Finite Group Databasebeta

← #530622

Subgroups — #530622

Order 512.

2670 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#389C211yesmaximal, characteristic
2#14682C211yesmaximal, characteristic
3#14460C214yesmaximal, complemented
4#14459C214yesmaximal, complemented
5#14266C212yesmaximal, complemented
6#14006C212yesmaximal, complemented
7#13982C212yesmaximal, complemented
8#13924C212yesmaximal
9#13923C212yesmaximal
10#13907C211yesmaximal, characteristic
11#13907C211yesmaximal, characteristic
12#13883C212yesmaximal
13#13883C212yesmaximal
14C22⋊C4×C4⋊C4C211yesmaximal, characteristic
15C22⋊C4×C4⋊C4C211yesmaximal, characteristic
16#13783C212yesmaximal, complemented
17#13753C211yesmaximal, characteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C42C2211yescharacteristic
22C2×#389C4116yescomplemented
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#382C2212yes—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×#376C2212yes—
43C22×#95C2212yes—
44#1445C2214yes—
45#1445C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2214yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2214yes—
68#1434C2214yes—
69#1434C2212yes—
70#1434C2212yes—
71#1434C2212yes—
72#1434C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2211yescharacteristic
82C22⋊#95C2211yescharacteristic
83C4×#95C2212yes—
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4C2×C418yes—
94C22×C4⋊C4C2×C418yes—
95C22×C4⋊C4C2×C4116yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C42—44no—
117C22×C42C2×C418yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2314yes—
123C22×C42C2314yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#389—28no—
127#389—44no—
128#389C2×C418yes—
129#389C2×C418yes—
130#389C2×C418yes—
131#389C2×C418yes—
132#389C2×C4116yes—
133#389C2×C4116yes—
134#382—44no—
135#380—24no—
136#380—24no—
137#380—44no—
138#380—44no—
139C4×C4⋊C4—28no—
140C4×C4⋊C4—44no—
141C4×C22⋊C4—24no—
142C4×C22⋊C4—44no—
143C4×C22⋊C4—44no—
144C4×C22⋊C4Q818yes—
145#376—44no—
146C2×#95C2×C4116yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C25C2411yescharacteristic
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—44no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4C22×C418yes—
199C23×C4C22×C414yes—
200C23×C4C22×C418yes—
201C23×C4C22×C414yes—
202C23×C4C22×C414yes—
203C23×C4C2×Q814yes—
204C23×C4#4114yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—232no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—416no—
234C2×C4⋊C4—432no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C22×C418yes—
251C2×C4⋊C4C22×C414yes—
252C2×C4⋊C4C22×C414yes—
253C2×C4⋊C4C22×C418yes—
254C2×C4⋊C4C22×C418yes—
255C2×C4⋊C4C22×C418yes—
256C2×C4⋊C4C22×C418yes—
257C2×C4⋊C4C22×C418yes—
258C2×C4⋊C4C22×C414yes—
259C2×C4⋊C4#4118yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4C2×Q814yes—
277C2×C22⋊C4C2×Q814yes—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—28no—
286C2×C42—416no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42C22×C414yes—
295C2×C42C22×C414yes—
296C2×C42C22×C414yes—
297C2×C42C22×C414yes—
298C2×C42C2×Q818yes—
299#95—24no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—28no—
304#95—42no—
305#95—44no—
306#95—42no—
307#95—44no—
308#95—42no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—42no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95C22×C418yes—
327#95C22×C418yes—
328#95C22×C418yes—
329#95C22×C418yes—
330#95C2×Q818yes—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24C23×C412yes—
336C24C22×Q812yes—
337C24C2×#4112yes—
338C24H(5,𝔽2)12yes—
339C24H(5,𝔽2)12yes—
340C242−1+412yes—
341C242−1+412yes—
342C24C2511yescharacteristic
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—216no—
351C22×C4—216no—
352C22×C4—28no—
353C22×C4—216no—
354C22×C4—28no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
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—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—216no—
396C22×C4—216no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—416no—
433C22×C4—416no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4C4×Q8116yes—
441C22×C4C23×C412yes—
442C22×C4C23×C412yes—
443C22×C4C23×C414yes—
444C22×C4C23×C414yes—
445C22×C4C23×C414yes—
446C22×C4C23×C412yes—
447C22×C4C23×C412yes—
448C22×C4C23×C414yes—
449C22×C4C23×C412yes—
450C22×C4C23×C412yes—
451C22×C4C22×Q814yes—
452C22×C4C22×Q814yes—
453C22×C4C22×Q812yes—
454C22×C4C22×Q812yes—
455C22×C4C2×#4118yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4112yes—
459C22×C4H(5,𝔽2)14yes—
460C4⋊C4—216no—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—88no—
472C4⋊C4—88no—
473C4⋊C4#117116yes—
474C22⋊C4—28no—
475C22⋊C4—28no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C42—28no—
481C42—28no—
482C42—216no—
483C42—44no—
484C42—44no—
485C42—48no—
486C42—88no—
487C42—88no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23C2×C4×Q814yes—
505C23#51814yes—
506C23#51914yes—
507C23#54214yes—
508C23#54214yes—
509C23#54414yes—
510C23#54414yes—
511C23C24×C411yescharacteristic
512C23C23×Q811yescharacteristic
513C23C22×#4112yes—
514C23C22×#4111yescharacteristic
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×H(5,𝔽2)11yescharacteristic
517C23C2×H(5,𝔽2)12yes—
518C23C2×2−1+411yescharacteristic
519C23C2×2−1+411yescharacteristic
520C23#58512yes—
521C23#58512yes—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—232no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—432no—
559C2×C4—416no—
560C2×C4—416no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—816no—
612C2×C4C2×#11718yes—
613C2×C4C2×#11714yes—
614C2×C4C2×C4×Q814yes—
615C2×C4C2×C4×Q814yes—
616C2×C4#51814yes—
617C2×C4#55318yes—
618C2×C4#55418yes—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—28no—
626C22—28no—
627C22—28no—
628C22—28no—
629C22#233118yes—
630C22C22×#11711yescharacteristic
631C22C22×C4×Q811yescharacteristic
632C22C2×#51811yescharacteristic
633C22C2×#51811yescharacteristic
634C22C2×#51911yescharacteristic
635C22#342811yescharacteristic
636C22#342811yescharacteristic
637C22C2×#54211yescharacteristic
638C22C2×#54211yescharacteristic
639C22C2×#54411yescharacteristic
640C22C2×#54411yescharacteristic
641C22C2×#55312yes—
642C22C2×#55412yes—
643C22#350412yes—
644C22#351212yes—
645C22#351312yes—
646C22#351312yes—
647C22#351612yes—
648C22#351612yes—
649C22#354112yes—
650C22#354112yes—
651C22#354212yes—
652C22#354212yes—
653C4—24no—
654C4—28no—
655C4—42no—
656C4—42no—
657C4—44no—
658C4—44no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—82no—
665C4—84no—
666C4—82no—
667C4—84no—
668C4#234518yes—
669C2—28no—
670C2C2×#233111yescharacteristic
671C2C2×#234511yescharacteristic
672C2#3419011yescharacteristic
673C2#3419211yescharacteristic
674C2#3421711yescharacteristic
675C2#3436311yescharacteristic
676C2#3437011yescharacteristic
677C2#3810012yes—
678C2#3810112yes—
679C2#3812412yes—
680C2#3812512yes—
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
3. #14460C2semidirect
4. #14459C2semidirect
5. #14266C2semidirect
6. #14006C2semidirect
7. #13982C2semidirect
16. #13783C2semidirect
22. C2×#389C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3892
#146822
#144602
#144592
#142662
#140062
#139822
#139242
#139232
#139072
#138832
C22⋊C4×C4⋊C42
#137832
#137532