Finite Group Databasebeta

← #183141

Subgroups — #183141

Order 512.

1376 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#117C211yesmaximal, characteristic
2#20472C211yesmaximal, characteristic, complemented
3#20470C211yesmaximal, characteristic, complemented
4C2×#1490C211yesmaximal, characteristic
5C2×#1478C211yesmaximal, characteristic
6#8183C211yesmaximal, characteristic, complemented
7#8181C211yesmaximal, characteristic, complemented
8#8179C211yesmaximal, characteristic, complemented
9#8177C211yesmaximal, characteristic, complemented
10#8174C211yesmaximal, characteristic, complemented
11#8172C211yesmaximal, characteristic, complemented
12#8170C211yesmaximal, characteristic, complemented
13C4×D4⋊C8C211yesmaximal, characteristic, complemented
14#7900C211yesmaximal, characteristic, complemented
15#7896C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C22×#117—21no—
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2354C412yes—
23#2352—22no—
24D4×C4⋊C4C412yes—
25#2344C412yes—
26#2342—22no—
27D4×C22⋊C4C412yes—
28#2293C2211yescharacteristic
29#2293C2211yescharacteristic
30#2290—21no—
31#2290—21no—
32#2285C2211yescharacteristic
33#2280—21no—
34C42×D4C2211yescharacteristic
35C42×D4C2211yescharacteristic
36C4×#117C2211yescharacteristic
37#1771—22no—
38#1767C2211yescharacteristic, complemented
39#1767C2211yescharacteristic, complemented
40C4×C4⋊C8C2211yescharacteristic, complemented
41C4×C4⋊C8C2211yescharacteristic, complemented
42#1753—22no—
43#1750—21no—
44#1750—21no—
45#1746—21no—
46#1744C2211yescharacteristic, complemented
47#1744C2211yescharacteristic, complemented
48#1726—21no—
49C4×#97C2211yescharacteristic, complemented
50C42×C8C2211yescharacteristic, complemented
51#1490C418yescomplemented
52#1478C418yescomplemented
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61#1279C2211yescharacteristic, complemented
62#1279C2211yescharacteristic, complemented
63#1279C2211yescharacteristic, complemented
64#1279C2211yescharacteristic, complemented
65#1279C2211yescharacteristic, complemented
66#1279C2211yescharacteristic, complemented
67#1279C2211yescharacteristic, complemented
68#1279C2211yescharacteristic, complemented
69C23×D4C2×C411yescharacteristic
70C24×C4—21no—
71C22×M16—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4C2×C414yes—
77C2×C4×D4C2×C414yes—
78C2×C4×D4C2×C412yes—
79C2×C4×D4C2×C412yes—
80C2×C4×D4C2×C411yescharacteristic
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4C2311yescharacteristic
83C2×C4×D4C2311yescharacteristic
84C2×#117—21no—
85C2×#117—22no—
86C2×#117—22no—
87C2×#117—21no—
88C2×#117—21no—
89C2×#117—44no—
90C2×#117—44no—
91C2×#117D411yescharacteristic
92C2×#117D411yescharacteristic
93C22×C4⋊C4—22no—
94C22×C4⋊C4C2×C411yescharacteristic
95C22×C22⋊C4—21no—
96C22×C22⋊C4—22no—
97C22×C42—21no—
98C22×C42—21no—
99C22×C42C2311yescharacteristic
100#432—22no—
101#432C2×C414yescomplemented
102#431—22no—
103#431—22no—
104#431C2×C414yescomplemented
105#431C2×C414yescomplemented
106C2×C4⋊C8—21no—
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C4×M16—22no—
115C4×M16C2×C414yescomplemented
116C2×#97—21no—
117C2×#97—21no—
118C2×#97—22no—
119C2×#97—21no—
120C2×#97—21no—
121C2×C4×C8—21no—
122C2×C4×C8—22no—
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129#390C2×C412yes—
130#390C2×C412yes—
131#389C2×C412yes—
132#389C2×C412yes—
133#388—22no—
134#388—22no—
135#388—22no—
136#388—22no—
137#387—22no—
138#387—22no—
139#386—22no—
140#386—22no—
141#385—22no—
142#385—22no—
143#384C2×C412yes—
144#382—21no—
145#381—22no—
146#380—21no—
147#380—22no—
148#380—22no—
149#380C2×C412yes—
150#379—22no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4C2×C412yes—
153C4×C4⋊C4C2×C412yes—
154C4×C4⋊C4C2×C412yes—
155C4×C4⋊C4D412yes—
156C4×C4⋊C4C2311yescharacteristic
157C4×C4⋊C4C2311yescharacteristic
158C4×C22⋊C4—22no—
159C4×C22⋊C4—22no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4—21no—
164C4×C22⋊C4C2×C412yes—
165C4×C22⋊C4D412yes—
166#376D411yescharacteristic
167#376D411yescharacteristic
168C43C2311yescharacteristic
169C43C2311yescharacteristic
170#336—22no—
171#336—22no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176D4⋊C8C2×C414yes—
177D4⋊C8C2×C414yes—
178D4⋊C8C2×C414yes—
179D4⋊C8C2×C414yes—
180D4⋊C8C2×C414yescomplemented
181D4⋊C8C2×C414yescomplemented
182D4⋊C8C2×C414yescomplemented
183D4⋊C8C2×C414yescomplemented
184C25—21no—
185C22×D4—24no—
186C22×D4—21no—
187C22×D4—22no—
188C22×D4—42no—
189C22×D4C4214yes—
190C22×D4C22×C411yescharacteristic
191C22×D4C22×C411yescharacteristic
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4C22×C411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C2×M16—24no—
209C2×M16—44no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4—24no—
225C4×D4—24no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4—42no—
231C4×D4—42no—
232C4×D4C4214yes—
233C4×D4C4214yes—
234C4×D4C22×C412yes—
235C4×D4C22×C412yes—
236C4×D4C22×C412yes—
237C4×D4C22×C412yes—
238#117—22no—
239#117—22no—
240#117—41no—
241#117—41no—
242#117—42no—
243#117—42no—
244#117—48no—
245#117—48no—
246#117—42no—
247#117—42no—
248#117—42no—
249#117—42no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—48no—
257C2×C4⋊C4C4214yes—
258C2×C4⋊C4C22⋊C412yes—
259C2×C4⋊C4C22⋊C412yes—
260C2×C4⋊C4C22×C412yes—
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4C22⋊C412yes—
279C2×C22⋊C4C22⋊C412yes—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42C22×C412yes—
289C2×C42C22×C412yes—
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C412yes—
292C2×C42C22×C412yes—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42#4112yes—
302C2×C42#4112yes—
303C2×C42C2411yescharacteristic
304C4⋊C8—24no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8C22×C412yes—
310C4⋊C8C22×C412yes—
311C4⋊C8C22×C412yes—
312C4⋊C8C22×C412yes—
313C22⋊C8—24no—
314C22⋊C8—24no—
315C22⋊C8—24no—
316#97—24no—
317#97—24no—
318#97—42no—
319#97—44no—
320#97—42no—
321C4×C8—24no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—42no—
327C4×C8—44no—
328C4×C8—42no—
329C4×C8C22×C412yes—
330C4×C8C22×C412yes—
331C4×C8C22×C412yes—
332C4×C8C22×C412yes—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—21no—
339#95—21no—
340#95C22⋊C412yes—
341#95C22⋊C412yes—
342#95C22⋊C412yes—
343#95C22⋊C412yes—
344C24—21no—
345C24—21no—
346C24—24no—
347C24—44no—
348C24—42no—
349C24—42no—
350C24—82no—
351C24C2×C22⋊C411yescharacteristic
352C2×D4—28no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—44no—
360C2×D4—48no—
361C2×D4—44no—
362C2×D4C2×C4212yes—
363C2×D4C2×C4212yes—
364C2×D4#13114yes—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—44no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4C2×C4212yes—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C412yes—
410C22×C4C2×C22⋊C412yes—
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C412yes—
415C22×C4#11712yes—
416C22×C4C4×D412yes—
417C22×C4C4×D412yes—
418C22×C4C4×D412yes—
419C22×C4C4×D412yes—
420C22×C4C23×C411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423M16—48no—
424C2×C8—24no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—82no—
457C4⋊C4C2×C4212yes—
458C4⋊C4C2×C4212yes—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—42no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—82no—
473C22⋊C4—82no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—42no—
485C42—42no—
486C42—44no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—44no—
492C42C2×C4212yes—
493C42#11712yes—
494C42C4×D412yes—
495C42C4×D412yes—
496C42C4×D412yes—
497C42C4×D412yes—
498C42AGL(1,ℤ/8)12yes—
499C42AGL(1,ℤ/8)12yes—
500C42C23×C411yescharacteristic
501C42C23×C411yescharacteristic
502C42C2×#4111yescharacteristic
503C42C2×#4111yescharacteristic
504C23—24no—
505C23—22no—
506C23—21no—
507C23—21no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—48no—
512C23—48no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—84no—
521C23—88no—
522C23—88no—
523C23C4×C22⋊C412yes—
524C23C22×C22⋊C411yescharacteristic
525D4—22no—
526D4—22no—
527D4—22no—
528D4—24no—
529D4—48no—
530D4—42no—
531D4#40514yes—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—88no—
581C2×C4—88no—
582C2×C4C4×C22⋊C412yes—
583C2×C4C4×C22⋊C412yes—
584C2×C4C4×C22⋊C412yes—
585C2×C4#41814yes—
586C2×C4C22×C4211yescharacteristic
587C2×C4C2×#11711yescharacteristic
588C2×C4C2×C4×D411yescharacteristic
589C2×C4C2×C4×D411yescharacteristic
590C2×C4C2×C4×D411yescharacteristic
591C2×C4C2×C4×D411yescharacteristic
592C2×C4C2×#13112yes—
593C2×C4#56812yes—
594C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
595C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
596C2×C4#57511yescharacteristic
597C2×C4#57511yescharacteristic
598C8—24no—
599C8—42no—
600C8—42no—
601C8—42no—
602C8—42no—
603C8—42no—
604C8—42no—
605C8—42no—
606C22—22no—
607C22—24no—
608C22—22no—
609C22—22no—
610C22—44no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—816no—
620C22—82no—
621C22—82no—
622C22C2×C4×C22⋊C411yescharacteristic
623C22#288012yes—
624C22C2×#41811yescharacteristic
625C22#289611yescharacteristic
626C22#290112yes—
627C4—24no—
628C4—22no—
629C4—21no—
630C4—21no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—82no—
639C4—82no—
640C4C2×#40511yescharacteristic
641C4#287311yescharacteristic
642C4#292412yes—
643C4C4×AGL(1,ℤ/8)11yescharacteristic
644C4C4×AGL(1,ℤ/8)11yescharacteristic
645C4C4×AGL(1,ℤ/8)11yescharacteristic
646C4C4×AGL(1,ℤ/8)11yescharacteristic
647C2—42no—
648C2—44no—
649C2—82no—
650C2#1751112yes—
651C2#1751312yes—
652C2#2037911yescharacteristic
653C2#2040511yescharacteristic
654C2C4×#41811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC23×D4C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20472C2semidirect
3. #20470C2semidirect
6. #8183C2semidirect
7. #8181C2semidirect
8. #8179C2semidirect
9. #8177C2semidirect
10. #8174C2semidirect
11. #8172C2semidirect
12. #8170C2semidirect
13. C4×D4⋊C8C2semidirect
14. #7900C2semidirect
15. #7896C2semidirect
38. #1767C22semidirect
39. #1767C22semidirect
40. C4×C4⋊C8C22semidirect
41. C4×C4⋊C8C22semidirect
46. #1744C22semidirect
47. #1744C22semidirect
49. C4×#97C22semidirect
50. C42×C8C22semidirect
51. #1490C4semidirect
52. #1478C4semidirect
61. #1279C22semidirect
62. #1279C22semidirect
63. #1279C22semidirect
64. #1279C22semidirect
65. #1279C22semidirect
66. #1279C22semidirect
67. #1279C22semidirect
68. #1279C22semidirect
101. #432C2×C4semidirect
104. #431C2×C4semidirect
105. #431C2×C4semidirect
115. C4×M16C2×C4semidirect
180. D4⋊C8C2×C4semidirect
181. D4⋊C8C2×C4semidirect
182. D4⋊C8C2×C4semidirect
183. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
D4×#1172
#204722
#204702
C2×#14902
C2×#14782
#81832
#81812
#81792
#81772
#81742
#81722
#81702
C4×D4⋊C82
#79002
#78962