Finite Group Databasebeta

← #368611

Subgroups — #368611

Order 512.

1604 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1982C211yesmaximal, characteristic
2C2×#1957C211yesmaximal, characteristic
3C2×#1925C212yesmaximal, complemented
4C2×C4×#107C211yesmaximal, characteristic, complemented
5#14154C211yesmaximal, characteristic
6C4⋊C42C211yesmaximal, characteristic, complemented
7#11189C212yesmaximal, complemented
8#11123C212yesmaximal, complemented
9#10931C212yesmaximal, complemented
10#10849C211yesmaximal, characteristic, complemented
11#10834C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
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×#384C2211yescharacteristic
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1982—24no—
32#1957C418yescomplemented
33#1925—28no—
34C4×#107—24no—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1438—22no—
42#1436—22no—
43C4⋊#95—21no—
44C4⋊#95C2212yes—
45C23×D4—21no—
46C23×D4—21no—
47C23×D4C2312yes—
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×#127—22no—
52C2×#127—22no—
53C2×#127—22no—
54C2×#127—22no—
55C2×#127—22no—
56C2×#127C2×C414yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C42—21no—
71C22×C42—21no—
72C22×C42—22no—
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75C2×#107—22no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#107D414yescomplemented
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—42no—
91C2×#102—42no—
92C2×#102—42no—
93C2×#102—42no—
94C2×#102C2×C418yes—
95C2×#102C2×C418yes—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8C2×C414yes—
99#390—24no—
100#390—24no—
101#390—42no—
102#390—42no—
103#390—44no—
104#390—44no—
105#384—22no—
106#384C2×C414yes—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—42no—
114C4×C4⋊C4—44no—
115C4×C4⋊C4D414yes—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95D412yes—
122#340—24no—
123#340—22no—
124#340—22no—
125#340—42no—
126#336—24no—
127#336—42no—
128C25—22no—
129C22×D4—24no—
130C22×D4—24no—
131C22×D4—22no—
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—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—81no—
153C22×D4—81no—
154C22×D4—81no—
155C22×D4—81no—
156C22×D4C22×C414yes—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8C22×C412yes—
185C22×C8C22×C412yes—
186C22×C8C2×D412yes—
187C22×C8#4112yes—
188#127—28no—
189#127—28no—
190#127—44no—
191#127—44no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
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—21no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—81no—
225C2×C4⋊C4—81no—
226C2×C4⋊C4D814yes—
227C2×C4⋊C4C22×C414yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4#4112yes—
233C2×C4⋊C4#4112yes—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—82no—
243C2×C22⋊C4—82no—
244C2×C22⋊C4—82no—
245C2×C22⋊C4—82no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—44no—
256C2×C42—42no—
257C2×C42—44no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—82no—
261C2×C42C22×C412yes—
262C2×C42C2×D412yes—
263C2×C42#4112yes—
264C2×C42#4112yes—
265C2×C42#4112yes—
266C2×C42#4112yes—
267#107—28no—
268#107—28no—
269C4⋊C8—44no—
270#102—216no—
271#102—48no—
272#102—48no—
273#102—48no—
274C4×C8—28no—
275#95—24no—
276#95—24no—
277#95—44no—
278#95—42no—
279#95—44no—
280#95—82no—
281C24—24no—
282C24—42no—
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—
296C24C22×D411yescharacteristic
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—42no—
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—48no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—84no—
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—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
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—22no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4C4×D412yes—
392C22×C4C4×D412yes—
393C22×C4#12112yes—
394C22×C4#12112yes—
395C22×C4#12312yes—
396C22×C4#12312yes—
397C22×C4C2×D814yes—
398C22×C4C2×D812yes—
399C22×C4#13512yes—
400C22×C4AGL(1,ℤ/8)12yes—
401C22×C4#13712yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4H(5,𝔽2)11yescharacteristic
413C22×C4H(5,𝔽2)11yescharacteristic
414C22×C4H(5,𝔽2)11yescharacteristic
415C22×C42−1+411yescharacteristic
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8C4×D414yes—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—28no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—42no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—42no—
457C4⋊C4—48no—
458C4⋊C4—44no—
459C4⋊C4—84no—
460C4⋊C4#12114yes—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—48no—
470C42—88no—
471C42—84no—
472C42#11714yes—
473C42C4×D414yes—
474C42#12414yes—
475C42#12614yes—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—82no—
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—
511C23C2×C4×D411yescharacteristic
512C23C2×#12111yescharacteristic
513C23C2×#12311yescharacteristic
514C23D4211yescharacteristic
515C23#54611yescharacteristic
516C23#54811yescharacteristic
517C23#55011yescharacteristic
518C23C22×D811yescharacteristic
519C23C2×#13511yescharacteristic
520C23C2×AGL(1,ℤ/8)11yescharacteristic
521C23C2×#13711yescharacteristic
522C23#57611yescharacteristic
523C23#57611yescharacteristic
524C23#57711yescharacteristic
525C23#57711yescharacteristic
526D4—44no—
527D4—44no—
528D4—44no—
529D4—44no—
530D4—816no—
531D4—82no—
532D4—816no—
533D4—82no—
534D4—82no—
535D4—82no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
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—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—82no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—82no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4#38814yes—
592C2×C4#38814yes—
593C2×C4C4×D814yes—
594C2×C4#44014yes—
595C2×C4#45914yes—
596C2×C4#46414yes—
597C2×C4#48014yes—
598C2×C4#48214yes—
599C2×C4C2×#11711yescharacteristic
600C2×C4C2×C4×D411yescharacteristic
601C2×C4C2×C4×D411yescharacteristic
602C2×C4C4×#4111yescharacteristic
603C2×C4#51811yescharacteristic
604C2×C4#52011yescharacteristic
605C2×C4C2×#12111yescharacteristic
606C2×C4#52511yescharacteristic
607C2×C4C2×#12411yescharacteristic
608C2×C4C2×#12611yescharacteristic
609C2×C4#52911yescharacteristic
610C2×C4#53211yescharacteristic
611C2×C4#53511yescharacteristic
612C2×C4#53711yescharacteristic
613C2×C4#54011yescharacteristic
614C2×C4#54211yescharacteristic
615C2×C4#54711yescharacteristic
616C2×C4#55011yescharacteristic
617C2×C4#55011yescharacteristic
618C2×C4#55311yescharacteristic
619C2×C4#55311yescharacteristic
620C2×C4#55511yescharacteristic
621C2×C4#55611yescharacteristic
622C2×C4#55611yescharacteristic
623C2×C4#55911yescharacteristic
624C2×C4#55911yescharacteristic
625C2×C4#55911yescharacteristic
626C2×C4#55911yescharacteristic
627C8—24no—
628C8—42no—
629C8—82no—
630C22—84no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22—162no—
638C22—162no—
639C22—162no—
640C22—162no—
641C22—162no—
642C22—162no—
643C22—162no—
644C22—162no—
645C22C2×#38811yescharacteristic
646C22#236611yescharacteristic
647C22#236711yescharacteristic
648C22#241411yescharacteristic
649C22#241611yescharacteristic
650C22#244511yescharacteristic
651C22#244811yescharacteristic
652C22C2×C4×D811yescharacteristic
653C22C2×#44011yescharacteristic
654C22#294811yescharacteristic
655C22#294911yescharacteristic
656C22C2×#45911yescharacteristic
657C22C2×#46411yescharacteristic
658C22#304511yescharacteristic
659C22#304511yescharacteristic
660C22C2×#48011yescharacteristic
661C22C2×#48211yescharacteristic
662C22#309611yescharacteristic
663C22#309611yescharacteristic
664C22#327211yescharacteristic
665C22#327811yescharacteristic
666C22D4×D811yescharacteristic
667C22#328711yescharacteristic
668C22#329411yescharacteristic
669C22#330411yescharacteristic
670C22#332511yescharacteristic
671C22#333311yescharacteristic
672C22#335811yescharacteristic
673C22#336511yescharacteristic
674C22#336611yescharacteristic
675C22#337511yescharacteristic
676C22#339111yescharacteristic
677C22#339611yescharacteristic
678C22#339711yescharacteristic
679C22#340011yescharacteristic
680C4—24no—
681C4—44no—
682C4—44no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—84no—
687C4—82no—
688C4#192518yescomplemented
689C4C4×#12111yescharacteristic
690C4#232411yescharacteristic
691C4#235411yescharacteristic
692C4#235711yescharacteristic
693C4#251211yescharacteristic
694C4#251311yescharacteristic
695C4#254811yescharacteristic
696C4#259211yescharacteristic
697C2—164no—
698C2#1402211yescharacteristic
699C2C2×#192511yescharacteristic
700C2#2106511yescharacteristic
701C2#2318511yescharacteristic
702C2#2319011yescharacteristic
703C2#2321611yescharacteristic
704C2#2322311yescharacteristic
705C2#2462611yescharacteristic
706C2#2463311yescharacteristic
707C2#2472611yescharacteristic
708C2#2473311yescharacteristic
709C2#2623011yescharacteristic
710C2#2623711yescharacteristic
711C2#2631611yescharacteristic
712C2#2632311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C22×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×#1925C2semidirect
4. C2×C4×#107C2semidirect
6. C4⋊C42C2semidirect
7. #11189C2semidirect
8. #11123C2semidirect
9. #10931C2semidirect
10. #10849C2semidirect
11. #10834C2semidirect
32. #1957C4semidirect
80. C2×#107D4semidirect
688. C4#1925semidirect

Maximal subgroups

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

subgroupindex
C2×#19822
C2×#19572
C2×#19252
C2×C4×#1072
#141542
C4⋊C422
#111892
#111232
#109312
#108492
#108342