Finite Group Databasebeta

← #414672

Subgroups — #414672

Order 512.

1664 conjugacy classes of subgroups.

The subgroup lattice has 771 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#479C211yesmaximal, characteristic
2C22×#441C211yesmaximal, characteristic
3C2×#2565C211yesmaximal, characteristic
4C2×#1992C211yesmaximal, characteristic
5C2×#1803C211yesmaximal, characteristic
6#13926C211yesmaximal, characteristic
7#11325C211yesmaximal, characteristic
8#11323C211yesmaximal, characteristic
9#11124C211yesmaximal, characteristic
10#11122C211yesmaximal, characteristic
11#11052C211yesmaximal, characteristic
12#11050C211yesmaximal, characteristic
13#11024C211yesmaximal, characteristic
14#11024C211yesmaximal, characteristic
15#10835C211yesmaximal, characteristic
16C23×Q16C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#479—24no—
21C2×#479—24no—
22C2×#479—24no—
23C2×#441—24no—
24C2×#441—24no—
25C2×#441—24no—
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#97C2211yescharacteristic
37#2565—24no—
38#2565—24no—
39C2×#398—22no—
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#387—22no—
45C2×#384C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#1992—24no—
50#1803—24no—
51C2×#340C2211yescharacteristic
52C2×#340C2211yescharacteristic
53C2×#340C2211yescharacteristic
54C2×#340C2211yescharacteristic
55C2×#340C2211yescharacteristic
56C2×#340C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61#1441—22no—
62C4⋊#95—22no—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2211yescharacteristic
65C23×Q8C2311yescharacteristic
66C23×Q8C2311yescharacteristic
67C22×Q16—22no—
68C22×Q16—22no—
69C22×Q16—22no—
70C22×Q16—22no—
71C22×Q16—22no—
72C22×Q16—22no—
73C22×Q16Q814yes—
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C2×#125—22no—
79C2×#125—22no—
80C2×#125—22no—
81C2×#125—22no—
82C2×#125—22no—
83C2×#125D414yes—
84C2×C4×Q8—22no—
85C2×C4×Q8—22no—
86C2×C4×Q8—22no—
87C2×C4×Q8—22no—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—22no—
93C2×C4×Q8—22no—
94C2×C4×Q8D414yes—
95C2×C4×Q8D414yes—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4D412yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C42—21no—
111C22×C42—21no—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#479—232no—
116#441—232no—
117C2×#107—22no—
118C2×#107—22no—
119C2×#107—22no—
120C2×#106—22no—
121C2×#106—22no—
122C2×#106—22no—
123C2×#106—22no—
124C2×#106—22no—
125C2×#106—22no—
126C2×C4⋊C8—22no—
127C2×C4⋊C8—22no—
128C2×C4⋊C8D414yes—
129C2×#103—22no—
130C2×#103—22no—
131C2×#103—22no—
132C2×#103—22no—
133C2×#103—22no—
134C2×#103—22no—
135C2×#103—22no—
136C2×#103—22no—
137C2×#103—22no—
138C2×#103—22no—
139C2×#103—22no—
140C2×#103—22no—
141C2×#103—22no—
142C2×#103—22no—
143C2×#103—22no—
144C2×#103—22no—
145C2×#103—41no—
146C2×#103—41no—
147C2×#97—22no—
148C2×#97—22no—
149C2×#97D414yes—
150#398—216no—
151#391—22no—
152#391—22no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391—41no—
158#391—41no—
159#387—216no—
160#384—22no—
161#384—22no—
162#384—22no—
163#384—22no—
164#382—22no—
165#382—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C2×#95—22no—
169C2×#95—22no—
170C2×#95—22no—
171C2×#95—22no—
172C2×#95—22no—
173C2×#95—22no—
174C2×#95—22no—
175C2×#95D412yes—
176#340—22no—
177#340—22no—
178#340—22no—
179#340—22no—
180#340—22no—
181#340—22no—
182#336—22no—
183#336—22no—
184#336—41no—
185#336—41no—
186C22×Q8—21no—
187C22×Q8—21no—
188C22×Q8—21no—
189C22×Q8—21no—
190C22×Q8—21no—
191C22×Q8—21no—
192C22×Q8—21no—
193C22×Q8—21no—
194C22×Q8—21no—
195C22×Q8—21no—
196C22×Q8—21no—
197C22×Q8—21no—
198C22×Q8—21no—
199C22×Q8—21no—
200C22×Q8—21no—
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—21no—
204C22×Q8—21no—
205C22×Q8—21no—
206C22×Q8—21no—
207C22×Q8—21no—
208C22×Q8C2×D412yes—
209C22×Q8C2×D412yes—
210C22×Q8C2×Q812yes—
211C22×Q8C2×Q812yes—
212C22×Q8#4112yes—
213C22×Q8#4112yes—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4C2×D412yes—
223C23×C4C2×D412yes—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2411yescharacteristic
227C2×Q16—28no—
228C2×Q16—28no—
229C2×Q16—28no—
230C2×Q16—28no—
231C2×Q16—28no—
232C2×Q16—28no—
233C2×Q16—28no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8C2×D412yes—
257C22×C8C2×D412yes—
258C22×C8C2×D412yes—
259C22×C8C2×D412yes—
260C22×C8C2×Q812yes—
261C22×C8#4112yes—
262#125—28no—
263#125—28no—
264#125—44no—
265#125—44no—
266C4×Q8—28no—
267C4×Q8—28no—
268C4×Q8—28no—
269C4×Q8—28no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—44no—
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—21no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—82no—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—42no—
347C2×C42—42no—
348C2×C42—42no—
349C2×C42—42no—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42#4112yes—
356#107—28no—
357#106—28no—
358#106—28no—
359C4⋊C8—28no—
360#103—28no—
361#103—28no—
362#103—28no—
363#103—28no—
364#103—44no—
365#103—44no—
366#97—28no—
367#95—28no—
368#95—28no—
369#95—28no—
370#95—24no—
371#95—44no—
372#95—44no—
373#95—82no—
374#95—82no—
375C24C22×D411yescharacteristic
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—41no—
407C2×Q8—41no—
408C2×Q8—41no—
409C2×Q8—41no—
410C2×Q8—41no—
411C2×Q8—41no—
412C2×Q8—41no—
413C2×Q8—41no—
414C2×Q8—41no—
415C2×Q8—41no—
416C2×Q8—41no—
417C2×Q8—41no—
418C2×Q8—41no—
419C2×Q8—41no—
420C2×Q8—41no—
421C2×Q8—41no—
422C2×Q8—41no—
423C2×Q8—41no—
424C2×Q8—41no—
425C2×Q8—41no—
426C2×Q8—41no—
427C2×Q8—41no—
428C2×Q8—41no—
429C2×Q8—41no—
430C2×Q8C22⋊Q814yes—
431C2×Q8C22⋊Q814yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4#12012yes—
494C22×C4#12012yes—
495C22×C4#12112yes—
496C22×C4#12112yes—
497C22×C4#12712yes—
498C22×C4#12712yes—
499C22×C4#13714yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×Q811yescharacteristic
506C22×C4C2×#4111yescharacteristic
507C22×C4C2×#4111yescharacteristic
508C22×C4C2×#4111yescharacteristic
509C22×C4C2×#4111yescharacteristic
510C22×C4H(5,𝔽2)11yescharacteristic
511C22×C42−1+411yescharacteristic
512C22×C42−1+411yescharacteristic
513C22×C42−1+411yescharacteristic
514Q16—232no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—41no—
531C2×C8—41no—
532C2×C8—41no—
533C2×C8—41no—
534C2×C8—41no—
535C2×C8—41no—
536C2×C8—41no—
537C2×C8—41no—
538C2×C8—41no—
539C2×C8—41no—
540C2×C8—41no—
541C2×C8—41no—
542C2×C8C22⋊Q814yes—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—48no—
552C4⋊C4—416no—
553C4⋊C4—41no—
554C4⋊C4—41no—
555C4⋊C4—41no—
556C4⋊C4—41no—
557C4⋊C4—41no—
558C4⋊C4—41no—
559C4⋊C4—41no—
560C4⋊C4—41no—
561C4⋊C4—48no—
562C4⋊C4—48no—
563C4⋊C4—48no—
564C4⋊C4—48no—
565C4⋊C4—41no—
566C4⋊C4—41no—
567C4⋊C4—41no—
568C4⋊C4—41no—
569C4⋊C4—41no—
570C4⋊C4—41no—
571C4⋊C4—41no—
572C4⋊C4—41no—
573C4⋊C4—84no—
574C4⋊C4—84no—
575C42—41no—
576C42—41no—
577C42—41no—
578C42—41no—
579C42—41no—
580C42—41no—
581C42—41no—
582C42—41no—
583C42—48no—
584C42—48no—
585C42#12014yes—
586C42#12014yes—
587C42#12114yes—
588C42#12114yes—
589C23C2×#12011yescharacteristic
590C23C2×#12111yescharacteristic
591C23C2×#12711yescharacteristic
592C23#54611yescharacteristic
593C23#54711yescharacteristic
594C23#54811yescharacteristic
595C23D4×Q811yescharacteristic
596C23C2×#13712yes—
597C23#57512yes—
598C23#57712yes—
599C23#57812yes—
600Q8—24no—
601Q8—24no—
602Q8—24no—
603Q8—24no—
604Q8—42no—
605Q8—42no—
606Q8—42no—
607Q8—42no—
608Q8—42no—
609Q8—42no—
610Q8—42no—
611Q8—42no—
612Q8—42no—
613Q8—42no—
614Q8—42no—
615Q8—42no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—42no—
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—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4—81no—
665C2×C4—81no—
666C2×C4—81no—
667C2×C4—81no—
668C2×C4—82no—
669C2×C4—82no—
670C2×C4—82no—
671C2×C4—82no—
672C2×C4—82no—
673C2×C4—82no—
674C2×C4—81no—
675C2×C4—81no—
676C2×C4—81no—
677C2×C4—81no—
678C2×C4—81no—
679C2×C4—81no—
680C2×C4—82no—
681C2×C4—82no—
682C2×C4#39214yes—
683C2×C4#39214yes—
684C2×C4C2×#12011yescharacteristic
685C2×C4C2×#12011yescharacteristic
686C2×C4C2×#12111yescharacteristic
687C2×C4C2×#12111yescharacteristic
688C2×C4C2×C22⋊Q811yescharacteristic
689C2×C4C2×C22⋊Q811yescharacteristic
690C2×C4C2×C22⋊Q811yescharacteristic
691C2×C4#52511yescharacteristic
692C2×C4#52511yescharacteristic
693C2×C4#52511yescharacteristic
694C2×C4#53611yescharacteristic
695C2×C4#53611yescharacteristic
696C2×C4#53611yescharacteristic
697C2×C4#53711yescharacteristic
698C2×C4#53711yescharacteristic
699C2×C4#53711yescharacteristic
700C2×C4#54611yescharacteristic
701C2×C4#54711yescharacteristic
702C2×C4#54811yescharacteristic
703C2×C4#54811yescharacteristic
704C2×C4#54811yescharacteristic
705C2×C4D4×Q811yescharacteristic
706C2×C4D4×Q811yescharacteristic
707C2×C4#55011yescharacteristic
708C2×C4#55211yescharacteristic
709C2×C4#55211yescharacteristic
710C2×C4#55411yescharacteristic
711C2×C4#55611yescharacteristic
712C8—24no—
713C8—42no—
714C8—42no—
715C8—42no—
716C22C2×#39211yescharacteristic
717C22#240711yescharacteristic
718C22#240911yescharacteristic
719C22#241511yescharacteristic
720C22#241711yescharacteristic
721C22#254011yescharacteristic
722C22#254211yescharacteristic
723C22#300712yes—
724C22#301312yes—
725C22#303712yes—
726C22#304612yes—
727C22#315211yescharacteristic
728C22#315311yescharacteristic
729C22#315411yescharacteristic
730C22#315611yescharacteristic
731C22#331212yes—
732C22#331712yes—
733C22#334412yes—
734C22#334912yes—
735C22#337112yes—
736C22#337412yes—
737C22#338712yes—
738C22#339012yes—
739C4—24no—
740C4—24no—
741C4—44no—
742C4—44no—
743C4—44no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4—82no—
749C4—161no—
750C4—161no—
751C4#240711yescharacteristic
752C4#240911yescharacteristic
753C4#241511yescharacteristic
754C4#241711yescharacteristic
755C4#253611yescharacteristic
756C4#253811yescharacteristic
757C4#261611yescharacteristic
758C4#284811yescharacteristic
759C2#1517711yescharacteristic
760C2#2342711yescharacteristic
761C2#2342911yescharacteristic
762C2#2413912yes—
763C2#2434812yes—
764C2#2469112yes—
765C2#2482612yes—
766C2#2842311yescharacteristic
767C2#2843111yescharacteristic
768C2#2854611yescharacteristic
769C2#2855011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C22×#4792
C22×#4412
C2×#25652
C2×#19922
C2×#18032
#139262
#113252
#113232
#111242
#111222
#110522
#110502
#110242
#108352