Finite Group Databasebeta

← #388052

Subgroups — #388052

Order 512.

1590 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
1C2×#1954C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3C4×#386C211yesmaximal, characteristic
4#11255C212yesmaximal, complemented
5#11042C212yesmaximal, complemented
6#11042C212yesmaximal, complemented
7#10876C212yesmaximal, complemented
8#10714C212yesmaximal, complemented
9#10712C212yesmaximal, complemented
10C24×C8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C2×#386C411yescharacteristic
21C2×#386C411yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C2212yes—
24#1954—24no—
25#1869—24no—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1437C412yes—
35#1436C412yes—
36C22⋊#95C411yescharacteristic
37C22⋊#95C411yescharacteristic
38C4×#95C2212yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C23×C8—22no—
45C23×C8—24no—
46C23×C8—41no—
47C23×C8—41no—
48C23×C8—42no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4×D4—41no—
54C2×C4×D4—41no—
55C2×C4×D4—41no—
56C2×C4×D4—41no—
57C2×C4×D4D414yes—
58C2×C4×D4Q814yes—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2×C412yes—
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C42—41no—
67C22×C42—41no—
68C22×C42—42no—
69C22×C42—42no—
70C22×C42C2×C412yes—
71C22×C42C2312yes—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C2×C4⋊C8—24no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—41no—
77C2×C4⋊C8—41no—
78C2×C4⋊C8—42no—
79C2×C4⋊C8—42no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—41no—
91C2×C22⋊C8—41no—
92C2×C4×C8—22no—
93C2×C4×C8—41no—
94C2×C4×C8—41no—
95#386—24no—
96#386C818yescomplemented
97C4×C22⋊C4—41no—
98C4×C22⋊C4—42no—
99C4×C22⋊C4—41no—
100C4×C22⋊C4—42no—
101C2×#95C2×C412yes—
102C2×#95C2×C412yes—
103C2×#95C2×C412yes—
104C2×#95C2×C412yes—
105C2×#95C2×C412yes—
106C2×#95C2×C412yes—
107C2×#95C2×C412yes—
108#336—24no—
109#336—24no—
110#336—42no—
111#336—42no—
112#336—42no—
113#336—42no—
114#336—42no—
115#336—42no—
116C25C22×C411yescharacteristic
117C25C22×C411yescharacteristic
118C22×D4—41no—
119C22×D4—41no—
120C22×D4—41no—
121C22×D4—41no—
122C22×D4C4⋊C414yes—
123C22×D4C2×C814yes—
124C22×D4M1614yes—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—42no—
144C23×C4—41no—
145C23×C4—41no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—41no—
149C23×C4—41no—
150C23×C4C2×C814yescomplemented
151C23×C4C22×C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C412yes—
156C23×C4C22×C412yes—
157C23×C4C22×C412yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×Q812yes—
162C23×C4C2×Q812yes—
163C23×C4C2411yescharacteristic
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—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—44no—
191C22×C8—48no—
192C22×C8—48no—
193C22×C8—44no—
194C22×C8—44no—
195C22×C8#4114yes—
196C22×C8#4114yes—
197C4×D4—44no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4—44no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4C4⋊C414yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4C4⋊C414yes—
219C2×C22⋊C4C4⋊C414yes—
220C2×C22⋊C4C2×C818yes—
221C2×C22⋊C4C2×C814yescomplemented
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
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—22no—
237C2×C42—22no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—48no—
242C2×C42—44no—
243C2×C42—44no—
244C2×C42C2×D412yes—
245C2×C42C2×Q812yes—
246C4⋊C8—44no—
247C4⋊C8—48no—
248C22⋊C8—48no—
249C22⋊C8—48no—
250C22⋊C8—48no—
251C22⋊C8—44no—
252C4×C8—44no—
253#95—24no—
254#95—24no—
255#95—42no—
256#95—42no—
257#95—42no—
258#95—42no—
259#95C2×C818yescomplemented
260C24—22no—
261C24—22no—
262C24—21no—
263C24—22no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—21no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24—21no—
275C24—21no—
276C24C2×C4⋊C412yes—
277C24C2×C4⋊C412yes—
278C24C22×C812yes—
279C24C22×C812yes—
280C24C2×M1612yes—
281C24C2×M1612yes—
282C24C23×C411yescharacteristic
283C2×D4—44no—
284C2×D4—44no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—44no—
288C2×D4—44no—
289C2×D4C4⋊C8116yescomplemented
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
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—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
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—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—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—48no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—48no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4C2×C4⋊C412yes—
378C22×C4C2×C4⋊C412yes—
379C22×C4C2×C4⋊C412yes—
380C22×C4C2×C4⋊C412yes—
381C22×C4C22×C814yes—
382C22×C4C22×C814yes—
383C22×C4C22×C812yes—
384C22×C4C2×M1612yes—
385C22×C4#13114yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×Q811yescharacteristic
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×C4H(5,𝔽2)12yes—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—416no—
403C2×C8—42no—
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—
425C4⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—48no—
429C22⋊C4—44no—
430C22⋊C4—48no—
431C22⋊C4—44no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—48no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—42no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—24no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—22no—
458C23—24no—
459C23—22no—
460C23—24no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23C2×C4⋊C814yes—
490C23C2×C4⋊C814yes—
491C23C22×C4⋊C411yescharacteristic
492C23C4×#4112yes—
493C23#51811yescharacteristic
494C23#51811yescharacteristic
495C23#51812yes—
496C23C23×C811yescharacteristic
497C23C22×M1611yescharacteristic
498C23C2×#13112yes—
499C23#56811yescharacteristic
500C23#56811yescharacteristic
501C23#56812yes—
502D4—416no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
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—24no—
523C2×C4—24no—
524C2×C4—416no—
525C2×C4—416no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4C2×C4⋊C814yes—
567C2×C4#53411yescharacteristic
568C2×C4#53411yescharacteristic
569C2×C4#53812yes—
570C2×C4#53812yes—
571C2×C4#54012yes—
572C2×C4#54012yes—
573C2×C4#54311yescharacteristic
574C2×C4#54311yescharacteristic
575C2×C4#54612yes—
576C2×C4#54612yes—
577C2×C4#55111yescharacteristic
578C2×C4#55312yes—
579C2×C4#55412yes—
580C2×C4#55412yes—
581C2×C4#55911yescharacteristic
582C2×C4#56012yes—
583C2×C4#56011yescharacteristic
584C2×C4H(3,𝔽4)11yescharacteristic
585C8—44no—
586C8—44no—
587C8—82no—
588C8—82no—
589C22—28no—
590C22—28no—
591C22—24no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22#231511yescharacteristic
612C22#233312yes—
613C22#234912yes—
614C22#237411yescharacteristic
615C22#237511yescharacteristic
616C22C22×C4⋊C811yescharacteristic
617C22#290511yescharacteristic
618C22#290612yes—
619C22C8×#4112yes—
620C22#296712yes—
621C22#296912yes—
622C22#297012yes—
623C22#297212yes—
624C22#297211yescharacteristic
625C22#297211yescharacteristic
626C22#297311yescharacteristic
627C22#297311yescharacteristic
628C22#297412yes—
629C22#297612yes—
630C22#297711yescharacteristic
631C22#297811yescharacteristic
632C22#297812yes—
633C22#298112yes—
634C4—24no—
635C4—44no—
636C4—44no—
637C4—44no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—82no—
642C4#268411yescharacteristic
643C4#268812yes—
644C4#270911yescharacteristic
645C4#274811yescharacteristic
646C4#278012yes—
647C4#280411yescharacteristic
648C2—44no—
649C2—44no—
650C2#1451311yescharacteristic
651C2#2193011yescharacteristic
652C2#2193711yescharacteristic
653C2#2232912yes—
654C2#2233512yes—
655C2#2269612yes—
656C2#2269912yes—
657C2#2329411yescharacteristic
658C2#2330211yescharacteristic
659C2#2330611yescharacteristic
660C2#2330711yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ 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. #11255C2semidirect
5. #11042C2semidirect
6. #11042C2semidirect
7. #10876C2semidirect
8. #10714C2semidirect
9. #10712C2semidirect
96. #386C8semidirect
150. C23×C4C2×C8semidirect
221. C2×C22⋊C4C2×C8semidirect
259. #95C2×C8semidirect
289. C2×D4C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#18692
C4×#3862
#112552
#110422
#108762
#107142
#107122