Finite Group Databasebeta

← #413132

Subgroups — #413132

Order 512.

1570 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2067C212yesmaximal, complemented
2C2×#1982C211yesmaximal, characteristic
3C2×#1957C211yesmaximal, characteristic
4C2×#1947C211yesmaximal, characteristic, complemented
5C2×#1939C211yesmaximal, characteristic, complemented
6#14154C211yesmaximal, characteristic
7#14151C211yesmaximal, characteristic, complemented
8#11123C212yesmaximal, complemented
9#11076C212yesmaximal, complemented
10#11066C212yesmaximal, complemented
11C4×#340C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C4×C8C2211yescharacteristic
21C2×#390—21no—
22C2×#390—21no—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#389—21no—
26C2×#389C2212yes—
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#2067—28no—
32#1982—24no—
33#1957—24no—
34#1947—24no—
35#1939—24no—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#336C2212yes—
42#1441—22no—
43#1438—22no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C23×D4—21no—
47C23×D4—21no—
48C23×D4C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C2×#127—22no—
53C2×#127—22no—
54C2×#127—22no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127Q814yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C42—21no—
73C22×C42—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×#107—24no—
77C2×#107—24no—
78C2×#107—42no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—24no—
83C2×C4⋊C8—42no—
84C2×C4⋊C8D414yes—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—42no—
94C2×#102—42no—
95C2×#102—42no—
96C2×#102—42no—
97C2×C4×C8—22no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100#390—24no—
101#390—24no—
102#390—42no—
103#390—42no—
104#390—44no—
105#390—44no—
106#389—24no—
107#389—42no—
108#389—44no—
109#384—22no—
110#384D414yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95D412yes—
123#340—22no—
124#340—24no—
125#340—22no—
126#340—22no—
127#340D414yescomplemented
128#336—24no—
129C25—22no—
130C22×D4—24no—
131C22×D4—24no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—81no—
154C22×D4—81no—
155C22×D4—81no—
156C22×D4—81no—
157C22×D4C2×Q814yes—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8C2×D414yes—
187C22×C8C2×D412yes—
188C22×C8#4112yes—
189#127—28no—
190#127—28no—
191#127—44no—
192#127—44no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—81no—
230C2×C4⋊C4—81no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4D814yes—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4#4112yes—
237C2×C4⋊C4#4112yes—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—82no—
247C2×C22⋊C4—82no—
248C2×C22⋊C4—82no—
249C2×C22⋊C4—82no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×Q812yes—
265C2×C42#4112yes—
266C2×C42#4112yes—
267C2×C42#4112yes—
268#107—48no—
269C4⋊C8—28no—
270C4⋊C8—48no—
271#102—48no—
272#102—48no—
273#102—48no—
274#102—48no—
275C4×C8—44no—
276#95—24no—
277#95—44no—
278#95—42no—
279#95—42no—
280#95—44no—
281#95—82no—
282C24—24no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24C22×D411yescharacteristic
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—84no—
331C2×D4—84no—
332C2×D4—84no—
333C2×D4—84no—
334C2×D4—84no—
335C2×D4—84no—
336C2×D4—84no—
337C2×D4—84no—
338C2×D4—84no—
339C2×D4—84no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4#12112yes—
393C22×C4#12112yes—
394C22×C4#12112yes—
395C22×C4#12112yes—
396C22×C4C22⋊Q812yes—
397C22×C4C22⋊Q812yes—
398C22×C4C2×D814yes—
399C22×C4C2×D812yes—
400C22×C4#13512yes—
401C22×C4AGL(1,ℤ/8)12yes—
402C22×C4#13712yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×Q811yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4H(5,𝔽2)11yescharacteristic
414C22×C4H(5,𝔽2)11yescharacteristic
415C22×C4H(5,𝔽2)11yescharacteristic
416C22×C42−1+411yescharacteristic
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—22no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—42no—
429C2×C8—41no—
430C2×C8—42no—
431C2×C8—41no—
432C2×C8—42no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—42no—
437C2×C8—41no—
438C4⋊C4—28no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—42no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—84no—
462C4⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—42no—
471C42—84no—
472C42—84no—
473C42#12114yes—
474C42C22⋊Q814yes—
475C42C22⋊Q814yes—
476C42#12614yes—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23C2×#12111yescharacteristic
513C23C2×#12111yescharacteristic
514C23C2×C22⋊Q811yescharacteristic
515C23D4211yescharacteristic
516C23#54611yescharacteristic
517C23#54811yescharacteristic
518C23#55011yescharacteristic
519C23C22×D811yescharacteristic
520C23C2×#13511yescharacteristic
521C23C2×AGL(1,ℤ/8)11yescharacteristic
522C23C2×#13711yescharacteristic
523C23#57611yescharacteristic
524C23#57611yescharacteristic
525C23#57711yescharacteristic
526C23#57711yescharacteristic
527D4—44no—
528D4—44no—
529D4—44no—
530D4—44no—
531D4—816no—
532D4—816no—
533D4—82no—
534D4—82no—
535D4—82no—
536D4—82no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4#39614yes—
593C2×C4#39614yes—
594C2×C4#45914yes—
595C2×C4#46414yes—
596C2×C4#46614yes—
597C2×C4#46814yes—
598C2×C4#47414yes—
599C2×C4#47814yes—
600C2×C4C2×#12111yescharacteristic
601C2×C4C2×C22⋊Q811yescharacteristic
602C2×C4C2×C22⋊Q811yescharacteristic
603C2×C4C2×#12611yescharacteristic
604C2×C4#52911yescharacteristic
605C2×C4#52911yescharacteristic
606C2×C4#52911yescharacteristic
607C2×C4#53211yescharacteristic
608C2×C4#53211yescharacteristic
609C2×C4#53211yescharacteristic
610C2×C4#54011yescharacteristic
611C2×C4#54011yescharacteristic
612C2×C4#54011yescharacteristic
613C2×C4#54211yescharacteristic
614C2×C4#54211yescharacteristic
615C2×C4#54211yescharacteristic
616C2×C4D4211yescharacteristic
617C2×C4D4211yescharacteristic
618C2×C4#54611yescharacteristic
619C2×C4#54611yescharacteristic
620C2×C4#54811yescharacteristic
621C2×C4#54811yescharacteristic
622C2×C4D4×Q811yescharacteristic
623C2×C4#55011yescharacteristic
624C2×C4#55011yescharacteristic
625C2×C4#55411yescharacteristic
626C2×C4#55411yescharacteristic
627C2×C4#55411yescharacteristic
628C8—42no—
629C8—44no—
630C8—42no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22—84no—
638C22—162no—
639C22—162no—
640C22—162no—
641C22—162no—
642C22—162no—
643C22—162no—
644C22—162no—
645C22—162no—
646C22C2×#39611yescharacteristic
647C22#242611yescharacteristic
648C22#242611yescharacteristic
649C22#242811yescharacteristic
650C22#242811yescharacteristic
651C22#245211yescharacteristic
652C22#245411yescharacteristic
653C22C2×#45911yescharacteristic
654C22C2×#46411yescharacteristic
655C22#304111yescharacteristic
656C22#304311yescharacteristic
657C22C2×#46611yescharacteristic
658C22C2×#46811yescharacteristic
659C22#306211yescharacteristic
660C22#306211yescharacteristic
661C22C2×#47411yescharacteristic
662C22C2×#47811yescharacteristic
663C22#308111yescharacteristic
664C22#308111yescharacteristic
665C22#327411yescharacteristic
666C22#327811yescharacteristic
667C22D4×D811yescharacteristic
668C22#328411yescharacteristic
669C22#329411yescharacteristic
670C22#330511yescharacteristic
671C22#332411yescharacteristic
672C22#333311yescharacteristic
673C22#335811yescharacteristic
674C22#336111yescharacteristic
675C22#336811yescharacteristic
676C22#337511yescharacteristic
677C22#339111yescharacteristic
678C22#339411yescharacteristic
679C22#339911yescharacteristic
680C22#340011yescharacteristic
681C4—24no—
682C4—44no—
683C4—82no—
684C4—84no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4#206718yescomplemented
690C4#242611yescharacteristic
691C4#242811yescharacteristic
692C4#245211yescharacteristic
693C4#245211yescharacteristic
694C4#245411yescharacteristic
695C4#245411yescharacteristic
696C4#264411yescharacteristic
697C4#286011yescharacteristic
698C2—164no—
699C2#1514111yescharacteristic
700C2C2×#206711yescharacteristic
701C2#2361411yescharacteristic
702C2#2527211yescharacteristic
703C2#2527711yescharacteristic
704C2#2529211yescharacteristic
705C2#2530311yescharacteristic
706C2#2536411yescharacteristic
707C2#2536911yescharacteristic
708C2#2538411yescharacteristic
709C2#2539511yescharacteristic
710C2#2646011yescharacteristic
711C2#2646311yescharacteristic
712C2#2652611yescharacteristic
713C2#2652911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
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
1. C2×#2067C2semidirect
4. C2×#1947C2semidirect
5. C2×#1939C2semidirect
7. #14151C2semidirect
8. #11123C2semidirect
9. #11076C2semidirect
10. #11066C2semidirect
11. C4×#340C2semidirect
127. #340D4semidirect
689. C4#2067semidirect

Maximal subgroups

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

subgroupindex
C2×#20672
C2×#19822
C2×#19572
C2×#19472
C2×#19392
#141542
#141512
#111232
#110762
#110662
C4×#3402