Finite Group Databasebeta

← #430978

Subgroups — #430978

Order 512.

1664 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2092C211yesmaximal, characteristic
3C2×#2058C211yesmaximal, characteristic, complemented
4C2×#2057C211yesmaximal, characteristic, complemented
5C2×#1864C211yesmaximal, characteristic
6C2×#1862C211yesmaximal, characteristic
7C2×#1847C211yesmaximal, characteristic, complemented
8#15234C211yesmaximal, characteristic, complemented
9#11371C211yesmaximal, characteristic, complemented
10#11324C211yesmaximal, characteristic, complemented
11#11323C211yesmaximal, characteristic, complemented
12#11123C211yesmaximal, characteristic
13#11121C211yesmaximal, characteristic
14#10927C211yesmaximal, characteristic
15#10875C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8—22no—
18C22×C4×D4C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#97C2211yescharacteristic
30#2542—24no—
31#2542—24no—
32C2×#399—22no—
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41#2092—24no—
42#2058—24no—
43#2057—24no—
44#1864—24no—
45#1862—24no—
46#1847—24no—
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C2×#340C2211yescharacteristic
51C2×#340C2211yescharacteristic
52C2×#340C2211yescharacteristic
53C2×#336C2211yescharacteristic
54C2×#336C2211yescharacteristic
55C2×#336C2211yescharacteristic
56C2×#336C2211yescharacteristic
57C2×#336C2211yescharacteristic
58#1442—22no—
59#1441—22no—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2211yescharacteristic
63C23×Q8—21no—
64C23×D4C2311yescharacteristic
65C24×C4—21no—
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—41no—
75C2×C4⋊Q8—41no—
76C2×C22⋊Q8—216no—
77C2×C22⋊Q8—48no—
78C2×C22⋊Q8—48no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×C4×D4—22no—
83C2×C4×D4—41no—
84C2×C4×D4—41no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4D412yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4—21no—
99C22×C22⋊C4—22no—
100C22×C22⋊C4—22no—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C2×#107—22no—
105C2×#107—22no—
106C2×#107—22no—
107C2×#107—22no—
108C2×#107—22no—
109C2×#107—22no—
110C2×#107—41no—
111C2×#107—41no—
112C2×#107—41no—
113C2×#106—22no—
114C2×#106—22no—
115C2×#106—22no—
116C2×#106—22no—
117C2×#106—22no—
118C2×#106—22no—
119C2×#106—41no—
120C2×#106—41no—
121C2×#106—41no—
122C2×#102—22no—
123C2×#102—22no—
124C2×#102—22no—
125C2×#102—22no—
126C2×#102—22no—
127C2×#102—22no—
128C2×#102—22no—
129C2×#102—22no—
130C2×#102—41no—
131C2×#102—41no—
132C2×#102Q814yescomplemented
133C2×#102Q814yescomplemented
134C2×#97—22no—
135C2×#97—22no—
136C2×#97—22no—
137#399—216no—
138#390—22no—
139#390—22no—
140#390—22no—
141#390D414yes—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#384—22no—
147#384—22no—
148#384—22no—
149#384D414yes—
150#382—22no—
151#382—22no—
152C4×C4⋊C4—22no—
153C4×C4⋊C4—22no—
154C2×#95—22no—
155C2×#95—21no—
156C2×#95—22no—
157C2×#95—21no—
158C2×#95—22no—
159C2×#95—22no—
160C2×#95D412yes—
161#340—22no—
162#340—22no—
163#340—22no—
164#340—22no—
165#340—22no—
166#340—22no—
167#336—22no—
168#336—22no—
169#336—22no—
170#336D414yes—
171#336D414yes—
172C25—21no—
173C22×Q8—24no—
174C22×Q8—44no—
175C22×Q8—42no—
176C22×Q8—42no—
177C22×Q8—44no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4C2×D412yes—
190C22×D4C2×Q812yes—
191C22×D4C2×Q812yes—
192C23×C4—24no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—42no—
201C23×C4—44no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4C2×D412yes—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D412yes—
207C23×C4C2×D411yescharacteristic
208C23×C4C2411yescharacteristic
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8C2×D412yes—
230C22×C8C2×D412yes—
231C22×C8C2×D412yes—
232C22×C8C2×D412yes—
233C22×C8C2×Q812yes—
234C22×C8C2×Q812yes—
235C22×C8#4112yes—
236C22×C8#4112yes—
237C4⋊Q8—44no—
238C4⋊Q8—44no—
239C4⋊Q8—44no—
240C4⋊Q8—44no—
241C22⋊Q8—464no—
242C4×D4—44no—
243C4×D4—44no—
244C4×D4—44no—
245C4×D4—44no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×Q812yes—
303C2×C4⋊C4C2×Q812yes—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331#107—44no—
332#107—44no—
333#107—44no—
334#106—44no—
335#106—44no—
336#106—44no—
337#102—28no—
338#102—28no—
339#102—44no—
340#102—44no—
341#97—28no—
342#95—24no—
343#95—44no—
344#95—42no—
345#95—42no—
346#95—44no—
347#95—82no—
348#95—82no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—41no—
353C24—41no—
354C24—41no—
355C24—41no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24C22×D411yescharacteristic
361C2×Q8—416no—
362C2×Q8—48no—
363C2×Q8—84no—
364C2×Q8—88no—
365C2×Q8—84no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4C4⋊Q814yes—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—48no—
419C22×C4—416no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—84no—
451C22×C4—88no—
452C22×C4—84no—
453C22×C4#12014yes—
454C22×C4#12712yes—
455C22×C4#12712yes—
456C22×C4AGL(1,ℤ/8)14yes—
457C22×C4AGL(1,ℤ/8)14yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×Q811yescharacteristic
463C22×C4C22×Q811yescharacteristic
464C22×C4C2×#4111yescharacteristic
465C22×C4C2×#4111yescharacteristic
466C22×C4H(5,𝔽2)11yescharacteristic
467C22×C4H(5,𝔽2)11yescharacteristic
468C22×C42−1+411yescharacteristic
469C22×C42−1+411yescharacteristic
470C22×C42−1+411yescharacteristic
471C22×C42−1+411yescharacteristic
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8C22⋊Q814yescomplemented
499C2×C8C22⋊Q814yescomplemented
500C4⋊C4—216no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—416no—
512C4⋊C4—416no—
513C4⋊C4—48no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C22⋊C4—48no—
535C22⋊C4—416no—
536C22⋊C4—416no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—41no—
549C23—24no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—42no—
561C23—42no—
562C23—82no—
563C23—82no—
564C23—82no—
565C23—82no—
566C23—81no—
567C23—81no—
568C23—81no—
569C23—81no—
570C23—82no—
571C23—82no—
572C23C2×#12011yescharacteristic
573C23C2×#12711yescharacteristic
574C23#53511yescharacteristic
575C23#54711yescharacteristic
576C23#54711yescharacteristic
577C23D4×Q811yescharacteristic
578C23D4×Q811yescharacteristic
579C23C2×AGL(1,ℤ/8)12yes—
580C23C2×AGL(1,ℤ/8)12yes—
581C23#57712yes—
582C23#57812yes—
583Q8—816no—
584D4—24no—
585D4—24no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—42no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—816no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—82no—
648C2×C4—82no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4#49614yes—
652C2×C4C2×C22⋊Q811yescharacteristic
653C2×C4C2×C22⋊Q811yescharacteristic
654C2×C4#52511yescharacteristic
655C2×C4C2×C4⋊Q811yescharacteristic
656C2×C4#53611yescharacteristic
657C2×C4#53611yescharacteristic
658C2×C4#53611yescharacteristic
659C2×C4#53611yescharacteristic
660C2×C4#53911yescharacteristic
661C2×C4#53911yescharacteristic
662C2×C4#54411yescharacteristic
663C2×C4#54411yescharacteristic
664C2×C4D4211yescharacteristic
665C2×C4D4211yescharacteristic
666C2×C4D4211yescharacteristic
667C2×C4D4211yescharacteristic
668C2×C4#54711yescharacteristic
669C2×C4#54711yescharacteristic
670C2×C4#54711yescharacteristic
671C2×C4#54711yescharacteristic
672C2×C4D4×Q811yescharacteristic
673C2×C4D4×Q811yescharacteristic
674C2×C4D4×Q811yescharacteristic
675C2×C4D4×Q811yescharacteristic
676C2×C4#55411yescharacteristic
677C2×C4#55411yescharacteristic
678C2×C4#56311yescharacteristic
679C2×C4#56311yescharacteristic
680C8—24no—
681C8—24no—
682C8—42no—
683C8—42no—
684C22—44no—
685C22—48no—
686C22—84no—
687C22—84no—
688C22—84no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22#240611yescharacteristic
696C22#243311yescharacteristic
697C22#243411yescharacteristic
698C22#254211yescharacteristic
699C22#254211yescharacteristic
700C22#272011yescharacteristic
701C22#272011yescharacteristic
702C22#300312yes—
703C22#301312yes—
704C22C2×#49611yescharacteristic
705C22#315111yescharacteristic
706C22#315411yescharacteristic
707C22#315611yescharacteristic
708C22#321712yes—
709C22#322512yes—
710C22#334312yes—
711C22#334512yes—
712C22#335312yes—
713C22#335412yes—
714C22#338412yes—
715C22#338412yes—
716C22#338612yes—
717C22#338612yes—
718C4—24no—
719C4—44no—
720C4—44no—
721C4—42no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—161no—
728C4—161no—
729C4#244011yescharacteristic
730C4#244011yescharacteristic
731C4#244911yescharacteristic
732C4#244911yescharacteristic
733C4#267711yescharacteristic
734C4#267711yescharacteristic
735C4#268011yescharacteristic
736C4#268011yescharacteristic
737C2—84no—
738C2#1543011yescharacteristic
739C2#2410012yes—
740C2#2572511yescharacteristic
741C2#2573711yescharacteristic
742C2#2584412yes—
743C2#2854011yescharacteristic
744C2#2854111yescharacteristic
745C2#2854511yescharacteristic
746C2#2854811yescharacteristic
747C2#3236212yes—
748C2#3236412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2058C2semidirect
4. C2×#2057C2semidirect
7. C2×#1847C2semidirect
8. #15234C2semidirect
9. #11371C2semidirect
10. #11324C2semidirect
11. #11323C2semidirect
15. #10875C2semidirect
132. C2×#102Q8semidirect
133. C2×#102Q8semidirect
498. C2×C8C22⋊Q8semidirect
499. C2×C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#20922
C2×#20582
C2×#20572
C2×#18642
C2×#18622
C2×#18472
#152342
#113712
#113242
#113232
#111232
#111212
#109272
#108752