Finite Group Databasebeta

← #582877

Subgroups — #582877

Order 512.

2420 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2#14658C212yesmaximal
3#14650C212yesmaximal, complemented
4#14642C214yesmaximal, complemented
5#14600C212yesmaximal, complemented
6#14506C212yesmaximal
7#14368C214yesmaximal, complemented
8#14266C212yesmaximal, complemented
9#14129C212yesmaximal, complemented
10#14045C212yesmaximal, complemented
11#13993C212yesmaximal, complemented
12#13974C212yesmaximal, complemented
13#13907C212yesmaximal, complemented
14#13871C211yesmaximal, characteristic
15#13847C211yesmaximal, characteristic
16C22×#123C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2486C4116yescomplemented
21C2×#398C2212yes—
22C2×#397C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#388C2214yes—
28C2×#381C2212yes—
29C2×#380C2214yes—
30C2×#380C2212yes—
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×#376C2211yescharacteristic
39#1445C2211yescharacteristic
40#1445C2211yescharacteristic
41#1445C2214yes—
42#1444C2214yes—
43#1443C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1440C2214yes—
52#1440C2212yes—
53#1439C2212yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2211yescharacteristic
59#1438C2211yescharacteristic
60#1438C2212yes—
61#1437C2212yes—
62#1437C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2212yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C22⋊#95C2214yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×#123—48no—
81C2×#123C2×C4116yes—
82C2×C4×D4—42no—
83C2×C4×D4—42no—
84C2×C4×D4C2×C418yes—
85C2×#117—44no—
86C2×#117—42no—
87C22×C4⋊C4—42no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4C2×C418yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2314yes—
117#398C2×C4116yes—
118#397C2×C4116yes—
119#389—44no—
120#389—44no—
121#389—44no—
122#389C2×C4116yes—
123#388—48no—
124#381C2×C4116yes—
125#380—48no—
126#380—44no—
127#380—42no—
128#380C2×C4116yes—
129#380C2×C418yes—
130C4×C4⋊C4—42no—
131C4×C4⋊C4—44no—
132C4×C22⋊C4—44no—
133C4×C22⋊C4—44no—
134#376—42no—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2311yescharacteristic
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C25C2411yescharacteristic
164C25C2411yescharacteristic
165C22×D4—24no—
166C22×D4—41no—
167C22×D4—41no—
168C22×D4—42no—
169C22×D4C22×C414yes—
170C23×C4—28no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—44no—
182C23×C4C22×C414yes—
183C23×C4C22×C414yes—
184C23×C4C22×C418yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2414yes—
191C23×C4C2411yescharacteristic
192C23×C4C2414yes—
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198#123—432no—
199C4×D4—416no—
200#117—416no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—416no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C22×C418yes—
231C2×C4⋊C4C22×C418yes—
232C2×C4⋊C4C22×C418yes—
233C2×C4⋊C4C22×C414yes—
234C2×C4⋊C4C22×C4116yes—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4C22×C418yes—
262C2×C22⋊C4C22×C414yes—
263C2×C22⋊C4C22×C414yes—
264C2×C22⋊C4C22×C4116yes—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—44no—
274C2×C42—42no—
275C2×C42—44no—
276C2×C42—44no—
277C2×C42—44no—
278C2×C42C22×C414yes—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—42no—
286#95—41no—
287#95—41no—
288#95—42no—
289#95—42no—
290#95—42no—
291#95—42no—
292#95—41no—
293#95—41no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—42no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95C22×C4116yes—
305#95C22×C418yes—
306#95C22×C4116yes—
307C24—22no—
308C24—21no—
309C24—21no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—21no—
317C24—21no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24C23×C412yes—
323C24C23×C412yes—
324C24H(5,𝔽2)14yes—
325C24C2511yescharacteristic
326C2×D4—28no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—48no—
330C2×D4—48no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—216no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—88no—
414C22×C4—88no—
415C22×C4C23×C412yes—
416C22×C4C23×C412yes—
417C22×C4C23×C418yes—
418C22×C4C23×C412yes—
419C22×C4C23×C418yes—
420C22×C4C23×C412yes—
421C22×C4C23×C412yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C4H(5,𝔽2)12yes—
424C22×C4H(5,𝔽2)14yes—
425C22×C42−1+414yes—
426C22×C42−1+414yes—
427C4⋊C4—216no—
428C4⋊C4—216no—
429C4⋊C4—48no—
430C4⋊C4—44no—
431C4⋊C4—416no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—88no—
436C4⋊C4—88no—
437C4⋊C4—88no—
438C22⋊C4—216no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—416no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—88no—
448C22⋊C4—84no—
449C42—28no—
450C42—48no—
451C42—48no—
452C42—48no—
453C42—88no—
454C23—22no—
455C23—24no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—22no—
468C23—24no—
469C23—41no—
470C23—42no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—42no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23#51814yes—
496C23C24×C411yescharacteristic
497C23C2×H(5,𝔽2)11yescharacteristic
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×H(5,𝔽2)12yes—
500C23C2×H(5,𝔽2)12yes—
501C23C2×2−1+412yes—
502C23C2×2−1+412yes—
503C23#58512yes—
504C23#58512yes—
505D4—416no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—416no—
548C2×C4—416no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—816no—
585C2×C4#51814yes—
586C2×C4#52018yes—
587C2×C4#56014yes—
588C2×C4#56218yes—
589C2×C4#56314yes—
590C22—24no—
591C22—28no—
592C22—42no—
593C22—44no—
594C22—44no—
595C22—42no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22C2×#51811yescharacteristic
609C22C2×#51811yescharacteristic
610C22C2×#52012yes—
611C22#342811yescharacteristic
612C22C4×2−1+412yes—
613C22C2×#56011yescharacteristic
614C22C2×#56212yes—
615C22C2×#56311yescharacteristic
616C22#353612yes—
617C22#353912yes—
618C22#354112yes—
619C22#354112yes—
620C22#354212yes—
621C22#354212yes—
622C22#354212yes—
623C22#354412yes—
624C22#355011yescharacteristic
625C22#355111yescharacteristic
626C22#355512yes—
627C22#355912yes—
628C22#356512yes—
629C4—42no—
630C4—42no—
631C4—48no—
632C4—42no—
633C4—44no—
634C4—42no—
635C4—48no—
636C4—82no—
637C4—84no—
638C4—82no—
639C4—82no—
640C4—84no—
641C4#237618yes—
642C2—44no—
643C2—82no—
644C2C2×#237611yescharacteristic
645C2#3436311yescharacteristic
646C2#3441311yescharacteristic
647C2#3446012yes—
648C2#3446212yes—
649C2#5309712yes—
650C2#5309812yes—
651C2#5313312yes—
652C2#5313412yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #14650C2semidirect
4. #14642C2semidirect
5. #14600C2semidirect
7. #14368C2semidirect
8. #14266C2semidirect
9. #14129C2semidirect
10. #14045C2semidirect
11. #13993C2semidirect
12. #13974C2semidirect
13. #13907C2semidirect
20. #2486C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
#146582
#146502
#146422
#146002
#145062
#143682
#142662
#141292
#140452
#139932
#139742
#139072
#138712
#138472