Finite Group Databasebeta

← #530579

Subgroups — #530579

Order 512.

2958 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2345C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic
3C2×#2283C211yesmaximal, characteristic
4#14685C211yesmaximal, characteristic
5#14387C214yesmaximal, complemented
6#14385C214yesmaximal, complemented
7#14266C212yesmaximal, complemented
8#14046C212yesmaximal, complemented
9#14042C212yesmaximal, complemented
10#13924C211yesmaximal, characteristic
11#13907C211yesmaximal, characteristic
12#13907C212yesmaximal
13#13882C212yesmaximal
14#13874C212yesmaximal
15#13874C212yesmaximal
16C22⋊C4×C4⋊C4C211yesmaximal, characteristic
17#13783C212yesmaximal, complemented
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C42C2211yescharacteristic
22#2345—28no—
23C2×#389C4116yescomplemented
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2214yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#387C2214yes—
34C2×#387C2214yes—
35#2283—28no—
36C2×#383C2212yes—
37C2×#382C2212yes—
38C2×#381C2211yescharacteristic
39C2×#381C2211yescharacteristic
40C2×#381C2212yes—
41C2×#380C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×#376C2212yes—
55C22×#95C2212yes—
56#1445C2214yes—
57#1445C2214yes—
58#1443C2214yes—
59#1443C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77#1434C2212yes—
78#1434C2214yes—
79#1434C2214yes—
80#1434C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—48no—
95C22×C4⋊C4C2×C418yes—
96C22×C4⋊C4C2×C418yes—
97C22×C4⋊C4C2×C4116yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C42—24no—
121C22×C42—24no—
122C22×C42C2×C418yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2314yes—
130C22×C42C2314yes—
131#389—28no—
132#389—28no—
133#389—48no—
134#389C2×C4116yes—
135#389C2×C418yes—
136#389C2×C418yes—
137#389C2×C4116yes—
138#389C2×C418yes—
139#389C2×C418yes—
140#387—216no—
141#387—48no—
142#383—28no—
143#382—28no—
144#381—24no—
145#381—24no—
146#381—44no—
147#380—24no—
148#380—24no—
149#380—28no—
150#380—24no—
151#380—24no—
152#380—44no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—28no—
156C4×C22⋊C4—24no—
157C4×C22⋊C4—28no—
158C4×C22⋊C4—44no—
159C4×C22⋊C4D418yes—
160#376—28no—
161C2×#95C2×C4116yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C25C2411yescharacteristic
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—24no—
209C23×C4—44no—
210C23×C4—42no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4C22×C414yes—
214C23×C4C22×C418yes—
215C23×C4C22×C414yes—
216C23×C4C22×C414yes—
217C23×C4C22×C418yes—
218C23×C4C2×D414yes—
219C23×C4#4114yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C2×C4⋊C4—232no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—232no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—432no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C22×C414yes—
270C2×C4⋊C4C22×C418yes—
271C2×C4⋊C4C22×C418yes—
272C2×C4⋊C4C22×C414yes—
273C2×C4⋊C4C22×C414yes—
274C2×C4⋊C4C22×C418yes—
275C2×C4⋊C4C22×C418yes—
276C2×C4⋊C4C22×C418yes—
277C2×C4⋊C4C22×C418yes—
278C2×C4⋊C4C22×C414yes—
279C2×C4⋊C4#4118yes—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4C2×D414yes—
297C2×C22⋊C4C2×D414yes—
298C2×C42—232no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—28no—
313C2×C42—28no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42C22×C414yes—
319C2×C42C22×C414yes—
320C2×C42C22×C414yes—
321C2×C42C22×C414yes—
322C2×C42C2×D418yes—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—24no—
335#95—28no—
336#95—28no—
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#95—44no—
346#95—44no—
347#95—44no—
348#95C22×C418yes—
349#95C22×C418yes—
350#95C22×C418yes—
351#95C22×C418yes—
352#95C2×D418yes—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24C23×C412yes—
358C24C22×D412yes—
359C24C2×#4112yes—
360C24H(5,𝔽2)12yes—
361C242−1+412yes—
362C242−1+412yes—
363C242−1+412yes—
364C24C2511yescharacteristic
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—216no—
381C22×C4—28no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—216no—
423C22×C4—216no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—48no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—88no—
466C22×C4—88no—
467C22×C4C4×D4116yes—
468C22×C4C23×C412yes—
469C22×C4C23×C412yes—
470C22×C4C23×C414yes—
471C22×C4C23×C414yes—
472C22×C4C23×C414yes—
473C22×C4C23×C412yes—
474C22×C4C23×C414yes—
475C22×C4C23×C412yes—
476C22×C4C23×C412yes—
477C22×C4C23×C412yes—
478C22×C4C22×D414yes—
479C22×C4C22×D414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C2×#4118yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4114yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)14yes—
487C22×C42−1+414yes—
488C4⋊C4—216no—
489C4⋊C4—28no—
490C4⋊C4—216no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—88no—
502C4⋊C4—88no—
503C4⋊C4#117116yes—
504C22⋊C4—28no—
505C22⋊C4—28no—
506C22⋊C4—44no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—48no—
510C42—28no—
511C42—28no—
512C42—28no—
513C42—28no—
514C42—216no—
515C42—48no—
516C42—88no—
517C42—88no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23C2×C4×D414yes—
535C23#51914yes—
536C23#52014yes—
537C23#53614yes—
538C23#53714yes—
539C23#54114yes—
540C23#54214yes—
541C23C24×C411yescharacteristic
542C23C23×D411yescharacteristic
543C23C22×#4112yes—
544C23C22×#4111yescharacteristic
545C23C2×H(5,𝔽2)12yes—
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×2−1+411yescharacteristic
548C23C2×2−1+411yescharacteristic
549C23C2×2−1+412yes—
550C23C2×2−1+411yescharacteristic
551C23#58512yes—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—232no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—432no—
590C2×C4—416no—
591C2×C4—416no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—816no—
645C2×C4C2×#11718yes—
646C2×C4C2×#11714yes—
647C2×C4C2×C4×D414yes—
648C2×C4C2×C4×D414yes—
649C2×C4#51814yes—
650C2×C4#51914yes—
651C2×C4#54718yes—
652C2×C4#55218yes—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—28no—
659C22—28no—
660C22—24no—
661C22—28no—
662C22—28no—
663C22#232318yes—
664C22C22×#11711yescharacteristic
665C22C22×C4×D411yescharacteristic
666C22C2×#51811yescharacteristic
667C22C2×#51911yescharacteristic
668C22C2×#51911yescharacteristic
669C22C2×#52011yescharacteristic
670C22#342811yescharacteristic
671C22C2×#53611yescharacteristic
672C22C2×#53711yescharacteristic
673C22C2×#54111yescharacteristic
674C22C2×#54211yescharacteristic
675C22C2×#54712yes—
676C22C2×#55212yes—
677C22#348812yes—
678C22#349412yes—
679C22#351612yes—
680C22#351612yes—
681C22#351912yes—
682C22#352012yes—
683C22#355212yes—
684C22#355212yes—
685C22#355912yes—
686C22#356312yes—
687C4—24no—
688C4—28no—
689C4—24no—
690C4—42no—
691C4—42no—
692C4—44no—
693C4—44no—
694C4—44no—
695C4—42no—
696C4—44no—
697C4—44no—
698C4—82no—
699C4—82no—
700C4—84no—
701C4—84no—
702C4#234418yes—
703C2—28no—
704C2C2×#232311yescharacteristic
705C2C2×#234411yescharacteristic
706C2#3419211yescharacteristic
707C2#3419411yescharacteristic
708C2#3420411yescharacteristic
709C2#3437111yescharacteristic
710C2#3437811yescharacteristic
711C2#3804012yes—
712C2#3804212yes—
713C2#3806612yes—
714C2#3806812yes—
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
5. #14387C2semidirect
6. #14385C2semidirect
7. #14266C2semidirect
8. #14046C2semidirect
9. #14042C2semidirect
17. #13783C2semidirect
23. C2×#389C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23452
C22×#3892
C2×#22832
#146852
#143872
#143852
#142662
#140462
#140422
#139242
#139072
#138822
#138742
C22⋊C4×C4⋊C42
#137832