Finite Group Databasebeta

← #382758

Subgroups — #382758

Order 512.

1410 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
1#34012C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic
3C4×#380C211yesmaximal, characteristic
4#11370C212yesmaximal, complemented
5#11359C212yesmaximal, complemented
6#11201C214yesmaximal
7#10852C214yesmaximal, complemented
8C23×C42C2211yescharacteristic
9C2×#423C2212yes—
10C2×#407C2212yes—
11#2285C411yescharacteristic, complemented
12#2285C411yescharacteristic, complemented
13#2285C411yescharacteristic, complemented
14#2285C411yescharacteristic, complemented
15#2285C2211yescharacteristic
16#2285C2211yescharacteristic
17#2283—21no—
18#2283—21no—
19#2283—21no—
20#2283—21no—
21#2283C412yescomplemented
22#2283C412yescomplemented
23#2280—21no—
24#2280—21no—
25#2280C2212yes—
26C2×#380—21no—
27C4×#117—21no—
28C4×#117—21no—
29C4×#117C2212yes—
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C22⋊C4C2212yes—
32#1797—21no—
33#1797—21no—
34#1797—21no—
35#1797—21no—
36C2×#339C2218yes—
37C2×#339C2218yes—
38#1728C2212yes—
39C4⋊#95—21no—
40#1434—21no—
41C22⋊#95—21no—
42C4×#95C2212yes—
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16C2314yes—
47C2×#117—21no—
48C2×#117—21no—
49C2×#117—21no—
50C2×#117—21no—
51C2×#117C2312yes—
52C2×#117C2312yes—
53C22×C4⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C42—28no—
56C22×C42C2×C411yescharacteristic
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42C2314yes—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69#423—28no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72#407—44no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75#384C2×C411yescharacteristic
76#384C2×C411yescharacteristic
77#384C2×C411yescharacteristic
78#384C2×C411yescharacteristic
79#383—22no—
80#383—22no—
81#382—21no—
82#382—21no—
83#382—21no—
84#382—21no—
85#382C2×C412yes—
86#382C2×C412yes—
87#381—22no—
88#381—22no—
89#381—22no—
90#380—28no—
91#380—21no—
92#380—21no—
93#380—21no—
94#380—21no—
95#380—21no—
96#380—21no—
97#380—21no—
98#380—21no—
99#380C2×C412yes—
100#380C2×C412yes—
101#380C2×C412yes—
102#380C2×C412yes—
103#379—22no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—28no—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C412yes—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—44no—
124#376—22no—
125#376—22no—
126#376—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C43—22no—
132#339—48no—
133#339—48no—
134#336—22no—
135#336—22no—
136C25C2×D411yescharacteristic
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C2×M16—28no—
168C2×M16—24no—
169C2×M16—28no—
170C2×M16—24no—
171C22×C8—24no—
172#117—24no—
173#117—24no—
174#117—24no—
175#117—24no—
176#117—44no—
177#117—44no—
178#117—44no—
179#117—44no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C412yes—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—28no—
222C2×C42—24no—
223C2×C42—28no—
224C2×C42—24no—
225C2×C42C22⋊C411yescharacteristic
226C2×C42C22⋊C411yescharacteristic
227C2×C42C22⋊C411yescharacteristic
228C2×C42C22⋊C411yescharacteristic
229C2×C42C22⋊C411yescharacteristic
230C2×C42C22⋊C411yescharacteristic
231C2×C42C22⋊C411yescharacteristic
232C2×C42C22⋊C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42#4118yes—
242C4⋊C8—44no—
243C4⋊C8—44no—
244C22⋊C8—44no—
245C22⋊C8—44no—
246#95—28no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—44no—
260#95—44no—
261#95—44no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—41no—
269C24—41no—
270C24C2×C22⋊C411yescharacteristic
271C24C2×C22⋊C411yescharacteristic
272C24C22×D411yescharacteristic
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—28no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—416no—
330C22×C4—416no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4#10414yes—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4#12314yes—
353C22×C4#12314yes—
354C22×C4C23×C411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C2×#4114yes—
357C22×C4H(5,𝔽2)12yes—
358C22×C4H(5,𝔽2)12yes—
359C22×C42−1+412yes—
360C22×C42−1+412yes—
361M16—48no—
362M16—48no—
363C2×C8—44no—
364C2×C8—44no—
365C2×C8—44no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—416no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C22⋊C4—416no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—44no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—44no—
400C42—44no—
401C42—42no—
402C42—42no—
403C42—44no—
404C42—44no—
405C42—44no—
406C42—48no—
407C42—48no—
408C42#11714yes—
409C42#11714yes—
410C42#12314yes—
411C42#12314yes—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—24no—
417C23—24no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23C2×#10412yes—
439C23#42112yes—
440C23C22×C22⋊C411yescharacteristic
441C23C2×#12312yes—
442C23#53511yescharacteristic
443C23#53611yescharacteristic
444C23#53712yes—
445C2×C4—28no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—48no—
488C2×C4—48no—
489C2×C4—48no—
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—84no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4—84no—
506C2×C4#38112yes—
507C2×C4#38112yes—
508C2×C4#38112yes—
509C2×C4#38112yes—
510C2×C4C2×#10412yes—
511C2×C4C2×#10412yes—
512C2×C4#42112yes—
513C2×C4#42112yes—
514C2×C4C2×#11712yes—
515C2×C4#52012yes—
516C2×C4#52012yes—
517C2×C4C2×#12312yes—
518C2×C4#53511yescharacteristic
519C2×C4#53611yescharacteristic
520C2×C4#53712yes—
521C2×C4#54014yes—
522C2×C4#54114yes—
523C2×C4#54214yes—
524C2×C4#54214yes—
525C8—84no—
526C22—22no—
527C22—22no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—48no—
541C22—48no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22C2×#38111yescharacteristic
551C22#231111yescharacteristic
552C22#231111yescharacteristic
553C22#262412yes—
554C22#262512yes—
555C22C22×#10411yescharacteristic
556C22C2×#42111yescharacteristic
557C22C2×#42111yescharacteristic
558C22C2×#42111yescharacteristic
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—88no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4#182614yes—
574C4#231414yes—
575C4#262412yes—
576C4#262512yes—
577C2—22no—
578C2—22no—
579C2—42no—
580C2—42no—
581C2#1434311yescharacteristic
582C2C2×#182611yescharacteristic
583C2#2070511yescharacteristic
584C2#2185611yescharacteristic
585C2#2185711yescharacteristic
586C2#2189911yescharacteristic
587C2#2190011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11370C2semidirect
5. #11359C2semidirect
7. #10852C2semidirect
11. #2285C4semidirect
12. #2285C4semidirect
13. #2285C4semidirect
14. #2285C4semidirect
21. #2283C4semidirect
22. #2283C4semidirect

Maximal subgroups

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

subgroupindex
#340122
#206262
C4×#3802
#113702
#113592
#112012
#108522