Finite Group Databasebeta

← #674890

Subgroups — #674890

Order 512.

2522 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×#2552C212yesmaximal
2C2×#2552C211yesmaximal, characteristic
3C2×C4×#125C211yesmaximal, characteristic
4#15629C214yesmaximal
5#15517C214yesmaximal
6#15221C212yesmaximal
7#15151C212yesmaximal
8#15149C212yesmaximal
9#15137C212yesmaximal
10#14712C211yesmaximal, characteristic
11#14151C211yesmaximal, characteristic
12#14054C212yesmaximal
13#14046C212yesmaximal
14#13910C211yesmaximal, characteristic
15#13910C212yesmaximal
16C4⋊C42C211yesmaximal, characteristic
17C4×#389C211yesmaximal, characteristic
18C22×#125C2212yes—
19C22×#125C2211yescharacteristic
20#2552—28no—
21#2552—216no—
22C2×#398C2214yes—
23C2×#398C2214yes—
24C2×#398C2214yes—
25C2×#398C2214yes—
26C4×#125—28no—
27C2×#389C2211yescharacteristic
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2214yes—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#387C2212yes—
42C2×#387C2212yes—
43C2×#384C2214yes—
44C2×#384C2212yes—
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C43C2211yescharacteristic
59#1445C2214yes—
60#1445C2214yes—
61#1444C2214yes—
62#1444C2214yes—
63#1444C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1442C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1436C2212yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2214yes—
84C4⋊#95C2212yes—
85C4×#95C2212yes—
86C2×#125—24no—
87C2×#125—28no—
88C2×#125—28no—
89C2×#125—24no—
90C2×#125—28no—
91C2×#125D418yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2314yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131#398—216no—
132#398—216no—
133#398—216no—
134#398—48no—
135#389—24no—
136#389—24no—
137#389—28no—
138#389—28no—
139#389—216no—
140#389—24no—
141#389—24no—
142#389—28no—
143#389—28no—
144#389—28no—
145#389—44no—
146#389—44no—
147#389—44no—
148#389—44no—
149#387—28no—
150#387—28no—
151#384—28no—
152#384—216no—
153#384—28no—
154#384—24no—
155#384Q818yes—
156#382—24no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—28no—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4D418yes—
165C4×C4⋊C4Q818yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C43—24no—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210#125—232no—
211#125—432no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D418yes—
281C2×C4⋊C4C2×D418yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×Q814yes—
286C2×C4⋊C4C2×Q818yes—
287C2×C4⋊C4C2×Q818yes—
288C2×C4⋊C4C2×Q818yes—
289C2×C4⋊C4C2×Q814yes—
290C2×C4⋊C4#4114yes—
291C2×C4⋊C4#4114yes—
292C2×C4⋊C4#4114yes—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—28no—
308C2×C42—24no—
309C2×C42—28no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—44no—
321C2×C42—41no—
322C2×C42—41no—
323C2×C42C2×D414yes—
324C2×C42C2×D418yes—
325C2×C42C2×D414yes—
326C2×C42C2×Q818yes—
327C2×C42C2×Q814yes—
328C2×C42C2×Q814yes—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—24no—
343#95—24no—
344#95—28no—
345#95—28no—
346#95—24no—
347#95—24no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95C2×D418yes—
354#95C2×Q818yes—
355C24C2511yescharacteristic
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×Q814yes—
455C22×C4C22×Q818yes—
456C22×C4C22×Q814yes—
457C22×C4C22×Q812yes—
458C22×C4C22×Q812yes—
459C22×C4C22×Q812yes—
460C22×C4C22×Q812yes—
461C22×C4C22×Q812yes—
462C22×C4C22×Q812yes—
463C22×C4C2×#4114yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C42−1+412yes—
472C22×C42−1+412yes—
473C22×C42−1+412yes—
474C22×C42−1+412yes—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—216no—
481C4⋊C4—216no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—416no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—416no—
489C4⋊C4—416no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—84no—
496C4⋊C4—88no—
497C4⋊C4—88no—
498C4⋊C4—88no—
499C4⋊C4—88no—
500C4⋊C4#121116yes—
501C4⋊C4#125116yescomplemented
502C42—28no—
503C42—216no—
504C42—216no—
505C42—416no—
506C42—44no—
507C42—44no—
508C42—48no—
509C42—44no—
510C42—44no—
511C42—44no—
512C42—48no—
513C42C4⋊Q8116yes—
514C23C23×D411yescharacteristic
515C23C23×D411yescharacteristic
516C23C23×Q811yescharacteristic
517C23C23×Q811yescharacteristic
518C23C22×#4111yescharacteristic
519C23C22×#4111yescharacteristic
520C23C22×#4111yescharacteristic
521C23C2×H(5,𝔽2)11yescharacteristic
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×2−1+411yescharacteristic
524C23C2×2−1+411yescharacteristic
525C23C2×2−1+411yescharacteristic
526C23C2×2−1+411yescharacteristic
527C23#58511yescharacteristic
528C23#58511yescharacteristic
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4C2×#12118yes—
632C2×C4C2×#12114yes—
633C2×C4C2×#12518yes—
634C2×C4C2×#12514yes—
635C2×C4C2×C4⋊Q814yes—
636C2×C4C2×C4⋊Q814yes—
637C2×C4C2×C4⋊Q814yes—
638C2×C4#53614yes—
639C2×C4#54414yes—
640C2×C4#54714yes—
641C2×C4D4×Q814yes—
642C2×C4D4×Q814yes—
643C2×C4#55014yes—
644C2×C4#55414yes—
645C2×C4#55714yes—
646C2×C4#55714yes—
647C2×C4#55714yes—
648C22C22×#12111yescharacteristic
649C22C22×#12511yescharacteristic
650C22C22×C4⋊Q811yescharacteristic
651C22C2×#53611yescharacteristic
652C22C2×#54411yescharacteristic
653C22C2×#54711yescharacteristic
654C22C2×D4×Q811yescharacteristic
655C22C2×D4×Q811yescharacteristic
656C22C2×#55011yescharacteristic
657C22D4×#4111yescharacteristic
658C22D4×#4111yescharacteristic
659C22C2×#55411yescharacteristic
660C22C2×#55711yescharacteristic
661C22C2×#55711yescharacteristic
662C22C2×#55711yescharacteristic
663C22#348211yescharacteristic
664C22#348311yescharacteristic
665C22#348811yescharacteristic
666C22#348911yescharacteristic
667C22#349811yescharacteristic
668C22#349911yescharacteristic
669C22#350111yescharacteristic
670C22#350211yescharacteristic
671C22#350311yescharacteristic
672C22#350411yescharacteristic
673C22#351511yescharacteristic
674C22#351511yescharacteristic
675C22#351511yescharacteristic
676C22#351711yescharacteristic
677C22#352211yescharacteristic
678C22#352311yescharacteristic
679C22#354411yescharacteristic
680C22#355211yescharacteristic
681C22#355411yescharacteristic
682C22#355511yescharacteristic
683C4—28no—
684C4—28no—
685C4—24no—
686C4—44no—
687C4—42no—
688C4—48no—
689C4—42no—
690C4—44no—
691C4—42no—
692C4—44no—
693C4—42no—
694C4—82no—
695C4—84no—
696C4—82no—
697C4—82no—
698C4#254618yes—
699C4#255218yescomplemented
700C2C2×#254611yescharacteristic
701C2C2×#255211yescharacteristic
702C2#3556611yescharacteristic
703C2D4×#12511yescharacteristic
704C2#3625911yescharacteristic
705C2#3644711yescharacteristic
706C2#3645511yescharacteristic
707C2#3892111yescharacteristic
708C2#3913911yescharacteristic
709C2#3918311yescharacteristic
710C2#3936911yescharacteristic
711C2#5945811yescharacteristic
712C2#5946711yescharacteristic
713C2#5966711yescharacteristic
714C2#5967211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
501. C4⋊C4#125semidirect
699. C4#2552semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×C4×#1252
#156292
#155172
#152212
#151512
#151492
#151372
#147122
#141512
#140542
#140462
#139102
C4⋊C422
C4×#3892