Finite Group Databasebeta

← #496331

Subgroups — #496331

Order 512.

1283 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2#20726C211yesmaximal, characteristic
3#20722C211yesmaximal, characteristic
4#13220C214yesmaximal, complemented
5#13217C214yesmaximal, complemented
6#12688C214yesmaximal
7C22×#123C2212yes—
8C22×C4×D4C2211yescharacteristic
9C23×C4⋊C4C411yescharacteristic, complemented
10#2920—21no—
11#2486—21no—
12#2486—21no—
13#2486—21no—
14#2486—21no—
15#2486C412yescomplemented
16#2486C412yescomplemented
17#2486C2212yes—
18#2486C2212yes—
19C2×#398—21no—
20C2×#397—21no—
21C2×#389—21no—
22#2280—21no—
23C2×#381C412yescomplemented
24C2×#380—21no—
25C2×#380C411yescharacteristic, complemented
26C2×#352C22116yes—
27#1821—21no—
28#1819—21no—
29#1800—21no—
30#1800C2212yes—
31#1799—21no—
32#1799C2212yes—
33#1783C2212yes—
34#1781C2212yes—
35C4×#100—21no—
36C4×#100C2212yes—
37C4×#99—21no—
38C4×#99C2212yes—
39C23×D4C2311yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C412yes—
42C24×C4D411yescharacteristic
43C2×#123—22no—
44C2×#123—22no—
45C2×#123—48no—
46C2×#123C2×C414yes—
47C2×#123C2314yes—
48C2×C4×D4—41no—
49C2×C4×D4—41no—
50C2×C4×D4—41no—
51C2×C4×D4—41no—
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4D411yescharacteristic
55C2×C4×D4D411yescharacteristic
56C2×C4×D4D411yescharacteristic
57C2×C4×D4D411yescharacteristic
58C2×C4×D4C2311yescharacteristic
59C2×C4×D4C2311yescharacteristic
60C2×#117—41no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—41no—
68C22×C4⋊C4—41no—
69C22×C4⋊C4—41no—
70C22×C4⋊C4—41no—
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4D411yescharacteristic
74C22×C22⋊C4—22no—
75C22×C22⋊C4C2×C412yes—
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×#100C2314yes—
86C2×#99—22no—
87C2×#99C2314yes—
88#407—22no—
89C4×M16—22no—
90#398—22no—
91#398—22no—
92#398—22no—
93#398—22no—
94#397—22no—
95#397—22no—
96#397—22no—
97#397—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#381—22no—
103#381—22no—
104#381—41no—
105#381C2×C414yes—
106#381C2×C414yes—
107#380—21no—
108#380—21no—
109#380—22no—
110#380—22no—
111#380—22no—
112#380—22no—
113#380—22no—
114#380—41no—
115#380C2×C412yes—
116#380C2×C412yes—
117#379—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—41no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123#352—416no—
124#343—22no—
125#343—22no—
126#342—22no—
127#342—22no—
128C25C22×C411yescharacteristic
129C25C2×D411yescharacteristic
130C22×D4—24no—
131C22×D4—41no—
132C22×D4—41no—
133C22×D4—42no—
134C22×D4C22×C411yescharacteristic
135C22×D4C2×D411yescharacteristic
136C22×D4C2×D411yescharacteristic
137C22×D4C2411yescharacteristic
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—42no—
153C23×C4—42no—
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×C414yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C2×M16—24no—
170#123—216no—
171#123—48no—
172#123—48no—
173#123—48no—
174C4×D4—41no—
175C4×D4—41no—
176C4×D4—41no—
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—
190#117—84no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—48no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C22⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—48no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4C22⋊C411yescharacteristic
235C2×C22⋊C4C22⋊C411yescharacteristic
236C2×C22⋊C4C22⋊C411yescharacteristic
237C2×C22⋊C4C22⋊C411yescharacteristic
238C2×C22⋊C4C22×C414yes—
239C2×C22⋊C4C22×C411yescharacteristic
240C2×C22⋊C4C22×C411yescharacteristic
241C2×C22⋊C4C2×D411yescharacteristic
242C2×C22⋊C4C2×D411yescharacteristic
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—42no—
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C4⋊C8—42no—
253C4⋊C8—42no—
254#100—28no—
255#100—44no—
256#99—28no—
257#99—44no—
258C22⋊C8—42no—
259C22⋊C8—42no—
260#97—42no—
261C4×C8—42no—
262#95—24no—
263#95—24no—
264#95—22no—
265#95—22no—
266#95—24no—
267#95—24no—
268#95—42no—
269#95—42no—
270C24—22no—
271C24—21no—
272C24—21no—
273C24—22no—
274C24—22no—
275C24—21no—
276C24—21no—
277C24—41no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24C2×C22⋊C411yescharacteristic
283C24C2×C22⋊C411yescharacteristic
284C24C2×C22⋊C411yescharacteristic
285C24C23×C411yescharacteristic
286C24C22×D411yescharacteristic
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—48no—
299C2×D4—48no—
300C2×D4H(5,𝔽2)14yes—
301C22×C4—24no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
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—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—41no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—82no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4—88no—
364C22×C4—88no—
365C22×C4#9911yescharacteristic
366C22×C4#9911yescharacteristic
367C22×C4#9911yescharacteristic
368C22×C4#9911yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379M16—44no—
380M16—44no—
381C2×C8—44no—
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—
396C4⋊C4—84no—
397C22⋊C4—28no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—44no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—84no—
412C22⋊C4—84no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—84no—
422C23—21no—
423C23—21no—
424C23—22no—
425C23—22no—
426C23—24no—
427C23—24no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—42no—
438C23—42no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—42no—
444C23—48no—
445C23—48no—
446C23—42no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—48no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—81no—
457C23—81no—
458C23—84no—
459C23C2×#9911yescharacteristic
460C23C2×#9911yescharacteristic
461C23#41012yes—
462C23C22×C22⋊C411yescharacteristic
463C23#51812yes—
464C23#53512yes—
465D4—44no—
466D4—44no—
467D4—44no—
468D4—44no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
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—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—88no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—88no—
525C2×C4—88no—
526C2×C4—84no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4#35214yescomplemented
530C2×C4C2×#9911yescharacteristic
531C2×C4C2×#9911yescharacteristic
532C2×C4C2×#9911yescharacteristic
533C2×C4C2×#9911yescharacteristic
534C2×C4#53512yes—
535C8—84no—
536C22—24no—
537C22—24no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—48no—
553C22—48no—
554C22—42no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—81no—
565C22—84no—
566C22—84no—
567C22—81no—
568C22C2×#35212yes—
569C22#212312yes—
570C22#230911yescharacteristic
571C22C22×#9911yescharacteristic
572C22C2×#41011yescharacteristic
573C4—22no—
574C4—21no—
575C4—21no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—84no—
584C4—84no—
585C4—164no—
586C4C2×#35212yes—
587C4C2×#35212yes—
588C2—21no—
589C2—21no—
590C2—22no—
591C2—22no—
592C2—42no—
593C2—42no—
594C2—42no—
595C2—81no—
596C2#2179311yescharacteristic
597C2C22×#35211yescharacteristic
598C2C2×#212311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13220C2semidirect
5. #13217C2semidirect
9. C23×C4⋊C4C4semidirect
15. #2486C4semidirect
16. #2486C4semidirect
23. C2×#381C4semidirect
25. C2×#380C4semidirect
529. C2×C4#352semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
#207262
#207222
#132202
#132172
#126882