Finite Group Databasebeta

← #204617

Subgroups — #204617

Order 512.

1536 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic
3C2×#1511C214yesmaximal
4#14685C211yesmaximal, characteristic
5#9683C214yesmaximal, complemented
6#9679C214yesmaximal, complemented
7C22×C4⋊Q8C411yescharacteristic, complemented
8C22×C22⋊Q8C2212yes—
9C23×C42C2211yescharacteristic
10C2×#407C2212yes—
11C2×C4×M16—21no—
12#2857—21no—
13#2857—21no—
14#2857—21no—
15#2857—21no—
16#2857C412yescomplemented
17#2857C412yescomplemented
18#2857C2212yes—
19#2857C2212yes—
20C2×#391C412yescomplemented
21C2×#389—21no—
22C2×#389—21no—
23C2×#384C411yescharacteristic, complemented
24C2×#381C2212yes—
25C2×#380—21no—
26C2×#380—21no—
27C2×#380—21no—
28#1749—21no—
29#1749C2212yes—
30C4×C22⋊C8—21no—
31C4×C22⋊C8C2212yes—
32#1728—21no—
33#1511—216no—
34C2×#328C22116yes—
35C4⋊#95—21no—
36C4⋊#95C2212yes—
37#1434—21no—
38#1434C2212yes—
39C22⋊#95—21no—
40C23×Q8C2×C412yes—
41C24×C4D411yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—44no—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2×C412yes—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—28no—
53C2×C22⋊Q8—28no—
54C2×C22⋊Q8C2×C414yes—
55C2×C22⋊Q8C2314yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2×C412yes—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C42—44no—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78#407—24no—
79#407D418yes—
80C2×C22⋊C8—22no—
81C2×C22⋊C8C2314yes—
82C4×M16—44no—
83C2×#97—22no—
84C2×C4×C8—22no—
85#391—22no—
86#391—22no—
87#391C2×C414yes—
88#391C2×C414yes—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#384—21no—
98#384—21no—
99#384C2×C412yes—
100#384C2×C412yes—
101#381—28no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—22no—
110#380—44no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95C2314yes—
118#336—22no—
119#336—22no—
120#328—232no—
121C25C2×D411yescharacteristic
122C22×Q8—22no—
123C22×Q8—28no—
124C22×Q8—44no—
125C22×Q8—44no—
126C22×Q8C22×C414yes—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D414yes—
156C23×C4#4114yes—
157C23×C4C2411yescharacteristic
158C2×M16—28no—
159C2×M16—44no—
160C2×M16—44no—
161C22×C8—24no—
162C4⋊Q8—44no—
163C4⋊Q8—44no—
164C4⋊Q8—44no—
165C4⋊Q8—44no—
166C22⋊Q8—216no—
167C22⋊Q8—28no—
168C22⋊Q8—28no—
169C22⋊Q8—48no—
170C22⋊Q8—44no—
171C22⋊Q8—44no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—48no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4C22×C414yes—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22⋊C411yescharacteristic
217C2×C42C22⋊C411yescharacteristic
218C2×C42C22⋊C411yescharacteristic
219C2×C42C22⋊C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C22⋊C8—28no—
229C22⋊C8—24no—
230C22⋊C8C2×D418yes—
231#97—44no—
232C4×C8—44no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—28no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—44no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—41no—
247C24—41no—
248C24C2×C22⋊C411yescharacteristic
249C24C2×C22⋊C411yescharacteristic
250C24#12412yes—
251C24#12712yes—
252C24C22×D411yescharacteristic
253C2×Q8—28no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8—44no—
259C2×Q8—816no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
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—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—41no—
309C22×C4—416no—
310C22×C4—416no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4#9914yes—
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C4×D414yes—
336C22×C4C4×D414yes—
337C22×C4#12114yes—
338C22×C4#12114yes—
339C22×C4#12412yes—
340C22×C4#12712yes—
341C22×C4#13714yes—
342C22×C4#13714yes—
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D412yes—
346C22×C4C2×#4112yes—
347M16—416no—
348C2×C8—28no—
349C2×C8—44no—
350C2×C8—44no—
351C4⋊C4—28no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—48no—
357C4⋊C4—44no—
358C4⋊C4—48no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—88no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—48no—
367C22⋊C4—88no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—24no—
395C23—21no—
396C23—21no—
397C23—24no—
398C23—21no—
399C23—21no—
400C23—24no—
401C23—24no—
402C23—42no—
403C23—42no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23#39012yes—
409C23#39012yes—
410C23C2×#9912yes—
411C23#41012yes—
412C23C22×C22⋊C411yescharacteristic
413C23C2×#12411yescharacteristic
414C23C2×#12711yescharacteristic
415C23C2×#13712yes—
416C23C2×#13712yes—
417Q8—88no—
418Q8—88no—
419C2×C4—28no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—28no—
439C2×C4—28no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—88no—
470C2×C4—88no—
471C2×C4—84no—
472C2×C4—84no—
473C2×C4—84no—
474C2×C4—84no—
475C2×C4—84no—
476C2×C4—84no—
477C2×C4—84no—
478C2×C4#39012yes—
479C2×C4#39012yes—
480C2×C4C2×#9912yes—
481C2×C4C2×#9912yes—
482C2×C4#41912yes—
483C2×C4#41912yes—
484C2×C4#41912yes—
485C2×C4#41912yes—
486C2×C4#42112yes—
487C2×C4#42112yes—
488C2×C4C2×C4×D412yes—
489C2×C4C2×#12112yes—
490C8—48no—
491C22—21no—
492C22—22no—
493C22—21no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—21no—
504C22—21no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—48no—
509C22—48no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22#196414yes—
515C22C2×#39011yescharacteristic
516C22C22×#9911yescharacteristic
517C22C2×#41011yescharacteristic
518C22C2×#41911yescharacteristic
519C22C2×#41911yescharacteristic
520C22C2×#42111yescharacteristic
521C22C2×#42111yescharacteristic
522C22#313912yes—
523C22#315212yes—
524C4—22no—
525C4—22no—
526C4—22no—
527C4—22no—
528C4—22no—
529C4—22no—
530C4—22no—
531C4—22no—
532C4—22no—
533C4—41no—
534C4—41no—
535C4—84no—
536C4—84no—
537C4—164no—
538C4#151114yescomplemented
539C2—22no—
540C2—22no—
541C2—24no—
542C2—42no—
543C2C2×#151111yescharacteristic
544C2#1756211yescharacteristic
545C2#1757811yescharacteristic
546C2#1759011yescharacteristic
547C2#2120211yescharacteristic
548C2C2×#196411yescharacteristic
549C2#2125211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9683C2semidirect
6. #9679C2semidirect
7. C22×C4⋊Q8C4semidirect
16. #2857C4semidirect
17. #2857C4semidirect
20. C2×#391C4semidirect
23. C2×#384C4semidirect
538. C4#1511semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C4×#4072
C2×#15112
#146852
#96832
#96792