Finite Group Databasebeta

← #368376

Subgroups — #368376

Order 512.

1858 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D8C211yesmaximal, characteristic
2C2×#1970C211yesmaximal, characteristic
3C2×#1970C211yesmaximal, characteristic
4C2×#1942C211yesmaximal, characteristic, complemented
5C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
6#14018C212yesmaximal, complemented
7#11076C212yesmaximal, complemented
8#11066C212yesmaximal, complemented
9#10862C211yesmaximal, characteristic, complemented
10#10862C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal, complemented
12C23×D8C2211yescharacteristic
13C22×C4×D4C2212yes—
14C2×C4×D8—24no—
15C2×C4×D8—24no—
16C2×C4×D8—24no—
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#389C2212yes—
30C2×#385—22no—
31C2×C4×C4⋊C4C2211yescharacteristic
32#1970—24no—
33#1970C418yescomplemented
34#1942—24no—
35C8×C4⋊C4—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1439—22no—
42#1439—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—22no—
46C23×D4C2312yes—
47C24×C4—22no—
48C22×D8—22no—
49C22×D8—22no—
50C22×D8—22no—
51C22×D8—22no—
52C22×D8C2×C414yes—
53C22×D8D414yes—
54C22×D8Q814yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C2×C4×D4—24no—
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—42no—
64C2×C4×D4—42no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C4×D8—232no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107C2×C414yescomplemented
88C2×#107D414yes—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—42no—
98C2×#102—42no—
99C2×#102—42no—
100C2×#102C2×C418yescomplemented
101C2×C4×C8—22no—
102C2×C4×C8—22no—
103C2×C4×C8—22no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108C2×C4×C8C2×C414yescomplemented
109C2×C4×C8D414yes—
110#390—24no—
111#390—24no—
112#390—42no—
113#390C2×C418yescomplemented
114#389—24no—
115#389—42no—
116#385—48no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95D411yescharacteristic
125C2×#95D411yescharacteristic
126#340—24no—
127#340—42no—
128#340—42no—
129#336—22no—
130#336—22no—
131C25—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×D4C22×C414yes—
144C22×D4C2×D414yes—
145C22×D4C2×Q814yes—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—82no—
155C23×C4—82no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
163C2×D8—28no—
164C2×D8—28no—
165C2×D8—28no—
166C2×D8—28no—
167C2×D8—28no—
168C2×D8—28no—
169C2×D8C4⋊C4116yescomplemented
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8C22×C412yes—
189C22×C8C22×C412yes—
190C22×C8C2×D412yes—
191C22×C8C2×D412yes—
192C22×C8C2×D412yes—
193C22×C8C2×D412yes—
194C22×C8C2×Q812yes—
195C22×C8#4112yes—
196C22×C8#4112yes—
197C22×C8#4112yes—
198C4×D4—216no—
199C4×D4—48no—
200C4×D4—48no—
201C4×D4—48no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4D814yescomplemented
227C2×C4⋊C4C22×C414yes—
228C2×C4⋊C4C2×D414yes—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
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—82no—
241C2×C22⋊C4—82no—
242C2×C22⋊C4—82no—
243C2×C22⋊C4—82no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42C22×C412yes—
259C2×C42C2×D412yes—
260C2×C42#4112yes—
261C2×C42#4112yes—
262#107—28no—
263#107—28no—
264#107—44no—
265C4⋊C8—28no—
266#102—216no—
267#102—48no—
268#102—48no—
269C4×C8—28no—
270C4×C8—28no—
271C4×C8—44no—
272#95—22no—
273#95—22no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—82no—
278#95—82no—
279C24—24no—
280C24—24no—
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—
294C24C22×D411yescharacteristic
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—24no—
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×D4C2×C4⋊C418yes—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—416no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—88no—
378C22×C4—88no—
379C22×C4—88no—
380C22×C4—88no—
381C22×C4—88no—
382C22×C4—88no—
383C22×C4C4×D412yes—
384C22×C4C4×D412yes—
385C22×C4#12112yes—
386C22×C4#12112yes—
387C22×C4C2×D812yes—
388C22×C4C2×D812yes—
389C22×C4C2×D812yes—
390C22×C4#13512yes—
391C22×C4#13512yes—
392C22×C4#13512yes—
393C22×C4C23×C411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×Q811yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4H(5,𝔽2)11yescharacteristic
404C22×C4H(5,𝔽2)11yescharacteristic
405C22×C4H(5,𝔽2)11yescharacteristic
406C22×C42−1+411yescharacteristic
407D8—232no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8C2×C4⋊C414yes—
430C2×C8C4×D414yes—
431C2×C8C4×D414yes—
432C2×C8#12014yes—
433C2×C8C22⋊Q814yes—
434C2×C8C22⋊Q814yes—
435C2×C8#12314yes—
436C4⋊C4—28no—
437C4⋊C4—28no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—88no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—88no—
454C22⋊C4—88no—
455C22⋊C4—88no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—41no—
468C23—28no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
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—
503C23C2×C4×D411yescharacteristic
504C23C2×#12111yescharacteristic
505C23#53211yescharacteristic
506C23D4211yescharacteristic
507C23D4211yescharacteristic
508C23D4×Q811yescharacteristic
509C23#55011yescharacteristic
510C23C22×D811yescharacteristic
511C23C2×#13511yescharacteristic
512C23C2×#13511yescharacteristic
513C23C2×#13511yescharacteristic
514C23#57611yescharacteristic
515C23#57611yescharacteristic
516C23#57611yescharacteristic
517C23#57811yescharacteristic
518D4—28no—
519D4—28no—
520D4—44no—
521D4—44no—
522D4—44no—
523D4—44no—
524D4—44no—
525D4—44no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—832no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4C4×D814yescomplemented
589C2×C4#46614yes—
590C2×C4C22×C4⋊C411yescharacteristic
591C2×C4C2×C4×D411yescharacteristic
592C2×C4C2×C4×D411yescharacteristic
593C2×C4C4×#4111yescharacteristic
594C2×C4#51811yescharacteristic
595C2×C4#52011yescharacteristic
596C2×C4C2×#12011yescharacteristic
597C2×C4C2×C22⋊Q811yescharacteristic
598C2×C4C2×C22⋊Q811yescharacteristic
599C2×C4C2×#12311yescharacteristic
600C2×C4#52511yescharacteristic
601C2×C4#52911yescharacteristic
602C2×C4#53511yescharacteristic
603C2×C4#53711yescharacteristic
604C2×C4#54011yescharacteristic
605C2×C4#54211yescharacteristic
606C2×C4#54611yescharacteristic
607C2×C4#54611yescharacteristic
608C2×C4#54611yescharacteristic
609C2×C4#54711yescharacteristic
610C2×C4#54811yescharacteristic
611C2×C4#55011yescharacteristic
612C2×C4#55411yescharacteristic
613C2×C4#55411yescharacteristic
614C2×C4#55411yescharacteristic
615C2×C4#55611yescharacteristic
616C2×C4#55911yescharacteristic
617C2×C4#55911yescharacteristic
618C8—42no—
619C8—42no—
620C8—42no—
621C8#38518yescomplemented
622C22—48no—
623C22—48no—
624C22—48no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22C4×#12111yescharacteristic
638C22D4×C4⋊C411yescharacteristic
639C22#236611yescharacteristic
640C22#240811yescharacteristic
641C22#244511yescharacteristic
642C22#245211yescharacteristic
643C22#253811yescharacteristic
644C22C2×C4×D811yescharacteristic
645C22C4×#13511yescharacteristic
646C22#294811yescharacteristic
647C22#295111yescharacteristic
648C22C2×#46611yescharacteristic
649C22#305011yescharacteristic
650C22#305811yescharacteristic
651C22#306111yescharacteristic
652C22#316811yescharacteristic
653C22#316911yescharacteristic
654C22#317511yescharacteristic
655C22#317611yescharacteristic
656C22D4×D811yescharacteristic
657C22D4×D811yescharacteristic
658C22#328011yescharacteristic
659C22#328011yescharacteristic
660C22#328011yescharacteristic
661C22#328311yescharacteristic
662C22#328511yescharacteristic
663C22#328711yescharacteristic
664C22Q8×D811yescharacteristic
665C22#338011yescharacteristic
666C22#338011yescharacteristic
667C22#338011yescharacteristic
668C22#339111yescharacteristic
669C22#339311yescharacteristic
670C22#339611yescharacteristic
671C22#339611yescharacteristic
672C4—24no—
673C4—24no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—84no—
679C4—84no—
680C4—162no—
681C4C2×#38511yescharacteristic
682C4#232411yescharacteristic
683C4#234911yescharacteristic
684C4#236711yescharacteristic
685C4#240711yescharacteristic
686C4#244711yescharacteristic
687C4#245411yescharacteristic
688C4#258011yescharacteristic
689C2—88no—
690C2#1401811yescharacteristic
691C2C4×#46611yescharacteristic
692C2#2152411yescharacteristic
693C2C4⋊C4×D811yescharacteristic
694C2#2268211yescharacteristic
695C2#2318511yescharacteristic
696C2#2318611yescharacteristic
697C2#2426611yescharacteristic
698C2#2426711yescharacteristic
699C2#2625211yescharacteristic
700C2#2625411yescharacteristic
701C2#2648211yescharacteristic
702C2#2648411yescharacteristic
703C2#2826311yescharacteristic
704C2#2826711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
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
4. C2×#1942C2semidirect
5. C2×C8×C4⋊C4C2semidirect
6. #14018C2semidirect
7. #11076C2semidirect
8. #11066C2semidirect
9. #10862C2semidirect
10. #10862C2semidirect
11. #10788C2semidirect
33. #1970C4semidirect
87. C2×#107C2×C4semidirect
100. C2×#102C2×C4semidirect
108. C2×C4×C8C2×C4semidirect
113. #390C2×C4semidirect
169. C2×D8C4⋊C4semidirect
226. C2×C4⋊C4D8semidirect
588. C2×C4C4×D8semidirect
621. C8#385semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D82
C2×#19702
C2×#19422
C2×C8×C4⋊C42
#140182
#110762
#110662
#108622
#107882