Finite Group Databasebeta

← #364050

Subgroups — #364050

Order 512.

1540 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34012C211yesmaximal, characteristic
2C2×#1728C211yesmaximal, characteristic
3C4×#381C211yesmaximal, characteristic
4#11370C212yesmaximal, complemented
5#11359C212yesmaximal, complemented
6#10808C214yesmaximal, complemented
7#10732C214yesmaximal
8C23×M16C2212yes—
9C23×C42C2211yescharacteristic
10#2285C411yescharacteristic, complemented
11#2285C411yescharacteristic, complemented
12#2285C411yescharacteristic, complemented
13#2285C411yescharacteristic, complemented
14#2285C2211yescharacteristic
15#2285C2211yescharacteristic
16#2283—21no—
17#2283—21no—
18#2283—21no—
19#2283—21no—
20#2283C412yescomplemented
21#2283C412yescomplemented
22#2280—21no—
23#2280—21no—
24#2280C2212yes—
25C2×#381—21no—
26C4×#117—21no—
27C4×#117—21no—
28C4×#117C2212yes—
29C2×C4×C22⋊C4C2212yes—
30C2×#339C2218yes—
31C2×#339C2218yes—
32#1728—21no—
33#1728—21no—
34#1728C2212yes—
35#1728C2212yes—
36C2×#336—21no—
37C2×#336—21no—
38#1434—21no—
39#1434—21no—
40C22⋊#95—21no—
41C4×#95C2212yes—
42C4×#95C2212yes—
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16—28no—
48C22×M16—28no—
49C22×M16—44no—
50C22×M16C2314yes—
51C23×C8—22no—
52C2×#117—21no—
53C2×#117—21no—
54C2×#117—21no—
55C2×#117—21no—
56C2×#117C2312yes—
57C2×#117C2312yes—
58C22×C22⋊C4—22no—
59C22×C42—28no—
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42C2314yes—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73#384C2×C411yescharacteristic
74#384C2×C411yescharacteristic
75#384C2×C411yescharacteristic
76#384C2×C411yescharacteristic
77#383—22no—
78#383—22no—
79#382—21no—
80#382—21no—
81#382—21no—
82#382—21no—
83#382C2×C412yes—
84#382C2×C412yes—
85#381—28no—
86#381—22no—
87#381—22no—
88#381—22no—
89#380—21no—
90#380—21no—
91#380—21no—
92#380—21no—
93#380—21no—
94#380—21no—
95#380—21no—
96#380—21no—
97#380C2×C412yes—
98#380C2×C412yes—
99#380C2×C412yes—
100#380C2×C412yes—
101#379—22no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
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⋊C4C2×C412yes—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C412yes—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—28no—
121#376—22no—
122#376—22no—
123#376—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C43—22no—
130#339—216no—
131#339—48no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136C25C2×D411yescharacteristic
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
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×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4#4118yes—
165C23×C4C2411yescharacteristic
166C2×M16—28no—
167C2×M16—24no—
168C2×M16—28no—
169C2×M16—28no—
170C2×M16—432no—
171C2×M16—44no—
172C2×M16—44no—
173C2×M16#4118yes—
174C22×C8—28no—
175C22×C8—24no—
176C22×C8—44no—
177C22×C8—44no—
178#117—28no—
179#117—24no—
180#117—24no—
181#117—44no—
182#117—44no—
183#117—44no—
184#117—44no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C412yes—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
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—24no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—28no—
225C2×C42—28no—
226C2×C42—24no—
227C2×C42—24no—
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×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244#95—28no—
245#95—28no—
246#95—22no—
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—44no—
259#95—44no—
260#95—44no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—41no—
266C24—41no—
267C24C2×C22⋊C411yescharacteristic
268C24C2×C22⋊C411yescharacteristic
269C24#12314yes—
270C24C22×D411yescharacteristic
271C22×C4—22no—
272C22×C4—22no—
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—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—216no—
287C22×C4—216no—
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—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
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—28no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—416no—
326C22×C4—416no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4#10414yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4#11714yes—
345C22×C4#11714yes—
346C22×C4#12314yes—
347C22×C4#12314yes—
348C22×C4#12314yes—
349C22×C4C23×C411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C2×#4114yes—
352C22×C4C2×#4114yes—
353C22×C4H(5,𝔽2)12yes—
354C22×C42−1+412yes—
355M16—216no—
356M16—416no—
357M16—48no—
358C2×C8—28no—
359C2×C8—416no—
360C2×C8—44no—
361C2×C8—44no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C22⋊C4—416no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—44no—
398C42—44no—
399C42—44no—
400C42—42no—
401C42—42no—
402C42—44no—
403C42—44no—
404C42—48no—
405C42—48no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—28no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—28no—
421C23—21no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23#38112yes—
428C23#38112yes—
429C23C2×#10412yes—
430C23#42112yes—
431C23C22×C22⋊C411yescharacteristic
432C23C2×#12312yes—
433C23#53212yes—
434C23#53712yes—
435C2×C4—28no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—28no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—28no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—21no—
457C2×C4—21no—
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—44no—
469C2×C4—44no—
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—48no—
479C2×C4—48no—
480C2×C4—48no—
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—84no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4#38112yes—
498C2×C4#38112yes—
499C2×C4C2×#10412yes—
500C2×C4C2×#10412yes—
501C2×C4#42112yes—
502C2×C4#42112yes—
503C2×C4C2×#11712yes—
504C2×C4C4×#4112yes—
505C2×C4#52012yes—
506C2×C4C2×#12312yes—
507C2×C4#52512yes—
508C2×C4#53511yescharacteristic
509C2×C4#53611yescharacteristic
510C2×C4#55118yes—
511C2×C4#55218yes—
512C8—48no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—416no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22#182614yes—
534C22C2×#38111yescharacteristic
535C22#229711yescharacteristic
536C22#231111yescharacteristic
537C22#249814yes—
538C22C22×#10411yescharacteristic
539C22C2×#42111yescharacteristic
540C22C2×#42111yescharacteristic
541C22C2×#42111yescharacteristic
542C4—22no—
543C4—22no—
544C4—22no—
545C4—22no—
546C4—22no—
547C4—22no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—88no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4—82no—
557C4#234314yes—
558C4#248212yes—
559C4#248412yes—
560C2—24no—
561C2—24no—
562C2—42no—
563C2#1395811yescharacteristic
564C2C2×#182611yescharacteristic
565C2#2070511yescharacteristic
566C2#2138211yescharacteristic
567C2#2138311yescharacteristic
568C2#2185611yescharacteristic
569C2#2185711yescharacteristic
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
6. #10808C2semidirect
10. #2285C4semidirect
11. #2285C4semidirect
12. #2285C4semidirect
13. #2285C4semidirect
20. #2283C4semidirect
21. #2283C4semidirect

Maximal subgroups

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

subgroupindex
#340122
C2×#17282
C4×#3812
#113702
#113592
#108082
#107322