Finite Group Databasebeta

← #805085

Subgroups — #805085

Order 512.

3194 conjugacy classes of subgroups.

The subgroup lattice has 768 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2361C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3#16400C212yesmaximal, complemented
4#16015C214yesmaximal, complemented
5#15327C214yesmaximal, complemented
6#15197C211yesmaximal, characteristic, complemented
7#15156C211yesmaximal, characteristic, complemented
8#15135C211yesmaximal, characteristic, complemented
9#15092C212yesmaximal, complemented
10#14825C212yesmaximal, complemented
11#14045C212yesmaximal, complemented
12#14018C212yesmaximal, complemented
13#13970C212yesmaximal
14#13920C211yesmaximal, characteristic, complemented
15#13907C212yesmaximal, complemented
16#13872C212yesmaximal, complemented
17C4×#399C211yesmaximal, characteristic, complemented
18C22×#121C2211yescharacteristic
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24C2×#400C2212yes—
25C2×#399C2211yescharacteristic
26C2×#399C2212yes—
27#2361—28no—
28D4×C4⋊C4—28no—
29C2×#390C2211yescharacteristic
30C2×#390C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#388C2212yes—
36C2×#386C2212yes—
37C2×#386C2214yes—
38C2×#385C2212yes—
39C2×#385C2214yes—
40C2×#385C2214yes—
41C2×#384C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2214yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51#1445C2211yescharacteristic
52#1445C2212yes—
53#1445C2212yes—
54#1444C2212yes—
55#1444C2211yescharacteristic
56#1444C2214yes—
57#1443C2214yes—
58#1442C2214yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2212yes—
65#1440C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2211yescharacteristic
69#1439C2212yes—
70#1439C2214yes—
71#1439C2214yes—
72#1438C2214yes—
73#1436C2214yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C22⋊#95C2214yes—
81C22⋊#95C2214yes—
82C22⋊#95C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2211yescharacteristic
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C24×C4C2314yes—
91C2×#121—24no—
92C2×#121—24no—
93C2×#121Q818yes—
94C2×C4×D4—28no—
95C2×C4×D4—28no—
96C2×C4×D4—28no—
97C2×C4×D4—28no—
98C2×C4×D4—44no—
99C2×C4×D4—44no—
100C2×C4×D4—44no—
101C2×C4×D4D418yes—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—48no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#400—44no—
140#399—24no—
141#399—44no—
142#390—24no—
143#390—28no—
144#389—28no—
145#389—28no—
146#389—24no—
147#389—44no—
148#388—28no—
149#386—28no—
150#386—48no—
151#385—28no—
152#385—216no—
153#385—48no—
154#384—24no—
155#380—28no—
156#380—44no—
157#380—48no—
158#380—44no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—44no—
162C4×C22⋊C4—24no—
163C4×C22⋊C4—44no—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C25C2412yes—
191C25C2411yescharacteristic
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—42no—
199C22×D4—42no—
200C22×D4C2×D414yes—
201C22×D4C2×Q818yes—
202C22×D4C2×Q814yes—
203C22×D4#4114yes—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—42no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4C2×D418yes—
225C23×C4C2×Q814yes—
226C23×C4C2411yescharacteristic
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241#121—232no—
242C4×D4—232no—
243C4×D4—432no—
244C4×D4—432no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—432no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—416no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×Q814yes—
289C2×C4⋊C4#4118yes—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—41no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4C2×D418yes—
322C2×C22⋊C4C2×Q818yes—
323C2×C22⋊C4#4114yes—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—42no—
336C2×C42C2×D414yes—
337#95—22no—
338#95—22no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—24no—
343#95—28no—
344#95—22no—
345#95—22no—
346#95—28no—
347#95—24no—
348#95—24no—
349#95—44no—
350#95—42no—
351#95—42no—
352#95—42no—
353#95—42no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—42no—
361#95—44no—
362#95—44no—
363C24—21no—
364C24—24no—
365C24—22no—
366C24—22no—
367C24—21no—
368C24—22no—
369C24—21no—
370C24—22no—
371C24—22no—
372C24—24no—
373C24—24no—
374C24—22no—
375C24—24no—
376C24—22no—
377C24—21no—
378C24—22no—
379C24C22×D414yes—
380C24C22×Q814yes—
381C24C22×Q812yes—
382C24C2×#4114yes—
383C24C2×#4112yes—
384C24H(5,𝔽2)12yes—
385C24C2511yescharacteristic
386C2×D4—216no—
387C2×D4—216no—
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—216no—
391C2×D4—216no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4C22⋊Q8116yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—28no—
434C22×C4—216no—
435C22×C4—216no—
436C22×C4—216no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—21no—
475C22×C4—48no—
476C22×C4—416no—
477C22×C4—416no—
478C22×C4—416no—
479C22×C4—48no—
480C22×C4—416no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—48no—
484C22×C4—416no—
485C22×C4—48no—
486C22×C4—84no—
487C22×C4—88no—
488C22×C4—88no—
489C22×C4—88no—
490C22×C4—88no—
491C22×C4—88no—
492C22×C4—88no—
493C22×C4C22×D412yes—
494C22×C4C22×D414yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×Q812yes—
498C22×C4C22×Q814yes—
499C22×C4C22×Q812yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4118yes—
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4114yes—
505C22×C4H(5,𝔽2)14yes—
506C22×C4H(5,𝔽2)12yes—
507C22×C4H(5,𝔽2)14yes—
508C22×C42−1+414yes—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—48no—
512C4⋊C4—416no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—44no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—88no—
522C4⋊C4—84no—
523C4⋊C4—88no—
524C4⋊C4#123116yes—
525C22⋊C4—216no—
526C22⋊C4—216no—
527C22⋊C4—28no—
528C22⋊C4—44no—
529C22⋊C4—416no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—416no—
534C22⋊C4—84no—
535C22⋊C4—88no—
536C22⋊C4—88no—
537C42—28no—
538C42—48no—
539C42—44no—
540C42—48no—
541C42—48no—
542C23—22no—
543C23—24no—
544C23—24no—
545C23—22no—
546C23—24no—
547C23—28no—
548C23—24no—
549C23—24no—
550C23—28no—
551C23—22no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—22no—
556C23—24no—
557C23—24no—
558C23—22no—
559C23—24no—
560C23—24no—
561C23—28no—
562C23—24no—
563C23—22no—
564C23—24no—
565C23—28no—
566C23—24no—
567C23—24no—
568C23—41no—
569C23—41no—
570C23—42no—
571C23—44no—
572C23—41no—
573C23—41no—
574C23—44no—
575C23—42no—
576C23—44no—
577C23—42no—
578C23—44no—
579C23—42no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—42no—
584C23—42no—
585C23—44no—
586C23C2×C22⋊Q818yes—
587C23#55414yes—
588C23C23×D411yescharacteristic
589C23C23×Q811yescharacteristic
590C23C22×#4111yescharacteristic
591C23C22×#4112yes—
592C23C22×#4111yescharacteristic
593C23C2×H(5,𝔽2)12yes—
594C23C2×H(5,𝔽2)11yescharacteristic
595C23C2×H(5,𝔽2)11yescharacteristic
596C23C2×H(5,𝔽2)12yes—
597C23C2×2−1+412yes—
598C23#58511yescharacteristic
599D4—232no—
600D4—432no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—432no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—416no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—41no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—41no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—42no—
677C2×C4—44no—
678C2×C4—42no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—44no—
684C2×C4—42no—
685C2×C4—816no—
686C2×C4—832no—
687C2×C4C2×C22⋊Q814yes—
688C2×C4C2×#12318yes—
689C2×C4C2×#12314yes—
690C2×C4#53514yes—
691C2×C4#54614yes—
692C2×C4#55318yes—
693C2×C4#55418yes—
694C22—24no—
695C22—28no—
696C22—28no—
697C22—28no—
698C22—28no—
699C22—48no—
700C22—44no—
701C22—44no—
702C22—42no—
703C22—48no—
704C22—44no—
705C22—42no—
706C22—44no—
707C22—48no—
708C22—42no—
709C22—44no—
710C22—42no—
711C22—42no—
712C22—44no—
713C22—42no—
714C22—48no—
715C22—44no—
716C22C22×C22⋊Q811yescharacteristic
717C22C22×#12311yescharacteristic
718C22C2×#53511yescharacteristic
719C22C2×#54611yescharacteristic
720C22C2×#55312yes—
721C22C2×#55411yescharacteristic
722C22C2×#55412yes—
723C22#347511yescharacteristic
724C22#348112yes—
725C22#348411yescharacteristic
726C22#348811yescharacteristic
727C22#349911yescharacteristic
728C22#350111yescharacteristic
729C22#350312yes—
730C22#351612yes—
731C22#351911yescharacteristic
732C22#351912yes—
733C22#352012yes—
734C22#352012yes—
735C22#352212yes—
736C22#353712yes—
737C22#354212yes—
738C22#354812yes—
739C4—24no—
740C4—28no—
741C4—44no—
742C4—42no—
743C4—44no—
744C4—42no—
745C4—42no—
746C4—44no—
747C4—82no—
748C4—82no—
749C4—81no—
750C4—84no—
751C4—82no—
752C4—84no—
753C4#248718yes—
754C2—44no—
755C2—48no—
756C2C2×#248711yescharacteristic
757C2#3573911yescharacteristic
758C2#3599611yescharacteristic
759C2#3615411yescharacteristic
760C2#3623611yescharacteristic
761C2#3895111yescharacteristic
762C2#4248011yescharacteristic
763C2#4494312yes—
764C2#4504612yes—
765C2#5940412yes—
766C2#5945412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×D4×C4⋊C4C2semidirect
3. #16400C2semidirect
4. #16015C2semidirect
5. #15327C2semidirect
6. #15197C2semidirect
7. #15156C2semidirect
8. #15135C2semidirect
9. #15092C2semidirect
10. #14825C2semidirect
11. #14045C2semidirect
12. #14018C2semidirect
14. #13920C2semidirect
15. #13907C2semidirect
16. #13872C2semidirect
17. C4×#399C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23612
C2×D4×C4⋊C42
#164002
#160152
#153272
#151972
#151562
#151352
#150922
#148252
#140452
#140182
#139702
#139202
#139072
#138722
C4×#3992