Finite Group Databasebeta

← #387632

Subgroups — #387632

Order 512.

1496 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1870C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3C4×#390C211yesmaximal, characteristic
4#11202C212yesmaximal, complemented
5#11042C212yesmaximal, complemented
6#11042C212yesmaximal, complemented
7#10795C212yesmaximal, complemented
8#10715C212yesmaximal, complemented
9#10714C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C2×#390C412yes—
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C22⋊C4C2212yes—
21C2×C4×C22⋊C4C2212yes—
22C2×C43C2211yescharacteristic
23#1870—24no—
24#1869—24no—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1437C411yescharacteristic
35#1437C411yescharacteristic
36#1437C411yescharacteristic
37#1437C411yescharacteristic
38C4⋊#95C412yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C22×M16—22no—
43C22×M16—24no—
44C22×M16—41no—
45C22×M16—41no—
46C22×M16—42no—
47C23×C8—22no—
48C23×C8—24no—
49C23×C8—41no—
50C23×C8—41no—
51C23×C8—42no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×C4×D4—41no—
57C2×C4×D4—41no—
58C2×C4×D4—41no—
59C2×C4×D4—41no—
60C2×C4×D4D414yes—
61C2×C4×D4Q814yes—
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C412yes—
70C22×C42C2×C412yes—
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2312yes—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×C4⋊C8—42no—
77C2×C4⋊C8—42no—
78C2×C4⋊C8—42no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—42no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8—41no—
88C2×C22⋊C8—41no—
89C2×C22⋊C8—41no—
90C2×C22⋊C8—41no—
91#390—28no—
92C4×C4⋊C4—41no—
93C4×C4⋊C4—41no—
94C4×C22⋊C4—42no—
95C4×C22⋊C4—42no—
96C4×C22⋊C4—42no—
97C4×C22⋊C4—42no—
98C2×#95C2×C412yes—
99C2×#95C2×C412yes—
100C2×#95C2×C412yes—
101C2×#95C2×C412yes—
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C43—41no—
105C43—41no—
106#336—24no—
107#336—24no—
108#336—24no—
109#336—24no—
110#336—42no—
111#336—42no—
112#336—42no—
113#336—42no—
114#336—42no—
115C25C22×C411yescharacteristic
116C25C22×C411yescharacteristic
117C22×D4—24no—
118C22×D4—24no—
119C22×D4—41no—
120C22×D4—41no—
121C22×D4C4⋊C414yes—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4C22×C412yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C412yes—
143C23×C4C22×C412yes—
144C23×C4C22×C412yes—
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×Q812yes—
149C23×C4C2×Q812yes—
150C23×C4C2411yescharacteristic
151C2×M16—44no—
152C2×M16—48no—
153C2×M16—48no—
154C2×M16—44no—
155C2×M16—44no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
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—44no—
183C22×C8—48no—
184C22×C8—48no—
185C22×C8—44no—
186C22×C8—44no—
187C22×C8#4114yes—
188C22×C8#4114yes—
189C4×D4—44no—
190C4×D4—44no—
191C4×D4—44no—
192C4×D4—44no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4C4⋊C414yes—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4C4⋊C414yes—
215C2×C22⋊C4C4⋊C414yes—
216C2×C22⋊C4M1618yescomplemented
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—42no—
237C2×C42—41no—
238C2×C42—41no—
239C2×C42—42no—
240C2×C42C2×D412yes—
241C2×C42C2×Q812yes—
242C4⋊C8—48no—
243C22⋊C8—48no—
244C22⋊C8—48no—
245C22⋊C8—44no—
246C22⋊C8—44no—
247#95—42no—
248#95—42no—
249#95—42no—
250#95—42no—
251#95—41no—
252#95—41no—
253C24—22no—
254C24—22no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24C2×C4⋊C412yes—
269C24C2×C4⋊C412yes—
270C24C2×M1614yes—
271C24C23×C411yescharacteristic
272C2×D4—28no—
273C2×D4—28no—
274C2×D4—44no—
275C2×D4—48no—
276C2×D4—48no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
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—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
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—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
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—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4C2×C4⋊C412yes—
354C22×C4C2×C4⋊C412yes—
355C22×C4C2×C4⋊C412yes—
356C22×C4C2×C4⋊C412yes—
357C22×C4C2×M1614yes—
358C22×C4C2×M1614yes—
359C22×C4#13114yes—
360C22×C4#13114yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×Q811yescharacteristic
363C22×C4C2×#4112yes—
364C22×C4C2×#4112yes—
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C4H(5,𝔽2)11yescharacteristic
368C22×C4H(5,𝔽2)11yescharacteristic
369C22×C4H(5,𝔽2)12yes—
370C22×C4H(5,𝔽2)12yes—
371M16—416no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—416no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C4⋊C4—44no—
402C4⋊C4—48no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—48no—
406C22⋊C4—48no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—48no—
414C42—44no—
415C42—44no—
416C42—42no—
417C42—42no—
418C42—42no—
419C42—42no—
420C42—42no—
421C42—42no—
422C42—42no—
423C42—42no—
424C23—22no—
425C23—24no—
426C23—22no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—22no—
435C23—24no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—42no—
440C23—41no—
441C23—41no—
442C23—42no—
443C23—42no—
444C23—41no—
445C23—41no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—41no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23#42314yes—
466C23C22×C4⋊C411yescharacteristic
467C23C4×#4112yes—
468C23#51811yescharacteristic
469C23#51811yescharacteristic
470C23#51812yes—
471C23C22×M1612yes—
472C23C2×#13112yes—
473C23C2×#13112yes—
474C23#56812yes—
475D4—416no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4#42414yes—
536C2×C4#53411yescharacteristic
537C2×C4#53411yescharacteristic
538C2×C4#53812yes—
539C2×C4#53812yes—
540C2×C4#54012yes—
541C2×C4#54012yes—
542C2×C4#54311yescharacteristic
543C2×C4#54311yescharacteristic
544C2×C4#54612yes—
545C2×C4#54612yes—
546C2×C4#55111yescharacteristic
547C2×C4#55111yescharacteristic
548C2×C4#55412yes—
549C2×C4#55412yes—
550C2×C4#55512yes—
551C2×C4#56011yescharacteristic
552C2×C4#56011yescharacteristic
553C2×C4#56012yes—
554C8—44no—
555C8—44no—
556C8—82no—
557C8—82no—
558C22—28no—
559C22—24no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—42no—
564C22—44no—
565C22—42no—
566C22—44no—
567C22—42no—
568C22—44no—
569C22—42no—
570C22—44no—
571C22—42no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22#231511yescharacteristic
577C22#233312yes—
578C22#234912yes—
579C22#237211yescharacteristic
580C22#238011yescharacteristic
581C22C2×#42311yescharacteristic
582C22C2×#42411yescharacteristic
583C22#290612yes—
584C22#296512yes—
585C22#296612yes—
586C22#296912yes—
587C22#297012yes—
588C22#297312yes—
589C22#297412yes—
590C22#297412yes—
591C22#297412yes—
592C22#297711yescharacteristic
593C22#297811yescharacteristic
594C22#297912yes—
595C22#298112yes—
596C22#298112yes—
597C4—24no—
598C4—44no—
599C4—44no—
600C4—44no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—82no—
605C4#269711yescharacteristic
606C4#269711yescharacteristic
607C4#269912yes—
608C4#274811yescharacteristic
609C4#274811yescharacteristic
610C4#278412yes—
611C2—44no—
612C2—82no—
613C2#1448611yescharacteristic
614C2#2193211yescharacteristic
615C2#2193711yescharacteristic
616C2#2233312yes—
617C2#2233512yes—
618C2#2269812yes—
619C2#2270112yes—
620C2#2327012yes—
621C2#2335612yes—
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. #11202C2semidirect
5. #11042C2semidirect
6. #11042C2semidirect
7. #10795C2semidirect
8. #10715C2semidirect
9. #10714C2semidirect
216. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#18702
C2×#18692
C4×#3902
#112022
#110422
#107952
#107152
#107142