Finite Group Databasebeta

← #582869

Subgroups — #582869

Order 512.

2476 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2#14674C212yesmaximal, complemented
3#14642C214yesmaximal, complemented
4#14634C212yesmaximal, complemented
5#14587C212yesmaximal, complemented
6#14508C212yesmaximal, complemented
7#14360C214yesmaximal
8#14264C212yesmaximal, complemented
9#14133C212yesmaximal, complemented
10#14005C212yesmaximal, complemented
11#13993C212yesmaximal, complemented
12#13973C212yesmaximal, complemented
13#13907C212yesmaximal, complemented
14#13906C211yesmaximal, characteristic
15#13839C211yesmaximal, 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×#384C2211yescharacteristic
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#1444C2214yes—
42#1444C2214yes—
43#1443C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1441C2211yescharacteristic
49#1441C2211yescharacteristic
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1440C2212yes—
54#1439C2211yescharacteristic
55#1439C2211yescharacteristic
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2212yes—
61#1439C2212yes—
62#1438C2212yes—
63#1437C2214yes—
64#1437C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73#1434C2214yes—
74C22⋊#95C2214yes—
75C4×#95C2212yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×#123—44no—
81C2×#123—44no—
82C2×#123C2×C4116yes—
83C2×C4×D4—42no—
84C2×C4×D4—42no—
85C2×C4×D4C2×C418yes—
86C2×#117—44no—
87C2×#117—42no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4C2×C418yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2314yes—
120#398C2×C4116yes—
121#397C2×C4116yes—
122#390—44no—
123#389—48no—
124#389—44no—
125#389—44no—
126#389—44no—
127#389C2×C4116yes—
128#387—48no—
129#384—42no—
130#381C2×C4116yes—
131#380—48no—
132#380—44no—
133#380—44no—
134#380C2×C4116yes—
135#380C2×C418yes—
136C4×C4⋊C4—42no—
137C4×C4⋊C4—44no—
138C4×C22⋊C4—44no—
139C4×C22⋊C4—42no—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2311yescharacteristic
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C25C2411yescharacteristic
167C25C2411yescharacteristic
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4C22×C414yes—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4C22×C414yes—
190C23×C4C22×C414yes—
191C23×C4C22×C418yes—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205#123—432no—
206C4×D4—416no—
207#117—416no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
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—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—416no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4C22×C418yes—
244C2×C4⋊C4C22×C4116yes—
245C2×C4⋊C4C22×C418yes—
246C2×C4⋊C4C22×C418yes—
247C2×C4⋊C4C22×C414yes—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4C22×C414yes—
275C2×C22⋊C4C22×C414yes—
276C2×C22⋊C4C22×C418yes—
277C2×C22⋊C4C22×C4116yes—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—42no—
291C2×C42—44no—
292C2×C42C22×C414yes—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—28no—
297#95—44no—
298#95—42no—
299#95—42no—
300#95—42no—
301#95—41no—
302#95—41no—
303#95—42no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95C22×C4116yes—
317#95C22×C418yes—
318#95C22×C4116yes—
319C24—22no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—22no—
334C24—22no—
335C24—21no—
336C24—22no—
337C24C23×C412yes—
338C24C23×C412yes—
339C24H(5,𝔽2)12yes—
340C242−1+412yes—
341C24C2511yescharacteristic
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—44no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
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—22no—
390C22×C4—22no—
391C22×C4—22no—
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—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—84no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4C23×C412yes—
436C22×C4C23×C412yes—
437C22×C4C23×C418yes—
438C22×C4C23×C418yes—
439C22×C4C23×C412yes—
440C22×C4C23×C412yes—
441C22×C4C23×C412yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C42−1+414yes—
448C22×C42−1+414yes—
449C4⋊C4—216no—
450C4⋊C4—232no—
451C4⋊C4—216no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—84no—
459C4⋊C4—88no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C4⋊C4—84no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—48no—
466C22⋊C4—416no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C42—28no—
474C42—48no—
475C42—48no—
476C42—84no—
477C42—88no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—24no—
495C23—41no—
496C23—42no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23#52014yes—
524C23C24×C411yescharacteristic
525C23C2×H(5,𝔽2)11yescharacteristic
526C23C2×H(5,𝔽2)11yescharacteristic
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×H(5,𝔽2)11yescharacteristic
529C23C2×H(5,𝔽2)12yes—
530C23C2×H(5,𝔽2)11yescharacteristic
531C23C2×2−1+412yes—
532C23C2×2−1+411yescharacteristic
533C23C2×2−1+412yes—
534C23#58512yes—
535D4—416no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—416no—
579C2×C4—416no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—816no—
616C2×C4#51814yes—
617C2×C4#51814yes—
618C2×C4#52018yes—
619C2×C4#56014yes—
620C2×C4#56218yes—
621C2×C4#56314yes—
622C22—24no—
623C22—28no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—44no—
629C22—44no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—44no—
637C22—44no—
638C22—81no—
639C22—81no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22—82no—
644C22C2×#51811yescharacteristic
645C22C2×#51811yescharacteristic
646C22C2×#52012yes—
647C22C2×#52011yescharacteristic
648C22C4×2−1+412yes—
649C22C2×#56011yescharacteristic
650C22C2×#56212yes—
651C22C2×#56311yescharacteristic
652C22#353611yescharacteristic
653C22#353611yescharacteristic
654C22#353711yescharacteristic
655C22#353711yescharacteristic
656C22#353812yes—
657C22#354112yes—
658C22#354212yes—
659C22#354412yes—
660C22#354612yes—
661C22#354712yes—
662C22#354812yes—
663C22#355011yescharacteristic
664C22#355111yescharacteristic
665C22#355212yes—
666C22#356312yes—
667C4—24no—
668C4—42no—
669C4—42no—
670C4—48no—
671C4—42no—
672C4—48no—
673C4—44no—
674C4—82no—
675C4—82no—
676C4—84no—
677C4—82no—
678C4—84no—
679C4#237618yes—
680C2—44no—
681C2—82no—
682C2C2×#237611yescharacteristic
683C2#3437911yescharacteristic
684C2#3441311yescharacteristic
685C2#3446512yes—
686C2#3448012yes—
687C2#5296611yescharacteristic
688C2#5296811yescharacteristic
689C2#5298212yes—
690C2#5300711yescharacteristic
691C2#5301711yescharacteristic
692C2#5310112yes—
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. #14674C2semidirect
3. #14642C2semidirect
4. #14634C2semidirect
5. #14587C2semidirect
6. #14508C2semidirect
8. #14264C2semidirect
9. #14133C2semidirect
10. #14005C2semidirect
11. #13993C2semidirect
12. #13973C2semidirect
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
#146742
#146422
#146342
#145872
#145082
#143602
#142642
#141332
#140052
#139932
#139732
#139072
#139062
#138392