Finite Group Databasebeta

← #530358

Subgroups — #530358

Order 512.

3414 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2349C211yesmaximal, characteristic
2C2×#2322C212yesmaximal
3C2×C4×#123C211yesmaximal, characteristic
4C22×#382C211yesmaximal, characteristic, complemented
5#14685C211yesmaximal, characteristic, complemented
6#14404C214yesmaximal, complemented
7#14403C214yesmaximal, complemented
8#14263C212yesmaximal, complemented
9#14044C212yesmaximal, complemented
10#14016C212yesmaximal
11#13965C211yesmaximal, characteristic, complemented
12#13960C211yesmaximal, characteristic
13#13918C212yesmaximal, complemented
14#13915C212yesmaximal, complemented
15#13896C212yesmaximal, complemented
16#13889C212yesmaximal, complemented
17#13802C211yesmaximal, characteristic, complemented
18C22×#123C2211yescharacteristic
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C42C2211yescharacteristic
23#2349—28no—
24#2322—216no—
25C4×#123C4116yescomplemented
26C2×#389C2214yes—
27C2×#389C2212yes—
28C2×#388C2214yes—
29C2×#388C2212yes—
30C2×#387C2214yes—
31C2×#387C2212yes—
32C2×#386C2212yes—
33C2×#386C2211yescharacteristic
34C2×#385C2214yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2211yescharacteristic
40C2×#382—28no—
41C2×#382C2211yescharacteristic
42C2×#382C2212yes—
43C2×#382C2214yes—
44C2×#382C2212yes—
45C2×#381C2212yes—
46C2×#381C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C22×#95C2211yescharacteristic
56C22×#95C2211yescharacteristic
57C22×#95C2212yes—
58C22×#95C2211yescharacteristic
59#1444C2214yes—
60#1444C2214yes—
61#1444C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2214yes—
67#1441C2212yes—
68#1439C2212yes—
69#1439C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2214yes—
75C4⋊#95C2214yes—
76C4⋊#95C2212yes—
77#1434C2212yes—
78#1434C2212yes—
79#1434C2212yes—
80#1434C2212yes—
81C22⋊#95C2214yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C4×#95C2212yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2312yes—
94C24×C4C2311yescharacteristic
95C2×#123C2×C418yes—
96C2×#123D418yes—
97C2×#123Q818yes—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—28no—
101C2×C4×D4—28no—
102C2×C4×D4—44no—
103C2×C4×D4C2×C418yes—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—44no—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C42—24no—
130C22×C42—24no—
131C22×C42C2×C418yescomplemented
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2314yes—
137C22×C42C2312yes—
138#389—216no—
139#389—28no—
140#388—216no—
141#388C2×C4116yescomplemented
142#387—216no—
143#387C2×C4116yescomplemented
144#386—28no—
145#386C2×C418yes—
146#385—28no—
147#385—216no—
148#385—28no—
149#385—28no—
150#385—44no—
151#385C2×C418yescomplemented
152#382—24no—
153#382—28no—
154#382—28no—
155#382—216no—
156#381—28no—
157#381C2×C418yescomplemented
158#380—24no—
159#380—24no—
160#380—24no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4C2×C4116yescomplemented
163C4×C22⋊C4—28no—
164C4×C22⋊C4C2×C4116yescomplemented
165C4×C22⋊C4C2×C418yescomplemented
166C2×#95—28no—
167C2×#95—24no—
168C2×#95—24no—
169C2×#95D418yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2311yescharacteristic
197C2×#95C2311yescharacteristic
198C2×#95C2312yes—
199C25C2411yescharacteristic
200C25C2411yescharacteristic
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4C22×C414yes—
206C22×D4C2×D414yes—
207C22×D4C2×Q814yes—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—42no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4C22×C414yes—
234C23×C4C22×C414yes—
235C23×C4C22×C414yes—
236C23×C4C22×C414yes—
237C23×C4C2×D414yes—
238C23×C4C2×D418yes—
239C23×C4C2×D414yes—
240C23×C4C2×Q814yes—
241C23×C4#4114yes—
242C23×C4C2412yes—
243C23×C4C2414yes—
244C23×C4C2414yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257#123C4⋊C4164yescomplemented
258C4×D4—232no—
259C4×D4—432no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—432no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4C22×C414yes—
288C2×C4⋊C4C22×C418yes—
289C2×C4⋊C4C22×C418yes—
290C2×C4⋊C4C2×D418yes—
291C2×C4⋊C4C2×Q818yes—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4C22×C414yes—
312C2×C22⋊C4C22×C414yes—
313C2×C22⋊C4C22×C414yes—
314C2×C22⋊C4C22×C418yes—
315C2×C22⋊C4C22×C414yes—
316C2×C22⋊C4C22×C418yes—
317C2×C22⋊C4C2×D414yes—
318C2×C22⋊C4C2×D418yes—
319C2×C22⋊C4C2×Q814yes—
320C2×C22⋊C4C2×Q818yes—
321C2×C42—232no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—28no—
331C2×C42—42no—
332C2×C42—44no—
333C2×C42—44no—
334C2×C42—44no—
335C2×C42C22×C414yes—
336C2×C42C22×C418yes—
337C2×C42C22×C418yes—
338C2×C42C22×C418yes—
339#95—22no—
340#95—28no—
341#95—28no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—22no—
347#95—28no—
348#95—28no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—24no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95C22×C418yes—
361#95C22×C418yes—
362#95C22×C418yes—
363#95C22×C418yes—
364#95C22×C418yes—
365#95C2×D414yes—
366#95C2×D414yes—
367#95C2×D418yes—
368C24—24no—
369C24—24no—
370C24—21no—
371C24—21no—
372C24—21no—
373C24—21no—
374C24—24no—
375C24—24no—
376C24C23×C412yes—
377C24C23×C412yes—
378C24C22×D412yes—
379C24C22×D412yes—
380C24C22×D412yes—
381C24C22×Q812yes—
382C24C22×Q812yes—
383C24C2×#4112yes—
384C24H(5,𝔽2)12yes—
385C242−1+412yes—
386C24C2511yescharacteristic
387C2×D4—28no—
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—28no—
391C2×D4—28no—
392C2×D4—28no—
393C2×D4C2×C4⋊C4116yes—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—216no—
436C22×C4—28no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—88no—
479C22×C4—88no—
480C22×C4—88no—
481C22×C4—88no—
482C22×C4—88no—
483C22×C4—88no—
484C22×C4C2×C4⋊C4116yes—
485C22×C4C4×D4116yes—
486C22×C4C23×C414yes—
487C22×C4C23×C412yes—
488C22×C4C23×C414yes—
489C22×C4C23×C412yes—
490C22×C4C23×C414yes—
491C22×C4C23×C414yes—
492C22×C4C23×C412yes—
493C22×C4C23×C412yes—
494C22×C4C23×C412yes—
495C22×C4C22×D414yes—
496C22×C4C22×D414yes—
497C22×C4C22×D414yes—
498C22×C4C22×D412yes—
499C22×C4C22×D414yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×Q814yes—
503C22×C4C22×Q814yes—
504C22×C4C22×Q812yes—
505C22×C4C2×#4112yes—
506C22×C4C2×#4112yes—
507C4⋊C4—28no—
508C4⋊C4—48no—
509C4⋊C4—416no—
510C4⋊C4—416no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4—88no—
516C4⋊C4—84no—
517C4⋊C4—88no—
518C4⋊C4C2×C4⋊C4132yes—
519C22⋊C4—28no—
520C22⋊C4—28no—
521C22⋊C4—28no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—88no—
527C22⋊C4C2×C4⋊C4116yes—
528C22⋊C4C2×C4⋊C4132yes—
529C42—216no—
530C42—216no—
531C42—28no—
532C42—48no—
533C42—48no—
534C42—88no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—28no—
542C23—28no—
543C23—28no—
544C23—28no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—28no—
554C23—28no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23C22×C4⋊C414yes—
562C23C22×C4⋊C414yes—
563C23C2×C4×D414yes—
564C23#52014yes—
565C23D4214yes—
566C23#54714yes—
567C23D4×Q814yes—
568C23#55414yes—
569C23C24×C411yescharacteristic
570C23C23×D411yescharacteristic
571C23C23×D411yescharacteristic
572C23C23×Q811yescharacteristic
573C23C22×#4111yescharacteristic
574C23C2×H(5,𝔽2)11yescharacteristic
575C23C2×2−1+411yescharacteristic
576C23#58514yes—
577C23#58514yes—
578D4—232no—
579C2×C4—232no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—416no—
619C2×C4—416no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—88no—
657C2×C4—816no—
658C2×C4—816no—
659C2×C4—816no—
660C2×C4C22×C4⋊C418yes—
661C2×C4C22×C4⋊C418yes—
662C2×C4C22×C4⋊C414yes—
663C2×C4C2×C4×D414yes—
664C2×C4C2×C4×D414yes—
665C22—24no—
666C22—28no—
667C22—28no—
668C22—24no—
669C22—28no—
670C22—28no—
671C22—28no—
672C22—28no—
673C22—24no—
674C22—28no—
675C22—42no—
676C22—44no—
677C22—42no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22D4×C4⋊C418yes—
683C22C23×C4⋊C411yescharacteristic
684C22C22×C4×D411yescharacteristic
685C22C2×#52011yescharacteristic
686C22#342812yes—
687C22#342812yes—
688C22#344912yes—
689C22#345512yes—
690C22C2×D4211yescharacteristic
691C22C2×#54711yescharacteristic
692C22C2×D4×Q811yescharacteristic
693C22C2×#55411yescharacteristic
694C22#348512yes—
695C22#349112yes—
696C22#350014yes—
697C22#350414yes—
698C22#354514yes—
699C22#355314yes—
700C4—28no—
701C4—24no—
702C4—28no—
703C4—24no—
704C4—24no—
705C4—42no—
706C4—44no—
707C4—44no—
708C4—42no—
709C4—44no—
710C4—44no—
711C4—82no—
712C4—82no—
713C4—84no—
714C4—84no—
715C2—28no—
716C2—44no—
717C2#3406511yescharacteristic
718C2C2×D4×C4⋊C411yescharacteristic
719C2#3418412yes—
720C2#3419911yescharacteristic
721C2#3439012yes—
722C2#3933312yes—
723C2#3934512yes—
724C2#3958912yes—
725C2#3960112yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C22×#382C2semidirect
5. #14685C2semidirect
6. #14404C2semidirect
7. #14403C2semidirect
8. #14263C2semidirect
9. #14044C2semidirect
11. #13965C2semidirect
13. #13918C2semidirect
14. #13915C2semidirect
15. #13896C2semidirect
16. #13889C2semidirect
17. #13802C2semidirect
25. C4×#123C4semidirect
131. C22×C42C2×C4semidirect
141. #388C2×C4semidirect
143. #387C2×C4semidirect
151. #385C2×C4semidirect
157. #381C2×C4semidirect
162. C4×C4⋊C4C2×C4semidirect
164. C4×C22⋊C4C2×C4semidirect
165. C4×C22⋊C4C2×C4semidirect
257. #123C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23492
C2×#23222
C2×C4×#1232
C22×#3822
#146852
#144042
#144032
#142632
#140442
#140162
#139652
#139602
#139182
#139152
#138962
#138892
#138022