Finite Group Databasebeta

← #558318

Subgroups — #558318

Order 512.

2552 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2#14699C212yesmaximal, complemented
3#14652C212yesmaximal, complemented
4#14610C212yesmaximal, complemented
5#14550C214yesmaximal, complemented
6#14507C212yesmaximal, complemented
7#14360C214yesmaximal
8#14129C212yesmaximal, complemented
9#14046C212yesmaximal, complemented
10#13993C212yesmaximal, complemented
11#13973C212yesmaximal, complemented
12#13907C212yesmaximal, complemented
13#13871C211yesmaximal, characteristic
14#13828C212yesmaximal, complemented
15C4×#390C211yesmaximal, characteristic
16C22×#123C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2486C4116yescomplemented
21C2×#398C2212yes—
22C2×#397C2212yes—
23C2×#390C2212yes—
24C2×#389C2214yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#387C2214yes—
30C2×#381C2212yes—
31C2×#380C2214yes—
32C2×#380C2212yes—
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C43C2211yescharacteristic
41#1443C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1440C2212yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1438C2212yes—
53#1437C2211yescharacteristic
54#1437C2214yes—
55#1437C2212yes—
56#1437C2211yescharacteristic
57#1437C2214yes—
58#1437C2211yescharacteristic
59#1437C2211yescharacteristic
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2212yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73#1434C2214yes—
74#1434C2214yes—
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⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2314yes—
119C22×C42C2311yescharacteristic
120#398C2×C4116yes—
121#397C2×C4116yes—
122#390—28no—
123#389—48no—
124#389—44no—
125#389—44no—
126#389—44no—
127#389C2×C4116yes—
128#387—48no—
129#381C2×C4116yes—
130#380—216no—
131#380—44no—
132#380—42no—
133#380C2×C4116yes—
134#380C2×C418yes—
135C4×C4⋊C4—42no—
136C4×C4⋊C4—44no—
137C4×C22⋊C4—44no—
138C4×C22⋊C4—44no—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C43—42no—
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—24no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—44no—
184C23×C4C22×C414yes—
185C23×C4C22×C414yes—
186C23×C4C22×C418yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2414yes—
190C23×C4C2414yes—
191C23×C4C2411yescharacteristic
192C23×C4C2414yes—
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200#123—432no—
201C4×D4—416no—
202#117—416no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
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—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—416no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C22×C418yes—
233C2×C4⋊C4C22×C4116yes—
234C2×C4⋊C4C22×C418yes—
235C2×C4⋊C4C22×C418yes—
236C2×C4⋊C4C22×C414yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
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—44no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4C22×C414yes—
263C2×C22⋊C4C22×C414yes—
264C2×C22⋊C4C22×C418yes—
265C2×C22⋊C4C22×C4116yes—
266C2×C22⋊C4#4118yes—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—44no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42—41no—
282C2×C42—41no—
283C2×C42—42no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42C22×C414yes—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—44no—
293#95—44no—
294#95—42no—
295#95—42no—
296#95—42no—
297#95—42no—
298#95—41no—
299#95—41no—
300#95—44no—
301#95—44no—
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—21no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—21no—
324C24—21no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24C23×C412yes—
329C24C23×C412yes—
330C24C2×#4114yes—
331C24C2511yescharacteristic
332C2×D4—28no—
333C2×D4—28no—
334C2×D4—44no—
335C2×D4—48no—
336C2×D4—48no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—216no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—216no—
386C22×C4—216no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—88no—
417C22×C4—88no—
418C22×C4C23×C412yes—
419C22×C4C23×C412yes—
420C22×C4C23×C418yes—
421C22×C4C23×C418yes—
422C22×C4C23×C412yes—
423C22×C4C23×C412yes—
424C22×C4C23×C412yes—
425C22×C4C2×#4114yes—
426C22×C4C2×#4114yes—
427C22×C4H(5,𝔽2)14yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C4H(5,𝔽2)12yes—
430C22×C4H(5,𝔽2)14yes—
431C22×C42−1+414yes—
432C22×C42−1+414yes—
433C4⋊C4—216no—
434C4⋊C4—232no—
435C4⋊C4—216no—
436C4⋊C4—48no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—88no—
442C4⋊C4—88no—
443C4⋊C4—88no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—48no—
447C22⋊C4—416no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—88no—
454C22⋊C4—84no—
455C42—28no—
456C42—48no—
457C42—44no—
458C42—44no—
459C42—48no—
460C42—48no—
461C42—48no—
462C42—88no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23C4×#4114yes—
505C23C24×C411yescharacteristic
506C23C22×#4112yes—
507C23C2×H(5,𝔽2)11yescharacteristic
508C23C2×H(5,𝔽2)11yescharacteristic
509C23C2×H(5,𝔽2)12yes—
510C23C2×H(5,𝔽2)12yes—
511C23C2×2−1+412yes—
512C23C2×2−1+412yes—
513C23#58512yes—
514D4—416no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—416no—
570C2×C4—416no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—816no—
593C2×C4#51814yes—
594C2×C4#51814yes—
595C2×C4#52018yes—
596C2×C4#56014yes—
597C2×C4#56218yes—
598C2×C4#56314yes—
599C22—28no—
600C22—24no—
601C22—44no—
602C22—42no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—42no—
607C22—44no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—42no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22C2×C4×#4111yescharacteristic
618C22C2×#51811yescharacteristic
619C22C2×#51811yescharacteristic
620C22C2×#52012yes—
621C22C4×2−1+412yes—
622C22#350512yes—
623C22#350912yes—
624C22#351512yes—
625C22#351912yes—
626C22#351912yes—
627C22#352012yes—
628C22C2×#56011yescharacteristic
629C22C2×#56212yes—
630C22C2×#56311yescharacteristic
631C22#353811yescharacteristic
632C22#354011yescharacteristic
633C22#354612yes—
634C22#354712yes—
635C22#354712yes—
636C22#354812yes—
637C22#355212yes—
638C4—24no—
639C4—44no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—44no—
644C4—42no—
645C4—48no—
646C4—48no—
647C4—82no—
648C4—84no—
649C4—84no—
650C4#237618yes—
651C2—44no—
652C2—82no—
653C2#3424611yescharacteristic
654C2#3431412yes—
655C2C2×#237611yescharacteristic
656C2#3441211yescharacteristic
657C2#3448012yes—
658C2#4564112yes—
659C2#4565612yes—
660C2#4618012yes—
661C2#4633912yes—
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. #14699C2semidirect
3. #14652C2semidirect
4. #14610C2semidirect
5. #14550C2semidirect
6. #14507C2semidirect
8. #14129C2semidirect
9. #14046C2semidirect
10. #13993C2semidirect
11. #13973C2semidirect
12. #13907C2semidirect
14. #13828C2semidirect
20. #2486C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
#146992
#146522
#146102
#145502
#145072
#143602
#141292
#140462
#139932
#139732
#139072
#138712
#138282
C4×#3902