Finite Group Databasebeta

← #214228

Subgroups — #214228

Order 512.

1326 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34935C211yesmaximal, characteristic
2#21373C211yesmaximal, characteristic, complemented
3#20629C211yesmaximal, characteristic, complemented
4#9076C212yesmaximal, complemented
5#9072C212yesmaximal, complemented
6#8944C212yesmaximal, complemented
7#8939C212yesmaximal, complemented
8#8709C212yesmaximal, complemented
9#8310C212yesmaximal, complemented
10C2×#525—21no—
11C2×C4×#41C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C2×#431C2211yescharacteristic, complemented
14C2×#423C2211yescharacteristic, complemented
15C22×C4⋊C8C2211yescharacteristic, complemented
16C2×#408—21no—
17C22×C4×C8C2211yescharacteristic, complemented
18#2538C412yes—
19#2537C412yes—
20#2536—21no—
21#2535—21no—
22#2534—21no—
23#2534—21no—
24#2486C411yescharacteristic
25#2486C411yescharacteristic
26#2408C411yescharacteristic
27#2408C411yescharacteristic
28#2392—21no—
29#2392—21no—
30C4×#121C2212yes—
31C4×#121C2212yes—
32#2285C2211yescharacteristic
33#2280—21no—
34#1984—21no—
35#1954—21no—
36#1869—21no—
37#1869—21no—
38#1798—21no—
39#1798—21no—
40#1797C2212yes—
41#1797C2212yescomplemented
42C4×C22⋊C8C2212yes—
43C4×C22⋊C8C2212yescomplemented
44#1729—21no—
45C2×D4⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49#1279C2212yescomplemented
50#1279C2212yescomplemented
51#1279C2212yescomplemented
52#1279C2212yescomplemented
53C22×#41D411yescharacteristic
54C23×D4C2×C411yescharacteristic
55C24×C4—21no—
56C22×M16—21no—
57C23×C8—21no—
58C23×C8—21no—
59#525—44no—
60#525—44no—
61C2×#123—21no—
62C2×#123—21no—
63C2×C22⋊Q8—22no—
64C2×#121—22no—
65C2×#121C2×C412yes—
66C2×#121C2×C412yes—
67C2×#120—21no—
68C2×#120—21no—
69C4×#41—41no—
70C4×#41—41no—
71C4×#41—41no—
72C4×#41—41no—
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4—41no—
80C2×C4×D4—41no—
81C2×C4×D4—41no—
82C2×C4×D4—41no—
83C2×C4×D4C2×C412yes—
84C2×C4×D4C2×C411yescharacteristic
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2×C411yescharacteristic
87C2×C4×D4D411yescharacteristic
88C2×C4×D4D411yescharacteristic
89C2×C4×D4D411yescharacteristic
90C2×C4×D4C2312yes—
91C2×#117—21no—
92C2×#117D411yescharacteristic
93C2×#117D411yescharacteristic
94C2×#117D411yescharacteristic
95C22×C4⋊C4C2×C411yescharacteristic
96C22×C22⋊C4—21no—
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#431—41no—
101#431—41no—
102#423—41no—
103#423—41no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—41no—
108C2×C4⋊C8—41no—
109C2×C4⋊C8C2312yes—
110C2×C4⋊C8C2312yes—
111#408—44no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8—22no—
117C2×C4×C8—21no—
118C2×C4×C8—41no—
119C2×C4×C8—41no—
120C2×C4×C8C2312yes—
121C2×C4×C8C2312yes—
122#398—21no—
123#398—21no—
124#397—21no—
125#397—21no—
126#396—22no—
127#396—22no—
128#394—21no—
129#394—21no—
130#394—22no—
131#394—22no—
132#392—21no—
133#392—21no—
134#390C2×C412yes—
135#390C2×C412yes—
136#389C2×C412yes—
137#389C2×C412yes—
138#388—22no—
139#388—22no—
140#386—22no—
141#386—22no—
142#384C2×C412yes—
143#382—21no—
144#381—21no—
145#381—21no—
146#380—21no—
147#380—22no—
148#380—22no—
149#380C2×C411yescharacteristic
150#380C2×C411yescharacteristic
151#380C2×C412yes—
152#380C2×C412yes—
153#379—21no—
154#379—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4C2312yes—
157C4×C22⋊C4—22no—
158C4×C22⋊C4—22no—
159#336—22no—
160#336—22no—
161#336—22no—
162#336—22no—
163D4⋊C8—42no—
164D4⋊C8—42no—
165D4⋊C8—42no—
166D4⋊C8—42no—
167C25—21no—
168C2×#41—41no—
169C2×#41—41no—
170C2×#41—41no—
171C2×#41—41no—
172C2×#41D814yes—
173C2×#41SD1614yes—
174C22×Q8C22⋊C411yescharacteristic
175C22×D4—24no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—21no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4—42no—
182C22×D4C22⋊C411yescharacteristic
183C22×D4C22⋊C411yescharacteristic
184C22×D4C22⋊C411yescharacteristic
185C22×D4C22×C412yes—
186C23×C4—21no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4C22×C411yescharacteristic
196C23×C4C22×C411yescharacteristic
197C23×C4C22×C411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C2×M16—42no—
205C2×M16—42no—
206C2×M16—42no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—44no—
212C22×C8—42no—
213C22×C8—42no—
214C22×C8—42no—
215C22×C8—42no—
216#123—44no—
217#123—44no—
218C22⋊Q8—48no—
219#121—24no—
220#121—24no—
221#121—42no—
222#121—42no—
223#121—48no—
224#120—44no—
225#120—44no—
226C4×Q8—41no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×Q8—41no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—42no—
235C4×D4—42no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—41no—
240C4×D4—41no—
241C4×D4—41no—
242C4×D4—41no—
243C4×D4—41no—
244C4×D4—41no—
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—41no—
248C4×D4—42no—
249C4×D4—42no—
250C4×D4—42no—
251C4×D4—42no—
252C4×D4—44no—
253C4×D4—44no—
254C4×D4—44no—
255C4×D4—44no—
256#117—41no—
257#117—41no—
258#117—41no—
259#117—41no—
260#117—41no—
261#117—41no—
262#117—42no—
263#117—42no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4C22⋊C411yescharacteristic
272C2×C4⋊C4C22⋊C411yescharacteristic
273C2×C4⋊C4C22⋊C411yescharacteristic
274C2×C4⋊C4C22⋊C411yescharacteristic
275C2×C4⋊C4C22⋊C411yescharacteristic
276C2×C4⋊C4C22⋊C411yescharacteristic
277C2×C4⋊C4C22×C412yes—
278C2×C4⋊C4C22×C412yes—
279C2×C4⋊C4C22×C412yes—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4C22⋊C411yescharacteristic
295C2×C22⋊C4C22⋊C411yescharacteristic
296C2×C22⋊C4C22⋊C411yescharacteristic
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C22⋊C4C22⋊C411yescharacteristic
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—22no—
313C2×C42C22×C411yescharacteristic
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2411yescharacteristic
323C4⋊C8—22no—
324C4⋊C8—22no—
325C4⋊C8—22no—
326C4⋊C8—22no—
327C4⋊C8—42no—
328C4⋊C8—42no—
329C4⋊C8—42no—
330C22⋊C8—44no—
331C22⋊C8—44no—
332C22⋊C8—42no—
333C22⋊C8—44no—
334C22⋊C8—44no—
335C4×C8—22no—
336C4×C8—22no—
337C4×C8—22no—
338C4×C8—22no—
339C4×C8—42no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—21no—
345#95—21no—
346#95—22no—
347C24—22no—
348C24—44no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24—82no—
353C24C2×C22⋊C411yescharacteristic
354C24C2×C22⋊C411yescharacteristic
355C24C2×C22⋊C411yescharacteristic
356#41—44no—
357#41—44no—
358#41—44no—
359#41—44no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8#10214yes—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—44no—
375C2×D4—42no—
376C2×D4—44no—
377C2×D4—48no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—82no—
384C2×D4—82no—
385C2×D4#10214yes—
386C2×D4#10214yes—
387C2×D4#10214yes—
388C2×D4#13114yes—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×D812yes—
451C22×C4C2×D812yes—
452C22×C4C2×D812yes—
453C22×C4C2×SD1612yes—
454C22×C4C2×SD1612yes—
455C22×C4C2×SD1612yes—
456C22×C4C23×C411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459M16—84no—
460C2×C8—24no—
461C2×C8—42no—
462C2×C8—48no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—84no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—82no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—42no—
529C42—42no—
530C42H(5,𝔽2)11yescharacteristic
531C42H(5,𝔽2)11yescharacteristic
532C42H(5,𝔽2)11yescharacteristic
533C42H(5,𝔽2)11yescharacteristic
534C23—21no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—48no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—88no—
555C23—84no—
556C23—88no—
557C23—84no—
558C23C2×#10212yes—
559C23C2×#10212yes—
560C23C2×#10212yes—
561C23C22×C22⋊C411yescharacteristic
562Q8—44no—
563D4—24no—
564D4—24no—
565D4—44no—
566D4—44no—
567D4—44no—
568D4—48no—
569D4—42no—
570D4—42no—
571D4—88no—
572D4—88no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—88no—
629C2×C4—84no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4C2×#10212yes—
635C2×C4C2×#10212yes—
636C2×C4C2×#10212yes—
637C2×C4#51811yescharacteristic
638C2×C4#51811yescharacteristic
639C2×C4#53411yescharacteristic
640C2×C4#53411yescharacteristic
641C2×C4#53411yescharacteristic
642C2×C4#53411yescharacteristic
643C2×C4C2×#13112yes—
644C2×C4#56812yes—
645C2×C4C22×D811yescharacteristic
646C2×C4C22×SD1611yescharacteristic
647C2×C4#57511yescharacteristic
648C2×C4#57511yescharacteristic
649C8—44no—
650C8—82no—
651C8—82no—
652C8—82no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—816no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22#230811yescharacteristic
677C22#288012yes—
678C22C22×#10211yescharacteristic
679C22#289611yescharacteristic
680C22#290112yes—
681C4—21no—
682C4—21no—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—41no—
691C4—41no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4#297612yes—
696C4#297711yescharacteristic
697C4#297811yescharacteristic
698C4#318611yescharacteristic
699C4#318611yescharacteristic
700C4#318711yescharacteristic
701C4#318711yescharacteristic
702C2—42no—
703C2—42no—
704C2—42no—
705C2—84no—
706C2#1750712yes—
707C2#1751312yes—
708C2#2174411yescharacteristic
709C2#2175811yescharacteristic
710C2#2177511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#34935C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21373C2semidirect
3. #20629C2semidirect
4. #9076C2semidirect
5. #9072C2semidirect
6. #8944C2semidirect
7. #8939C2semidirect
8. #8709C2semidirect
9. #8310C2semidirect
13. C2×#431C22semidirect
14. C2×#423C22semidirect
15. C22×C4⋊C8C22semidirect
17. C22×C4×C8C22semidirect
41. #1797C22semidirect
43. C4×C22⋊C8C22semidirect
49. #1279C22semidirect
50. #1279C22semidirect
51. #1279C22semidirect
52. #1279C22semidirect

Maximal subgroups

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

subgroupindex
#349352
#213732
#206292
#90762
#90722
#89442
#89392
#87092
#83102