Finite Group Databasebeta

← #558264

Subgroups — #558264

Order 512.

2630 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×#2486C211yesmaximal, characteristic
2#14697C212yesmaximal, complemented
3#14650C212yesmaximal, complemented
4#14600C212yesmaximal, complemented
5#14542C214yesmaximal, complemented
6#14507C212yesmaximal, complemented
7#14359C214yesmaximal
8#14132C212yesmaximal, complemented
9#14006C212yesmaximal, complemented
10#13993C212yesmaximal, complemented
11#13974C212yesmaximal, complemented
12#13907C212yesmaximal, complemented
13#13905C211yesmaximal, characteristic
14#13828C212yesmaximal, complemented
15C4×#386C211yesmaximal, characteristic
16C22×#123C2212yes—
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2486C4116yescomplemented
21C2×#398C2212yes—
22C2×#397C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#388C2214yes—
28C2×#386C2212yes—
29C2×#385C2214yes—
30C2×#382C2211yescharacteristic
31C2×#381C2212yes—
32C2×#380C2214yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41#1445C2212yes—
42#1445C2214yes—
43#1443C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1440C2214yes—
51#1439C2214yes—
52#1438C2212yes—
53#1437C2211yescharacteristic
54#1437C2214yes—
55#1437C2212yes—
56#1437C2212yes—
57#1437C2211yescharacteristic
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2211yescharacteristic
63#1436C2214yes—
64#1436C2214yes—
65#1436C2211yescharacteristic
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C4×#95C2212yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2312yes—
81C2×#123—48no—
82C2×#123C2×C4116yes—
83C2×C4×D4—42no—
84C2×C4×D4—42no—
85C2×C4×D4C2×C418yes—
86C22×C4⋊C4—42no—
87C22×C4⋊C4—42no—
88C22×C4⋊C4C2×C418yescomplemented
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C42—44no—
109C22×C42—42no—
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2314yes—
117#398C2×C4116yescomplemented
118#397C2×C4116yes—
119#389—44no—
120#389—44no—
121#389—44no—
122#389C2×C4116yescomplemented
123#388—48no—
124#386—28no—
125#385—48no—
126#382—42no—
127#381C2×C4116yescomplemented
128#380—216no—
129#380—44no—
130#380—44no—
131#380C2×C4116yes—
132#380C2×C418yescomplemented
133C4×C4⋊C4—42no—
134C4×C4⋊C4—44no—
135C4×C22⋊C4—44no—
136C4×C22⋊C4—42no—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C25C2411yescharacteristic
166C25C2411yescharacteristic
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—41no—
170C22×D4—41no—
171C22×D4C22×C414yes—
172C23×C4—28no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—44no—
191C23×C4C22×C414yes—
192C23×C4C22×C414yes—
193C23×C4C22×C418yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207#123—432no—
208C4×D4—416no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—416no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C22×C418yes—
236C2×C4⋊C4C22×C418yes—
237C2×C4⋊C4C22×C418yes—
238C2×C4⋊C4C22×C414yes—
239C2×C4⋊C4C22×C4116yes—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—44no—
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—44no—
261C2×C22⋊C4C22×C4116yes—
262C2×C22⋊C4C22×C414yes—
263C2×C22⋊C4C22×C414yes—
264C2×C22⋊C4C22×C418yes—
265C2×C22⋊C4#4118yes—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—41no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—41no—
278C2×C42—416no—
279C2×C42—44no—
280C2×C42—42no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42C22×C414yes—
285#95—28no—
286#95—28no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—44no—
291#95—44no—
292#95—42no—
293#95—42no—
294#95—42no—
295#95—41no—
296#95—41no—
297#95—42no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—42no—
302#95—42no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95C22×C4116yes—
311#95C22×C418yes—
312#95C22×C4116yes—
313C24—22no—
314C24—22no—
315C24—21no—
316C24—22no—
317C24—22no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24C23×C412yes—
328C24C23×C412yes—
329C24C2×#4114yes—
330C24H(5,𝔽2)14yes—
331C24C2511yescharacteristic
332C2×D4—28no—
333C2×D4—28no—
334C2×D4—44no—
335C2×D4—48no—
336C2×D4—48no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—216no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4C23×C412yes—
427C22×C4C23×C412yes—
428C22×C4C23×C418yes—
429C22×C4C23×C418yes—
430C22×C4C23×C412yes—
431C22×C4C23×C412yes—
432C22×C4C23×C412yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4114yes—
435C22×C4H(5,𝔽2)12yes—
436C22×C4H(5,𝔽2)12yes—
437C22×C4H(5,𝔽2)14yes—
438C22×C42−1+414yes—
439C22×C42−1+414yes—
440C4⋊C4—216no—
441C4⋊C4—216no—
442C4⋊C4—48no—
443C4⋊C4—44no—
444C4⋊C4—416no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—88no—
448C4⋊C4—88no—
449C4⋊C4—88no—
450C22⋊C4—232no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—88no—
458C22⋊C4—88no—
459C42—28no—
460C42—44no—
461C42—44no—
462C42—48no—
463C42—48no—
464C42—88no—
465C42—84no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23C4×#4114yes—
507C23#51814yes—
508C23C24×C411yescharacteristic
509C23C22×#4112yes—
510C23C2×H(5,𝔽2)11yescharacteristic
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×H(5,𝔽2)12yes—
513C23C2×H(5,𝔽2)12yes—
514C23C2×2−1+412yes—
515C23C2×2−1+412yes—
516C23#58512yes—
517D4—416no—
518C2×C4—28no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—432no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—416no—
575C2×C4—416no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—816no—
598C2×C4#51814yes—
599C2×C4#52018yes—
600C2×C4#56014yes—
601C2×C4#56218yes—
602C2×C4#56314yes—
603C22—28no—
604C22—28no—
605C22—24no—
606C22—24no—
607C22—44no—
608C22—42no—
609C22—42no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22C2×C4×#4111yescharacteristic
622C22C2×#51811yescharacteristic
623C22C2×#51811yescharacteristic
624C22C2×#52012yes—
625C22C4×2−1+412yes—
626C22#350512yes—
627C22#350912yes—
628C22#351512yes—
629C22#351912yes—
630C22#351912yes—
631C22#352012yes—
632C22C2×#56011yescharacteristic
633C22C2×#56212yes—
634C22C2×#56311yescharacteristic
635C22#353612yes—
636C22#354112yes—
637C22#354212yes—
638C22#354412yes—
639C22#355011yescharacteristic
640C22#355111yescharacteristic
641C22#355912yes—
642C4—42no—
643C4—42no—
644C4—44no—
645C4—44no—
646C4—42no—
647C4—44no—
648C4—48no—
649C4—48no—
650C4—42no—
651C4—84no—
652C4—82no—
653C4—84no—
654C4#237618yes—
655C2—44no—
656C2—44no—
657C2#3424611yescharacteristic
658C2#3431412yes—
659C2C2×#237611yescharacteristic
660C2#3441311yescharacteristic
661C2#3446012yes—
662C2#4562312yes—
663C2#4563112yes—
664C2#4564812yes—
665C2#4568812yes—
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
2. #14697C2semidirect
3. #14650C2semidirect
4. #14600C2semidirect
5. #14542C2semidirect
6. #14507C2semidirect
8. #14132C2semidirect
9. #14006C2semidirect
10. #13993C2semidirect
11. #13974C2semidirect
12. #13907C2semidirect
14. #13828C2semidirect
20. #2486C4semidirect
88. C22×C4⋊C4C2×C4semidirect
117. #398C2×C4semidirect
122. #389C2×C4semidirect
127. #381C2×C4semidirect
132. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
#146972
#146502
#146002
#145422
#145072
#143592
#141322
#140062
#139932
#139742
#139072
#139052
#138282
C4×#3862