Finite Group Databasebeta

← #530718

Subgroups — #530718

Order 512.

3102 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2344C212yesmaximal
2C22×#389C211yesmaximal, characteristic
3#14685C211yesmaximal, characteristic
4#14446C212yesmaximal, complemented
5#14441C212yesmaximal, complemented
6#14435C214yesmaximal, complemented
7#14280C212yesmaximal, complemented
8#14083C214yesmaximal, complemented
9#13882C212yesmaximal
10#13878C212yesmaximal
11#13874C212yesmaximal
12#13870C212yesmaximal
13#13870C212yesmaximal
14#13790C212yesmaximal, complemented
15#13756C211yesmaximal, characteristic
16C22×#117C2211yescharacteristic
17C22×#117C2212yes—
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2344—216no—
24C2×#389C4116yescomplemented
25C2×#389C2214yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#387C2214yes—
33C2×#387C2214yes—
34C2×#387C2214yes—
35C2×#381C2212yes—
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#380C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2214yes—
43C2×C4×C22⋊C4C2212yes—
44C22×#95C2212yes—
45#1445C2214yes—
46#1444C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2214yes—
68#1434C2212yes—
69#1434C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C4×#95C2212yes—
76C4×#95C2212yes—
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C2×#117—24no—
83C2×#117—28no—
84C2×#117—42no—
85C2×#117—42no—
86C2×#117—44no—
87C2×#117—44no—
88C2×#117—42no—
89C2×#117—42no—
90C2×#117D418yes—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4—42no—
96C22×C4⋊C4C2×C4116yes—
97C22×C4⋊C4C2×C418yes—
98C22×C4⋊C4C2×C418yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C42—42no—
120C22×C42—42no—
121C22×C42C2×C418yes—
122C22×C42C2314yes—
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#389—216no—
131#389C2×C4116yes—
132#389C2×C4116yes—
133#389C2×C418yes—
134#389C2×C418yes—
135#389C2×C418yes—
136#389C2×C418yes—
137#387—216no—
138#387—216no—
139#387—216no—
140#381—28no—
141#381—28no—
142#381—44no—
143#381—44no—
144#380—28no—
145#380—42no—
146#380—42no—
147C4×C4⋊C4—216no—
148C4×C22⋊C4—28no—
149C2×#95C2×C4116yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C25C2411yescharacteristic
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—24no—
197C23×C4—44no—
198C23×C4—42no—
199C23×C4C22×C418yes—
200C23×C4C22×C414yes—
201C23×C4C22×C418yes—
202C23×C4C22×C414yes—
203C23×C4C22×C414yes—
204C23×C4C2×D414yes—
205C23×C4#4114yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2414yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220#117—416no—
221#117—432no—
222#117—416no—
223C2×C4⋊C4—264no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—232no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—416no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4C22×C414yes—
260C2×C4⋊C4C22×C418yes—
261C2×C4⋊C4C22×C418yes—
262C2×C4⋊C4C22×C418yes—
263C2×C4⋊C4C22×C418yes—
264C2×C4⋊C4C22×C418yes—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C418yes—
268C2×C4⋊C4C22×C414yes—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4#4118yes—
271C2×C22⋊C4—24no—
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—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4C2×D418yes—
283C2×C42—28no—
284C2×C42—28no—
285C2×C42—24no—
286C2×C42—28no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—416no—
294C2×C42—41no—
295C2×C42—41no—
296C2×C42—41no—
297C2×C42—41no—
298C2×C42—41no—
299C2×C42—41no—
300C2×C42—41no—
301C2×C42—41no—
302C2×C42—44no—
303C2×C42C22×C414yes—
304C2×C42C22×C414yes—
305C2×C42C22×C414yes—
306C2×C42C22×C414yes—
307C2×C42C2×D418yes—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95—42no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95C22×C418yes—
331#95C22×C418yes—
332#95C22×C418yes—
333#95C22×C418yes—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24C23×C412yes—
339C24C22×D412yes—
340C24C2×#4112yes—
341C24H(5,𝔽2)12yes—
342C24H(5,𝔽2)12yes—
343C24H(5,𝔽2)12yes—
344C242−1+412yes—
345C24C2511yescharacteristic
346C22×C4—216no—
347C22×C4—216no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—28no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—216no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
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—48no—
432C22×C4—44no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4C4×D4116yes—
442C22×C4C23×C412yes—
443C22×C4C23×C414yes—
444C22×C4C23×C414yes—
445C22×C4C23×C414yes—
446C22×C4C23×C414yes—
447C22×C4C23×C412yes—
448C22×C4C23×C412yes—
449C22×C4C23×C412yes—
450C22×C4C23×C412yes—
451C22×C4C23×C412yes—
452C22×C4C22×D418yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4118yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C42−1+414yes—
460C22×C42−1+414yes—
461C4⋊C4—216no—
462C4⋊C4—28no—
463C4⋊C4—216no—
464C4⋊C4—216no—
465C4⋊C4—216no—
466C4⋊C4—216no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—416no—
472C4⋊C4—48no—
473C4⋊C4—416no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4#117116yescomplemented
477C22⋊C4—216no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C42—44no—
482C42—44no—
483C42—44no—
484C42—44no—
485C42—48no—
486C42—416no—
487C42—48no—
488C42—416no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23C2×C4×D414yes—
506C23#51814yes—
507C23#52014yes—
508C23#53414yes—
509C23#53714yes—
510C23#53814yes—
511C23#53914yes—
512C23C24×C411yescharacteristic
513C23C23×D411yescharacteristic
514C23C22×#4112yes—
515C23C22×#4111yescharacteristic
516C23C2×H(5,𝔽2)11yescharacteristic
517C23C2×H(5,𝔽2)11yescharacteristic
518C23C2×H(5,𝔽2)11yescharacteristic
519C23C2×2−1+412yes—
520C23C2×2−1+412yes—
521C23C2×2−1+411yescharacteristic
522C23#58512yes—
523C2×C4—232no—
524C2×C4—232no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—416no—
569C2×C4—432no—
570C2×C4—432no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4C2×#11718yes—
609C2×C4C2×#11714yes—
610C2×C4C2×C4×D414yes—
611C2×C4C2×C4×D414yes—
612C2×C4#51914yes—
613C2×C4#51914yes—
614C22—28no—
615C22—28no—
616C22—24no—
617C22—28no—
618C22—28no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22#232518yes—
625C22C22×#11711yescharacteristic
626C22C22×C4×D411yescharacteristic
627C22C2×#51811yescharacteristic
628C22C2×#51911yescharacteristic
629C22C2×#51911yescharacteristic
630C22C2×#52011yescharacteristic
631C22#342811yescharacteristic
632C22C2×#53411yescharacteristic
633C22C2×#53711yescharacteristic
634C22C2×#53811yescharacteristic
635C22C2×#53911yescharacteristic
636C22D4×#4112yes—
637C22#347512yes—
638C22#349212yes—
639C22#349212yes—
640C22#349912yes—
641C22#350912yes—
642C22#352212yes—
643C22#352312yes—
644C22#353512yes—
645C22#353512yes—
646C22#355912yes—
647C22#356312yes—
648C4—24no—
649C4—24no—
650C4—24no—
651C4—28no—
652C4—24no—
653C4—24no—
654C4—44no—
655C4—44no—
656C4—48no—
657C4—42no—
658C4—44no—
659C4—84no—
660C4—84no—
661C4—84no—
662C2—28no—
663C2C2×#232511yescharacteristic
664C2D4×#11711yescharacteristic
665C2#3417811yescharacteristic
666C2#3419611yescharacteristic
667C2#3420911yescharacteristic
668C2#3435911yescharacteristic
669C2#3437811yescharacteristic
670C2#3842512yes—
671C2#3842712yes—
672C2#3846512yes—
673C2#3846712yes—
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
4. #14446C2semidirect
5. #14441C2semidirect
6. #14435C2semidirect
7. #14280C2semidirect
8. #14083C2semidirect
14. #13790C2semidirect
24. C2×#389C4semidirect
476. C4⋊C4#117semidirect

Maximal subgroups

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

subgroupindex
C2×#23442
C22×#3892
#146852
#144462
#144412
#144352
#142802
#140832
#138822
#138782
#138742
#138702
#137902
#137562