Finite Group Databasebeta

← #365002

Subgroups — #365002

Order 512.

1874 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#467C211yesmaximal, characteristic
2C22×C4×Q16C211yesmaximal, characteristic, complemented
3C2×#1969C211yesmaximal, characteristic, complemented
4C2×#1808C211yesmaximal, characteristic
5C2×C8×C22⋊C4C211yesmaximal, characteristic
6#13969C212yesmaximal
7#11190C212yesmaximal, complemented
8#10970C212yesmaximal
9#10862C211yesmaximal, characteristic, complemented
10#10862C211yesmaximal, characteristic, complemented
11#10789C212yesmaximal, complemented
12C23×Q16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C22×C4×Q8C2212yes—
16C2×#467—24no—
17C2×#467—24no—
18C2×#467C418yescomplemented
19C2×C4×Q16—24no—
20C2×C4×Q16—24no—
21C2×C4×Q16—24no—
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#103C2212yes—
26C22×#103C2212yes—
27C22×#103C2212yes—
28C22×C22⋊C8C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C2×#391C2212yes—
32C2×#380C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#1969—24no—
35#1808—24no—
36C8×C22⋊C4—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1441—22no—
43#1439—22no—
44#1437—22no—
45#1436—22no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×Q8C2312yes—
49C24×C4C2311yescharacteristic
50C22×Q16—22no—
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—22no—
54C22×Q16C2×C414yes—
55C22×Q16D414yes—
56C22×Q16D414yes—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—22no—
63C23×C8C2×C414yes—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—42no—
72C2×C22⋊Q8—42no—
73C2×C22⋊Q8C2×C418yes—
74C2×C4×Q8—24no—
75C2×C4×Q8—24no—
76C2×C4×Q8—24no—
77C2×C4×Q8—24no—
78C2×C4×Q8—42no—
79C2×C4×Q8—42no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4D411yescharacteristic
88C22×C22⋊C4D411yescharacteristic
89C22×C42—22no—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#467—232no—
93C4×Q16—232no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107—22no—
97C2×#107—22no—
98C2×#107—22no—
99C2×#107—22no—
100C2×#107—22no—
101C2×#107C2×C414yes—
102C2×#107D414yes—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—42no—
111C2×#103C2×C418yes—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117C2×C4×C8—22no—
118C2×C4×C8—22no—
119C2×C4×C8—22no—
120C2×C4×C8D414yes—
121#391—24no—
122#391—42no—
123#380—24no—
124#380—42no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95D411yescharacteristic
135C2×#95D411yescharacteristic
136#340—24no—
137#340—42no—
138#340—42no—
139#336—22no—
140#336—22no—
141C25C2×D411yescharacteristic
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8C22×C414yes—
154C22×Q8C2×D414yes—
155C22×Q8C2×D414yes—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4C22×C412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4#4112yes—
179C23×C4C2411yescharacteristic
180C2×Q16—28no—
181C2×Q16—28no—
182C2×Q16—28no—
183C2×Q16—28no—
184C2×Q16—28no—
185C2×Q16—28no—
186C2×Q16C22⋊C4116yescomplemented
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
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—28no—
206C22×C8—28no—
207C22×C8—28no—
208C22×C8—44no—
209C22×C8—44no—
210C22×C8—44no—
211C22×C8—44no—
212C22×C8C22×C412yes—
213C22×C8C22×C412yes—
214C22×C8C2×D412yes—
215C22×C8C2×D412yes—
216C22×C8C2×D412yes—
217C22×C8C2×D412yes—
218C22×C8C2×D412yes—
219C22×C8#4112yes—
220C22×C8#4112yes—
221C22×C8#4112yes—
222C22⋊Q8—216no—
223C22⋊Q8—48no—
224C22⋊Q8—48no—
225C22⋊Q8—48no—
226C4×Q8—216no—
227C4×Q8—48no—
228C4×Q8—48no—
229C4×Q8—48no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C22×C414yes—
253C2×C4⋊C4C2×D414yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—82no—
265C2×C22⋊C4Q1614yescomplemented
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—44no—
277C2×C42—44no—
278C2×C42—44no—
279C2×C42C2×D412yes—
280C2×C42#4112yes—
281#107—28no—
282#107—28no—
283#107—44no—
284#103—216no—
285#103—216no—
286#103—48no—
287C22⋊C8—28no—
288C4×C8—28no—
289C4×C8—44no—
290#95—22no—
291#95—22no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—82no—
298#95—82no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—21no—
311C24C4×D412yes—
312C24C2×Q1612yes—
313C24#13512yes—
314C24C22×D411yescharacteristic
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8C2×C22⋊C418yes—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
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—21no—
357C22×C4—21no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
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—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
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—42no—
409C22×C4—42no—
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—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4C4×D412yes—
428C22×C4#12112yes—
429C22×C4#12112yes—
430C22×C4C2×Q1612yes—
431C22×C4C2×Q1612yes—
432C22×C4#13512yes—
433C22×C4#13512yes—
434C22×C4C23×C411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4H(5,𝔽2)11yescharacteristic
445C22×C4H(5,𝔽2)11yescharacteristic
446C22×C4H(5,𝔽2)11yescharacteristic
447C22×C42−1+411yescharacteristic
448Q16—232no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
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—416no—
471C2×C8C2×C22⋊C414yes—
472C2×C8C4×D414yes—
473C2×C8C4×D414yes—
474C2×C8#12014yes—
475C2×C8#12114yes—
476C2×C8#12114yes—
477C2×C8#12314yes—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—416no—
483C4⋊C4—416no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—416no—
495C22⋊C4—88no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—416no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23C4×Q1614yescomplemented
540C23C2×C4×D411yescharacteristic
541C23C2×#12111yescharacteristic
542C23#53211yescharacteristic
543C23D4211yescharacteristic
544C23D4211yescharacteristic
545C23D4211yescharacteristic
546C23#54711yescharacteristic
547C23C22×Q1611yescharacteristic
548C23C2×#13511yescharacteristic
549C23C2×#13511yescharacteristic
550C23C2×#13511yescharacteristic
551C23#57611yescharacteristic
552C23#57811yescharacteristic
553C23#57811yescharacteristic
554C23#57811yescharacteristic
555Q8—28no—
556Q8—28no—
557Q8—44no—
558Q8—44no—
559Q8—44no—
560Q8—44no—
561Q8—44no—
562Q8—44no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—28no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4#46714yescomplemented
634C2×C4C22×C22⋊C411yescharacteristic
635C2×C4C2×C4×D411yescharacteristic
636C2×C4C2×C4×D411yescharacteristic
637C2×C4C4×#4111yescharacteristic
638C2×C4#51811yescharacteristic
639C2×C4#52011yescharacteristic
640C2×C4C2×#12011yescharacteristic
641C2×C4C2×#12111yescharacteristic
642C2×C4C2×#12111yescharacteristic
643C2×C4C2×#12311yescharacteristic
644C2×C4#52511yescharacteristic
645C2×C4#52911yescharacteristic
646C2×C4#53511yescharacteristic
647C2×C4#53711yescharacteristic
648C2×C4#54011yescharacteristic
649C2×C4#54211yescharacteristic
650C2×C4#54611yescharacteristic
651C2×C4#54611yescharacteristic
652C2×C4#54611yescharacteristic
653C2×C4#54611yescharacteristic
654C2×C4#54611yescharacteristic
655C2×C4#54711yescharacteristic
656C2×C4#54711yescharacteristic
657C2×C4#54811yescharacteristic
658C2×C4#55011yescharacteristic
659C2×C4#55311yescharacteristic
660C2×C4#55311yescharacteristic
661C2×C4#55511yescharacteristic
662C8—42no—
663C8—42no—
664C8—42no—
665C8#38618yes—
666C22—24no—
667C22—24no—
668C22—24no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22C4×#12111yescharacteristic
682C22D4×C22⋊C411yescharacteristic
683C22#236111yescharacteristic
684C22#240811yescharacteristic
685C22#242211yescharacteristic
686C22#242811yescharacteristic
687C22#253711yescharacteristic
688C22C2×C4×Q1611yescharacteristic
689C22C4×#13511yescharacteristic
690C22#295011yescharacteristic
691C22#295111yescharacteristic
692C22C2×#46711yescharacteristic
693C22#305011yescharacteristic
694C22#305911yescharacteristic
695C22#306111yescharacteristic
696C22#316811yescharacteristic
697C22#317011yescharacteristic
698C22#317511yescharacteristic
699C22#317711yescharacteristic
700C22#328011yescharacteristic
701C22#328311yescharacteristic
702C22#328511yescharacteristic
703C22#328511yescharacteristic
704C22#328511yescharacteristic
705C22#328511yescharacteristic
706C22#328511yescharacteristic
707C22D4×Q1611yescharacteristic
708C22D4×Q1611yescharacteristic
709C22D4×Q1611yescharacteristic
710C22#328711yescharacteristic
711C22#328711yescharacteristic
712C22#333611yescharacteristic
713C22#333611yescharacteristic
714C22#333811yescharacteristic
715C22#334211yescharacteristic
716C4—24no—
717C4—42no—
718C4—42no—
719C4—42no—
720C4—48no—
721C4—42no—
722C4—84no—
723C4—84no—
724C4—162no—
725C4C2×#38611yescharacteristic
726C4#232411yescharacteristic
727C4#233911yescharacteristic
728C4#236311yescharacteristic
729C4#240711yescharacteristic
730C4#242011yescharacteristic
731C4#242711yescharacteristic
732C4#253511yescharacteristic
733C2—44no—
734C2#1397011yescharacteristic
735C2C4×#46711yescharacteristic
736C2#2152511yescharacteristic
737C2C22⋊C4×Q1611yescharacteristic
738C2#2245111yescharacteristic
739C2#2302411yescharacteristic
740C2#2302611yescharacteristic
741C2#2426611yescharacteristic
742C2#2426811yescharacteristic
743C2#2503311yescharacteristic
744C2#2503511yescharacteristic
745C2#2541211yescharacteristic
746C2#2541511yescharacteristic
747C2#2822611yescharacteristic
748C2#2822811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×Q16C2semidirect
3. C2×#1969C2semidirect
7. #11190C2semidirect
9. #10862C2semidirect
10. #10862C2semidirect
11. #10789C2semidirect
18. C2×#467C4semidirect
186. C2×Q16C22⋊C4semidirect
265. C2×C22⋊C4Q16semidirect
539. C23C4×Q16semidirect
633. C2×C4#467semidirect

Maximal subgroups

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

subgroupindex
C22×#4672
C22×C4×Q162
C2×#19692
C2×#18082
C2×C8×C22⋊C42
#139692
#111902
#109702
#108622
#107892