311yescharacteristic
589C8—44no—
590C8—82no—
591C22—28no—
592C22—28no—
593C22—24no—
594C22—24no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22#230811yescharacteristic
614C22#233912yes—
615C22#268411yescharacteristic
616C22#268612yes—
617C22#271711yescharacteristic
618C22C22×#10311yescharacteristic
619C22#289611yescharacteristic
620C22#289712yes—
621C22#318711yescharacteristic
622C22#318711yescharacteristic
623C22#318911yescharacteristic
624C22#318911yescharacteristic
625C22#319812yes—
626C22#319911yescharacteristic
627C22#320011yescharacteristic
628C22#329512yes—
629C22#329912yes—
630C22#331912yes—
631C22#332112yes—
632C22#333512yes—
633C22#333812yes—
634C22#334712yes—
635C22#334812yes—
636C4—24no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—84no—
641C4—84no—
642C4—84no—
643C4—162no—
644C4#233712yes—
645C4#237211yescharacteristic
646C4#237911yescharacteristic
647C4#268411yescharacteristic
648C4#268612yes—
649C4#271711yescharacteristic
650C2—44no—
651C2—44no—
652C2#1447311yescharacteristic
653C2#2175911yescharacteristic
654C2#2177011yescharacteristic
655C2#2248712yes—
656C2#2249212yes—
657C2#3126111yescharacteristic
658C2#3126311yescharacteristic
659C2#3132212yes—
660C2#3132412yes—
661C2#3225911yescharacteristic
662C2#3226111yescharacteristic
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
4. #11384C2semidirect
5. #11200C2semidirect
6. #11200C2semidirect
7. #10807C2semidirect
8. #10730C2semidirect
9. #10721C2semidirect
17. #2563C4semidirect
59. C2×C4⋊Q8C2×C4semidirect
103. #380C2×C4semidirect
105. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#18692
#138722
#113842
#112002
#108072
#107302
#107212