Finite Group Databasebeta

← #387417

Subgroups — #387417

Order 512.

1692 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#124C211yesmaximal, characteristic
2C2×#1870C211yesmaximal, characteristic
3C2×#1869C211yesmaximal, characteristic
4#11380C212yesmaximal, complemented
5#11147C212yesmaximal, complemented
6#11147C212yesmaximal, complemented
7#10795C212yesmaximal, complemented
8#10715C212yesmaximal, complemented
9#10714C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×#124C411yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C4×#124—24no—
20C4×#124—24no—
21C2×#391C411yescharacteristic
22C2×#390C411yescharacteristic
23C2×#388C412yes—
24C2×#388C412yes—
25C2×#382C411yescharacteristic
26C2×C4×C22⋊C4C2212yes—
27C2×C4×C22⋊C4C2212yes—
28C2×C43C2211yescharacteristic
29#1870—24no—
30#1869—24no—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C23×Q8C2×C411yescharacteristic
42C23×D4C2×C411yescharacteristic
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C22×M16—22no—
46C22×M16—24no—
47C22×M16—41no—
48C22×M16—41no—
49C22×M16—42no—
50C23×C8—22no—
51C23×C8—24no—
52C23×C8—41no—
53C23×C8—41no—
54C23×C8—42no—
55C23×C8C2312yes—
56C23×C8C2312yes—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C2×#124—24no—
60C2×#124—28no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—41no—
64C2×C4×Q8—41no—
65C2×C4×Q8—41no—
66C2×C4×Q8—41no—
67C2×C4×D4—41no—
68C2×C4×D4—41no—
69C2×C4×D4—41no—
70C2×C4×D4—41no—
71C2×C4×D4D414yes—
72C2×C4×D4D414yes—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2×C412yes—
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C412yes—
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C412yes—
85C22×C42C2312yes—
86C22×C42C2312yes—
87C22×C42C2311yescharacteristic
88C2×C22⋊C8—24no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8—41no—
97C2×C22⋊C8—41no—
98C2×C22⋊C8—41no—
99C2×C22⋊C8—41no—
100#391—24no—
101#390—24no—
102#388—28no—
103#388—28no—
104#382—24no—
105C4×C22⋊C4—24no—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—24no—
108C4×C22⋊C4—24no—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C43—22no—
114C43—22no—
115#336—24no—
116#336—24no—
117#336—24no—
118#336—24no—
119#336—42no—
120#336—42no—
121#336—42no—
122#336—42no—
123#336—42no—
124#336—42no—
125C25C22×C411yescharacteristic
126C25C22×C411yescharacteristic
127C22×Q8—22no—
128C22×Q8—24no—
129C22×Q8—24no—
130C22×Q8—41no—
131C22×Q8—41no—
132C22×D4—24no—
133C22×D4—24no—
134C22×D4—41no—
135C22×D4—41no—
136C22×D4C22⋊C414yes—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
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×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C412yes—
158C23×C4C22×C412yes—
159C23×C4C22×C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2411yescharacteristic
166C2×M16—44no—
167C2×M16—48no—
168C2×M16—48no—
169C2×M16—44no—
170C2×M16—44no—
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—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—44no—
198C22×C8—48no—
199C22×C8—48no—
200C22×C8—44no—
201C22×C8—44no—
202C22×C8#4114yes—
203C22×C8#4114yes—
204#124—232no—
205C4×Q8—44no—
206C4×Q8—44no—
207C4×Q8—44no—
208C4×Q8—44no—
209C4×D4—44no—
210C4×D4—44no—
211C4×D4—44no—
212C4×D4—44no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4C22⋊C414yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4C22⋊C414yes—
238C2×C22⋊C4C22⋊C414yes—
239C2×C22⋊C4M1618yescomplemented
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—42no—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C22⋊C8—48no—
266C22⋊C8—48no—
267C22⋊C8—44no—
268C22⋊C8—44no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273C24—22no—
274C24—21no—
275C24—21no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—21no—
282C24—21no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24C2×C22⋊C412yes—
289C24C2×C22⋊C412yes—
290C24C2×M1614yes—
291C24C23×C411yescharacteristic
292C2×Q8—28no—
293C2×Q8—48no—
294C2×Q8—48no—
295C2×Q8—44no—
296C2×Q8—44no—
297C2×D4—28no—
298C2×D4—28no—
299C2×D4—44no—
300C2×D4—48no—
301C2×D4—48no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×M1614yes—
383C22×C4C2×M1614yes—
384C22×C4#13114yes—
385C22×C4#13114yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C2×#4112yes—
389C22×C4C2×#4112yes—
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C4H(5,𝔽2)11yescharacteristic
394C22×C4H(5,𝔽2)12yes—
395C22×C42−1+412yes—
396M16—416no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—416no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
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—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—48no—
429C22⋊C4—216no—
430C22⋊C4—216no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C42—28no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—48no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—42no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—44no—
450C23—22no—
451C23—24no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—22no—
461C23—24no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—42no—
466C23—41no—
467C23—41no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23#40714yes—
492C23C22×C22⋊C411yescharacteristic
493C23C4×#4112yes—
494C23#51811yescharacteristic
495C23#51811yescharacteristic
496C23#52012yes—
497C23C22×M1612yes—
498C23C2×#13112yes—
499C23C2×#13112yes—
500C23#56812yes—
501Q8—416no—
502D4—416no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4#40814yes—
563C2×C4#53411yescharacteristic
564C2×C4#53411yescharacteristic
565C2×C4#53411yescharacteristic
566C2×C4#53411yescharacteristic
567C2×C4#53812yes—
568C2×C4#53812yes—
569C2×C4#54212yes—
570C2×C4#54212yes—
571C2×C4#54612yes—
572C2×C4#54612yes—
573C2×C4#54712yes—
574C2×C4#54712yes—
575C2×C4#55111yescharacteristic
576C2×C4#55111yescharacteristic
577C2×C4#55512yes—
578C2×C4H(3,𝔽4)11yescharacteristic
579C2×C4#56212yes—
580C2×C4#56311yescharacteristic
581C8—44no—
582C8—44no—
583C8—82no—
584C8—82no—
585C22—28no—
586C22—24no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—42no—
591C22—44no—
592C22—44no—
593C22—42no—
594C22—42no—
595C22—44no—
596C22—44no—
597C22—42no—
598C22—42no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22#230811yescharacteristic
604C22#233712yes—
605C22#233912yes—
606C22#237211yescharacteristic
607C22#237911yescharacteristic
608C22C2×#40711yescharacteristic
609C22C2×#40811yescharacteristic
610C22#288012yes—
611C22#296512yes—
612C22#296612yes—
613C22#296912yes—
614C22#297012yes—
615C22#297412yes—
616C22#297412yes—
617C22#297911yescharacteristic
618C22#298011yescharacteristic
619C22#298112yes—
620C22#298912yes—
621C22#299012yes—
622C22#299312yes—
623C22#299412yes—
624C4—24no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—42no—
631C4—82no—
632C4#268411yescharacteristic
633C4#268411yescharacteristic
634C4#268612yes—
635C4#268612yes—
636C4#271711yescharacteristic
637C4#271711yescharacteristic
638C2—44no—
639C2—82no—
640C2#1447311yescharacteristic
641C2#2174011yescharacteristic
642C2#2174511yescharacteristic
643C2#2238912yes—
644C2#2239112yes—
645C2#2247112yes—
646C2#2247412yes—
647C2#2327012yes—
648C2#2334912yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11380C2semidirect
5. #11147C2semidirect
6. #11147C2semidirect
7. #10795C2semidirect
8. #10715C2semidirect
9. #10714C2semidirect
239. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1242
C2×#18702
C2×#18692
#113802
#111472
#107952
#107152
#107142