Finite Group Databasebeta

← #517538

Subgroups — #517538

Order 512.

2518 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#117C211yesmaximal, characteristic
2#14689C211yesmaximal, characteristic, complemented
3#14687C211yesmaximal, characteristic, complemented
4#14682C211yesmaximal, characteristic
5#14682C211yesmaximal, characteristic
6#14352C212yesmaximal, complemented
7#14351C212yesmaximal
8#13946C212yesmaximal, complemented
9#13945C212yesmaximal, complemented
10#13944C212yesmaximal
11#13943C212yesmaximal
12#13885C212yesmaximal, complemented
13#13884C212yesmaximal, complemented
14#13876C212yesmaximal
15#13875C212yesmaximal
16#13788C212yesmaximal
17#13787C212yesmaximal, complemented
18C4×#382C211yesmaximal, characteristic, complemented
19C4×#382C211yesmaximal, characteristic, complemented
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C2×#388C2214yes—
24C2×#387C2214yes—
25C2×#384C2212yes—
26C2×#382C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#382C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C4×#117C4116yescomplemented
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×#376C2211yescharacteristic
45C2×#376C2211yescharacteristic
46C2×C43C2211yescharacteristic
47C2×C43C2211yescharacteristic
48#1445C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1440C2214yes—
55#1439C2214yes—
56#1437C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70#1434C2212yes—
71#1434C2212yes—
72#1434C2212yes—
73#1434C2212yes—
74#1434C2212yes—
75#1434C2212yes—
76#1434C2214yes—
77#1434C2214yes—
78#1434C2212yes—
79#1434C2212yes—
80#1434C2212yes—
81#1434C2212yes—
82#1434C2212yes—
83#1434C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2212yes—
87C22⋊#95C2212yes—
88C4×#95C2212yes—
89C4×#95C2212yes—
90C4×#95C2212yes—
91C4×#95C2212yes—
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C2×#117—28no—
96C2×#117—28no—
97C2×#117C2×C418yes—
98C2×#117C2×C418yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C42—24no—
116C22×C42—24no—
117C22×C42C2×C418yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2314yes—
130C22×C42C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#388—216no—
137#387—216no—
138#384—44no—
139#382—24no—
140#382—24no—
141#382—24no—
142#382—24no—
143#382—44no—
144#380—44no—
145#380—44no—
146C4×C4⋊C4—44no—
147C4×C4⋊C4—44no—
148C4×C4⋊C4C2×C418yes—
149C4×C4⋊C4C2×C418yes—
150C4×C4⋊C4C2×C418yes—
151C4×C4⋊C4C2×C418yes—
152C4×C22⋊C4C2×C418yes—
153C4×C22⋊C4C2×C418yes—
154C4×C22⋊C4C2×C418yes—
155C4×C22⋊C4C2×C418yes—
156#376C2×C418yes—
157#376C2×C418yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2311yescharacteristic
187C2×#95C2311yescharacteristic
188C2×#95C2311yescharacteristic
189C2×#95C2311yescharacteristic
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C43C2×C418yes—
193C43C2×C418yes—
194C25C2411yescharacteristic
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4C22×C414yes—
210C23×C4C22×C414yes—
211C23×C4C22×C414yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230#117—232no—
231#117—232no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C22×C414yes—
254C2×C4⋊C4C22×C414yes—
255C2×C4⋊C4C22×C414yes—
256C2×C4⋊C4C22×C414yes—
257C2×C4⋊C4#4118yes—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4C22×C414yes—
268C2×C22⋊C4C22×C414yes—
269C2×C22⋊C4C22×C414yes—
270C2×C22⋊C4C22×C414yes—
271C2×C22⋊C4#4118yes—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—232no—
291C2×C42—28no—
292C2×C42—28no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42C22×C414yes—
302C2×C42C22×C414yes—
303C2×C42C22×C414yes—
304C2×C42C22×C414yes—
305C2×C42C22×C414yes—
306C2×C42C22×C414yes—
307C2×C42C22×C414yes—
308C2×C42C22×C414yes—
309C2×C42C22×C414yes—
310C2×C42C22×C414yes—
311C2×C42C22×C414yes—
312C2×C42C22×C414yes—
313C2×C42C22×C414yes—
314C2×C42C22×C414yes—
315C2×C42C22×C414yes—
316C2×C42C22×C414yes—
317C2×C42#4114yes—
318C2×C42#4114yes—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—44no—
334#95—44no—
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#95—44no—
344#95—44no—
345#95C22×C414yes—
346#95C22×C414yes—
347#95C22×C414yes—
348#95C22×C414yes—
349#95C22×C414yes—
350#95C22×C414yes—
351#95C22×C414yes—
352#95C22×C414yes—
353C24—22no—
354C24—21no—
355C24—21no—
356C24—21no—
357C24—21no—
358C24—21no—
359C24—21no—
360C24—22no—
361C24—22no—
362C24C23×C412yes—
363C24C2×#4114yes—
364C24C2511yescharacteristic
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—28no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4C23×C412yes—
456C22×C4C23×C412yes—
457C22×C4C23×C412yes—
458C22×C4C23×C412yes—
459C22×C4C23×C412yes—
460C22×C4C23×C412yes—
461C22×C4C23×C412yes—
462C22×C4C23×C412yes—
463C22×C4C23×C412yes—
464C22×C4C23×C412yes—
465C22×C4C23×C412yes—
466C22×C4C23×C412yes—
467C22×C4C23×C412yes—
468C22×C4C23×C412yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4114yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C42−1+412yes—
483C22×C42−1+412yes—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—88no—
492C4⋊C4—88no—
493C4⋊C4#117116yes—
494C22⋊C4—28no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C22⋊C4#117116yes—
500C42—28no—
501C42—28no—
502C42—28no—
503C42—28no—
504C42—28no—
505C42—28no—
506C42—28no—
507C42—28no—
508C42—28no—
509C42—28no—
510C42—28no—
511C42—44no—
512C42—44no—
513C42—44no—
514C42—44no—
515C42—88no—
516C42—88no—
517C42#117116yes—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23C2×#11714yes—
541C23C24×C411yescharacteristic
542C23C22×#4111yescharacteristic
543C23C22×#4111yescharacteristic
544C23C22×#4112yes—
545C23C22×#4112yes—
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×2−1+411yescharacteristic
549C23C2×2−1+411yescharacteristic
550C23#58512yes—
551C23#58512yes—
552C2×C4—232no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—416no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—416no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4C2×#11714yes—
644C2×C4C2×#11714yes—
645C2×C4C2×#11714yes—
646C2×C4C2×#11714yes—
647C2×C4C2×#11714yes—
648C2×C4C2×#11714yes—
649C2×C4C2×#11714yes—
650C2×C4C2×#11714yes—
651C2×C4#52014yes—
652C2×C4#52014yes—
653C2×C4#54014yes—
654C2×C4#54114yes—
655C2×C4#54214yes—
656C2×C4#54214yes—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—42no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22C22×#11711yescharacteristic
669C22C22×#11711yescharacteristic
670C22C22×#11711yescharacteristic
671C22C2×#52011yescharacteristic
672C22C2×#52011yescharacteristic
673C22#342811yescharacteristic
674C22#342811yescharacteristic
675C22#344412yes—
676C22#344412yes—
677C22C2×#54011yescharacteristic
678C22C2×#54111yescharacteristic
679C22C2×#54211yescharacteristic
680C22C2×#54211yescharacteristic
681C22#345512yes—
682C22#345512yes—
683C22#351412yes—
684C22#351412yes—
685C22#351412yes—
686C22#351512yes—
687C22#351512yes—
688C22#351512yes—
689C22#351712yes—
690C22#351812yes—
691C4—24no—
692C4—24no—
693C4—24no—
694C4—24no—
695C4—24no—
696C4—24no—
697C4—24no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—84no—
705C4—84no—
706C4—84no—
707C4—84no—
708C4#231418yes—
709C2—44no—
710C2#3401211yescharacteristic
711C2C2×#231411yescharacteristic
712C2#3404811yescharacteristic
713C2#3419111yescharacteristic
714C2#3419111yescharacteristic
715C2#3419111yescharacteristic
716C2#3419311yescharacteristic
717C2#3513512yes—
718C2#3513612yes—
719C2#3513712yes—
720C2#3514212yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14689C2semidirect
3. #14687C2semidirect
6. #14352C2semidirect
8. #13946C2semidirect
9. #13945C2semidirect
12. #13885C2semidirect
13. #13884C2semidirect
17. #13787C2semidirect
18. C4×#382C2semidirect
19. C4×#382C2semidirect
33. C4×#117C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1172
#146892
#146872
#146822
#143522
#143512
#139462
#139452
#139442
#139432
#138852
#138842
#138762
#138752
#137882
#137872
C4×#3822