Finite Group Databasebeta

← #190359

Subgroups — #190359

Order 512.

1173 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34190C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3#17362C211yesmaximal, characteristic
4#8779C212yesmaximal, complemented
5#8778C212yesmaximal, complemented
6#8777C212yesmaximal, complemented
7#8776C212yesmaximal, complemented
8#8051C212yesmaximal, complemented
9C2×#1284C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11C22×#99—21no—
12C2×#407C2212yes—
13#2357C2211yescharacteristic
14#2357C2211yescharacteristic
15#2353C411yescharacteristic, complemented
16#2353C411yescharacteristic, complemented
17#2353C2211yescharacteristic
18#2353C2211yescharacteristic
19#2349C411yescharacteristic, complemented
20#2349C411yescharacteristic, complemented
21#2344—21no—
22#2342—21no—
23#2342—21no—
24D4×C22⋊C4—21no—
25#2319—21no—
26#2318—21no—
27#2318—21no—
28#2283C411yescharacteristic, complemented
29#2283C411yescharacteristic, complemented
30#2280C411yescharacteristic, complemented
31#2280C2211yescharacteristic
32C2×#381C411yescharacteristic, complemented
33C2×#381C2211yescharacteristic
34#2275—21no—
35C2×C4×C22⋊C4C2212yes—
36C4×#99—21no—
37C4×#99—21no—
38C4×#99C2212yes—
39C4×#99C2212yes—
40C2×#342—21no—
41#1465C2212yes—
42#1465C2212yes—
43#1463—21no—
44#1462—21no—
45#1459—21no—
46#1459—21no—
47#1297C2214yescomplemented
48#1297C2214yescomplemented
49#1284C2214yescomplemented
50#1284C2214yescomplemented
51C23×D4D411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4C2311yescharacteristic
54C22×M16—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C2×#117—21no—
71C22×C4⋊C4D411yescharacteristic
72C22×C22⋊C4—22no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42—22no—
78C22×C42C2×C411yescharacteristic
79C2×#99—28no—
80C2×#99—22no—
81C2×#99—22no—
82#407—22no—
83#407—22no—
84#407—22no—
85#407C2314yes—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88#390—21no—
89#390—21no—
90#389—21no—
91#389—21no—
92#388—21no—
93#388—21no—
94#388—21no—
95#388—21no—
96#388—21no—
97#388—21no—
98#388—21no—
99#388—21no—
100#387—21no—
101#387—21no—
102#387—21no—
103#387—21no—
104#386—21no—
105#386—21no—
106#386—21no—
107#386—21no—
108#385—21no—
109#385—21no—
110#385—21no—
111#385—21no—
112#383—21no—
113#383C2×C411yescharacteristic, complemented
114#383C2×C411yescharacteristic, complemented
115#382—21no—
116#382C2×C411yescharacteristic, complemented
117#382C2×C411yescharacteristic, complemented
118#381—21no—
119#381—21no—
120#381—21no—
121#381—21no—
122#381C2×C411yescharacteristic
123#381C2×C411yescharacteristic
124#381C2×C411yescharacteristic
125#381C2×C411yescharacteristic
126#381C2×C411yescharacteristic, complemented
127#381C2×C411yescharacteristic, complemented
128#381C2311yescharacteristic
129#381C2311yescharacteristic
130#380—21no—
131#380—21no—
132#380—21no—
133#380—21no—
134#380C2×C411yescharacteristic
135#380C2×C411yescharacteristic
136#380C2×C411yescharacteristic, complemented
137#380C2×C411yescharacteristic, complemented
138#379—21no—
139#379—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4C2×C411yescharacteristic, complemented
144C4×C4⋊C4C2×C411yescharacteristic, complemented
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—22no—
154C4×C22⋊C4—22no—
155C4×C22⋊C4—22no—
156C4×C22⋊C4C2×C411yescharacteristic, complemented
157C4×C22⋊C4C2×C411yescharacteristic, complemented
158C4×C22⋊C4C2314yes—
159C4×C22⋊C4C2311yescharacteristic
160C4×C22⋊C4C2311yescharacteristic
161#376—21no—
162#376C2×C411yescharacteristic, complemented
163#376C2×C411yescharacteristic, complemented
164C2×#95—21no—
165C2×#95—21no—
166C2×#95—22no—
167C2×#95C2×C411yescharacteristic
168C2×#95C2×C411yescharacteristic
169C2×#95C2×C411yescharacteristic, complemented
170C2×#95C2×C411yescharacteristic, complemented
171#342—22no—
172#342—22no—
173#324—22no—
174#324—22no—
175#323—22no—
176#323—22no—
177C25C22⋊C411yescharacteristic
178C25C2×D411yescharacteristic
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—24no—
182C22×D4—42no—
183C22×D4C22⋊C411yescharacteristic
184C22×D4C22⋊C411yescharacteristic
185C22×D4C2×D411yescharacteristic
186C22×D4C2×D411yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—42no—
196C23×C4—82no—
197C23×C4—82no—
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22⋊C411yescharacteristic
203C23×C4C22×C411yescharacteristic
204C23×C4C22×C411yescharacteristic
205C23×C4C22×C411yescharacteristic
206C23×C4C22×C411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4#4112yes—
211C23×C4#4112yes—
212C23×C4#4112yes—
213C23×C4#4112yes—
214C23×C4C2411yescharacteristic
215C2×M16—24no—
216C2×M16—42no—
217C2×M16—84no—
218C22×C8—42no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—44no—
236C4×D4—44no—
237#117—82no—
238#117—82no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4C22⋊C411yescharacteristic
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—82no—
278C2×C22⋊C4—82no—
279C2×C22⋊C4C22⋊C411yescharacteristic
280C2×C22⋊C4C22⋊C411yescharacteristic
281C2×C22⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C22⋊C4C2×D411yescharacteristic
284C2×C22⋊C4C2×D411yescharacteristic
285C2×C22⋊C4C2×D411yescharacteristic
286C2×C22⋊C4C2×D411yescharacteristic
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—24no—
300C2×C42—42no—
301C2×C42—84no—
302C2×C42C22⋊C411yescharacteristic
303C2×C42C22⋊C411yescharacteristic
304C2×C42C22×C411yescharacteristic
305C2×C42C22×C411yescharacteristic
306#99—28no—
307#99—28no—
308C22⋊C8—24no—
309C22⋊C8—24no—
310C22⋊C8—42no—
311C22⋊C8—42no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—24no—
333#95—42no—
334#95C22×C411yescharacteristic
335#95C22×C411yescharacteristic
336#95C22×C411yescharacteristic
337#95C22×C411yescharacteristic
338#95C22×C411yescharacteristic
339#95C22×C411yescharacteristic
340#95C22×C411yescharacteristic
341#95C22×C411yescharacteristic
342C24—22no—
343C24—22no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—42no—
348C24—41no—
349C24—42no—
350C24—81no—
351C24—81no—
352C24C2×C22⋊C411yescharacteristic
353C24C2×C22⋊C411yescharacteristic
354C24C2×C22⋊C411yescharacteristic
355C24C2×C22⋊C411yescharacteristic
356C24#12312yes—
357C24#12312yes—
358C24C22×D411yescharacteristic
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4#10414yes—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—48no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4—81no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—84no—
432C22×C4—84no—
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4#11711yescharacteristic
442C22×C4#11711yescharacteristic
443C22×C4#11711yescharacteristic
444C22×C4#11711yescharacteristic
445C22×C4#11712yes—
446C22×C4#11712yes—
447C22×C4#12312yes—
448C22×C4#12312yes—
449C22×C4#12312yes—
450C22×C4#12312yes—
451C22×C4#12312yes—
452C22×C4#12312yes—
453C22×C4C23×C411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457M16—84no—
458M16—84no—
459C2×C8—44no—
460C2×C8—84no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—28no—
482C22⋊C4—48no—
483C22⋊C4—44no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—81no—
493C22⋊C4—81no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—81no—
499C22⋊C4—81no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—42no—
509C42—42no—
510C42—81no—
511C42—81no—
512C42—81no—
513C42—81no—
514C42—81no—
515C42—81no—
516C42—84no—
517C42—84no—
518C23—24no—
519C23—22no—
520C23—24no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—24no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—81no—
548C23—81no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23#38111yescharacteristic
554C23#38112yes—
555C23#38111yescharacteristic
556C23C2×#10412yes—
557C23C2×#10412yes—
558C23#42112yes—
559C23C22×C22⋊C411yescharacteristic
560C23C2×#12312yes—
561D4—28no—
562D4—44no—
563D4—44no—
564D4—44no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—48no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—816no—
638C2×C4—162no—
639C2×C4—162no—
640C2×C4—162no—
641C2×C4—162no—
642C2×C4#38111yescharacteristic
643C2×C4#38111yescharacteristic
644C2×C4#38112yes—
645C2×C4C2×#10412yes—
646C2×C4C2×#11711yescharacteristic
647C2×C4C2×#11711yescharacteristic
648C2×C4C2×#12312yes—
649C8—84no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—22no—
654C22—21no—
655C22—22no—
656C22—22no—
657C22—21no—
658C22—22no—
659C22—22no—
660C22—21no—
661C22—21no—
662C22—22no—
663C22—42no—
664C22—42no—
665C22—48no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—81no—
676C22—81no—
677C22—84no—
678C22—84no—
679C22—81no—
680C22—81no—
681C22—84no—
682C22#182614yes—
683C22#182812yes—
684C22C2×#38111yescharacteristic
685C22C22×#10411yescharacteristic
686C22C2×#42111yescharacteristic
687C4—22no—
688C4—22no—
689C4—21no—
690C4—21no—
691C4—21no—
692C4—21no—
693C4—22no—
694C4—41no—
695C4—41no—
696C4—41no—
697C4—41no—
698C4—41no—
699C4—81no—
700C4—81no—
701C4—81no—
702C4—81no—
703C4—81no—
704C4—81no—
705C4—81no—
706C4—81no—
707C4—88no—
708C2—21no—
709C2—21no—
710C2—22no—
711C2—24no—
712C2—42no—
713C2—41no—
714C2—42no—
715C2—81no—
716C2C2×#182611yescharacteristic
717C2#2070811yescharacteristic
718C2C2×#182811yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8779C2semidirect
5. #8778C2semidirect
6. #8777C2semidirect
7. #8776C2semidirect
8. #8051C2semidirect
9. C2×#1284C2semidirect
15. #2353C4semidirect
16. #2353C4semidirect
19. #2349C4semidirect
20. #2349C4semidirect
28. #2283C4semidirect
29. #2283C4semidirect
30. #2280C4semidirect
32. C2×#381C4semidirect
47. #1297C22semidirect
48. #1297C22semidirect
49. #1284C22semidirect
50. #1284C22semidirect
113. #383C2×C4semidirect
114. #383C2×C4semidirect
116. #382C2×C4semidirect
117. #382C2×C4semidirect
126. #381C2×C4semidirect
127. #381C2×C4semidirect
136. #380C2×C4semidirect
137. #380C2×C4semidirect
143. C4×C4⋊C4C2×C4semidirect
144. C4×C4⋊C4C2×C4semidirect
156. C4×C22⋊C4C2×C4semidirect
157. C4×C22⋊C4C2×C4semidirect
162. #376C2×C4semidirect
163. #376C2×C4semidirect
169. C2×#95C2×C4semidirect
170. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341902
C2×C4×#992
#173622
#87792
#87782
#87772
#87762
#80512
C2×#12842