Finite Group Databasebeta

← #182650

Subgroups — #182650

Order 512.

1412 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34203C211yesmaximal, characteristic
2#20379C211yesmaximal, characteristic
3C2×#1520C211yesmaximal, characteristic
4C2×#1508C211yesmaximal, characteristic
5#13801C211yesmaximal, characteristic
6#8221C212yesmaximal, complemented
7#8219C212yesmaximal, complemented
8#8217C212yesmaximal, complemented
9C4×#328C212yesmaximal, complemented
10#8016C211yesmaximal, characteristic
11#8012C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C2×#519C411yescharacteristic, complemented
14C22×#117C2211yescharacteristic
15C2×#407C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C2×#405—21no—
18#2362—21no—
19#2362—21no—
20#2360—21no—
21#2360—21no—
22#2344—22no—
23#2342—22no—
24#2323C412yescomplemented
25#2323C412yescomplemented
26#2316C411yescharacteristic, complemented
27#2310C411yescharacteristic, complemented
28C4×C22⋊Q8C2212yes—
29C4×C22⋊Q8C2212yes—
30#2280C2211yescharacteristic
31C2×#380C2211yescharacteristic
32#2274C411yescharacteristic, complemented
33C2×C4×C4⋊C4—21no—
34C2×C4×C22⋊C4C2211yescharacteristic
35#1750—22no—
36#1749C2212yes—
37C4×C22⋊C8C2212yes—
38#1729—21no—
39#1520C418yescomplemented
40#1508C418yescomplemented
41C2×#328C2212yes—
42C2×#328C2212yes—
43C2×#328C2212yes—
44C2×#328C2212yes—
45#1437—22no—
46C4⋊#95—21no—
47#1434C2212yes—
48C4×#95C2212yes—
49#1294C2211yescharacteristic
50#1294C2212yes—
51#1294C2211yescharacteristic
52#1294C2211yescharacteristic
53#1294C2212yes—
54#1294C2211yescharacteristic
55C23×Q8C2×C411yescharacteristic
56C24×C4C2311yescharacteristic
57C22×M16—21no—
58C23×C8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8C2×C414yes—
64C2×C22⋊Q8C2×C414yes—
65C2×C22⋊Q8C2×C412yes—
66C2×C22⋊Q8C2312yes—
67#519—44no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8C2×C412yes—
71C2×C4×Q8C2×C412yes—
72C2×#117—22no—
73C2×#117—44no—
74C2×#117C2×C411yescharacteristic
75C2×#117C2×C411yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117C2×C411yescharacteristic
78C2×#117C2×C412yes—
79C2×#117D411yescharacteristic
80C2×#117D412yes—
81C2×#117D411yescharacteristic
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4D411yescharacteristic
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4D412yes—
91C22×C42—21no—
92C22×C42—22no—
93C22×C42C2312yes—
94#407—22no—
95#407C2×C414yes—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8C2×C414yes—
100C2×C22⋊C8C2311yescharacteristic
101C2×C22⋊C8C2311yescharacteristic
102C2×C22⋊C8C2312yes—
103#405—44no—
104C2×#97—22no—
105C2×C4×C8—22no—
106#391—21no—
107#391—21no—
108#391C2×C412yes—
109#391C2×C412yes—
110#389—21no—
111#389—21no—
112#389—22no—
113#389—22no—
114#389C2×C412yes—
115#389C2×C412yes—
116#388—22no—
117#388—22no—
118#388—22no—
119#388—22no—
120#387—22no—
121#387—22no—
122#387—22no—
123#387—22no—
124#385—22no—
125#385—22no—
126#381—21no—
127#381—22no—
128#380—22no—
129#380—22no—
130#380C2×C411yescharacteristic
131#380C2×C411yescharacteristic
132#380C2×C414yes—
133#380C2311yescharacteristic
134#380C2311yescharacteristic
135#379—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—44no—
141C4×C4⋊C4C2×C412yes—
142C4×C4⋊C4C2×C412yes—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—22no—
149C4×C22⋊C4C2×C414yes—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95—22no—
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156#336—22no—
157#336—22no—
158#328C2×C418yes—
159#328C2×C418yes—
160#328C2×C418yes—
161#328C2×C418yes—
162C25C2×D411yescharacteristic
163C22×Q8—21no—
164C22×Q8—22no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8C4214yescomplemented
168C22×Q8C22×C412yes—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—21no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C412yes—
179C23×C4C22×C412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4#4112yes—
186C23×C4#4112yes—
187C23×C4C2411yescharacteristic
188C2×M16—24no—
189C2×M16—42no—
190C2×M16—42no—
191C22×C8—24no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—22no—
195C22×C8—42no—
196C22×C8—42no—
197C22⋊Q8—24no—
198C22⋊Q8—24no—
199C22⋊Q8—24no—
200C22⋊Q8—24no—
201C22⋊Q8—24no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C22⋊Q8C4218yescomplemented
205C22⋊Q8C22×C414yes—
206C22⋊Q8C22×C414yes—
207C4×Q8—44no—
208C4×Q8—44no—
209C4×Q8—44no—
210C4×Q8—44no—
211#117—24no—
212#117—24no—
213#117—44no—
214#117—44no—
215#117—48no—
216#117—42no—
217#117—42no—
218#117—42no—
219#117—42no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C4214yescomplemented
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22⋊C412yes—
238C2×C4⋊C4C22⋊C411yescharacteristic
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C22⋊C412yes—
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22×C412yes—
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—81no—
263C2×C22⋊C4—81no—
264C2×C22⋊C4C22⋊C412yes—
265C2×C22⋊C4C22⋊C412yes—
266C2×C22⋊C4C22⋊C412yes—
267C2×C22⋊C4C22⋊C412yes—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—44no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—44no—
279C2×C42—44no—
280C2×C42—82no—
281C2×C42C22×C412yes—
282C2×C42C22×C412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C22⋊C8—24no—
286C22⋊C8—24no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—22no—
290C22⋊C8C22×C412yes—
291C22⋊C8C22×C412yes—
292C22⋊C8C22×C414yes—
293#97—44no—
294C4×C8—44no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—21no—
300#95—21no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—22no—
310#95—44no—
311#95—44no—
312#95—82no—
313#95C22×C412yes—
314#95C22×C412yes—
315#95C22×C414yes—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—41no—
321C24—41no—
322C24—41no—
323C24—41no—
324C24C2×C22⋊C411yescharacteristic
325C24C2×C22⋊C411yescharacteristic
326C24C4×D412yes—
327C24C4×D412yes—
328C24C22×D411yescharacteristic
329C2×Q8—22no—
330C2×Q8—24no—
331C2×Q8—48no—
332C2×Q8—48no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8C2×C4214yes—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
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—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4C2×C4212yes—
396C22×C4C2×C4212yes—
397C22×C4C2×C22⋊C412yes—
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C412yes—
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4#11712yes—
404C22×C4#11712yes—
405C22×C4C4×D412yes—
406C22×C4C4×D412yes—
407C22×C4C4×D412yes—
408C22×C4C4×D412yes—
409C22×C4C4×D412yes—
410C22×C4C4×D412yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C23×C411yescharacteristic
413C22×C4C23×C411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417M16—48no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—24no—
421C2×C8—41no—
422C2×C8—42no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—48no—
427C2×C8—42no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—28no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—42no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—48no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—82no—
446C4⋊C4—82no—
447C4⋊C4#9914yes—
448C4⋊C4C2×C4214yes—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—28no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—48no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C22⋊C4—84no—
466C42—24no—
467C42—24no—
468C42—42no—
469C42—42no—
470C42—48no—
471C42—44no—
472C42—44no—
473C42—42no—
474C42—42no—
475C42—82no—
476C42—82no—
477C42—88no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—81no—
503C23—81no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23—81no—
508C23—81no—
509C23—81no—
510C23C4×C22⋊C412yes—
511C23C4×C22⋊C412yes—
512C23C22×C22⋊C411yescharacteristic
513C23C2×C4×D411yescharacteristic
514C23C2×C4×D411yescharacteristic
515C23#57711yescharacteristic
516C23#57711yescharacteristic
517C23#57811yescharacteristic
518C23#57811yescharacteristic
519Q8—44no—
520Q8—44no—
521Q8—48no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
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—24no—
547C2×C4—24no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—48no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—41no—
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—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—84no—
590C2×C4—82no—
591C2×C4—81no—
592C2×C4—82no—
593C2×C4—81no—
594C2×C4—82no—
595C2×C4—81no—
596C2×C4—82no—
597C2×C4—81no—
598C2×C4C4×C22⋊C412yes—
599C2×C4C4×C22⋊C412yes—
600C2×C4C2×#9914yes—
601C2×C4C2×#9912yes—
602C2×C4#41012yes—
603C2×C4C22×C4211yescharacteristic
604C2×C4C2×#11711yescharacteristic
605C2×C4C2×C4×D411yescharacteristic
606C2×C4C2×C4×D411yescharacteristic
607C8—42no—
608C8—42no—
609C8—44no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22C2×C4×C22⋊C411yescharacteristic
641C22C22×#9911yescharacteristic
642C22C2×#41011yescharacteristic
643C22#289811yescharacteristic
644C22#289811yescharacteristic
645C22#290111yescharacteristic
646C22#290111yescharacteristic
647C22#294711yescharacteristic
648C22#294711yescharacteristic
649C22#294911yescharacteristic
650C22#295011yescharacteristic
651C4—21no—
652C4—21no—
653C4—21no—
654C4—22no—
655C4—21no—
656C4—24no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—84no—
664C4—82no—
665C4—81no—
666C4—81no—
667C4—81no—
668C4—81no—
669C4C4×#9914yes—
670C2—21no—
671C2—21no—
672C2—21no—
673C2—21no—
674C2—42no—
675C2—42no—
676C2#1760511yescharacteristic
677C2#1760611yescharacteristic
678C2#1760811yescharacteristic
679C2#1761211yescharacteristic
680C2C2×C4×#9911yescharacteristic
681C2#2040511yescharacteristic
682C2#2041711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8221C2semidirect
7. #8219C2semidirect
8. #8217C2semidirect
9. C4×#328C2semidirect
13. C2×#519C4semidirect
24. #2323C4semidirect
25. #2323C4semidirect
26. #2316C4semidirect
27. #2310C4semidirect
32. #2274C4semidirect
39. #1520C4semidirect
40. #1508C4semidirect
167. C22×Q8C42semidirect
204. C22⋊Q8C42semidirect
235. C2×C4⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
#342032
#203792
C2×#15202
C2×#15082
#138012
#82212
#82192
#82172
C4×#3282
#80162
#80122