Finite Group Databasebeta

← #365780

Subgroups — #365780

Order 512.

1712 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#435C211yesmaximal, characteristic
2C2×#1955C211yesmaximal, characteristic
3C2×#1940C211yesmaximal, characteristic, complemented
4C2×#1831C211yesmaximal, characteristic, complemented
5C4×#390C211yesmaximal, characteristic
6#11042C212yesmaximal, complemented
7#11042C212yesmaximal, complemented
8#10835C211yesmaximal, characteristic, complemented
9#10715C212yesmaximal, complemented
10#10714C212yesmaximal, complemented
11#10693C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#435—24no—
16C2×#435—24no—
17C2×#435—24no—
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2212yes—
24C22×C22⋊C8C2212yes—
25C22×#97C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C2×#390C412yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2212yes—
30C2×C4×C22⋊C4C2212yes—
31C2×C43C2211yescharacteristic
32#1955—24no—
33#1940—24no—
34#1831—24no—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1437C411yescharacteristic
43#1437C411yescharacteristic
44#1437C411yescharacteristic
45#1437C411yescharacteristic
46C4⋊#95C412yes—
47C23×D4C2×C411yescharacteristic
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—24no—
54C22×M16—42no—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—24no—
59C23×C8—42no—
60C23×C8C2312yes—
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2312yes—
65C23×C8C2311yescharacteristic
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4D414yes—
71C2×C4×D4Q814yes—
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C412yes—
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C412yes—
82C22×C42C2×C411yescharacteristic
83C22×C42C2312yes—
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86#435—232no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—42no—
94C2×C4⋊C8—42no—
95C2×C4⋊C8D414yes—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—42no—
107C2×C22⋊C8—42no—
108C2×#97—22no—
109C2×#97—22no—
110C2×#97—22no—
111C2×#97—22no—
112C2×#97—22no—
113C2×#97D414yes—
114#390—28no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C22⋊C4—42no—
118C4×C22⋊C4—42no—
119C4×C22⋊C4—42no—
120C4×C22⋊C4—42no—
121C2×#95C2×C411yescharacteristic
122C2×#95C2×C411yescharacteristic
123C2×#95C2×C412yes—
124C2×#95C2×C412yes—
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127C43—22no—
128C43—22no—
129#336—24no—
130#336—24no—
131#336—24no—
132#336—24no—
133#336—24no—
134#336—42no—
135#336—42no—
136C25C22×C411yescharacteristic
137C25C22×C411yescharacteristic
138C22×D4—24no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—42no—
142C22×D4C4⋊C414yes—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4C22×C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C412yes—
163C23×C4C22×C412yes—
164C23×C4C22×C412yes—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×Q812yes—
170C23×C4C2×Q812yes—
171C23×C4C2411yescharacteristic
172C2×M16—28no—
173C2×M16—48no—
174C2×M16—48no—
175C2×M16—44no—
176C2×M16—44no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—28no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
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—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—48no—
207C22×C8—48no—
208C22×C8—44no—
209C22×C8—44no—
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8C2×D412yes—
213C22×C8C2×D412yes—
214C22×C8#4114yes—
215C22×C8#4114yes—
216C4×D4—28no—
217C4×D4—28no—
218C4×D4—28no—
219C4×D4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4C4⋊C414yes—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4C4⋊C414yes—
242C2×C22⋊C4C4⋊C414yes—
243C2×C22⋊C4M1618yescomplemented
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—22no—
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—42no—
262C2×C42—42no—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×Q812yes—
267C2×C42#4112yes—
268C2×C42#4112yes—
269C4⋊C8—28no—
270C4⋊C8—48no—
271C4⋊C8—44no—
272C22⋊C8—28no—
273C22⋊C8—28no—
274C22⋊C8—48no—
275C22⋊C8—48no—
276#97—28no—
277#97—44no—
278#95—22no—
279#95—22no—
280#95—42no—
281#95—42no—
282#95—42no—
283#95—42no—
284C24—21no—
285C24—21no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—21no—
292C24—21no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24C2×C4⋊C412yes—
300C24C2×C4⋊C412yes—
301C24C2×M1614yes—
302C24C23×C411yescharacteristic
303C2×D4—28no—
304C2×D4—28no—
305C2×D4—44no—
306C2×D4—48no—
307C2×D4—48no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
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—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
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×C4C2×C4⋊C412yes—
382C22×C4C2×C4⋊C412yes—
383C22×C4C2×C4⋊C412yes—
384C22×C4C2×C4⋊C412yes—
385C22×C4C4×D412yes—
386C22×C4C4×D412yes—
387C22×C4C4×D412yes—
388C22×C4C4×D412yes—
389C22×C4C2×M1614yes—
390C22×C4C2×M1614yes—
391C22×C4#13114yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×Q811yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4112yes—
399C22×C4C2×#4112yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C4H(5,𝔽2)12yes—
402M16—416no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—24no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—416no—
418C2×C8—42no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8#12414yes—
438C4⋊C4—28no—
439C4⋊C4—48no—
440C22⋊C4—28no—
441C22⋊C4—28no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C42—28no—
447C42—48no—
448C42—44no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42—42no—
453C42—42no—
454C42—42no—
455C42—42no—
456C42—42no—
457C42#12014yes—
458C42C22⋊Q814yes—
459C42C22⋊Q814yes—
460C42#12314yes—
461C23—24no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—24no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—22no—
476C23—41no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23#42314yes—
503C23C22×C4⋊C411yescharacteristic
504C23C2×C4×D411yescharacteristic
505C23C2×C4×D411yescharacteristic
506C23C4×#4112yes—
507C23#51812yes—
508C23C22×M1612yes—
509C23C2×#13112yes—
510C23#56812yes—
511C23#56812yes—
512D4—416no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4#38514yes—
572C2×C4#38514yes—
573C2×C4C2×#12011yescharacteristic
574C2×C4C2×C22⋊Q811yescharacteristic
575C2×C4C2×C22⋊Q811yescharacteristic
576C2×C4C2×#12311yescharacteristic
577C2×C4C2×#12411yescharacteristic
578C2×C4#52912yes—
579C2×C4#53211yescharacteristic
580C2×C4#53511yescharacteristic
581C2×C4#53511yescharacteristic
582C2×C4#54012yes—
583C2×C4#54612yes—
584C2×C4#54612yes—
585C2×C4#54612yes—
586C2×C4#55012yes—
587C2×C4#55312yes—
588C2×C4#55412yes—
589C2×C4#55412yes—
590C2×C4#55512yes—
591C8—24no—
592C8—44no—
593C8—42no—
594C8—42no—
595C8—42no—
596C8—82no—
597C22—24no—
598C22—28no—
599C22—44no—
600C22—44no—
601C22—42no—
602C22—44no—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—44no—
610C22—44no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22C2×#38511yescharacteristic
616C22C4×#12411yescharacteristic
617C22#232811yescharacteristic
618C22#234912yes—
619C22#236312yes—
620C22C2×#42311yescharacteristic
621C22#290511yescharacteristic
622C22#290612yes—
623C22#292911yescharacteristic
624C22#293011yescharacteristic
625C22D4×M1612yes—
626C22D4×M1612yes—
627C22#293512yes—
628C22#296512yes—
629C22#296712yes—
630C22#296812yes—
631C22#297012yes—
632C22#297312yes—
633C22#297412yes—
634C22#298012yes—
635C22#298112yes—
636C4—24no—
637C4—24no—
638C4—44no—
639C4—44no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—82no—
644C4#242012yes—
645C4#243111yescharacteristic
646C4#243511yescharacteristic
647C4#245412yes—
648C4#256311yescharacteristic
649C4#256611yescharacteristic
650C2—44no—
651C2—82no—
652C2#1397911yescharacteristic
653C2#2083711yescharacteristic
654C2#2083811yescharacteristic
655C2#2163212yes—
656C2#2226612yes—
657C2#2269812yes—
658C2#2270212yes—
659C2#2308512yes—
660C2#2308612yes—
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
3. C2×#1940C2semidirect
4. C2×#1831C2semidirect
6. #11042C2semidirect
7. #11042C2semidirect
8. #10835C2semidirect
9. #10715C2semidirect
10. #10714C2semidirect
11. #10693C2semidirect
243. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C22×#4352
C2×#19552
C2×#19402
C2×#18312
C4×#3902
#110422
#108352
#107152
#107142
#106932