Finite Group Databasebeta

← #190819

Subgroups — #190819

Order 512.

1456 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C42×#41C211yesmaximal, characteristic
2#20808C211yesmaximal, characteristic, complemented
3C4×#431C211yesmaximal, characteristic, complemented
4#8708C212yesmaximal, complemented
5#8564C212yesmaximal, complemented
6#8563C212yesmaximal, complemented
7#8312C212yesmaximal
8C4×Q8⋊C8C212yesmaximal, complemented
9C4×D4⋊C8C212yesmaximal, complemented
10C2×C4×#41C411yescharacteristic
11C2×C4×#41C411yescharacteristic
12C2×C4×#41C2211yescharacteristic
13C2×#431—21no—
14C2×#431—21no—
15C42×Q8C411yescharacteristic
16C42×Q8C411yescharacteristic
17C42×Q8C2211yescharacteristic
18C42×Q8C2211yescharacteristic
19C42×D4—21no—
20C42×D4—21no—
21C42×D4—21no—
22C42×D4—21no—
23C42×D4C411yescharacteristic
24C42×D4C411yescharacteristic
25C42×D4C2211yescharacteristic
26C42×D4C2211yescharacteristic
27C4×#117—21no—
28C4×#117—21no—
29C4×#117—21no—
30C4×#117C411yescharacteristic
31C4×#117C411yescharacteristic
32C4×#117C2211yescharacteristic
33C2×C43—21no—
34C2×C43C2211yescharacteristic
35#1842C2212yes—
36#1831—21no—
37#1797—21no—
38#1797C2212yes—
39C4×C4⋊C8—21no—
40C4×C4⋊C8—21no—
41C4×C4⋊C8C2212yes—
42C4×C4⋊C8C2212yes—
43C4×C22⋊C8—21no—
44C4×C22⋊C8C2212yes—
45#1744—21no—
46C42×C8C2212yes—
47C2×Q8⋊C8C2214yes—
48C2×D4⋊C8C2214yes—
49#1279C2214yes—
50#1279C2214yes—
51C22×#41C2×C411yescharacteristic
52C4×#41—24no—
53C4×#41—24no—
54C4×#41—24no—
55C4×#41—24no—
56C4×#41—24no—
57C4×#41C818yescomplemented
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×#117—21no—
77C2×#117—21no—
78C2×#117—21no—
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C411yescharacteristic
81C2×#117C2×C411yescharacteristic
82C22×C42—21no—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42—21no—
87C22×C42—21no—
88C22×C42—21no—
89C22×C42C2×C411yescharacteristic
90C22×C42C2×C411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42C2311yescharacteristic
96#431—28no—
97#431—44no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8C2314yes—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8C2314yes—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150#376—21no—
151#376—21no—
152#376—21no—
153C43—21no—
154C43—21no—
155C43—21no—
156C43—21no—
157C43—21no—
158C43—21no—
159C43—21no—
160C43C2×C411yescharacteristic
161C43C2×C411yescharacteristic
162C43D411yescharacteristic
163C43D411yescharacteristic
164C43D411yescharacteristic
165C43D411yescharacteristic
166C43C2311yescharacteristic
167C43C2311yescharacteristic
168#336—22no—
169#336—22no—
170#336—22no—
171#336—22no—
172Q8⋊C8—28no—
173D4⋊C8—28no—
174C2×#41—22no—
175C2×#41—22no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—22no—
179C2×#41C2×C814yes—
180C22×Q8C22×C411yescharacteristic
181C22×D4—21no—
182C22×D4C22×C411yescharacteristic
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22⋊C411yescharacteristic
193C23×C4C22⋊C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C22×C8—24no—
198C22×C8—24no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8C2×C814yes—
208C4×Q8C2×C814yes—
209C4×Q8M1614yes—
210C4×Q8#4114yes—
211C4×D4—28no—
212C4×D4—28no—
213C4×D4—28no—
214C4×D4—28no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—28no—
218C4×D4—28no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4C2×C814yes—
226C4×D4C2×C814yes—
227C4×D4M1614yes—
228C4×D4#4114yes—
229#117—28no—
230#117—28no—
231#117—22no—
232#117—22no—
233#117—22no—
234#117—22no—
235#117—22no—
236#117—28no—
237#117C2×C814yes—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C42—28no—
260C2×C42—28no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—28no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—42no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42C22⋊C411yescharacteristic
298C2×C42C22⋊C411yescharacteristic
299C2×C42C22⋊C411yescharacteristic
300C2×C42C22⋊C411yescharacteristic
301C2×C42C22⋊C411yescharacteristic
302C2×C42C22⋊C411yescharacteristic
303C2×C42C22⋊C411yescharacteristic
304C2×C42C22⋊C411yescharacteristic
305C2×C42C22⋊C411yescharacteristic
306C2×C42C22⋊C411yescharacteristic
307C2×C42C22⋊C411yescharacteristic
308C2×C42C22⋊C411yescharacteristic
309C2×C42C2×C814yes—
310C2×C42M1614yes—
311C2×C42C22×C411yescharacteristic
312C2×C42C22×C411yescharacteristic
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×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2411yescharacteristic
327C4⋊C8—28no—
328C4⋊C8—24no—
329C4⋊C8—24no—
330C4⋊C8—44no—
331C4⋊C8—44no—
332C22⋊C8—28no—
333C22⋊C8—44no—
334C4×C8—28no—
335C4×C8—24no—
336C4×C8—24no—
337C4×C8—44no—
338C4×C8—44no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351C24—21no—
352C24C2×C22⋊C411yescharacteristic
353#41—28no—
354#41—28no—
355#41—28no—
356#41—28no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8#11712yes—
361C2×Q8C22×C812yes—
362C2×Q8C2×M1612yes—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4#11712yes—
370C2×D4C22×C812yes—
371C2×D4C2×M1612yes—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—41no—
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×C4C22⋊C814yes—
423C22×C4C22⋊C814yes—
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C22×C812yes—
436C22×C4C2×M1612yes—
437C22×C4C23×C411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C2×C8—28no—
441C2×C8—44no—
442C2×C8—44no—
443C2×C8—44no—
444C2×C8—44no—
445C2×C8—44no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—24no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4#11712yes—
459C4⋊C4#11712yes—
460C4⋊C4C22×C812yes—
461C4⋊C4C22×C812yes—
462C4⋊C4C2×M1612yes—
463C4⋊C4C2×M1612yes—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C42—24no—
474C42—24no—
475C42—24no—
476C42—24no—
477C42—21no—
478C42—21no—
479C42—22no—
480C42—24no—
481C42—22no—
482C42—21no—
483C42—21no—
484C42—22no—
485C42—21no—
486C42—21no—
487C42—24no—
488C42—24no—
489C42—24no—
490C42—22no—
491C42—24no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—41no—
503C42—41no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42—41no—
508C42—41no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—42no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42C22⋊C814yes—
518C42C22⋊C814yes—
519C42#10414yes—
520C42C22×C812yes—
521C42C22×C812yes—
522C42C2×M1612yes—
523C42C2×M1612yes—
524C42C2×M1612yes—
525C42C2×M1612yes—
526C42#13514yes—
527C42AGL(1,ℤ/8)12yes—
528C42#13712yes—
529C42C2×#4112yes—
530C42C2×#4112yes—
531C23—22no—
532C23—21no—
533C23—22no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C2×C22⋊C812yes—
545C23C22×C22⋊C411yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—22no—
549Q8#43114yescomplemented
550D4—216no—
551D4—22no—
552D4—22no—
553D4—22no—
554D4#43114yescomplemented
555C2×C4—216no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—41no—
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—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—44no—
614C2×C4—48no—
615C2×C4C2×C22⋊C812yes—
616C2×C4C2×C22⋊C812yes—
617C2×C4C2×C22⋊C812yes—
618C2×C4C2×C22⋊C812yes—
619C2×C4C2×C22⋊C812yes—
620C2×C4#40712yes—
621C2×C4#40712yes—
622C2×C4#41612yes—
623C2×C4#41612yes—
624C2×C4#41712yes—
625C2×C4#41712yes—
626C2×C4C2×#10412yes—
627C2×C4C2×#10412yes—
628C2×C4C2×#10412yes—
629C2×C4#42112yes—
630C2×C4C2×#11711yescharacteristic
631C2×C4C2×#11711yescharacteristic
632C2×C4#52512yes—
633C2×C4#52512yes—
634C2×C4C23×C811yescharacteristic
635C2×C4C22×M1611yescharacteristic
636C2×C4C22×M1611yescharacteristic
637C2×C4C22×M1611yescharacteristic
638C2×C4C2×#13512yes—
639C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
640C2×C4C2×#13711yescharacteristic
641C8—48no—
642C8—84no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—24no—
647C22—41no—
648C22—42no—
649C22—41no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—41no—
654C22—42no—
655C22—41no—
656C22—42no—
657C22#228011yescharacteristic
658C22C22×C22⋊C811yescharacteristic
659C22C2×#40711yescharacteristic
660C22C2×#41611yescharacteristic
661C22C2×#41711yescharacteristic
662C22C22×#10411yescharacteristic
663C22C2×#42111yescharacteristic
664C4—22no—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—24no—
672C4—41no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—41no—
679C4—42no—
680C4#147614yes—
681C4#147714yes—
682C4#148914yes—
683C4C2×#43111yescharacteristic
684C4C2×#43111yescharacteristic
685C4#292011yescharacteristic
686C4#292011yescharacteristic
687C4#310612yes—
688C4#310712yes—
689C2—42no—
690C2—44no—
691C2C2×#147611yescharacteristic
692C2C2×#147711yescharacteristic
693C2#1744311yescharacteristic
694C2C2×#148911yescharacteristic
695C2#2071411yescharacteristic
696C2#2073711yescharacteristic
697C2#2074711yescharacteristic
subgroupG/H
centerC2×C42C2×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C42 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20808C2semidirect
3. C4×#431C2semidirect
4. #8708C2semidirect
5. #8564C2semidirect
6. #8563C2semidirect
8. C4×Q8⋊C8C2semidirect
9. C4×D4⋊C8C2semidirect
57. C4×#41C8semidirect
549. Q8#431semidirect
554. D4#431semidirect

Maximal subgroups

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

subgroupindex
C42×#412
#208082
C4×#4312
#87082
#85642
#85632
#83122
C4×Q8⋊C82
C4×D4⋊C82