Finite Group Databasebeta

← #482378

Subgroups — #482378

Order 512.

1211 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2#20726C211yesmaximal, characteristic
3#17362C211yesmaximal, characteristic
4#11881C214yesmaximal, complemented
5#11803C214yesmaximal, complemented
6#11780C214yesmaximal, complemented
7C22×#123—21no—
8C22×C4×D4C2211yescharacteristic
9C23×C4⋊C4C411yescharacteristic
10#2920—21no—
11C2×#407—21no—
12#2486—21no—
13#2486—21no—
14#2486—21no—
15#2486—21no—
16#2486C412yes—
17#2486C412yes—
18#2486C2212yes—
19#2486C2212yes—
20C2×#398C2212yes—
21C2×#397C412yes—
22C2×#389—21no—
23C2×#381—21no—
24C2×#380—21no—
25C2×#380C411yescharacteristic
26#1821C2212yes—
27#1800—21no—
28#1800C2212yes—
29#1783—21no—
30C4×#100—21no—
31C4×#100C2212yes—
32#1465—21no—
33#1465C2212yes—
34#1463C2212yes—
35#1462—21no—
36#1459—21no—
37#1459C2212yes—
38#1346C22116yescomplemented
39C23×D4D411yescharacteristic
40C24×C4—21no—
41C24×C4C2×C411yescharacteristic
42C24×C4D411yescharacteristic
43C22×M16—41no—
44C2×#123—22no—
45C2×#123—22no—
46C2×#123—22no—
47C2×#123—22no—
48C2×#123—48no—
49C2×C4×D4—41no—
50C2×C4×D4—41no—
51C2×C4×D4—41no—
52C2×C4×D4—41no—
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4D411yescharacteristic
56C2×C4×D4D411yescharacteristic
57C2×C4×D4D411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×C4×D4C2311yescharacteristic
60C2×C4×D4C2311yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—41no—
67C22×C4⋊C4—41no—
68C22×C4⋊C4—41no—
69C22×C4⋊C4—41no—
70C22×C4⋊C4C2×C412yes—
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C22⋊C4—22no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4D411yescharacteristic
78C22×C42D411yescharacteristic
79#432—41no—
80#432—41no—
81#431—41no—
82#431—41no—
83#423—22no—
84C2×#100—22no—
85C2×#100—22no—
86#407—22no—
87#407—22no—
88#407—22no—
89#407—41no—
90#407—41no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—41no—
93C4×M16—22no—
94#398—22no—
95#398—22no—
96#398C2×C414yes—
97#398C2314yes—
98#397—22no—
99#397—22no—
100#397C2×C414yes—
101#397C2×C414yes—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#381—22no—
107#381—22no—
108#381—22no—
109#381—22no—
110#380—21no—
111#380—21no—
112#380—22no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380C2×C412yes—
117#380C2×C412yes—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95C2×C412yes—
121#343—22no—
122#343C2314yes—
123#324—22no—
124#324C2314yes—
125#323—22no—
126#323—22no—
127C25C22⋊C411yescharacteristic
128C25C22⋊C411yescharacteristic
129C22×D4—24no—
130C22×D4—41no—
131C22×D4—41no—
132C22×D4—42no—
133C22×D4C22⋊C411yescharacteristic
134C22×D4C22⋊C411yescharacteristic
135C22×D4C2×D411yescharacteristic
136C22×D4C2×D411yescharacteristic
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C2×M16—24no—
169C2×M16—42no—
170C2×M16—42no—
171C2×M16—84no—
172C22×C8—42no—
173#123—48no—
174#123—48no—
175#123—48no—
176#123—48no—
177C4×D4—41no—
178C4×D4—41no—
179C4×D4—41no—
180C4×D4—41no—
181C4×D4—41no—
182C4×D4—41no—
183C4×D4—41no—
184C4×D4—41no—
185C4×D4—41no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—41no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C22×C414yes—
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2411yescharacteristic
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—48no—
233C2×C22⋊C4C22⋊C411yescharacteristic
234C2×C22⋊C4C22⋊C411yescharacteristic
235C2×C22⋊C4C22⋊C411yescharacteristic
236C2×C22⋊C4C22⋊C411yescharacteristic
237C2×C22⋊C4C22×C411yescharacteristic
238C2×C22⋊C4C22×C411yescharacteristic
239C2×C22⋊C4C2×D411yescharacteristic
240C2×C22⋊C4C2×D411yescharacteristic
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C4⋊C8—42no—
250C4⋊C8—42no—
251#100—44no—
252#100—44no—
253C22⋊C8—24no—
254C22⋊C8—42no—
255C22⋊C8—42no—
256C22⋊C8—42no—
257C22⋊C8—42no—
258C22⋊C8—42no—
259#97—42no—
260C4×C8—42no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—22no—
265#95—24no—
266#95C22×C414yes—
267C24—21no—
268C24—21no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—21no—
273C24—21no—
274C24—41no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24C2×C22⋊C411yescharacteristic
280C24C2×C22⋊C411yescharacteristic
281C24C2×C22⋊C411yescharacteristic
282C24C2×C22⋊C411yescharacteristic
283C24C2×C22⋊C411yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—48no—
296C2×D4—48no—
297C2×D4AGL(1,ℤ/8)14yes—
298C22×C4—24no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—41no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4#9911yescharacteristic
362C22×C4#9911yescharacteristic
363C22×C4#9911yescharacteristic
364C22×C4#9911yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375M16—44no—
376M16—44no—
377M16—84no—
378M16—84no—
379C2×C8—44no—
380C2×C8—44no—
381C2×C8—84no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊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—48no—
406C22⋊C4—44no—
407C22⋊C4—48no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C23—22no—
417C23—21no—
418C23—21no—
419C23—22no—
420C23—24no—
421C23—24no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—42no—
438C23—48no—
439C23—48no—
440C23—42no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—48no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—81no—
451C23—81no—
452C23—84no—
453C23C2×#9911yescharacteristic
454C23C2×#9911yescharacteristic
455C23#41812yes—
456C23#41812yes—
457C23#42112yes—
458C23C22×C22⋊C411yescharacteristic
459D4—44no—
460D4—44no—
461D4—44no—
462D4—44no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—88no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—88no—
519C2×C4—88no—
520C2×C4#35214yes—
521C2×C4C2×#9911yescharacteristic
522C2×C4C2×#9911yescharacteristic
523C2×C4C2×#9911yescharacteristic
524C2×C4C2×#9911yescharacteristic
525C2×C4C2×AGL(1,ℤ/8)12yes—
526C8—84no—
527C8—84no—
528C22—24no—
529C22—24no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—48no—
544C22—42no—
545C22—48no—
546C22—42no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—81no—
557C22—84no—
558C22—84no—
559C22—81no—
560C22C2×#35212yes—
561C22#213512yes—
562C22C22×#9911yescharacteristic
563C22C2×#41811yescharacteristic
564C22C2×#42111yescharacteristic
565C4—21no—
566C4—21no—
567C4—22no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—84no—
576C4—84no—
577C4C2×#35212yes—
578C4C2×#35212yes—
579C2—21no—
580C2—22no—
581C2—21no—
582C2—22no—
583C2—42no—
584C2—42no—
585C2—42no—
586C2—81no—
587C2#1757711yescharacteristic
588C2C22×#35211yescharacteristic
589C2C2×#213511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11881C2semidirect
5. #11803C2semidirect
6. #11780C2semidirect
38. #1346C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
#207262
#173622
#118812
#118032
#117802