Finite Group Databasebeta

← #494010

Subgroups — #494010

Order 512.

1074 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#536C211yesmaximal, characteristic
2#20728C211yesmaximal, characteristic
3#20723C211yesmaximal, characteristic
4#12855C212yesmaximal
5#12794C214yesmaximal, complemented
6#12692C212yesmaximal
7#12621C214yesmaximal, complemented
8C2×#536C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10C22×C4×Q8C411yescharacteristic, complemented
11#2920—21no—
12#2327—21no—
13#2326—21no—
14#2323—21no—
15#2323—21no—
16#2322C412yescomplemented
17#2322C2212yes—
18#2317C411yescharacteristic, complemented
19#2311C2211yescharacteristic
20#2310C411yescharacteristic, complemented
21C4×C4⋊Q8—21no—
22C4×#124—21no—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8—21no—
25C4×#123C412yescomplemented
26C4×#123C2212yes—
27#2283—21no—
28C4×#117C411yescharacteristic, complemented
29C2×#352C2218yes—
30#1822—21no—
31#1820—21no—
32#1801—21no—
33#1800C2212yes—
34#1799—21no—
35#1799C2212yes—
36#1785C2212yes—
37#1782C2212yes—
38C4×#101—21no—
39C4×#100C2212yes—
40C4×#99—21no—
41C4×#99C2212yes—
42#1390C2218yes—
43C22×#41C2311yescharacteristic
44C23×Q8C2×C411yescharacteristic
45#536—28no—
46C2×C4⋊Q8—21no—
47C2×#124—21no—
48C2×#123C2×C412yes—
49C2×#123C2312yes—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C4×#41—41no—
53C4×#41—41no—
54C4×#41—41no—
55C4×#41—41no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—41no—
61C2×C4×Q8—41no—
62C2×C4×Q8—41no—
63C2×C4×Q8—41no—
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C411yescharacteristic
75C2×#117D411yescharacteristic
76C2×#117D411yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C412yes—
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C412yes—
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83#432—41no—
84#432—41no—
85#431—41no—
86#431—41no—
87#423—22no—
88C2×#101—22no—
89C2×#100C2314yes—
90C2×#99—22no—
91C2×#99C2314yes—
92#407—22no—
93C4×M16—22no—
94#391—21no—
95#391—21no—
96#391—21no—
97#391—21no—
98#391—21no—
99#390—21no—
100#389—22no—
101#389—22no—
102#389—21no—
103#389—21no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#386C2×C412yes—
113#386C2312yes—
114#385—22no—
115#385—22no—
116#385C2×C412yes—
117#385C2×C412yes—
118#384—21no—
119#383—41no—
120#382—21no—
121#382—41no—
122#381—22no—
123#381C2×C412yes—
124#381C2×C412yes—
125#380—21no—
126#380—21no—
127#380—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—41no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4C2×C412yes—
137C4×C22⋊C4C2×C412yes—
138#376—21no—
139#376—41no—
140C43—21no—
141#352—216no—
142#343—22no—
143#343—22no—
144#342—22no—
145#342—22no—
146C2×#41—24no—
147C2×#41—41no—
148C2×#41—41no—
149C2×#41—42no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—24no—
153C22×Q8—41no—
154C22×Q8—41no—
155C22×Q8—42no—
156C22×Q8C2×D411yescharacteristic
157C22×D4C22×C411yescharacteristic
158C22×D4C2×D411yescharacteristic
159C22×D4C2411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C412yes—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C2×M16—24no—
172C4⋊Q8—28no—
173#124—28no—
174#123—24no—
175#123—24no—
176#123—24no—
177#123#4118yes—
178C22⋊Q8—28no—
179C22⋊Q8—28no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8—42no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4—41no—
206#117—24no—
207#117—41no—
208#117—41no—
209#117—41no—
210#117—41no—
211#117—41no—
212#117—41no—
213#117—42no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4C22⋊C411yescharacteristic
241C2×C22⋊C4C22⋊C411yescharacteristic
242C2×C22⋊C4C22×C412yes—
243C2×C22⋊C4C22×C412yes—
244C2×C22⋊C4C22×C411yescharacteristic
245C2×C22⋊C4C22×C411yescharacteristic
246C2×C22⋊C4C2×D411yescharacteristic
247C2×C22⋊C4C2×D411yescharacteristic
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C4⋊C8—42no—
280C4⋊C8—42no—
281#101—44no—
282#100—28no—
283#99—28no—
284#99—44no—
285C22⋊C8—42no—
286C22⋊C8—42no—
287#97—42no—
288C4×C8—42no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—42no—
300#95—42no—
301#95—42no—
302#95—42no—
303C24C2×C22⋊C411yescharacteristic
304C24C23×C411yescharacteristic
305C24C22×D411yescharacteristic
306#41—48no—
307#41—48no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—41no—
313C2×Q8—42no—
314C2×Q8—48no—
315C2×Q8—48no—
316C2×Q8—42no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—42no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4C2×#4114yes—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4—82no—
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×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4#11714yes—
381C22×C4C22×D411yescharacteristic
382M16—44no—
383M16—44no—
384C2×C8—44no—
385C4⋊C4—24no—
386C4⋊C4—28no—
387C4⋊C4—28no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—84no—
404C22⋊C4—28no—
405C22⋊C4—22no—
406C22⋊C4—28no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—42no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4#11714yes—
419C42—24no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—44no—
439C42—44no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—82no—
444C42—84no—
445C42#9911yescharacteristic
446C42#9911yescharacteristic
447C42#9911yescharacteristic
448C42#9911yescharacteristic
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23C22×C22⋊C411yescharacteristic
463C23C2×#11712yes—
464C23#52512yes—
465Q8—44no—
466Q8—44no—
467Q8—44no—
468Q8—48no—
469D4—44no—
470D4—44no—
471D4—44no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—41no—
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—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—84no—
531C2×C4C2×#9911yescharacteristic
532C2×C4C2×#9911yescharacteristic
533C2×C4C2×#9911yescharacteristic
534C2×C4C2×#9911yescharacteristic
535C2×C4C2×#9911yescharacteristic
536C2×C4C2×#9911yescharacteristic
537C2×C4#41012yes—
538C2×C4#52512yes—
539C8—84no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—81no—
556C22—81no—
557C22#228011yescharacteristic
558C22C22×#9911yescharacteristic
559C22C2×#41011yescharacteristic
560C4—21no—
561C4—21no—
562C4—21no—
563C4—22no—
564C4—21no—
565C4—21no—
566C4—22no—
567C4—44no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—44no—
576C4—82no—
577C4—82no—
578C4—164no—
579C4#212312yes—
580C2—21no—
581C2—21no—
582C2—42no—
583C2—42no—
584C2—81no—
585C2#2072211yescharacteristic
586C2C2×#212311yescharacteristic
587C2#3334711yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #12794C2semidirect
7. #12621C2semidirect
10. C22×C4×Q8C4semidirect
16. #2322C4semidirect
18. #2317C4semidirect
20. #2310C4semidirect
25. C4×#123C4semidirect
28. C4×#117C4semidirect

Maximal subgroups

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

subgroupindex
C4×#5362
#207282
#207232
#128552
#127942
#126922
#126212