Finite Group Databasebeta

← #530338

Subgroups — #530338

Order 512.

3822 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#380C211yesmaximal, characteristic
2#14681C211yesmaximal, characteristic
3#14453C212yesmaximal, complemented
4#14452C212yesmaximal, complemented
5#14372C214yesmaximal, complemented
6#14297C212yesmaximal, complemented
7#14107C214yesmaximal, complemented
8#13903C212yesmaximal
9#13882C212yesmaximal, complemented
10#13874C212yesmaximal, complemented
11#13870C212yesmaximal, complemented
12#13868C212yesmaximal
13#13799C212yesmaximal, complemented
14#13780C212yesmaximal, complemented
15C4×#379C211yesmaximal, characteristic
16C25×C4C2211yescharacteristic
17C22×#117C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C2×#387C2214yescomplemented
22C2×#381C2212yes—
23C2×#381C2212yes—
24C2×#381C2212yes—
25C2×#381C2212yes—
26C2×#381C2212yes—
27C2×#380C4116yescomplemented
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×#379C2212yes—
36C2×#379C2211yescharacteristic
37C2×#379C2211yescharacteristic
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43C22×#95C2212yes—
44#1445C2214yescomplemented
45#1443C2214yescomplemented
46#1442C2214yescomplemented
47#1441C2214yescomplemented
48#1441C2214yescomplemented
49#1441C2214yescomplemented
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1436C2214yescomplemented
59#1436C2214yescomplemented
60#1436C2214yescomplemented
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63#1436C2214yescomplemented
64#1434C2214yescomplemented
65#1434C2214yescomplemented
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C4×#95C2214yescomplemented
76C26C2311yescharacteristic
77C24×C4—28no—
78C24×C4—44no—
79C24×C4—42no—
80C24×C4—42no—
81C24×C4C2×C418yes—
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C2×#117—28no—
91C2×#117—44no—
92C2×#117—44no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4—42no—
96C22×C4⋊C4C2×C418yes—
97C22×C4⋊C4C2×C418yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4—44no—
107C22×C22⋊C4—44no—
108C22×C22⋊C4C2×C4116yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C42C2314yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2314yes—
127#387—48no—
128#381—28no—
129#381—28no—
130#381—44no—
131#381—44no—
132#381—44no—
133#380—44no—
134#380C2×C4116yes—
135#380C2×C4116yes—
136#380C2×C418yes—
137#380C2×C418yes—
138#380C2×C418yes—
139#380C2×C418yes—
140#379—28no—
141#379—42no—
142#379—42no—
143C4×C22⋊C4—28no—
144C4×C22⋊C4—44no—
145C4×C22⋊C4—42no—
146C4×C22⋊C4—44no—
147C4×C22⋊C4—42no—
148C2×#95C2×C4116yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C25—24no—
174C25—24no—
175C25—24no—
176C25—24no—
177C25—22no—
178C25—22no—
179C25—42no—
180C25—42no—
181C25C22×C414yes—
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—232no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—416no—
207C23×C4—416no—
208C23×C4—42no—
209C23×C4—41no—
210C23×C4—41no—
211C23×C4—41no—
212C23×C4—41no—
213C23×C4—42no—
214C23×C4—41no—
215C23×C4—41no—
216C23×C4—42no—
217C23×C4—41no—
218C23×C4—41no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—432no—
222C23×C4—432no—
223C23×C4C22×C418yes—
224C23×C4C22×C414yes—
225C23×C4C22×C414yes—
226C23×C4C22×C414yes—
227C23×C4C22×C414yes—
228C23×C4C22×C418yes—
229C23×C4C22×C414yes—
230C23×C4C22×C414yes—
231C23×C4#4118yes—
232C23×C4C2414yes—
233C23×C4C2414yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2414yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244#117—432no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—416no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C22×C418yes—
263C2×C4⋊C4C22×C414yes—
264C2×C4⋊C4C22×C414yes—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C418yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—432no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4C22×C418yes—
307C2×C22⋊C4C22×C418yes—
308C2×C22⋊C4C22×C418yes—
309C2×C22⋊C4C22×C418yes—
310C2×C22⋊C4#4118yes—
311C2×C42—28no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42—44no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95C22×C418yes—
344#95C22×C418yes—
345#95C22×C418yes—
346#95C22×C418yes—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—232no—
352C24—28no—
353C24—28no—
354C24—28no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—28no—
370C24—28no—
371C24—44no—
372C24—44no—
373C24—48no—
374C24—48no—
375C24—416no—
376C24—416no—
377C24—48no—
378C24—48no—
379C24—416no—
380C24—48no—
381C24—48no—
382C24C23×C412yes—
383C24C23×C412yes—
384C24C23×C412yes—
385C24C2×#4114yes—
386C24C2×#4114yes—
387C24H(5,𝔽2)12yes—
388C24H(5,𝔽2)14yes—
389C24H(5,𝔽2)12yes—
390C242−1+414yes—
391C24C2511yescharacteristic
392C22×C4—216no—
393C22×C4—216no—
394C22×C4—28no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—216no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—4128no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—88no—
493C22×C4—88no—
494C22×C4#117116yescomplemented
495C22×C4C23×C412yes—
496C22×C4C23×C414yes—
497C22×C4C23×C414yes—
498C22×C4C23×C414yes—
499C22×C4C23×C414yes—
500C22×C4C23×C412yes—
501C22×C4C23×C412yes—
502C22×C4C23×C412yes—
503C22×C4C2×#4118yes—
504C22×C4C2×#4114yes—
505C22×C4C2×#4114yes—
506C22×C4H(5,𝔽2)14yes—
507C4⋊C4—216no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—48no—
513C4⋊C4—416no—
514C4⋊C4—88no—
515C22⋊C4—216no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—416no—
521C22⋊C4—416no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C42—416no—
530C42—84no—
531C42—84no—
532C42—88no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—28no—
538C23—24no—
539C23—24no—
540C23—28no—
541C23—28no—
542C23—24no—
543C23—24no—
544C23—28no—
545C23—232no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—28no—
559C23—24no—
560C23—24no—
561C23—28no—
562C23—28no—
563C23—28no—
564C23—24no—
565C23—24no—
566C23—416no—
567C23—416no—
568C23—416no—
569C23—416no—
570C23—464no—
571C23—464no—
572C23—416no—
573C23—416no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—44no—
579C23—42no—
580C23—42no—
581C23—44no—
582C23—42no—
583C23—42no—
584C23—44no—
585C23—44no—
586C23C2×#11714yes—
587C23C4×#4114yes—
588C23#51814yes—
589C23#51814yes—
590C23#51914yes—
591C23#53818yes—
592C23#54018yes—
593C23C24×C411yescharacteristic
594C23C22×#4112yes—
595C23C22×#4112yes—
596C23C2×H(5,𝔽2)11yescharacteristic
597C23C2×H(5,𝔽2)12yes—
598C23C2×H(5,𝔽2)11yescharacteristic
599C23C2×H(5,𝔽2)12yes—
600C23C2×2−1+412yes—
601C23#58512yes—
602C2×C4—232no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—28no—
631C2×C4—28no—
632C2×C4—28no—
633C2×C4—416no—
634C2×C4—416no—
635C2×C4—416no—
636C2×C4—416no—
637C2×C4—416no—
638C2×C4—432no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—816no—
679C2×C4C2×#11714yes—
680C2×C4C2×#11714yes—
681C2×C4#51814yes—
682C22—216no—
683C22—216no—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—24no—
688C22—24no—
689C22—24no—
690C22—24no—
691C22—24no—
692C22—24no—
693C22—24no—
694C22—24no—
695C22—24no—
696C22—464no—
697C22—48no—
698C22—48no—
699C22—48no—
700C22—48no—
701C22—42no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22—42no—
709C22#233318yes—
710C22C22×#11711yescharacteristic
711C22C2×C4×#4111yescharacteristic
712C22C2×#51811yescharacteristic
713C22C2×#51811yescharacteristic
714C22C2×#51811yescharacteristic
715C22C2×#51911yescharacteristic
716C22#342811yescharacteristic
717C22C2×#53812yes—
718C22C2×#54012yes—
719C22#347514yes—
720C22#349012yes—
721C22#349012yes—
722C22#349214yes—
723C22#352014yes—
724C22#353512yes—
725C22#353512yes—
726C22#354214yes—
727C4—24no—
728C4—24no—
729C4—24no—
730C4—48no—
731C4—44no—
732C4—44no—
733C4—44no—
734C4—44no—
735C4—42no—
736C4—84no—
737C4—84no—
738C4—84no—
739C2—28no—
740C2—44no—
741C2—44no—
742C2C2×#233311yescharacteristic
743C2#3410911yescharacteristic
744C2#3417611yescharacteristic
745C2#3417811yescharacteristic
746C2#3424511yescharacteristic
747C2#3435911yescharacteristic
748C2#3436311yescharacteristic
749C2#3790914yes—
750C2#3791114yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14453C2semidirect
4. #14452C2semidirect
5. #14372C2semidirect
6. #14297C2semidirect
7. #14107C2semidirect
9. #13882C2semidirect
10. #13874C2semidirect
11. #13870C2semidirect
13. #13799C2semidirect
14. #13780C2semidirect
21. C2×#387C22semidirect
27. C2×#380C4semidirect
44. #1445C22semidirect
45. #1443C22semidirect
46. #1442C22semidirect
47. #1441C22semidirect
48. #1441C22semidirect
49. #1441C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. #1434C22semidirect
65. #1434C22semidirect
75. C4×#95C22semidirect
494. C22×C4#117semidirect

Maximal subgroups

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

subgroupindex
C22×#3802
#146812
#144532
#144522
#143722
#142972
#141072
#139032
#138822
#138742
#138702
#138682
#137992
#137802
C4×#3792