Finite Group Databasebeta

← #524521

Subgroups — #524521

Order 512.

2522 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic
2C2×#2320C212yesmaximal
3#14714C211yesmaximal, characteristic
4#14400C214yesmaximal
5#14395C214yesmaximal
6#14261C212yesmaximal
7#14155C211yesmaximal, characteristic
8#14142C212yesmaximal
9#14042C212yesmaximal
10#13981C212yesmaximal
11#13935C212yesmaximal
12#13921C212yesmaximal
13#13911C212yesmaximal
14#13910C211yesmaximal, characteristic
15C4⋊C42C211yesmaximal, characteristic
16C4×#389C211yesmaximal, characteristic
17#13755C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20#2365C4116yescomplemented
21#2320—216no—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32C2×#387C2214yes—
33C2×#387C2214yes—
34C2×#387C2214yes—
35C2×#384C2211yescharacteristic
36C2×#384C2212yes—
37C2×#383C2214yes—
38C2×#382C2214yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×#376C2212yes—
48C2×C43C2211yescharacteristic
49#1445C2214yes—
50#1444C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1436C2212yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2211yescharacteristic
79#1434C2212yes—
80#1434C2214yes—
81#1434C2212yes—
82#1434C2214yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2312yes—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8C2×C418yes—
89C2×C4×Q8—28no—
90C2×C4×Q8—28no—
91C2×C4×Q8C2×C4116yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2314yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2314yes—
122C22×C42C2311yescharacteristic
123#391—28no—
124#391—28no—
125#391—44no—
126#391—44no—
127#391C2×C4116yes—
128#389—28no—
129#389—24no—
130#389—44no—
131#389C2×C4116yes—
132#389D418yes—
133#387—216no—
134#387—216no—
135#387C2×C4132yes—
136#384—24no—
137#384—44no—
138#383—216no—
139#382—216no—
140C4×C4⋊C4—28no—
141C4×C4⋊C4—24no—
142C4×C4⋊C4—24no—
143C4×C4⋊C4—44no—
144C4×C4⋊C4—44no—
145C4×C4⋊C4C2×C418yes—
146C4×C4⋊C4C2×C418yes—
147C4×C4⋊C4C2×C4116yes—
148#376—28no—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C43—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8C22×C418yes—
190C23×C4C2412yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C4⋊Q8—232no—
208C4×Q8—264no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C22×C4116yes—
253C2×C4⋊C4C22×C418yes—
254C2×C4⋊C4C22×C418yes—
255C2×C4⋊C4C22×C414yes—
256C2×C4⋊C4C22×C414yes—
257C2×C4⋊C4C22×C414yes—
258C2×C4⋊C4C22×C414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4#4114yes—
263C2×C42—24no—
264C2×C42—28no—
265C2×C42—24no—
266C2×C42—28no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—28no—
291C2×C42—41no—
292C2×C42—41no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42C22×C4116yes—
296C2×C42C22×C418yes—
297C2×C42C22×C418yes—
298C2×C42C22×C414yes—
299C2×C42C2×D414yes—
300C2×C42#4114yes—
301C2×C42#4114yes—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—42no—
319#95—44no—
320#95—42no—
321#95—42no—
322#95—42no—
323#95—42no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95C22×C418yes—
331#95C22×C418yes—
332#95C22×C4116yes—
333#95C22×C4116yes—
334#95C2×D418yes—
335C24C2511yescharacteristic
336C2×Q8—216no—
337C2×Q8—216no—
338C2×Q8—216no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—48no—
342C2×Q8—48no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4C23×C414yes—
434C22×C4C23×C414yes—
435C22×C4C23×C414yes—
436C22×C4C23×C418yes—
437C22×C4C23×C412yes—
438C22×C4C23×C414yes—
439C22×C4C23×C412yes—
440C22×C4C23×C412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C42−1+412yes—
457C22×C42−1+412yes—
458C22×C42−1+412yes—
459C22×C42−1+412yes—
460C22×C42−1+412yes—
461C22×C42−1+412yes—
462C4⋊C4—216no—
463C4⋊C4—216no—
464C4⋊C4—232no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—88no—
477C4⋊C4—84no—
478C4⋊C4—88no—
479C4⋊C4C4×D4116yes—
480C42—216no—
481C42—232no—
482C42—216no—
483C42—28no—
484C42—44no—
485C42—44no—
486C42—44no—
487C42—44no—
488C42—48no—
489C42—48no—
490C42—44no—
491C42—88no—
492C42#117116yes—
493C23C24×C411yescharacteristic
494C23C23×D411yescharacteristic
495C23C22×#4111yescharacteristic
496C23C22×#4111yescharacteristic
497C23C22×#4111yescharacteristic
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×H(5,𝔽2)11yescharacteristic
500C23C2×2−1+411yescharacteristic
501C23C2×2−1+411yescharacteristic
502C23C2×2−1+411yescharacteristic
503C23C2×2−1+411yescharacteristic
504C23C2×2−1+411yescharacteristic
505C23C2×2−1+411yescharacteristic
506C23#58511yescharacteristic
507C23#58511yescharacteristic
508Q8—432no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
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×C4C2×#11714yes—
613C2×C4C2×#11714yes—
614C2×C4C2×#11714yes—
615C2×C4C2×C4×D418yes—
616C2×C4C2×C4×D414yes—
617C2×C4#51914yes—
618C2×C4#52014yes—
619C2×C4#52014yes—
620C2×C4#54114yes—
621C2×C4#54114yes—
622C2×C4#54214yes—
623C2×C4#54214yes—
624C2×C4#54714yes—
625C2×C4#55014yes—
626C2×C4#55514yes—
627C2×C4#55614yes—
628C22C22×#11711yescharacteristic
629C22C22×C4×D411yescharacteristic
630C22C2×#51911yescharacteristic
631C22C2×#52011yescharacteristic
632C22C2×#52011yescharacteristic
633C22C4×2−1+411yescharacteristic
634C22C4×2−1+411yescharacteristic
635C22C2×#54111yescharacteristic
636C22C2×#54111yescharacteristic
637C22C2×#54211yescharacteristic
638C22C2×#54211yescharacteristic
639C22C2×#54711yescharacteristic
640C22C2×#55011yescharacteristic
641C22C2×#55511yescharacteristic
642C22C2×#55611yescharacteristic
643C22#349511yescharacteristic
644C22#349511yescharacteristic
645C22#349611yescharacteristic
646C22#349711yescharacteristic
647C22#349911yescharacteristic
648C22#349911yescharacteristic
649C22#350911yescharacteristic
650C22#351011yescharacteristic
651C22#351511yescharacteristic
652C22#351811yescharacteristic
653C22#352011yescharacteristic
654C22#352311yescharacteristic
655C22#354611yescharacteristic
656C22#355011yescharacteristic
657C22#355211yescharacteristic
658C22#355211yescharacteristic
659C22#355411yescharacteristic
660C22#355511yescharacteristic
661C22#355511yescharacteristic
662C22#355611yescharacteristic
663C4—28no—
664C4—24no—
665C4—44no—
666C4—44no—
667C4—42no—
668C4—42no—
669C4—48no—
670C4—44no—
671C4—42no—
672C4—44no—
673C4—42no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—84no—
678C4#232018yescomplemented
679C4#235418yes—
680C2C2×#232011yescharacteristic
681C2C2×#235411yescharacteristic
682C2#3425411yescharacteristic
683C2#3430811yescharacteristic
684C2#3431111yescharacteristic
685C2#3448111yescharacteristic
686C2#3448811yescharacteristic
687C2#3894011yescharacteristic
688C2#3894311yescharacteristic
689C2#3898411yescharacteristic
690C2#3898611yescharacteristic
691C2#3935011yescharacteristic
692C2#3935311yescharacteristic
693C2#3938511yescharacteristic
694C2#3938711yescharacteristic
subgroupG/H
center, Frattini, socle, prime radical, cosocleC24C25
commutatorC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #2365C4semidirect
678. C4#2320semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
C2×#23202
#147142
#144002
#143952
#142612
#141552
#141422
#140422
#139812
#139352
#139212
#139112
#139102
C4⋊C422
C4×#3892
#137552