Finite Group Databasebeta

← #387418

Subgroups — #387418

Order 512.

1552 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2398C211yesmaximal, characteristic
2C2×#1870C211yesmaximal, characteristic
3C2×#1869C211yesmaximal, characteristic
4#11380C212yesmaximal, complemented
5#11147C212yesmaximal, complemented
6#11147C212yesmaximal, complemented
7#10796C212yesmaximal, complemented
8#10715C212yesmaximal, complemented
9#10714C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18#2398—24no—
19#2398—24no—
20C2×#400C412yes—
21C2×#397C411yescharacteristic
22C2×#397C411yescharacteristic
23C2×#394C412yes—
24C2×#393C411yescharacteristic
25C2×#393C411yescharacteristic
26C2×C4×C22⋊C4C2212yes—
27C2×C4×C22⋊C4C2212yes—
28C2×#376C2211yescharacteristic
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×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—41no—
62C2×C4×Q8—41no—
63C2×C4×Q8—41no—
64C2×C4×Q8—41no—
65C2×C4×D4—41no—
66C2×C4×D4—41no—
67C2×C4×D4—41no—
68C2×C4×D4—41no—
69C2×C4×D4D414yes—
70C2×C4×D4D414yes—
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C42C2312yes—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83C2×C22⋊C8—24no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—42no—
91C2×C22⋊C8—41no—
92C2×C22⋊C8—41no—
93C2×C22⋊C8—41no—
94C2×C22⋊C8—41no—
95#400—28no—
96#397—24no—
97#397—24no—
98#394—28no—
99#393—24no—
100#393—24no—
101C4×C22⋊C4—24no—
102C4×C22⋊C4—24no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—24no—
105#376—22no—
106#376—22no—
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114#336—24no—
115#336—24no—
116#336—24no—
117#336—24no—
118#336—42no—
119#336—42no—
120#336—42no—
121#336—42no—
122#336—42no—
123#336—42no—
124C25C22×C411yescharacteristic
125C25C22×C411yescharacteristic
126C22×Q8—22no—
127C22×Q8—24no—
128C22×Q8—41no—
129C22×Q8—41no—
130C22×Q8—42no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—41no—
134C22×D4—41no—
135C22×D4—41no—
136C22×D4—41no—
137C22×D4C22⋊C414yes—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4C22×C412yes—
155C23×C4C22×C412yes—
156C23×C4C22×C412yes—
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2411yescharacteristic
167C2×M16—44no—
168C2×M16—48no—
169C2×M16—48no—
170C2×M16—44no—
171C2×M16—44no—
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—22no—
198C22×C8—44no—
199C22×C8—48no—
200C22×C8—48no—
201C22×C8—44no—
202C22×C8—44no—
203C22×C8#4114yes—
204C22×C8#4114yes—
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—24no—
214C2×C4⋊C4—22no—
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—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4C22⋊C414yes—
239C2×C22⋊C4C22⋊C414yes—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C22⋊C8—48no—
259C22⋊C8—48no—
260C22⋊C8—44no—
261C22⋊C8—44no—
262#95—22no—
263#95—22no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269C24—22no—
270C24—21no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—21no—
276C24—21no—
277C24—22no—
278C24—21no—
279C24—21no—
280C24—22no—
281C24—22no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—22no—
287C24C2×C22⋊C412yes—
288C24C2×C22⋊C412yes—
289C24#13112yes—
290C24#13112yes—
291C24C23×C411yescharacteristic
292C2×Q8—48no—
293C2×Q8—44no—
294C2×Q8—44no—
295C2×Q8—48no—
296C2×Q8—44no—
297C2×D4—28no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—44no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
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—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4C2×C22⋊C412yes—
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C22⋊C412yes—
388C22×C4C2×C22⋊C412yes—
389C22×C4#13114yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C2×#4112yes—
393C22×C4C2×#4112yes—
394C22×C4H(5,𝔽2)11yescharacteristic
395C22×C4H(5,𝔽2)11yescharacteristic
396C22×C4H(5,𝔽2)11yescharacteristic
397C22×C4H(5,𝔽2)11yescharacteristic
398C22×C4H(5,𝔽2)12yes—
399C22×C42−1+412yes—
400M16—416no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—416no—
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—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—48no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—42no—
444C42—42no—
445C42—42no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C23—22no—
452C23—24no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—42no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23#40814yes—
497C23C22×C22⋊C411yescharacteristic
498C23C4×#4112yes—
499C23#51811yescharacteristic
500C23#51811yescharacteristic
501C23#52012yes—
502C23C2×#13111yescharacteristic
503C23C2×#13111yescharacteristic
504C23C2×#13112yes—
505C23#56811yescharacteristic
506C23#56811yescharacteristic
507C23#56812yes—
508Q8—416no—
509D4—416no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
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—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
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—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4#53411yescharacteristic
573C2×C4#53411yescharacteristic
574C2×C4#53411yescharacteristic
575C2×C4#53411yescharacteristic
576C2×C4#53812yes—
577C2×C4#53812yes—
578C2×C4#54212yes—
579C2×C4#54212yes—
580C2×C4#54612yes—
581C2×C4#54612yes—
582C2×C4#54712yes—
583C2×C4#54712yes—
584C2×C4#55111yescharacteristic
585C2×C4#55111yescharacteristic
586C2×C4#55512yes—
587C2×C4H(3,𝔽4)11yescharacteristic
588C2×C4#56212yes—
589C2×C4#56311yescharacteristic
590C8—44no—
591C8—44no—
592C8—82no—
593C8—82no—
594C22—28no—
595C22—24no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—44no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—44no—
607C22—44no—
608C22—42no—
609C22—42no—
610C22—81no—
611C22—81no—
612C22—82no—
613C22—82no—
614C22—81no—
615C22—81no—
616C22#230811yescharacteristic
617C22#233712yes—
618C22#233912yes—
619C22#237211yescharacteristic
620C22#237911yescharacteristic
621C22C2×#40811yescharacteristic
622C22#287911yescharacteristic
623C22#288012yes—
624C22#296612yes—
625C22#296712yes—
626C22#296812yes—
627C22#296912yes—
628C22#297611yescharacteristic
629C22#297611yescharacteristic
630C22#297611yescharacteristic
631C22#297612yes—
632C22#297611yescharacteristic
633C22#297711yescharacteristic
634C22#297811yescharacteristic
635C22#299212yes—
636C22#299312yes—
637C22#299412yes—
638C22#299512yes—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—44no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4#268411yescharacteristic
648C4#268411yescharacteristic
649C4#268612yes—
650C4#268612yes—
651C4#271711yescharacteristic
652C4#271711yescharacteristic
653C2—44no—
654C2—82no—
655C2#1447311yescharacteristic
656C2#2174111yescharacteristic
657C2#2174411yescharacteristic
658C2#2239012yes—
659C2#2239212yes—
660C2#2247012yes—
661C2#2247212yes—
662C2#2327511yescharacteristic
663C2#2327711yescharacteristic
664C2#2334811yescharacteristic
665C2#2335411yescharacteristic
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. #10796C2semidirect
8. #10715C2semidirect
9. #10714C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23982
C2×#18702
C2×#18692
#113802
#111472
#107962
#107152
#107142