Finite Group Databasebeta

← #219772

Subgroups — #219772

Order 512.

1404 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2452C211yesmaximal, characteristic
2#22401C211yesmaximal, characteristic
3D4×C22⋊C8C211yesmaximal, characteristic
4C2×#1472C212yesmaximal, complemented
5#9650C211yesmaximal, characteristic, complemented
6#9650C211yesmaximal, characteristic, complemented
7#9085C212yesmaximal, complemented
8#9081C212yesmaximal, complemented
9#8558C211yesmaximal, characteristic, complemented
10#8308C212yesmaximal, complemented
11C4×#324C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C411yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×C4×D4C2211yescharacteristic
15C2×#436—21no—
16C2×C8×D4—21no—
17C2×#407C2212yes—
18C22×C22⋊C8C2212yes—
19#2452—21no—
20#2452—21no—
21#2452—22no—
22#2452—22no—
23#2452—21no—
24#2452—21no—
25C2×#396C412yes—
26C2×#393C412yes—
27C2×#390C411yescharacteristic
28C2×#390C411yescharacteristic
29C2×#389C411yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#385C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1954—21no—
34#1954—21no—
35#1870—22no—
36#1869—22no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1779C2212yes—
40#1779C2212yes—
41C4×C22⋊C8C2211yescharacteristic
42C4×C22⋊C8C2211yescharacteristic
43#1472—28no—
44C2×#324C2212yes—
45C2×#324C2211yescharacteristic
46C2×#324C2212yes—
47C2×#324C2212yes—
48C2×#324C2212yes—
49C2×#324C2212yes—
50C2×#324C2211yescharacteristic
51#1280C2212yes—
52#1280C2211yescharacteristic
53#1280C2211yescharacteristic
54C23×Q8C2×C412yes—
55C23×D4C2×C411yescharacteristic
56C24×C4C2312yes—
57C22×M16—22no—
58C23×C8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—44no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—42no—
70C2×C22⋊Q8—42no—
71C2×C4×D4—44no—
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D412yes—
75C2×C4×D4D412yes—
76C2×C4×D4D411yescharacteristic
77C2×C4×D4D411yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C4⋊C4C2×C412yes—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2×C412yes—
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4C2×C412yes—
87C22×C22⋊C4C2×C412yes—
88C22×C22⋊C4C2×C412yes—
89C22×C22⋊C4C2×C412yes—
90C22×C22⋊C4C2×C412yes—
91C22×C42C2×C411yescharacteristic
92C22×C42C2×C411yescharacteristic
93C22×C42C2311yescharacteristic
94#436—44no—
95C8×D4—44no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98#407—24no—
99#407—42no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—42no—
108C2×C22⋊C8C2312yes—
109C2×C22⋊C8C2312yes—
110C2×C22⋊C8C2311yescharacteristic
111C2×C22⋊C8C2311yescharacteristic
112C2×C22⋊C8C2311yescharacteristic
113C2×C22⋊C8C2311yescharacteristic
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116#396—22no—
117#396—22no—
118#396—22no—
119#396—22no—
120#393—22no—
121#393—22no—
122#393—22no—
123#393—22no—
124#390—21no—
125#390—22no—
126#390—21no—
127#390—21no—
128#390—22no—
129#390—21no—
130#389—21no—
131#389—22no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389—21no—
137#385—22no—
138#385—22no—
139#385—22no—
140#385—22no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C2×#95C2×C411yescharacteristic
146C2×#95C2×C412yes—
147C2×#95C2×C412yes—
148C2×#95C2311yescharacteristic
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#324—24no—
154#324—22no—
155#324—24no—
156#324—24no—
157#324—42no—
158#324—42no—
159#324D414yescomplemented
160C25C22×C412yes—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—24no—
164C22×Q8—42no—
165C22×Q8—44no—
166C22×Q8—82no—
167C22×D4—24no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—42no—
171C22×D4C22⋊C411yescharacteristic
172C22×D4C22⋊C412yes—
173C22×D4C22⋊C411yescharacteristic
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—44no—
179C23×C4—82no—
180C23×C4C22×C412yes—
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C412yes—
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C412yes—
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D412yes—
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4C2411yescharacteristic
194C2×M16—44no—
195C2×M16—44no—
196C2×M16—82no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—44no—
204C22×C8—44no—
205C22×C8—82no—
206C4⋊Q8—48no—
207C4⋊Q8—44no—
208C4⋊Q8—44no—
209C22⋊Q8—44no—
210C22⋊Q8—44no—
211C22⋊Q8—44no—
212C22⋊Q8—44no—
213C22⋊Q8—44no—
214C22⋊Q8—44no—
215C22⋊Q8—44no—
216C22⋊Q8—44no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—44no—
222C4×D4—44no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
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—24no—
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—22no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C22⋊C411yescharacteristic
253C2×C4⋊C4C22⋊C412yes—
254C2×C4⋊C4C22⋊C411yescharacteristic
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4#4112yes—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—82no—
273C2×C22⋊C4C22⋊C412yes—
274C2×C22⋊C4C22⋊C412yes—
275C2×C22⋊C4C22⋊C412yes—
276C2×C22⋊C4C22⋊C412yes—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C4⋊C8—42no—
290C4⋊C8—42no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—44no—
302C22⋊C8—44no—
303C22⋊C8—44no—
304C22⋊C8—44no—
305C22⋊C8—82no—
306C22⋊C8—82no—
307C22⋊C8C2×D412yes—
308C22⋊C8C2×D412yes—
309C4×C8—42no—
310C4×C8—42no—
311#95—21no—
312#95—21no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—21no—
318#95—21no—
319C24—24no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24C2×C22⋊C412yes—
330C24C2×C22⋊C412yes—
331C24C23×C411yescharacteristic
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—82no—
337C2×Q8—82no—
338C2×Q8—88no—
339C2×Q8—88no—
340C2×D4—24no—
341C2×D4—22no—
342C2×D4—24no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—42no—
350C2×D4#10114yes—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—41no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C412yes—
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C412yes—
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C4×D412yes—
421C22×C4C4×D412yes—
422C22×C4#13112yes—
423C22×C4#13112yes—
424C22×C4#13112yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4H(5,𝔽2)11yescharacteristic
430C22×C4H(5,𝔽2)11yescharacteristic
431M16—88no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—88no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4#9914yes—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C42—22no—
485C42—22no—
486C42—41no—
487C42—41no—
488C42—42no—
489C42—42no—
490C42—42no—
491C42—42no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23—82no—
525C23C2×#10114yes—
526C23C22×C22⋊C411yescharacteristic
527C23C2×C4×D411yescharacteristic
528C23#51811yescharacteristic
529C23C2×#13111yescharacteristic
530C23C2×#13111yescharacteristic
531C23C2×#13111yescharacteristic
532C23#56811yescharacteristic
533Q8—88no—
534Q8—88no—
535D4—28no—
536D4—48no—
537D4—44no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
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—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4#40812yes—
608C2×C4#40812yes—
609C2×C4C2×#9914yes—
610C2×C4C2×#9912yes—
611C2×C4C2×#10112yes—
612C2×C4D4211yescharacteristic
613C2×C4D4211yescharacteristic
614C2×C4#54611yescharacteristic
615C2×C4#54611yescharacteristic
616C8—42no—
617C8—42no—
618C8—81no—
619C8—81no—
620C8—82no—
621C8—82no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—84no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22D4×C22⋊C411yescharacteristic
653C22C2×#40811yescharacteristic
654C22#288011yescharacteristic
655C22C22×#9911yescharacteristic
656C22#288311yescharacteristic
657C22C22×#10111yescharacteristic
658C22#288911yescharacteristic
659C22#293111yescharacteristic
660C22#293511yescharacteristic
661C22#297911yescharacteristic
662C22#298111yescharacteristic
663C4—21no—
664C4—21no—
665C4—21no—
666C4—21no—
667C4—24no—
668C4—22no—
669C4—41no—
670C4—41no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—82no—
676C4—82no—
677C4—81no—
678C4#147214yescomplemented
679C2—21no—
680C2—21no—
681C2—21no—
682C2—21no—
683C2—44no—
684C2—82no—
685C2C2×#147211yescharacteristic
686C2#1738811yescharacteristic
687C2#1741411yescharacteristic
688C2#1742611yescharacteristic
689C2#2240211yescharacteristic
690C2D4×#9911yescharacteristic
691C2D4×#10111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1472C2semidirect
5. #9650C2semidirect
6. #9650C2semidirect
7. #9085C2semidirect
8. #9081C2semidirect
9. #8558C2semidirect
10. #8308C2semidirect
11. C4×#324C2semidirect
159. #324D4semidirect
678. C4#1472semidirect

Maximal subgroups

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

subgroupindex
C2×#24522
#224012
D4×C22⋊C82
C2×#14722
#96502
#90852
#90812
#85582
#83082
C4×#3242