Finite Group Databasebeta

← #412786

Subgroups — #412786

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2094C211yesmaximal, characteristic
2C2×#2092C211yesmaximal, characteristic
3C2×#1949C211yesmaximal, characteristic, complemented
4C2×#1948C211yesmaximal, characteristic, complemented
5C2×#1804C211yesmaximal, characteristic
6C2×#1802C211yesmaximal, characteristic
7#14354C211yesmaximal, characteristic, complemented
8#13785C211yesmaximal, characteristic
9#11373C211yesmaximal, characteristic, complemented
10#11372C211yesmaximal, characteristic, complemented
11#11325C211yesmaximal, characteristic, complemented
12#11323C211yesmaximal, characteristic, complemented
13#11051C211yesmaximal, characteristic
14#11049C211yesmaximal, characteristic
15#10833C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×#376C2211yescharacteristic
37#2094—24no—
38#2092—24no—
39#1949—24no—
40#1948—24no—
41#1804—24no—
42#1802—24no—
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C2×#340C2211yescharacteristic
51C2×#336C2211yescharacteristic
52C2×#336C2211yescharacteristic
53C2×#336C2211yescharacteristic
54C2×#336C2211yescharacteristic
55#1444—22no—
56#1444—22no—
57#1437—22no—
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2211yescharacteristic
60C22⋊#95—22no—
61C23×D4C2311yescharacteristic
62C24×C4—21no—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4Q814yes—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×#107—22no—
92C2×#107—22no—
93C2×#107—41no—
94C2×#106—22no—
95C2×#106—22no—
96C2×#106—22no—
97C2×#106—22no—
98C2×#106—22no—
99C2×#106—22no—
100C2×#106—41no—
101C2×#106—41no—
102C2×#106—41no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8D414yes—
108C2×C4⋊C8D414yes—
109C2×#102—22no—
110C2×#102—22no—
111C2×#102—22no—
112C2×#102—22no—
113C2×#102—22no—
114C2×#102—22no—
115C2×#102—22no—
116C2×#102—22no—
117C2×#102—41no—
118C2×#102—41no—
119C2×#102—41no—
120C2×#102—41no—
121#390—22no—
122#390—22no—
123#390D414yes—
124#390Q814yes—
125#384—22no—
126#384—22no—
127#384—22no—
128#384—22no—
129#384—41no—
130#384—41no—
131#382—22no—
132#382—22no—
133#382—22no—
134#382—22no—
135#382—41no—
136#382—41no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—22no—
139#376—22no—
140#376—22no—
141C2×#95—21no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—21no—
145C2×#95—21no—
146C2×#95—22no—
147C2×#95—22no—
148C2×#95D412yes—
149C2×#95D412yes—
150#340—22no—
151#340—22no—
152#340—22no—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—22no—
157#340—22no—
158#336—22no—
159#336—22no—
160#336—22no—
161#336—22no—
162C25—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4C2×D412yes—
175C22×D4C2×Q812yes—
176C22×D4C2×Q812yes—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—24no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—82no—
188C23×C4—82no—
189C23×C4C2×D412yes—
190C23×C4C2×D412yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2411yescharacteristic
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
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×C8C2×D412yes—
219C22×C8C2×D412yes—
220C22×C8C2×D412yes—
221C22×C8C2×D412yes—
222C4×D4—28no—
223C4×D4—28no—
224C4×D4—44no—
225C4×D4—44no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
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—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—82no—
270C2×C4⋊C4—82no—
271C2×C4⋊C4—82no—
272C2×C4⋊C4—82no—
273C2×C4⋊C4—82no—
274C2×C4⋊C4—82no—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×Q812yes—
277C2×C4⋊C4C2×Q812yes—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—82no—
284C2×C22⋊C4—82no—
285C2×C22⋊C4—82no—
286C2×C22⋊C4—82no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—42no—
296C2×C42—82no—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2×Q812yes—
301C2×C42C2×Q812yes—
302C2×C42#4112yes—
303C2×C42#4112yes—
304C2×C42#4112yes—
305C2×C42#4112yes—
306C2×C42#4112yes—
307#107—44no—
308#106—44no—
309#106—44no—
310#106—44no—
311C4⋊C8—28no—
312C4⋊C8—28no—
313#102—44no—
314#102—44no—
315#102—44no—
316#102—44no—
317#95—24no—
318#95—24no—
319#95—44no—
320#95—44no—
321#95—42no—
322#95—42no—
323#95—42no—
324#95—82no—
325#95—82no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—41no—
330C24—41no—
331C24—41no—
332C24—41no—
333C24—42no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24C22×D411yescharacteristic
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4C4⋊Q814yes—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—48no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—88no—
419C22×C4—88no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4#12114yes—
423C22×C4C22⋊Q814yes—
424C22×C4C22⋊Q812yes—
425C22×C4C22⋊Q812yes—
426C22×C4#12312yes—
427C22×C4#12312yes—
428C22×C4#12712yes—
429C22×C4#12712yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×Q811yescharacteristic
434C22×C4C22×Q811yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4H(5,𝔽2)11yescharacteristic
441C22×C4H(5,𝔽2)11yescharacteristic
442C22×C42−1+411yescharacteristic
443C22×C42−1+411yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4—88no—
503C4⋊C4—88no—
504C22⋊C4—48no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—84no—
516C42C22⋊Q814yes—
517C42C22⋊Q814yes—
518C42#12314yes—
519C42#12314yes—
520C23—24no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—42no—
532C23—42no—
533C23—82no—
534C23—82no—
535C23—81no—
536C23—81no—
537C23—81no—
538C23—81no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23C2×#12111yescharacteristic
544C23C2×C22⋊Q811yescharacteristic
545C23C2×C22⋊Q811yescharacteristic
546C23C2×#12311yescharacteristic
547C23C2×#12711yescharacteristic
548C23#53511yescharacteristic
549C23#53611yescharacteristic
550C23#57612yes—
551C23#57712yes—
552C23#57712yes—
553C23#57712yes—
554D4—24no—
555D4—24no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—816no—
591C2×C4—82no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4#39914yes—
621C2×C4#39914yes—
622C2×C4C2×C22⋊Q811yescharacteristic
623C2×C4C2×C22⋊Q811yescharacteristic
624C2×C4C2×#12311yescharacteristic
625C2×C4C2×#12311yescharacteristic
626C2×C4C2×C4⋊Q811yescharacteristic
627C2×C4#53211yescharacteristic
628C2×C4#53311yescharacteristic
629C2×C4#53311yescharacteristic
630C2×C4#54211yescharacteristic
631C2×C4#54211yescharacteristic
632C2×C4#54411yescharacteristic
633C2×C4#54411yescharacteristic
634C2×C4D4211yescharacteristic
635C2×C4D4211yescharacteristic
636C2×C4#54711yescharacteristic
637C2×C4#54711yescharacteristic
638C2×C4#54711yescharacteristic
639C2×C4#54711yescharacteristic
640C2×C4D4×Q811yescharacteristic
641C2×C4D4×Q811yescharacteristic
642C2×C4#55011yescharacteristic
643C2×C4#55011yescharacteristic
644C2×C4#55211yescharacteristic
645C2×C4#55211yescharacteristic
646C2×C4#55311yescharacteristic
647C2×C4#55311yescharacteristic
648C2×C4#55411yescharacteristic
649C2×C4#55411yescharacteristic
650C8—42no—
651C8—42no—
652C8—42no—
653C8—42no—
654C22—44no—
655C22—48no—
656C22—84no—
657C22—82no—
658C22—84no—
659C22—82no—
660C22—84no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22C2×#39911yescharacteristic
666C22#248611yescharacteristic
667C22#248711yescharacteristic
668C22#254111yescharacteristic
669C22#254211yescharacteristic
670C22#262811yescharacteristic
671C22#285711yescharacteristic
672C22#304212yes—
673C22#304512yes—
674C22#308112yes—
675C22#308112yes—
676C22#308412yes—
677C22#308412yes—
678C22#309612yes—
679C22#309912yes—
680C22#315411yescharacteristic
681C22#315411yescharacteristic
682C22#315411yescharacteristic
683C22#315511yescharacteristic
684C22#322112yes—
685C22#322412yes—
686C22#324112yes—
687C22#324412yes—
688C4—24no—
689C4—24no—
690C4—44no—
691C4—44no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—161no—
696C4—161no—
697C4—161no—
698C4—161no—
699C4#242211yescharacteristic
700C4#242211yescharacteristic
701C4#242311yescharacteristic
702C4#242311yescharacteristic
703C4#244011yescharacteristic
704C4#244011yescharacteristic
705C4#244111yescharacteristic
706C4#244111yescharacteristic
707C2—84no—
708C2#1513511yescharacteristic
709C2#2373511yescharacteristic
710C2#2373811yescharacteristic
711C2#2765412yes—
712C2#2768812yes—
713C2#2845211yescharacteristic
714C2#2845511yescharacteristic
715C2#2852411yescharacteristic
716C2#2852511yescharacteristic
717C2#2999012yes—
718C2#3320512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1949C2semidirect
4. C2×#1948C2semidirect
7. #14354C2semidirect
9. #11373C2semidirect
10. #11372C2semidirect
11. #11325C2semidirect
12. #11323C2semidirect
15. #10833C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20942
C2×#20922
C2×#19492
C2×#19482
C2×#18042
C2×#18022
#143542
#137852
#113732
#113722
#113252
#113232
#110512
#110492
#108332