Finite Group Databasebeta

← #388980

Subgroups — #388980

Order 512.

1790 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×#1869C212yesmaximal, complemented
2C2×#1869C212yesmaximal, complemented
3C4×#392C211yesmaximal, characteristic
4#11298C212yesmaximal, complemented
5#11147C211yesmaximal, characteristic, complemented
6#10964C212yesmaximal, complemented
7#10921C211yesmaximal, characteristic, complemented
8#10921C212yesmaximal, complemented
9#10714C212yesmaximal, complemented
10C24×C8C2212yes—
11C22×C4×D4C2211yescharacteristic
12C22×C4×D4C2212yes—
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19C22×C4×C8C2211yescharacteristic
20C2×#392C411yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C2212yes—
23#1869—28no—
24#1869—28no—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33#1444C411yescharacteristic
34#1439C412yes—
35#1439C411yescharacteristic
36#1438C412yes—
37#1438C411yescharacteristic
38C4×#95C2211yescharacteristic
39C23×D4C2×C411yescharacteristic
40C23×D4C2×C412yes—
41C24×C4C2311yescharacteristic
42C24×C4C2312yes—
43C23×C8—24no—
44C23×C8—24no—
45C23×C8—24no—
46C23×C8—42no—
47C23×C8—42no—
48C23×C8—42no—
49C23×C8—42no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4×D4—24no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—42no—
59C2×C4×D4—42no—
60C2×C4×D4—41no—
61C2×C4×D4—41no—
62C2×C4×D4—42no—
63C2×C4×D4—42no—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C412yes—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2312yes—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8—42no—
85C2×C22⋊C8—41no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—41no—
91C2×C22⋊C8D418yescomplemented
92C2×C4×C8—42no—
93C2×C4×C8—41no—
94#392C818yescomplemented
95C4×C22⋊C4—42no—
96C4×C22⋊C4—41no—
97C4×C22⋊C4—42no—
98C4×C22⋊C4—41no—
99C2×#95C2×C412yes—
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C412yes—
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C412yes—
105C2×#95C2×C411yescharacteristic
106#336—24no—
107#336—24no—
108#336—24no—
109#336—24no—
110#336—42no—
111#336—42no—
112#336—41no—
113#336—42no—
114C25C22×C411yescharacteristic
115C25C22×C412yes—
116C22×D4—24no—
117C22×D4—24no—
118C22×D4—24no—
119C22×D4—24no—
120C22×D4—41no—
121C22×D4—42no—
122C22×D4—42no—
123C22×D4—41no—
124C22×D4—42no—
125C22×D4—42no—
126C22×D4C2×C814yes—
127C22×D4C2×C818yescomplemented
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C412yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D414yes—
157C23×C4#4114yes—
158C23×C4C2411yescharacteristic
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—22no—
183C22×C8—22no—
184C22×C8—48no—
185C22×C8—48no—
186C22×C8—48no—
187C22×C8—48no—
188C22×C8—48no—
189C22×C8—48no—
190C22×C8—48no—
191C22×C8C2×D414yes—
192C22×C8C2×D414yes—
193C22×C8#4114yes—
194C4×D4—44no—
195C4×D4—48no—
196C4×D4—44no—
197C4×D4—48no—
198C4×D4—48no—
199C4×D4—48no—
200C4×D4—48no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4C2×C814yescomplemented
226C2×C22⋊C4C2×C818yescomplemented
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—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—21no—
241C22⋊C8—48no—
242C22⋊C8—44no—
243C22⋊C8—48no—
244C22⋊C8—48no—
245C22⋊C8—48no—
246C22⋊C8—44no—
247C4×C8—44no—
248#95—42no—
249#95—41no—
250#95—42no—
251#95—42no—
252#95—41no—
253#95—41no—
254#95C2×C814yescomplemented
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—21no—
273C24—22no—
274C24—21no—
275C24—21no—
276C24—22no—
277C24C4×D414yes—
278C24C22×C812yes—
279C24C22×C814yes—
280C24#13114yes—
281C24C23×C411yescharacteristic
282C2×D4—216no—
283C2×D4—216no—
284C2×D4—44no—
285C2×D4—48no—
286C2×D4—48no—
287C2×D4—44no—
288C2×D4—48no—
289C2×D4—44no—
290C2×D4—48no—
291C2×D4—48no—
292C2×D4—48no—
293C2×D4—48no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
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—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
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—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
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—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4C4×D414yes—
380C22×C4C22×C812yes—
381C22×C4C22×C814yes—
382C22×C4C22×C812yes—
383C22×C4#13112yes—
384C22×C4C22×D412yes—
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C4H(5,𝔽2)12yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—432no—
402C2×C8—42no—
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—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—44no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—44no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C42—42no—
431C42—42no—
432C42—42no—
433C42—42no—
434C42—41no—
435C42—42no—
436C42—41no—
437C42—41no—
438C42—42no—
439C42—41no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—42no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23—42no—
494C23—42no—
495C23—41no—
496C23—41no—
497C23C8×D418yes—
498C23C2×C4×D412yes—
499C23C4×#4111yescharacteristic
500C23#51811yescharacteristic
501C23#51812yes—
502C23#51811yescharacteristic
503C23C23×C811yescharacteristic
504C23C2×#13112yes—
505C23C2×#13111yescharacteristic
506C23#56811yescharacteristic
507C23#56812yes—
508C23#56811yescharacteristic
509D4—416no—
510D4—432no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4#53412yes—
570C2×C4#53512yes—
571C2×C4#53812yes—
572C2×C4#53812yes—
573C2×C4#53812yes—
574C2×C4#54012yes—
575C2×C4D4211yescharacteristic
576C2×C4#54612yes—
577C2×C4#54612yes—
578C2×C4#54612yes—
579C2×C4#55111yescharacteristic
580C2×C4#55312yes—
581C2×C4#55312yes—
582C2×C4#56012yes—
583C2×C4#56011yescharacteristic
584C2×C4#56011yescharacteristic
585C8—44no—
586C8—44no—
587C8—44no—
588C8—82no—
589C22—28no—
590C22—28no—
591C22—28no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—82no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—82no—
614C22—81no—
615C22#232112yes—
616C22#233311yescharacteristic
617C22#235811yescharacteristic
618C22#236412yes—
619C22#238011yescharacteristic
620C22C2×C8×D412yes—
621C22#293012yes—
622C22#293312yes—
623C22#293512yes—
624C22C8×#4111yescharacteristic
625C22#296711yescharacteristic
626C22#296812yes—
627C22#297211yescharacteristic
628C22#297212yes—
629C22#297211yescharacteristic
630C22#297612yes—
631C22#297612yes—
632C22#297611yescharacteristic
633C22#297712yes—
634C22#297711yescharacteristic
635C22#298012yes—
636C22#298011yescharacteristic
637C22#298111yescharacteristic
638C4—44no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—82no—
644C4—81no—
645C4—81no—
646C4#267812yes—
647C4#268812yes—
648C4#270312yes—
649C4#277812yes—
650C2—48no—
651C2—82no—
652C2#1453311yescharacteristic
653C2#2202812yes—
654C2#2203712yes—
655C2#2232911yescharacteristic
656C2#2234011yescharacteristic
657C2#2284211yescharacteristic
658C2#2284511yescharacteristic
659C2#2311612yes—
660C2#2312412yes—
661C2#2335511yescharacteristic
662C2#2335811yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC2×#392C4
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
1. C2×#1869C2semidirect
2. C2×#1869C2semidirect
4. #11298C2semidirect
5. #11147C2semidirect
6. #10964C2semidirect
7. #10921C2semidirect
8. #10921C2semidirect
9. #10714C2semidirect
91. C2×C22⋊C8D4semidirect
94. #392C8semidirect
127. C22×D4C2×C8semidirect
225. C2×C22⋊C4C2×C8semidirect
226. C2×C22⋊C4C2×C8semidirect
254. #95C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#18692
C4×#3922
#112982
#111472
#109642
#109212
#107142